Thermodynamics of Brownian Ratchets - Duke …reif/courses/molcomplectures/Molecular... ·...

ThermodynamicsofBrownianRatchets

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,minustheamountofworkdonebythesystemonitssurroundings.àThelawofconservaKonofenergycanbestated:Theenergyofanisolatedsystemisconstant.

•  Second:Theentropyofanyisolatedsystemnotinthermalequilibriumalmostalwaysincreases.

•  Third:Theentropyofasystemapproachesaconstantvalueasthetemperatureapproacheszero,whichimpliesthatitisimpossibleforanyproceduretobringasystemtotheabsolutezerooftemperatureinafinitenumberofsteps

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(

Feynman,(R.(P.,(Leighton,(R.(B.,(&(Sands,(M.(L.((1965).(The$Feynman$Lectures$on$Physics:$Mechanics,$radia7on,$and$heat((Vol.(1).(AddisonKWesley.(

hLp://www.hpcgears.com/products/ratchets_pawls.htm(

The(Ratchet(As(An(Engine(Ratchet,(pawl(and(spring.(

Feyman’sRatchetFeynman’s(Brownian)ratchet:whatisitandwhatwasthepointofexplainingit?

(FromthefunexplanaKon@:h\p://gravityandlevity.wordpress.com/2010/12/07/feynmans-ratchet-and-the-perpetual-moKon-gambling-scheme/)

1.  Rod2.  Paddlewheel:thefinsonthelee3.  Apawl:thatyellowthingthatforcestheratchetnottogobackintheotherdirecKon4.  Aratchet:thatsawtoothedgearontheright5.  Aspring/bearing(m)lowfricKon,toallowrotaKon.

2

Forward rotation

energy to lift the pawl

( ) 1/1 τθε LfB eZf +−−= Boltzmann factor for

work provided by vane

θε L+

ε

energy to rotate wheelby one tooth

T2

θ

T1

fBfν ratching rate with ν

attempt frequency

energy provided to ratchet ε

θL work done on load

θν Lf fB power delivered

Backward rotation

energy to lift the pawl

2/1 τε−−= eZf bB Boltzmann factor for tooth slip

θε L+

ε

energy given to vane

T1

bBfν slip rate with

attempt frequency ν

θL work provided by load

T2

L

θ

ε

Equilibrium and reversibility

ratching rate = slip rate fB

bB ff =

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:

2

1

2

1

ττ

ε

θε=

+=

−eqL

dqdq

isentropic process

1 21 2

1 2

0dq dq

dS dSτ τ

+ = + =

1

2

1eqLτ

θ ετ

= −

1τ

2τ

Ratchet Brownian motor

Angular velocity of ratchet: ( )

−=−=Ω

−+

−21 τ

ε

τ

θε

θνθν eeffL

bB

fB

( )

− →Ω

−−== 121 τθ

τε

τττ θνL

eeL

−→Ω

−−= 210 τ

ε

τ

ε

θν eeLWithout load:

Equal temperatures:

L

Ωrectification

2τ

1τ

Escherichia coli ATP synthase H. Wang and G. Oster (Nature 396:279-282 1998)

Feyman’sRatchetFeynman’s(Brownian)ratchet:whatisitandwhatwasthepointofexplainingit?Inonesentence:ThefirstmoralistheSecondLawitself:itisimpossibletoextractdirectedmo9onfromarandomprocess(asingleheatreservoir).Anyonewhoclaimstheycandosoiseithermistakenoracharlatan.


Inotherwords….Well,theBrownianratchetwasa‘thought’experiment,alongwithMaxwell’sdemon.Itwasusedhistoricallytoa\empttoexplainawaythe1)  potenKalexistenceofamolecularmachinethatcanperform….2)  perpetualmoKon,withoutviolaKngthelawsofthermodynamics,namelytheconservaKonof

energyandofmolecularmachines.Inotherwords,afraudJasFeynmanfound….andifanyonetriestoconvinceyouotherwise…justsay:But,bydecreeofthermodynamics,Feynman’sratchetcannotworkasaheatengine.ItplainlyviolatestheSecondLaw,whichsaysthatusefulworkcanonlybeobtainedbytheflowofenergyfromhightolowtemperature.Thisdevicepurportstogetenergyfromasingletemperaturereservoir:thatoftheairaroundit.Wheredoesitgowrong?(hint:BrownianmoFonalsoaffectsthepawl)


