calculating precise and accurate free energies in

Transcription

CALCULATING PRECISE AND ACCURATE FREE
ENERGIES IN BIOMOLECULAR SYSTEMS
a dissertation
submitted to the department of chemistry
and the committee on graduate studies
of stanford university
in partial fulfillment of the requirements
for the degree of
doctor of philosophy
Michael R. Shirts
November, 2004
c Copyright by Michael R. Shirts 2005
All Rights Reserved
ii
I certify that I have read this dissertation and that, in
my opinion, it is fully adequate in scope and quality as a
dissertation for the degree of Doctor of Philosophy.
Vijay S. Pande
(Principal Adviser)
Michael Levitt
Hans C. Andersen
Approved for the University Committee on Graduate
Studies.
iii
Preface
In this dissertation, I present a series of closely related studies on the development
of efficient and useful ligand binding prediction. Two of the chapters have been
previously published, and the other three are in the process of being submitted.
Chapters 1 and 2 discuss the problem of efficiently computing the free energies
of interaction both for arbitrary classical systems and for condensed phases systems
of importance in biomolecular systems, presenting evidence for greatly improved efficiencies than those usually obtained. In Chapters 3 and 4, I apply these methods
to evaluate the free energies of solvation of amino acid side chain analogs parameterized using a variety of biomolecular force fields, and with a variety of different water
molecules, obtaining precisions that are an order of magnitude better than obtained
previously, and demonstrating that current biomolecular models have significant deviations from experiment. In Chapter 5, I use the methods developed and practical
insights learned in the previous research to one of the most important challenges
in computational chemistry, computation of ligand binding energies in biomolecular
systems.
Chapter 1 is adapted from M. R, Shirts, E. Bair, G. Hooker, and V. S. Pande,
Phys. Rev. Let., 91, 140601 (2003). Bair and Hooker noticed the original connection
between the physical ratio of probability of forward and reverse simulations and the
statistic concept of logistic regression, located several important resources on the
underlying statistical concepts, and helped edit the paper, but the derivations and
the writing was performed by myself.
Chapter 3 is adapted from M. R. Shirts, J. W. Pitera, W. C. Swope, and V. S.
Pande, J. Chem. Phys., 2003, 5564–5575 (2003). Pitera and Swope rovided extensive
iv
background information, helped select many of the simulation conditions, derived a
key correction term, and provided extensive editing and corrections. All scientific
coding, data preparation, analysis, and original writing was done by myself.
Chapters 2 and 4 are adapted from M. R. Shirts and V. S. Pande (in preparation),
and Chapter 5 is adapted from M. R. Shirts, G. Jayachandran, C. D. Snow, and V. S.
Pande (in preparation). All scientific code changes, analysis, writing was performed
by myself, with the exception of some sections of Chapter 5. The structural analysis of
sampling of FKBP was done in conjuction with Chris Snow, and the preparation and
running of the FKBP simulations was done in conjunction with Guha Jayachandran,
as well as rewriting of many of the analysis scripts for improved performance.
v
Acknowledgements
Vijay Pande has been a wonderful advisor, always available for discussion and advice
while never trying to micromanage or dominate, and showing great care for his students’ professional development. I would also like to thank the entire Pande group
for their help and advise in research in general, and for their comments and advise
on this dissertation specifially. I would like to thank my reading committee, Michael
Levitt and Hans Andersen, for important comments and suggestions to that greatly
improved this disseration. I would especially like to acknowledge Guha Jayachandran
and Chris Snow for their significant work on the FKBP project, and Young Min Rhee
and for many instances of help and advice with running Folding@Home.
Bill Swope and Jed Pitera were incredible resources on the history and practice
of computer simulation, and always provided a critical eye on research papers, never
letting anything unscientific slip through. Eric Bair and Giles Hooker provided invaluable statistical insight, saving many months of research. Erik Lindahl helped
immensely in my extensive changes in GROMACS, as did Jay Ponder with similar
changes in TINKER. And this research would be much less interesting and exciting
without the hundreds of thousands of users of Folding@Home who have volunteered
their computers for scientific advancement. Many other scientists have provided important advice and explanations, such as Chris Jarzynski, Benoit Roux, Xiao-li Meng,
Teresa Head-Gordon, Tom Wandless, Hidei Fujitani, and Hans Horn, among others.
I would like to especially thank the Fannie and John Hertz Foundation for generously funding the majority of my graduate career, as well as the Stanford Graduate
Fellowship.
I would like to thank the entire community of the Stanford Wards of the Church of
vi
Jesus Christ of Latter-day Saints for their incredible friendship and support throughout the entire process of my graduate career, and especially in the last stages of
dissertation writing. My family has helped me throughout the process, especially my
mother and father who have been a neverending source of support and guidance.
vii
Contents
Preface
iv
Acknowledgements
vi
1 Maximum likelihood methods
1
1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
Maximum likelihood estimator derivation . . . . . . . . . . . . . . . .
6
1.3
Variance estimates from maximum likelihood methods . . . . . . . . .
9
1.4
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2 Efficiency of free energy methods
13
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.2
Comparison of theoretical variance and bias estimates . . . . . . . . .
18
2.2.1
Underlying theory of free energy estimates . . . . . . . . . . .
18
2.2.2
Limiting moment bias and variances . . . . . . . . . . . . . .
21
2.3
Averages of the forward and reverse simulations . . . . . . . . . . . .
22
2.4
Bennett acceptance ratio . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.5
Thermodynamic integration . . . . . . . . . . . . . . . . . . . . . . .
25
2.6
Analytically solvable models: offset harmonic wells . . . . . . . . . .
27
2.7
Analytically solvable problems: nested harmonic wells . . . . . . . . .
29
2.7.1
Variance and bias with EXP . . . . . . . . . . . . . . . . . . .
30
2.7.2
Thermodynamic Integration . . . . . . . . . . . . . . . . . . .
31
viii
2.7.3
2.8
2.9
Analytically solvable models: offset harmonic wells of different
curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
Intermediate states . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
2.8.1
Variance and bias of Gaussians with intermediates . . . . . . .
34
2.8.2
Intermediates in the case of nested harmonic wells . . . . . . .
36
Direct computations on systems relevant to biomolecules . . . . . . .
37
2.9.1
Free energy of methane solvation from non-equilibrium work .
38
2.9.2
Free energies of solvation of methane from equilibrium simulations 39
2.10 Larger molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
2.11 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
2.12 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
2.13 Appendix: Bias of BAR . . . . . . . . . . . . . . . . . . . . . . . . .
44
3 Comparison of force fields
56
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
3.2
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
3.2.1
Computational methods . . . . . . . . . . . . . . . . . . . . .
63
3.2.2
Free energy calculations . . . . . . . . . . . . . . . . . . . . .
69
3.3
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
3.4
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
3.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
4 Comparison of water models
103
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.2
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.2.1
Computational methods . . . . . . . . . . . . . . . . . . . . . 108
4.2.2
Free energy calculations . . . . . . . . . . . . . . . . . . . . . 113
4.3
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.4
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
ix
5 Ligand binding with distributed computing
134
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.2
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.3
5.4
5.5
5.2.1
Computational methods . . . . . . . . . . . . . . . . . . . . . 140
5.2.2
Free energy calculations . . . . . . . . . . . . . . . . . . . . . 144
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.3.1
Precision of results . . . . . . . . . . . . . . . . . . . . . . . . 149
5.3.2
Free energy as a function of time . . . . . . . . . . . . . . . . 151
5.3.3
Sampling of degrees of freedom . . . . . . . . . . . . . . . . . 152
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.4.1
Sampling and precision . . . . . . . . . . . . . . . . . . . . . . 157
5.4.2
Accuracy
5.4.3
Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . 161
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Bibliography
175
x
Chapter 1
Equilibrium free energies from
nonequilibrium measurements
using maximum likelihood methods
1
CHAPTER 1. MAXIMUM LIKELIHOOD METHODS
Abstract
We present a maximum likelihood argument for the Bennett acceptance
ratio method, and derive a simple formula for the variance of free energy
estimates generated using this method. This derivation of the acceptance
ratio method, using a form of logistic regression, a common statistical
technique, allows us to shed additional light on the underlying physical
and statistical properties of the method. For example, we demonstrate
that the acceptance ratio method yields the lowest variance for any estimator of the free energy which is unbiased in the limit of large numbers
of measurements.
2
1.1
3
Introduction
Finding the free energy difference between different states of a physical system is of
great general interest in many scientific fields, including drug design [RE01], basic
statistics [GM98], and even non-perturbative quantum chromodynamics [SS98]. It is
of interest to the experimental community as well as the theoretical and computational
communities [LDS+ 02]. Recently, there has been increased interest in determining
the uncertainty and bias in any attempt to extract free energies from a suitable set of
data [GM98, Hum02, HJ01a, KC98, ZW02b, ZW02a, LK99, LK01a, LK01b, LSK03,
SPSP03].
We can separate the calculation of precise and accurate free energy differences into
two non-overlapping problems. First, we must generate a number, n, of statistically
uncorrelated measurements of the system, the particular measurement depending on
the type of free energy estimation performed. Second, we must extract a free energy
estimate from these n measurements, along with estimates for the statistical bias
and variances of our estimate. The generation of accurate estimates for statistical
uncertainty and statistical bias are vital for any such free energy estimate to be
useful. This first problem reduces to sampling the system in the proper manner, and
will not be dealt with here. We will assume that we are already in possession of a set
of n uncorrelated measurements of the proper observable for our method, and address
only the statistical issues related to the extraction of free energy estimates from these
measurements.
There are a variety of commonly used methods for finding the free energy of a
physical change in system. Many of these can be expressed finding the equilibrium free
energy from nonequilibrium work distributions. Thermodynamic perturbation theory
(TPT) or free energy perturbation (FEP) [Zwa54] estimates free energy differences by
exponentially averaging potential energy differences between a reference state sampled
at equilibrium and a target state [Zwa54]. However, FEP can be seen as a special
case of “fast growth” [Hum02, HJ01a], nonequilibrium exponential work averaging,
as the energy difference is the infinitely fast adiabatic work of transition between
the two states [Jar97, Cro00]. “Slow growth” thermodynamic integration has been
4
shown to have high intrinsic biases and is unreliable as originally implemented [PK89,
SM91a]. However, the “free energies” obtained from these simulations are actually
measurements from a nonequilibrium work distribution and an ensemble of values
can therefore also be used to obtain correct free energies [HJ01a]. The rest of this
paper will focus on the generalized nonequilibrium problem.
Assume there are two different equilibrium states defined on a phase space by
energy functions U0 (~q) and U1 (~q). Let ∆F be the free energy between these states,
defined as the log of the ratio of the partition functions associated with U0 (~q) and
U1 (~q). We can associate a work with the process of changing energy functions from
U0 to U1 or vice versa while the system is maintained in temperature equilibrium
with the surroundings. By sampling initial conditions from equilibrium, we obtain
a distribution in either direction of such work values. For infinitely fast switching,
these distributions are simply of ±∆U = ±(U1 − U0 ) canonically sampled from the
initial state.
It has long been known that the exponential average of equilibrium energy differences between two states yields the free energy difference between the states [Zwa54].
More recently, Jarzynski demonstrated that distribution of nonequilibrium work values can yield an equilibrium free energy by taking the exponential average of the
set of nonequilibrium work values [Jar97]. However, the exponential average of a set
of data X = {xi , · · · , xn }, defined as −(1/β) ln hexp (−βX)i (where β = 1/kT ), is a
statistic that is both inherently noisy and biased, even if the spread of the data is only
moderately larger than kT . The results of exponential averaging strongly depend on
the behavior at the tails of the distribution, which, by definition, are not as well sampled as the rest of the distribution. Previous studies have explored and demonstrated
the poor behavior of exponential averaging for small sample sizes [KC98, ZW02b,
ZW02a, LK99, LK01a, LK01b, LSK03].
In an examination of free energy estimation between two states sampled at equilibrium, Bennett [Ben76] demonstrated that it is possible to use the information
contained in both the forward and reverse distributions of the potential energy difference together in a manner which was significantly better than estimates using only
energy difference data from a simulation performed on one direction. This derivation
5
can trivially be generalized to the nonequilibrium work case, replacing ∆U with the
nonequilibrium work [Cro00]. Bennett, in the FEP case, and Crooks, in the general
case, showed that the equation:
exp(−β∆F ) =
hf (W )iF
hf (−W ) exp(−βW )iR
(1.1)
is true for any function f (W ), where we define the measurement from the initial
state to the final state as the “forward” direction (denoted by the subscript F ) and
the measurement from the final state to the initial state as the “reverse” direction
(denoted by the subscript R).
Bennett then minimized the statistical variance with respect to this function f (W )
to find that f (W ) = (1 + nF /nR exp(β(W − ∆F ))−1 is the function which minimizes
the variance in this free energy estimate, where nF and nR are the number of simulations in the forward and reverse direction, respectively. The free energy difference ∆F
can easily be found by iterative methods [Ben76]. Although this method is known
and referenced in the literature, it is strangely seldom used in practice despite its
theoretical advantages.
We have found that it is possible to derive this acceptance ratio method motivated
by an entirely different reasoning – the maximum likelihood of the free energy difference given a set of work measurements. This derivation has theoretical advantages.
Maximum likelihood estimators are particularly well-behaved estimators, as they can
be shown under relatively weak conditions to be asymptotically efficient, meaning
that no other asymptotically unbiased estimator can have lower variance [LC98].
The term “asymptotically unbiased” means that the estimate becomes unbiased as
the number of measurements goes to infinity. We therefore are able to show that the
acceptance ratio method is the best asymptotically unbiased estimate possible given a
set of nonequilibrium work values in the forward and reverse directions, unrestricted
by the functional form of the free energy estimate such as Eq. (1.1).
1.2
6
Maximum likelihood estimator derivation
We start from the fact that [Cro00]:
PF (W )
ln
PR (−W )
!
= β(W − ∆F )
(1.2)
where PF (W ) and PR (W ) are probability distributions for the work of nonequilibrium processes from two the states in opposing directions, arbitrarily labeled as F
(“forward”) and R (“reverse”). In order to simplify the notation, and without loss of
generality, we will replace the reverse work distribution with the equivalent distribution formed by substituting −W for W . PR (W ) will therefore refer to the distribution
of the negative of the reverse work.
Eq. 1.2 can be recognized as specific case of logistic regression, a common statistical technique usually used for epidemiological outcome prediction [HTF01], and we
will use several important results from this field in our derivation. Logistic regression
models are solved by maximum likelihood methods [HTF01, NM89], which we apply
here. Our specific problem is simplified because the exact value for β is given as an
input to the simulation; we need only estimate the free energy.
The ratio in Eq. 1.2 can then be written as P (W |F )/P (W |R) where P (W |F ) is
the conditional probability of a work value given that it is a forward measurement
and P (W |R) is the conditional probability of a work value given that it is is a reverse
measurement. We would like to compute the likelihood of a free energy estimate
of given a number of work measurements which are specified as either forward or
reverse. Although either P (W |R) or P (W |F ) can be eliminated, we are left with
one independent continuous free energy distribution, and writing either P (W |F ) or
P (W |R) in a closed form is system dependent. Although the problem can be solved
by finding the likelihood over the this continuous distribution without the need for
a closed form [And72, PP79], it is possible to rewrite this problem in a much more
tractable way.
Using the rules of conditional probabilities, and the fact that P (F |W )+P (R|W ) =
7
1, we rewrite this probability distribution as follows:
P (W |F )
=
P (W |R)
P (F |W )P (W )
P (F )
P (R|W )P (W )
P (R)
=
P (F |W )P (R)
P (F |W ) P (R)
=
P (R|W )P (F )
1 − P (F |W ) P (F )
(1.3)
We note that P (R)/P (F ) = nR /nF , where nF and nR are the number of forward and
reverse measurements respectively. We define the constant M = kT ln(nF /nR ) and
rewrite Eq. (1.2) as:
ln
P (F |W )
= β(M + W − ∆F )
1 − P (F |W )
(1.4)
Given Eq. (1.4) and an estimate for ∆F , we can rewrite the probability of a single
measurement P (F |Wi ) as:
P (F |Wi ) =
1
1 + exp(−β(M + Wi − ∆F ))
(1.5)
1
1 + exp(β(M + Wi − ∆F ))
(1.6)
Similarly,
P (R|Wi ) =
We now have expressions for the probabilities P (F |Wi ) and P (R|Wi ) given a value of
the free energy ∆F without introducing any additional parameters and eliminating
the need to parameterize a continuous distribution. We can then estimate the free
energy that would maximize the likelihood of having found the specified distribution
of forward and reverse probabilities at the specified values of W .
Given a value for the free energy ∆F , the overall likelihood L of obtaining the
given measurements can be expressed as a joint probability of obtaining the forward
measurements of work at the specified work values times the joint probability of
obtaining the reverse measurements at the specified work values:
L(∆F ) =
nF
Y
i=1
P (F |Wi )
nR
Y
j=1
P (R|Wj )
(1.7)
=
nF
Y
nR
Y
1
1
i=1 1 + exp(−β(M + Wi − ∆F )) j=1 1 + exp(β(M + Wi − ∆F ))
8
(1.8)
The most likely value of ∆F is the value that maximizes the likelihood, but it is usually
easier to solve the equivalent problem of maximizing the log likelihood. Taking first
the log of the likelihood and then taking the derivative with respect to ∆F and setting
it equal to zero we obtain:
nF
nR
X
∂ ln L(∆F ) X
−β
β
=
+
=0
∂∆F
i=1 1 + exp(β(M + Wi − ∆F )) j=1 1 + exp(−β(M + Wj − ∆F ))
(1.9)
or
nF
X
nR
X
1
1
−
=0
i=1 1 + exp(β(M + Wi − ∆F ))
j=1 1 + exp(−β(M + Wj − ∆F ))
(1.10)
The left side of Eq. (1.10) is a strictly increasing function in ∆F , and has limits of −nR
as ∆F → −∞ and nF as ∆F → ∞, so we are guaranteed that we have one unique
root. This value of the free energy difference ∆F is the maximum likelihood estimate
(MLE) of the measured data. This is the likelihood with a fixed probability for forward
and reverse measurements obtained from the ratio P (F |W )/P (R|W ), not a fixed
number of each type of measurement, obtained from the ratio P (W |F )/P (W |R).
However, it has been shown that these two approaches to logistic regression yield the
same values for the parameters being estimated [And72, PP79].
Eq. (1.10) is exactly equivalent to the Bennett acceptance ratio method, as can
be seen by comparison to Eq. 12a and Eq. 12b of Bennett’s paper describing the
method [Ben76]. In other words, the Bennett acceptance ratio yields the free energy
which, given a series of work measurements in the forward and reverse direction,
maximizes the chance these work values would be observed.
As mentioned before, the MLE is an asymptotically efficient estimator if certain
weak conditions are met. These conditions are that there exists a unique root in
the likelihood equation for every n, and that the third derivative with respect to
the parameter (here, the free energy ∆F ) is bounded [LC98]. These conditions are
satisfied in this case.
1.3
9
Variance estimates from maximum likelihood
methods
For a given method for estimating the free energy, it is important to know what
the uncertainty of the estimate is. The variance of the MLE of some parameter θ
asymptotically converges to 1/I(θ) as n → ∞, where I(θ) is the Fisher information
of the joint distribution of the work values with respect to the parameter θ [LC98].
The Fisher information is defined as the negative of the expectation value of the
second derivative of the log likelihood with respect to the parameter θ [LC98]. In our
case, the parameter is ∆F , the free energy difference, and the estimate of the Fisher
information given finite sampling is:
∂ 2 ln L(∆F )
−
(∂∆F )2
*
+


1
β2  X

=
2 i=ntot 1 + cosh(β(M + Wi − ∆F ))
(1.11)
where this sum goes over both the forward and reverse work measurements.
However, this is the variance estimate for a fixed probability of forward and reverse
simulations, not a fixed number of simulations in the forward and reverse direction,
and includes evaluations of free energy over the cases that nF is not the specified
number. We must include this restriction in order to obtain the correct variance
for the maximum likelihood estimate for the fixed number case. Anderson showed
that this difference can be compensated for by subtracting 1/nF + 1/nR from the
variance [And72], meaning that this overall variance can then be rewritten:
*
1 
1
2
β ntot
2 + 2 cosh(β(M + Wi − ∆F ))
+−1

