The Physics and Physical Chemistry of Molecular Machines R

DOI:10.1002/cphc.201600184 Reviews

Very Important Paper The Physics and Physical Chemistry of Molecular Machines R. Dean Astumian,*[a] Shayantani Mukherjee,*[b] and Arieh Warshel*[b]

ChemPhysChem 2016, 17,1719 –1741 1719 2016 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Reviews

The concept of a“power stroke”—a free-energy releasing con- chine. The gating of the chemical free energy occurs through formationalchange—appears in almostevery textbook that chemicalstate dependentconformational changes of the mo- deals with the molecular details of muscle, the flagellar rotor, lecular machine that, in turn, are capable of generating direc- and many other biomolecular machines.Here, it is shown by tional mechanical motions. In strongcontrasttothis general using the constraints of microscopicreversibility that the conclusion for molecular machines driven by catalysis of power stroke modelisincorrect as an explanation of how achemical reaction, apowerstroke maybe(and often is) an chemicalenergy is used by amolecular machine to do me- essential component for amolecular machine driven by exter- chanical work. Instead, chemically driven molecular machines nal modulation of pH or redox potential or by light.This differ- operating under thermodynamic constraints imposedbythe ence between optical and chemical driving properties arises reactantand product concentrationsinthe bulk functionasin- from the fundamental symmetry difference between the phys- formationratchets in which the directionalityand stopping ics of optical processes, governed by the Bose–Einstein rela- torque or stopping force are controlled entirely by the gating tions, and the constraints of microscopic reversibility for ther- of the chemical reactionthat provides the fuel for the ma- mally activated processes.

1. Introduction function, that is, the assumption that the velocity (NOT acceleration) of each relevant degree of freedomisproportionalto One of the most important features of aliving system is its the force that causes it. This is the regime in which Onsager ability to harvest energy from the environmenttodowork and derived his reciprocal relations,[16] and is also the regime in to form structure. These tasks are accomplished in biological which the Onsager–Machlup thermodynamic action theory[17] [1,2] d2~r systemsbymolecular machines such as myosin and kinesin, is valid. The inertialforce m dt2 is very small (negligible) in com- [3] [4] d~r the FoF1 ATPsynthase, the bacterial flagellar motor, the ribo- parisontothe viscousdrag force g dt,and hence does not some,[5] and various DNA and RNA processing enzymes,[6] appear in Equation (1). Thisregime of motionwas explored among many others. Recent work has described great progress beautifully by Purcell in his paper “Life at Low Reynold’s [18] in accomplishing the synthetic imitationofsome of these re- Number”. The effect of the solvent is modelled in termspffiffiffiffiffiffi of markable devices.[7–13] At first glance,itwould seem that aphys- the viscous drag coefficient, g,and thermal noise, 2D~fðÞt , ical theory formolecular machines must be extremelycompli- with afluctuation dissipation relationbetween the viscous cated and requires astrong focus on the fact that the chemical drag coefficient and the amplitude of thermal noise, gD ¼ kBT. driving forces that provide the energy to fuel the machines are All energies in this paper are given in units of the thermal [14] very far from thermodynamic equilibrium. In fact, however, energy kBT,where kB is Boltzmann’s constant and T is the the “physics” of achemically driven molecular machine—its Kelvin temperature. equationofmotion—is very simple:[15] It is important to notethat Equation (1) describes amechanical equilibrium theory—the average net force is zero, d~r pffiffiffiffiffiffi hg d~r þrUðÞ~r i¼0, about which there is Gaussiandistributed ¼ÀgÀ1rUðÞþ~r 2D ~fðÞt ð1Þ dt dt thermalnoise ~fðÞt .All of the information about how the structure relates to the mechanism is contained in the energy func- In Equation (1), ~risthe vector comprising the relevant de- tion UðÞ~r .Inthe zero noise limit, the system would inexorably grees of freedom of the machine, g is the coefficientofviscous find alocal energy minimumand remain there forever. The friction, ~fðÞt is randomthermalnoise, the components of transitions between minimathat are necessary for the molecu- which are given by independent normalized Gaussian distribu- lar machine to carry out its functionrequirethermal noise, and tions, and ÀrUðÞ~r is the force due to the gradientofasingle, hence all chemically driven molecular machinesinwater,the time-independent, potentialenergysurface, UðÞ~r . functions of which are described by Equation (1), are properly Equation (1) reflects an important assumption aboutthe termed“Brownian Motors”.[19–21] regime of motion in whichamolecular machine carries out its The Langevin equation [Eq. (1)] expresses completely the “physics” of achemically driven molecular machine.Many au- [a] Prof. R. D. Astumian thors, however,seem to be looking for adescription in terms Department of Physics, University of Maine of classicalmechanics,[6] and this is what cannotbegiven, for Orono, ME 04469 (USA) E-mail:[email protected] the simple reason that the problem of mechano-chemical cou- [b] Dr.S.Mukherjee, Prof. A. Warshel pling by an enzyme is NOT aproblem of classical mechanics. It Department of Chemistry makesalmost as little sense to seek amechanical description University of Southern California, Los Angeles of the coupling between achemical reaction and the motion California (USA) of amolecular machine in water as it does to seek amechanical E-mail:[email protected] [email protected] description of the diffraction of an electron. Quantum mechan- The ORCID identification number(s) for the author(s) of this article can ics is of course fundamentally not mechanicalbut rather prob- be found under http://dx.doi.org/10.1002/cphc.201600184. abilistic,whereas the thermodynamics and kinetics of molecu- An invited contribution to aSpecialIssue on Molecular Machines lar machines are only practically probabilistic ratherthan me-

