Thermodynamics of Brownian Ratchets

ThermodynamicsofBrownian
Ratchets
ReemMokhtar
Thermodynamics
1.  Firstlawofthermodynamics:Youcan’twin.
2.  Secondlawofthermodynamics:Youcan’t
evenbreakeven.
3.  Thirdlawofthermodynamics:Youcan’tstay
outofthegame.
Source:Voet,Donald;Voet,JudithG.(2010-11-22).Biochemistry,4thEdiKon(BIOCHEMISTRY
(VOET))(Page52).JohnWiley&Sons,Inc.
Thermodynamics(
2nd(law,(by(Sadi(Carnot(in(1824.(
(
•  Zeroth:(If(two(systems(are(in(thermal(equilibrium(with(
a(third(system,(they(must(be(in(thermal(equilibrium(
with(each(other.((
•  First:(Heat(and(work(are(forms(of(energy(transfer.(
•  Second:(The(entropy(of(any(isolated(system(not(in(
thermal(equilibrium(almost(always(increases.(
•  Third:(The(entropy(of(a(system(approaches(a(constant(
value(as(the(temperature(approaches(zero.(
(
hLp://en.wikipedia.org/wiki/Laws_of_thermodynamics(
Thermodynamics
•  First:thechangeintheinternalenergyofaclosedsystemisequal
totheamountofheatsuppliedtothesystem,minustheamountof
workdonebythesystemonitssurroundings.
àThelawofconservaKonofenergycanbestated:Theenergyof
anisolatedsystemisconstant.
•  Second:Theentropyofanyisolatedsystemnotinthermal
equilibriumalmostalwaysincreases.
•  Third:Theentropyofasystemapproachesaconstantvalueasthe
temperatureapproacheszero,whichimpliesthatitisimpossible
foranyproceduretobringasystemtotheabsolutezeroof
temperatureinafinitenumberofsteps
Source:Wikipedia,h\p://en.wikipedia.org/wiki/Laws_of_thermodynamics
What(is(a(Ratchet?(
A(device(that(allows(a(sha9(to(turn(only(one(way(
hLp://www.hpcgears.com/products/ratchets_pawls.htm(
Feynman,(R.(P.,(Leighton,(R.(B.,(&(Sands,(M.(L.((1965).(The$Feynman$Lectures$on$Physics:$
Mechanics,$radia7on,$and$heat((Vol.(1).(AddisonKWesley.(
The(Ratchet(As(An(Engine(
Ratchet,(pawl(and(spring.(
Feyman’sRatchet
Feynman’s(Brownian)ratchet:whatisitandwhatwasthepoint
ofexplainingit?
1. 
2. 
3. 
4. 
5. 
Rod
Paddlewheel:thefinsonthelee
Apawl:thatyellowthingthatforcestheratchetnottogobackintheotherdirecKon
Aratchet:thatsawtoothedgearontheright
Aspring/bearing(m)lowfricKon,toallowrotaKon.
(FromthefunexplanaKon@:
h\p://gravityandlevity.wordpress.com/2010/12/07/feynmans-ratchet-and-the-perpetualmoKon-gambling-scheme/)
Forward rotation
ε
Lθ
ε + Lθ
−1 −(ε +L θ )/ τ1
f
fB = Z e
ν f Bf
ν f Bf Lθ
ε
energy to lift the pawl
work done on load
energy to rotate wheel
by one tooth
Boltzmann factor for
work provided by vane
ratching rate with ν
attempt frequency
θ
T2
power delivered
energy provided to ratchet
T1
Backward rotation
ε
energy to lift the pawl
Lθ
work provided by load
ε + Lθ
energy given to vane
−1 −ε / τ 2
b
fB = Z e
b
ν fB
Boltzmann factor for
tooth slip
ε
θ
L
T2
slip rate with
attempt frequency ν
T1
Equilibrium and reversibility
ratching rate = slip rate
f Bb = f Bf
Reversible process by increasing the
load infinitesimally from equilibrium
Leq. This forces a rotation leading to
heating of reservoir 1 with
dq1=ε+Leq ? and cooling of
reservoir 2 as dq2=−ε:
ε + Leq θ τ1
dq1
=
=
− dq 2
ε
τ2
dq1 dq2
+
= dS1 + dS2 = 0
τ1 τ 2
isentropic process
 τ1 
Leq θ =  −1ε
 τ2 
τ2
τ1
Ratchet Brownian motor
ε + Lθ
Angular velocity of ratchet:
Without load:
Lθ
−


τ1=τ 2 =τ
τ
Ω(L )  → θν e  e −1


ε
−
τ
Equal temperatures:
Ω
ε
−
 − τ1
f
b
Ω = θν (fB − f B ) = θν  e
− e τ2


ε
ε
−
− 

=0
Ω  L→
θν  e τ1 − e τ 2 




τ1
rectification
L
τ2




Feyman’sRatchet
Feynman’s(Brownian)ratchet:whatisitandwhatwasthepoint
ofexplainingit?