ntot ntot 
−
+
nF
nR
(1.12)
where the average in the above equations is over all work measurements, both forward
and reverse. Anderson’s derivation is rather technical, and we provide an alternate
derivation as Supplementary Material to the original paper [SBHP03]. Comparing
the methods obtained by the maximum likelihood and Bennett’s original variance
calculation, we find that Eq. (1.12) can be identified with Eq. 10b in Ref. [Ben76], by
10
recognizing that ρ1 /ρ0 is simply equal to the factor exp(β(W − ∆F )).
In addition to the variance, the statistical bias of an estimator is important. Although we know that the MLE is asymptotically unbiased, we still need some sense of
the bias for finite numbers of measurements. Typically, the bias of a MLE is proportional to n−1 [NM89]. Since the standard error is proportional to n−1/2 , in general the
statistical bias will not be significant with the respect to the statistical uncertainty.
1.4
Discussion
There are strong connections between the acceptance ratio method and exponential
averaging. In the limit that nF nR , Eq. (1.10) becomes:
nR
i=nF nF
P
exp(−β(Wi − ∆F )) −
P
i=nR
1=0
nR hexp(−β(W − ∆F ))iF = nR
hexp(−βW )iF = exp(−β∆F )
(1.13)
Assuming that nR nF , we find that hexp(βW )iR = exp(β∆F ), and we recover
the well-known fact that the free energy is the expectation value of the exponential
average of either the forward or the reverse distribution. Thus, one can interpret
the exponential average as the maximum likelihood estimator for the free energy
difference in the limit that samples are only drawn from one distribution. No other
asymptotically unbiased estimate that draws from only one distribution can have a
lower variance.
However, in an analysis of the computational efficiency of the acceptance ratio
method as the ratio nF /nR is varied, Bennett showed that the optimally efficient
ratio was always relatively close to one [Ben76]. The limit of nR nF and nF nR is significantly less efficient, sometimes by three or four orders of magnitude.
Essentially, there is only one independent distribution, as the forward and reverse
distributions are related by Eq. (1.2). Above ∆F , the forward distribution will be
more statistically accurate in estimating this independent distribution, and below
∆F , the reverse distribution will be the most statistically accurate. It will therefore
11
always be a better to partition the available simulations or experiments between both
directions rather than explore the distribution of work values in one direction for
twice as long. Therefore, the acceptance ratio method will always be preferable to
FEP. We note that if FEP is performed in both the forward and reverse directions, as
is typically the case, then the same data can be used in the acceptance ratio method
– separate simulations need not be run.
The fact that the MLE is the asymptotically unbiased estimator with the minimum
variance does not necessarily guarantee that is the best estimator by all measures.
It is sometimes possible to find an estimator that is not asymptotically biased, but
has significantly smaller variance, resulting in a smaller mean squared error for a
finite number of measurements. Preliminary evidence indicates, for example, that
the probability distributions PR (W ) and PF (W ) can be smoothed by such methods
as convolution with a kernel and used in Eq. (1.2) to obtain mean squared errors
similar to or smaller than acceptance ratio estimates in some cases [N1A].
An important future question for applications is the nature of the related distributions PF (W ) and PR (W ). These distributions will depend on the underlying system
and process and cannot be computed or even necessarily estimated a priori. However,
an understanding of which features of the switching process lead to minimum variance distributions in the work will greatly facilitate the design of minimum variance
pathways and intermediate states between the end states, and therefore improve the
ability to calculate free energies with low bias and variance. Some work has been performed in toward understanding the features of the distribution in the case of TPT
[LK99, LK01a, LK01b], but the problem is still unresolved in the general case.
1.5
Conclusions
We have demonstrated that the Bennett acceptance ratio method can be interpreted
in terms of the maximum likelihood estimate of the free energy difference given a set
of nonequilibrium work walues in the forward and reverse directions. This extends
Bennett’s work by showing that the acceptance ratio method is the minimum variance
asymptotically unbiased estimate regardless of the functional form used. We have also
12
derived an expression for the variance of the Bennett acceptance ratio method using
the same methods. We note use of the Bennett acceptance ratio will always yield lower
variance than exponential averaging. We also note that the conclusions presented in
this paper are valid for the general case of work distributions from an initial state to
a final state, not only the specific case of energy differences from an equilibrium state
to another state.
Chapter 2
Comparison of efficiency and bias
of free energies computed by
exponential averaging, the Bennett
acceptance ratio, and
thermodynamic integration
13
CHAPTER 2. EFFICIENCY OF FREE ENERGY METHODS
Abstract
Recent work has demonstrated the Bennett acceptance ratio method is
the best asymptotically unbiased method for determining the equilibrium
free energy using work distributions collected from both equilibrium and
non-equilibrium simulations. However, it is not immediately clear what
the actual advantage of this acceptance ratio method over other methods
in realistic conditions. In this study, we first review theoretical limits of
the bias and variance of exponential averaging (EXP), thermodynamic
integration (TI), and the Bennett acceptance ratio (BAR). In the process, we present a new simple schema for estimating variance and bias of
many estimators, and demonstrate the connections between BAR and the
weighted histogram analysis method (WHAM). Next, a series of analytically solvable toy problems is examined to shed more light on the relative
performance in terms of the bias and efficiency of these three methods. Finally, sample problems of the insertion/deletion of both a Lennard-Jones
particle and a much larger molecules in TIP3P water are examined by
various methods. In all tests of atomistic systems, free energies obtained
with BAR have significantly lower bias and smaller variance than when
using EXP, especially when the width of the distribution is large. For
example, BAR can extract as much information from multiple fast, farfrom-equilibrium simulations as from fewer simulations near equilibrium,
which EXP cannot. Although TI and sometimes even EXP can be slightly
more efficient in idealized toy problems, in realistic atomistic situations,
BAR is significantly more efficient than all other methods tested.
14
2.1
15
Introduction
Finding the free energy difference between different states of a physical system is
of great general interest in many scientific fields, from drug design [RE01], to basic
statistics [GM98], to even non-perturbative quantum chromodynamics [SS98]. It is of
interest to the experimental community as well as the theoretical and computational
communities [LDS+ 02]. Recently, there has been increased interest in determining
the uncertainty and bias in any attempt to extract free energies from a suitable sets of
data [GM98, Hum02, HJ01a, KC98, ZW02b, ZW02a, LK01a, LK01b, LK99, LSK03,
SPSP03]. As these calculations or experiments are in general expensive, requiring
significantly more effort than simply measuring a single ensemble-averaged observable,
there is also great interest in maximizing the efficiency of such measurements.
There are essentially two disjoint problems that must be solved in order to calculate free energy differences precisely and accurately. First, we must generate a
number, n, of statistically uncorrelated measurements of the system, the particular
measurement depending on the type of free energy estimation performed. Second, we
must extract a free energy estimate from these n measurements, along with estimates
for the statistical bias and variances of our estimate. The generation of accurate estimates for statistical uncertainty and statistical bias are vital for any such free energy
estimate to be of use. This first problem reduces to sampling the system in the proper
(usually Boltzmann) manner, and will not be discussed further in this report. We will
assume that we are already in possession of a set of n uncorrelated measurements of
the proper observable for our method, and deal only with the statistical issues related
to the extraction of free energy estimates from these measurements.
In many cases, two two systems have so little overlap in phase space that useful
estimation of the free energy becomes impossible with simulations solely at the end
states. It is simple, however, to define a series of intermediate states, determine the
energies between these intermediate states, and sum these intermediate energies to
obtain a final answer. If the data is being collected by simulation, the intermediates states need not be physically realizable, only amenable to simulation. Only the
end states matter. This choice of a pathway of intermediate states can also greatly
16
influence the precision and accuracy of the free energy obtained. We will discuss
the question of pathway choice to some extent in this report, though we will not
fully explore the choice of maximal efficient pathway for arbitrary problems, instead
examining the efficiency assuming a suitable pathway has been found.
There are several commonly used methods for finding the free energy of a physical
change in a system. In “multicanonical” thermodynamic integration [SM91a], now
usually referred to simply as thermodynamic integration (TI), a parameterization
(in some variable λ) from the initial state to the final state is introduced. The
equilibrium ensemble average of the derivative of the Hamiltonian with respect to
this parameterization is then computed at a number of points along this pathway,
and integrated numerically to obtain the free energy difference.
“Slow growth”, in which the numeric integral of hdH/dλi over different equilibrium states is approximated by a single integral in which λ goes from 0 to 1 in a
time dependent manner, has been shown to have high intrinsic biases and generally
yields very poor results [PK89, SM91a], although the “free energies” obtained from
these simulations can be treated as measurements from non-equilibrium work distributions and exponentially averaged to obtain correct free energies [HJ01a]. This
can be done by means of the recently discovered Jarzynski equality, which specifies that if the non-equilibrium work W to take a number of systems in thermal
equilibrium from an initial Hamiltonian to a different final Hamiltonian is averaged
over the entire Boltzmann-weighted initial ensemble, the free energy (or equilibrium
work) between the two states ∆F = −β − 1 hexp (−βW )i [Jar97]. This remarkable
result is independent of the path, and depends only on the thermal equilibrium of
the inital state, and the process moving the states from one Hamiltonian to the
other obeying detailed balance. Using this relationship, other methods such “Fast
growth” [Hum02, HJ01a], and thermodynamic perturbation theory (TPT) or free
energy perturbation (FEP) [Zwa54] reduce to finding the free energy difference between two equilibrium states given a distribution of non-equilibrium work differences
between the states by exponential averaging [Jar97, Cro00]. FEP was originally developed in the context of estimating free energy differences by exponentially averaging
potential energy differences between a reference state sampled at equilibrium and a
17
target state [Zwa54]. However, this potential energy difference is simply equivalent
to the work of an infinitely fast adiabatic transition between the two states, and thus
can be interpreted under Jarzynski’s relationship as well. FEP and TPT are sometimes used to describe all these methods of finding differences between systems with
“perturbed” Hamiltonians, though most commonly they refer to finding free energies
through exponential averaging.
Recent work has shown that a neglected method, Bennett’s acceptance ratio
method [Ben76], is theoretically always more efficient than exponential averaging
in computing the free energy given a set of work values in both the forward and
reverse directions. [SBHP03, LSK03]. Working from different principles, an alternate
minimum variance algorithm called the weighted histogram analysis method has been
developed [KBS+ 92, SR01]. WHAM is usually used to compute potentials of mean
force or other thermodynamic observables along a reaction coordinate. From a histogram of intermediate simulations. However, in order to compute these observables,
the free energies between the states much be estimated. The expression for the free
energies reduces to the Bennett acceptance ratio method in the case of determining
the free energy between two states. However, despite the theoretical advantages,
many, and perhaps most, researchers do not use either of these methods in computation of free energies for chemical processes, most likely not because they are not
aware existence of these methods, but because they do not realize the advantages
these methods in practice.
In this paper, we will compare both theoretical and experimental variances and
biases of exponential averaging, Bennett acceptance ratio and thermodynamic integration. We will apply these methods to examine their variances and biases in
practice, first to a series of illustrative toy models, many of which yield analytical
results, and more importantly in a variety of simulations of free energies of sample
molecular systems of types that may be of practical use for chemical and biochemical
experiments.
In doing so, we will introduce some new derivations of these variances that are
significantly simpler than previous derivations, and illustrate connections to other
18
important methods, such the weighted histogram analysis method. We will sometimes use the abbreviations EXP for exponential averaging, TI for thermodynamic
integration, BAR for the Bennett Acceptance Ratio, and WHAM for the weighted
histogram analysis method.
2.2
Comparison of theoretical variance and bias
estimates
2.2.1
Underlying theory of free energy estimates
Assume there are two states defined by energy functions on a phase space, U0 (~q)
and U1 (~q). Let ∆F be the free energy between the states, defined as the log of the
ratio of the partition functions associated with U0 (~q) and U1 (~q). We can associate
a work with the process of changing energy functions from U0 to U1 or visa versa
while the system is maintained in temperature equilibrium with the surroundings.
By sampling initial conditions from equilibrium, we obtain a distribution in either
direction of such work values. For infinitely fast switching, these distributions are
simply of ±∆U = ±(U1 − U0 ) canonically sampled from the initial state.
It has long been known that the free energy difference between two states can
be computed by taking the exponential average of these energy differences [Zwa54],
where the exponential average of a set of data, X = {xi , · · · , xn }, is defined as
−(1/β) ln hexp (−βX)i, where β = 1/kT . Jarzynski demonstrated that distribution of non-equilibrium work values over the canonical ensemble of stating states also
yields an equilibrium free energy [Jar97], and indeed that the equilibrium exponential
average is a special case in the limit of infinitely fast changes.
However, the exponential average over a distribution is a statistic that is both
inherently noisy and biased, even if the spread of the data is only moderately larger
than kT . Exponential averaging depends a great deal on the behavior at the tails
of the distribution, which, by definition, are not as well sampled as the rest of the
distribution, and the results of exponential averaging will therefore will have both
high variance and bias. Previous studies have explored and demonstrated the poor
19
behavior of exponential averaging for small sample sizes [KC98, ZW02b, ZW02a,
LK01a, LK01b, LK99, LSK03].
Bennett showed that the value of ∆F which satisfied the equation:
nR
X
1
1
−
=0
j=1 1 + exp(−β(M + Wj − ∆F ))
i=1 1 + exp(β(M + Wi − ∆F ))
nF
X
(2.1)
where M = kt ln nf /nr , with nf and nr the number of values from the forward and
reverse distributions of work, respectively, minimized the variance in the estimation
of the free energy among free energy estimates of the form:
exp(−β∆F ) =
hf (W )iF
hf (−W ) exp(−βW )iR
(2.2)
where f (W ) is an arbitrary function. The left side of Eq. 2.1 is a monotonically
decreasing function for all ∆F , and is unbounded for both positive and negative ∆F ,
so we are guaranteed that we have one unique root for the free energy.
The overall variance of this estimate can be written as:
"*
1 
2
β ntot
1
2 + 2 cosh(β(M + Wi − ∆F ))
+#−1
!
ntot ntot 
−
+
nf
nr
(2.3)
where the average in the above equations is over all work measurements, both forward
and reverse. [SBHP03]
We have shown in a previous paper [SBHP03] that the Bennett acceptance ratio
can be understood as the maximum likelihood estimator of the free energy given a set
of forward and reverse non-equilibrium work measurements, starting from the basic
relation [Cro00]:
PF (W )
ln
PR (−W )
!
= β(W − ∆F )
(2.4)
where PF (W ) and PR (W ) are probability distributions for the work of non-equilibrium
processes from two the states in opposing directions, arbitrarily labeled as F (“forward”) and R (“reverse”). It is therefore the minimum variance estimator among
20
all asymptotically unbiased estimators, i.e., those estimators that become unbiased
in the limit of an infinite number of observations. No other asymptotically unbiased
estimator can extract a more precise value for ∆F information from a given set of
data.
We also demonstrated that EXP in one direction is the maximum likelihood estimator in the limit of absence of knowledge about the other direction. [SBHP03]
However, the lack of knowledge about the other direction greatly limits the precision
of this method, and using measurements from both distributions is always guaranteed
to be better [Ben76, SBHP03]. Various studies have explored and demonstrated the
poor behavior of exponential averaging in the limit of small numbers of measurements
measurements. [KC98, ZW02b, ZW02a, LK01a, LK01b, LK99, LSK03], and at least
one has confirmed the superiority of the acceptance ratio to TPT [LSK03] in a limited
number of physically relevant cases.
It is also now clear that WHAM between two states can be reduced to BAR. To
observe this, we take the iterative equations for WHAM for a two state problem with
two Hamiltonians:
*
exp(−βF0 ) =
1+
n1
n0
n0
n1
1
exp(βF0 − β(H0 − H1 ))
*
exp(−βF1 ) =
1+
+
1
exp(βF1 − β(H1 − H0 ))
*
+
0
+
n1
n0
n0
n1
1
(2.5)
exp(βF0 − β(H0 − H1 )) 1
*
+
0
1+
1+
+
1
exp(βF1 − β(H1 − H0 ))
+1
Eliminating F0 and F1 by replacement with ∆F = F1 − F0 and applying the relationship Eq. 2.4 yields the Bennett acceptance ratio formula Eq. 2.2. It appears
that the general formulation for WHAM for more than two states is not applicable
for general “fast-growth” simulations, as it does not not take into account the special relationships between paired forward and reverse work distributions expressed in
Eq. 2.4, but it is very possible that later research will discover a WHAM-like relationship between multiple work distributions, as the subject has not been sufficiently
explored.
2.2.2
21
Limiting moment bias and variances
Although complicated methods have been presented for the asymptotic bias of EXP [ZW02b],
we provide a extremely simple method that gives the dominant terms for both the
variance and bias of, known in statistics as the limiting moment approach [LC98].
This method is trivially generalizable to a wide class of other estimators.
Suppose that we have a random variable X, sampled n times, such that hXi = ξ
and var (X) = σ 2 . We define X = n−1
Pn
i=1
Xn . It has been shown [LC98] that if
we have a function h, given certain weak constraints on h, that the finite n bias in a
function of the mean of X is:
D
E
h(X) − h(ξ) =
σ 2 00
h (ξ) + O(n−2 )
2n
(2.6)
In the case that h = ln(X), all the constraints on h(X) required are satisfied, and we
obtain:
D
E
ln(X) − ln(ξ) = −
σ2 1
+ O(n−2 )
2n ξ 2
(2.7)
These equations can be obtained by examining the Taylor expansion of h(X). In the
case of exponential averaging, our random variable is not the work W , but instead
the exponential of the work, exp(−βW ). Let us define σ̂ 2 as the variance of X =
exp(−β(W −∆F )) for the forward case and X = exp(β(W −∆F )) for the reverse case.
This is different from exponential averaging only by a additive constant, and thus does
not affect the variance. With this choice, h00 (W ) = (h0 )2 = 1 when W = ∆F . We can
then write Eq. 2.7 as:
hln(exp (−βX))i − β∆F =
σ̂ 2
+ O(n−2 )
2n
(2.8)
The term σ̂ 2 /2n is the large n limit of the bias in the exponential average. This is
the same large n limit for the bias term derived by Zuckerman and Woolf [ZW02b].
We note that although the method presented here does not give information about
higher powers in n−1 , it is significantly simpler and trivially generalizable. In general,
statistics will be unreliable unless we have collected enough data so that the n−2 and
22
lower terms are negligible, so the fact that this gives only the n−1 term is not a serious
limitation of the large n limit approximation.
The variance from the mean of this same function can be expressed as:
var h(X) =
σ2 0
[h (ξ)]2 + O(n−2 )
n
(2.9)
In the case of h = ln(X), and X = exp(β(W − ∆F )), this reduces to:
var h(X) =
σ̂ 2
+ O(n−2 )
n
(2.10)
We again obtain a result identical to that of Zuckerman and Woolf [ZW02b], and
we similarly see that, to order n, the variance of the exponential average is half that
of the bias. We can observe that the reason for this particularly simple result is
that h00 (ξ) = [h0 (ξ)]2 for the case that h(x) = ln x. The uncertainty, or standard
error, is O(n−1/2 ) while the bias is O(n−1 ). This brings out an important point —
in most cases of asymptotically unbiased estimators, the variance will be a much
greater source of error in the calculations than the bias, as the uncertainty it will be
of O(n−1/2 ), whereas the bias is of O(n−1 ).
2.3
Averages of the forward and reverse simulations
From Eq. 2.4, we note that since the exponential averages of the forward and reverse
distributions have opposite signs, the bias will be in opposite directions signs for
the two distributions. Occasionally, the free energy from forward and reverse EXP
simulations are summed in order to remove some of the bias from the two individual
runs. In general, although the biases are in opposite directions, since the biases
are directly proportional to the variances, which can be drastically different for the
forward and reverse simulations, and therefore will not cancel.
The bias and variance of the average of the forward and reverse cases, ∆Fsum can
thus be computed as:
23
bias(β∆Fsum ) =
σ̂ 2
σ̂F2
− R + O(n−2 )
8nF
8nR
(2.11)
var (β∆Fsum ) =
σ̂F2
σ̂ 2
+ R + O(n−2 )
4nF
4nR
(2.12)
As has been noted [LK99, LK01a, LK01b], the simple average of the exponential
average is not the ideal combination of the forward and reverse work. Is there a way
to combine the forward and reverse exponential averages which will give a smaller
variance? If we estimate the free energy by the sum of the forward variance times
a factor a and the reverse variance by the factor 1 − a (such that the two weighting
factors sum to 1), we find that the variance will be minimized by
a=
nF σ̂R2
var (β∆F )R
=
2
2
nF σ̂R + nR σ̂F
var (β∆F )R + var (β∆F )F
(2.13)
This is, therefore, an improved combination compared to the simple average. We
note that summing together the biases for the forward and reverse cases using this
weighting and Eq. 2.8, the bias of order n−1 vanishes. So this weighting is also
the minimum square error linear combination of the forward and reverse exponential
averages.
Given σ̂F and σ̂R , what is the best choice of nF and nR given the constraint that
nF + nR = n? In other words, what is the best way to maximize computational
efficiency in choosing to run simulations in the forward or the reverse direction?
Substituting the weighting faction in Eq. 2.13 into the expression for the variance
and substituting n − nF for nR , we find an expression for the variance:
var (β∆F ) =
σ̂F2 σ̂R2
nF σ̂R2 + (n − nF )σ̂F2
(2.14)
to minimize with respect to nF . There are no extrema in the interval (0, n) and therefore the variance is minimized when all the trial runs are of the direction with the
smallest variance σ̂ 2 . The ideal division between forward and reverse distribution measurement in exponential averaging is therefore to sample both directions sufficiently
to roughly estimate the variances σ̂ 2 , and then sample only from the distribution with
24
the smallest variance. Kofke and co-workers have demonstrated that this sampling is
usually worse when sampling from the region of lower entropy [Kof02, LK01b, LK01a].
2.4
Bennett acceptance ratio
However, as is noted earlier in this paper, the best way to utilize a number of forward
and reverse work measurements is usually not to exponentially average them and
properly weight them, but to use the same data in BAR. In some cases, it may
only be possible or reasonable to run free energy simulations in both directions —
for example, where a large number of different ligand free energies are computed
simultaneously, in one step, from a non-physical intermediate state whose binding
free energy is precisely known. In this case, using only EXP is clearly preferable if it
is possible to converge accurately.
A limiting moment analysis of the Bennett acceptance ratio method is not directly
possible, as it is an implicit function of ∆F . Of course, we already have a perfectly
serviceable estimate for the variance, but we have no such estimate for the bias. It
is important, of course, to note that the bias is not simply 1/2 times the variance,
since the estimate is not simply the logarithm of an average. But there is a close
relationship to the variance. We now present an expression for the leading 1/n term
of the bias of BAR, which can be written in the form:
bias(β∆FBAR ) =
K
var(β∆FBAR )
2
(2.15)
where K is of order O(n0 ) in the number of measurements (see Appendix for derivation). Indeed, in most situations (as demonstrated in our sample problems), this K
is much less than 1. Whenever the forward and reverse distributions are symmetric,
for example, K is exactly zero.
2.5
25
Thermodynamic integration
What about the bias and variance of thermodynamic integration? TI is significantly
harder to compare theoretically with the other methods presented here, as the variance and bias that are obtained are not easily related to the variance and bias estimates for either EXP or for BAR. In TI, we first choose some pathway (here parameterized by the variable λ) between our initial state and final state, which we denote by
U (λ), where U (0) is the initial state and U (1) is the final state. There are an infinite
number of ways to define paths. Once we define a path, regardless of the choice, we
can compute the free energy as:
∆F =
Z
1
*
0
+
dU (λ)
dλ
dλ
(2.16)
and the variance of this free energy estimate, computing our averages at fixed λ, as:
var (∆F ) =
Z
0
1
!
Z 1
dU (λ)
var
dλ =
dλ
0
*
dU (λ)
dλ
!2 +
*
dU (λ)
−
dλ
+2
dλ
(2.17)
Inherently, there is no bias in this equation, if we could truly sample from the
equilibrium distribution along λ continuously. But to compute fixed-λ averages with
a finite number of samples, we must select finite values of λ to simulate at.
Most chemical and biological simulations are therefore performed at fixed values
of λ, and a free energy is generated by some sort of numerical integration. It is in
this step that the bias is usually introduced. The variance of simple averages is wellbehaved, in that relatively few uncorrelated measurements are needed to get a decent
estimate of the averages. Cursory analysis would therefore indicate that TI would
be preferable, if we could somehow sample along the entire “reaction coordinate” λ
profile. However, if we are running just two endpoint simulations, or are running just
a few intermediates, it is also easy to see how curvature along this pathway could
result in extremely poor free energy estimates where the other methods might be
preferable.
Though it is not commonly know in the chemical and physics community, it is
26
also correct to compute this average as an expectation value over both the ensemble
and λ [GM98]. This also gives the free energy without bias; however the variance will
then be:
var (∆F ) =
Z
0
1
*
dU (λ)
dλ
!2 +
dλ −
Z
0
1
*
dU (λ)
dλ
+!2
dλ =
Z
0
1
*
dU (λ)
dλ
!2 +
dλ − ∆F 2
(2.18)
which will always be somewhat larger. There have been some attempts to sample
along λ in the chemical literature [KB96, BGB00] However, there are several difficulties in this. Such methods are non-trivial to implement into various chemical
and biomolecular codes, and not particularly well-understood in comparison to other
sampling methods of a single Hamiltonian. The correlation times for re-equilibration
for most methods of sampling along λ are usually extremely long, and it is difficult
to devise schemes that sample along all of the λ range for the relevant Hamiltonians.
Additionally, this second variance is always larger than the fixed λ variance, so it may
not be an ideal method, though it bears further investigation in some cases.
The bias of EXP and BAR are inversely proportional to the number of samples
collected at each intermediate state. In the continuous λ case, the bias will always
be zero; whereas in the case of discrete λ case, bias of TI is of order one in the
number of samples; bias can only be reduced by running simulations at additional
intermediate points. An alternate way to decrease the bias for discrete λ sampling is
to use a higher order integration algorithm that the trapezoidal rule usually used –
for example, Simpson’s rule, or Gaussian quadrature [RM93]. While these algorithms
scale much better in the number of intermediates, they perform much more poorly
when the variance is high, as the limiting behavior depends on derivatives that are
much less well determined by the data than the function itself [SVG94]. Gaussian
quadrature also requires the variances, or at least estimates for the variances be known
beforehand in order to determine at which values of λ to collect data, which is usually
not practical. Simpson’s rule requires intervals that are equally spaced, a requirement
that may be at odds with functions where the curvature is concentrated primarily in
one region. For these reasons, we will only briefly touch on the use of these alternate
27
numerical integration methods for the use of general free energy calculations, though
in some specialized cases they may be appropriate.
2.6
Analytically solvable models: offset harmonic
wells
What results do these estimates yield for model systems — for example, in perhaps the
simplest case, where the distribution of work in both directions is a Gaussian? Such a
distribution can be generated by actual potential energy functions; for example, two
harmonic wells with identical curvature a, separated by some constant b, will result
twin distributions PR (W ) and PF (W ), where W is the potential energy difference
between the two, which are both Gaussians. If we have potential energy functions
HA = a(x + c/2)2 and HB = a(x − c/2)2 (see Fig. 2.1a), the energy difference W
at any point x will be simply 2acx. Assume we are sampling from the a(x + c/2)2
surface. The normalized probability of any state will be simply:
s
P (x)dx =
aβ
exp(−βa(x − c/2)2 )dx
π
(2.19)
As there is a one-to-one correspondence between x values and W values, making the
substitution W = 2acx yields:
s
P (W ) =
β
(W − ac2 )2
exp −β
4πac2
4ac2
!!
(2.20)
PR (W ) and PF (W ) will be Gaussians with variance 2kT ac2 , separated by 2kT ac2 ,
symmetric across zero.
Generally, then, let us assume that P0→1 (W ) is a normalized Gaussian, with mean
µ and variance σ 2 (σ in units of β 2 ). The free energy is therefore:
∆F = −β
−1
ln
Z
1
(W − µ)2
√
exp −
exp (−βW ) dW = µ − βσ 2 /2 (2.21)
2
2
2σ
2πσ
!
28
Remembering Eq. 2.4, we see that if P0→1 (W ) is Gaussian, with the given mean
and variance, then P1→0 (−W ) must also be Gaussian, with identical variance, with
the means between the two distributions differing by σ 2 . Without loss of generality,
we can set the free energy to be zero, and set the means of the two forward and
reverse Gaussians to be −βσ 2 /2 and βσ 2 /2 respectively.
What are the variance and biases of the free energy of a Gaussian distribution
with exponential averaging? With the theory developed earlier, we find them by
evaluating the function σ̂ 2 described above. With the symmetric distributions, the
free energy must be zero, so:
D
E
exp (−W )2 = hexp (−2W )i = exp σ 2
(2.22)
and the limiting term in the variance of EXP with n samples will be:
var(β∆F ) ≈
σ̂ 2
exp σ 2 − 1
=
n
n
(2.23)
with, as usual, bias equal to half the variance. As we can see, this has particularly
poor properties for σ > 1. If we were to average results from both the forward and
reverse directions, the variance would be unchanged (as both forward and reverse give
the same variance), but the bias would be zero, as the two estimates are perfectly
symmetric.
What are the variance and bias of BAR in this case? We have an equation for the
leading term of the variance (Eq. 2.3), but it is not particularly amenable for analytical
work. Gelman and Meng present an expression which demonstrates that the variance
for BAR with Gaussians is approximately proportional to σ 2 (exp(σ 2 /8) − 1) for large
σ [GM98], but we know that it in most cases it will be less than for EXP with the
same number of observations, as it extracts the maximum amount of information
from both the forward and reverse distributions.
The limiting behavior of the bias, in the case of the twin Gaussians symmetric
around zero is identically zero in the case of equal numbers of forward and reverse
simulations, much different from the case of EXP for Gaussians.
Next we examine the bias and variance of TI. For now, we will assume the most
29
“natural” case, that sampling can be done at any point along the λ path — later, we
will address the case of intermediate points. One simple path definition is simply the
linear interpolation between potential functions:
U (λ) = λUA + (1 − λ)UB = λa(x + c/2)2 + (1 − λ)a(x − c/2)2
(2.24)
In this case, it is simple to compute that hdH/dλi is ac2 (1 − 2λ), and the variance
(where σ 2 = 2kT ac2 ) is σ 2 + σ 4 /12, leading to a total variance of (1/n)σ 2 + σ 4 /12).
There are, of course, an infinite number of other possible paths. perhaps the
simplest make the constant c λ-dependent, so that U (λ) = a(x − (1 − 2λc)/2)2 ,
shifting the location of the harmonic well, but not its curvature. In this case, each
intermediate case is a Gaussian with the same curvature and hence same free energy,
so hdU/dλi = ∆F = 0. Then the variance is 2kT ac2 = σ 2 for all λ, leading to an
overall variance of (1/n)σ 2 . It is actually possible to construct pathways that have
even lower variance for this case [GM98], but even in the simplest cases it they are
difficult to derive.
In Fig. 2.2, we compare, as a function of σ in units of kT , the variance of EXP,
BAR, and TI. We plot the standard error (i.e., the square root of the variance) for
√
both EXP and BAR, as a function of σm in units of kT , and multiplied by n.
Clearly, as σ > 1, BAR becomes much more efficient than EXP. TI is even better
than BAR in this case, though not as drastically except at higher values of σ.
2.7
Analytically solvable problems: nested harmonic wells
Let us take the case of a model yielding slightly more complicated, non-symmetric
behavior. We take two 2-D harmonic potential energy functions, Ua = a(y 2 + y 2 ) and
Ub = b(x2 + y 2 ) (See Fig. 2.1b). The free energy can be easily computed for arbitrary
d-dimension harmonic wells to be d/(2β) log (a/b), yielding β −1 log (a/b) in this case.
Let us compute the forward and reverse differences in potential energy between
these two distributions, assuming a > b for simplicity. We will call the direction
30
in which we move to a higher energy surface “forward” (PF (W )), and the opposite
direction “reverse” (PR (W )).
If we are simulation on the lower surface b, then energy difference W = (a−b)(x2 +
y 2 ) corresponds to a state with energy b(x2 + y 2 ), which of course has normalized
probability (from the Boltzmann distribution):
P (r)dr =
βb
βb
exp −βb(x2 + y 2 ) dxdy =
exp −βb(x2 + y 2 ) 2πrdr
π
π
(2.25)
Since there is still a one-to-one correspondence between differences in energy and
states, we can substitute W = (a − b)r2 and dW = 2(a − b)rdr to obtain:
βb
−βbW
PF (W ) =
exp
a−b
a−b
!
βa
−βaW
PR (W ) =
exp
a−b
a−b
!
(2.26)
Similarly,
(2.27)
Applying EXP to these equations yields the correct free energy difference from b to
a, β −1 ln (a/b).
2.7.1
Variance and bias with EXP
Evaluating the variance of hexp β(W − ∆F )i, we obtain
1 (a − b)2
β 2 n 2ab − b2
1 (a − b)2
var (PR (W )) = 2
β n 2ab − a2
var (PF (W )) =
(2.28)
(2.29)
with the biases one half of this variance. Surprisingly, the variance and bias are only
defined for PR (W ) if a < 2b. If the upper well is more than twice as narrow as to
the lower well, the spread of values leads to a density that does not converge. This
is perfectly mathematically acceptable. This means that if a ≥ 2b, the asymptotic
variance will not simply be proportional to n−1 , but will be more complicated, and
31
usually, significantly larger. This demonstrates some serious consequences of the use
of EXP in determining free energies. This also demonstrates a clear example of the
fact, as noted before, the distribution from higher entropy (the well of lower curvature)
to yields a better EXP than the reverse distribution. Not surprisingly, BAR still fails
to yield an analytical form, but we graph the results of all three in Fig. 2.3.
2.7.2
Thermodynamic Integration
Using the simple linear interpolation, we have the potential energy function U (λ) =
λUA + (1 − λ)UB = λar2 + (1 − λ)br2 , and dH/dλ = (a − b)r2 . Then
hdH/dλi =
var (dH/dl) =
a−b
β(λa + (1 − λ)b)
(2.30)
(a − b)2
β 2 (λa + (1 − λ)b)2
(2.31)
yielding an overall total variance (integrating uniformly along λ) of:
(a − b)2
β 2 ab
(2.32)
Of course, there is nothing to prevent us from sampling more in areas of higher
variance, and less in areas of low variance — TI is still unbiased in this case. The
minimum variance is obtained by choosing a density of sampling that is proportional
to var (dH/dl)−1/2 at each λ value, yielding a final variance of β −2 log (a/b)2 . Of
course, this requires knowing beforehand the variance as a function of λ! One could
construct a situation, of course, where the sampling is dynamically reweighted to
different values of λ as data is collected. Care must be taken to ensure that this
dynamic sampling is done over long enough time intervals, or some regions of high
variance may not be sufficiently sampled. In practice, therefore, this ideal weighting
is not achievable.
If we take another pathway, for example, U (λ) = aL b1−L r2 , then we obtain
var (dU/dλ) = β −2 log (a/b)2 for all λ, yielding at total variance of β −2 log (a/b)2 ,
the same as the reweighted variance of above.
32
In Fig. 2.3, we compare the total variance as a function of a/b (with b = kT ), of
forward and reverse EXP, BAR, and the two pathways for TI. We see that reverse
EXP is extremely poor, with variance going to infinity when a > 2b. Forward EXP
is the most efficient free energy method in this case, a surprising fact, confirmed
with numerical simulations, given that the proof of the Bennett method finds the
minimum for all asymptotically unbiased estimators for given a set of forward and
reverse measurements. But it appears that the data from the low-information content
reverse sampling is making BAR worse than it would otherwise be. In any case, the
efficiency of BAR is very close to forward EXP, with both TI pathways close behind.
In Fig. 2.4, we compare the total bias as a function of a/b (with b = kT ) of the
forward and reverse EXP and BAR. We defer the discussion of the bias of TI to a
later discussion of the role of intermediates, as continuous sampling in λ is free of
bias. The negative of the reverse EXP and BAR bias are shown in the graph. We
note that the bias of BAR is extremely low over all the entire interval. The variance
of both EXP are simply the 1/2 the variance, so they have identical relative behavior
as in the previous graph.
2.7.3
Analytically solvable models: offset harmonic wells of
different curvature
It seems from the above cases that is may be relatively easy to construct such potentials. We give an example of one such other very simple analytically solvable problem
that demonstrates the difficulties in general, and show some of the nonintuitive nature
of these probability densities.
Take U (x) = a(x − c)2 + bx2 . This represents two harmonic wells, offset, and
with different curvature (Figure 2.1c). Again, we assume a > b and that call the
direction in which we move to the higher curvature surface “forward” (PF (W )), and
the opposite direction “reverse” (PR (W )). The free energy can easily be compute to
be the same as non-offset wells, namely β −1 log (a/b).
The energy difference W = (a − b)x2 − 2ac + c2 now corresponds to two possible
states, solving a quadratic equation for x. Both states are equally valid. Transforming
33
the probability density as done previously, we end up with normalized probability
densities:

PF (W ) = 
βb
q
4π abc2 + (a − b)W


+ exp 
−βb −c +

PR (W ) = 
ac +
βa
q
4π abc2 + (a − b)W

q

1/2 
2 
2
ac − abc − (a − b)W 



 
exp −βb −c +
a−b
q
abc2 − (a − b)W
a−b
2 

 


q

1/2 
2 
2 − (a − b)W
ac
−
abc



 
exp −βa −c +

a−b
q

2 
2
ac + abc − (a − b)W 

 
exp −βa −c +

+
a−b
(2.33)
which is, obviously much more complicated than any previous distribution equation
derived in this paper. The behavior of these probability densities is sufficiently interesting to warrant a bit of further examination. Looking at Fig. 2.5, the probability
curves interpolate between the Gaussians of the first toy problem, and the (1-D variations) of the logarithmic functions of the second toy problem.
In general, we need to find a reverse function to map from each sampled state to
the corresponding work value. As the complexity of the system increases, the number
of energy states which correspond to a given one-dimensional work difference increases
dramatically. With biomolecules in the condensed phase, any such equations become
completely unusable at best, and most cases unwritable. For non-equilibrium work
distributions, the computation of these work distributions cannot even be attempted
with these methods, as there are no equilibrium weightings of the individual states
to work from.
2.8
34
Intermediate states
The above analyses assumed either simulations were sampled with no intermediate
state, or in the case of TI, that any intermediate state is accessible to sampling.
These situation are frequently either not possible, or far from ideal. In the case of
BAR and EXP, the states of interest are frequently too far apart in free energy to
be accessible from each other with sampling from endpoints. In TI, the equilibration
times to switch between intermediate states are far too long, and so we must run
multiple simulations at a specified number of fixed λ intermediate states.
We will make these analysis assuming that that we are collecting our n samples
at over m states (intermediates plus the initial and final state). We will use the
same pathways as are described for TI in the examples above. For simplicity, we
will use equally spaced intermediates, although other spacings may prove preferable
in practice. In general, for well-chosen paths, using multiple intermediate states will
result in the measurement of free energies between states with more overlap, and
that therefore have better convergence properties. However, since with m states, we
will need to measure m − 1 free energies, whose variance and bias will generally be
additive, and each of these states will be sampled with only 1/m as many data points,
if the statistic of interest converges too slowly with increasing intermediate number,
then more intermediates can actually mean a greater variance or bias.
2.8.1
Variance and bias of Gaussians with intermediates
For simplicity, we first approach the case of harmonic potential wells with the simple
linear interpolation of the potential function, as in Eqn. 2.24. If we assume we are
measuring an energy difference between states with λA and λB , using the methods
developed earlier in this paper, we obtain W = 2acx(λB − λA ), and normalized
distribution:
s
PA (W ) =