ReformulatedThegoalofthesethoughtexperimentswas…Toprovethenonexistence/existenceofaperpetualmo9onmachineofthesecondkindOr“amachinewhichspontaneouslyconvertsthermalenergyintomechanicalwork.”Whenthethermalenergyisequivalenttotheworkdone,thisdoesnotviolatethelawofconserva9onofenergy.Howeveritdoesviolatethemoresubtlesecondlawofthermodynamics(seealsoentropy).ThesignatureofaperpetualmoKonmachineofthesecondkindisthatthereisonlyoneheatreservoirinvolved,whichisbeingspontaneouslycooledwithoutinvolvingatransferofheattoacoolerreservoir.Thisconversionofheatintousefulwork,withoutanysideeffect,isimpossible,accordingtothesecondlawofthermodynamics.

FromWikipedia:h\p://en.wikipedia.org/wiki/Perpetual_MoKon_Machine#ClassificaKon

Rephrased1.  Fallacy:“Oh,youforgottotakeintoaccountfricKon,”they’llsay,

andthenthey’llgiveyouashortlectureontheFirstLawofthermodynamics.“Energyisneithercreatednordestroyed,”they’llsay.

2.  Thetruth,however,isthatmostperpetualmoKonmachinesthatyouarelikelytoencounterdonotviolateenergyconserva9on.Rather,thetrickyandpersistentscienKfic“scams”violatethemuchmorenebulousSecondLawofThermodynamics,whichsays(inoneofitsformulaKons):

–  Itisimpossibleforadevicetoreceiveheatfromasinglereservoiranddoanetamountofwork.


Whatareyoureallytryingtosay…AlthoughatfirstsighttheBrownianratchetseemstoextractusefulworkfromBrownianmoKon,Feynmandemonstratedthatiftheen;redeviceisatthesametemperature,theratchetwillnotrotateconKnuouslyinonedirecKonbutwillmoverandomlybackandforth,andthereforewillnotproduceanyusefulwork.AsimplewaytovisualizehowthemachinemightfailistorememberthatthepawlitselfwillundergoBrownianmo9on.ThepawlthereforewillintermiFentlyfailbyallowingtheratchettoslipbackwardornotallowingittoslipforward.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 !!"

motor protein conformational change: µs"decay of temperature gradient over 10 nm: ns"

wrong "model"

τrel ≈ Cl2/(4π2κ)"

Brownian Ratchet" (A.F. Huxley 57)"

Cargo"Thermal motion"

Track"

Net "transport"

perpetuum mobile? Not if ATP is used to switch the off-rate."

Motor"

Ifyoureallywantedthethingtowork…

•  YoucouldmakeT2->T1,suchthatthetemperatureistakenfromthetemperaturegradient….

•  BUTYOUJUSTDIDTHEOPPOSITEOFWHATYOUSAIDYOUWANTEDTODO!!–  YouareusingTWOthermalreservoirs!There’salsosomeheatwasted….–  Waitaminute…thislookslikeaminiheatengineà“asystemthatperformsthe

conversionofheatorthermalenergytomechanicalworkbybringingaworkingsubstancefromahightemperaturestatetoalowertemperaturestate”

–  …..whichcomplieswiththesecondlawofthermodynamics…whichsayswhat?•  Itisimpossibleforadevicetoreceiveheatfromasinglereservoiranddoanetamountof

work.

Sohowdoesthishelpus…Thismeanswehavetobasicallyintroducethisdirectedenergysomehow,inlayterms(becauseI’mnotaphysicist,Ilikelayterms).WhichisexactlywhatscienKstsdidrecentlyàalmost40yearsaeerFeynman’sfamoustalk!

5

Maxwell’s demon

W. Smoluchowski (1941):No automatic, permanently effective perpetual motion machine van violate the second law bytaking advantage of statistical fluctuations (Feynman: the demon is getting hot). Such device might perhaps function if operated by intelligentbeings.

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.

Quantum demon? (ask Milena Grifoni)

Fluctuations of µm-sized trapped colloidal particles

physics.okstate.edu/ackerson/vackerson/

G.M. Wang et al., Phys. Rev. Lett. 89(2002)050601

optvoptF

( )0

1 t

t opt optv F s dsτ

Σ = ⋅∫

Entropy production

Noise ratchet

Wait!Whatwasallthatbusinesswiththedemon?