Inonesentence:
ThefirstmoralistheSecondLawitself:itisimpossibleto
extractdirectedmo9onfromarandomprocess(asingleheat
reservoir).Anyonewhoclaimstheycandosoiseither
mistakenoracharlatan.
(FromthefunexplanaKon@:
h\p://gravityandlevity.wordpress.com/2010/12/07/feynmans-ratchet-and-the-perpetualmoKon-gambling-scheme/)
Inotherwords….
Well,theBrownianratchetwasa‘thought’experiment,alongwithMaxwell’sdemon.
Itwasusedhistoricallytoa\empttoexplainawaythe
1) 
potenKalexistenceofamolecularmachinethatcanperform….
2) 
perpetualmoKon,withoutviolaKngthelawsofthermodynamics,namelytheconservaKonof
energyand
ofmolecularmachines.
Inotherwords,afraudJasFeynmanfound….andifanyonetriestoconvinceyouotherwise…
justsay:
But,bydecreeofthermodynamics,Feynman’sratchetcannotworkasaheatengine.Itplainlyviolates
theSecondLaw,whichsaysthatusefulworkcanonlybeobtainedbytheflowofenergyfromhighto
lowtemperature.Thisdevicepurportstogetenergyfromasingletemperaturereservoir:thatofthe
airaroundit.
Wheredoesitgowrong?(hint:BrownianmoFonalsoaffectsthepawl)
(FromthefunexplanaKon@:
h\p://gravityandlevity.wordpress.com/2010/12/07/feynmans-ratchet-and-the-perpetualmoKon-gambling-scheme/)
Reformulated
Thegoalofthesethoughtexperimentswas…
Toprovethenonexistence/existenceofaperpetualmo9onmachineofthesecondkind
Or
“amachinewhichspontaneouslyconvertsthermalenergyintomechanicalwork.”
Whenthethermalenergyisequivalenttotheworkdone,thisdoesnotviolatethelawof
conserva9onofenergy.
Howeveritdoesviolatethemoresubtlesecondlawofthermodynamics(seealsoentropy).
ThesignatureofaperpetualmoKonmachineofthesecondkindisthatthereisonlyone
heatreservoirinvolved,whichisbeingspontaneouslycooledwithoutinvolvingatransfer
ofheattoacoolerreservoir.
Thisconversionofheatintousefulwork,withoutanysideeffect,isimpossible,accordingto
thesecondlawofthermodynamics.
FromWikipedia:
h\p://en.wikipedia.org/wiki/Perpetual_MoKon_Machine#ClassificaKon
Rephrased
1.  Fallacy:“Oh,youforgottotakeintoaccountfricKon,”they’llsay,
andthenthey’llgiveyouashortlectureontheFirstLawof
thermodynamics.“Energyisneithercreatednordestroyed,”
they’llsay.
2.  Thetruth,however,isthatmostperpetualmoKonmachinesthat
youarelikelytoencounterdonotviolateenergyconserva9on.
Rather,thetrickyandpersistentscienKfic“scams”violatethe
muchmorenebulousSecondLawofThermodynamics,whichsays
(inoneofitsformulaKons):
–  Itisimpossibleforadevicetoreceiveheatfromasinglereservoir
anddoanetamountofwork.
(FromthefunexplanaKon@:
h\p://gravityandlevity.wordpress.com/2010/12/07/feynmans-ratchet-and-the-perpetualmoKon-gambling-scheme/)
Whatareyoureallytryingtosay…
AlthoughatfirstsighttheBrownianratchetseemstoextractuseful
workfromBrownianmoKon,Feynmandemonstratedthatiftheen;re
deviceisatthesametemperature,theratchetwillnotrotate
conKnuouslyinonedirecKonbutwillmoverandomlybackandforth,
andthereforewillnotproduceanyusefulwork.