β
W − ac2 (1 − 2λA )(λB − λA )
−aβ
exp
4πac2 (λa − λb )
4ac(λA − λB )
!2 

(2.34)
35
As this is still a Gaussian, but now with variance 2kT πac2 (λA − λB )2 , and average
ac2 (λA − λB )(1 − 2λA ), we can apply the same equations we did before. With equally
spaced intermediates, the variance of each distribution will be simply m−2 the original
σ 2 variance, meaning that the overall variance of EXP will be (with n/m observations
at each of m total states):
var (∆Fm ) = (m2 /n)(exp(σ 2 /m2 ) − 1)
(2.35)
In the c ∝ λ pathway, then
W = a(x−(1−2λA )c/2)2 −a(x−(1−2λB )c/2)2 = 2acx(λA −λB )+ac2 (λB −λA )(1−(λB +λA ))
(2.36)
However, the normalized distribution again ends up being a Gaussian with different
mean than in the previous example, but the same variance, 2kT ac2 (λA −λB )2 , yielding
the same overall variance as EXP, shown in Eq. 2.35.
Increasing the intermediates will always improve sampling for EXP, although the
efficiency falls off relatively quickly. In the limit if large m, the variance converges to
the T I case, σ 2 (see Fig. 2.6). The variance and bias of BAR, again, are not possible
to express analytically. Because BAR scales slightly greater than exponential as
discussed earlier (though an exponential with a much slower rate in increase than
EXP), it will actually reach a minimum with number of intermediates, and starts
increasing (Fig. 2.6).
We note that for no intermediates, we will need to run two sets BAR, but only
one for EXP. But the number of intermediates that needs to be run is m + 1/m, so
the more intermediates need to be run, the smaller this advantage of EXP. However,
if double wide sampling is used for EXP [JR85], where exponential averaging is run
from intermediate n to n − 1 and n + 1 and from intermediate n + 2 to intermediate
n + 1 and n + 3, this 2:1 advantage remains.
The variance for TI will be independent of the number of intermediates, since the
variance is the same at all points. In the linear case, bias would be identically zero for
any number of intermediate states as long as the states λ = 0 and λ = 1 are sampled
36
from. In the c ∝ λ case, the bias will be zero for any number of samples. This is,
of course, a special, even degenerate case, as the variance is the same at all λ and
hdH/dλi is a straight line in both cases (Fig. 2.6).
2.8.2
Intermediates in the case of nested harmonic wells
We now turn to the more complicated case of intermediates in nested harmonic wells
discussed earlier. Using a linear interpolation with λ, and writing k(λ) = b + λ(a − b)
we compute the distributions from λA to λB as:
βk(λA )
−βk(λA )W
PF (W ) =
exp
(λA − λB )(a − b)
(λA − λB)(a − b)
!
βk(λB )
−βk(λA )W
PR (W ) =
exp
(λA − λB )(a − b)
(λA − λB)(a − b)
!
(2.37)
Similarly,
(2.38)
yielding an overall variance of:
var (∆F (λA → λB )) =
(a − b)(λA − λB )
k(λA )(k(λA ) − 2(a − b)(λA − 2λB ))
(2.39)
There is no analytical results as a function of the number m states, but we can graph
the results. The formulas for BAR and TI are similarly non-analytical, so we will
examine the graphical comparisons for all methods. We use a ratio of a/b of 20, and
note that the variance will only be defined for reverse EXP when m > 20, and is
therefore not shown.
In Fig. 2.7, we see that only in the case of TI and reverse EXP does adding
intermediates improve the variance. Forward EXP and BAR are virtually the same,
with TI needing significantly more intermediates to converge. Examining the bias of
EXP, BAR, and TI (Fig. 2.8), we see that the bias of BAR is almost always small, and
it converges quickly to zero. TI bias starts out quite high, but rapidly converges to
zero as well. The bias of forward EXP, however, increases slightly from the minimum
obtained with only end states.
37
We also examine the logarithmic interpolation pathway between nested harmonics. In Fig. 2.9, we see that the variance of TI is independent of the number of states
sampled from. Forward EXP and BAR dip slightly below the TI variance, and then
plateau to the same, while reverse EXP asymptotically approaches this variance as
well. All variance converge to approximately the same value with increasing intermediates as with the linear path, but do so much more quickly as a function of the
number of intermediates used.
For the bias of the logarithmic path (no figure), we remember that the bias of
the forward and reverse EXP are just 1/2 the variance, and that the logarithmic
interpolation for TI always has no bias. The bias of BAR goes to zero with increasing
intermediate states, even more quickly than in the linear interpolation case.
The improved behavior of the logarithmic interpolation pathway makes sense, as
neighbors will tend to be more evenly spaced than with linear interpolation. For
example, if a = 20 and b = 1 as above, in the logarithmic case a single intermediate
√
will have prefix 20, whereas in the linear case, the intermediate will have prefix
21/2. Since the free energy is kT log (a/b), the logarithmic intermediate has a free
energy equally between the end states, where in the linear case the intermediate is
much close to the a end state.
2.9
Direct computations on systems relevant to
biomolecules
Theoretical analysis and toy problems are not sufficient to elucidate all the subtleties of free energy computations for more complex systems, so we turn to data
collected from simulations for more guidance and examples. We will first look at a
the free energy of Lennard-Jones sphere solvated in TIP3P model, first then using
work distributions generated from slow- and fast-growth simulations, and then work
distributions generated from EXP data with a number of different intermediates.
Next, we will examine the deletion/insertion of a model of 3-methylindole, the
core molecule of the side chain of the amino acid tryptophan. This system was
38
chosen primarily because of its use in other studies [SPSP03] — the primary interest
for this study is simply its large size, with 19 atoms, making it a difficult test case
and rigid structure, making it easier to remove the question of sampling uncorrelated
measurements.
2.9.1
Free energy of methane solvation from non-equilibrium
work
Our first test physical system is a Lennard-Jones sphere using the united atom
methane parameters of OPLS ( = 0.294 kcal/mol, σ = 3.73 Å) [JMS84] inside a
box of 216 TIP3P water molecules [JCM+ 83]. It is modified slightly in that inside
of 0.8σ, the Lennard-Jones term is replaced by a quadratic function, chosen such
that it is continuous and has a derivative at 0.8σ [HJ01a]. For kT = 0.592 kcal/mol,
(corresponding to 298K) the region inside 0.8σ is almost never occupied, so it is very
similar to the original Lennard-Jones. This formulation of Van der Waals attractions
was chosen so that insertions from zero interaction do not have numerical instabilities
resulting from the r−12 term as r goes to 0. Separate simulations estimated that the
free energy difference of implementing this modification to be about -0.02 kcal/mol
± 0.02 kcal mol.
The pathway in λ between the two states (here, the presence and absence of
the Lennard-Jones particle) used here is simply λ2 times the intramolecular energy
between the solute and the water, also the pathway used in [HJ01a], and not much
more complicated than the linear path described eariler. We will determine the
free energy of solvation of this Lennard-Jones sphere using the methods presented
above. For a reference, we first used exhaustive TI (71 total lambda values, 1.0 ns
at each value repeated in 4-fold replica, for ≈280 ns total sampling time) to obtain
2.654 ± 0.007 kcal/mol.
We compute the work required to insert and remove the Lennard-Jones sphere
from the box of water over a range of times from 500 ps to 0.1 ps. We fix the total
simulation time at 10.0 ns in each direction. We then compute the free energy from
BAR, EXP in the forward direction, EXP in the reverse direction, average of the two
39
EXP, and the optimally weighted EXP (using the weighting in Eq. 2.13), presented
in Table 2.1.
Uncertainties for all measurements were determined both by the bootstrap method [ET93],
using 1000 bootstrap samples, and the analytical variance and bias estimates presented in the theory section. We find that the bootstrap and the analytical uncertainties (with distribution variances estimated from the data) agree to within 15%
for all measurements, usually with significantly better agreement. In Table 2.1, we
therefore present only the analytic uncertainties. The forward and reverse EXP estimates use only half the data than the BAR, the average EXP, and the optimal EXP
averages do.
We see in Table 2.1 that BAR is uniformly good over all sets of data, although the
variance grows slightly as the work distributions are taken further from equilibrium
(i.e., from shorter simulations). Forward EXP is relatively good over most of the
range, but the uncertainties are higher using faster growth simulations. Reverse EXP
are also good for slow growth work distributions, but become very poor for fast growth
simulations, indicating that we are not sampling well from the reverse distribution.
The average and optimal exponential average combination have good behavior for
slow growth, with values close to the actual value and with low variances. However,
as the reverse average becomes poor, the average and optimal combination values
suffer as well.
2.9.2
Free energies of solvation of methane from equilibrium
simulations
To augment this data, a series of equilibrium simulations were performed, each simulating at a different set of fixed λs along the same pathway described in the nonequilibrium work section. 20 ns total time was run for each series, divided among the
different λ values equally, so that sets of more closely spaced λs were run for less
time at each λ. We ran 2, 3, 4, 5, 6, 8, and 10 values of λ, using equal spacing from
0.0 to 1.0, inclusive. Results from BAR and forward and reverse EXP are shown in
40
Table 2.2. We see that BAR is essentially correct over all the intervals, even including one-step insertion/deletion. Only forward EXP (insertion) gets close to the right
value in one step, but with a much higher variance. Even if we are two assume an
efficiency gained by double wide sampling, as discussed earlier, the uncertainty would
√
decrease by at most a factor of 1/ 2. This would mean an uncertainty in this case of
0.18 kcal/mol for one-step insertion, still significantly higher than the 0.08 kcal/mol
obtained using BAR. TI has a strong numerical bias that results in incorrect over all
the numbers of intervals. The curvature of hdH/dλi near λ = 0 (full deletion) is such
that a reallocation of λ sampling near the origin would have given a lower overall
bias for larger numbers of states, but clearly BAR is much more effective over a much
larger range of allocation of independent measurements.
We also reanalyze the EXP data for the one step insertion/deletion (the first line
in Table 2.2) to illustrate the nature of the variance and bias (as determined by
bootstrap sampling) of BAR compared to EXP. We compare only to forward EXP,
as the reverse distribution is not well sampled enough in this case.
It is clear from Table 2.4 that BAR estimate is uniformly better than the forward
EXP. Comparable results (both with respect to bias and variance) can be obtained
with only 10,000 × 2 measurements using the acceptance ratio estimate as with all
100,000 samples with exponential averaging, indicating that in this case, the acceptance ratio is five times more efficient for an equivalent precision. Since we know for
the agreement for large n that the analytical and bootstrap bias and variances agree,
the deviations for small n seem to indicate that we are now getting into the range
that the additional terms in the Taylor series expansion in Eq. 2.7 are significant.
2.10
Larger molecules
We now examine the case of the solvation of 3-methylindole, an analog of the side
chain of the amino acid tryptophan. This is the largest molecule examined in an
earlier study of force fields and small molecule solvation [SPSP03]. The molecule
was solvated in TIP3P water, and the free energy to take it to the gas phase was
measured. The complete methodological details are recorded in that paper, but we
41
review the key features here. A pathway was constructed that involved first turning
off the charges with a linearly in λ, and the Lennard-Jones intermolecular interactions
were turned off with a “soft-core” interaction [BMvS+ 94], which smoothes out the
infinity produced by the r−12 term as λ → 0. This pathway was previously shown to
be more efficient than pathway linear in λ or powers of λ. We used a total of 21 states
along the Coulomb pathway and 41 states along the Lennard-Jones pathway (both
numbers including end states), and sampled five runs of 1.0 ns each using molecular
dynamics.
Taking the OPLS-AA model of the molecule (several force field parameterizations
were used), the final value of the solvation energy was −3.69 ± 0.03 kcal/mol [N2B].
We neglect any long-range correction for Lennard-Jones attractive energy [SPSP03],
which is independent of the free energy method used. Analysis after the fact revealed
that approximately this number of intermediate states was required to eliminate bias
larger than the uncertainty. For example, using TI using only total states (chosen to
be sample areas of high curvature more than areas of low curvature) [N2A], we obtain
a free energy of −2.94 ± 0.26 kcal/mol, significantly different than our previous, more
accurate, estimate. The importance of a low curvature is especially evident in this
very realistic example — the charging energy using only three points is −6.11 ± 0.04
versus −5.77±0.02 for 21 points over a relatively flat curve, a smaller error than in the
energy the desolvation of the uncharged molecule, where we get 3.17±0.25 using seven
states 2.08 ± 0.04 using seven states over a function with much higher curvature (see
Fig. 2.10). We note that using Simpson’s rule to integrate the Coulombic part yields
5.76 ± 0.05, within uncertainty of the correct answer; however, it may be somewhat of
a coincidence in this case. For sampling along smoother curves where equal spacing is
reasonable, therefore, an increased order algorithm like Simpson’s rule may sometimes
prove useful.
How do BAR and EXP in both the insertion and deletion directions compare?
Using exactly these 8 states, insertion EXP gives a free energy of −3.44 ± 0.09,
whereas deletion EXP yields 6.01 ± 0.18. BAR gives −3.68 ± 0.02 kcal/mol, almost
indistinguishable to the TI result achieved with more than 7 times as much simulation
data. Although there may be some cancellation of error to achieve results so tight,
42
the results from BAR are still clearly better for the simulation time used than any
other method. These numbers are presented for comparison in Table 2.3. In this
case, BAR provides a clear advantage to TI and to either version of EXP.
2.11
Discussion
It appears that in realistic situations, BAR estimate will almost always be better than
EXP. For near-equilibrium measurements, this difference may be marginal, but for
larger free energy / phase space differences, this advantage can become appreciable,
as seen in all three examples. However, the example of nested Gaussians demonstrates that occasionally obtaining the free energy from EXP can be more efficient.
Care must be taken to identify such situations, and more study may need to be done
to understand when this can occur. Additionally, we have seen that in toy problems, TI is frequently the best behaved estimator. However, in realistic chemical and
biomolecular situations, pathways with moderate amounts of curvature can make TI
very inefficient. For charging free energies, which have a relative smooth profile, TI
may compete with BAR for efficiency.
There are a number of additional questions, beyond the scope of this study, that
must be addressed in order to effectively apply these methods to quantitative calculation of free energies. First, although we have presented formulas for estimating
the variance and bias due to limited sampling of the free energy methods, these estimates themselves are subject to sample size bias and can therefore not always be
depended upon to provide accurate uncertainty estimates. We note that although
the free energy depends on logarithms of exponential averages, the variance of the
free energy depends on exponential averages themselves and are therefore much more
dependent on good sampling than the free energy estimates themselves. It is possible that in cases where the distributions are particularly wide, alternative estimators
may be necessary to calculate the variance and bias to a sufficient level of precision
to determine the usefulness of a free energy estimate.
Recent work [YZ04] has used bias estimates obtained to correct the free energy
from EXP. However, since bias correction is inherently error prone, it remains unclear
43
if it is actually preferable to BAR. For extremely large energy differences, perhaps
this sort of block averaging bias correction could be applied to BAR.
Another important question for applications is the nature of the distributions
PF (W ) and PR (W ). There is, of course, only one independent distribution, as the two
are exactly related by Eq. 2.4. This distribution will depend on the underlying process
and cannot be computed or even estimated a priori. However, an understanding of
what features of the switching process lead to narrow distributions in the work will
greatly facilitate the design of ideal pathways and intermediates between the end
states being studied. Some work has been performed at understanding this difference
in the case of TPT [LK01a, LK01b], but the problem is still unresolved in any sort
of general case more complicated than the toy problems presented here.
Additionally, the fact that BAR is the asymptotically unbiased estimator with the
minimum variance does not necessarily guarantee that is the best estimator by all
measures. It is, in many cases, possible to find an estimator that is more inherently
biased, but has significantly smaller variances, resulting in a much smaller mean
squared error. Statistically, the most important figure of merit is the mean square
error, the sum of the variance and the square of the bias. Preliminary evidence
indicates, for example, that the probability distributions P (W )R and P (−W )F can
be smoothed by such methods as convolution with a kernel and used in Eq. 2.4 to
obtain significantly smaller mean squared errors in some cases [N1A].
As mentioned above, there choice of a path is vitally important for estimating
large free energies. We have seen that improved choice of path decreases the bias
and variance of all methods. There are many more questions to be asked about
improved pathways for given biomolecular simulations, and it a subject that has not
been sufficiently addressed in the literature.
2.12
Conclusions
We have demonstrated that the Bennett acceptance can be interpreted in terms of
the maximum likelihood estimate of the free energy difference given a set of work data
from the forward and reverse directions. We have derived expressions of the bias and
44
variance for both the Bennett acceptance ratio method and thermodynamic averaging
that are simpler and more general than previous methods. We have also demonstrated
that simple averaging of forward and reverse exponential averaging is rarely the best
use of available computational time, and in using exponential averaging, the free
energy of the minimum variance distribution should always be used, although determining which distribution is the minimum variance distribution can prove difficult
with a finite number of samples. We also note that the Bennett acceptance ratio will
almost always have lower variance and bias than exponential averaging, especially in
realistic simulations. We note that the conclusions presented in this paper are valid
for the general case of work distributions from an initial state to a final state, not
only the specific case of energy differences from an equilibrium state to another state
and for general work, as in the case of TPT.
We have found in a variety of realistic sample problems relevant to biomolecular
and chemical studies that BAR is always better than EXP in computing free energies,
frequently significantly better when the free energies to be measured are moderately
large compared with kT . Unless the curvature in hdH/dλi is extremely low (which
can be the case in electrostatic charging), BAR is much more efficient than TI as
well. BAR is actually simpler to use than TI, as it does not require code to compute
analytical derivatives to be implemented, merely that the potential energy be evaluated for the mutable part of the system at a variety of λ values. Because of this,
BAR (or an equivalent method like WHAM) is recommended as the default method
for computing free energies of biomolecular and chemical processes.
2.13
Appendix: Bias of BAR
This derivation will follow loosely from the derivation provided in by Mardia, et
al. [MST99]. Let us express the free energy estimate obtained from a finite number
of samples nf and nr from Eq. 2.1 as dfn , and the infinite estimate as ∆F . The bias
will be h∆Fn − ∆F i, where the average is over all possible realizations of the nf and
nr samples.
We will express the finite nf ,nr expression for the derivative of the log likelihood
45
as Ln (W ), with L(W ) as the infinite-n limit (with W simply a place-holder variable).
By construction, Ln (∆Fn ) = 0, and L(∆F ) = 0.
Because BAR is an asymptotically efficient estimator of the free energy [LC98], the
distribution of estimates of the free energy at large n tends to a normal distribution
with variance as given in Eq. 2.3. We can then estimate to first order:
∆Fn − ∆F ≈ var (∆F ) Ln (∆F ) + O(n−2 )
(2.40)
We can then express Ln (∆Fn ) as a Taylor series in ∆Fn :
1
0 = Ln (∆Fn ) = Ln (∆F )+(∆Fn −∆F )L0n (∆F )+ (∆Fn −∆F )2 L00n df +O((∆Fn −∆F )3 )
2
(2.41)
Solving for ∆Fn − ∆F , we get:
1
∆Fn − ∆F ≈ var (∆F ) (∆Fn − ∆F )L0n (∆F ) + (∆Fn − ∆F )2 L00n df + O(n−2 )
2
(2.42)
We could solve directly for ∆Fn − ∆F , but this will make our average more
difficult to compute. So we will estimate again by substituting var (∆F ) Ln (∆F ) for
∆Fn − ∆F as per Eq. 2.40, yielding:
1
∆Fn − ∆F ≈ var (∆F ) var (∆F ) Ln (∆F )L0n (∆F ) + (∆Fn − ∆F )2 L00n df + O(n−2 )
2
(2.43)
Taking the expectation value of this over all realizations of nf and nr , var (∆F ) is a
constant, and will go through the expectation operation. Additionally, the covariance
of ∆Fn − ∆F )2 and L00n (df ) will be O(n−2 ), so we can take the expectations separately
and multiply, yielding:
h∆Fn − ∆F i ≈ var (∆F )
var (∆F ) hLn (∆F )L0n (∆F )i
46
1
+ var (∆F ) hL00n df i +O(n−2 )
2
(2.44)
1
h∆Fn − ∆F i ≈ var (∆F )2 (2 hLn (∆F )L0n (∆F )i + hL00n df i) + O(n−2 )
2
(2.45)
Let us examine this equation. var (∆F )2 is of order n−2 , while hL00n df i is of order
n−1 . Since Ln ∆F = 0 + O(1/n), the dominant term in the Ln (∆F )L0n (∆F ) term
is the covariance of Ln (∆F )L0n (∆F ), which is proportional n−1 . We can therefore
rewrite this as:
1
h∆Fn − ∆F i ≈ var (∆F ) (var (∆F ) (2 hLn (∆F )L0n (∆F )i + hL00n df i)) + O(n−2 )
2
(2.46)
which can be written as:
1
h∆Fn − ∆F i ≈ var (∆F ) K
2
(2.47)
K = var (∆F ) (2 hLn (∆F )L0n (∆F )i + hL00n (∆F )i)
(2.48)
where
where K is independent of n by the analysis above. We note that:
Ln (∆F ) =
X
nf
X
1
1
−
(2.49)
1 + exp(β(M + W − ∆F )) nf 1 + exp(−β(M + W − ∆F ))
L0n (∆F ) =
X
nr ,nf
1
2 + 2 cosh(β(M + W − ∆F )
(2.50)
47
sinh(M + W − ∆F )
2 + 2 cosh(β(M + W − ∆F ))2
(2.51)
L00n (∆F ) =
X
nr ,nf
We note that Ln (∆F ) and L00n (∆F ) is antisymmetric around M − ∆F , whereas
L0n (∆F ) is symmetric. The components of PR (W ) and PF (W ) that are symmetric
will therefore cancel each other out. If nf = nr and therefore M = 0, then if PR (W )
and PF (W ) are symmetric (as in the case of Gaussian distributions), K and therefore
the leading term in the bias will be zero.
Offset Harmonics with Different Curvature
Energy
Energy
Nested Harmonics
Energy
Offset Harmonics
48
Generalized Coordinate
A
B
C
Figure 2.1: Three analytical problems from this paper: A: offset harmonic wells
where U1 (x) = a(x + c/2)2 and U2 (x) = a(x − c/2)2 , B: nested harmonic wells where
U1 (x, y) = a(x2 + y 2 ) and U2 (x, y) = b(x2 + y 2 ), with a > b (cross section shown), and
C: offset harmonics with different curvatures, with U1 (x) = a(x−c)2 and U2 (x) = bx2 .
Standard Error for EXP, BAR, and TI
Standard Error in KT
20
15
10
5
0
1
2
Σ in kT
3
4
Figure 2.2:
Standard error (square root of nvar (∆F )) versus σ (2kT ac2 ) of offset harmonic wells
for EXP (solid), BAR (dashed), and TI (dotted). BAR is significantly better than
EXP for almost all values of σ, and TI is somewhat better than BAR, although TI
takes advantage of certain simplicities in this toy problem.
49
Table 2.1: Free energies of hydration computed by both “fast-” and “slow-”growth.
The first column is the length of the MD runs in ps used to generate the distribution
of work values, and the second column is the number of repetitions used in the forward
and backward directions, for a total of 20 ns simulation. The third through seventh
columns are the free energy estimates and analytic uncertainty estimates from the
acceptance ratio, forward exponential averaging, reverse exponential averaging, the
average of forward and reverse, and the optimal combination of forward and reverse
exponential averages as per Eq. 2.14. All values in kcal/mol. The answer using
exhaustive TI (71 total lambda values, 1.0 ns at each value, 4 copies, for ≈280 ns
total) is 2.654 ± 0.007 kcal/mol.
time
500
200
100
50
20
10
5
2
1
0.5
0.2
0.1
n
20
50
100
200
500
1000
2000
5000
10000
20000
50000
100000
BAR
2.67±0.03
2.64±0.03
2.58±0.04
2.69±0.04
2.61±0.04
2.67±0.04
2.62±0.04
2.61±0.05
2.63±0.05
2.64±0.05
2.73±0.05
2.66±0.05
Forward EXP
2.70±0.04
2.67±0.04
2.65±0.06
2.69±0.08
2.67±0.09
2.49±0.11
2.75±0.13
2.59±0.20
3.04±0.12
2.63±0.19
2.58±0.28
2.33±0.22
Reverse EXP
2.65±0.05
2.65±0.06
2.58±0.10
2.67±0.08
2.36±0.07
2.24±0.08
2.69±0.36
2.51±0.41
1.49±0.30
0.14±0.10
-0.39±0.11
-0.81±0.12
Average
2.68±0.03
2.66±0.04
2.61±0.06
2.68±0.06
2.52±0.06
2.37±0.07
2.72±0.19
2.55±0.23
2.26±0.16
1.38±0.11
1.10±0.15
0.76±0.12
Optimal FEP Sum
2.68±0.02
2.67±0.02
2.63±0.04
2.68±0.04
2.49±0.04
2.32±0.05
2.74±0.10
2.58±0.13
2.82±0.09
0.64±0.06
0.01±0.08
-0.05±0.08
50
Table 2.2: Free energies of hydration computed with EXP. First column (N ) is the
number of EXP runs used. The number of ∆λ intervals is therefore N − 1. Columns
are the free energy estimate using the acceptance ratio, the exponential average in
the forward direction, and exponential average in the reverse direction. Uncertainties
are from the analytical estimate for the variance. The answer using exhaustive TI (71
total lambda values, 1.0 ns at each value, 4 copies, for ≈280 ns total) is 2.654 ± 0.007
kcal/mol. All free energies were estimated by running ≈20 ns total for each simulation,
divide equally among the states used. The potential energy difference was output
every 0.1 ps.
Runs
2
3
4
5
6
8
10
BAR
2.66 ± 0.08
2.63 ± 0.04
2.60 ± 0.02
2.68 ± 0.01
2.67 ± 0.01
2.67 ± 0.01
2.68 ± 0.01
EXP Forward
2.61 ± 0.26
2.66 ± 0.09
2.54 ± 0.05
2.62 ± 0.04
2.73 ± 0.02
2.68 ± 0.02
2.69 ± 0.02
EXP Reverse
-1.65 ± 0.06
-0.10 ± 0.12
1.34 ± 0.14
2.76 ± 0.13
2.47 ± 0.05
2.48 ± 0.08
2.52 ± 0.03
TI
-2.91 ± 0.003
-1.48 ± 0.008
-0.25 ± 0.01
1.24 ± 0.02
1.77 ± 0.02
2.41 ± 0.02
2.40 ± 0.02
Table 2.3: Free energies of hydration of 3-methylindole, in kcal/mol. Methodology of
simulations are as described in the previous study [SPSP03]
Method
TI
TI
FEP (A → 0)
FEP (0 → A)
FEP (Ave)
BAR
States
61
8
8
8
8
8
Coul
-5.77 ± 0.01
-6.11 ± 0.04
-5.57 ± 0.07
-5.68 ± 0.04
-5.63 ± 0.04
-5.69 ± 0.01
LJ
2.08 ± 0.02
3.17 ± 0.25
2.13 ± 0.07
-0.32 ± 0.18
0.90 ± 0.10
2.01 ± 0.02
Total
-3.69 ± 0.03
-2.94 ± 0.25
-3.44 ± 0.09
-6.01 ± 0.19
-4.72 ± 0.11
-3.68 ± 0.02
51
Table 2.4: Sample number dependence on the variance and bias of free energy calculations. The first column is n, the number of samples from the 100,000 sample set
used to estimate the free energy at a time, ranging from the whole set to 100. The
second column is b, the number of bootstrap samples used to compute the statistics.
The third and eighth columns are the average of the acceptance ratio or exponential
average estimate of n samples selected from the full set b times. Thus, all work values
are used on average an equal number of times. The fourth and ninth columns are
the analytic variance estimates from the full set, and the sixth and eleventh columns
are analytic bias estimates. The fifth and tenth columns are the bootstrap variance
estimates, and the seventh and twelfth columns are the bootstrap bias estimates.
n
100000
80000
50000
40000
25000
20000
10000
8000
5000
4000
2500
2000
1000
800
500
400
250
200
100
bootstraps
1000
1250
2000
2500
4000
5000
10000
12500
20000
25000
40000
50000
100000
125000
200000
250000
400000
500000
1000000
∆F
2.67
2.67
2.67
2.67
2.68
2.68
2.71
2.72
2.74
2.76
2.80
2.83
2.97
3.03
3.15
3.21
3.32
3.37
3.48
Acceptance Ratio
AVar BVar ABias
0.08
0.08
0.004
0.09
0.09
0.01
0.11
0.11
0.01
0.13
0.13
0.01
0.16
0.16
0.02
0.18
0.18
0.02
0.25
0.26
0.04
0.28
0.30
0.05
0.36
0.38
0.08
0.40
0.42
0.10
0.51
0.52
0.17
0.57
0.57
0.21
0.80
0.72
0.42
0.90
0.75
0.52
1.14
0.79
0.84
1.27
0.79
1.05
1.61
0.77
1.67
1.80
0.75
2.09
2.54
0.65
4.18
BBias
0.003
0.004
0.003
0.01
0.01
0.02
0.04
0.05
0.08
0.10
0.14
0.17
0.31
0.36
0.49
0.55
0.65
0.70
0.82
∆F
2.68
2.70
2.73
2.77
2.88
2.97
3.36
3.57
4.12
4.51
5.50
6.11
8.63
9.66
12.38
13.90
17.64
19.69
27.12
AVar
0.26
0.30
0.37
0.42
0.53
0.59
0.84
0.93
1.18
1.32
1.67
1.87
2.64
2.95
3.73
4.18
5.28
5.90
8.35
Forward
BVar ABias
0.28
0.03
0.33
0.07
0.43
0.12
0.49
0.15
0.66
0.24
0.75
0.29
1.23
0.59
1.48
0.74
2.03
1.18
2.36
1.47
3.11
2.35
3.55
2.94
5.17
5.89
5.78
7.36
7.36
11.77
8.18
14.72
10.03 23.55
10.90 29.44
13.64 58.87
BBias
0.07
0.09
0.12
0.16
0.27
0.36
0.75
0.96
1.51
1.90
2.89
3.50
6.01
7.05
9.77
11.29
15.03
17.08
24.51
52
Standard Error for EXP, BAR, and two TI paths
3
2
1
2
3
4
5
6
ab
Figure 2.3:
Standard error (square root of nvar (∆F )) versus a/b, the ratio of the curvatures
of nested harmonic wells for reverse EXP (thick solid), forward EXP (solid), linear
TI (long-dashed), logarithmic interpolation TI (dotted), and BAR (short-dashed).
Surprisingly, forward EXP is slightly better than BAR, and contrary to the previous
example, both methods beat TI.
Bias for EXP and BAR
3
Bias in KT
2
1
2
3
4
5
6
7
8
ab
Figure 2.4:
Bias times n (number of samples) versus a/b, the ratio of the curvatures of nested
harmonic wells for reverse EXP (thick solid), forward EXP (dashed), and BAR(thin
solid). The reverse and BAR biases are actually the negative of the bias. The bias
for BAR is very close to zero for for all a/b, with the bias of forward EXP increasing
linearly with increasing a/b, and the bias of reverse EXP exploding to infinity as
a/b → 2.
53
Work probability densities for offset, differently curved harmonic wells
Probability Density
1
0.8
0.6
0.4
0.2
0
1
Work in kT
2
3
Figure 2.5: Probability of forward (dashed) and reverse (solid) work distributions,
for (thick lines) a = 1.5, b = 1.3, and c = 0.4, and (thin lines) a = 1.3, b = 1.0
and c = 0.4. Relatively small changes in a and b drastically change the shape of the
distributions, roughly interpolating between the 1-D logistic curves of the second toy
problem, and the Gaussians of the first toy problem.
20
15
10
5
2
4
6
Number of intermediates
8
10
Figure 2.6: Standard error of EXP (solid), BAR (dashed), and TI (dotted) estimates
of the free energy, as a function of number of intermediates, from Gaussians with
variance σ 2 = 25. EXP converges to the TI result, but BAR reaches a minimum
before increasing in variance.
54
10
5
5
10
Number of states
15
20
Figure 2.7: Standard error of TI (solid), BAR (dotted), and EXP (dashed) estimates
of the free energy, as a function of number of intermediates, from nested Gaussians
with a/b = 20, using a linear path. Error from EXP and BAR increase slightly with
increase in intermediates, while TI initially starts high and decreased to a similar
value.
Bias for EXP, BAR, and TI
10
Bias in KT
8
6
4
2
5
10
Number of states
15
20
Figure 2.8: Bias of forward EXP (solid), BAR (dotted), and TI (dashed) estimates
of the free energy, as a function of number of intermediates, from nested Gaussians
with a/b = 20, using a linear path. Error from forward EXP increases slightly with
number of intermediates, while TI decreased quickly from relatively high values, and
BAR remains low in all cases.
55
5
4
3
5
10
Number of states
15
20
Figure 2.9: Standard Error of reverse EXP (solid), forward EXP, (dashed), BAR
(dotted), and TI (solid horizontal) estimates of the free energy, as a function of
number of intermediates, from nested Gaussians with a/b = 20, using a logarithmic
path. All values go to their asymptotic limit (approximately the same in all cases)
much more quickly than the linear path. Reverse EXP is the largest, while BAR and
forward EXP reach a minimum and then gradually increase.
dHdΛ
80
60
kcalmol
40
20
0
-20
-40
-60
0
0.2
0.4
0.6
0.8
1
Λ
Figure 2.10: hdH/dλiC (thick line) and hdH/dλiLJ (thin line) as a function of λ for
3-methylindole, the small molecule amino acid side chain analog of tryptophan, using
the OPLS-AA force field. Uncertainties are multiplied by 5.0 in order to guide the
eye, as they would otherwise be difficult to distinguish from the lines themselves.
Chapter 3
Extremely precise free energy
calculations of amino acid side
chain analogs: Comparison of
common molecular mechanics force
fields for proteins
56
CHAPTER 3. COMPARISON OF FORCE FIELDS
Abstract
Quantitative free energy computation involves both using a model that
is sufficiently faithful to the experimental system under study (accuracy)
and establishing statistically meaningful measures of the uncertainties resulting from finite sampling (precision). We use large-scale distributed
computing to access sufficient computational resources to extensively sample molecular systems and thus reduce statistical uncertainty of measured
free energies. In order to examine the accuracy of a range of common
models used for protein simulation, we calculate the free energy of hydration of 15 amino acid side chain analogs derived from recent versions of
the OPLS-AA, CHARMM, and AMBER parameter sets in TIP3P water
using thermodynamic integration. We achieve a high degree of statistical
precision in our simulations, obtaining uncertainties for the free energy
of hydration of 0.02 to 0.05 kcal/mol, which are in general an order of
magnitude smaller than those found in other studies. Notably, this level
of precision is comparable to that obtained in experimental hydration free
energy measurements of the same molecules. Root mean square differences from experiment over the set of molecules examined using AMBER-,
CHARMM-, and OPLS-AA-derived parameters were 1.35 kcal/mol, 1.31
kcal/mol, and 0.85 kcal/mol, respectively. Under the simulation conditions used, these force fields tend to uniformly underestimate solubility of
all the side chain analogs. The relative free energies of hydration between
amino acid side chain analogs were closer to experiment but still exhibited significant deviations. Although extensive computational resources
may be needed for large numbers of molecules, sufficient computational
resources to calculate precise free energy calculations for small molecules
are accessible to most researchers.
57
3.1
58
Introduction
An important goal of computational chemistry is the accurate prediction of free energies in molecular systems. The calculation of free energies is a computationally
intensive task, even for small systems, because it requires sampling of all thermally
relevant configurations of the system. Extensive work over the last 20 years has gone
into developing the theoretical and computational apparatus to perform free energy
computations, and these calculations can now be performed for many systems with
moderate levels of accuracy [CP02]. However, the accuracy of current calculations
greatly limits their applicability, especially for predictions of experimental observables
such as drug binding affinities. For example, the threshold for pharmaceutically useful
predictions is an accuracy of approximately 0.5-1.0 kcal/mol, implying a binding constant correct to within an order of magnitude of the experimental value. Obtaining
such precision in a repeatable and consistent manner is an unsolved problem [CP02].
Also, it is unclear to what extent uncertainties presented in many published studies
truly capture the statistical uncertainty inherent in free energy simulations [PC95].
There are two main problems in the quantitative prediction of interaction free
energies. First, there are the discrepancies between the models used for simulation
and the experimentally measured reality. This lack of accuracy means the model may
not adequately represent the experimental system under study. Second, the phase
space of the model must be sufficiently sampled to capture all thermally relevant
contributions to the ensemble average of the observables of interest. Otherwise, the
results will lack the necessary precision, and independent calculations will most likely
lead to differing answers. In general, insufficient attention has been given to separating
the differences between free energies computed from simulation and those derived
experimentally into contributions from inaccuracies in the model and the lack of
precision in the measurement of ensemble averages. Only if sufficient precision is
obtained in a statistically well-defined manner is it possible to design models that are
sufficiently accurate for the application at hand.
59
Difficulties in obtaining high-precision measurements In order for the free
energy of a physical or chemical process to be correctly computed by simulation techniques, thermally relevant configurations of the system must be thoroughly explored.
This is a difficult challenge for the simulation of large biological systems with many
degrees of freedom and long dynamical correlation times. Long, “brute-force” molecular dynamics simulations, typically on the nanosecond timescale, are currently insufficient to capture the long time scales of typical biological events, which are frequently
on the microsecond or millisecond timescale. Without proper statistical uncertainty
analysis, reported uncertainties can be significantly smaller than the spread in values
that would likely be generated under repeated independent executions of the simulation. One common practice in reporting computed free energies is to assign as an
uncertainty the hysteresis in the free energy as the “mutation” is performed forward
and backward. This is not actually a valid uncertainty estimate, as it is related to systematic error, not the statistical uncertainty. Moreover, it is frequently smaller than
the actual statistical uncertainty of the measurement [PK89, CKP96]. Additionally,
observables from molecular dynamics and Monte Carlo simulations can falsely appear
to have converged if these observables are coupled to degrees of freedom with long
correlation times [Ber91]. There is a need for rigorous analysis in the determination
and reporting of the uncertainty due to limited sampling.
Accuracy of models Current generation biomolecular force fields such as AMBER(ff94) [CCB+ 95], CHARMM22 [ADMBB+ 98], GROMOS [vGBE+ 96], and OPLSAA [JMTR96] are in many cases successful in capturing the qualitative behavior of
protein structure and dynamics [PB02, KM02, HOVG02]. These force fields for proteins were primarily built and parameterized to match small molecule data, both
experimental and theoretical. Parameters were usually tuned to give accurate fits to
quantum mechanical energy barriers, as well as to reproduce enthalpies of vaporization
and densities for pure liquids [CCB+ 95, ADMBB+ 98, vGBE+ 96, JMTR96]. However,
computational limitations usually did not allow parameters to be further tuned to also
reproduce accurate free energies of solvation in water, a thermodynamic measure that
is likely to be very relevant for biological processes. Other molecular force fields such
60
as TraPPE [CS99, CPS01, WMS00], NERD [NEdP98], and GIBBS99 [EP99] have incorporated phase equilibria data that implicitly include free energies of transfer, but
as yet do not include the molecular diversity necessary for biomolecular simulation.
It must also be kept in mind when performing a simulation that simply specifying a set of force field parameters is not sufficient to define a computational model.
Anything that affects the Hamiltonian changes the model, and thus will affect the
end results. For example, there are a large number of choices of simulation methodology, such as choice of boundary conditions, truncation or tapering of non-bonded
interactions, and constraint of bonds or other degrees of freedom, all of which can
change the Hamiltonian. Each of the force fields mentioned above was parameterized using certain assumptions about simulation procedures and parameters. Using
different simulation conditions than those used to parameterize the force field has
consequences for the observed results.
Purposes of present study In this paper, we have attempted to separate these
two sources of deviation from experiment for a simple test system, the hydration
of neutral amino acid side chain analogs. The correspondence between amino acids
and amino acid side chain analogs is shown in Table 3.1. We chose this system
for a variety of reasons.
First, experimental data are available for direct com-
parison [WACS81, BNM84, CGML81, HM75, AWFC90, VGSW99]. Second, these
molecules are simple enough that we expect to be able to obtain good statistical
sampling, thus allowing us to obtain a lower bound on the amount of the sampling
necessary for larger, biological systems. Third, a variety of well-defined, extensively
tested, and widely-used potential sets are available for these molecules – in this paper,
we use AMBER(ff94), CHARMM22, and OPLS-AA [N3A]. If we succeed in obtaining
sufficient sampling of these systems, and thus obtain high precision values, remaining deviations from experiment must arise from inaccuracies in these models – where
“models” refers both to the force field parameters and to the choice of simulation
protocols.
There are other possible choices of a test system, such as partition coefficients
of blocked amino acids between water and a hydrophobic solvent or the free energy
61
of hydration of whole or blocked amino acids, but these alternate test systems have
serious disadvantages. Partition coefficients are likely better measures of force field
utility for proteins than direct hydration, and extensive experimental data is available. Force field parameterizations of non-aqueous solvents (such as 1-octanol, cyclohexane, chloroform, and other solvents for which experiments have been performed)
have been proposed and partially explored [DK95, FK98, TVG94, EMK97, CPS01].
However, models for these solvents are less developed than are the various water
models, especially for solvent-solute simulations, and very few simulations have used
protein force field parameters in conjunction with these solvents. Since the solvent
model is just as important as the solute model in computing the free energy of solvation, we believe that computation of partition coefficients would be more a test
of these less-developed non-aqueous solvent models than of the more mature solute
models [N3B]. Additionally, published protein/solute parameters have generally been
developed for use with high dielectric solvents like water and may not be as accurate
in lower dielectric solvents. In the case of whole amino acid solvation, experimental
results for solvation of the entire amino acids are very difficult to obtain, as isolated
amino acids are extremely soluble in water in zwitterionic or even blocked forms. As
additional support for computing the free energies of hydration as a test of protein
force fields, the free energy of hydration of these amino acid side chain analogs is
highly correlated to such measures as Chothia’s criterion of hydrophobicity [Cho76].
(with R = 0.90) as measured from the distribution of amino acids within globular
proteins [WACS81, RW88]
When evaluating models, one is torn between using more recent simulation methods with the best proven scientific validity, the methods used for the actual parameterization of the model, and the most commonly used methods in the literature – which
may all be somewhat different. For example, most force fields (for both proteins and
water) were parameterized using simulations that employed finite-ranged non-bonded
interactions, which suggests that one should use the corresponding truncation protocol for non-bonded interactions in order to be faithful to the parameterization of the
model. However, truncation schemes for electrostatic interactions have been shown to
lead to qualitatively incorrect results, [BC92]. as compared with more sophisticated
62
and increasingly more common methods such as Ewald summation or the use of a
reaction field. Our goal here is to compare models and we choose to be faithful, where
possible, to the methods used in the original parameterizations. Additionally, despite
their limitations, cutoffs are still frequently used in simulations of proteins.
For sampling, we chose to use molecular dynamics as the most direct method for
isobaric-isothermal Boltzmann sampling of condensed phase molecular systems. We
realize that this method is not the most computationally efficient in general. But it
is sufficient for the problem at hand, as well as being simple to implement, frequently
utilized, well-studied, and well-understood.
Some studies have examined the free energy of hydration of a few representative
molecules using standard force fields in explicit water simulations [CNOJ93, VM02,
KDMJ94b, JN93, HW98, CS00]. However, there has not been a comprehensive attempt to compare common force fields for their accuracy in predicting the hydration
free energies for a set of molecules relevant to protein simulations employing explicit
representations of water. One reason has been a historical problem of adapting force
fields to a diversity of molecular dynamics software packages. The code used in this
study is a modified version of TINKER [tin], which supports a variety of common
force fields. Another major reason for this lack of extensive comparison is a lack
of sufficient computational resources to evaluate free energies of large numbers of
molecules with sufficient precision. Using the power of distributed computing [SP00],
we are able to access sufficient computer time for such large computational tasks.
These results can serve as one possible test of the applicability of current protein
models studied with explicit representation of water molecules. The present work
could guide the development of more accurate force fields. It demonstrates that it is
now easier to incorporate very important yet computationally intensive observables
such as hydration free energy into force field parameterization efforts. Free energy
simulations are useful parameter verification methods as they quantify the effects of
both the enthalpy and entropy of molecular interactions.
3.2
3.2.1
63
Methods
Computational methods
The Folding@Home distributed computing infrastructure [SP00] was used to perform the free energy calculations presented in this paper. Folding@Home (at http://
folding.stanford.edu) is a collection of approximately 200,000 volunteered computers
across the world running the Folding@Home client program. Volunteered computers automatically download calculations to be performed and return the processed
results to servers at Stanford University. Folding@Home has previously been used
to perform massively parallel calculations successfully simulating the folding kinetics
of small peptides and proteins [SNPG02, ZSP01, PBC+ 03, ZSK+ 02, LSSP02] and to
design ensembles of new protein sequences which fit to specified structures [LEDP02].
The main data set alone (3 force field parameter sets × 15 amino acid side chain
analogs × 5 trials × 1.2 ns × 61 λ values) represents approximately 140 CPU-years
on a 1 GHz Celeron, and all of the simulations together represent closer to 200 CPU
years. The majority (greater than 95%) of data was collected in less than two months,
approximately equivalent to an 800 processor cluster of similar machines working
uninterrupted for the same time period. These calculations used only a small fraction
of Folding@Home’s total capacity at any given time.
The simulations were performed using a modified version of the molecular dynamics suite TINKER (v3.8) [tin]. Our modifications included adding Andersen pressure
and temperature control [And80] and a variety of methods for free energy computation using thermodynamic integration (such as “soft-core” λ dependence [BMvS+ 94]),
as well as adding some performance optimizations.
Parameters for the amino acid side chain analogs were generated from the versions
of AMBER(ff94) [CCB+ 95], CHARMM22 [ADMBB+ 98], and OPLS-AA(1996) [JMTR96],
force fields included in the TINKER v3.8 distribution. CHARMM22 is the most recent CHARMM version for proteins (CHARMM27 parameters are only different for
nucleic acids and lipids). ff94 and ff96 in Amber differ only in peptide backbone
64
torsion angle parameters [KDC+ 97], and ff98 did not change any protein parameters from earlier parameter sets, so these differences are not relevant here. OPLSAA [JMTR96] parameters used date from October 1997 and are similar to the most
recent OPLS-AA/L [KFRJ01], only differing somewhat in the torsion angle and the
sulfur Lennard-Jones parameters. In this paper, we will refer to these parameter
sets as CHARMM, AMBER, and OPLS-AA, recognizing that there are variants of
each. We studied all the neutral amino acid side chains except for those of proline
and glycine. In addition, we examined analogs of both neutral protonation states of
His (referred to as Hid and Hie analogs, where the proton is attached to the δ and nitrogens, respectively).
We have omitted the side chain analogs of amino acids carrying a formal charge
at neutral pH in our computations. A finite-ranged treatment for non-bonded interactions is ultimately inappropriate for the inherently long-ranged Coulombic interactions of charged species. Complex corrections to free energies from finite-ranged
simulations are required to recover approximations to the full electrostatic free energy of charged species, and errors in these approximations are somewhat uncontrolled [RM98, SB88]. Experimental data does exist for the free energy of hydration
of these charged amino acid side chain analogs at alternate pH values where they are
neutral [WACS81], but not all of the force fields have parameters defined for these
neutral species.
Some modifications to the side chain β-carbon parameters were necessary in order
to use the protein force field parameters for the side chain analogs. For the non-bonded
parameters for these atoms, the Lennard-Jones term was taken from the standard
aliphatic carbons (either CH3 or CH2 , depending on the amino acid) of each parameter
set. The partial charge of this atom was determined for each amino acid side chain
analog by reducing the partial charge on the β-carbon by the amount of the charge
of one hydrogen atom (see Figure 3.2). If bonded parameters (bonds, angles, and
torsions) were already defined in the parameter set for atoms of the resulting type and
connectivity, these were used. Otherwise, the bonded terms originally assigned that
atom were retained. TINKER input parameter and coordinate files for all the amino
acid side chain analogs in all parameter sets used are provided as Supplementary
65
Material to the original paper [SPSP03].
In the case of the AMBER parameter set, additional changes were necessary.
CHARMM and OPLS-AA are formulated to have electostatically neutral side chain
moieties, while AMBER charges are only neutral for the entire residue. To create a
neutral side chain analog with the AMBER parameters, we followed the procedure
for adding an additional hydrogen described in the previous paragraph, and then
added the charge on the β-carbon necessary to make the overall side chain analog
neutral. The average of this last correction to the partial charges on the β-carbon
across the AMBER parameter set was -0.0725, with root mean square difference
0.021, and maximum difference of -0.1197 (for the leucine analog). These changes
result in a slightly different dipole moment for these side chain analog atoms than
might occur if the same atoms were in an intact residue, or alternatively if the charges
were rederived with ab initio calculations, and will therefore slightly change the free
energy of solvation. To understand the effect in hydration free energy that this dipole
moment difference would create, we performed a separate simulation on the Leu
analog in which the correction to the charges to achieve neutrality was performed by
adjusting the charges on the three hydrogens attached to the β-carbon instead of the
charges on the β-carbon itself, with the assumption that most other choices would
fall between these extremes. The difference in hydration free energies obtained was
of the size of the uncertainty (as discussed at length in the Results section). This
demonstrates that differences between this simulation and simulations with other
possible variants of the AMBER parameters are likely to be statistically insignificant.
The original TIP3P water model [JCM+ 83] was used, which is rigid and includes
no Lennard-Jones term on the hydrogens. This is slightly different than the TIP3Plike water model used to develop the CHARMM parameter set, which does have
Lennard-Jones terms on the hydrogens [ADMBB+ 98]. TIP3P (including slight variants, like the CHARMM TIP3P mentioned above) is probably the most common
water model used in explicit water simulations, and was selected to best capture typical current methods, even though TIP4P and TIP5P have been shown to reproduce
some properties of water more effectively than TIP3P [MJ00].
Thermodynamic integration was used to compute the hydration free energy (full
66
details below), with molecular dynamics used to compute the ensemble average of
dH/dλ at a number of values of λ. For each value of λ, simulations using a 2 fs
time step were equilibrated for 200 ps, followed by 1.0 ns data collection. Since each
simulation was carried out in five-fold replica, a total of 5 ns of simulation time was
collected at each value of λ. Integration of the equations of motion were performed
using the velocity Verlet algorithm [SABW82], and all bonds were constrained using
RATTLE [And83] with a distance constraint of 10−6 Å. To allow for better load
balancing in the Folding@Home distributed computing environment, simulations were
divided up into blocks of 100 or 200 ps. The only side effect of this windowing is on
the computation of the correlation times of the system, as we discuss below.
Andersen pressure control[And80] was used to produce controlled pressure simulations. Andersen pressure control (in conjunction with Andersen temperature control) rigorously implements an isobaric-isothermal (NPT) ensemble. The pressure
was calculated using a molecule-based virial, with the molecular centers defined as
the molecular center of mass. Isotropic scaling of the cubic volume at each step
was performed by scaling the coordinates of molecular centers, not the atomic positions [RC86, RC83, CR86]. A piston mass of 0.0004 g mol−1 Å
−4
was used. The
thermodynamic observables are formally independent of the choice of the piston mass.
Therefore, as long as the piston is heavy enough that the simulation is stable with
the choice of timestep, and light enough so that that the timescale for the volume
fluctuations (which were of order 1 ps for these simulations) is much shorter than the
simulation length, the value of the piston mass is unimportant. Periodic boundary
conditions were used and the simulation cell was constrained to a cubic shape. At
all times, the edge length of the simulation cell was more than twice the range of the
finite-ranged potentials used.
All simulations were carried out at 298 K. Andersen temperature control was implemented by reassignment of all velocities from the Maxwell-Boltzmann distribution
at periodic intervals, which in the limit of long time is rigorously equivalent to an
isothermal ensemble [And80]. In the 1.0 ns simulations presented here, velocities were
reassigned every 500 steps (1.0 ps), resulting in 1000 separate reassignments. The average kinetic energy of the simulations was checked to verify that it was in agreement
67
with the control temperature of 298 K.
Statistical uncertainties, δA, for all observables, A, were calculated by using the
variance, hA2 ic , and correlation time of the observable, τA , over the entire T = 1.0
ns simulation. The uncertainty in the mean for the observable A is then given by
the formula δA =
q
q
hA2 ic 2τA /T [SABW82, AT87]. δA corresponds to one standard
deviation, σ, and all results are reported as hAi ± 1σ.
The normalized fluctuation autocorrelation functions were calculated with a resolution of 20 fs within the window of each 100 or 200 ps block sent to each participating
Folding@Home machine. Correlation times for the components (Coulombic or Van
der Waals) of dH/dλ were computed by numerically integrating the autocorrelation
functions of each of these observables by the trapezoid rule [N3D]. It is possible
there are correlations comparable to or longer than 100 or 200 ps in the components
of dH/dλ, thus making this windowing inaccurate. To ensure the validity of the
windowing procedure and to identify the possible presence of long-time scale correlations, values were sampled every 0.2 ps for the entire 1.0 ns run (for a total of
5000 samples) for 20 of the 45 molecule/force field combinations, and autocorrelation
times were computed for these entire 1.0 ns runs for comparison with the windowed
correlation times. These times were found to deviate little, as discussed below.
A single simulation is not statistically relevant; it is usually necessary to run
multiple independent simulations to verify that the computed uncertainty estimates
from single simulations are reasonable. For each combination of force field and amino
acid side chain analog, we ran five separate simulations and examined the deviations
of the individual results from their mutual average to decide if the simulations had
truly converged.
Charge-neutral group-centered tapered cutoffs were used for the non-bonded interactions of both solvent and solute. In all cases, the smallest (in number of atomic
sites) possible groups of contiguous atom centers which had zero total partial charge
were selected, which resulted in different group definitions depending on the parameter set used. The center of each group was chosen as the atom closest to the geometric
center of the group - if two atoms are equally close, the atom with the largest absolute
68
value charge was chosen as the group center. We used combining rules for LennardJones terms and exclusion rules for 1,4 terms appropriate for each force field.
Interactions between atoms using group switching are defined as follows. We
first compute the standard Lennard-Jones term (representing the Van der Waals
interactions) and Coulombic term for each atom pair i and j located at ~ri and ~rj :