OK,thisisanotherthoughtexperiment(toomuchthinkinghere)createdbyphysicistJamesMaxwell.Whyonearthwouldhedothat?Well,hewastryingtosaythatthesecondlawofthermodynamicswasn’tABSOLUTELYcertainàitonlyhasstaKsKcalcertainty.So,again,ittriestoviolatethissecondlaw(thisisgetngpersonal).

FromWikipedia:h\p://en.wikipedia.org/wiki/Maxwell's_demonAnotherfunexplanaKonath\p://alienryderflex.com/maxwells_demon.shtml

What?Imaginethatyouhavetwocompartments:onewithhotandtheothercoldwater.Ifbothcompartmentsweretobemixedtogether,you’dreachanequilibrium,right?OK,butwhatifyouhadthisli\lepeskyhypotheKcalbeing(let’scallitademon)thatcanopenandallowmoleculesthroughtoreversethismixing/equilibriumprocess,butwithoutexpendingenergyandseparaKnghotfromcoldbyonlyallowingmoleculesho\er-than-averagethroughtoonesidethrougha‘door’.

Checkoutthenewexperimentthat‘convertsinformaKontopureenergy’h\p://www.livescience.com/8944-maxwell-demon-converts-informaKon-energy.html

Right,theviolaKonagain.Inotherwords,goingfromadisordered(higherentropy)toanorderedsystem(lowerentropy)withoutexpendingenergy.Thisviolatesthesecondlawofthermodynamics…whichsayswhat?

à  entropyshouldnotdecreaseinanisolatedsystemà  Itisimpossibleforadevicetoreceiveheatfromasinglereservoiranddoanetamountofwork.à  itisimpossibletoextractdirectedmoFonfromarandomprocess(asingleheatreservoir).à  ItisimpossibletoextractanamountofheatQHfromahotreservoiranduseitalltodoworkW.

SomeamountofheatQCmustbeexhaustedtoacoldreservoir.Thisprecludesaperfectheatengine.

Checkoutthenewexperimentthat‘convertsinformaKontopureenergy’h\p://www.livescience.com/8944-maxwell-demon-converts-informaKon-energy.html

ard motion of tiny particles within translucent pollen grainssuspended in water.[7] An explanation of the phenomenon—now known as Brownian motion or movement—was providedby Einstein in one[8] of his three celebrated papers of 1905 andwas proven experimentally[9] by Perrin over the nextdecade.[10] Scientists have been fascinated by the implicationsof the stochastic nature of molecular-level motion ever since.The random thermal fluctuations experienced by moleculesdominate mechanical behavior in the molecular world. Eventhe most efficient nanoscale machines are swamped by itseffect. A typical motor protein consumes ATP fuel at a rate of100–1000 molecules every second, which corresponds to amaximum possible power output in the region of 10!16 to10!17 W per molecule.[11] When compared with the randomenvironmental buffeting of approximately 10!8 W experi-enced by molecules in solution at room temperature, it seemsremarkable that any form of controlled motion is possible.[12]

When designing molecular machines it is important toremember that the presence of Brownian motion is aconsequence of scale, not of the nature of the surroundings.It cannot be avoided by putting a molecular-level structure ina near-vacuum, for example. Although there would be fewrandom collisions to set such a Brownian particle in motion,

there would be little viscosity to slow it down. These effectsalways cancel each other out, and as long as a temperature canbe defined for an object it will undergo Brownian motionappropriate to that temperature (which determines thekinetic energy of the particle) and only the mean-free pathbetween collisions is affected by the concentration ofparticles. In the absence of any other molecules, heat wouldstill be transmitted from the hot walls of the container to theparticle by electromagnetic radiation, with the randomemission and absorption of the photons producing theBrownian motion. In fact, even temperature is not aparticularly effective modulator of Brownian motion sincethe velocity of the particles depends on the square root of thetemperature. So to reduce random thermal fluctuations to10% of the amount present at room temperature, one wouldhave to drop the temperature from 300 K to 3 K.[12,13] It seemssensible, therefore, to try to utilize Brownian motion whendesigning molecular machines rather than make structuresthat have to fight against it. Indeed, the question of how to(and whether it is even possible to) harness the inherentrandom motion present at small length scales to generatemotion and do work at larger length scales has vexedscientists for some considerable time.