Asimplewaytovisualizehowthemachinemightfailistoremember
thatthepawlitselfwillundergoBrownianmo9on.Thepawl
thereforewillintermiFentlyfailbyallowingtheratchettoslip
backwardornotallowingittoslipforward.
è FeynmandemonstratedthatifT1=T2,then
pawlfailurerate=rateofforwardratcheKng,
è Sothatnonetmo9onresultsoverlongenoughperiods
FromWikipedia:
h\p://en.wikipedia.org/wiki/Brownian_ratchet
The Feynman Thermal Ratchet"
Pforward~exp(-Δε/kT1)"
Pbackward~exp(-Δε/kT2)"
"
works only if T1>T2 !!"
τrel ≈ Cl2/(4π2κ)"
motor protein conformational change: µs"
decay of temperature gradient over 10 nm: ns"
wrong "
model"
Brownian Ratchet"
(A.F. Huxley 57)"
Cargo"
Net "
transport"
Thermal motion"
Motor"
Track"
perpetuum mobile? Not if ATP is used to switch the off-rate."
Ifyoureallywantedthethingto
work…
• 
YoucouldmakeT2->T1,suchthatthetemperatureistakenfromthe
temperaturegradient….
• 
BUTYOUJUSTDIDTHEOPPOSITEOFWHATYOUSAIDYOUWANTEDTODO!!
–  YouareusingTWOthermalreservoirs!There’salsosomeheatwasted….
–  Waitaminute…thislookslikeaminiheatengineà“asystemthatperformsthe
conversionofheatorthermalenergytomechanicalworkbybringingaworking
substancefromahightemperaturestatetoalowertemperaturestate”
–  …..whichcomplieswiththesecondlawofthermodynamics…whichsayswhat?
•  Itisimpossibleforadevicetoreceiveheatfromasinglereservoiranddoanetamountof
work.
Sohowdoesthishelpus…
Thismeanswehavetobasicallyintroducethisdirectedenergy
somehow,inlayterms(becauseI’mnotaphysicist,Ilikelay
terms).
WhichisexactlywhatscienKstsdidrecentlyàalmost40years
aeerFeynman’sfamoustalk!
Maxwell’s demon
W. Smoluchowski (1941):
No automatic, permanently effective perpetual
motion machine van violate the second law by
taking advantage of statistical fluctuations
(Feynman: the demon is getting hot). Such device
might perhaps function if operated by intelligent
beings.
W.H. Zurek, Nature 341(1989)119:
The second law is safe from intelligent
beings as long as their abilities to process
information are subject to the same laws
as these of universal Turing machines.
Wait!Whatwasallthatbusinesswith
thedemon?
OK,thisisanotherthoughtexperiment(toomuchthinkinghere)createdbyphysicist
JamesMaxwell.
Whyonearthwouldhedothat?
Well,hewastryingtosaythatthesecondlawofthermodynamicswasn’tABSOLUTELY
certainàitonlyhasstaKsKcalcertainty.
So,again,ittriestoviolatethissecondlaw(thisisgetngpersonal).
FromWikipedia:h\p://en.wikipedia.org/wiki/Maxwell's_demon
AnotherfunexplanaKonath\p://alienryderflex.com/maxwells_demon.shtml
What?
Imaginethatyouhavetwocompartments:onewithhotandtheothercoldwater.
Ifbothcompartmentsweretobemixedtogether,you’dreachanequilibrium,right?
OK,butwhatifyouhadthisli\lepeskyhypotheKcalbeing(let’scallitademon)that
canopenandallowmoleculesthroughtoreversethismixing/equilibriumprocess,but
withoutexpendingenergyandseparaKnghotfromcoldbyonlyallowingmolecules
ho\er-than-averagethroughtoonesidethrougha‘door’.
Checkoutthenewexperimentthat‘convertsinformaKontopureenergy’
h\p://www.livescience.com/8944-maxwell-demon-converts-informaKon-energy.html
Right,theviolaKonagain.
Inotherwords,goingfromadisordered(higherentropy)toanorderedsystem(lower
entropy)withoutexpendingenergy.