σij
qi qj
+ 4ij 
U (rij ) =
rij
rij
!12
σij
−
rij
!6 

(3.1)
where rij = |~ri −~rj |, qi is the partial charge on atom i, and ij and σij are the standard
Lennard-Jones parameters. If the beginning of the switching region (moving outwards
from rij = 0) is defined as rswitch and the end of the switch region is defined as roff ,
then the interaction energy between neutral group A, with nA atomic sites and group
center located at rA , and neutral group B, with nB atomic sites and group center
located at rB , is defined as:
UAB = S(rAB )
nA X
nB
X
U (rij )
(3.2)
i=1 j=1
where rAB = |~rA − ~rB |, and S(rAB ) is defined as [SB94]:
S(rAB ) =


1







for 0 < rAB < rswitch
2 −r 2 )2 (r 2 +2r 2 −3r 2
(roff
AB
AB
off
switch )
2 −r 2
3
(roff
)
switch
for rswitch < rAB < roff
0
for rAB > roff
(3.3)
where A and B are the centers of the groups to which atoms i and atoms j respectively
belong.
This function of r2 satisfies the conditions S(roff ) = 0, S(rswitch ) = 1, and
dS(roff )/dr = dS(rswitch )/dr = 0. We note than in order to strictly conserve energy,
S(r) would also have to satisfy the conditions that d2 S(roff )/dr2 = d2 S(rswitch )/dr2 =
0. However, trial constant energy simulations with 900 TIP3P molecules using 2 fs
timesteps exhibited no statistically meaningful drift in total energy, even for simulations as long as 200 ps. Short trajectories from the same starting point were run
69
with varying time step size, and the standard deviation of the total energy (or enthalpy, in the case of the isobaric-isoenthalpic (NPH) ensemble) was quadratic in the
timestep size (as checked down to 0.125 fs), consistent with the inherent accuracy
of the velocity Verlet algorithm [SABW82]. For the main data set, all solvent-solute
and solute-solute interactions were tapered with rswitch = 10Å and roff = 12Å, while
the solvent-solvent interactions were tapered from rswitch = 8Å to roff = 10Å [N3G].
To prepare the simulation cell, a cubic box of 900 TIP3P water molecules was
taken from a larger, previously equilibrated water box and was further equilibrated
under NPT conditions described above for an additional 200 ps. The average volume
3
over this equilibration run was 2.700 ± 0.003 × 104 Å . We obtained a density of
0.997 ± 0.001 g cm−3 for pure TIP3P water, in good agreement with the experimental
density of water at 298 K [Lid01]. Our computed density is only slightly different
than the 1.001 ± 0.001 g cm−3 previously reported [JJ98]. We believe this difference is
most likely attributable to different truncation schemes for long-range interactions, as
well as an apparent slight system-size dependence of densities and other observables
under periodic boundary conditions [N3C]. However, the pair distribution functions
of the O– O, O– H, and H– H atoms pairs agree well with previously published
results [JJ98] as does the heat capacity (19.9 ± 0.2 in this study versus 20.0 ± 0.6
cal/mol K). The amino acid side chain analogs were then solvated by placing them in
the center of this box and removing all water molecules with atomic centers within
1.5 Å of any atomic center of the solute. An extremely short (200 steps) NVT
(isochoric-isothermal) simulation with very frequent velocity reassignments was then
used to remove highly unfavorable interactions. A further 200 ps NPT equilibration
simulation was performed for each value of λ prior to the 1.0 ns of data collection.
3.2.2
Free energy calculations
Experimental free energies of hydration for weakly soluble solutes are determined from
concentration measurements made on two phase systems, where one phase consists
of a vapor with a partial pressure Ps of some solute molecule of type s, and the other
phase consists of an aqueous solution with a number density concentration for solute
70
molecules of ρls . When such a two phase system is at equilibrium with respect to
transfer of molecules of type s between the phases, the solvation free energy is given
by[WACS81, BNM84]:
∆Gsolv = kT ln(Ps /ρls kT )
(3.4)
Free energies of hydration are computed by simulation by decoupling a molecule of
the solute (here, amino acid side chain analogs) from the solvent by thermodynamic
integration (TI), using the identity:
∆Gsim =
Z
0
1
*
dH(λ)
dλ
dλ
+
(3.5)
where H is parameterized Hamiltonian, and where the coupled state (λ = 1) corresponds to a simulation where the solute is fully interacting with the solvent and the
uncoupled state (λ = 0) corresponds to a simulation where the solute does not interact with the solvent. We demonstrate in the Appendix that the relationship between
∆Gsolv and ∆Gsim is given by:
∆Gsolv = ∆Gsim − kT ln(V ∗ /V1 )
(3.6)
where V1 is the mean volume at full coupling (λ = 1) and V ∗ is the mean volume of
a box of pure solvent, with the same number of solvent molecules as in the coupled
simulation. We will show below that the term that depends on the ratio of these
volumes is negligible with respect to the statistical uncertainty in our calculations of
∆Gsim , and can thus be safely ignored. We feel that the rigorous derivation of this
relationship that is covered in the Appendix is particularly useful and general. For
example, there are no assumptions that the solution must be dilute or ideal.
We have chosen to use the “multiconfiguration” TI formulation [SM91a] where
equilibrium runs at a variety of fixed λ values are performed and then integrated
numerically, as opposed to a so-called “slow growth” method, with λ changing continuously throughout the simulation, as the latter methodology introduces uncontrolled
systematic error into the calculation [PK89].
71
We parameterize only the non-bonded interactions between the solute and solvent
molecules by λ, i.e., the nonbonded potential function is of the form:
U (λC , λLJ ) = Us + Uw + Uw−w + Us−w (λC , λLJ )
(3.7)
where Us is the intramolecular potential energy of the solute, Uw is the intramolecular
potential energy of the waters (zero for TIP3P water), Uw−w is the intermolecular
energy of the water-water interactions, and Us−w is the intermolecular energy of the
solute-water interactions. There are a wide variety of choices for parameterization
in λ from the initial to the final Hamiltonian, but the choice of a good path for a
given system is still an unsolved problem [CP02, BMvS+ 94, DK96, PVG02]. Simply
multiplying the difference between the initial and final Hamiltonians by λ, although
analytically correct, has the disadvantage of producing discontinuities in hdH/dλi at
r = 0 and λ = 0 (and thus extremely high variances in dH/dλ anywhere near r = 0
and λ = 0) due to the 1/r term, as well as other unphysical behavior like the “fusion”
of bare charges. This can be avoided by first decoupling the Coulombic interaction
while the Lennard-Jones term is still present. However, we are then left with a
similar discontinuity in the 1/r12 term. Although these terms are only repulsive, they
still cause such problems such as the “fence post” effect, with a near-delta function
excluding all other molecules from the region occupied by the “mutating” ones for all
λ > 0. Volumes near these atoms are never sampled at any value of λ > 0, and thus
numerical integration of Eq. (3.5) will not give good results. The discontinuities are
also present in the potential, leading to instabilities in the dynamical integration for
small values of λ.
We avoid these singularities by using the following expression for the λ-dependent
non-bonded interaction energy:
Us−w (λC , λLJ ) =
XX
i
λC
j
qi q j
rij
!
+λ4LJ 4ij
1
1
−
(3.8)
2
6
2
[αLJ (1 − λLJ ) + (rij /σij ) ]
αLJ (1 − λLJ )2 + (rij /σij )6
72
where the sum i is over all solute atoms, and the sum j is over all solvent atoms.
Eq. (3.8) includes the standard Coulombic term with a linear dependence on λ,
but also incorporates the “soft-core” parameterization of Beutler et al. [BMvS+ 94,
PVG02] in which the Lennard-Jones term goes to zero in a well-behaved manner with
λ. The additional parameter αLJ is a positive constant which controls this transition,
and is equal to 0.5 for all simulations in the study. In applying the transformation
from solute fully coupled with the solvent (λC = 1, λLJ = 1) to the solute fully decoupled from the solvent (λC = 0, λLJ = 0), we first decouple solvent-solute Coulombic
interactions, and then the solvent-solute Lennard-Jones terms. This procedure avoids
the singularities in the potential discussed earlier. It is essential to use this or a similarly well-behaved function in order to get accurate results when atomic sites are
being “grown” into or removed from the solvent.
It is tempting to assign the free energy resulting from the turning off the Coulombic potential as the “electrostatic contribution to” the free energy, and the turning
off of the Lennard-Jones term as the “Van der Waals contribution to” the free energy
of solvation. However, we note that it is impossible to directly assign a Coulombic or
Van der Waals partitioning of the total free energy, as these are both path-dependent
terms [MVG94]. Instead, the free energy of the Coulombic decoupling can more accurately be identified as the “charging” or “discharging” free energy of the solute
(though this is not completely correct, as the intramolecular Coulombic interactions
are not turned off), and the Van der Waals contribution as the free energy of “insertion” of the bonded assembly of uncharged Lennard-Jones spheres that make up the
molecule into the simulation cell.
The Coulombic interactions were decoupled by reducing λC in steps of 0.05. For
the Van der Waals terms, preliminary investigations showed that uneven spacing in
λLJ could give better results than even spacing, as closer spacing is desired when the
curvature of hdH/dλi is larger. Therefore, the following 41 values of λLJ were used:
0.00, 0.10, 0.20, 0.22, 0.24, 0.26, 0.28, 0.30, 0.32, 0.34, 0.36, 0.38, 0.40, 0.41, 0.42,
0.43, 0.44, 0.45, 0.46, 0.47, 0.48, 0.49, 0.50, 0.52, 0.54, 0.56, 0.58, 0.61, 0.64, 0.67,
0.70, 0.73, 0.76, 0.79, 0.82, 0.85, 0.88, 0.91, 0.94, 0.97, and 1.00. Larger λ spacing
gave hydration free energies that deviated from those free energies obtained with the
73
given λ spacing by a significant fraction of the statistical uncertainty (as discussed
below). Numeric integration was carried out by the trapezoid rule. Representative
curves of hdH/dλi versus λ are shown in Figure 3.5.
Each set of 61 total λ values for each of the 15 molecules was simulated in five-fold
replica. For each the five replicas, the same initial coordinates were used, but each
replicated simulation was started with different random number seeds for the velocity
reassignment used in the thermal control protocol. Ideally, we would start with
something approximating a Boltzmann-weighted, uncorrelated ensemble of starting
states, but the correlation times for dH/dλ are sufficiently short that the five replicas
are essentially uncorrelated after the 200 ps equilibration, with the possible exception
of some torsional degrees of freedom discussed later in this study.
We also computed long range Lennard-Jones corrections to the solvation free energy, an estimate for total attractive energy from the Lennard-Jones term neglected
from outside the cutoff regions [AT87, WFH02]. These long range corrections to the
solute-solvent interaction were computed after the simulation, and were not included
in the calculation of the pressure or the dynamics. Long-range corrections were not
applied at any point to the solvent-solvent interactions. The corrections included both
the tapered and group-based nature of the the intermolecular interactions. If we assume a TIP3P-like model of water with only one Lennard-Jones term per molecule,
then for each solute the long range contribution can be calculated by numerically
integrating
ELRC =
X
8πρij
i
Z
π
θ=0
Z
∞
r= rswitch