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 ofparticle motion to do work at any scale. The zeroth law ofthermodynamics tells us about the nature of equilibrium, thefirst law is concerned with the total energy of a system, whilethe third law sets the limits against which absolute measure-ments can be made. Whenever energy changes hands,however, (body to body or form to form) it is the secondlaw of thermodynamics which comes into play. This lawprovides the link between the fundamentally reversible lawsof physics and the clearly irreversible nature of the universe inwhich we exist. Furthermore, it is the second law ofthermodynamics, with its often counterintuitive consequen-ces, that governs many of the important design aspects of howto harness Brownian motion to make molecular-levelmachines. Indeed, the design of tiny machines capable ofdoing work was the subject of several celebrated historical“thought-machines” intended to test the very nature of thesecond law of thermodynamics.[14–18]

1.2.1.1.Maxwell’s Demon

Scottish physicist James Clerk Maxwell played a majorrole (along with Ludwig Boltzmann) in developing the kinetictheory of gases, which established the relationship betweenheat and particle motion and gave birth to the concept ofstatistical mechanics. In doing so, Maxwell realized theprofundity of the statistical nature of the second law ofthermodynamics which had recently been formulated[19] byRudolf Clausius andWilliam Thomson (later Lord Kelvin). Inan attempt to illustrate this feature, Maxwell devised thethought experiment which has come to be known asMaxwell!s demon.[14,15,20]

Figure 1. The fundamental difference between a “switch” and a“motor” at the molecular level. Both translational and rotary switchesinfluence a system as a function of the switch state. They switchbetween two or more, often equilibrium, states. Motors, however,influence a system as a function of the trajectory of their componentsor a substrate. Motors function repetitively and progressively on asystem; the affect of a switch is undone by resetting the machine.a) Rotary switch. b) Rotary motor. c) Translational switch. d) Transla-tional motor or pump.

Molecular DevicesAngewandte

Chemie

75Angew. Chem. Int. Ed. 2007, 46, 72 – 191 ! 2007 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.angewandte.org

Carter, N. J., & Cross, R. A. (2005). Mechanics of the kinesin step. Nature, 435(7040), 308–312."

designed to illustrate how the irreversible second law ofthermodynamics arises from the fundamentally reversiblelaws of motion.[18] Feynman!s device (Figure 4b) consists of aminiature axle with vanes attached to one end, surrounded bya gas at temperature T1. At the other end of the axle is aratchet and pawl system, held at temperature T2. The questionposed by the system is whether the random oscillationsproduced by gas molecules bombarding the vanes can berectified by the ratchet and pawl so as to get net motion in onedirection. Exactly analogous to Smoluchowski!s trapdoor,Feynman showed that if T1=T2 then the ratchet and pawlcannot extract energy from the thermal bath to do work.Feynman!s major contribution from the perspective ofmolecular machines, however, was to take the analysis onestage further: if such a system cannot use thermal energy to dowork, what is required for it to do so? Feynman showed thatwhen the ratchet and pawl are cooler than the vanes (that is,T1>T2) the system does indeed rectify thermal motions andcan do work (Feynman suggested lifting a hypothetical fleaattached by a thread to the ratchet).[25] Of course, the machinenow does not threaten the second law of thermodynamics asdissipation of heat into the gas reservoir of the ratchet andpawl occurs, so the temperature difference must be main-tained by some external means. Although insulating amolecular-sized system from the outside environment is

difficult (and temperature gradients cannot be maintainedover molecular-scale distances, see Section 1.3), what thishypothetical construct provides is the first example of aplausible mechanism for a molecular motor—whereby therandom thermal fluctuations characteristic of this size regimeare not fought against but instead are harnessed and rectified.The key ingredient is the external addition of energy to thesystem, not to generate motion but rather to continually orcyclically drive the system away from equilibrium, therebymaintaining a thermally activated relaxation process thatdirectionally biases Brownian motion towards equilibrium.[26]

This profound idea is the key to the design of molecular-levelsystems that work through controling Brownian motion and isexpanded upon in Section 1.4.