Thisviolatesthesecondlawofthermodynamics…whichsayswhat?
à 
à 
à 
à 
entropyshouldnotdecreaseinanisolatedsystem
Itisimpossibleforadevicetoreceiveheatfromasinglereservoiranddoanetamountofwork.
itisimpossibletoextractdirectedmoFonfromarandomprocess(asingleheatreservoir).
ItisimpossibletoextractanamountofheatQHfromahotreservoiranduseitalltodoworkW.
SomeamountofheatQCmustbeexhaustedtoacoldreservoir.Thisprecludesaperfectheatengine.
Checkoutthenewexperimentthat‘convertsinformaKontopureenergy’
h\p://www.livescience.com/8944-maxwell-demon-converts-informaKon-energy.html
Angewandte
Molecular Devices
Chemie
there would be little viscosity to slow it down. These effects
always cancel each other out, and as long as a temperature can
be defined for an object it will undergo Brownian motion
appropriate to that temperature (which determines the
kinetic energy of the particle) and only the mean-free path
between collisions is affected by the concentration of
particles. In the absence of any other molecules, heat would
still be transmitted from the hot walls of the container to the
particle by electromagnetic radiation, with the random
emission and absorption of the photons producing the
Brownian motion. In fact, even temperature is not a
particularly effective modulator of Brownian motion since
the velocity of the particles depends on the square root of the
temperature. So to reduce random thermal fluctuations to
10 % of the amount present at room temperature, one would
have to drop the temperature from 300 K to 3 K.[12, 13] It seems
sensible, therefore, to try to utilize Brownian motion when
designing molecular machines rather than make structures
that have to fight against it. Indeed, the question of how to
(and whether it is even possible to) harness the inherent
random motion present at small length scales to generate
motion and do work at larger length scales has vexed
scientists for some considerable time.
Figure 1. The fundamental difference between a “switch” and a
“motor” at the molecular level. Both translational and rotary switches
influence a system as a function of the switch state. They switch
between two or more, often equilibrium, states. Motors, however,
influence a system as a function of the trajectory of their components
or a substrate. Motors function repetitively and progressively on a
system; the affect of a switch is undone by resetting the machine.
a) Rotary switch. b) Rotary motor. c) Translational switch. d) Translational motor or pump.
1.2.1. The Brownian Motion “Thought Machines”
The laws of thermodynamics govern how systems gain,
process, and release energy and are central to the use of
particle motion to do work at any scale. The zeroth law of
thermodynamics tells us about the nature of equilibrium, the
first law is concerned with the total energy of a system, while
the third law sets the limits against which absolute measurements can be made. Whenever energy changes hands,
Carter, N. J., & Cross,
R. A. (2005). Mechanics of the kinesin step.
however, (body to body or form to form) it is the second
Nature, 435(7040),law
308–312."
of thermodynamics which comes into play. This law
provides the link between the fundamentally reversible laws
of physics and the clearly irreversible nature of the universe in
ard motion of tiny particles within translucent pollen grains
[7]
Chemie
Figure 3. Szilard’s engine which utilizes a “pressure demon”.[17]
a) Initially a single Brownian particle occupies a cylinder with a piston
at either end. A frictionless partition is put in place to divide the
container into two compartments (a!b). b) The demon then detects
the particle and determines in which compartment it resides. c) Using
this information, the demon is able to move the opposite piston into
position without meeting any resistance from the particle. d) The
partition is removed, allowing the “gas” to expand against the piston,
doing work against any attached load (e). To replenish the energy used
by the piston and maintain a constant temperature, heat must flow
into the system. To complete the thermodynamic cycle and reset the
machine, the demon’s memory of where the particle was must be
erased (f!a). To fully justify the application of a thermodynamic
concept such as entropy to a single-particle model, a population of
Szilard devices is required. The average for the ensemble over each of
these devices can then be considered to represent the state of the
system, which is comparable to the time average of a single multiparticle system at equilibrium, in a fashion similar to the statistical
mechanics derivation of thermodynamic quantities.