 q
r2 sin2 θ + (r cos θ − r0 )2
σij
− q
2
σij
r2 sin θ + (r cos θ − r0 )2
12

6 
2
 
 S(r)r sin θdθdr
(3.9)
where i runs over the solute atoms, r is the distance from solute atom i, rswitch is the
radius at which tapering begins, ρ is the number density of the solvent molecules,
ij and σij are the standard Lennard-Jones parameters between solute atom i and
74
the solvent molecule Lennard-Jones site of type j, and S(r) is the switching function
described in Eq. (3.3). r0 is the distance from the atom i to the center of the group to
which the atom belongs, and is determined by each solute-force field combination. It is
treated as a constant for the purposes of computing this correction, as for virtually all
of the neutral groups, these distances are fixed by the bond constraints. This derivation of the correction assumes that the solvent radial distribution function g(rij ) = 1
in the tapering region and beyond, which is an adequate assumption for TIP3P. The
pressure would be slightly changed if a similar correction was also included during the
simulation, but since this correction is only applied to the solute-solvent interactions,
the effect of this pressure change on the density or on the hydration free energy is
expected to be negligible.
3.3
Results
The main results of this study are a set of highly precise hydration free energy values of
the fifteen neutral amino acid side chain analogs modeled by three different parameter
sets. The averages over the five replicas are shown in Tables 3.2, 3.3, and 3.4, and
Figure 3.1. The results of all five simulations for each amino acid analog are included
in the Supplementary Material in the original paper [SPSP03]. Additionally, we report
on a series of calculations that examine the effect of changes in certain simulation
parameters and protocols on the accuracy and precision of the results.
From Figure 3.3, one of the first things we notice is that virtually all side chain
analogs are not sufficiently soluble relative to experiment. Large, polar side chain
analogs are particularly inaccurate, off by 1-2 kcal/mol. After adding the long range
Lennard-Jones correction, the average absolute error from experiment for AMBER,
CHARMM, and OPLS-AA were 1.22 kcal/mol, 1.06 kcal/mol, and 0.75 kcal/mol
and the corresponding root mean square (RMS) errors from experiments were 1.35
kcal/mol, 1.31 kcal/mol, and 0.85 kcal/mol (from Tables 3.2, 3.3, and 3.4).
For some purposes, the relative hydration free energies are more relevant for simulation than the absolute hydration free energies. For example, one may be concerned
75
with the relative preference for different amino acids to be solvent-exposed on a protein surface, or one may try to calculate the change of free energy of binding of an
inhibitor when there is a mutation in the protein. In this case, the errors are somewhat
smaller than are the errors in absolute hydration free energy. For the ∆∆G of Ala →
X (where X is any other side chain analog), AMBER has an absolute error of 0.67,
and an RMS error of 0.85, while CHARMM has an absolute/RMS error of 0.71/0.98,
and OPLS-AA has absolute/RMS errors of 0.51/0.60 (all values in kcal/mol), with
the Lennard-Jones correction included. These relative free energy results are shown
in Figures 3.6 and 3.7. They can be computed from Tables 3.2, 3.3, and 3.4, and
are also included as a separate table in the Supplementary Material of the original
paper [SPSP03]. For ∆∆G’s of all 105 mutations of X → Y (where X and Y are any
two of the amino acid side chain analogs), we get absolute/RMS errors for AMBER,
CHARMM, and OPLS-AA of 0.68/0.89, 0.90/1.20, 0.58/0.75, with the long range
Lennard-Jones correction included. We note that the long range correction improves
the values of relative as well as the values of the absolute hydration free energy for
all three force fields.
We also computed the free energy of hydration of a single molecule of TIP3P
water solvated in a box of TIP3P water. This calculation used 8/10 Å tapered
cutoffs for all water molecules (including the “solute”). The calculated free energy
was −6.15 ± 0.02 kcal/mol. With the long range Lennard-Jones correction included,
it becomes −6.21 ± 0.02 kcal/mol. This compares to experimental values of −6.33
kcal/mol [AWFC90] and −6.32 kcal/mol [BNM84]. Interestingly, this value is closer
to experiment than most of the amino acid side chain analogs, demonstrating that
parameter sets may be optimized for some interactions (solvent-solvent) but not for
others (solvent-solute).
In this study, we use parameters that are derived from protein force fields, and
compare them with small molecule experimental data. It is important to address both
whether the modified parameters are significantly different from the original protein
parameters, and whether it is reasonable to compare these protein force field-derived
parameters to the small molecule data.
The explicit changes in the published parameter sets were the truncations of the
76
amino acids to the side chain analogs, and the changes in the β-carbon charge needed
to produce molecular electrostatic neutrality for AMBER. The side chain parameters
in these force fields were derived primarily from condensed phase and quantum mechanical data of the same or very similar small molecules as are studied in this paper
as side chain analogs [CCB+ 95, ADMBB+ 98, vGBE+ 96, JMTR96]. Essentially, our
method of truncation (see Figure 3.2) is the opposite of the method used to construct
the side chain parameters in the first place (the exception being the case of AMBER partial charges). Comparison to small molecule experimental free energy data
is reasonable as it is consistent with the methodology used to construct the force
fields.
The effect of the additional charge changes for AMBER can be estimated. We
examined the effect of different charge reassignment schemes in the Leu analog, which
had the largest β-carbon partial charge correction. We found that the first charge
distribution of the Leu analog (the one used in Table 3.2) has a ∆Gsolv = 3.13 ±
0.03 kcal/mol, whereas the alternate-charge Leu analog, where the charges of the
hydrogens of the β-carbon were changed instead of the charge of the β-carbon itself,
has ∆Gsolv = 3.07±0.03 kcal/mol (both of them before the Van der Waals correction,
which is identical for both). Since the difference is small, we believe that these changes
do not affect the conclusions reached in this paper.
We must take into account in our calculations differences between the end states
of the simulated system and the experimental system. This was derived in Eq. (3.6),
and involves only the average volume V1 of the fully coupled system, and average
volume V ∗ of a box of pure solvent containing the same number of solvent molecules
as the coupled system. We estimate V ∗ by finding the average volume of a box of
900 TIP3P waters, and scaling linearly to the actual number of waters in each solute
simulation. The largest value for the kT ln(V ∗ /V1 ) correction in all three force fields
was for the Leu analog, at 0.011 kcal/mol, approximately 40% of the uncertainty in the
measurement. In all cases, this correction was between 0.011 and 0.003 kcal/mol, and
between 10% and 30% of the uncertainties reported in Tables 3.2, 3.3, and 3.4. The
uncertainties in these correction terms are negligible, approximately 10−4 kcal/mol.
Because of the magnitude of this term, we can therefore safely neglect it in our
77
calculations.
Another consideration related to the reproducibility and transferability of these results is their robustness to changes in the non-bonded interaction truncation protocol.
Ideally, we would like to perform the calculations under the same conditions as those
used to develop the parameters and/or the conditions under which they are typically
used, sets of conditions that are not necessarily the same. For example, OPLS-AA
was developed using quadratically tapered atomic spherical cutoffs [JMTR96], and
AMBER was developed with residue-based abrupt cutoffs [CCB+ 95], which result in
discontinuous interaction energies and forces at the cutoff boundary. We adopted the
tapered cutoffs for solvent/solute interactions which are similar to those implemented
in CHARMM [SB94], though we chose 10/12 Å tapered truncations, longer than the
range used in the parameterization of CHARMM [ADMBB+ 98].
However, it is important to note that the methods used to make non-bonded
interactions finite ranged can significantly affect the free energy or other simulation
results [LKN02, MN02]. In order to examine how different non-bonded truncation
schemes can affect free energy calculations, we first examined the difference between
group-based and residue-based tapering schemes in the case of five side chain analogs
(Ile, Thr, Tyr, Gln, and Trp). Atom-based schemes were not examined as they can
produce anomalous simulation results unless they employ extremely long truncation
or tapering distances [SABW82, SB94, Lea96]. We compared the OPLS-AA results
(from the simulations shown in Table 3.4) produced using the group-based finite
ranged potentials to another set of results produced using “residue”-based cutoffs
(where the entire side chain analog was treated as one group). For this test, we used
only every other λ value, for a total of 31, as the extremely high precision is not as
vital in this comparison and the errors in numerical integration will mostly cancel out,
but in all other respects these simulations were run using the same parameters and
protocols as those in the main data set. In Table 3.5, we see that group and residuebased truncation schemes generally have differences that are statistically significant,
but are still very small (about 0.1 kcal/mol). The exception is the Gln analog, which
has a difference in the Coulombic energy of 0.42 ± 0.03 kcal/mol. The difference
between group-based and residue-based truncation schemes may be expected to be
78
the greatest in the Gln analog, as it contains several neutral groups with large partial
charges inside each group. We thus see that there may be issues with comparing
results produced with residue and group-based truncation schemes.
We also examined the effect of using different interaction truncation ranges, again
using only 31 values of λ. In Table 3.7, we see that different ranges of tapered
cutoffs can have a drastic difference on the computed free energies, up to 1.56 ±
0.07 kcal/mol in the case of the Trp analog. However, the difference in free energy
between different cutoff ranges is due primarily to the attractive Lennard-Jones terms
at long range. If we add the Lennard-Jones correction term, adjusted for the different
truncation range, the difference in the Van der Waals interactions becomes essentially
negligible relative to the uncertainties of the simulations. The long range LennardJones correction thus becomes useful not only in improving the match to experiment
for the hydration free energies, but in eliminating almost all of the effect of the
truncation of the Lennard-Jones term at different cutoff lengths. However, the effect
of changing cutoff lengths has a non-monotonic effect on the Coulombic component
of the free energy of hydration. For the Ile analog, it is negligible, and for the
Trp analog, it is relatively small, but for the Gln analog, with large partial charges
and a large dipole moment, it can be up to 0.46 ± 0.07 kcal/mol. Many published
simulation results use cutoff schemes shorter than our 10/12 Å tapered cutoff, which
will most likely result in values even less favorable for solvation and thus further from
the experimental results than the ones described in this paper.
To make reasonable comparisons to experiment and to other simulations, we also
must understand the precision of these results. One source of numerical error is
from the numerical integration with respect to λ of hdH/dλi to produce the overall
free energy change. We can examine this by looking at how much the calculated
free energy changes when we omit some values of hdH/dλi. First, we examine the
convergence of the Coulombic contribution with increasing sampling of λC . For the
results in Table 3.4, we used a λC spacing of 0.05. If we instead were to use a λC
spacing of 0.1, over the 15 side chain analogs of OPLS-AA data set, we get an RMS
difference of 0.010 kcal/mol for the Coulombic contribution to the hydration free
energy from the ∆λ = 0.05 results, with a maximum absolute difference of 0.025
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kcal/mol for the Asn analog. For the Asn analog, this is approximately the same as
the calculated uncertainty in the Coulombic component, while for the other analogs
it is less, suggesting that λ spacing of 0.1 is slightly too large given the computed
uncertainty, while 0.05 is probably sufficient, as the integration error is quadratic
in ∆λ. If we increase to a spacing of 0.2, the RMS difference in the Coulombic
component increases to 0.045 kcal/mol, with a maximum of 0.096 kcal/mol, again
for the Asn analog, indicating that this spacing is clearly insufficient. We can see
the shape of the Coulombic components of hdH/dλi for the Trp and Ala analogs in
Figure 3.5. Because of the smooth shape, it may be possible to achieve a higher level of
accuracy in the Coulombic part with fewer values of λ using a higher order integration
scheme [RM93]. However, higher order integration schemes can be problematic due
to the difficulty of determining precise higher derivatives in λ [SVG94].
Next, we look at the convergence with respect to the number of sampled hdH/dλi
values of the Van der Waals contributions to the free energy. When we use the data
at only every other value of λLJ (of the initial 41 values specified above), we obtain an
RMS difference of 0.09 kcal/mol, with maximum of 0.160 kcal/mol for the Trp analog.
This is approximately three times greater than the uncertainty in the computed value
of the Van der Waals component of solvation, suggesting that this spacing over the
entire λLJ range is not sufficient for the precision that is possible. Since using simple
trapezoidal integration yields an error that converges with the square of the spacing,
the original (41 λLJ ) spacing scheme is most likely sufficiently within the uncertainty
of the simulation. Using only every fourth λLJ value yields even worse results, with
a RMS difference of 0.337 kcal/mol, and maximum error of 0.58 kcal/mol, again for
the Trp analog. The shape of the curve of hdH/dλi of the Van der Waals component
is also shown in Figure 3.5 for the Trp and Ala analogs. We see from the curve of
the hdH/dλi that there is a peak that requires such close spacing in λ for adequate
integration. For other molecules, such as the Ala analog, the curvature is much less,
so that such close spacing is not necessarily required.
It also is important in free energy simulations to compute the uncertainty due to
limited sampling in a systematic way. In this study, we have done this by computing
the autocorrelation function and correlation time of the components of dH/dλ and
80
using this information to compute statistical uncertainty estimates for hdH/dλi. We
looked at the behavior of the correlation times in the simulations which included both
windowed and full 1.0 ns correlation functions, which were performed for 20 out of the
total 45 molecule/force field combinations. These latter simulations included some of
the larger molecules, like the Trp and Tyr side chain analogs. The average (over all
runs and λ values) of the windowed correlation times of the Coulombic component
of dH/dλ was 0.16 ps with a 0.10 ps standard deviation, and all were less than 0.55
ps. For the Coulombic component correlation times from the full 1.0 ns simulations,
the average was 0.23 ps, with a 0.11 ps standard deviation, with all being less than
0.86 ps. For the windowed correlation times of the Van der Waals terms, the average
was 0.35 ps, with a standard deviation of 0.26 ps. All were less than 1.5 ps. For
the full 1.0 ns Van der Waals component correlation times, the average was 0.42 ps
with a standard deviation of 0.28 ps, and all were less than 2.3 ps. These correlation
times were of course λ and molecule dependent, so these numbers cannot be used
as predictions for future studies. But it is clear that we are running, at minimum,
hundreds of correlation times in order to collect hdH/dλi at each λ, so the statistical
analysis is indeed meaningful – if, of course, all the relevant degrees of freedom are
sampled.
It was found that in all cases, the uncertainties determined from the 1.0 ns correlation function were between 10% and 20% larger than the 100 ps window uncertainties.
We note the 200 fs resolution of the autocorrelation function from the entire 1.0 ns
of data overestimates the contribution to the correlation time from the part of the
correlation function before 200 fs, and thus may account for a portion of the somewhat longer estimates for τA . Since the correlation time contributes to the overall
uncertainty in the square root, the effects on correlation time estimation from the
windowing are negligible in the small molecule systems under study.
If these uncertainty estimates results are valid, we would expect that deviations
from the average over the five separate simulations roughly obeys a normal distribution with a variance given by our uncertainty estimate. There are 45 × 5 each
Coulombic and Van der Waals values determined in the main data set, and we can
compare them from the 45 averages for each combination of molecule and force field.
81
We find that for the Coulombic values, 180 (80.0%) are within one standard deviation from the average, 220 (97.7%) within two standard deviations, and all but two
(99.1%) within three standard deviations. If anything, this seems to suggest that our
estimated uncertainties are slightly too large, as the percentages found within the first
three standard deviations of the normal distribution are 68.3%, 95.5%, and 99.7%,
respectively. For the Van der Waals values, 155 (68.9%) are within one standard
deviation, 215 (95.6%) are within two standard deviations, and all but one (99.6%)
are within three standard deviations. We can also look at the distribution of all
hdH/dλi values at all values of λ used to calculate ∆Gsolv . Here, we see that for
the Coulombic component the breakdown in terms of percentage within (1σ,2σ,3σ) is
(72.0%,96.8%,99.9%), (using 45 × 21 × 5 = 4725 values) and for the Van der Waals
component (using 45 × 40 × 5 = 9000 values, excluding λ = 0 where hdH/dλi is
constrained to be zero), we get (71.2%,96.3%,99.7%). In general, we seem to be fairly
close to a normal distribution, suggesting that our calculated uncertainty, based on
the correlation times of our observables, is really what it purports to be.
Even if our observations seem to behave as expected statistically, we must also
ensure that we are sampling all the relevant degrees of freedom. In general, this
is a difficult problem, but for smaller molecules like those examined here it is not
insurmountable, even by the brute force methods used in this study. To test the
extent of conformational sampling in these systems, we ran separate 1.0 ns simulations
of the Ala, Ile, Phe, Ser, and Gln side chain analogs using the OPLS-AA force field,
under the same conditions as the Folding@Home simulations. These simulations were
run fully coupled, i.e., with both the Coulombic and Van der Waals components at
λ = 1. At smaller values of λ, most barriers to motion should be smaller, and thus
sampling should in general occur even more rapidly. There are some exceptions –
for example, intramolecular hydrogen bonding in vapor phase – but there are no
intramolecular hydrogen bonds or other similar intramolecular attractions possible
within the molecules in this study. We anticipate that these fully coupled simulations
are therefore either representative of or slower in sampling these degrees of freedom
than simulations run at other values of λ.
The degrees of freedom that contribute most to hdH/dλi are those associated
82
with the rearrangement of the solvent. To evaluate the sampling of these degrees of
freedom, we computed the residence time of water molecules in the hydration shell
for the Phe and Ala analogs. We defined a water molecule as being “in the hydration
shell” if the distance between the oxygen atom of the water and any of the solute
carbon atoms is less than 4.5Å. We found a mean time of residence in the shell of
0.56±0.68 ps for the Ala analog, and 0.69±1.05 ps for the Phe analog, which are over
three orders of magnitude less than the overall simulation time of 1.0 ns. We also
computed the correlation function of occupation of the hydration shell, averaging
over all water molecules which are in the hydrophobic shell at some point during
the simulation, and integrating this function to obtain the correlation time. This
correlation time also captures returns to the hydration shell after the first departure.
For the Ala, this time was 7.4 ps, and for Phe, it was 13.0 ps. During the Ala analog
simulation, 767 of 897 water molecules spent some time in the solvation shell, while
during the Phe analog simulation, 803 of 892 did, indicating extensive sampling, as
the hydration shell itself covers only about 5% of the simulation cell volume. It is clear
that these simulations are indeed running tens and hundreds of times the correlation
times of any of the observables that are correlated to the free energy components that
we are trying to measure.
For the Ser analog, we examined the lifetime of the hydrogen bonds between
the water molecules and the OH moiety. The hydrogen bonds are defined as pairs
of molecules with O– O distance less than 3.5 Å and
6
O– H– O < 110 degrees,
and with this definition, the residence time is 0.22 ± 0.54 ps, also orders of magnitude
smaller than the simulation time. The correlation time of hydrogen-bonding molecules
(computed in the same manner as the correlation time of hydration shell occupation)
was longer, 2.3 ps, and 610 of the 897 water molecules hydrogen bonded with the solute
over the course of the simulation. In all cases, it appears we can have confidence that
these solvent degrees of freedom are well sampled in our simulations.
In general, the slowest degrees of freedom in molecular dynamics simulations are
those associated with torsional motion. To explore the sampling of these degrees
of freedom in a long “brute-force” calculation, we looked at selected heavy-atom
torsional angles, specifically the C–C–C–C angle in the Ile analog, the C–C–C–O
83
angle in the Gln analog, and the C–C–S–C angle in the Met analog. We computed
the value of cos χ (where χ is one of the three torsional angles described above) as
well as its correlation function. The 1.0 ns time courses of cos χ of the three angles
are shown in Figure 3.4.
For the Gln analog, we find that there is fairly good sampling for this torsional
angle. For the Met analog, the anti and the gauche states are only moderately sampled
– we note approximately 12 transitions back and forth during the run, which may not
be enough to adequately sample the true thermodynamic probability of the different
states. More worrisome, for the Ile analog, there is only one transition between the
gauche and anti conformations, and this lack of sampling could very well result in
biases in the results.
Previous data suggests that the difference in hydration free energy of n-butane
(the Ile side chain analog) fixed in the gauche and anti conformations is around 0.3
kcal/mol [KDMJ94b] which is certainly larger than the precision of our calculations
here, though difficult to compare to as no uncertainties are provided in that study.
In order to assess the size of errors in free energy due to a lack of sufficient sampling
of the dihedral angle, we examined the difference in hydration free energy of the Ile
conformations separately. One way to do this is to rigidly restrain anti and gauche
conformations and evaluate the free energy of hydration of both, but the distribution
of the torsional angle for Ile shown in Figure 3.4 shows a fairly broad angular distribution, so relative free energies of fixed conformations are not the most relevant
observable. We instead constructed a restraining potential of the form:
0.01
H(ψ) = exp
(0.5 + cos ψ)2
!
−1
(3.10)
and then added it to the torsional term. This serves to produce an impenetrable
barrier between the anti and gauche configurations, as it goes to infinity at ±60
degrees from the anti configuration, while adding less than 20% of kT to any state
outside of the range of ±(60±20) degrees. We can see from Figure 3.4 that this
excluded region is essentially never occupied even without the restraining potential.
We performed a set of simulations with the OPLS-AA Ile side chain analog which
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differed from the simulations whose results are presented in Table 3.4 only in the
presence of the restraining potential. We examined only the Van der Waals contribution to the free energy of hydration, as the Coulombic contribution was only about
0.01 kcal/mol, and any changes in this value with respect to the torsional angle are
expected to be smaller than even this amount.
We found that the anti conformation had a Van der Waals component of the free
energy of solvation of 3.12 ± 0.03 kcal/mol, while the corresponding value for the
gauche conformation is 3.07 ± 0.03 kcal/mol. These values compare with the computed value in Table 3.4 of 3.10±0.03 kcal/mol. These free energy results demonstrate
that for n-butane, the free energy of hydration is almost completely independent of
the conformation. It is possible, however, that molecules with large dipole differences
between torsional conformations could have a larger hydration free energy conformational dependence. Additionally, the fact that some dihedral angles have such
long correlation times implies that accurate calculations for even moderately more
complex systems will be challenging, and require more complicated sampling algorithms [SFK98, WS00, CE00, RKdP03, MBZT99, UJ03].
There are many studies of free energies of solvation of models of small molecules,
and it is important to compare to these previous studies where possible. Unfortunately, comparisons between these simulations and the current simulations can prove
extremely tricky. The majority of previous studies were performed using earlier parameter sets (such as CHARMM19 or “united-atom” OPLS), or different solvent
models (such as TIP4P or SPC) and thus are not expected to be directly comparable
to the present study. There are also many simulation parameters (such as the number
of water molecules, methods of truncation of long range interactions, and inclusion of
long range corrections to cutoffs) that can drastically affect the results. A complete
review of the wealth of solvation free energy simulations is outside the scope of the
present paper, so we will focus in our comparisons instead on a few studies performed
under similar conditions, and that should be expected to give similar results.
One recent study of free energies of solvation using OPLS-AA is closely comparable
to the present study. Wescott et al. reported a thorough study of small, straight-chain
hydrocarbons, using the OPLS-AA force field and TIP3P water [WFH02]. These
85
results include a long-range correction, so are expected to be very comparable to
the study described in this paper. They obtain values of 2.35 kcal/mol for methane
(the Ala side chain analog), 2.40 kcal/mol for propane (the Val analog), and 2.59
kcal/mol for n-butane (the Ile analog), compared to our results of 2.31, 2.59, and 2.73,
respectively. Unfortunately, no uncertainty estimates are given, so it is impossible
to judge if these results are within the mutual uncertainty of the two studies, but
they are relatively close, within 0.2 kcal/mol for all molecules. Wescott et al. also
agree qualitatively that the models’ free energies of solvation are uniformly too high.
Inspection of Figure 3a in this study also indicates that the area of greatest positive
curvature in hdH/dλi is not well sampled in λ compared to the present study (see
Figure 3.5), meaning their reported results are most likely somewhat too negative.
There are significantly fewer discussions in the literature of hydration free energy
calculations for the amino acid side chain analogs with CHARMM and AMBER.
The only directly comparable result (i.e., using the same charge and Lennard-Jones
parameters) found using the CHARMM22 force field was for methane (Ala analog),
with a value of 2.10 ± 0.25 kcal/mol [YM98]. Our calculation was slightly outside
this uncertainty range (2.44 ± 0.02 kcal/mol), but again, a variety of differences in
simulation parameters make exact comparison difficult.
Although the GROMOS96 potential set was not examined in this study, a recent publication makes it possible to draw some comparisons [VM02]. This work
used SPC water, rather than TIP3P water, but as GROMOS96 was designed with
SPC water [vGBE+ 96], the GROMOS96/SPC combination represents the fairest test.
From the data in Table 2 of Villa and Mark [VM02], we obtain an average absolute
difference from experiment of 2.2 kcal/mol, and a RMS difference from experiment of
2.7 kcal/mol, significantly larger than the errors for the parameter sets studied above.
These results were even more positve with respect to experiment than those in the
present study. Reported uncertainties for these simulations spanned the range from
0.15 kcal/mol (for Ala) to 0.5 kcal/mol (for Tyr), approximately 5-10 times those of
the present study. These results were even more positive relative to experiment that
the force fields used in this study.
3.4
86
Discussion
In the previous section, we presented the results of our simulations and analyzed the
precision of the results and the agreement with experimental measurements. In this
section, we discuss the utility and obtainability of high precision results, as well as
what these results can tell us about the models and about simulations using these
models.
How accurate do we expect the hydration free energies computed using these parameter sets to be? The parameter sets used in this study were not parameterized to
reproduce free energies of hydration because, until recently, it was too computationally intensive to so do in an iterative manner with sufficient precision [N3E], Instead,
Lennard-Jones terms were generally derived to reproduce bulk properties using condensed phase simulations of pure liquids, and charges were generated from quantum
mechanical calculations in isolated (gas phase) molecules, with some constraints and
scaling factors used to account for the solvent environment [CCB+ 95, ADMBB+ 98,
JMTR96], In this light, there is no a priori reason to expect the free energies correspond particularly well to experimental results. Indeed, it is encouraging that all
three parameter sets are qualitatively close to the experimental values.
These uniformly high values are not unexpected. As noted above, Wescott et al.
reports free energies for OPLS-AA/TIP3P that are significantly higher than experiment. Other simulations using OPLS-AA and TIP4P water have given free energies
that were uniformly higher than experiment [KDMJ94b, CS00], though slightly less
so. This is reasonable, as TIP4P has a slightly deeper Lennard-Jones minimum, which
should result in greater attraction.
Given the high precision free energy values obtained in this study, we must ask the
unusual question if the experimental data have sufficient precision to provide accurate
comparison. A number of studies have reported free energies of hydration for many of
the amino acid side chain analogs studied here [WACS81, BNM84, CGML81, HM75,
AWFC90, VGSW99] as shown in Table 3.6. We note good agreement among these
experimental studies, and we also note the variation among experimental measurements is comparable to the uncertainties in our simulated results [N3F]. In addition
87
to the variation shown in Table 3.6, Plyasunov et al. examined a large number of
literature values of the free energy of solvation for several of these molecules [PS00].
For toluene (Phe), 46 literature values were found, with a standard deviation of 0.06
kcal/mol. For methanol (Ser), 14 values were found, with a standard deviation of
0.05 kcal/mol, and for ethanol (Thr), 17 values were found with a standard deviation
of 0.04 kcal/mol. In light of this variation, it appears for these molecules that the
uncertainty in the computed free energies approaches the experimental uncertainty.
However, it is important to note that for more complex systems of interest, such as the
free energy of binding in ligand–protein complexes, both calculations and experiments
are very likely to be less precise.
This study also draws attention to the sensitivity of results to changes in the
conditions of the simulation. In this paper, we studied the effect on computed free
energies of changes in parameter sets, inclusion or exclusion of long range LennardJones corrections, and different methods of implementing finite-ranged interactions.
As has also been noted by other researchers [SB88, LKN02, MN02]. many of these
effects can result in significant differences in calculated values. It is only with high
precision results that these effects can emerge from the statistical noise inherent in
free energy simulations.
We call particular attention to the variation in results caused by the differences in
methods to make non-bonded interactions finite-ranged. Although the Lennard-Jones
terms are quite small outside the cutoff ranges, they are attractive everywhere, and
thus can contribute significantly to the solvation free energy [JMS84]. It has been
previously suggested that this correction can significantly affect the results of a simulation [WFH02, LW96]. As non-bonded potential sets are effective pair potentials,
force field designers are of course free to include or exclude a long range correction
in their parameterization. The Lennard-Jones functional form is an approximation
to Van der Waals forces in any case, and consistency between the development and
application of the force fields is the important point. The OPLS-AA parameter
set was parameterized including a long range correction [JMTR96, JMS84], whereas
CHARMM and AMBER were not. However, we find that including this long range
correction improves hydration free energy results for all three parameter sets, as
88
shown in Tables 3.2, 3.3, and 3.4. Perhaps most importantly, as noted in Table 3.7,
applying a proper long range correction to the Van der Waals term eliminates much
of the difference in hydration free energies that results from truncation schemes with
different distance ranges.
The differences that the same finite-ranged schemes cause in the electrostatic
terms can presumably be eliminated by moving to methods like Ewald summation
or reaction field schemes for computing the electrostatic interactions. Other studies
have pointed out additional reasons for moving away from finite-ranged treatments of
electrostatic interactions [BC92, LKN02]. As cutoffs for Van der Waals terms are still
necessary when using methods that eliminate the need for finite-ranged Coulombic
forces, like Ewald summation, the use of a correction term can remove the Van der
Waals truncation scheme as a parameter which arbitrarily affects the the solvation
free energy. Moving to treatments of long ranged interactions that require fewer useradjusted arbitrary parameters will greatly improve the ability to compare simulations
from different simulation studies and to develop parameters for force fields that are
more easily transferable.
In general, previous simulations with a variety of models reproduced experimental
values of free energy within or near the uncertainty of the studies performed, though
this is not always true for all molecules and all sets. However, the uncertainties that
the studies list (usually 0.2–0.5 kcal/mol) are sufficiently large as to make comparisons
to experiments difficult. True differences of the models from experiment are masked
by such large uncertainties, making model improvement difficult. Uncertainties of
this magnitude may rapidly accumulate for simulations of larger numbers of atoms.
Most prior simulations computed only relative free energies of hydration, or computed absolute free energies by summing relative free energies of transformation relative to a single absolute energy calculation. We emphasize that the calculated free
energies in this paper are all absolute free energies, using simulations in which entire
molecules are completely decoupled from the solvent. Complete solvent-solute decoupling, rather than mutating between two solutes, entails a significantly larger change
in the Hamiltonian, and is thus harder to do with small error. It requires greater care
to ensure that the non-bonded interaction terms change in a well-behaved manner
89
with respect to λ, and requires more intermediate steps from starting to final Hamiltonian. However, it is a significantly more flexible method, not requiring a series of
structurally similar molecules.
We would like to emphasize several steps that are required in order to obtain high
precision, apart from simply utilizing more computational power. A proper statistical
uncertainty analysis should always be conducted in order to quantify reproducibility.
Computation of correlation times and variances for the dH/dλ terms are required,
and this information is relatively straightforward to collect from a simulation. To
obtain highly precise results, methods should be chosen that give small variance in
hdH/dλi. The choice of λ parameterization greatly affects this variance. This paper
is not intended as a comprehensive guide to such parameterizations. However, the
decoupling of electrostatic and Van der Waals interactions and the use of a soft-core
potential in atomic site introduction/removal results in magnitudes of variance of
hdH/dλi that make possible the level of precision presented here. It is also very
desirable to run multiple copies of each simulation, starting from initial conditions
that are as uncorrelated as possible. If the variation between the multiple runs does
not agree with the statistical uncertainty estimates, then insufficient sampling has
been performed. This establishes a useful necessary but not sufficient condition on
the sampling required for a computed free energy to converge. In order to strictly
guarantee that sufficient sampling has taken place, one must monitor all the relevant
degrees of freedom and ensure they are extensively sampling their allowed phase space.
For small molecules, this is relatively straightforward, as detailed above. However,
for even small proteins, this becomes very difficult, and it is as yet unclear how this
can be done in a general case.
The high level of precision obtained in this paper will not be necessary for all uses
of free energy calculations. However, high precision measurements are necessary in
order to understand the magnitude and source of deviations from experiment, and
to understand the subtle effects of changing simulation conditions on the free energy
results. High precision measurements are also necessary to study the effectiveness of
new methodologies, and to judge small differences between molecules.
Although determining free energies for large numbers of small molecule systems
90
may require special computational resources like Folding@Home, obtaining the equivalent degree of precision for a few molecules is within the reach of most academic and
industrial research groups. The computational power required to produce the precision in these studies for one molecule with one force field would be equivalent to about
10 dual processor 1.5 GHz Athlons running for less than 2 days (about a week for the
five-fold replication performed in this study). In retrospect, there are also many ways
to improve the computational efficiency of the simulations, aside from hardware and
software optimizations. For example, the simulation cells were conservatively large,
and a speedup factor of at least two could be obtained with a smaller cell. Sixty-one
λ points were used, but analysis afterward showed that at least 10 of the Coulombic
λ-points were unnecessary for obtaining the desired precision for these molecules. For
the smaller molecules, the number of intermediate Coulombic and Van der Waals λ
points could be reduced even further. If a slightly larger uncertainty is acceptable
(say, 0.05 to 0.1 kcal/mol), both the total simulation time required and the number
of λ values needed could be reduced even more. In this case, and given the correlation times of these molecules, it would be possible to reduce the total collection time
to 200-300 ps and still include enough uncorrelated data points necessary for good
statistical validation. Under these conditions, it would be possible to evaluate several molecules per day with the cluster described above. For absolute solvation free
energies of small molecules with few torsional degrees of freedom in explicit water,
uncertainties less than 0.1 kcal/mol are easily obtainable by most researchers. It is
important to note that these should not be taken as general guidelines for all free
energy simulations. The appropriate spacing in λ and the necessary simulation time
required need to be determined individually for each system.
It is still to be determined how the required computational effort to obtain sufficiently precise free energies scales with solute size and complexity, especially for those
solutes with many torsional barriers and many stable conformations. The current
study suggests that simply using straightforward long-running molecular dynamics
simulations may not be sufficient, as torsional degrees of freedom are not well-sampled.
This means that more sophisticated sampling schemes, such as replica exchange and
other multicanonical methods [SKO00, MSO01, WS00, CE00, RKdP03, MBZT99,
91
UJ03], umbrella sampling [TV77, BSK99], or locally enhanced sampling [SFK98]
may be necessary for more complex solutes like biomolecules.
Although the free energies of hydration of amino acid side chain analogs are one
measure of fitness of a force field, it should be emphasized that there are other measures of fitness that are also important. For example, Price and Brooks [PB02] have
run explicit water protein simulations with these same force fields, monitoring such
observables as the RMS deviation from experimental structures, the radius of gyration, and the solvent-accessible surface area, finding no significant difference between
AMBER, CHARMM, and OPLS-AA. Other studies have compared conformational
and interaction energies of a variety of force fields [BCM+ 97, Hal99].
This study leaves some important questions unanswered. One is the sensitivity
of these calculated free energies of hydration to the choice of other very commonly
used water models, such as SPC, TIP4P, and TIP5P. Each model has enough differences that discrepancies in solute-solvent energetics and kinetics are likely [Kin98,
JJ98]. Another is whether improved results are possible with improved treatments
of Coulombic forces, such as reaction field and Ewald summation methods. Comparisons of solvation free energies with simulations using different water models and
Ewald summation are currently underway.
Our results have some implications for simulations using parameter sets such as
AMBER, CHARMM, and OPLS-AA to estimate free energies. We cannot expect that
calculations performed on more complicated systems, such as those used to compute
ligand-protein binding free energies, will be any more accurate than the hydration
free energies (or at least the relative hydration free energies) of the respective small
constituents. It may be possible for sufficiently precise calculations to be more accurate than this limit, but that is dependent on fortuitous compensating errors between
ligand-water and ligand-protein interactions. Our results may also have implications
for the utility of these force fields for predictions of protein structure, stability, and
dynamics. However, it is also true that many computed observables may be relatively insensitive to the details of the potentials that produced the differences from
experiment noted in this paper.
This study also reveals the power that distributed computing can provide. Thanks
92
to the rapid increase in available computer power in the last decade, we can afford
somewhat more costly algorithms when we are striving to obtain precise results with
rigorous approaches, we can apply these algorithms to large numbers of molecules,
and we can collect more comprehensive data. We also note that distributed resources
such as Folding@Home may be of great use in the development of potential sets. First,
it becomes much easier to explore multidimensional parameter space in a systematic
way, and second, it becomes possible to use a wider variety of computationally intensive observables (such as free energies of solvation or partition coefficients) in the
actual development of parameters.
3.5
Conclusions
We have demonstrated that free energies of hydration for popular computational
models can be computed with a precision of three significant figures in a reasonable
amount of time, at least in the case of solutes with low numbers of degrees of torsional
freedom. This degree of precision is at least an order of magnitude better than most
previous studies, and the uncertainties reported have been rigorously computed and
verified statistically. The computational precision we have obtained is approximately
equal to the experimental precision for the systems examined, and can be obtained for
the absolute free energy of hydration, not simply for relative free energies of hydration.
If the model under study is close enough to experiment, this degree of precision is
sufficient for virtually all practical uses.
Current generation parameter sets can be off as much as 2 kcal/mol per side chain
analog, and therefore may be insufficient to compute free energies that accurately
compare to experimental results for many applications. The three studied potential sets are fairly similar, and deviate from experimental hydration free energies in
similar ways, though to somewhat different degrees. However, we have found that
these results can be highly dependent on simulation protocols such as the choice of
implementation of finite-range interactions, making sweeping generalizations about
suitability of potential sets impossible.
It is clear from the merely adequate sampling of the torsional degrees of freedom
93
in these simple systems that more complicated systems, with many correlated torsional terms producing large barriers between physically relevant states, will present
difficulties and may be largely impossible to study adequately with simple molecular dynamics. In order to sufficiently sample these systems it will require increased
computational power, smarter sampling schemes, and a clear separation and understanding of the errors arising from insufficient sampling versus improperly represented
energetics.
This paper also demonstrates that the pathway to accurate and reliable methods
to compute the free energies of interaction of ligand/macromolecular models may
be clearer than previously thought. This is especially true in the light of new distributed computing techniques, which provide the greatly increased computational
power needed for both the development of improved parameter sets and the sufficient
sampling of complicated models. We expect that this will be enormously useful to
both the scientific study of macromolecular interaction and to the commercial development of useful drugs.
94
Hydration Free Energy of Amino Acid Side Chain Analogs
Ala Val Leu Ile Ser Thr Phe Tyr Cys Met Asn Gln Trp Hid Hie
2
kcal / mol
0
-2
-4
-6
Experiment
AMBER
-8
CHARMM
OPLS-AA
-10
Figure 3.1: Free energies of solvation for 15 neutral amino acid side chain analogs,
for the AMBER(ff94), CHARMM22, and OPLS-AA force fields, in kcal/mol. Each
reported value is the average of five independent trials, and includes the long range
Van der Waals correction as computed by Eq. 3.9. Statistical uncertainties are not
shown, as they would be too small to read relative to the resolution of the plot, but
are listed in Tables 3.2, 3.3, and 3.4.
-0.683
0.418
OH
H
0.06
H
C
-0.683
OH
H
H
0.06
0.06
C
0.06
0.085
0.145
Backbone
0.418
H
0.06
Figure 3.2: Differences in partial charges between (left) the OPLS-AA parameter set
serine side chain and (right) the OPLS-AA-derived parameters used for the side chain
analog in this study. Only the β-carbon changes in charge.
95
Hydration Free Energy of Amino Acid Side Chain Analogs:
Differences from Experiment
2.5
AMBER
CHARMM
2
kcal / mol
OPLS-AA
1.5
1
0.5
0
Figure 3.3: Difference of free energy of solvation of the 15 amino acid side chain
analogs from experiment, as listed in Tables 3.2, 3.3, and 3.4, and shown in Figure 3.1.
All values in kcal/mol.
Heavy Atom Dihedral Angle χ over 1.0 ns
1
1
Ile
Cos χ
0
0
-1
1
0
-1
1
Met
0
-1
1
-1
1
0
-1
0
Gln
200
400
600
800
-1
1000
ps
Figure 3.4: Cosine of the heavy atom dihedral angle of the three amino acid side chain
analogs studied in 1.0 ns simulations, with λ = 1 (i.e. fully coupled to the solvent).
The Ile analog (C–C–C–C dihedral) is on top, the Met analog (C–C–S–C dihedral)
is in the middle, and the Gln analog (C–C–C–O dihedral) is on the bottom. We see
that the Met analog dihedral angle is not sampled well, and the Ile analog dihedral
angle is very poorly sampled.
96
dH / dλ
80
60
a
kcal / mol
40
20
b
d
0
c
-20
-40
-60
0
0.2
0.4
λ
0.6
0.8
1
Figure 3.5: hdH/dλC i and hdH/dλLJ i as a function of λ, for the amino acid side
chain analogs of Trp and Ala using the OPLS-AA force field. These two side chain
analogs were chosen as they are extrema of the behaviors of the other analogs. The
Van der Waals component of Trp is curve a, the Van der Waals component of Ala is
curve b, the Coulombic component of Trp is curve c, and the Coulombic component