ChemPhysChem 2016, 17,1719 –1741 www.chemphyschem.org 1720 2016 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Reviews chanical. In afull molecular dynamics simulation involving all modelthe system in terms of Newton’s equations for longer degrees of freedom of both the protein and of the molecules than afew picoseconds is doomed to failure. in the solution, the dynamics would be described by Newton’s The principle of microscopic reversibility[23] provides asolid equations of motion in which acceleration and not velocity foundation for development of athermodynamic theory for appear,but the impracticality of aclassical mechanical descrip- molecular machines. For an over-dampedsystem described by tion in terms of Newton’s equations (or Lagrange’s or Hamil- Equation (1), Bier et al.[24] used the Onsager–Machlup thermo- ton’s) is overwhelming. There are 1018–1020 collisions[22] each dynamic action theory[17] to deriveEquation (2): second between water molecules and amolecular machine ÂÃ"# like myosin, the flagellar motor, or kinesin, and any attemptto 0 ~rðÞt 00 0 00 P ~r ðÞt ¼ 0 ! ~r ðt ¼ t Þ 0 00 Pð~r !~r Þ hif ¼ eUðÞ~r ÀUðÞ~r ~ 0 00 0 rtðÞf Àt 00 Pð~r ~r Þ P ~r ðÞt ¼ tf ÀÀ ~r ðt ¼ 0Þ Dean Astumian received his Ph.D. from the University of Texas at Arlington. ð2Þ Following staff positions at the NIH and NIST,hemoved to the University for motion on apotential energy surface UðÞ~r where ~r 0 and ~r 00 of Chicago as Assistant then Associate are two arbitrary points. Both the numerator and the denomi- Professor,and then as Full Professor to nator on the left hand side of Equation (2) depend on the path the University of Maine. He is afellow rtðÞand on the interval tf ,but the ratio depends on neither of the American Physical Society,and and is astate function that depends only on the difference in he received the Galvani Prize of the the energies of the initial and final states and on the tempera- Bioelectrochemical Society,aHumboldt ture, which is subsumed in our energy units kBT.The ratio in Prize in 2009, and the Feynman Prize the third identity has been enclosed in brackets as only the in 2011. His research focus is on kinetic ratio makes sense—thenotation Pðr0 ! r00Þ alone makes little mechanisms and thermodynamics of sense withoutspecification of rtðÞand of tf .Because the ratio molecular motors. is astate function, the identity holds also for the ratio of the integralsofthe numerator anddenominator over all rtðÞand Shayantani Mukherjee obtained her tf .Note that Equation (2) can also be very easily derived by Ph.D. from Jadavpur University in using the principle of detailed balance at equilibrium. Al- Computational Biology while working thoughthis derivation uses knowledge of the behavior of the as aResearch Fellow at the Saha Insti- system at equilibrium, Equation (2) itself, while requiring me- tute of Nuclear Physics in Kolkata, chanicalequilibrium, holds arbitrarily far from thermodynamic India. She then moved to Michigan equilibrium.[25] State University as aPost Doctoral Although the physics and physical chemistry of chemically Fellow and is currently aResearch As- driven molecular machines are actually quite simple, the mech- sociate at the University of Southern anism by which these tiny machines work is contrary to our California. Her research interests are in macroscopic experience, to the way light-driven molecular ma- elucidating structure–function relation- chinesare shown to work, and to expectations based on ex- ships of biological macromolecules, perimental responses following externalchanges in the envi- molecular motors, and kinetics and thermodynamics of complex ronment.Itisthe deviation from what we perceive to be cellular processes. “commonsense”, and even from what seemstobesupported by experimental observation, that leads to much confusion in Arieh Warshel was born in 1940 in Kib- the literature. To firm up the understanding of what Equa- butz Sde-Nahum, Israel. He received tions (1) and (2) tell us about the mechanism of molecular aB.Sc. from the Technion in 1966 and motors and rotors, let us consider aspecific example involving aPh.D. from the Weizmann Institute in recentcomputational work on the F1 ATPase, acomponent of 1969. He is currently Distinguished the FoF1 ATPsynthase found in the mitochondria. This molecu- Professor and the Dana and David lar machine uses aproton electrochemical gradient to provide Dornsife Chair in Chemistry at the Uni- the energy for synthesis of ATP, the major storageform of versity of Southern California. Dr.War- chemicalenergy in acell.[3] shel has pioneered computer simulations of the functions of biological molecules, and he and his group con- 2. F1 ATPase, an ATP-Driven Molecular Rotor tinue to push the frontiers in the area. In principle, there may be many degrees of freedomincorpo- Dr.Warshel received numerous awards including the Nobel Prize in rated in the vector ~rinEquation (1), but as Bier and Astu- Chemistry in 2013. He is amember of the National Academy of Sci- mian,[26] and contemporaneouslyMarcello Magnasco,[27] sug- ence and an honorary member of the Royal Society of Chemistry gested (see also ref. [28]), and as elaborated by Keller and Bus- and the holder of several honorary degrees. tamante,[29] the problem often reducestocoupling between