Figure 4. a) Smoluchowski’s trapdoor: an “automatic” pressure
demon (the directionally discriminating behavior is carried out by a
wholly mechanical device, a trapdoor which is intended to open when
hit from one direction but not the other).[16] Like the pressure demon
shown in Figure 2 b, Smoluchowski’s trapdoor aims to transport
particles selectively from the left compartment to the right. However,
in the absence of a mechanism whereby the trapdoor can dissipate
energy it will be at thermal equilibrium with its surroundings. This
means it must spend much of its time open, unable to influence the
transport of particles. Rarely, it will be closed when a particle
approaches from the right and will open on collision with a particle
coming from the left, thus doing its job as intended. Such events are
balanced, however, by the door snapping shut on a particle from the
right, pushing it into the left chamber. Overall, the probability of a
particle moving from left to right is equal to that for moving right to
left and so the trapdoor cannot accomplish its intended function
adiabatically. b) Feynman’s ratchet and pawl.[18] It might appear that
Brownian motion of the gas molecules on the paddle wheel in the
right-hand compartment can do work by exploiting the asymmetry of
the teeth on the cog of the ratchet in the left-hand compartment.
While the spring holds the pawl between the teeth of the cog, it does
indeed turn selectively in the desired direction. However, when the
pawl is disengaged, the cog wheel need only randomly rotate a tiny
amount in the other direction to move back one tooth whereas it must
rotate randomly a long way to move to the next tooth forward. If the
paddle wheel and ratchet are at the same temperature (that is, T1 = T2)
6 T2 then the system will
these rates cancel out. However, if T1 ¼
directionally rotate, driven solely by the Brownian motion of the gas
molecules. Part (b) reprinted with permission from Ref. [18].
Carter, N. J., & Cross, R. A. (2005). Mechanics of the kinesin step.
Nature, 435(7040), 308–312."
irreversible second law of
designed to illustrate how the
thermodynamics arises from the fundamentally reversible
laws of motion.[18] Feynman!s device (Figure 4 b) consists of a
ton
cts
sing
nto
on,
used
w
he
f
h of
l
w of
sible
of a
d by
is a
tion
ions
n be
Figure 4. a) Smoluchowski’s trapdoor: an “automatic” pressure
demon (the directionally discriminating behavior is carried out by a
Angewandte
wholly mechanical device, a trapdoor which is intended to open when
Molecular Devices
Chemie
[16]
hit from one direction but not the other). Like the pressure demon
shown in Figure 2 b, Smoluchowski’s trapdoor aims to transport
particles selectively from the left compartment to the right. However,
in the absence of a mechanism whereby the trapdoor can dissipate
energy it will be at thermal equilibrium with its surroundings. This
means it must spend much of its time open, unable to influence the
transport of particles. Rarely, it will be closed when a particle
approaches from the right and will open on collision with a particle
coming from the left, thus doing its job as intended. Such events are
balanced, however, by the door snapping shut on a particle from the
right, pushing it into the left chamber. Overall, the probability of a
particle moving from left to right is equal to that for moving right to Figure 4. a) Smoluchowski’s trapdoor: an “automatic” pressure
demon (the directionally discriminating behavior is carried out by a
left and so the trapdoor cannot accomplish its intended function
wholly mechanical device, a trapdoor which is intended to open when
adiabatically. b) Feynman’s ratchet and pawl.[18] It might appear that hit from one direction but not the other).[16] Like the pressure demon
Brownian motion of the gas molecules on the paddle wheel in the
shown in Figure 2 b, Smoluchowski’s trapdoor aims to transport
[17]
Figure 3. Szilard’s
which by
utilizes
a “pressure
demon”.
right-hand compartment
canengine
do work
exploiting
the
asymmetry of particles selectively from the left compartment to the right. However,
in the absence of a mechanism whereby the trapdoor can dissipate
a) Initially a single Brownian particle occupies a cylinder with a piston
the teeth on the
cog of the ratchet in the left-hand compartment.
energy it will be at thermal equilibrium with its surroundings. This
at either end. A frictionless partition is put in place to divide the
means it must spend much of its time open, unable to influence the
While the spring
holds
the
pawl
between
the
teeth
of
the
cog,
it
does
container into two compartments (a!b). b) The demon then detects
the particle and
determines
in which
compartment
it resides.
c) Using
indeed turn selectively
in the
desired
direction.
