# Lecture : Inhomogeneous Solvation Theory with

## Transcription

Lecture : Inhomogeneous Solvation Theory with
```Statistical Thermodynamics
Lecture : Inhomogeneous Solvation Theory with
Application to WaterMap
Di Cui
[email protected]
Statistical Thermodynamics
Spring 2013
Overview of the Lecture
Part I: Basic theory of Inhomogeneous Solvation Theory (IST)
Solvation Energy
One Body Solvation Entropy
Two Body Solvation Entropy
Part II: Application of IST – WaterMap
Factor Xa – ligand binding
Part III: Application of IST – GIST
Solvation Free Energy of Simple Solutes
Statistical Thermodynamics
Spring 2013
Solvation Free Energy
ΔG
pure solvent
solution
1.  Solvation free energy is the amount of free energy associated with
dissolving a solute molecule into solvent.
2.  It corresponds to the free energy change for turning on the solute-solvent
interaction potential.
3.  It can be obtained from free energy perturbation (FEP), a number of
intermediate states needs to be introduced to connect the end-points.
4.  It can also be obtained using solution theory with simulation only at endpoints.
Statistical Thermodynamics
Spring 2013
Framework for Solvation Free Energy Calculation
Molecular dynamics simulation using
potential functions
Distribution functions of solutions
Δµ =
∫ d xρ (x)#\$(e(x) − Ts(x)) − (e
0
− Ts0 )%& Solution theory to calculate solvation
free energy
Statistical Thermodynamics
Spring 2013
Inhomogeneous Solvation Theory (IST)
Viewing the solution as inhomogeneous with
solute considered “fixed” at the origin,
generating an external field that creates solvent
density fluctuations around.
A formalism that isolates the effect of the solute
on the structure of the the solvent next to it
without having to deal with the large amounts of
unperturbed solvent. ΔGsolv = ΔEsolv − TΔSsolv
ΔGsolv is solvation free energy, ΔEsolv is solvation energy, ΔSsolv is
solvation entropy.
The next step is to write the solvation energy and entropy in terms of
correlation functions of end-points.
Lazaridis T (1998) J Phys Chem B 102:3531-3541
Statistical Thermodynamics
Spring 2013
Solvation Energy
ΔEsolv = Esw + Eww − E pure
Esw is the total solute-solvent interaction in solution, Eww is the total
solvent-solvent interaction in solution, Epure is the total interaction in
pure water.
1
Eww =
2
Esw =
∫∫
∫ ρ (x)U
sw
(x)dx
(2)
ρ (x, x1 )Uww(x, x1 )dxdx1
Usw and Uww are the solute-solvent and solvent-solvent interaction
potentials, x refers to the position and orientation of a solvent
molecule relative to the solute, ρ(x) is density distribution from solutesolvent, ρ(2)(x,x1) is density distribution from solute-solvent-solvent. Statistical Thermodynamics
Spring 2013
Solvation Entropy
ΔSsolv = Ssw + Sww − S pure = ΔSsw + ΔSww
Ssw accounts for the solute-water correlations, Sww for water-water
correlations in solution and Spure for water-water correlations in pure
water. Entropy density at location x, can be expressed based on GreenWallace expansion:
s(x) = −kB ∑
i=1
1
ρ (i) (x, x1,!, xi−1 )
( ) ∫ dx1 !dxi−1
ln δ g(i) (x, x1,!, x x−1 )
i!
ρ (x)
S=
∫ ρ (x)s(x)dx
S (1body) = ΔSsw = −kB ρ0 ∫ gsw (x) ln gsw (x)dx
orient
ΔSsw = ΔS trans
sw + ΔS sw
Wallace D (1987) J Chem Phys 87:2282-2284
Statistical Thermodynamics
Spring 2013
Entropy Terms Continue
ΔSsw = −kB
ΔSsw = −kB
ρ0
8π 2
ρ0
8π 2
∫∫ g
sw
(r).gsw (θ r )ln[gsw (r).gsw (θ r)]drdθ
∫∫ gsw (r).gsw (θ r )ln gsw (r)drdθ − kB
ρ0
ΔS = −kB 2
8π
trans
sw
∫∫ g
sw
ρ0
8π 2
∫∫ g
sw
(r).gsw (θ r )ln gsw (θ r)drdθ
(r).gsw (θ r )ln gsw (r)drdθ
= − kB ρ0 ∫ gsw (r) ln gsw (r)dr
ρ0
ΔS orient
=
−k
g (r).gsw (θ r )ln gsw (θ r)drdθ
sw
B
2 ∫∫ sw
8π
−kB
= ρ0 ∫ gsw (r)[ 2
8π
θ
∫g
= ρ0 ∫ gsw (r)S (r)dr
sw
(θ r) ln gsw (θ r)dθ ]dr
Sθ (r) =
−kB
8π 2
Gilson M (2012) J Chem Phys 137, 044101
∫g
sw
(θ r) ln gsw (θ r)dθ
Statistical Thermodynamics
Spring 2013
Application of IST - WaterMap
1.  WaterMap: mapping thermodynamic properties of water molecules
that solvate protein binding sites and using this data to understand
binding affinities.