of Ala is curve d (essentially coincident with the x-axis). The curves for all of the
other amino acid side chain analogs and force field combinations resemble the Trp
and Ala OPLS-AA curves for both components, differing mainly in magnitude. The
location of the maximum in the Van der Waals component differs between the amino
acid side chain analogs, but this maximum occurs in the range λ = 0.35 – 0.5 for all
of the molecule/force field sets under study. Uncertainties are multiplied by 5.0 in
order to guide the eye, as they would otherwise be difficult to distinguish from the
lines themselves.
97
Relative Hydration Free Energy of Amino Acid Side Chain Analogs : Ala
X Mutation
Val Leu Ile Ser Thr Phe Tyr Cys Met Asn Gln Trp Hid Hie
0
-2
kcal / mol
-4
-6
-8
Experiment
AMBER
-10
CHARMM
OPLS-AA
-12
Val Leu
Ile
Ser Thr Phe Tyr Cys Met Asn Gln Trp Hid Hie
Figure 3.6: ∆∆G of the mutation of Ala → X side chain analogs, computed by
subtracting the ∆G of hydration for the two amino acids. Note that the ∆∆Gs’
differences from experiment are smaller than for the ∆Gs’ shown in Tables 3.2, 3.3,
and 3.4 and Figure 3.1.
Difference from Experiment of Relative Hydration Free Energy of
Amino Acid Side Chain Analogs: Ala
X mutation
2
AMBER
1.5
CHARMM
kcal / mol
OPLS-AA
1
0.5
0
-0.5
Figure 3.7: Differences in ∆∆G of the mutation of Ala → X side chain analogs from
experiment. Note that the ∆∆Gs’ mean differences from experiment are closer than
for the ∆Gs’ shown in Tables 3.2, 3.3, and 3.4, and Figure 3.1.
98
Table 3.1: Correspondence between amino acids and the amino acid side chain analogs
used in this study.
Ala
Ile
Phe
Met
Trp
methane Val
propane Leu
iso-butane
n-butane Ser
methanol Thr
ethanol
toluene Tyr
p-cresol Cys
methanethiol
methyl ethyl sulfide Asn
acetamide Gln
propionamide
3-methylindole Hid 4-methylimidazole Hie 4-methylimidazole
Table 3.2: Free energies of hydration for 15 neutral amino acid side chain analogs,
for the AMBER(ff94) force field, in kcal/mol. Each reported value is the average of
five trials. LRC is the long range Van der Waals correction as computed by Eq. 3.9.
Uncertainties are the averaged uncertainty of the five runs, and represent one standard deviation from the mean (±1σ). Experimental values are from Wolfenden et
al. [WACS81].
Side Chain
Analog
Coulombic
Van der Waals
Total
LRC
Total (w/LRC)
Expt.
AMBER
Ala
Val
Leu
Ile
Ser
Thr
Phe
Tyr
Cys
Met
Asn
Gln
Trp
Hid
Hie
-0.00 ± <0.001
2.69 ± 0.02
2.68 ± 0.02
-0.02 ± 0.001
3.02 ± 0.03
3.00 ± 0.03
-0.05 ± 0.001
3.19 ± 0.03
3.13 ± 0.03
-0.00 ± 0.001
3.27 ± 0.03
3.27 ± 0.03
-6.13 ± 0.01
1.93 ± 0.02 -4.20 ± 0.02
-5.77 ± 0.01
2.21 ± 0.03 -3.56 ± 0.03
-2.35 ± 0.01
3.03 ± 0.04
0.68 ± 0.04
-6.19 ± 0.02
2.65 ± 0.04 -3.54 ± 0.04
-2.04 ± 0.01
2.37 ± 0.02
0.33 ± 0.02
-1.61 ± 0.01
2.96 ± 0.03
1.35 ± 0.03
-9.27 ± 0.02
1.78 ± 0.03 -7.49 ± 0.03
-9.35 ± 0.02
2.08 ± 0.03 -7.27 ± 0.04
-6.39 ± 0.02
2.35 ± 0.04 -4.05 ± 0.05
-9.51 ± 0.02
1.56 ± 0.03 -7.95 ± 0.04
-10.07 ± 0.02
1.57 ± 0.03 -8.50 ± 0.04
Average absolute error w/o LRC: 1.64
Average absolute error with LRC: 1.22
-0.12
2.57 ± 0.02
1.94
-0.31
2.69 ± 0.03
1.99
-0.41
2.72 ± 0.03
2.28
-0.42
2.84 ± 0.03
2.15
-0.17
-4.37 ± 0.02
-5.06
-0.27
-3.83 ± 0.03
-4.88
-0.58
0.10 ± 0.04
-0.76
-0.69
-4.23 ± 0.04
-6.11
-0.22
0.11 ± 0.02
-1.24
-0.44
0.91 ± 0.03
-1.48
-0.31
-7.80 ± 0.03
-9.68
-0.42
-7.69 ± 0.04
-9.38
-0.83
-4.88 ± 0.05
-5.88
-0.48
-8.43 ± 0.04 -10.27
-0.48
-8.98 ± 0.04 -10.27
RMS error with w/o LRC: 1.76
RMS error with LRC: 1.35
99
Table 3.3: Free energies of hydration for 15 neutral amino acid side chain analogs,
for the CHARMM22 force field, in kcal/mol. Each reported value is the average of
five trials. LRC is the long range Van der Waals correction as computed by Eq. 3.9.
Uncertainties are the averaged uncertainty of the five runs, and represent one standard deviation from the mean (±1σ). Experimental values are from Wolfenden et
al. [WACS81].
Side Chain
Analog
Coulombic
Van der Waals
Total
LRC
Total (w/LRC)
Expt.
CHARMM
Ala
Val
Leu
Ile
Ser
Thr
Phe
Tyr
Cys
Met
Asn
Gln
Trp
Hid
Hie
-0.03 ± <0.001
2.57 ± 0.02
2.55 ± 0.02
-0.05 ± 0.001
2.87 ± 0.03
2.82 ± 0.03
-0.04 ± 0.001
3.35 ± 0.03
3.31 ± 0.03
-0.06 ± 0.001
3.12 ± 0.03
3.05 ± 0.03
-6.66 ± 0.01
2.25 ± 0.02 -4.42 ± 0.02
-6.49 ± 0.01
2.53 ± 0.03 -3.96 ± 0.03
-1.69 ± 0.01
2.31 ± 0.04
0.62 ± 0.04
-6.10 ± 0.02
2.22 ± 0.04 -3.87 ± 0.04
-1.12 ± 0.004
1.41 ± 0.02
0.30 ± 0.02
-0.59 ± 0.004
2.14 ± 0.03
1.55 ± 0.03
-9.63 ± 0.01
2.04 ± 0.03 -7.59 ± 0.03
-9.48 ± 0.02
2.36 ± 0.03 -7.12 ± 0.03
-4.60 ± 0.01
1.77 ± 0.04 -2.84 ± 0.05
-11.70 ± 0.02
2.11 ± 0.03 -9.59 ± 0.04
-11.94 ± 0.02
2.08 ± 0.03 -9.86 ± 0.04
-0.11
2.44 ± 0.02
1.94
-0.29
2.52 ± 0.03
1.99
-0.36
2.94 ± 0.03
2.28
-0.38
2.67 ± 0.03
2.15
-0.17
-4.59 ± 0.02
-5.06
-0.26
-4.22 ± 0.03
-4.88
-0.53
0.09 ± 0.04
-0.76
-0.58
-4.46 ± 0.04
-6.11
-0.27
0.02 ± 0.02
-1.24
-0.47
1.08 ± 0.03
-1.48
-0.30
-7.89 ± 0.03
-9.68
-0.39
-7.51 ± 0.03
-9.38
-0.74
-3.57 ± 0.05
-5.88
-0.41
-10.00 ± 0.04 -10.27
-0.41
-10.27 ± 0.04 -10.27
RMS error with w/o LRC: 1.67
100
Table 3.4: Free energies of hydration for 15 neutral amino acid side chain analogs, for
the OPLS-AA force field, in kcal/mol. Each reported value is the average of five trials.
LRC is the long range Van der Waals correction as computed by Eq. 3.9. Uncertainties
are the averaged uncertainty of the five runs, and represent one standard deviation
from the mean (±1σ). Experimental values are from Wolfenden et al. [WACS81].
Side Chain
Analog
Coulombic
Van der Waals
Total
LRC
Total (w/LRC)
Expt.
OPLS-AA
Ala
Val
Leu
Ile
Ser
Thr
Phe
Tyr
Cys
Met
Asn
Gln
Trp
Hid
Hie
-0.00 ± <0.001
2.43 ± 0.02
2.42 ± 0.02
-0.00 ± 0.001
2.88 ± 0.03
2.87 ± 0.03
0.00 ± 0.001
3.06 ± 0.03
3.06 ± 0.03
-0.01 ± 0.001
3.10 ± 0.03
3.10 ± 0.03
-6.42 ± 0.01
2.23 ± 0.02 -4.19 ± 0.02
-6.32 ± 0.01
2.46 ± 0.03 -3.86 ± 0.03
-2.51 ± 0.01
2.50 ± 0.04 -0.01 ± 0.04
-7.01 ± 0.02
2.34 ± 0.04 -4.67 ± 0.04
-3.88 ± 0.01
2.51 ± 0.02 -1.37 ± 0.03
-3.88 ± 0.01
3.02 ± 0.03 -0.86 ± 0.03
-9.70 ± 0.02
1.50 ± 0.03 -8.20 ± 0.03
-9.79 ± 0.02
1.80 ± 0.03 -7.99 ± 0.04
-5.77 ± 0.02
2.08 ± 0.04 -3.69 ± 0.05
-9.99 ± 0.02
1.61 ± 0.03 -8.38 ± 0.04
-10.16 ± 0.02
1.60 ± 0.03 -8.56 ± 0.04
-0.11
2.31 ± 0.02
1.94
-0.28
2.59 ± 0.03
1.99
-0.37
2.69 ± 0.03
2.28
-0.37
2.73 ± 0.03
2.15
-0.17
-4.36 ± 0.02
-5.06
-0.25
-4.11 ± 0.03
-4.88
-0.53
-0.54 ± 0.04
-0.76
-0.58
-5.25 ± 0.04
-6.11
-0.22
-1.59 ± 0.03
-1.24
-0.42
-1.27 ± 0.03
-1.48
-0.32
-8.53 ± 0.03
-9.68
-0.41
-8.40 ± 0.04
-9.38
-0.74
-4.44 ± 0.05
-5.88
-0.49
-8.87 ± 0.04 -10.27
-0.49
-9.05 ± 0.04 -10.27
RMS error w/o LRC: 1.23
Table 3.5: The effect of using group-based or residue-based tapered cutoffs on calculated free energies of hydration for selected side chain analogs using the OPLS-AA
force field. Differences are somewhat greater than the statistical uncertainty, especially for the Gln analog. These results do not include the long range correction for
the Van der Waals terms. All energies are in kcal/mol.
Side Chain
Analog
Ile
Thr
Tyr
Gln
Trp
Coulombic
-0.01 ± 0.001
-6.42 ± 0.02
-7.07 ± 0.02
-9.39 ± 0.02
-5.79 ± 0.02
Residue
Van der Waals
3.25 ± 0.05
2.52 ± 0.04
2.64 ± 0.06
1.91 ± 0.05
2.28 ± 0.06
Total
3.24 ± 0.05
-3.90 ± 0.04
-4.43 ± 0.06
-7.48 ± 0.05
-3.51 ± 0.07
Coulombic
-0.01 ± 0.001
-6.32 ± 0.02
-7.01 ± 0.02
-9.81 ± 0.02
-5.78 ± 0.02
Group
Van der Waals
3.17 ± 0.05
2.51 ± 0.04
2.47 ± 0.05
1.90 ± 0.04
2.24 ± 0.06
Total
3.16 ± 0.05
-3.81 ± 0.04
-4.54 ± 0.06
-7.90 ± 0.05
-3.54 ± 0.07
101
Table 3.6: Six different experimental data sets of free energy of hydration of amino
acid side chain analogs. There is some variation between different studies, in most
cases of the same order of magnitude as the uncertainties in the computed hydration
free energies [N3F].
Side Chain
Analog
Ala
Val
Leu
Ile
Phe
Ser
Thr
Cys
Tyr
Asn
[WACS81]
[CGML81]
[HM75]
[AWFC90]
[PS00]
[VGSW99]
[BNM84]
1.94
1.99
2.28
2.15
-0.76
-5.06
-4.88
-1.24
-6.11
-9.68
2.00
1.96
2.32
2.08
-0.89
-5.11
-5.01
-1.24
-6.14
-9.71
1.95
1.99
2.29
2.15
-0.76
-5.07
-4.90
-1.24
-6.12
—
1.98
1.96
2.32
2.07
-0.79
-5.10
-5.00
-1.36
-6.14
—
1.98
2.02
2.32
2.18
-0.77
-5.14
-4.96
-1.26
—
—
2.00
1.95
2.25
2.07
-0.80
-5.05
-5.00
—
—
—
1.93
1.98
2.28
2.15
-0.77
—
—
—
—
—
102
Table 3.7: The effect of using different non-bonded interaction truncation ranges on
the calculated hydration free energy for selected amino acid side chain analogs using
the OPLS-AA force field. We note drastic differences in the simulated Van der Waals
energies, but these can mostly by accounted for by adding the analytic LennardJones long range correction (LRC) described in Eq. 3.9 in the text. Differences
in the Coulombic effect are lower, but occasionally significant and non-monotonic.
Uncertainties are not listed for space reasons, but are all very close to the uncertainties
for the corresponding molecules in Tables 3.2,3.3, and 3.4. All energies are in kcal/mol.
Side Chain
Analog
with LRC
VdW Total
2.81
2.80
2.81
2.80
2.84
2.83
2.80
2.80
2.80
2.79
Cutoffs
6/10
7/10
8/10
10/12
12/14
Coulombic
-0.01
-0.01
-0.01
-0.01
-0.00
VdW
3.77
3.62
3.52
3.17
3.02
Total
3.67
3.61
3.51
3.16
3.02
LRC
-0.97
-0.80
-0.68
-0.37
-0.22
Trp
6/10
7/10
8/10
10/12
12/14
-5.77
-5.86
-5.85
-5.78
-5.66
3.45
3.10
2.82
2.24
1.86
-2.32
-2.75
-3.03
-3.54
-3.80
-2.01
-1.66
-1.39
-0.74
-0.45
1.44
1.45
1.44
1.49
1.42
-4.33
-4.41
-4.42
-4.28
-4.24
Gln
6/10
7/10
8/10
10/12
12/14
-9.92
-10.00
-10.08
-9.81
-9.62
2.55
2.34
2.20
1.90
1.70
-7.38
-7.66
-7.89
-7.90
-7.91
-1.07
-0.89
-0.75
-0.41
-0.25
1.48
1.45
1.45
1.50
1.46
-8.45
-8.54
-8.63
-8.31
-8.16
Ile
Chapter 4
Solvation free energies of amino
acid side chains for common
molecular mechanics water models
103
CHAPTER 4. COMPARISON OF WATER MODELS
Abstract
Quantitative free energy computation involves both using a model that
is sufficiently faithful to the experimental system under study (accuracy)
and establishing statistically meaningful measures of the uncertainties resulting from finite sampling (precision). In order to examine the accuracy
of a range of common water models used for protein simulation for their
solute/solvent properties, we calculate the free energy of hydration at
298 K of fifteen amino acid side chain analogs derived from the OPLSAA parameter set in the TIP3P, TIP4P, SPC, SPC/E, TIP3P-MOD, and
TIP4P-Ew water models. We achieve a high degree of statistical precision
in our simulations, obtaining uncertainties for the free energy of hydration of 0.02 to 0.06 kcal/mol, equivalent to our previous study and to that
obtained in experimental hydration free energy measurements of the same
molecules. TIP3P-MOD, a model designed to give improved free energy
of hydration for methane, gives uniformly the closest match to experiment; also, the ability to accurately model pure water properties does not
necessarily predict ability to predict solute/solvent behavior.
We also evaluate the free energies of a number of novel modifications of
TIP3P designed as a proof of concept that it is possible to obtain much
better solute/solvent free energetic behavior without significantly changing pure water properties. We decrease the average error to zero while
reducing the RMS error below that of any of the published water models,
with measured liquid water properties remaining almost constant. This
demonstrates there is still both significant parameter flexibility and room
for improvement within current fixed charge biomolecular force fields.
Recent research in computational efficiency of free energy methods allow
us to perform simulations that previously required large scale distributed
computing, performing four times as much computational work in approximately a tenth of the computer time as a similar study a year ago, making
it amenable to running on a local cluster.
104
4.1
105
Introduction
An important goal of computational chemistry is the accurate prediction of free energies in molecular systems. The calculation of free energies is a computationally
intensive task, even for small systems, because it requires sampling of all thermally
relevant configurations of the system. Extensive work over the last 20 years has
gone into developing the theoretical and computational apparatus to perform free
energy computations, and these calculations can now be performed for many systems
with moderate levels of accuracy [CP02, Kol93, Jor04]. However, current calculations
have still not progressed to be generally useful in applications. For example, reliably
predicting of experimental observables such as drug binding affinities, which require
accuracies of 0.5-1.0 kcal/mol, is still not possible.
There are two main problems in the quantitative prediction of interaction free
energies. First, there are the discrepancies between the models used for simulation
and the experimentally measured reality. This lack of accuracy means the model may
not adequately represent the experimental system under study. Second, the phase
space of the model must be sufficiently sampled to capture all thermally relevant
contributions to the ensemble average of the observables of interest. Otherwise, the
results will lack the necessary precision, and independent calculations will most likely
lead to differing answers. Only if sufficient precision is obtained in a statistically
well-defined manner is it possible to design models that are sufficiently accurate for
the application at hand.
In a previous study of neutral amino acid side chain analogs [SPSP03], we demonstrated that precision equivalent to the experimental precision, at the level of 0.02-0.05
kcal/mol, is possible to achieve for small molecule energies of solvation. This was an
order of magnitude greater precision than for any other set of multiple free energies
of solvation. We demonstrated that current levels of computational resources (albeit
using the several thousand computers at at time with the Folding@Home [SP00] distributed computing environment) could achieve the time scales necessary to sample
the all of the relevant degrees of freedom for free energy of hydration. We found that
current biomolecular force fields (specifically OPLS-AA [JMTR96], CHARMM [ADMBB+ 98],
106
and AMBER [CCB+ 95], did not quantitatively match experimental measurements.
The hydration free energies obtained from simulations were uniformly too high compared to experiment, with an RMS error of ≈ 1 kcal/mol.
Purposes of present study In this paper, we extend the previous study by examining the free energy of solvation of the neutral amino acid side chain analogs as
a the function of the water model used for solvation. Amino acid side chain analogs
represent a natural test case for biomolecular interaction. First, experimental data
are available for direct comparison [WACS81, BNM84, CGML81, HM75, AWFC90,
VGSW99]. Second, the conformation space of these molecules is simple enough that
we expect to be able to obtain good statistical sampling, thus allowing us to obtain a
lower bound on the amount of the sampling necessary for larger, biological systems.
Our previous study [SPSP03] demonstrated that we were, indeed able to sample these
molecules, sufficient to obtain experimental precision. Third, a variety of well-defined,
extensively tested, and widely-used potential sets are available for these molecules –
in this paper, we use OPLS-AA, which had the best performance of free energy relative to experiment [JMTR96, KFRJ01, SPSP03]. If we succeed in obtaining sufficient
sampling of these systems, and thus obtain high precision values, remaining deviations from experiment must arise from inaccuracies in these models – where “models”
refers both to the force field parameters and to the choice of simulation protocols.
The correspondence between amino acids and amino acid side chain analogs is shown
in Table 4.1. In our previous paper, we analyzed other possible test systems, and
discussed why the amino acid side chain analogs were the most preferable.
When evaluating models, one is torn between using more recent simulation methods with the best proven scientific validity, the methods used for the actual parameterization of the model, and the most commonly used methods in the literature – which
may all be somewhat different. For example, most force fields (for both proteins and
water) were parameterized using simulations that employed finite-ranged non-bonded
interactions, which suggests that one should use the corresponding truncation protocol for non-bonded interactions in order to be faithful to the parameterization of the
model. However, truncation schemes for electrostatic interactions have been shown to
107
lead to qualitatively incorrect results [BC92], as compared with more sophisticated
and increasingly more common methods such as Ewald summation or the use of a
reaction field.
In the previous amino acid side chain analog paper, we demonstrated how the
group-based tapered cutoffs, even group based smoothed ones that do not have many
of the irregularities noted earlier [SB94], gave highly variable results depending on
the cutoff distance, a disadvantage for comparison between studies carried out by
different researchers [SPSP03]. In this study, we use methodology that meet, as much
as possible, two separate critera. First, we choose methods as physical as possible
– for example, that conserve energy, rigorously implement the desired ensemble, are
ergodic, and have a conserved quantity. This removes several sources of error and
irreproducibility. Secondly, we choose methods that are as free as possible of arbitrary,
non-physical parameters, such as long range cutoffs. Most importantly, we use Ewald
methods for evaluation of the long range Coulombic interactions, and a long range
correction for the Lennard-Jones cutoff. A greater discussion of these and other
methodological choices is included in the Methods section.
For sampling, we chose to use molecular dynamics as the most direct method for
isobaric-isothermal Boltzmann sampling of condensed phase molecular systems. We
realize that this method may not not the most computationally efficient in general.
But it is sufficient for the problem at hand, as well as being simple to implement,
frequently utilized, well-studied, and well-understood.
These results can serve as one possible test of the applicability of explicit fixedcharge water models with current classical protein models. The present work could
guide the development of more accurate force fields. It demonstrates that it is now
easier to incorporate very important yet computationally intensive observables such as
hydration free energy into force field parameterization efforts. Free energy simulations
thus serve as useful parameter verification methods as they quantify the effects of both
the enthalpy and entropy of molecular interactions.
4.2
4.2.1
108
Methods
The simulations were performed on the Stanford Bio-X cluster of 2.8 GHz Intel Xeon
processors Pentium IV. The simulations were performed using a modified version of
the molecular dynamics suite GROMACS (v3.1.4) [BvdSvD95, LHvdS01]. Our modifications included adding Andersen pressure and temperature control [And80], and velocity Verlet (which is required for correct pressure control implementation) [SABW82].
We also implemented SETTLE [MK92] for 4-point water models, and a variety of
modifications for free energy computation, mostly bookkeeping to allow for decoupling free energies to be computed through Bennett acceptance ratio, but also a
modified “soft-core” λ dependence, described later. Arrangements are being made to
include versions of these modifications in the next version of GROMACS. GROMACS
was compiled in double precision, instead of the usual single precision, as this was
determined to be necessary for converging the bond constraints to the desired relative
precision. Although GROMACS is optimized for single precision, on the Pentium IV
processors used and with the methods implemented, the time taken was only about
10% greater.
Parameters for the amino acid side chain analogs were generated from the OPLSAA(1996) [JMTR96] force fields, and were identical to those used our the previous
amino acid side chain paper, with the exception of the charge and Lennard-Jones sulfur parameters, which were obtained from the OPLS-AA/L modifications [KFRJ01].
We studied all the neutral amino acid side chain analogs except for those of proline
and glycine. In addition, we examined analogs of both neutral protonation states of
His (referred to as Hid and Hie analogs, where the proton is attached to the δ and nitrogens, respectively).
We have omitted the side chain analogs of amino acids carrying a formal charge at
neutral pH in our computations. Ewald methods provide a much better treatment of
charged species than the cutoffs used in our previous paper, as they rigorously treat
the long range Coulombic interactions in the periodic system. However, to simulate
a charged molecule, we must either add counterions or add a uniform neutralizing
109
background charge distribution. The effect of either of these on the free energy
of hydration becomes complicated to evaluate. Additionally, although experimental
data does exist for the free energy of hydration of these charged amino acid side chain
analogs, it is more complicated to compute and leads to larger uncertainties, as it
is the neutral species that enters the vapor state, and one must include the pKa as
well [WACS81, RW88].
Some modifications to the side chain β-carbon parameters were necessary in order
to use the protein force field parameters for the side chain analogs. For the nonbonded parameters for these atoms, the Lennard-Jones term was taken from the
standard aliphatic carbons (either CH3 or CH2 , depending on the amino acid) of each
parameter set. The partial charge of this atom was determined for each amino acid
side chain analog by reducing the partial charge on the β-carbon by the amount of the
charge of one hydrogen atom (see Figure 4.1). If bonded parameters (bonds, angles,
and torsions) were already defined in the parameter set for atoms of the resulting
type and connectivity, these were used. We discuss in our previous paper how these
assumptions are reasonable for OPLS-AA, and are in fact approximately the reverse
process used for building the amino acids from small molecules, as can be seen by
comparison to the OPLS-AA parameters. Input parameter and coordinate files for
all the amino acid side chain analogs in all parameter sets used will be provided as
Supplementary Materials to the paper.
Six different published water models were compared. The original TIP3P water model [JCM+ 83] was used, which is rigid and includes no Lennard-Jones term
on the hydrogens. This is slightly different than the TIP3P-like water model used
to develop the CHARMM parameter set, which does have Lennard-Jones terms on
the hydrogens [ADMBB+ 98]. TIP3P (including slight variants, like the CHARMM
TIP3P mentioned above) is probably the most common water model used in explicit
water simulations of biomolecules. TIP4P [JCM+ 83] was developed at the same time
as TIP3P, and most of the OPLS-AA parameters were developed in conjunction with
TIP4P. It is less common than TIP3P in general use, however, because the treatment
of the dummy site is somewhat complicated, and many code bases did not support
it for some time, or still do not support it now, delaying adoption. SPC [BPvG+ 81]
110
is an important early model still used by many simulations groups, especially in
Europe. SPC/E was an improved version [BPvG+ 81], designed to correct for the selfpolarization in water. TIP4P-Ew [HSP+ 04] was very recently designed by researchers
in the IBM Blue Gene project specifically for Ewald methods, and has demonstrated
extremely good fit to experimental data. TIP3P-MOD [SK95] is not in common usage, and involved modifying the Lennard-Jones and σ in order to try to improve the
solute-solvent interaction between methane and water without adversely affecting the
pure liquid properties of water. The various parameters and pure liquid properties
of these water models are compared in Table 4.2. There are, of course, many other
effective pair potential water models that have been developed, for both biological
and chemical systems, but a complete discussion of the subject is beyond the scope
of this paper.
The hydration free energies were computed using the Bennett acceptance ratio
method [SBHP03], with molecular dynamics used to sample the potential energy
differences between neighboring λ states. For each value of λ, simulations using a 2.0
fs time step were equilibrated for 1.0 ns, followed by 5.0 ns data collection. Potential
energy differences were output every 50 steps, or 0.1 ps. Integration of the equations
of motion were performed using the velocity Verlet algorithm, [SABW82] and all
bonds were constrained using RATTLE [And83] with a relative accuracy of 10−6 for
the small molecules, and SETTLE [MK92, HSP+ 04] for the water molecules.
Andersen pressure control[And80] was used to produce controlled pressure simulations. Andersen pressure control (in conjunction with Andersen temperature control)
rigorously implements an isobaric-isothermal (NPT) ensemble. The pressure was calculated using an atomic virial, with the molecular centers defined as the molecular
center of mass. Isotropic scaling of the cubic volume at each step was performed by
scaling the atomic positions, and then correcting the intramolecular distances with
RATTLE or SETTLE.
The Anderson piston mass was chosen to correspond with a compressibility with
4.5×10−5 bar−1 and a time constant of 1.67 ps. The thermodynamic observables
are formally independent of the choice of the piston mass. Therefore, as long as the
piston is heavy enough that the simulation is stable with the choice of time step, and
111
light enough so that that the time scale for the volume fluctuations (which were of
order 1 ps for these simulations) is much shorter than the simulation length, the value
of the piston mass is unimportant. Periodic boundary conditions were used and the
simulation cell was constrained to a cubic shape.
isothermal ensemble [And80]. In the 5.0 ns simulations presented here, velocities
were reassigned every 100 steps (0.2 ps), resulting in 25,000 separate reassignments.
The average kinetic energy of the simulations was checked to verify that it was in
agreement with the control temperature of 298 K.
Perhaps the most important methodological choices in molecular simulations are
those used to evaluate the long range interactions. In this study, a neighbor-list of 9.0
Å updated every 10 steps was used for the short-range interactions. Particle Mesh
Ewald [EPB+ 95] was used to evaluate the Coulombic interactions, with a real space
cutoff of 7.5 Å and a PME order of 6. The Fourier spacing was chosen to be as close to
1.0 Å as possible at each step given the box size and the need for integer numbers of
grid points. The relative tolerance between the long and short range Coulomb energy
was set to 10−6 , yielding a Gaussian width of 2.16831 Å. The Lennard-Jones energy
was computed with a switched, atomic-based cutoff between 7.5 Å and 7.0 Å. A long
range correction was added to the energy and pressure to eliminate the effect of this
finite range cutoff of the Lennard-Jones, including the effect of the switch [AT87].
These cutoffs are significantly shorter than used in many other simulations. They
were chosen as the cutoff parameters that made the simulations as fast as possible
while still changing the instantaneous potential energy by less than 0.05% and the
instantaneous pressure by less than 1% from much longer (up to 14 Å) short range
cutoffs. There is great value in simulation methodology that is almost completely
independent of cutoff parameters, as it removes arbitrary, non-physical variables that
can greatly affect observables from the comparisons between the results of different published simulations, and thus affect the validity of scientific conclusions and
comparisons.
112
The switching function S(r) is the standard GROMACS switching function, chosen so that the function and its derivative are continuous. We note than in order to strictly conserve energy, S(r) would also have to satisfy the conditions that
d2 S(roff )/dr2 = d2 S(rswitch )/dr2 = 0. However, trial constant energy simulations with
900 TIP3P molecules using 2 fs time steps exhibited no statistically meaningful drift
in total energy. Short trajectories from the same starting point were run with varying time step size, and the standard deviation of the total energy (or enthalpy, in the
case of the isobaric-isoenthalpic (NPH) ensemble) was quadratic in the time step size
(as checked down to 0.125 fs), consistent with the inherent accuracy of the velocity
Verlet algorithm [SABW82]. We therefore conclude that this switching function is
sufficiently accurate.
To prepare the simulation cell, a previously equilibrated cubic box of 900 water
molecules was taken, and the amino acid side chain analogs were then solvated by
placing them in the center of this box and removing all water molecules with atomic
centers within 2.0 Å of any atomic center of the solute. An extremely short (200-400
steps) NVT (isochoric-isothermal) simulation with very frequent velocity reassignments was then used to remove highly unfavorable interactions. A further 1.0 ns
NPT equilibration simulation was performed for each value of λ prior to the 5.0 ns
of data collection.
from single simulations are reasonable. For each free energy between a given two λ
values for each molecule, we estimated the uncertainty by dividing the full 5.0 ns
simulations into five 1.0 blocks, and computed the variance of the average over these
five simulations, by the standard formula
(hXi − X)2 /(n − 1). The total variance
P
for each molecular solvation free energy is computed as the sum of the variance for
each λ interval, with the standard error being the square root of the variance. This
analysis is not as sophisticated as in our previous study of amino acid side chain
analogs [SPSP03], and in general, this level of uncertainty analysis would not be
sufficient. However, in the previous study we demonstrated that these time scales
are sufficient to sample over the relevant degrees of freedom [N4A], and thus, making
113
a reasonable approximation that the change in simulation conditions does not affect
these correlations, this simpler uncertainty analysis is justified.
For the pure water simulations, 900 water molecules were used, and the averages were collected over a single 5.0 ns run. Statistical uncertainties, δA, for
the ensemble observables A in Table 4.2 were calculated by using the variance,
hA2 ic , and correlation time of the observable, τA , over the T = 5.0 ns simulation.
The uncertainty in the mean for the observable A is then given by the formula
δA =
q
q
hA2 ic 2τA /T [SABW82, AT87]. δA corresponds to one standard devia-
tion, σ, and all results are reported as hAi ± 1σ. Correlation times were computed
using GROMACS analysis tools.
4.2.2
Experimental free energies of hydration for weakly soluble solutes are determined from
concentration measurements made on two phase systems, where one phase consists
of a vapor with a partial pressure Ps of some solute molecule of type s, and the other
phase consists of an aqueous solution with a number density concentration for solute
molecules of ρls . When such a two phase system is at equilibrium with respect to
transfer of molecules of type s between the phases, the solvation free energy is given
by [WACS81, BNM84]:
∆Gsolv = kT ln(Ps /ρls kT )
(4.1)
In these simulations, H is the parameterized Hamiltonian, where the coupled
state (λ = 1) corresponds to a simulation where the solute is fully interacting with
the solvent and the uncoupled state (λ = 0) corresponds to a simulation where the
solute does not interact with the solvent. Free energies of hydration are computed
by simulation by decoupling a molecule of the solute (here, amino acid side chain
analogs) from the solvent by the Bennett acceptance ratio. We have found that the
Bennett acceptance ratio can be much more efficient than TI for classical force field
simulations in solution as described in Chapter 2 of this disseration.
We analyzed the correction needed to relate the actual solvation free energy ∆Gsolv
and the decoupling free energy ∆Gsim :
∆Gsolv = ∆Gsim − kT ln(V ∗ /V1 )
114
(4.2)
where V1 is the mean volume at full coupling (λ = 1) and V ∗ is the mean volume
of a box of pure solvent, with the same number of solvent molecules as in the coupled simulation [SPSP03]. In all cases, this correction was between 0.005 and 0.001
kcal/mol (roughly proportional to molecular size), an order of magnitude less than
the uncertainty, and so was not included.
We parameterized only the non-bonded interactions between the solute and solvent
molecules by λ, i.e., the non-bonded potential function is of the form:
U (λC , λLJ ) = Us + Uw + Uw−w + Us−w (λC , λLJ )
(4.3)
where Us is the intramolecular potential energy of the solute, Uw is the intramolecular
potential energy of the waters (zero for the classical models used in this study),
Uw−w is the intermolecular energy of the water-water interactions, and Us−w is the
intermolecular energy of the solute-water interactions. It was necessary to modify
the GROMACS code to make it possible to neglect the intramolecular terms when
changing parameters with respect to λ.
There are a wide variety of choices for paramerization in λ from the initial to
the final Hamiltonian, but the choice of a good path for a given system is still an
unsolved problem[CP02, BMvS+ 94, DK96, PVG02]. As discussed in our previous
paper [SPSP03], many of the common choices, such as linear interpolation between
Hamiltonians, can be extremely computationally inefficient, and have poor convergence properties at the end states. We avoided many of these these problems by using
the following expression for the λ-dependent non-bonded interaction energy:
XX
i
λC
j
qi q j
rij
!
+λLJ 4ij
1
1
−
(4.4)
6
2
[αLJ (1 − λLJ ) + (rij /σij ) ]
αLJ (1 − λLJ ) + (rij /σij )6
115
Eq. (4.4) includes the standard Coulombic term with a linear dependence on λ, but
also incorporates a “soft-core” parameterization, where the infinity in the LennardJones interaction is smoothed out as a function of λ. The parameter αLJ is a positive
constant which controls this transition, and is equal to 0.5 for all simulations in this
study. In our previous study, we used the function of Beutler, et al. [BMvS+ 94], but
some unpublished experimentation suggests that the form used here, with the leading
λ4 term is replaced by at λ1 term, and the αLJ (1−λLJ )2 term replaced by αLJ (1−λLJ ),
is more efficient, by perhaps 30%. In applying the transformation from solute fully
coupled with the solvent (λC = 1, λLJ = 1) to the solute fully decoupled from the
solvent (λC = 0, λLJ = 0), we first decouple solvent-solute Coulombic interactions,
and then the solvent-solute Lennard-Jones terms.
The Coulombic interactions were decoupled with λC values of 0.0, 0.5 and 1.0.
For the Lennard-Jones terms, preliminary investigations showed that uneven spacing
in λLJ could give better results than even spacing, as closer spacing is desired when
the curvature of hdH/dλi is larger. The λLJ values chosen were therefore 0.0, 0.2,
0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 1.0. We chose these values after testing determined
that they were sufficiently closely spaced to converge the free energy of the largest
molecule, the tryptophan analog, to within the desired uncertainty.
At each nth λ value, the potential energy using the Hamiltonian with the n −
1 and n + 1 λ values is computed, and the difference between these energies and
original potential energy is output. The Bennett acceptance algorithm is applied
to the distributions of potential energy differences from the from the n + 1 and n
states, sampled over both the n and n + 1 states [Ben76, SBHP03]. This process
extracts the free energy difference that best corresponds to these potential energy
difference distributions, and is further explained in the cited references. Since we
are computing the difference between energies in the isobaric-isothermal ensemble,
the free energy we are most interested in is the Gibbs free energy. We should add
to this potential energy difference a term proportional to the applied pressure times
the volume difference between the two states. This correction has not been applied,
and may be the source of some error. However, because this term is so small, this
116
correction is expected to be negligible.
We also computed long range Lennard-Jones corrections to the solvation free energy, an estimate for total attractive energy from the Lennard-Jones term neglected
from outside the cutoff regions [AT87, WFH02]. These long range corrections to the
solute-solvent interaction were computed after the simulation. This is in addition to
the long range dispersion correction performed during the simulation, which serves
only to correct the potential energy and pressure for the effect of the finite-range
Lennard-Jones cutoff, and was not included in the potential energy difference used
for the free energy computation.
For each solute the long range contribution to the free energy can be calculated
by numerically integrating:
ELRC =
X
i
16πρij
Z
∞
"
r= rswitch
σij
r
12
σij
−
r
6 #
S(r)r2 dr
(4.5)
where i runs over the solute atoms, r is the distance from solute atom i, ρ is the
number density of the solvent molecules, ij and σij are the standard Lennard-Jones
parameters between solute atom i and the solvent molecule Lennard-Jones site of type
j, and S(r) is the switching function used in GROMACS. In this study, there is only
one Lennard-Jones site per solvent molecule, so we can average over j. This derivation
of the correction assumes that the solvent radial distribution function g(rij ) = 1 in the
tapering region and beyond, which is an adequate assumption for the water models
used here, and for the size of the ligands.
4.3
Results
The main results of this study are a set of highly precise hydration free energy values
of the fifteen neutral amino acid side chain analogs solvated in the six previously
published water models. Additionally, we present free energies of hydration of the
side chain analogs in a series of modified water models that demonstrate that it is
possible to greatly improve solvent-solute free energies without significantly adversely
117
affecting the pure liquid properties of a water model. The free energies of the amino
acid side chains solvated with published water models is shown in Table 4.6, and the
free energies of the parameterized waters is shown in Table 4.7.
In Table 4.2, we present the model definitions as well as our data collected on
pure water using the simulation parameters used for the free energies of hydration
as defined in this paper. We see that all models, using the simulation conditions
described in the Methods section, are relatively accurate in predicting water properties
at 298 K. However, some models, such as TIP3P and SPC, do not give as accurate
predictions under the current conditions as in previously published papers [JCM+ 83,
JJ98, HSP+ 04] This is most likely because of differences of simulation conditions
between this and the previous studies; for example, Ewald methods generally result
in slightly lower densities than cutoffs [HSP+ 04].
The published values for TIP4P-Ew, which was developed using similar conditions
as ours with the same philosophy of maximum transferability of simulation parameters, yield encouragingly close results. Our density of 0.9954 ± 0.0002 g cm−3 agrees
exactly with their density of 0.9954 ± 0.0003 g cm−3 , and our (polarization corrected)
enthalpy of solvation of 10.565 ± 0.005 kcal/mol is only very slightly outside the extremely strict precision of the published value of 10.583 ± 0.004 [HSP+ 04]. These
simulations were performed on completely independent codes, with many differences
in input parameters. Some of these differences, with the previously published TIP4PEw study first and the current study second, are 512 vs. 900 water molecules, 1 vs. 2
fs time steps, velocity reassignment every 2.0 vs. 0.2 ps, 0.35 Å vs. 2.16 Å screening
Gaussian, and Lennard-Jones switched cutoffs of 9.0 Å to 9.5 Å versus 7.0 Å to 7.5
Å. However, since both simulations used methods that are relatively insensitive to
the input parameters (as described earlier in this paper), we believe this represents a
small success of the philosophy of parameterization for simulation conditions that are
not dependent on arbitrary parameters. As such methodologies become more widely
adapted, comparisons between simulation studies may become significantly easier.
In Table 4.3 are the free energy of solvation amino acids side chain analogs in
the six published water models. The average error and RMS error are computed in
two ways; first, as the flat average over all fifteen side chain analogs, and secondly,
118
as an average weighted by the naturally occurring frequency of the corresponding
side chains in proteins [MA88]. This averaging method is used because if used as a
validation for protein simulations, data relating to the accuracy of a less common side
chain, such as cystine, should not be as important as the data relating to leucine,
more than five times as common.
In both the weighted and unweighted averages in Table 4.3, all models underestimate solvation free energy to some extent. Indeed, all but TIP3P-MOD (which, although published, is not a water model in common usage) have solvation free energies
that are worse, on average, than TIP3P in comparison to experiment. Interestingly,
models that better capture pure liquid properties do not necessarily have improved
solute-solvent free energies. TIP4P-Ew is perhaps the most accurate three or four
point fixed charge model for representing water over a range of temperatures, thanks
to rigorous quantitative fitting to a range of experimental data [HSP+ 04]. However,
it has one of the worst fits of free energies of solvation to experiment of the models
in this study.
One important question is whether the experimental free energy solvation is actually the best number to compare the computational free energy to in order to asses the
model validity. For example, we do not necessarily expect classical fixed-charge models to be able to correctly model both the condensed phase and vapor phase behavior,
because of the change in polarization that we would expect to occur. Therefore, it
might be preferable to compare the simulation results to the experimental results with
a correction added to account for the change in polarization when transferring to the
vapor state, as the repolarization in the vapor phase is not important to the solutesolvent interactions. This decision was made in the parameterization of SPC/E and
TIP4P-Ew [BGS87, HSP+ 04], as can be seen in Table 4.2, these get the polarizationcorrected enthalpy of hydration is much closer to experiment than the non-corrected
enthalpy of hydration.
This correction for the enthalphy could also be applied to the free energy. Unfortunately, for arbitrary small molecules, these gas phase dipole moments have not
been well studied. For the relatively nonpolar molecules, this change is very small,
and the dipole readjustment terms are negligible. However, for the larger and more
119
polar molecules in this study, experimental numbers do not exist, or are extremely
imprecise, and a correction is therefore not possible. If this method were perused, it
might therefore be best to calculate the gas dipole moments from high level quantum
calculations.
It is useful to compare the free energies of hydration in TIP3P between the current
paper and our previous amino acid side chain paper. Because the sulfur parameters
have changed between the data sets, it is only valid to compare the other thirteen
molecules. Other than the change in sulfur parameters, any other differences will be
a result of the simulation conditions, with a hopefully small contribution from experimental error. Most important among the simulation differences is almost certainly the
treatment of long range interactions. We find an average difference of -0.19 kcal/mol
(with the new results being more favorable towards solvation), with a RMS difference
of 0.24 kcal/mol, and variance 0.14 kcal/mol. The differences ranged between 0.01
and -0.044 kcal/mol, and interestingly four of the five largest differences in magnitude belonged to the four hydrophobic side chain analogs of Val, Ile, Leu, and Phe.
Most likely, since the charges on these molecules themselves are relatively small, the
differences are due to changes in water structure in PME versus group-based tapered
cutoffs. In general these differences are rather low, which is somewhat encouraging,
in that different simulation methods yield fairly close free energies. However, it also
demonstrats the problems inherent in comparing results of studies run with different
simulation methodologies.
Perhaps the most important reason that current water and protein models do not
accurately predict free energies of solvation is simply that they were never parameterized to predict such observables. Free energies of solvation are computationally
expensive, and computational limitations of the time usually did not allow parameters to be further tuned to also reproduce accurate free energies of solvation in
water, a thermodynamic measure that is likely to be very relevant for biological processes. Current generation biomolecular force fields such as AMBER(ff94) [CCB+ 95],
CHARMM22 [ADMBB+ 98], GROMOS [vGBE+ 96], and OPLS-AA [JMTR96] are in
many cases successful in capturing the qualitative behavior of protein structure and
dynamics [PB02, KM02, HOVG02]. However, parameters were usually tuned to give
120
accurate fits to quantum mechanical energy barriers, as well as to reproduce enthalpies
of vaporization and densities for pure liquids. [CCB+ 95, ADMBB+ 98, vGBE+ 96,
JMTR96] Other molecular force fields such as TraPPE [CS99, CPS01, WMS00],
NERD [NEdP98], and GIBBS99 [EP99] have incorporated phase equilibria data that
implicitly include free energies of transfer, but as yet do not include the molecular
diversity necessary for biomolecular simulation.
Given that most current molecular force fields are not designed for predicting
solvation free energies, then it may be the case that there is sufficient flexibility within
the parameters to better predict the free energy of solvation while still not greatly
altering the pure liquid properties. So as a proof of concept, we decided to try to
reduce the average solvent-solute free energies while maintaining the solvent-solvent
and solute-solute properties.
We decided to adopt the philosophy that was used in the development of TIP3PMOD [SK95]. First, adjusting the Lennard-Jones parameters for the solute is not
an option, because the physical properties of the pure liquid solutes depend almost
entirely on these Lennard-Jones parameters. Adjusting the charges for the solutes was
not feasible either, as the making the charges sufficiently large to improve the solvation
free energy would lead to entirely unrealistic dipole moments for the hydrophobic
species.
Of course, because of the combining rules used, the solute-solvent behavior depends just as much on the solvent parameters as the solute properties. The precise
partial charges on the water are vital to maintain the correct water properties. The
Lennard-Jones parameters, however, are less significant. Because of the strength of
the Coulombic attractions, the average postion of the oxygen atoms of the water
molecules is within the r−12 Lennard-Jones repulsive core. Therefore, the depth of
the well is not nearly as important as the location of the repulsive wall. It should
therefore be possible to simultaneously increase , decreasing the depth of the well,
and decrease σ, keeping the repulsive core in approximately the same place, and thus
generally maintaining pure liquid properties.
TIP3P has = 0.15210 kcal/mol and σ = 3.15061Å, while TIP3P-MOD has
exactly the same geometry and partial charges, but a drastically different = 0.1900
121
kcal/mol and σ = 3.12171Å (see Table 4.5). Despite this difference, TIP3P-MOD
has an almost identical enthalpy of vaporization (10.109 kcal/mol for TIP3P-MOD,
10.091 kcal/mol for TIP3P-MOD), and a density that is actually closer, under the
current simulation conditions, to the experimental value of 0.9972 (0.9998 g cm−3 for
TIP3P-MOD, versus 0.9859 g cm−3 for TIP3P).
We therefore define a series of six models, M20, M21, M22, M23, M24, and M25,
identified by the significant digits of their value: M20 has = 0.20 kcal/mol, M21
has = 0.21 kcal/mol, and so on. We adjust the σ for each of these to fit as closely
as possible to the experimental density of water at 298 K, with a granularity of 0.001
Å in σ. The final parameters and pure liquid properties are shown in Table 4.4,
in comparison to TIP3P and TIP3P-MOD. This choice of density as the primary
variable for fitting to pure liquid properties is somewhat arbitrary. This was intended
as a proof of concept to demonstrate the parameter flexibility in current models,
not a production-level water model for extensive simulations, and fitting to only one
observable makes the exercise easier.
Looking at the pure liquid properties, we see that with all models M20-M25, we
are able to get a density that is within 0.009 g cm−3 of experiment for all models, and
are actually much closer to experiment than TIP3P, at 0.1% error versus 1.1% error.
The heat of vaporization is further from experiment (not surprisingly, since we were
not attempting to constrain it), but no model is further than 1.6% from the heat of
vaporization of TIP3P, the original model that all parameters but σ and are taken
from. The free energy of solvation is slightly further away from the experimental
value of -6.32 kcal/mol [BNM84] or -6.33 kcal/mol [AWFC90], but are still almost
within the uncertainty of the TIP3P and TIP3P-MOD. So we have a class of models
with very similar (or even improved, considering the density) pure liquid properties