However,
when
the transport of particles. Rarely, it will be closed when a particle
approaches from the right and will open on collision with a particle
this information, the demon is able to move the opposite piston into
pawl is disengaged,
the
cog
wheel
need
only
randomly
rotate
a
tiny
coming from the left, thus doing its job as intended. Such events are
position without meeting any resistance from the particle. d) The
balanced, however, by the door snapping shut on a particle from the
amount in thepartition
other is
direction
to
move
back
one
tooth
whereas
it
must
removed, allowing the “gas” to expand against the piston,
doing
work against
anymove
attached
To replenish
the energy If
used
rotate randomly
a long
way to
toload
the(e).
next
tooth forward.
the right, pushing it into the left chamber. Overall, the probability of a
particle moving from left to right is equal to that for moving right to
by the piston and maintain a constant temperature, heat must flow
) and so the trapdoor cannot accomplish its intended function
paddle wheel and
ratchet are at the same temperature (that is, T = T2left
into the system. To complete the thermodynamic cycle and reset the1
adiabatically. b) Feynman’s ratchet and pawl.[18] It might appear that
6 of T
the system
will
these rates cancel
out.
if T1 ¼
machine,
theHowever,
demon’s memory
where
the particle
was must
be
2 then
erased (f!a).
fully justify
of amotion
thermodynamic
directionally rotate,
drivenTo solely
by the
theapplication
Brownian
of the gas Brownian motion of the gas molecules on the paddle wheel in the
right-hand compartment can do work by exploiting the asymmetry of
concept such as entropy to a single-particle model, a population of
molecules. Part
(b) reprinted with permission from Ref. [18].
the teeth on the cog of the ratchet in the left-hand compartment.
Szilard devices is required. The average for the ensemble over each of
While the spring holds the pawl between the teeth of the cog, it does
these devices can then be considered to represent the state of the
indeed turn selectively in the desired direction. However, when the
system, which is comparable to the time average of a single multipawl is disengaged, the cog wheel need only randomly rotate a tiny
particle system at equilibrium, in a fashion similar to the statistical
difficult (and
temperature
gradients quantities.
cannot be maintainedamount in the other direction to move back one tooth whereas it must
mechanics
derivation of thermodynamic
rotate randomly a long way to move to the next tooth forward. If the
over molecular-scale distances, see Section 1.3), what thispaddle wheel and ratchet are at the same temperature (that is, T1 = T2)
6 T2 then the system will
these rates cancel out. However, if T1 ¼
hypotheticaldesigned
construct
provides
is irreversible
the first second
example
to illustrate
how the
law ofof a
directionally rotate, driven solely by the Brownian motion of the gas
Carter, N. J., & Cross, R. A. (2005). Mechanics of the kinesin step.
Nature, 435(7040), 308–312."
plausible mechanism
forarises
a molecular
motor—whereby
thermodynamics
from the fundamentally
reversible the
D. A. Leigh, F. Zerbetto, and E. R. Kay
he two sets
ched in the
d from left
may move
n also be
t shown in
ocity. Such
chets. This
evices such
of thermal
het characntial in one
, however,
nor in fact
a particle
iquid. This
en fluctuaf potential
continuum
ations and
n terms of
m has most
n Section 5
6, while in
which the
rnal direclatter. The
but rather
rranging a
nces which
ervals, can
lity of the
the ratchet
Figure 8. A temperature (or diffusion) ratchet.[39f ] a) The Brownian
particles start out in energy minima on the potential-energy surface
with the energy barriers @ kBT1. b) The temperature is increased so
that the height of the barriers is ! kBT2 and effectively free diffusion is
allowed to occur for a short time period (much less than required to
reach global equilibrium). c) The temperature is lowered to T1 once
more, and the asymmetry of the potential means that each particle is
statistically more likely to be captured by the adjacent well to the right
rather than the well to the left. d) Relaxation to the local energy
minimum (during which heat is emitted) leads to the average position
of the particles moving to the right. Repeating steps (b)–(d), progressively moves the Brownian particles further and further to the right.
Note the similarities between this mechanism and that of the on–off
ratchet shown in Figure 6.
Carter, N. J., & Cross, R. A. (2005). Mechanics of the kinesin step.
435(7040),
applyingNature,
temperature
variations 308–312."
to
therefore, it seems that
any process which involves an asymmetric potential-energy
surface could result in a ratchet effect.[47] In chemical terms,
which has particular relevance for explaining a
Fi
in
se
al
b
Figure 6. An example of a Figure
pulsating7.ratchet
mechanism—an
on–off
Another example of a pulsating
m
[39f ]
ratchet.
a) The Brownian particles start[42b]
out in energy minima on the
ing ratchet.