2.  This tool was developed by Friesner’s group in Columbia University,
who is founder of a software company called Schrödinger.
3.  Water displacement is a key step that affects protein-ligand binding.
4.  Usually, there is a gain in entropy as releasing binding site water
molecules into bulk.
Ligand
+
Friesner A (2007) PNAS 104:808-813
Friesner A (2008) JACS 130:2817-2831
Protein
Statistical Thermodynamics
Spring 2013
Displacement of Unstable Water
Enantiomers bind to protein Bcl-xL, R-enantiomer binds much more strongly R
For R, Kd = 0.0008(µM), displacement of 3
unstable water makes the binding affinity larger
S
For S, Kd = 0.252 (µM), without displacement of
unstable water, binding affinity is smaller Statistical Thermodynamics
Spring 2013
Mapping Thermodynamic Properties of Water
1.  Binding Cavity: The starting points for each simulation was proteinligand complex. The ligand was then removed and the vacated
volumes that water can fill is binding cavity.
2.  Density Profile: Throughout the MD simulation, monitoring the water
entered binding cavity and calculated the water density profile.
3.  Clustering Algorithm: Identifying subvolumes of the binding cavities
with high densities, defined as hydration sites.
4.  IST: Excess energy and excess entropy for each hydration sites can
be calculated based on IST.
Statistical Thermodynamics
Spring 2013
Example: Solvent Density Distribution and Clustering
Binding cavity of streptavidin and
typical solvating water configuration Solvent density shown in green and the
clustering of the density in wireframe
Statistical Thermodynamics
Spring 2013
More on Five-membered Water Ring
1.  The formation of five-membered water ring is energetically favorable
but entropically unfavorable.
2.  Only fleetingly observed in bulk water, but always observed in the
streptavidin binding cavity due to the topographical characteristics.
3.  The ring is enclosed above and below by hydrophobic groups, the
only orientations for water can maintain maximal number of hydrogen
bonds are those consistent with ring formation.
4.  Due to the high order of the water in the ring, displacement of the
water may contribute as large as -7 kcal/mol to the free energy.
Statistical Thermodynamics
Spring 2013
WaterMap Example: Factor Xa Ligand Binding
1.  Factor Xa: an important drug target in the thrombosis pathway,
several inhibitors are currently on clinical trials.
2.  Factor Xa inhibitors generally bind in an L-shaped conformation
3.  Using IST to calculate binding free energy differences ΔΔG between
pairs of Factor Xa ligands, not to compute the absolute binding free
energy ΔG of a given ligand and receptor.
4.  Displacement of water molecules is not the only factor determining
absolute binding free energy. (loss of entropy of binding the ligand;
interaction between ligand and protein)
5.  For congeneric ligands that differ by only small chemical
modifications, these additional contributions are small, properties of
the excluded solvent are dominant.
Friesner A (2008) JACS 130:2817-2831
Statistical Thermodynamics
Spring 2013
Factor Xa Hydration Site
contribute both energetically and entropically
contribute energetically Friesner A (2008) JACS 130:2817-2831
contribute entropically
Statistical Thermodynamics
Spring 2013
Scoring Function
ΔGbind =
∑E
rwd
(1−
lig,hs
−T
∑S
rwd
rlig − rhs
Rco
(1−
)Θ(Ehs − Eco )Θ(Rco − rlig − rhs )
rlig − rhs
)Θ(S hs − Sco )Θ(Rco − rlig − rhs )
Rco
If a heavy atom of a ligand overlapped with a hydration site, it displayed
the water from that site.
The less energetically or entropically favorable the expelled water, the
more favorable its contributions to the binding free energy.
A hydration site would contribute to the binding free energy if Shs or Ehs
were beyond the fitted entropy and energy cutoff parameters, SCO and ECO.
A flat reward value was given for such hydration site as Srwd and Erwd.
A fit cutoff distance (Rco) was used to determine whether ligand atom
displaced water from a hydration site.
lig,hs
1.
2.
3.
4.
Friesner A (2008) JACS 130:2817-2831
Statistical Thermodynamics
Congeneric Ligand Pairs
XLC
Spring 2013
XLD
-CH3 group here
displaced water in
hydration 13 ΔΔGscore = -2.85 kcal/mol ΔΔGexp = -2.94 kcal/mol Statistical Thermodynamics
Spring 2013
Comparison with Experiment
R2 = 0.81
Computed relative free energy difference using the five-parameter form of
scoring function vs experimental results of the 31 congeneric inhibitor pairs.