1.2.2.Machines That Operate at Low Reynolds Number

Whilst rectifying Brownian motionmay provide the key topowering molecular-level machines, it tells us nothing abouthow that power can be used to perform tasks at the nanoscaleand what tiny mechanical machines can and cannot be

Figure 3. Szilard’s engine which utilizes a “pressure demon”.[17]

a) Initially a single Brownian particle occupies a cylinder with a pistonat either end. A frictionless partition is put in place to divide thecontainer into two compartments (a!b). b) The demon then detectsthe particle and determines in which compartment it resides. c) Usingthis information, the demon is able to move the opposite piston intoposition without meeting any resistance from the particle. d) Thepartition is removed, allowing the “gas” to expand against the piston,doing work against any attached load (e). To replenish the energy usedby the piston and maintain a constant temperature, heat must flowinto the system. To complete the thermodynamic cycle and reset themachine, the demon’s memory of where the particle was must beerased (f!a). To fully justify the application of a thermodynamicconcept such as entropy to a single-particle model, a population ofSzilard devices is required. The average for the ensemble over each ofthese devices can then be considered to represent the state of thesystem, which is comparable to the time average of a single multi-particle system at equilibrium, in a fashion similar to the statisticalmechanics derivation of thermodynamic quantities.

Figure 4. a) Smoluchowski’s trapdoor: an “automatic” pressuredemon (the directionally discriminating behavior is carried out by awholly mechanical device, a trapdoor which is intended to open whenhit from one direction but not the other).[16] Like the pressure demonshown in Figure 2b, Smoluchowski’s trapdoor aims to transportparticles selectively from the left compartment to the right. However,in the absence of a mechanism whereby the trapdoor can dissipateenergy it will be at thermal equilibrium with its surroundings. Thismeans it must spend much of its time open, unable to influence thetransport of particles. Rarely, it will be closed when a particleapproaches from the right and will open on collision with a particlecoming from the left, thus doing its job as intended. Such events arebalanced, however, by the door snapping shut on a particle from theright, pushing it into the left chamber. Overall, the probability of aparticle moving from left to right is equal to that for moving right toleft and so the trapdoor cannot accomplish its intended functionadiabatically. b) Feynman’s ratchet and pawl.[18] It might appear thatBrownian motion of the gas molecules on the paddle wheel in theright-hand compartment can do work by exploiting the asymmetry ofthe 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 doesindeed turn selectively in the desired direction. However, when thepawl is disengaged, the cog wheel need only randomly rotate a tinyamount in the other direction to move back one tooth whereas it mustrotate randomly a long way to move to the next tooth forward. If thepaddle wheel and ratchet are at the same temperature (that is, T1=T2)these rates cancel out. However, if T1¼6 T2 then the system willdirectionally rotate, driven solely by the Brownian motion of the gasmolecules. Part (b) reprinted with permission from Ref. [18].


Chemie












Chemie











Chemie



potential does not have to be regular. As long as the two setsof energy minima and maxima are repetitively switched in thesame order, the particle will tend to be transported from leftto right in Figure 7, even though occasionally it may moveover a barrier in the wrong direction.

The basic pulsating ratchet requirements can also berealized in another way. A potential such as that shown inFigure 7a can be given a directional drift velocity. Suchsystems are often termed “traveling potential” ratchets. Thisprinciple is essentially the same as macroscopic devices suchas Archimedes! screw, yet in the presence of thermalfluctuations these systems exhibit Brownian ratchet charac-teristics and are closely related to tilting the potential in onedirection (as discussed in Section 1.4.2.2). Clearly, however,an asymmetric potential is not absolutely required, nor in factare thermal fluctuations; imagine, for example, a particle“surfing” on a traveling wave on the surface of a liquid. Thiscategory of mechanism is at the boundary between fluctua-tion-driven transport and transport as a result of potentialgradients, with the precise location on this continuumdependent on the importance of thermal fluctuations andspatial asymmetry under the conditions chosen. In terms ofchemical systems, the traveling-potential mechanism has mostrelevance for the field-driven processes discussed in Section 5and the self-propulsion mechanisms of Section 6, while inSection 7 we shall discuss some situations in which thebalance between random fluctuations and external direc-tional driving force is shifted strongly towards the latter. Thetraveling motion does not have to be continuous, but rathermay take place in discrete steps. Furthermore, arranging acontinuously traveling potential to “jump” distances whichare multiples of its period, at random or regular intervals, canbe used to escape from the inherent directionality of thetraveling-wave scheme and it has been shown that the ratchetdynamics are essentially unaffected. In the limiting situation,this can be reduced to dichotomous switching between twospatially shifted potentials which are otherwise identical,which is very similar to a rocking ratchet (see Section 1.4.2.2)and is also relevant to the unidirectional motors discussed inSection 2.2.