compared to TIP3P.
How do these models compare to the other models with respect to the free energy of solvation of the amino acid side chain analogs? This data is presented in
Table 4.5. We see that the increased well depth greatly increases the affinity of the
solute molecules for the solvent, such that for the M24 and M25 models, the average
122
error goes almost exactly to zero. Taking the amino acid frequency-weighted average, this is an average improvement of -0.50 kcal/mol, with almost no affect on the
pure liquid properties. This is accomplished while at the same time reducing the
frequency-weighted RMS from 0.51 kcal/mol to 0.31 kcal/mol, and improving the
slope of the fit between experiment and simulation.
4.4
Discussion
precision of the results and the agreement with experimental measurements. In this
section, we discuss the utility and obtainability of high precision results, as well as
what these results can tell us about the models and simulations using these models.
There are many studies of free energies of solvation of models of small molecules,
and it is important to compare to these previous studies where possible. Unfortunately, comparisons between these simulations and the current simulations can prove
extremely difficult. A variety of modified force fields and simulation parameters have
been used in the literature, which can lead to results that are significantly different. In
our previous study of amino acid side chain analogs [SPSP03], we reviewed a number
of studies, and we refer the reader to this paper for a discussion of those comparisons. In this paper, however, we will review several comparisons of free energies of
similar molecules that have been published in the last year, since the publication of
the previous study.
One important study is the parameterization of a new version of the GROMOS
force field, parameterized to fit to the free energy of solvation to within 0.2 kcal/mol
error [OVMG04], more accurate than OPLS-AA with any of the original published
water models. These parameters were developed by first fitting the charges and
Lennard-Jones parameters to capture pure liquid properties and solvation free energies in cyclohexane, and then fitting new charges to the solvation free energies in
water. The fact that reparameterizing the solutes, instead of the solvent, as in this
study, achieved similarly improved results reinforces the idea that there is sufficient
flexibility in the parameters to fit to new data. However, the fact that they were
123
unable to use the same charges for both water and cyclohexane solvation also demonstrates the limitations of fixed-charge models and may point to the need for future
polarizable force fields. Changing the solvent parameters at the same time will add
some flexibility as demonstrated in this study, but is not at all clear if it would be
sufficient.
Another recent study evaluated the free energies of solvation for OPLS-AA in SPC
and TIP4P water [MT03]. We found that most energies presented here were different
from ours by between 0.0 and 2.0 standard deviations of the mutual uncertainty, with
a tendency to be slightly less favorable for solvation. In general, this is quite good
agreement given the difference in simulation conditions. The uncertainties of this
study were approximately five times the uncertainty of the current study, so this is
mostly likely a result of differing conditions, and perhaps a slight underestimate of
the uncertainty.
How accurate do we expect the hydration free energies computed using these
parameter sets to be? OPLS-AA and other current force fields (with the exception of GROMOS 53A5 and 53A6, published in mid-2004 [OVMG04]) were not parameterized to reproduce free energies of hydration because, until recently, it was
too computationally intensive to so do in an iterative manner with sufficient precision [N4B]. Instead, Lennard-Jones terms were generally derived to reproduce
bulk properties using condensed phase simulations of pure liquids, and charges were
generated from quantum mechanical calculations in isolated (gas phase) molecules,
with some constraints and scaling factors used to account for the solvent environment [CCB+ 95, ADMBB+ 98, JMTR96]. In this light, there is no a priori reason to
expect the free energies correspond particularly well to experimental results.
Given the high precision free energy values obtained in this study, we must ask
the unusual question if the experimental data have sufficient precision to provide
accurate comparison. As described in the previous study [SPSP03], a meta-analysis of
a number of experimental results indicated that the experimental values are between
0.03 and 0.07 kcal/mol for each molecule. The level of uncertainty in this paper is
then approximately equal to the experimental uncertainty.
A companion problem to ask is whether simulations this precise are even necessary.
124
Certainly the high level of precision obtained in this paper will not be necessary for all
uses of free energy calculations. For many predictive purposes, such as ligand binding,
we only are interested in free energies of binding correct to 0.5-1.0 kcal/mol. However,
we note that these results presented are for small molecules, and the error increases
with size and molecular complexity. For larger molecules, many fewer uncorrelated
measurements are accessible, and more efficient methods and extensive computational
power as presented in this paper will be necessary to achieve precision levels necessary
for ligand binding.
Additionally, we must demand much higher precision in the parameterization of
models than we do for predictions or other measurements. Assuming Gaussian error
distribution, reporting a standard deviation implies that approximately 68% of the
measurements would fall within one standard deviation of the answer. However,
for parameterization, as the results would be used over and over again, if at all
possible, we would prefer a much greater confidence in a given interval. Not all
observables have the same dependence on the parameterizations; some mighy be
more sensitive to changes in parameters that the ones chosen in the parameterizaion.
For parameterization, we would therefore suggest that the computational experiments
be as accurate as the experimental numbers where at all possible or plausible. True
differences of the models from experiment are masked by such large uncertainties,
making model improvement difficult. Uncertainties of this magnitude may rapidly
accumulate for simulations of larger numbers of atoms.
Most prior simulations computed only relative free energies of hydration, or computed absolute free energies by summing relative free energies of transformation relative to a single absolute energy calculation. We emphasize that the calculated free
energies in this paper are all absolute free energies, using simulations in which entire
molecules are completely decoupled from the solvent. Complete solvent-solute decoupling, rather than mutating between two solutes, entails a significantly larger change
in the Hamiltonian, and is thus harder to do with small error. It requires greater care
to ensure that the non-bonded interaction terms change in a well-behaved manner
with respect to λ, and requires more intermediate steps from starting to final Hamiltonian. However, it is a significantly more flexible method, not requiring a series of
125
structurally similar molecules.
We would like to emphasize several steps that are required in order to obtain
high precision free energies, apart from simply utilizing more computational power.
A proper statistical uncertainty analysis should always be conducted in order to
quantify reproducibility, either computing correlation functions or at the very least
blocking the simulations at a time scale many times larger than the know correlation
time.
The pathway that is chosen is vital to ensure the best overlap possible between
intermediate states. This paper is not intended as a comprehensive guide to such
parameterizations. However, the decoupling of electrostatic and Van der Waals interactions and the use of a soft-core potential in atomic site introduction/removal
results in magnitudes of variance of the intermediate free energies that make possible
the level of precision presented here. It is also very desirable to run multiple copies of
each simulation, starting from initial conditions that are as uncorrelated as possible.
If the variation between the multiple runs does not agree with the statistical uncertainty estimates, then insufficient sampling has been performed. This establishes a
useful necessary but not sufficient condition on the sampling required for a computed
free energy to converge. In order to strictly guarantee that sufficient sampling has
taken place, one must monitor all the relevant degrees of freedom and ensure they are
extensively sampling their allowed phase space. For small molecules, this is relatively
straightforward, as detailed in our previous amino acid side chain study. However,
for even small proteins, this becomes very difficult, and it is as yet unclear how this
can be done in a general case.
In our previous study, we described how such free energy computations, suitable
for parameterization of force fields, are now accessible to most researchers. The
advances in free energy methodology presented in this paper make such computations
much more possible. For example, in the previous paper, we estimated the total
CPU time as approximately 70 CPU years. In this study, we were able to perform
approximately four times as many free energy computations on the same size system in
only 8.4 CPU years, performed entirely on a local cluster, for an efficiency of more than
30 times. Part of this speedup is due to running simulations a year and a half later,
126
thus benefiting from Moore’s law, and part of it is from using an efficient code like
GROMACS. But the majority of this speedup is from the use of a smoother pathway
with respect to λ, and using the Bennett acceptance ratio to extract energies between
states with relatively low overlap much more efficiently. Even greater efficiency could
be obtain for a small decrease in precision by running somewhat decreased length
of simulations, combined with slightly fewer intermediate states of λ, or a slightly
smaller water box.
We have not attempted to compute the entire matrix of water models versus force
fields, as it is sufficiently clear that current generations of fixed-charge force fields
in general do not fit solute-solvent interactions, and it would be preferable to update them to get this data closer. GROMOS 53a5 and 53a6 represent a promising
start [OVMG04]. But it has been noted that the best effective pair-potentials for
some situations are not necessarily the best for other situations. For example, alkane
parameters that have been optimized for gas-phase and neat liquid simulations can
be worse than earlier models in the ability to compute accurate free energies of hydration of solutes [YM98, KDMJ94a], and in the parameterization of GROMOS it
proved impossible to fit accurately to free energies of solvation in both water and
cyclohexane [OVMG04]. It seems likely that more complicated force fields may be
necessary for transferability to multiple chemical environments. For example, polarizable force fields that have the ability to incorporate some many-body effects
and to respond to different environments are being developed by many different
groups [TSBD00, BKZ+ 99, KSBF04, PB04, PC03, RP02].
We do not intend to propose the M20-M25 water models as ideal water models
for adoption by the simulation community. Although the data shown here demonstrates for many purposes they are likely to serve as well as TIP3P or SPC, they
have only been specifically parameterized to get the density and average free energies of hydration correct. There are a range of other important observables to get
correct, such as the heat capacity, isothermal compressibility, and radial and angular
distribution functions, and ideally the parameterization would be done over a range
of temperatures that are relevant to biological simulations. For example, the O-O
radial distribution functions obtained in this study of TIP3P-MOD and M20-25 all
127
have a shallow minimum close to 4.5 Å, significantly further out from experiment
than TIP3P, at 3.5 Å. TIP4P-Ew, on the other hand, which compares extremely well
with experimental X-ray scattering data [HSP+ 04], has a somewhat deeper minimum
at 3.3 Å.
More preferable would be to follow the philosophy used in the development of
TIP4P-Ew, and simultaneously quantitatively parameterize to various pure liquid
properties and solvent/solute properties, which could be either free energy calculations or simulations of mixtures. This study and the other recent studies [SPSP03,
OVMG04, Mac04] have demonstrated that current fixed-charge biological force fields
work much better than they have any right to, but still have many flaws. A clear
next step is the inclusion of polarizability [Mac04, PC03], but improving current fixedcharge force fields may solve many problems that are currently extant before needing
to invoke polarizability.
Our results have some implications for simulations using standard water models
such as the ones studied in this paper in estimating free energies. We cannot expect that calculations performed on more complicated systems, such as those used
to compute ligand-protein binding free energies, will be any more accurate than the
hydration free energies (or at least the relative hydration free energies) of the respective small constituents. It may be possible for sufficiently precise calculations to be
more accurate than this limit, but that is dependent on fortuitous compensating errors between ligand-water and ligand-protein interactions. Our results may also have
implications for the utility of these force fields for predictions of protein structure,
stability, and dynamics. However, it is also true that many computed observables may
be relatively insensitive to the details of the potentials that produced the differences
from experiment noted in this paper.
4.5
Conclusions
We have examined the accuracy of a range of common water models used for protein
simulation for their solute/solvent properties, calculating the free energy of hydration
of fifteen amino acid side chain analogs derived from the OPLS-AA parameter set in
128
the TIP3P, TIP4P, SPC, SPC/E, TIP3P-MOD, and TIP4P-Ew water models. We
achieve a high degree of statistical precision in our simulations, obtaining uncertainties
for the free energy of hydration of 0.02 to 0.06 kcal/mol, equivalent to our previous
study and to that obtained in experimental hydration free energy measurements of the
same molecules. This degree of precision is at least an order of magnitude better than
most previous studies, and if the model under study is close enough to experiment,
this degree of precision is sufficient for virtually all practical uses.
All models evaluated had free energies of hydration that were less favorable to
hydration than experiment. TIP3P-MOD, a rarely-used model designed to give improved free energy of hydration for methane, gave uniformly the closest match to
experiment. All others were worse than TIP3P, the model used in a previous high
precision study. Importantly, the ability to accurately model pure water properties
does not necessarily predict ability to predict solute/solvent behavior.
We also evaluated the free energies of a number of new modifications of TIP3P
designed as a proof of concept to show that it is possible to obtain much better solute/solvent free energetic behavior without significantly changing pure liquid properties. We are able to decrease the average error to zero while reducing the RMS error
below that of any of the published water models, with measured liquid water properties remaining almost constant, demonstrating there is still both significant parameter
flexibility and room for improvement within current fixed charge biomolecular force
fields.
Because of recent developments we have made in computational efficiency of free
energy calculations, we were able to perform simulations that previously required
large scale distributed computing on smaller local cluster. We performed four times
as much computational work in a litle more a tenth of the computer time as a similar
study a year ago, making it amenable to running on a local cluster. This paper
also demonstrates that the pathway to accurate and reliable methods to compute
the free energies of interaction of ligand/macromolecular models may be clearer than
previously thought. There is still some flexibility and room for improvement in current
generation fixed-charge force fields.
-0.683
0.418
OH
H
0.06
H
C
129
-0.683
0.418
OH
H
H
0.06
0.06
C
0.06
0.085
0.145
H
Backbone
0.06
Figure 4.1: Differences in partial charges between (left) the OPLS-AA parameter set
serine side chain and (right) the OPLS-AA-derived parameters used for the side chain
analog in this study. Only the β-carbon changes in charge.
Table 4.1: Correspondence between amino acids and the amino acid side chain analogs
used in this study, and the naturally occurring frequency of the amino acids in proteins [MA88]
Ala
Val
Ile
Leu
Phe
methane
propane
n-butane
iso-butane
toluene
8.3%
6.6%
5.2%
9.0%
3.9%
Ser
Thr
Tyr
Cys
Met
methanol
ethanol
p-cresol
methanethiol
methyl ethyl sulfide
6.9%
5.8%
5.8%
1.7%
2.4%
Asn
Gln
Hid
Hie
Trp
acetamide
propionamide
4-methylimidazole
4-methylimidazole
3-methylindole
4.4%
4.0%
2.2%
2.2%
1.3%
130
Table 4.2: Model input parameters and pure liquid properties for SPC, SPC/E,
TIP3P, TIP4P, TIP3P-MOD, and TIP4P-Ew water models. Uncertainties for the
density are all ≤ 0.0002 g cm−3 , for the heat of vaporization are ≤ 0.005 kcal/mol,
and for the free energies are ≤ 0.03 kcal/mol. Computation of the heat of vaporization, including the polarization correction, was done according to previously published
methods [HSP+ 04]. Water experimental data of -6.33 and -6.32 come from [AWFC90]
and [BNM84], respectively.
(kcal/mol)
σ (Å)
q
dOD (Å)
dOH (Å)
6 HOH
Dipole (Debye)
Density (g cm−3 )
∆Hvap (kcal/mol)
∆Hvap [pol]
∆G (kcal/mol)
SPC
0.15550
3.16557
0.41000
N/A
1.0000
109.47
2.27
0.9775
10.474
9.645
-6.16
SPC/E
0.15550
3.16557
0.42380
N/A
1.0000
109.47
2.35
0.9988
11.686
10.506
-7.05
TIP3P
0.15210
3.15061
0.41700
N/A
0.9572
104.52
2.35
0.9859
10.091
8.911
-6.10
TIP4P
0.15550
3.15365
0.52000
1.50
0.9572
104.52
2.177
0.9997
10.412
9.913
-6.11
TIP3P-MOD
0.19000
3.12171
0.41700
N/A
0.9572
104.52
2.35
0.9945
10.109
8.929
-6.17
TIP4P-Ew
0.16275
3.16435
0.52422
1.25
0.9572
104.52
2.321
0.9954
11.610
10.565
-6.98
Experiment
0.9972
10.518
10.518
-6.33,-6.32
Table 4.3: Free energy of solvation of amino acid side chain analogs in published water
models. Slope and Constant are from the linear fit from the flat averages over the
fifteen side chain analogs. All averages in kcal/mol. Uncertainties of the individual
free energy solvation used in the average are 0.02 to 0.06 kcal/mol.
TIP3P-MOD
TIP3P
SPC
TIP4P
SPC/E
TIP4P-Ew
Flat
Ave. Error RMS Error
0.33
0.51
0.67
0.79
0.69
0.82
0.84
0.97
0.90
1.01
0.90
1.02
Weighted
0.23
0.42
0.50
0.64
0.53
0.64
0.71
0.82
0.71
0.82
0.65
0.77
Slope
0.943
0.948
0.929
0.936
0.932
0.939
Constant
0.119
0.389
0.427
0.596
0.642
0.668
131
Table 4.4: , σ, and pure liquid properties for TIP3P, TIP3P-MOD, and novel
Lennard-Jones-modified waters M20-M25. All model parameters except for and
σ are the same as TIP3P, as shown in Table 4.2.Uncertainties in density are ≤ 0.0002
g cm−3 , for the heat of vaporization are ≤ 0.003 kcal/mol, and for the free energies
are ≤ 0.02 kcal/mol. Computation of the heat of vaporization was done according to
previously published methods [HSP+ 04], but without the polarization correction.
(kcal/mol)
σ (Å)
Density (g cm−3 )
∆Hvap (kcal/mol)
∆G (kcal/mol)
TIP3P
0.1521
3.15061
0.9859
10.091
-6.16
TIP3P-MOD
0.1900
3.12171
0.9998
10.109
-6.17
M20
0.20
3.120
0.9976
10.044
-6.12
M21
0.21
3.118
0.9963
9.998
-6.14
M22
0.22
3.115
0.9976
9.983
-6.12
M23
0.23
3.113
0.9970
9.961
-6.13
M24
0.24
3.111
0.9976
9.948
-6.12
M25
0.25
3.110
0.9969
9.928
-6.13
Table 4.5: Free energy of solvation of amino acid side chain analogs in TIP3P, TIP3PMOD, and the Lennard-Jones modified water parameters presented in Table 4.4.
Slope and Constant are from the linear fit from the flat averages over the fifteen side
chain analogs. All averages in kcal/mol. Uncertainties of the individual free energy
solvation used in the average are 0.02 to 0.06 kcal/mol.
TIP3P
TIP3P-MOD
M20
M21
M22
M23
M24
M25
Flat
0.67
0.79
0.33
0.51
0.25
0.43
0.22
0.43
0.11
0.38
0.08
0.38
0.01
0.38
0.02
0.39
Weighted
0.50
0.64
0.23
0.42
0.18
0.37
0.15
0.37
0.06
0.35
0.04
0.36
0.00
0.36
0.01
0.37
Slope
0.927
0.943
0.957
0.959
0.965
0.968
0.979
0.981
Constant
0.389
0.119
0.090
0.060
-0.026
-0.041
-0.066
-0.052
132
Table 4.6: Free energy of solvation of amino acid side chain analogs in published water
models in Table 4.2. Average deviations from experiment are presented in Table 4.3.
All result in kcal/mol. Uncertainties of the individual free energy solvation used in
the average are 0.02 to 0.06 kcal/mol, roughly proportional to molecular size. TIP4P,
SPC/E, and TIP4P-Ew have longer diffusion times, hence longer water decorrelation
times, and hence slightly larger uncertainties for each amino acid side chain analog.
Ala
Val
Ile
Leu
Phe
Ser
Thr
Tyr
Cys
Met
Asn
Gln
Hid
Hie
Trp
Experiment
1.94
1.99
2.15
2.28
-0.76
-5.06
-4.88
-6.11
-1.24
-1.48
-9.68
-9.38
-10.27
-10.27
-5.88
TIP3P-MOD
2.18
2.15
2.10
2.08
-1.31
-4.58
-4.42
-5.96
-0.83
-0.78
-8.88
-9.02
-9.36
-9.46
-5.55
TIP3P
2.24
2.34
2.43
2.27
-0.86
-4.51
-4.22
-5.46
-0.55
-0.35
-8.51
-8.63
-8.88
-9.08
-4.88
SPC
2.17
2.41
2.48
2.50
-0.79
-4.62
-4.43
-5.39
-0.50
-0.42
-8.53
-8.59
-8.71
-9.14
-4.70
TIP4P
2.25
2.55
2.68
2.71
-0.65
-4.71
-4.47
-5.06
-0.43
-0.07
-8.52
-8.49
-8.66
-9.10
-4.14
SPC/E
2.27
2.56
2.76
2.75
-0.54
-4.55
-4.37
-5.07
-0.33
-0.05
-8.32
-8.42
-8.55
-8.99
-4.30
TIP4P-Ew
2.27
2.63
2.82
2.64
-0.42
-4.65
-4.44
-4.80
-0.34
-0.08
-8.51
-8.63
-8.68
-8.96
-4.02
133
Table 4.7: Free energy of solvation of amino acid side chain analogs in TIP3P, TIP3PMOD, and the Lennard-Jones modified water parameters presented in Table 4.4.
Average deviations from experiment are presented in Table 4.5. All result in kcal/mol.
Uncertainties of the individual free energy solvation used in the average are 0.02 to
0.05 kcal/mol, roughly proportional to molecular size.
Ala
Val
Ile
Leu
Phe
Ser
Thr
Tyr
Cys
Met
Asn
Gln
Hid
Hie
Trp
Experiment
1.94
1.99
2.15
2.28
-0.76
-5.06
-4.88
-6.11
-1.24
-1.48
-9.68
-9.38
-10.27
-10.27
-5.88
TIP3P
2.24
2.34
2.43
2.27
-0.86
-4.51
-4.22
-5.46
-0.55
-0.35
-8.51
-8.63
-8.88
-9.08
-4.88
TIP3P-MOD
2.18
2.15
2.10
2.08
-1.31
-4.58
-4.42
-5.96
-0.83
-0.78
-8.88
-9.02
-9.36
-9.46
-5.55
M20
2.13
2.08
2.14
2.13
-1.32
-4.63
-4.45
-6.20
-0.86
-0.86
-9.05
-9.26
-9.50
-9.60
-5.63
M21
2.19
2.10
2.12
2.11
-1.45
-4.57
-4.51
-6.31
-0.96
-0.90
-9.02
-9.24
-9.48
-9.73
-5.76
M22
2.19
2.05
2.10
1.96
-1.55
-4.66
-4.53
-6.40
-1.09
-1.10
-9.21
-9.41
-9.58
-9.89
-5.93
M23
2.14
2.05
2.07
1.90
-1.60
-4.56
-4.58
-6.45
-1.06
-1.04
-9.25
-9.39
-9.74
-9.91
-6.02
M24
2.21
2.00
2.00
1.99
-1.65
-4.65
-4.58
-6.63
-1.09
-1.09
-9.36
-9.54
-9.86
-10.02
-6.17
M25
2.22
2.07
2.03
2.01
-1.64
-4.62
-4.57
-6.69
-1.10
-1.11
-9.32
-9.59
-9.87
-9.97
-6.19
Chapter 5
Computing the free energy of
binding of ligands to FKBP12
using worldwide distributed
computing
134
CHAPTER 5. LIGAND BINDING WITH DISTRIBUTED COMPUTING
Abstract
We calculate the absolute free energy of binding for nine ligands to the
FKBP12 protein with full atomistic molecular dynamics simulations with
explicit water, using the distributed computing resources of Folding@Home.
Because of the large amount of simulation time available, over 700,000
CPU days, we are able to compute the absolute, instead of relative, binding free energies, and remove many of the constraints such as making parts
of the system rigid typical in free energy simulations. We have obtain
binding free energies with clear predictive value, with RMS from fit of ∼1
kcal/mol, despite the size of the sampling problem and the lack of force
fields parameterized for condensed phase free energy calculations. We
also identify important barriers to further development of absolute binding calculations that would have been impossible to identify and classify
without the statistically significant levels of information that distributed
computing makes possible.
135
5.1
136
Introduction
An important goal of computational chemistry is the accurate prediction of free energies in molecular systems. The calculation of free energies is a computationally intensive task, even for small systems, because it requires correctly sampling the thermally
relevant configurations. Extensive work over the last 20 years has gone into developing the theoretical and computational apparatus of free energy computations, and
these calculations can now be performed for many systems with moderate levels of
accuracy [CP02, Kol93, Jor04, LJ97]. However, the accuracy of current calculations
has still not progressed enough to as useful, for example, in reliably predicting of
experimental observables such as drug binding affinities, which require accuracies of
0.5-1.0 kcal/mol.
There are two main barriers to the quantitative prediction of free energies of interaction. First, the phase space of the model must be sufficiently sampled to capture
all thermally relevant contributions to the ensemble average of the observables of
interest [RS04, Ber91, CKP96]. Otherwise, the results will lack the necessary precision, and independent calculations will most likely lead to differing answers. Only if
sufficient precision is obtained in a statistically well-defined manner is it possible to
design models that are sufficiently accurate for the application at hand. Second, there
are the discrepancies between the models used for simulation and the experimentally
measured reality. These force fields are both insufficiently verified with respect to
experiment and inherently approximate in their ability to replicate full physical phenomena [SPSP03, JMTR96, KFRJ01, ADMBB+ 98, CCB+ 95]. This lack of accuracy
means the model may not adequately represent the experimental system under study.
Many biological processes involve large-scale conformational rearrangement that are
usually thought to be beyond the current computational resources and methodologies,
and recent generations of force fields have not necessarily been tuned to solvent-solute
interactions [JMTR96, KFRJ01, ADMBB+ 98, CCB+ 95].
In previous studies of amino acid side chain analogs described in Chapters 3 and
4, we demonstrated that precision of 0.02-0.05 kcal/mol, equivalent to the precision
of the corresponding experiments, is possible to achieve for small molecule energy
137
of solvation. This was an order of magnitude greater precision than for any other
previously computed set of free energies of solvation. We compared the free energy
of hydration to experiment for three common biomolecular potential sets and six
water models, and demonstrated that the free energies of hydration were significantly
underestimated in all cases tested, with an RMS of ∼1 kcal/mol. The fact that these
free energies are so inaccurate for small molecules suggests that there may be some
problems in computing free energies for larger systems. Additionally, the long time
scales of the torsional degrees of freedom are worrisome for the calculation of free
energies in complex systems, such as ligand binding.
Initially, these small molecule free energies were computed using Folding@Home.
However, due to methodological improvements in the calculation of free energies that
improved the efficiency by over an order of magnitude, later computations were performed on local clusters. Due to the success in obtaining high precision of these
simulations, and the qualitative (if not quantitative) validation of current biomolecular force fields for free energy calculations of biological models in explicit water,
we have applied the principles learned in these simulations towards the much more
difficult challenge of the calculation of ligand binding free energies.
Previous ligand binding calculations Much time and effort has gone into the
prediction of ligand binding affinities, in both industry and academia because of the
direct application to rational drug design and atomistic understanding of biomolecular interactions [CP02, Kol93, LJ97, Jor04]. Because of the computational cost of
full atomistic simulations of ligand binding, the majority of the focus has gone into
QSAR (quantitative structure-activity relationship) and QSAR-like methods, which
are informatics-based methods. These methods require a training set of accurate
binding affinities, and the selection of sets of important surrogate properties that can
be correlated with the binding affinity. These methods have shown success in distinguishing good and poor ligands within a homologous class [Baj02, Kle00, Kub02].
138
However, they suffer from considerable training set bias and difficulties in selecting good variables on which to train, reducing the ability to systematically improvement or generalize these methods. At a somewhat higher level, some calculations have been of a hybrid variety, correlating thermodynamic and energetic data
from molecular simulations to binding affinities, such as extended LIE and MMPBSA [AqMS94, LTRJ99a, ZFG+ 01, LTRJ99b], which showing improved predictive
behavior, but still only within homologous classes, and with considerable training set
data.
Significantly less, but still considerable, work has gone into the computation of
free energies using atomistic, rigorously statistically mechanically based simulations.
Virtually all of the computations using explicit representation of water have taken
advantage of the thermodynamic cycle to compute relative free energies between
ligands or between mutant proteins [SM92, SM91b]. The free energy of a change
to “mutate” a small part of the ligand from one chemical identity to another, or
to mutate one residue to another, is computed both in complex and in isolation,
and the difference of free energy of binding between the two entities is computed
as the difference between these two changes. In the 1990’s, many research groups
made important advances in free energy calculations, especially in the computation
of relative free energies [Kol93, LJ97, RE01, Jor04]. However, enthusiasm waned
somewhat as when the remaining hurdles to useable results became clear [CP02].
Even the most rigorous binding calculations usually make approximations such as
keeping the backbone rigid, or solvating only the binding region with just a sphere of
water, approximations whose effects have never been fully quantified. Usually, such
simulations use the equivalent of only a few hundred picoseconds or a nanosecond
of simulation [RE01, LJ98, PJ01] to attempt to capture the relevant dynamics of
proteins. The errors reported are frequently similar in magnitude to the differences,
which makes accurate assessment of utility difficult.
It is certain that by measuring relative free energies, the effect of the many of
these approximations cancel out in the difference between the two mutations. It
is not at all clear how much the differences cancel, because of the complex nature
of binding phemonena and dynamics, and it is certain that in some cases, these
139
approximations may be a source of significant error in the overall binding strength.
Additionally, measuring small relative free energies is a method of somewhat limited
utility, because of the requirements of molecular similarity between the species being
compared. It would be much preferable to used methods for calculating absolute
binding free energies, if it was possible to do with any degree of efficiency.
Purposes of present study We have chosen FKBP12 as our model system for
binding affinity in this paper. There are a number of reasons for choosing this system. FKBP12 is important in cell signaling pathways. It is an extremely thermodynamically stable protein, with relatively little fluctuation [HLY+ 93], and the binding
pocket is hydrophobic, well defined, and stable. There are many experimental studies
for FKBP binding affinity, with many different ligands [HLY+ 93, DBF+ 96, SHMF96].
For this initial study, we took nine ligands with measured Ki,app ’s ranging from 2000
to 0.4 nm [HLY+ 93]. All affinities were measured by the same research group, published in the same paper, removing an important source of experimental uncertainty.
The ligands and the abbreviations are shown in Fig. 5.1, with binding constants and
free energies of binding in Table 5.1.
This study is novel in many ways. We are attempting to compute absolute free
energies of binding, for both moderately sized and very large ligands, rather than
relative free energies of binding of small changes. We are removing from the system
constraints that are usually applied, such as keeping the protein backbone rigid, and
simulating only a limited part of the system. With these relaxation of restrictions,
we are also using significantly more computational time than any other rigorous free
energy computation. We estimate that overall, this study used nearly 700,000 CPU
days, several orders of magnitude more than any other study to compute statistically
mechanical valid free energies of binding. Because of this computational power, we
are able to generate a data set that can be analyzed in a number of ways previously
impossible. If we succeed in obtaining sufficient sampling of these systems, and thus
obtain high precision values, remaining deviations from experiment must arise from
inaccuracies in these models – where “models” refers both to the force field parameters
and to the choice of simulation protocols. Conversely, if we are unable to obtain
140
sufficient sampling to compute the free energies in a repeatable manner, then we will
have put a lower limit on the sampling needed to compute free energies of ligand
binding in explicit water.
5.2
5.2.1
Methods
The Folding@Home distributed computing infrastructure [SP00] was used to perform the free energy calculations presented in this paper. Folding@Home (at http://
folding.stanford.edu) is a collection of approximately 200,000 volunteered computers
across the world running the Folding@Home client program. Volunteered computers automatically download calculations to be performed and return the processed
results to servers at Stanford University. Folding@Home has previously been used
to perform massively parallel calculations successfully simulating the folding kinetics
peptides and of small proteins [SNPG02, ZSP01, PBC+ 03, ZSK+ 02, LSSP02], design
ensembles of new protein sequences which fit to specified structures, [LEDP02], and
compute the free energies of solvation for large sets of small molecules [SPSP03].
All ligands were parameterized with AM1-BCC charges [JBJB00, JJB02], and all
other parameters from the 2002 version of AMBER General Force Field (GAFF) [WWC+ 04].
FKBP12 was parameterized using AMBER99-φ [SP04] This forcefield is identical to
AMBER (ff99) [WCK00] with the exception that the backbone φ angles are defined as
in AMBER (ff94) [CCB+ 95]. This has been shown to reproduce certain experimental
measurements more faithfully than AMBER99 [SP04]. One additional difference is
that 1,4 H–H intramolecular non-bonded interactions were not included in the intramolecular energy of the protein. As there are no polar-polar 1,4 H–H pairs on
proteins, this effect is expected to be relatively low; and most likely will slightly lower
torsional barriers. TIP3P [JCM+ 83] was used as the water model. 4 Cl- ions, again
using AMBER parameters, were included to neutralize the overall system.
To prepare initial complex models we used the appropriate crystal structures for
FK506 (1FKF), L8 (1FKG), and L9 (1FKH) and constructed models for the other
141
complexes in a conservative scheme. To generate the L5 model we deleted the cyclohexyl moiety from the L9 model. To generate the L3 model we further truncated
the phenylpropyl ester to an ethyl ester. To construct L2, we replaced the L3 tertpentyl group with a cyclohexyl group. To generate L6 we took the L5 model and
replaced the unsubstituted phenyl with a 3,4,5-trimethoxyphenyl. To generate L12
we started with the L13 complex (1FKI). L13 has a chain of 12 carbon atoms, while
L12 has only 10. We deleted the two methylene groups in the middle of this chain
and minimized the energy to link the chain once more. The final complex, L14, was
the most challenging model. It combines the key 1,3-diphenylpropyl subunit of L8
into an intramacrocyclic phenylene group. We aligned the L8 and L13 models in the
pipecolyl region and spliced the coordinates for the ester and methylene chain from
L13. The end of this chain was linked to the macrocycle phenyl group with a novel
ether and the entire ligand was steepest-descent energy minimized using the applicable GROMACS parameters described below. We generated two models for bound
L14 by building the ether from either meta position of the phenyl from the L8 model.
These initial conformations will be referred to as the CRYS models.
In order to understand the nature of the sampling using extensive molecular dynamics, we also present results generated from alternate initial coordinates. This set
of models was generated by novel modal Monte Carlo code (written by researchers at
Protein Mechanics, now a part of Locus Pharmaceuticals). These structures all had
the ligand in the binding pocket, but not necessarily in the same binding mode as
the models generated from crystal structures. Upon inspection, the models appeared
slightly less physical than the CRYS models; however, we are most interested in testing the hypothesis that our sampling can relax out of sub-optimally docked states, so
this is perfectly acceptable, and perhaps preferable as a test case. These models will
be referred to the ALTN models.
Unless otherwise noted, the methods and simulation parameters used here are
the same as used in a previous study of the free energies of solvation for amino acid
side chain parameters, using a modified version of GROMACS (v3.1.4) [BvdSvD95,
LHvdS01]. The parameters used here do not correspond in all details to the tested
methodologies for free energy calculations laid out earlier chapters; for example, the
142
thermostat and barostat, and the PME parameters are different for techical reasons
relating to difficulties of porting to a distributed computing environment. However,
we believe that the differences and/or error introduced are generally very small compared with other errors in the system, such as the sampling of the torsional degrees
of freedom.
For sampling, we chose to use molecular dynamics as the most direct method
for isobaric-isothermal Boltzmann sampling of condensed phase molecular systems.
Integration of the equations of motion were performed using the leapfrog Verlet algorithm algorithm [AEH87], and all bonds were constrained using LINCS [HBBF97]
with order 12 for the proteins and ligands, and SETTLE [MK92, HSP+ 04] for the
water molecules. We realize that this method is not the most computationally efficient in general. But it is sufficient for the problem at hand, as well as being simple
to implement, frequently utilized, well-studied, and well-understood.
isothermal ensemble [And80, N5B]. The Nose-Hoover algorithm was investigated, but
it was found that when the ligand was decoupled from the rest of the simulation, the
velocity was distributed to the translational degrees of freedom of decoupled ligand
in a non-ergodic way. Using Andersen temperature control, velocities were reassigned
every 50 steps (0.1 ps). The average kinetic energy of the simulations was checked to
verify that it was in agreement with the control temperature of 298 K [N5A]. The
pressure was constrained to 1.0 atm using the Berendsen pressure control algorithm,
with a time constant of 0.5 ps, and a compressibility of 4.5 × 10−5 .
Perhaps the most important methodological choices in molecular simulations are
those used to evaluate the long range interactions. In this study, a neighbor-list of
10.0 Å updated every 10 steps was used for the short range interactions. Particle
Mesh Ewald (PME) [EPB+ 95] was used to evaluate the Coulombic interactions, with
a real space cutoff of 9.0 Å and a PME order of 4. The Fourier spacing was chosen
to be as close to 1.2 Å as possible at each step given the box size and the need for
integer numbers of grid points. The relative tolerance between the long and short
143
range Coulomb energy was set to 10−8 , yielding a Gaussian width of 2.221 Å. The
Lennard-Jones energy was computed with a switched, atomic-based cutoff between
8.0 Å and 9.0 Å. A long range correction was added to the energy and pressure
to eliminate the effect of this finite range cutoff of the Lennard-Jones, including the
effect of the switch.
Unbound ligand simulations were conducted in a truncated octahedron box with
periodic boundary conditions. Each of the ligands was parameterized by placement in
a solvent box with minimum 5 Å between the largest ligand and the nearest cell wall,
resulting in a main unit cell length of 27.2 Å, and 473 to 504 TIP3P water molecules,
followed by steepest descent energy minimization. This was followed by molecular
dynamics equilibration of the solvent with all other atoms frozen for 200 ps, finally,
equilibration the entire system for 200 ps prior to running on Folding@Home. Conditions for the equilibrations were largely the same as those for the production runs;
Berendsen temperature control was used rather than Andersen, however. FKBP12ligand complex simulations were prepared as above also, with the exception that
minimum distance was 6 Å between the complex and the nearest unit cell wall on
either side, (resulting in main unit cell vectors of length between 56.9 and 60.8 Å, depending on the orientation of the protein and the docked position of the ligand. The
total number of water molecules was between 4343 and 5082 TIP3P water molecules.
Simulations were run on Folding@Home in work units of 1.0 ns (ligand) or 200
ps (complex) each. After completing each work unit, volunteered computers running
Folding@Home returned the work with a restart file saved at the end of the unit to
servers at Stanford, and the restart file was processed to generate a new work unit,
which was then assigned to a new volunteered client computer. The load was balanced
to keep the twenty copies of each system at approximately the same length, though
because of the intricacies of the heterogeneous environment, some simulations ended
up longer than others.
5.2.2
144
All free energies were computed using the Bennett acceptance ratio molecular dynamics used to sample the potential energy differences between different λ states.
For each value of λ, simulations using a 2.0 fs time step were equilibrated for 1.0
ns, followed by 5.0 ns data collection. Potential energy differences between different
intermediate states were output every 50 steps, or 0.1 ps.
At each nth λ value, the potential energy using the Hamiltonian with the n −
1 and n + 1 λ values is computed, and the difference between these energies and
original potential energy is output. The Bennett acceptance algorithm is applied
to the distributions of potential energy differences between the n + 1 and n states,
sampled over both the n and n + 1 states [Ben76, SBHP03]. This process computes
the free energy difference that statistically best corresponds to these potential energy
difference distributions, and is further explained in the cited references. We apply the
Bennett acceptance ratio to each 1.0 ns set (of 50,000 samples) of potential energy
differences in both directions in the ligand decoupling case, and each 200 ps set (of
10,000 samples) in the complex decoupling case. Averages of the free energy at each
λ value are computed over the 1.0 ns or 200 ps samples. For each λ value and ligand,
we ran twenty copies of the system, and averaged over these copies as well.
from single simulations are reasonable. For each free energy between two λ values, we
computed the variance of the average over these twenty simulations for each ligand,
by the standard formula
(hXi − X)2 /(n − 1). The total variance for each molecular
P
solvation free energy is computed as sum of the variance for each λ interval, with the
standard error being the square root of the variance. This assumes that the separate
clones are uncorrelated, which is not necessarily the case. Previous studies described
in the earlier chapters of this dissertation indicate that the water degrees of freedom
re-equilibrate on the time scale of picoseconds. However, the relative translation and
rotation of the ligand degrees of freedom and the torsional degrees of freedom take
much longer [SPSP03], sometimes on the order of nanoseconds.
In order to compute the absolute free energies, we compute the difference of two
145
free energies for each ligand. First, the free energy of solvation for each ligand is
computed, as described in previous chapters. Then, the free energy of turning off
the interactions between the ligand and the rest of the system, both the protein
and the solvent, is computed. The difference can be related to the experimental
binding energy. The theory behind this association is surprisingly complicated, and
has been discussed extensively [GGBM97, RNPS96, BTLK03], and we follow the
theory presented there. We will use the terms “solvent decoupling” and “complex
decoupling” to refer to the two calculations, for lack of better terminology.
We parameterized only the non-bonded interactions between the ligand and solvent or solvent plus complex molecules by λ. It was necessary to modify the GROMACS code to make it possible to neglect the intramolecular terms when changing
parameters with respect to λ. There are a wide variety of choices for paramerization
in λ from the initial to the final Hamiltonian, but the choice of a good path for a
given system is still an unsolved problem [CP02, BMvS+ 94, DK96, PVG02]. As discussed in previous chapters, many of the common choices, such as linear interpolation
between Hamiltonians, can be extremely computationally inefficient, and have poor
convergence properties at the end states. We avoided many of these these problems
by using the following expression for the λ-dependent non-bonded interaction energy:
XX
i
λC
j
+λLJ 4ij
qi q j
rij
(5.1)
1
1
−
6
2
[αLJ (1 − λLJ ) + (rij /σij ) ]
αLJ (1 − λLJ ) + (rij /σij )6
Eq. (5.1) includes the standard Coulombic term with a linear dependence on λ, but
also incorporates a “soft-core” parameterization, where the infinity in the LennardJones interaction is smoothed out as a function of λ. The parameter αLJ is a positive
constant which controls this transition, and is equal to 0.5 for all simulations in this
study. In our previous study, we used the function of Beutler, et al. [BMvS+ 94], but
some unpublished experimentation suggests that the form used here, with the leading
!
146
λ4 term is replaced by at λ1 term, and the αLJ (1−λLJ )2 term replaced by αLJ (1−λLJ ),
is more efficient, by perhaps 30%. In applying the transformation from solute fully
coupled with the solvent (λC = 1, λLJ = 1) to the solute fully decoupled from the
solvent (λC = 0, λLJ = 0), we first decouple solvent-solute Coulombic interactions,
and then the solvent-solute Lennard-Jones terms.
In both the solvent and complex decoupling simulations, the Coulombic interactions were decoupled with λC values of 0.0, 0.1, 0.25, 0.40, 0.55, 0.70, 0.85 and 1.0.
For the Lennard-Jones terms, preliminary investigations showed that uneven spacing
in λLJ could give better results than even spacing, as closer spacing is desired when
the curvature of hdH/dλi is larger. Therefore, the variably spaced λLJ were used:
0.00, 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.65, 0.70, 0.75, 0.80, 0.85, 0.90, 0.95, and
1.00. This spacing was determined to be sufficient after it was found to be sufficient
to converge the free energies of hydration of L2 and L20 in preliminary simulations.
Although the correlation times for rearrangement are much longer in the complex, the
width of the distributions should be approximately the same as with the hydration
free energies.
A problem with decoupling the ligand from the protein and water is that to
compute an accurate free energy from the entirely decoupled state to the next-least
coupled state, we need to sample over the entire simulation box, which is likely to
have a extremely long correlation time. This is not important in the ligand-only state,
as without the ligand, the solvent box is isotropic. To overcome this problem with the
complex, we use a formalism presented previously in various forms, by attaching the
decoupled ligand by a spring to the protein, and gradually turning this spring off as
the ligand is coupled. In the fully coupled state, the spring is turned off. The effect of
this spring in the ghost state can be computed analytically. We use a spring constant
2
of 0.239 kcal/mol/Å in the fully decoupled state, and the force constant is reduced
linearly throughout the Lennard-Jones decoupling simulations, becoming zero when
the Lennard-Jones intermolecular terms are fully on. The spring is attached on one
end at the central pipecolyl nitrogen in all ligands. The other end is a dummy point,
defined geometrically in relationship to the Cα of Ser38 and the Cα of Val55, at the
sides of the binding pocket and the C atom bound to N of Trp59, at the bottom
147
of the binding pocket. The location was chosen by taking the average position of
the attachment nitrogen over nanosecond simulations of the bound state, and was
extremely stable, with a variance of under 1Å. The analytical correction that must
be added to the difference between the two solvation free energies is expressed as
described previously [GGBM97, RNPS96, BTLK03]