For details of its operation
(F
potential-energy surface with the energy barriers @ kBT. b) The potenth
tial is turned off so that free Brownian motion powered diffusion is
ch
allowed to occur for a short time period (much less than required to
reach global equilibrium). c)
On turning
back on minima
again,
7
series
of the
twopotential
different
an
the asymmetry of the potential means that the particles have a greater
whichwell
a Brownian
particle his
probability of being trappedalong
in the adjacent
to the right rather
k
than the adjacent well to the
Note this step involves
raising
byleft.
sequentially
raising
andthelowerin
d
example of a pulsating ratchet mechanism—an
energy ofon–off
the particles. d) Relaxation to the local energy minima
maxima. The particle starts in N
(during
whichon
heat
is emitted) leads to the average position of the
) The Brownian particles start out in energy
minima
the
se
(Figure
7 asteps
or (b)–(d)
c). Raising
that energ
to the right.
Repeating
progressively
b) Themoving
potenergy surface with the energy barriers @ kBT.particles
m
moves the Brownian particles
further
further to wells
the right.
those
inand
adjacent
provides the
off so that free Brownian motion powered diffusion is
N. J.,required
& Cross,toR. A. (2005).
Mechanics
of the
step. motio
change
position
by!kinesin
Brownian
ccur for a short time period (muchCarter,
less than
Angew. Chem. Int. Ed. 2007, 46, 72 – 191
2007 Wiley-VCH Verlag
Nature, back
435(7040),
308–312." 7 e). By simultaneously (or beforeh
equilibrium). c) On turning the potential
on again,
try of the potential means that the particles have a greater
heights of the energy barrier to
ondo,
and
wnian
3 and
witch
rmed
l and
asymmetric potential-energy surface (comprising a periodic
atchet
d in a
on of
mplest
tooth
than
action
tchet,
f the
tchet,
ly or
nisms
). As
tered
ntials
lassic
ing a
off
Figure 7. Another example of a pulsating ratchet mechanism—a flashing ratchet.[42b] For details of its operation, see the text.
Carter, N. J., & Cross, R. A. (2005). Mechanics of the kinesin step.
series of two different minima and two different
maxima) 435(7040), 308–312."
Nature,
along which a Brownian particle is directionally transported
by sequentially raising and lowering each set of minima and
maxima. The particle starts in a green or orange well
A
Molecular Devices
Figure 9. A rocking ratchet.[39f] a) The Brownian particles start out in
energy minima on the potential-energy surface with the energy barriers
@ kBT. b) A directional force is applied to the left. c) An equal and
opposite directional force is applied to the right. d) Removal of the
force and relaxation to the local energy minimum leads to the average
position of the particles moving to the right. Repeating steps (b)–(d)
progressively moves the Brownian particles further and further to the
right.
Figure
10. A type of information
ratchet mechanism
Carter, N. J., & Cross, R. A. (2005).
Mechanics
of the kinesin
step. for trans
Brownian particle along a potential-energy surface.
D
cases Nature,
(imagine 435(7040),
applying the 308–312."
electric field
arrows indicate the transfer of information that signals the p
periodic force
discussed above for longer in one direction than the other),
but is less so for stochastic driving forces. In general, these
[39h, 42b, 45]
the particle. If the signal is distance-dependent—say, energy
from an excited state which causes lowering of an energy bar
Angewandte
Chemie
rt out in
gy barriers
al and
of the
he average
s (b)–(d)
her to the
tric field
he other),
ral, these
ratchets.
y ratchet
surface—
d globally
Figure 10. A type of information ratchet mechanism for transport of a
Brownian particle along a potential-energy surface.[39h, 42b, 45] Dotted
arrows indicate the transfer of information that signals the position of
the particle. If the signal is distance-dependent—say, energy transfer
from an excited state which causes lowering of an energy barrier—
then the asymmetry in the particle’s position between two barriers
provides the “information” which transports the particle directionally
along the potential energy surface.
Carter, N. J., & Cross, R. A. (2005). Mechanics of the kinesin step.
Nature, 435(7040), 308–312."