1.4.2.2. Tilting Ratchets[39f ]

In this category, the underlying intrinsic potential remainsunchanged and the detailed balance is broken by applicationof an unbiased driving force to the Brownian particle. Perhapsthe most apparent unbiased driving force is heat, and ratchetmechanisms based on periodic or stochastic temperaturevariations are generally termed “temperature” or “diffusion”ratchets. In its simplest form (Figure 8) this mechanism is verysimilar to the on–off ratchet. Initially the thermal energy islow so that the particles cannot readily cross the energybarriers. A sudden increase in temperature can be applied sothat kBT is much greater that the potential amplitude causingthe particles to diffuse as if over a virtually flat potential-energy surface (Figure 8b). Returning to the original, lower,temperature (Figure 8c) is equivalent to turning the potentialback on in Figure 6c and more particles will have moved tothe next well to the right than to the left. Under this scheme,

therefore, it seems that applying temperature variations toany process which involves an asymmetric potential-energysurface could result in a ratchet effect.[47] In chemical terms,this means that the rotation of a chiral group around acovalent bond can, at least in principle, be made unidirec-tional in such a manner. This concept is explored further inrelation to a molecular ratchet system in Section 2.1.2.

An unbiased driving force can also be achieved byapplying a directional force in a periodic manner so that,over time, the bias averages to zero, thus generating a“rocking” ratchet. The simplest form of this is shown inFigure 9. Periodic application (to the left and then right) of adriving force that allows the particle to surmount the barriers(for example, by applying an external field if the particle ischarged) results in transport. Motion over the steep barrier isagain most likely as it involves the shortest distance. Such amechanism is physically equivalent to tilting the ratchetpotential in one direction then the other. Of course, if thedriving force is strong enough and is constantly applied in onedirection or the other, thermal fluctuations are not necessary,which would then correspond to a power stroke.

Analogues of a rocking ratchet in which the applieddriving force is a form of stochastic noise are known as“fluctuating force” ratchets (in certain cases, also “correla-tion” ratchets). Finally, a tilting ratchet can be achieved in asymmetric potential if the perturbation itself results in spatialasymmetry (becoming very similar to the travelling-potentialmodels in Section 1.4.2.1). This may be rather clear for the

Figure 8. A temperature (or diffusion) ratchet.[39f ] a) The Brownianparticles start out in energy minima on the potential-energy surfacewith the energy barriers @ kBT1. b) The temperature is increased sothat the height of the barriers is ! kBT2 and effectively free diffusion isallowed to occur for a short time period (much less than required toreach global equilibrium). c) The temperature is lowered to T1 oncemore, and the asymmetry of the potential means that each particle isstatistically more likely to be captured by the adjacent well to the rightrather than the well to the left. d) Relaxation to the local energyminimum (during which heat is emitted) leads to the average positionof the particles moving to the right. Repeating steps (b)–(d), progres-sively moves the Brownian particles further and further to the right.Note the similarities between this mechanism and that of the on–offratchet shown in Figure 6.

D. A. Leigh, F. Zerbetto, and E. R. KayReviews

82 www.angewandte.org ! 2007 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Angew. Chem. Int. Ed. 2007, 46, 72 – 191


into two basic types—pulsating ratchets and tilting ratchets—and are the subject of a recent major review by Reimann,[39f]

and information ratchets, which are much less common in thephysics literature, but have been discussed by Parrondo,Astumian, and others.[39h,42b,45] Both energy ratchets andinformation ratchets bias the movement of a Browniansubstrate.[46] However, we will also show (Sections 1.4.3 and4.4) that they offer clues for how to go beyond a simple switchwith a chemical machine to enable tasks to be performedthrough the non-equilibrium control of conformational andco-conformational changes within molecular structures.

1.4.2.1. Pulsating Ratchets[39f ]

Pulsating ratchets are a general category of energy ratchetin which potential-energy minima and maxima are varied in aperiodic or stochastic fashion, independent of the position ofthe particle on the potential-energy surface. In its simplestform this can be considered as an asymmetric sawtoothpotential being repetitively turned on and off faster thanBrownian particles can diffuse over more than a small fractionof the potential energy surface (an “on–off” ratchet,Figure 6). The result is net directional transport of theparticles across the surface (left to right in Figure 6).