∆Gspring
3
= kT ln 
V
kT π
K
!3/2 

(5.2)
Where K is the spring constant, V is the volume of reference state. The aver3
age volume for a reference concentration of 1 M is 1.661 × 103 Å /molecule, so this
correction is therefore in all cases 3.35 kcal/mol, in the direction of a smaller binding constant. This accounts for the relative degrees of translational freedom without
needing to sample the entire simulations cell. Similar correction terms are presented
by Boresch that account for rotational restraints, but we made the assumption that
the rotational sampling was sufficient, given the amount of simulations time, an assumption analyzed later in this paper.
In order to define a bonding constant, we must define a bound state. A state
where the ligand has floated away from the protein does not correspond to experimental bound state in any experiment. In simulations of the complex, we therefore constrain the ligand by a hard harmonic spring with force constant of 5.975
2
kcal/mol/Å , when the central pipecolyl nitrogen reaches more than 12.0 Å from the
spring attachment point in all simulations. As discussed later, this pipecolyl nitrogen
remains within 1 to 2 Å from in the bound simulation, and within 8 Å in the unbound
(but string-constrained) state. This distance is never approached in the simulations,
so we consider this a reasonable definition of the bound state.
We also examined long range Lennard-Jones corrections to both deletion free energies, an estimate for total attractive energy from the Lennard-Jones term neglected
from outside the cutoff regions [AT87, SPSP03] In previous papers, this long range
correction was found to accurately correct for the presence of finite-range LennardJones cutoffs, to very high precision [SPSP03]. These long range corrections to the
solute-solvent interaction were computed after the simulation. This is in addition to
148
the long range dispersion correction performed during the simulation, which serves
only to correct the potential energy and pressure for the effect of the finite-range
Lennard-Jones cutoff, and was not included in the potential energy difference used
for the free energy computation.
To the lowest order, one might expect the long range corrections from the solvent
decoupling and the complex decoupling computations for each ligand to be approximately the same, thus canceling and having no effect on the final binding energy.
However, at closer inspection, there are reasons why this might not be a good approximation. The density of Lennard-Jones sites is not the same in the protein as
it is in bulk solvent. Additionally, the σ and for the average Lennard-Jones site
in the protein are different than for water. So we must make a more sophisticated
approximation. As it turns out, this difference results in a significant affect on the
binding free energy.
For each solute atom, the analytic long range contribution to the free energy of
solvation can be calculated by integrating:
ELRC =
X
i
16πρij
Z
∞
r= rswitch
"
σij
r
12
σij
−
r
6 #
S(r)r2 dr
(5.3)
where i runs over the solute atoms, r is the distance from solute atom i, ρ is the
number density of the solvent molecules, ij and σij are the standard Lennard-Jones
parameters between solute atom i and the solvent molecule Lennard-Jones site of type
j, and S(r) is the switching function used in GROMACS. The approximations in this
correction are that the solvent radial distribution function g(r) = 1 and is isotropic
in and beyond the tapering region, and that the system density is the same as the
bulk density of the solvent, ρ, and a single ij value for the surroundings. While
proven to be quite accurate for small molecule solvation [SPSP03], for a complex,
these approximation are not strictly applicable.
In Table 5.2, we compare the analytical correction to the long range LennardJones energy for the ligand in pure water with the difference in Lennard-Jones energy
computed with exact potential energy evaluations at different cutoffs distances. We
149
used cutoffs tapered from 8 Å to 9 Å with the energy computed with cutoffs tapered
from 24 Å to 25 Å, for each ligand-FKBP complex, and cutoffs tapered from 12 Å
to 12.5 Å for each ligand alone. This energy is averaged over one structure every
1.0 ns, over the the entire data set from 2.0 ns onward, for both the ligand and the
complexes. The variance in all cases is less than 0.18 kcal/mol for each different
ligand, so the average over the approximately 200 frames for each set of complexes is
∼0.01 kcal/mol. 25 Å was slightly less than half the minimum image distance, and
so it is not possible to go out any further. As can be seen this correction is highly
significant, of the same order of magnitude of the binding energy itself.
To quantify this approximation, we need to examine how far out this LennardJones correction needs to extend. We evaluate this by looking at the difference between the analytical correction and the exactly evaluated Lennard-Jones energies for
the largest ligand, L20, in solution and in the FKBP complex. This data is presented
in Fig. 5.2, and Table 5.2. As we see, the difference is non-negligible out to a significant distance. At 14 to 15 Å, the difference is still over 0.4 kcal/mol, a significant
difference if we are interested in precision of 0.5 kcal/mol. The size of these corrections has some significant implications for binding calculations and biomolecular
simulation in general, and these further implications are discussed later in this paper.
Because g(r) is not isotropic, even the exactly evaluated potential energies are not
a good estimate. The attractive energy between the ligand and protein adds an additional pull between the ligand and the protein that is not reflected in the Boltzmann
weighting of these states, so even the evaluated correction actually underestimates
the size of the correction.
5.3
5.3.1
Results
Precision of results
A vast amount of data was generated in these simulations: over 120 µs of molecular
dynamics data for the ligand decoupling simulations, over 30 µs for the ALTN complex
decoupling simulations, and over 40 µs for the CRYS complex decoupling systems.
150
This represents ∼600 ns per λ value per ligand for the ligand decoupling simulations,
with individual simulations averaging ∼20 ns in length and ∼150-200 ns per λ value
per ligand for the complexes, with typical lengths between 8 and 10 ns. Over 900,000
structures at intervals of 100 ps for the complexes and 200 ps for the ligands were
collected. It can be challenging to present this amount of data in the clearest possible
way.
The true test of convergence is if identical free energies (within the resolution of the
uncertainty) are obtained from independent initial samples. We test this assumption
with the CRYS and ALTN starting models sets. In Table 5.3, we present the free
energy of binding for experiment versus both sets. Preliminary results indicated that
it took at least 2.0 ns for free energies to plateau within each individual simulation, an
assumption we will discuss further later in this section, so the averages are computed
from 2.0 ns to the end of the simulations.
Before a further analysis of the data set, we examine the rationale of the inclusion
of the long range correction to the Lennard-Jones truncation. One could argue that
since the protein force field parameters were originally parameterized for short range
cutoffs, they should not be used in these simulations. However, since the non-bonded
parameters were never defined for ligand binding, or even protein simulations, it is
difficult to argue what may be the ideal assumption for such situation. But the fact
that small molecule solvation energies parameterized with short range cutoffs yield
free energies of solvation that are significantly closer to experiment when long range
Lennard-Jones corrections are included [SPSP03] suggest that this is a reasonable
correction to include. This long range correction is almost independent of the two
data sets, but does differ slightly, indicating some small differences in the structures
of the two sets that are still evident at longer distances.
Returning to Table 5.3, we see that although the free energies computed from the
two sets are highly correlated, they are not yet converged to within the uncertainty.
The uncertainty, as discussed earlier, is calculated by taking the standard deviation
over each λ interval and computing the standard deviation at each time point over
the twenty (or less than twenty, if not all sets had extended that far) parallel copies
to reach that time point. The errors were then propagated through the average over
151
time to obtain the final results. The two sets of results are actually highly correlated,
with slope 0.84 and R2 = 0.82, and an average absolute difference between the results
from the two sets, including long range correction of 0.84 kcal/mol. This is reasonable
precision, within the 1.0 kcal/mol difference that would be pharmaceutically useful.
This represents a significant achievement, especially given the enormous conformational space that is searched, and the uncertainty propagated by taking differences
between large solvent decoupling and ligand decoupling simulations.
However, this difference between computational sets is approximately an order of
magnitude greater than the calculated uncertainty, indicating that the two simulations
are not exploring all of the same conformations. Additionally, the largest absolute
error is for L2, the smallest ligand with the fewest rotatable bonds, which we would
expect would yield the best sampling and the smallest difference. This may indicate
that the smaller errors for the larger, more complicated ligands are due to insufficient
sampling.
5.3.2
Free energy as a function of time
In order to better understand these differences, we look at the sampling as a function
of time. We will examine two specific cases of the free energy of binding as a function
of time. First, we examine L14, which actually had three separate starting configurations, two from CRYS and one from ALTN. Secondly, we examine L2, which despite
being the smallest and apparently best sampled ligand, still suffers from disagreement
between the two initial structures, as seen in Table 5.3. We expect that the major
source of uncertainty in the ligand binding simulation comes from the complex decoupling simulations. The ligand simulations have run for two to three times as long,
and the ligand is less constrained than in the complex. We will therefore focus on
sources of uncertainty from the complex decoupling simulations.
The free energy of binding of L14 as a function of time for the two modeled initial
structures from CRYS and the structure from ALTN is presented in Fig. 5.3. The
ligand solvation energy for the entire simulation is computed. As can be seen, energies
rise sharply over the first 2 ns of the simulations, justifying the use of at least 2 ns
152
of equilibration time. However, even after that, the free energies appear to still be
rising. The energies of binding from different initial structures are distinguishable,
and appear to be slightly outside of the uncertainties. However, we note that the
ALTN structure always has a somewhat higher energy than the other two throughout
the course of the simulations.
In Fig. 5.4 are the total, charging, and Lennard-Jones solvation free energies of
binding for the CRYS and ALTN free energies. In Table 5.3, we see that the total
binding energy of the CRYS structure is changing over time, rising at least until the
4 ns mark, apparently reaching a steady state with a difference of about 2 kcal/mol.
The free energy of removing the uncharged Lennard-Jones framework is approximately the same for both structures over time, remaining roughly constant over the
entire simulation, or at least after the first 0.5 ns. However, the free energy of charging is very different, and is much more correlated to the overall binding energy, with
a much lower error than the total energy.
5.3.3
Sampling of degrees of freedom
It is important to examine the transitions of the degrees of freedom in order to understand the reasons for the two initial conditions differing by more than the computed
uncertainty. As mentioned before, the only degrees of freedom that are likely to suffer severe problems in sampling are the relative translation and orientation of the
ligand and protein, and the torsional degrees of freedom in the system. The degrees
of freedom of the water are expected to be on the order of hundreds of picoseconds
from previous simulations. Although some water molecules may be localized at various places in the interface, any such water trapping will almost certainly correspond
to slow motions of the protein and ligand, so we may safely omit the waters in our
analysis.
We first examine the relative motions of protein and ligand. We take as a test
case L9 from the CRYS set, one of the structures directly taken from the crystal
structure, and intermediate in the size among the ligands. All structural averages are
computed are over approximately 2000 structures, sampled over 100 ps intervals, from
153
the twenty trajectories run in parallel. Because of the heterogeneous nature of the
Folding@Home system, not all trajectories end at exactly the same place, resulting
in poorer statistics near the tail of graphed averages.
To look for equilibration in the ligand translational degrees of freedom we plot the
average and standard deviation of the distance between the attachment point and
the central pipecolyl nitrogen (Fig. 5.5). The spring attachment point was set from
nanosecond bound state simulations to the average location of the pipecolyl nitrogen,
and defined geometrically with respect two three protein atoms as described earlier,
and represents a useful surrogate for the average position.
For the fully coupled case, λCoulomb close to 0, we see the ligand remains very
close to the average position, with an average RMS distance of around 0.5 Å. The
spring is turned off in end state, so the localization is due entirely to protein-ligand
interactions. This tight binding is not necessarily a bad thing; if this is indeed the
dominant binding mode, the other position would contribute insignificantly to the
binding energy. But the fact that it does remain essentially localized on this time
scale measured raises questions that need to be addressed. In the decoupled state,
the RMS distance appears somewhat large for isotropic sampling. Examining the
distributions of distances indicates that there is still a significant amount of statistical
noise, indicating that the decoupled ligand, attached only by a spring to the rest of
the system, is moving quite slowly. However, the free energy contribution from the
interval beween this state and the next highest fluctuates quite rapidly, and the
torsional decorrelation show below indicate that this probably does not significantly
impact the solvation.
To measure the rotational sampling of the ligands we look at the angle formed
between the carbonyl carbon and pipecolyl nitrogen of the ligands and three spatially
separated anchor beta carbon atoms in the protein, Cys22, Thr27, and Ser77. We bin
the angles in 30 degree increments between 0 and 180, and re-weight the proportions
by the integral of 1/ sin(φ) over the interval to properly account for geometrical factors. In Fig. 5.6 and Fig. 5.7 we show how the distribution of these angles increasingly
randomizes as the coupling between ligand and protein decreases in L9.
For the fully coupled case, λCoulomb close to 0, we see very strongly only one
154
binding mode during the simulation. Reference to the structures indicates that L9 is
indeed not reorienting in the time scale. If this is indeed the dominant binding mode,
the other orientations would contribute insignificantly to the binding energy, and the
overall energy would still be close to the correct value for the model. However, the
lack of reorientation means we cannot rule out other binding modes.
Also importantly, in the decoupled state, as the ligand loses its interaction with
the rest of the system, it should also lose its orientation with respect to the protein.
In graphs of both φ1 and φ2 , we see that L9 indeed reaches this flat distribution
very effectively, indicating good rotational sampling in this state. Indeed, extensive
reorientation is evident in the last five or six λ states, indicating good orientational
sampling in the states where such reorientation is expected. The fact that reorientation is possible also provides evidence that this is a good free energy pathway, as
the relative orientation of the intermediate states interpolates well between the end
states.
As a side note of scientific interest, the change on binding modes with λ as the
Coulomb interaction is turned off may provide microscopic information about the importance of electrostatics in assisting binding orientation. The free energy of turning
off the charges for L9 is only 0.09 ± 0.04 kcal/mol, indicating that in this case the
electrostatics have little effect on the overall binding constant, but have a large effect
on fixing the final geometrical state.
Much more difficult to examine are the torsional degrees of freedom. We first
examine the torsional degrees of freedom for the protein collectively by examining
the Cα-RMSD from the crystal structure for the fully coupled and fully decoupled
ligand states, shown in Fig 5.8. We see that in the coupled state, the RMSD remains
approximately at 0.5 Å throughout the simulation, but clearly relaxes somewhat, by
perhaps 0.2 or 0.3 Å over the approximately 12 ns of simulation. Although the difference is slight, it suggests the system still requires significant time to truly approach
the equilibrium ensemble. On the brighter side, it indicates that the force field chosen
appears to be adequate for modeling the protein structure. In the uncoupled state,
this RMSD is somewhat greater, going over 1 Å, and the drift is more pronounced.
This additional drift is expected since the simulation was started from the coupled
155
bound state, but mostly plateaus after 2-3 ns. This greater RMSD and the larger
variance in RMSD may also reflect a slight lessening of the stability of FKBP in the
absence of ligand.
We look next at the torsional degrees of freedom of the ligand. The number of
rotatable bonds varies significantly between ligands, as shown in Table 5.1. Without
enumeration of the tenable ligand conformations it is difficult to precisely gauge
how completely the molecular dynamics simulations are sampling the available space.
Furthermore, with only twenty trajectories at 100 ps resolution at each value of λ, it
is difficult to quantitatively study the extent of the sampling.
We rely upon one useful principle; if the sampling is sufficient, continuing the
simulations should rarely if ever yield a new conformation. This is a necessary, though
not sufficient condition for complete sampling. To judge whether we have observed
a conformation before (either for the ligands or for the protein side chains in contact
with the ligands) we discretize the rotatable bond torsions into 60 degree bins (e.g.
between -30 and 30 degrees). All angles were measured using the GROMACS utility
g angle, the alpha carbon RMSD between the starting model and each snapshot was
measured using g rms. A conformation is then specified by the bin numbers of its
active torsions. Thus each ligand has a theoretical maximum 6N conformations where
N is the number of rotatable bonds from Table 5.1. Given the combinatorial nature
of this binning we expect the behavior of L2 with only 6 active torsions to differ
greatly from the behavior of FK506 with 26 active torsions.
Fig. 5.9 shows the fraction of conformations visited in each 100 ps time interval
that have not been observed previously in any trajectory (the entire set of trajectories,
over all λ values). L2 and L3 rarely find new conformations after 1 ns. This suggests
that any additional torsional transitions have significantly longer than the 10 ns time
scale reached in this simulation. In comparison, ligands L5, L8, L9, and L12 require
∼10 ns before the fraction of conformations sampled in the next 100 ps window drops
below 10%. Finally, trajectories for ligands L6, L14, and L20 are still visiting many
novel conformations after 10 ns. L20 does not revisit any conformational state until
almost 5 ns have passed.
It is surprising that L6 with 13 rotatable bonds seems to present a more difficult
156
sampling challenge than L12 with 18 rotatable bonds. It perhaps has to do with the
constrained nature of the large trisubstituted phenyl ring. Outside of L6 and L12,
the results are as expected given out combinatorial definition of a conformation and
the number of degrees of freedom per ligand
We present a similar analysis of the distribution of side chain χ1 values for eleven
side chains that line the binding pocket (Y26, F36, D37, F46, V55, I56, W59, Y82,
H87, I90, F99) in Fig. 5.10. Despite the large differences in size, the protein side
chains appear to equilibrate on a similar time scale for all ligands. All of the ensembles
discover less than 25% novel side chain combinations after ∼5 ns. These fractions
appear to be non-exponential, indicating a range of time scales, some of which are
longer than the 10 ns time scale of the simulation.
After having presented the sampling and precision of these simulations, we turn
to the comparison with experiment and accuracy. Differences from experiment will
depend not only upon the precision of the measurements, but on the accuracy of the
model in describing the ligand, protein, and solvent. Returning to Table 5.3, we see
that the energies of the CRYS set (which we expect to represent better initial binding
states) have moderate correlation, with a fit of slope 0.91 and offset 1.64 kcal/mol,
and R2 = 0.42. Excluding the two largest molecules, L14 and L20, which as we will
see, turn out to be clearly poorly sampled, increases the R2 to 0.57, without affecting
the slope. This is also shown in Fig. 5.11. The fact that the fit to experiment has
slope very close to 1.0 indicates that the relative free energies will be in general, fairly
good. Indeed, we can fit a line of slope 1.0 to the data with an overall root mean
square error of 1.79 kcal/mol overall, decreasing to 1.05 kcal/mol without L14 and
L20.
We hypothesized that this long range Lennard-Jones correction would improve
the correlation to experiment. Without this correction, the fit to experiment has
slope 0.71, with R2 decreased to 0.30, so this hypothesis appears to be correct. We
additionally hypothesized that the CRYS initial structures would yield better results
than the ALTN structures, which is confirmed; with the Lennard-Jones correction,
the fit of the ALTN set to experiment has slope 0.57 with R2 = 0.24; and without
the correction, this decreased to a slope of 0.40 with R2 = 0.15.
5.4
157
Discussion
sampling of the binding complexes and the agreement of binding with experimental
measurements. In this section, we discuss the utility and obtainability of high precision results, as well as what these results can tell us about the models and about
simulations using these models.
5.4.1
Sampling and precision
One extremely clear result is that for complex systems, the equilibrium times for the
systems will be much larger than is usually estimated. At least 2.0 ns was required
to re-equilibrate the system, and even this is likely to be a insufficient. Even for
the measurement of smaller relative free energy changes, the fluctuation time scales
of the protein and the ligand are the same as for larger free energy changes; some
of these structural fluctuations can be neglected without affecting the free energies
because of cancellation, but many will not. If a crystal structure is modeled and a
few hundred picoseconds or even a nanosecond of data is collected, it is likely to be
unrepresentative of the true equilibrium ensemble.
To obtain highly precise results, it is important to make the simulations as efficient
as possible. The pathway that is chosen is vital to ensure the best overlap possible
between intermediate states, as is the method to extract free energies from the neighboring state. The decoupling of electrostatic and Lennard-Jones interactions and the
use of a soft-core potential in atomic site introduction/removal as well as the use of
Bennett acceptance ratio results in magnitudes of variance of the intermediate free
energies that make possible the level of precision presented here.
It is also very desirable to run multiple copies of each simulation for a complex
system whose dynamics is unknown, starting from initial conditions that are as uncorrelated as possible. As demonstrated in this study, the variance between structurally
decorrelated independent states was much larger than the uncertainties computed
from independent runs started at the same configuration. This provides a clear indication than the sampling may be insufficient, and indicating the need for further
158
inspection of the microscopic details of the bound state.
In order to strictly guarantee that sufficient sampling has taken place, one must
monitor all the relevant degrees of freedom and ensure they are extensively sampling
their allowed phase space. For small molecules, this is relatively straightforward, as
detailed in our previous amino acid side chain study. However, for even small proteinligand systems like FKBP12, this becomes very difficult, as can be seen in this study,
and it is as yet unclear in the general case the ideal way to easily monitor this.
5.4.2
Accuracy
Even accounting for the lack of precision, how accurate do we expect the hydration
free energies computed using these parameter sets to be? Our previous studies have
demonstrated that the free energies of solvation of small molecules fragments of protein force fields have free energies that are considerably different, with an RMS of over
1 kcal/mol for the amino acid side chain analogs for AMBER. These were molecules
of no more than 20 atoms, whereas the ligands in this study are two to seven times
larger. The GAFF parameters are almost certainly less comparable to experiment, as
they were not rigorously optimized to individual molecules. In this light, there is no a
priori reason to expect the free energies correspond particularly well to experimental
results. Indeed, it is encouraging that the ligands are so close to the experimental
values.
It is also important to ask how accurate the experimental binding energies are,
and if the experimental data have sufficient precision to provide accurate comparison.
In general, binding free energies are very difficult to measure with any degree of
precision, and the values can vary between methods. We have tried to avoid some of
these problems by using data generated by a single group, and even from the same
set of experiments. The data presented here are Ki,app from an enzymatic activity
assay, and though the uncertainty likely represents the repeatability of measurements
by the same researchers, may not directly correlate to binding energies. There is not
a large amount of duplicated experimental data between research groups, but the few
comparisons are enlightening. Alternate measurements [LJ98] of ligands L5 and L8
159
yield 17 nmol versus 10 ± 1 nmol for L5 reported in [HLY+ 93], and 250 nmol versus
110 ± 20 nmol reported in [HLY+ 93] for L8. This represents free energy differences
of 0.3 and 0.5 kcal, respectively, indicating that the experimental uncertainty may be
larger than the number shown in Table 5.1 may show.
There are likewise few rigorous computational studies of the free energy [LJ98].
The one directly comparable study computed a relative free energy between L5 and L8
as 1.4 kcal/mol, similar to the two experimental free energies of 1.6 and 1.4 kcal/mol,
much better than the experimental free energies of 3.5 and 2.9 kcal/mol for CRYS and
ALTN respectively. It is important to note that the other four relative free energies
computed in the paper differ from experiment by 1.1, 1.9, 0.6, and 2.6 kcal/mol, so
given the problems of sampling free energies, is likely to result partly from cancellation
of error, as the conformational sampling is limited by a rigid protein side chain and
the computational resources applied.
There are some interesting qualitative differences between the charging and LennardJones solvation free energies in the current study. The Coulombic free energy differences are, on average, close to zero, with averages and standard deviations of 1.5 ±
1.6 kcal/mol for CRYS, and 0.8 kcal/mol ± 0.9 kcal/mol for ALTN, compared to
averages and standard deviations of -10.0 ± 1.2 kcal/mol for CRYS and -10.2 ± 1.1
kcal/mol for ALTN for the Lennard-Jones solvation energy. Approximately half of
the variance between ligands is in the charging step, despite this contributing little
to the overall binding. This can perhaps explained by reference to the rotational
distributions in Fig. 5.6 and Fig. 5.7. We see that decharging appears to relax the
angular distribution of the binding energy. This appears to indicate that the partial
charges are necessary to fix the ligand in the correct orientation, but not to provide
the overall attractive force to the complex.
Furthermore, the correlation between the Lennard-Jones solvation free energies of
the two sets is 0.86 with R2 = 70, versus 0.36 with R2 = 0.42 for the charging free
energy, and the average Lennard-Jones solvation energy differs by only 0.2 kcal/mol,
versus 0.8 kcal/mol for the charging energy. For almost all values of the LennardJones solvation process, the rotational sampling sampling appears to be much better
than in the charging phase, implying that current sampling is much closer to being
160
sufficient, and improving the sampling may only be necessary for the states closer to
full coupling.
Although the Lennard-Jones terms are quite small outside the cutoff ranges, they
are attractive everywhere, and thus can contribute significantly to the solvation free
energy [JMS84]. It has been previously shown that this correction can significantly
affect the results of a simulation, and improve the correlation to experiment [WFH02,
LW96, SPSP03, HSP+ 04]. One could argue that this correction should be neglected,
as the most of these potential sets were not originally parameterized using it. However,
in previous chapters we find that this long range correction greatly improves hydration
free energy results for all three parameter sets, and therefore is likely to improve the
binding free energy as well.
The difference in the ligand binding case is that the simple analytical correction,
which works very well with small molecule solvation, is not sufficient because of the
breakdown of the assumptions, as discussed in the Methods section. In Table 5.2 and
Fig. 5.2, we see that the difference between direct evaluation and analytic corrections
is significant as far out as 14 to 16 Å, a larger cutoff than is usually used in simulations.
A seemingly small difference of only 2 Å in the Lennard-Jones cutoff from 10 to 12 Å
could result in a difference in the calculated binding energy of a full kcal/mol. This
sort of lack of transferability makes it much more difficult to compare simulations
from different research groups. Moving to treatments of long ranged interactions that
require fewer user-adjusted arbitrary parameters will greatly improve the ability to
compare simulations from different researchers.
The results presented here certainly mean at least that longer Lennard-Jones
cutoffs than normal should be used in binding simulations, but may in fact mean it
may be necessary to implement something like Ewald summation for Lennard-Jones
interactions [EPB+ 95] for accurate solute-solvent interactions. But it is clear that in
the case of the FKBP binding simulations, the correction does significantly improves
the correlation of the two sample sets to experiment and too each other, even if it is
not ideal.
While discussing the shortcomings of the simulations presented in this study, it is
still important to note that despite all the failings in sampling, and the use of force
161
fields that have shown significant errors in free energy calculations, these simulations
are still predictive, with a RMS to fit for the relatively well sampled ligands of nearly
1 kcal/mol. We are also able to get quite high correlation in the free energies of
simulations run from very independent starting states.
5.4.3
Future Directions
This study leaves some important tasks on the road to quantitative ligand binding.
The most important improvement is greater sampling. It seems unlikely that using
less than an order of magnitude more sampling of simple molecular dynamics will
affect these results, based on the sampling results and divergence of the new sets.
Improved sampling methods will almost certainly be necessary, such as replica exchange and other multicanonical methods [SKO00, MSO01, WS00, CE00, RKdP03,
MBZT99, UJ03], which have proven useful for improved sampling in protein folding
simulations, for example [RC03, GO03, RP03].
One strong possibility for overcoming the difficulties in sampling is to use continuum solvation methods. Continuum (or implicit) models of solvation models are
a good compromise, greatly reducing computational time but still remaining an essentially physical method [GZL02, LZGF03]. Implicit solvent models have proven
to give both qualitative and quantitative results for protein kinetics and thermodynamics [ZSP01, SNPG02, PBC+ 03, SSR02]. Despite being a more drastic approximation, implicit solvent models may yield more accurate energies for solvent-solute
interactions than the more detailed explicit water simulations, because they are parameterized directly to solvent-solute properties, as opposed to neat liquid properties [FB04, GZL02].
Most importantly, implicit solvent methods can enable greatly improved sampling with respect to all-atom solvent models. Most obviously, the smaller number
of molecules reduces the computational load, extending the time scale of any simulation technique. However, the biggest advantage is the improved ability to make
large steps through conformational space. Explicit-water sampling is inherently limited when taking large-scale moves, because of the need to remove and re-equilibrate
162
water molecules when making large-scale rearrangements. Very recently, some rigorous absolute binding calculations have been performed with continuum solvation for
small model host-guest systems [KG02, PG02], and the resources of Folding@Home
can make possible the application of such techniques to larger, more practically useful
systems. But the granularity of water is certainly involved in binding in many circumstances, as demonstrated by the presence of crystallographic waters in or near the
binding site in many structures, meaning that continuum solvation may not always
be sufficient to compute binding affinities.
Another important direction of exploration is the use of improved pathways for
free energy computation. It is possible that the current method of double decoupling,
in which the solvent decoupling and complex decoupling simulations are used, is not
an ideal one. For example, it may be instead preferable to pull the ligand away
in small steps, and then perform an analytical correction when sufficiently far from
the protein. This could be done in a series of either equilibrium or non-equilibrium
simulations [HJ01b]. However, the limiting factor in all these simulations is most
likely to be the sampling of the individual states, as the time scales for equilibration
are likely to be similar in all methods.
It is likely that even with improved sampling and more efficient pathways, improved forcefields will be necessary for quantitatively accurate binding energies. We
cannot expect that calculations performed on complicated systems such as the computation of ligand-protein binding free energies to be any more accurate than the hydration free energies (or at least the relative hydration free energies) of their respective
small constituents. It therefore seems likely that more complicated, or at least more
precisely calibrated force fields may be necessary for transferability to multiple chemical environments. For example, polarizable force fields that have the ability to incorporate some many-body effects and to respond to different environments are being developed by many different groups [TSBD00, BKZ+ 99, KSBF04, PB04, PC03, RP02].
Based on the number of CPU years devoted to this project (over 200,000 CPU
days each for the ligand decoupling simulations, the CRYS set, and and the ALTN
set of complex decoupling simulations, for a total of nearly 700,000 CPU days), and
the lack of converged results obtained, one might reasonably ask if this vast resource
163
was optimally used to run simple molecular dynamics, as opposed to more complex
sampling methodologies or better, more time intensive force fields. We point out
that these simulations usually represented only between 2% to 5% of the total computational power of Folding@Home over a few months. The limiting factor in these
simulations was not the computational time, either in CPU time or wall clock time,
but the development and preparation time, making simple methodologies the clear
choice to start with. Additionally, the complexity of the problems was such that
we wanted to first examine the efficacy of the simplest methods of sampling, simple
molecular dynamics. The lessons learned in this exercise open the doors for more
sophisticated methods of sampling in the future.
This study reveals the power that distributed computing can provide. Thanks to
the rapid increase in available computer power in the last decade, we can afford to
attack much larger problems in computational chemistry, with many fewer restrictions on the degrees of freedom of the system. Although computations of this scale
are not currently available to most researchers, the degree of computations presented
here are will most likely become available in the near future. For example, Novartis already has an internal distributed computing network that has several thousand
nodes, similar to the resources used for this project [N5C]. Besides improvements in
individual processors, the total computational resources on the Internet is mostly untapped [SP00], and projects such as BOINC are developing tools to make distributed
computing projects possible for any researcher. Ligand binding computations of the
scale and nature as the one here will become commonplace in the near future.
5.5
Conclusions
This study represents perhaps the the largest concentration of computational resources brought to bear on the problem of ligand-protein binding. This has allowed
us to remove many restrictions and simplifications of the binding energy, and to
directly attack the calculation of absolute, as opposed to relative free energies.
It is clear from the relatively poor sampling of some of the ligand torsional degrees of freedom that truly precise ligand protein binding measurements will remain
164
exceedingly difficult to perform with simple molecular dynamics in the near term.
To sufficiently sample these systems it will require not only increased computational
power, but smarter sampling schemes, and to obtain quantitative binding affinities,
improvements in biomolecular parameterization will be needed as well.
In this study, we have obtained binding free energies with clear predictive value,
despite the size of the sampling problem to overcome and the lack of force fields
parameterized for condensed phase free energy calculations. We have also identified
important barriers to further development that would have been impossible without
the amount of statistical information that distributed computing has made possible
to collect. This represent a small but useful step towards both the scientific study
of macromolecular interaction and to the commercial development of useful drugs as
computational resources become more available.
L2
L5
L3
L6
L12
165
L8
L9
L14
L20
Figure 5.1: The structures of the nine ligands in this studies, and their identifiers.
L20 is FK506, the immunosuppresant that was the first discovered ligand of FKBP.
166
Table 5.1: Experimental binding data for FKBP12 Ligands, from [HLY+ 93]. Binding
constants are Ki,app in nmol, and free energies of binding are in kcal/mol, computed
from kT log(Ki,app ) Uncertainties are computed from 1/2(ln(x + y) − ln(x − y)), where
x = Ki,app and y is the uncertainty. Total number of atoms and the number of
rotatable bonds for each ligand is also listed.
Ligand
LG2
LG3
LG5
LG6
LG8
LG9
L12
L14
L20
Binding
∆FBinding
Atoms
Constant
2000 ± 300 -7.77 ± 0.09
46
600 ± 60
-8.42 ± 0.05
45
110 ± 20
-9.49 ± 0.11
58
12 ± 5
-10.80 ± 0.26
70
10 ± 1
-10.91 ± 0.06
68
7±2
-11.12 ± 0.17
74
30 ± 10
-10.25 ± 0.21
64
1 ± 0.4
-12.27 ± 0.25
85
0.3 ± 0.1 -12.98 ± 0.21
126
Rotatable
Bonds
6
7
10
13
11
11
18
21
26
Long Range Correction
12
Analytic
10
kcalmol
Ligand
8
Complex
6
Binding
4
2
10
12
14
16
18
Distance @ÞD
20
22
24
Figure 5.2: Long range correction, as computed analytically and by evaluation of
potential energy differences from structures. For comparison, the analytical correction
assuming g(r) = 1 and uniform water solvation is shown. The ligand correction is
assumed to be equal to the analytical correction outside 13 Å, which appears to be
valid. The correction to the binding energy is still significant (∼0.4 kcal/mol) out as
far as 14 to 15 Å.
167
Table 5.2: Long range correction, as computed analytically and by evaluation of
potential energy differences from structures. For comparison, the analytical correction
assuming g(r) = 1 and uniform water solvation is shown. The ligand correction is
assumed to be equal to the analytical correction outside 13 Å, which appears to be
valid. The correction to the binding energy is still significant (∼0.4 kcal/mol) out as
far as 14 to 15 Å.
Cutoff
8.0-9.0
8.5-9.5
9.0-10.0
9.5-10.5
10.0-11.0
11.0-12.0
12.0-13.0
14.0-15.0
16.0-17.0
18.0-19.0
20.0-21.0
22.0-23.0
24.0-25.0
Analytic
-9.62
-8.10
-6.89
-5.90
-5.10
-3.88
-3.02
-1.93
-1.31
-0.93
-0.68
-0.52
-0.40
Ligand
-9.06
-7.76
-6.68
-5.78
-5.03
-3.86
-3.02
-1.93
-1.31
-0.93
-0.68
-0.52
-0.40
Complex
-12.90
-10.94
-9.33
-7.99
-6.87
-5.14
-3.90
-2.35
-1.50
-1.01
-0.72
-0.53
-0.40
Difference
-3.84
-3.19
-2.65
-2.21
-1.84
-1.28
-0.88
-0.42
-0.19
-0.08
-0.03
-0.01
0.00
Binding Energy of L14
-2
0
2
4
6
8
10
-3
kcal/mol
-4
L14-ALTN
-5
L14-CRYSA
-6
L14-CRYSB
-7
-8
-9
time [ns]
Figure 5.3: Free energy of binding as a function of time for the two structures from
CRYS and one structure from ALTN. To aid the eye, the values at each nth value
are smoothed by weighting using the formula 0.25 ∆Gn−1 + 0.5∆Gn + 0.25∆Gn+1 .
168
Table 5.3: Experimental binding energy of the ligands compared with calculated
binding energy from both the CRYS and ALTN starting conformations. The value
of L14 for CRYS is the average of two both initial structures. LRC is the LennardJones long range correction, with direct evalation of the difference in selected frames.
Including the correction brings the binding energies significantly closer to experiment.
Uncertainties δ are calculated as indicated in the text, and obviously do not explain
the difference between the two sets of calculated free energies.
LG2
LG3
LG5
LG6
LG8
LG9
L12
L14
L20
Experiment
-7.77 ± 0.09
-8.42 ± 0.05
-9.49 ± 0.11
-10.80 ± 0.26
-10.91 ± 0.06
-11.12 ± 0.17
-10.25 ± 0.21
-12.27 ± 0.25
-12.98 ± 0.21
CRYS
LRC no LRC
-4.80
-2.96
-7.60
-5.77
-6.30
-4.17
-7.67
-5.29
-9.81
-7.34
-8.50
-5.92
-6.89
-4.44
-6.01
-3.10
-12.73
-8.89
ALTN
δ
LRC no LRC
δ
0.05 -6.61
-4.85
0.06
0.05 -6.79
-4.96
0.06
0.05 -7.87
-5.77
0.06
0.06 -6.46
-4.25
0.06
0.06 -10.79
-8.28
0.07
0.06 -8.62
-6.23
0.07
0.07 -7.45
-5.30
0.07
0.07 -6.38
-3.72
0.08
0.07 -12.59
-8.79
0.08
169
Binding Energy of LG2
-5
0
2
4
6
8
10
kcal/mol
-6
-7
LG2-ALTN
LG2-CRYS
-8
-9
-10
-11
time (ns)
Electrostatic Contribution to Binding of LG2
4
kcal/mol
3
2
LG2-ALTN
LG2-CRYS
1
0
0
2
4
6
8
10
-1
-2
time (ns)
Lennard-Jones Contribution to Binding of LG2
-7
kcal/mol
0
2
4
6
8
10
-8
LG2-ALTN
LG2-CRYS
-9
-10
time (ns)
Figure 5.4: Free energy of binding as a function of time for the structures of LG2
from CRYS and from ALTN.
170
Ligand Distance to Attachment
Distance [Å]
3
2
1
0
0
2
4
6
8
10
12
14
6
8
10
12
14
Distance [Å]
10
8
6
4
2
0
0
2
4
Time [ns]
Figure 5.5: Root mean square distance of central pipecolyl nitrogen to the spring
attachment point. The spring attachment point was set in long bound state simulations as the average location of this atom. In the fully coupled state, below, the
protein-ligand interactions are turned, but the spring is on.
Rotation and Coupling
"1 Distribution
1
0.75
0.5
0.25
0
0
0.25 0.55 0.85
Coulomb ! Value
0
0.2
0.4
0.6
0.7
0.8
0.9
1
Lennard-Jones ! Value
Figure 5.6: Distributions of φ1 , the angle defined by the shared carbonyl carbon
and pipecolyl carbon and Cβ of Cys22, as a function of the coupling parameter λ,
for LG9. Distributions are normalized for geometrical factors. At full coupling, the
ligand remains primarily in one binding mode. The fully uncoupled state has a flat
angular distribution, demonstrating good sampling of the uncoupled state.
171
Rotation and Coupling
"2 Distribution
1
0.75
0.5
0.25
0
0
0.25
0.55
0.85
0
0.2
Coulomb ! Value
0.4
0.6
0.7
0.8
0.9
1
Lennard-Jones ! Value
Figure 5.7: Distributions of φ2 , the angle defined by the shared carbonyl carbon
and pipecolyl carbon and Cβ of Thr22, as a function of the coupling parameter λ,
for LG9. Distributions are normalized for geometrical factors. At full coupling, the
ligand remains primarily in one binding mode. The fully uncoupled state has a flat
angular distribution, demonstrating good sampling of the uncoupled state.
Protein Relaxation
<RMSDc! > [Å]
1.5
1
0.5
0
2
4
6
8
10
12
14
6
8
10
12
14
<RMSDc! > [Å]
1.5
1
0.5
0
2
4
Time [ns]
Figure 5.8: C-α root-mean-square distance of protein from crystal structure as a
function of time of LG9.
172
Novel Torsion Combinations
1
LG2
0.5
LG3
LG5
0
0
5
10
1
LG6
0.5
LG8
LG9
0
0
5
10
1
L12
0.5
L14
L20
0
0
5
Time [ns]
10
Figure 5.9: Fraction of novel ligand discrete torsional rotamers as a function of time,
for all nine ligands. Rotamers are descretized in 60 degree bins. The larger ligands
are clearly covering a smaller fraction of side chain torsional space as a function of
time, and are still encountering significant number of new torsional conformations at
the end of the simulations.
173
Novel Torsion Combinations
1
LG2
0.5
LG3
LG5
0
0
5
10
1
LG6
0.5
LG8
LG9
0
0
5
10
1
0.5
L12
L14
L20
0
0
5
Time [ns]
10
Figure 5.10: Fraction of novel discrete side chain torsional rotamers for eleven side
chains that line the binding pocket, as a function of time, for all nine ligands. Side
chains are descretized in 60 degree bins. The larger ligands are clearly discovering
more side chains as a function of time.
174
Experimental vs. Computational
Binding Energies
-4
-14
-13
-12
-11
-10
-9
-8
-7
-5
-6
-6
Computed
-7
-8
-9
-10
-11
CRYS
Linear Fit
-12
-13
-14
Experiment
Figure 5.11: Correlation of experimental free energies to computed free energies for
the CRYS data set. Uncertainties of the error bars for experiment are shown, but the
not error bars for the computed values, as they are shown by the sampling results to
not be applicable. The linear fit has slope 0.91, with R2 = 0.57. The results fit to
experiment with slope 1.0 with 1.78 kcal/mol RMS error for all points, and slope 1.0
with 1.05 kcal/mol RMS error excluding the worst sampled ligands L14 and L20.
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