Friesner A (2008) JACS 130:2817-2831
Statistical Thermodynamics
Spring 2013
Choice of Subvolumes in IST
1.  An important decision in IST is the choice of subvolumes over which to
perform the calculations.
2.  In protein binding sites, water molecules commonly cluster in distinct
locations and the concept of hydration site is useful.
3.  For small molecule solvation, two approaches have been used to
account for the volume around the solute: the region surrounding the
solute was split into subshells at different distances or split into cubic
voxels on a Cartesian grid (GIST).
Statistical Thermodynamics
Spring 2013
Application of IST – GIST
1.  Grid Inhomogeneous Solvation Theory (GIST): discretization of
inhomogeneous solvation theory on a 3D grid, the spatial integrals in the
IST expressions are replaced by discrete sums over the voxels.
2.  Specifically, a spatial region R is discretized into voxels indexed by k,
centered at locations rk and having volumes Vk, each voxel has the same
volume and the density is treated as uniform over each voxel k.
R
sw
ΔE =
∑
ΔE sw (rk )
k∈R
ΔS
R,trans
sw
=
∑
k∈R
trans
sw
ΔS
(rk )
R,orient
ΔSsw
=
∑
k∈R
Gilson M (2012) J Chem Phys 137, 044101
orient
ΔSsw
(rk )
Statistical Thermodynamics
Spring 2013
GIST Example: CB7
Cucurbit[7]uril
Contour of ΔEsw orange: more
favorable, blue: less favorable
Contour of -TΔSswtrans tan: more
favorable, red: less favorable
Contour of ΔEww cyan: more
favorable, orange: less favorable
Contour of -TΔSsworient yellow: more
favorable, violet: less favorable
Gilson M (2012) J Chem Phys 137, 044101
Statistical Thermodynamics
Spring 2013
GIST Example : Results
ΔEsw
ΔEww
-TΔSswtrans -TΔSsworient
ΔG
-45.9
19.8
3.6
6.9
-15.7
(kcal/mol) (kcal/mol) (kcal/mol) (kcal/mol) (kcal/mol)
Statistical Thermodynamics
Spring 2013
GIST Example: Comparison with FEP
1.  Comparison of the solvation free energy for several small molecule
solutes calculated from GIST and FEP.
2.  While a direct comparison with experiment is interesting, the results rely
on the force field and particularly the water model that is used.
3.  A more useful comparison is with an equivalent computational technique,
decoupling test of the method from test of the parameters.
Huggins D (2013) JPCB 117, 8232-8244
Statistical Thermodynamics
Spring 2013
Results of GIST Calculations on Simple Solutes
solute
ΔEsw
(kcal/mol)
ΔEww
-TΔSswtrans -TΔSsworient -TΔSwwtrans -TΔSwworient
ΔG
(kcal/mol) (kcal/mol) (kcal/mol) (kcal/mol) (kcal/mol) (kcal/mol)
acetamide
-29.4
14.8
3.7
2.9
-0.4
1.2
-7.2
benzene
-15.8
8.3
4.7
2.7
-0.5
1.9
1.3
isobutane
-9.8
4.8
4.6
2.4
-0.4
1.8
3.2
methane
-3.6
1.2
2.3
1.6
-0.4
0.9
2.1
methanol
-19.7
9.7
3.1
2.5
-0.4
1.3
-3.4
Nmethylaceta
mide
-29.1
14.2
4.5
3.2
-0.6
1.6
-6.2
Huggins D (2013) JPCB 117, 8232-8244
Statistical Thermodynamics
Spring 2013
Solvation Free Energy Comparison
solute
Experiment
(kcal/mol)
FEP calculation
(kcal/mol)
GIST calculation
(kcal/mol)
acetamide
-9.7
-8.3
-7.2
benzene
-0.8
0.3
1.3
isobutane
2.3
3.1
3.2
methane
2.0
2.5
2.1
methanol
-5.1
-4.6
-3.4
Nmethylaceta
mide
-10.1
-6.8
-6.2
Huggins D (2013) JPCB 117, 8232-8244
Statistical Thermodynamics
Spring 2013
Solvation Free Energy Correlations
Huggins D (2013) JPCB 117, 8232-8244
Statistical Thermodynamics
Spring 2013
Summary
1.  Inhomogeneous Solvation Theory (IST) provide a framework for relating
distribution functions of solutions to their thermodynamic properties.
2.  IST can be applied to calculate the solvation free energy of solute
without introduction of intermediate states.
3.  In IST, systems are spatially decomposed to consider the contribution of
specific regions to the total solvation free energy.
4.  Water molecules at a protein receptor site that are most advantageous to
replace by a ligand can be identified to aid in the process of structure
based drug design.
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