More general than the special case of an on–off ratchet,any asymmetric periodic potential may be regularly orstochastically varied to give a ratchet effect (such mechanismsare generally termed “fluctuating potential” ratchets). Aswith the simple on–off ratchet, most commonly encounteredexamples involve switching between two different potentialsand are therefore often termed “flashing” ratchets. A classicexample, which has particular relevance for explaining a

number of biological processes[42b] as well as being the basisfor a [2]catenane rotary motor (see Section 4.6.3), is illus-trated in Figure 7. It consists in physical terms of anasymmetric potential-energy surface (comprising a periodic

series of two different minima and two different maxima)along which a Brownian particle is directionally transportedby sequentially raising and lowering each set of minima andmaxima. The particle starts in a green or orange well(Figure 7a or c). Raising that energy minimumwhile loweringthose in adjacent wells provides the impetus for the particle tochange position by Brownian motion (Figure 7b!7c or 7d!7e). By simultaneously (or beforehand) changing the relativeheights of the energy barrier to the next energy well, thekinetics of the Brownian motion in each direction aredifferent and the particle is transported from left to right.Note that the position of the particle does not influence thesequence in which (or when, or if) the energy minima andmaxima are changed. Furthermore, the switching of the

Figure 6. An example of a pulsating ratchet mechanism—an on–offratchet.[39f ] a) The Brownian particles start out in energy minima on thepotential-energy surface with the energy barriers @ kBT. b) The poten-tial is turned off so that free Brownian motion powered diffusion isallowed to occur for a short time period (much less than required toreach global equilibrium). c) On turning the potential back on again,the asymmetry of the potential means that the particles have a greaterprobability of being trapped in the adjacent well to the right ratherthan the adjacent well to the left. Note this step involves raising theenergy of the particles. d) Relaxation to the local energy minima(during which heat is emitted) leads to the average position of theparticles moving to the right. Repeating steps (b)–(d) progressivelymoves the Brownian particles further and further to the right.

Figure 7. Another example of a pulsating ratchet mechanism—a flash-ing ratchet.[42b] For details of its operation, see the text.


Chemie













Chemie












Chemie



periodic force cases (imagine applying the electric fielddiscussed above for longer in one direction than the other),but is less so for stochastic driving forces. In general, thesemechanisms are known as “asymmetrically tilting” ratchets.

1.4.2.3. Information Ratchets[39h,42b,45]

In the pulsating and tilting types of energy ratchetmechanisms, perturbations of the potential-energy surface—or of the particle!s interaction with it—are applied globallyand independent of the particle!s position, while the perio-dicity of the potential is unchanged. Information ratchets(Figure 10) transport a Brownian particle by changing theeffective kinetic barriers to Brownian motion depending onthe position of the particle on the surface. In other words, theheights of the maxima on the potential-energy surface changeaccording to the location of the particle (this requiresinformation to be transferred from the particle to the surface)whereas the potential-energy minima do not necessarily needto change at all. This switching does not require raising thepotential energy of the particle at any stage, rather the motioncan be powered with energy taken entirely from the thermalbath by using information about the position of the particle.This is directly analogous to the mechanism required ofMaxwell!s pressure demon (Figure 2b, Section 1.2.1.1), butdoes not break the second law of thermodynamics as therequired information transfer (actually, information era-sure[48]) has an intrinsic energy cost that has to be metexternally.

It appears to us that information-ratchet mechanisms ofrelevance to chemical systems can arise in at least three ways:1) a localized change to the intrinsic potential-energy surfacedepending on the position of the particle (Figure 10); 2) aposition-dependent change in the state of the particle whichalters its interaction with the potential-energy surface at thatpoint; or 3) switching between two different intrinsic periodicpotentials according to the position of the particle.[39h,49] Anexample of the first of these types, in which the systemresponds to the “information” from the particle by loweringthe energy barrier to the right-hand side (and only to theright-hand side) of the particle, is shown in Figure 10.

The particle starts in one of the identical-minima energywells (Figure 10a). The position of the particle lowers thekinetic barrier for passage to the adjacent right-hand well andit moves there by Brownian motion (10b!10c). At this pointit can sample two energy wells by Brownian motion, and arandom reinstatement of the barrier has a 50% chance ofreturning the particle to its starting position and a 50%chance of trapping it in the newly accessed well to the right

Figure 9. A rocking ratchet.[39f ] a) The Brownian particles start out inenergy minima on the potential-energy surface with the energy barriers@kBT. b) A directional force is applied to the left. c) An equal andopposite directional force is applied to the right. d) Removal of theforce and relaxation to the local energy minimum leads to the averageposition of the particles moving to the right. Repeating steps (b)–(d)progressively moves the Brownian particles further and further to theright.