It appears to us that information-ratchet mechanisms of
Kinesin
The two heads of the kinesindimer work in a coordinated manner to
move processivelyalong the microtubule. The catalytic core (blue) is
bound to a tubulinheterodimer (green, b-subunit; white, a-subunit)
along a microtubule protofilament (the cylindrical microtubule is
composed of 13 protofilament tracks). In solution, both kinesinheads
contain ADP in the active site (ADP release is rate-limiting in the
absence of microtubules). The chaotic motion of the kinesinmolecule
reflects Brownian motion. One kinesinhead makes an initial weak
binding interaction with the microtubule and then rearranges to
engage in a tight binding interaction. Only one k i n e s i nhead can readily
make this tight interaction with the microtubule, due to restrai nts
imposed by the coiled -coil and pre-stroke conformation of the neck
linker in the bound head. Microtubule binding releases ADP from the
attached head. ATP then rapidly enters the empty nucleotide bind ing
site, which triggers the neck linker to zipper onto the catalyti c core
(red to yellow transition). This action throws the detached head
forward and allows it to reach the next tubulinbinding site, thereby
creating a 2-head -bound intermediate in which the neck linkers in
the trailing and leading heads are pointing forward (post-stroke;
yellow) and backwards (pre-stroke; red) respectively. The trailing
head hydrolyzes the ATP (yellow flash of ADP -Pi), and reverts to a
weak microtubule binding state (indicated by the bouncing motion)
and releases phosphate (fading Pi). Phosphate release also causes the
unzippering of the neck linker (yellow to red transition). The exact
timing of the strong-to-weak microtubule binding transition and the
phosphate release step are not well-defined from current
experimental data. During the time when the trailing head execut es
the previously described actions, the leading head releases ADP, binds
ATP, and zippers its neck linker onto the catalytic core. This neck
linker motion throws the trailing head forward by 160 Å to the
vicinity of new tubulinbinding site. After a random diffusional search,
the new lead head docks tightly onto the binding site which comp letes
the 80 Å step of the motor. The movie shows two such 80 Å steps o f
the kinesinmotor. The surface features of the kinesin motor domains
and the microtubule protofilament were rendered from X -ray and
EM crystallographic structures by Graham Johnson (fiVth media:
www.fiVth.com) using the programs MolView, Strata Studio Pro and
Cinema 4D. PDB files used were human conventional kinesin
(prestroke, red: #1BG2) and rat conventional kinesin(poststroke,
yellow: #2KIN). In human conventional kinesin, the neck linker is
mobile and its located in the prestrokestate is estimated from cryo electron microscopy data. Transitions between states were
performed by performing computer-coordinated extrapolations
between the prestrokeand poststroke positions. The durations of the
events in this sequence were optimized for clarity and do not
necessarily reflect the precise timing of events in the ATPasecycle.
Three-bead assay "
with ncd"
Stepping and Stalling of a Single Kinesin Molecule "
300
6
240
200
5
224
150
4
3
100
2
50
~ 6 pN stall force" 1
0
-50
force [pN]
displacement [nm]
250
[nm]
7
208
192
176
time [s]
160
6 6.5 7 7.5 8 8.5 9
0
0
10
20
30
40
50
-1
60
~ 8 nm steps"
time [s]
Svoboda, Schmidt, Schnapp, Block, Nature (1993), 365: 721"
Randomness
parameter"
Δt=const."
clockwork "
r = 0!
r:= lim "
t -> ∞"
<x2(t)> - <x(t) > 2"
k" k" k" k" k"
1 -> 2 -> 3 -> 4 -> 5 -> 6"
d <x(t)>"
Δt exponentially
distributed"
Poisson process"
r = 1!
k
1 -> 2"
Myosin: averaged power strokes"
(Veigel et al. Nature 99, 398, 530)"
Myosin Power Stroke"
ADP"
+Pi"
Mechano-chemical cycle:"
attach"
M*ATP"
working stroke"
myosin"
M*ADP*Pi"
actin"
ADP"
detach"
M"
recovery stroke"
Pi"
Conformational Change of Single ncd Molecule"
MT displacement [nm]
10
2 µM ATP
5
0
-5
0
400
800
1200
1600
time [ms]
ADP*Pi"
ADP"
release"
DeCastro, Fondecave, Clarke, Schmidt, Stewart, Nature Cell Biology (2000), 2:724"