Figure 10. A type of information ratchet mechanism for transport of aBrownian particle along a potential-energy surface.[39h,42b,45] Dottedarrows indicate the transfer of information that signals the position ofthe particle. If the signal is distance-dependent—say, energy transferfrom an excited state which causes lowering of an energy barrier—then the asymmetry in the particle’s position between two barriersprovides the “information” which transports the particle directionallyalong the potential energy surface.


Chemie



3

ATP synthase, H. Wang and G. Oster (Nature 396, 279, 1998) Myosin

Muscle myosin is a dimerof two identical motor heads that are anchored to the thick filament (top) by a coiled-coil (gray rod extending to the upper right). The helical actinfilament is shown at the bottom (gray). Myosin's catalytic core is blue and its mechanical elements (converter, lever arm helix and surrounding light chains) are colored yellow or red. In the beginning of themovie, the myosin heads are in the prestroke ADP-Pi state (yellow) and the catalytic cores bind weakly to actin . Once a head docks properly onto an actinsubunit (green), phosphate (Pi) is released from the active site. Phosphate release increases the affinity of the myosin head for actin and swings the converter/lever arm to the poststroke, ADP state (transition from yellow to red). The swing of the lever arm moves the actinfilament by ~100 Å the exact distance may vary from cycle to cycle depending upon the initial prestroke binding configuration of the myosin on actin . After completing the stroke, ADP dissociates and ATP binds to the empty active site, which causes the catalytic core to detach from actin. The lever arm then recocks back to its prestroke state (transition from red to yellow). The surface features of the myosin head and the actin filament were rendered from X-ray crystal structures by Graham Johnson (fiVth media: www.fiVth.com) using the programs MolView, Strata Studio Pro and Cinema 4D. PDB files used were ADP-AlF4-smooth muscle myosin (prestroke, yellow: #1BR2) and nucleotide-free chicken skeletal myosin (poststroke, red: #2MYS). Transitions between myosin crystal structure states were performed by computer coordinated extrapolations between the known prestrokeand poststrokepositions.

http://www.sciencemag.org/feature/data/1049155.shl

KinesinThe 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 toengage 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 headforward 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 of 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 (f iVth 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.

Molecular gears

http://chem.iupui.edu/Research/Robertson/Robertson.html#Gears

Diffusion in asymmetric potentials

R. Dean Astumian, Science 1997 276: 917-922.

electrostatic potential

chemicalpotential

Driven Brownian ratchets

Three-bead assay "with ncd"

-50

0

50

100

150

200

250

300

-1

0

1

2

3

4

5

6

7

0 10 20 30 40 50 60

disp

lace

men

t [nm

]

force [pN]

time [s]

160

176

192

208

224

240

6 6.5 7 7.5 8 8.5 9

[nm

]

time [s]

Stepping and Stalling of a Single Kinesin Molecule "

~ 6 pN stall force"

~ 8 nm steps"

Svoboda, Schmidt, Schnapp, Block, Nature (1993), 365: 721"

<x2(t)> - <x(t) > 2"

Randomness parameter"

r:= lim "t -> ∞" d <x(t)>"

1 -> 2 -> 3 -> 4 -> 5 -> 6"k" k" k" k" k"

r = 0!

r = 1!

1 -> 2"k

Δt=const."

clockwork"

Δt exponentially distributed"

Poisson process"

Myosin: averaged power strokes"(Veigel et al. Nature 99, 398, 530)"

Myosin Power Stroke"

Mechano-chemical cycle:"

M*ATP"

M*ADP*Pi"

M"

attach"

working stroke"

detach"

recovery stroke"

ADP"+Pi"

ADP"

Pi"

actin"myosin"

Conformational Change of Single ncd Molecule"

-5

0

5

10

0 400 800 1200 1600

MT

displa

cem

ent [

nm]

time [ms]

2 µM ATP

release"

DeCastro, Fondecave, Clarke, Schmidt, Stewart, Nature Cell Biology (2000), 2:724"

ADP"ADP*Pi"

Thermodynamics of Brownian Ratchets - Duke …reif/courses/molcomplectures/Molecular... ·...

Documents

Transcript of Thermodynamics of Brownian Ratchets - Duke …reif/courses/molcomplectures/Molecular... ·...