ChemPhysChem 2016, 17,1719 –1741 www.chemphyschem.org 1721 2016 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Reviews only two effective coordinates:one describing the mechanical motion—rotation in the cases of the F1 ATPaseand the bacterial flagellar motor,ortranslation in the cases of kinesin,myosin V, and other molecular walkers; and one describing the chemicalreaction—proton transport or ATPhydrolysis. For the purposes of this discussion, let us consider amechanical coordinate q and achemical coordinate x such that ~rtðÞ¼ðÞqðÞt ; xðÞt T ,where superscript T denotes transpose. This [30] approachiswell illustrated by recent work on the F1 ATPase where Mukherjee and Warshel have computationally investi- gated the energy of F1 ATPase in many of its possible rotational and chemical state dependentconformations. Their key result is shown in the potential energy landscapeinFigure 1a, where the horizontal axis represents mechanical rotation of F1 ATPase, andthe vertical axis represents the transitions between the differentchemical states. The energy surfaceshown in Figure 1a is an equilibrium picture of the system—thereisnonet tilt along either the chemical or the mechanical coordinate. Even so, the landscapetells us much of what we need to know about the coupling. The deep zigzag energy valley in blue for the wells (states) and green for the saddle points (transition states) is the hallmark signature of aBrownian motor.[28] This pattern shows that under circumstances where it is more likely to bind ATPand release ADP and Pi (Pi= inorganic phosphate) than it is to bind ADP and Pi andrelease ATP(i.e.,where the chemical potential of ATPishigherthan that of ADP and Pi)there will be net clockwise (positive) rotationinthe absence of an applied torque. The mostprobable pathway through this energy landscape, indicated by the white dashed line, is aclear visual indi- cation of the coupling, even without explicitly including the chemicalfree energy released during the process. The motion on this two dimensional energy landscape can be described using the Langevin equation, Equation (1), where boundary conditions incorporate the effect of havingATP in excessand ADP in deficit of their equilibrium amounts. The mechanism can also be viewed in several other ways. In

Figure 1b the energy profiles for two chemical states, D1E2T3 and E1T2D3,are shown schematically as afunction of the rotational angle between À2408 and +2008 with chemical transitions betweenthe two states. This is atypical “ratchet” model Figure 1. a) Energy landscape for the F1 ATPase from the work of Warshel [30] in which mechanical motion is described along the horizontal and Mukherjee. The fundamental periods Dx and Dq for which UðÞ¼x; q UðÞx Æ iDx; q Æ jDq with i,j=any integers, are shown. b) A“ratch- axis and the effect of chemistry is modeled in terms of transi- et” representation in terms of two 1D energyprofiles with transitions be- tions between thesehorizontal energy landscapes for the me- tweenthem for the two chemical states D1E2T3 andE1T2D3 from À2408 to chanical motion.The behavior of the system can be modeled +2008 (the area enclosed in the brightgreen dashed box in Figure1a). The based on this description by reaction–diffusionequations. In remarkable and salientpoint is that the mechanism shown by the green arrows(clockwise rotationof1208)seems by commonsense to be far more early descriptions using this approach, the constants describ- likely than that shown by the red arrows (counterclockwiserotation by ing the transition rates between the potentials were assigned 1608), but, in fact, if e*=e these two processes are equallylikely,and if in consistency with the principle of microscopicreversibili- e*

Twoperiodsofthe kinetic lattice model representing the occursuncoupled to chemistry,and c/cR (futilecycling) in which chemistry energy minimaasstates (A/C, B) and the saddle points as tran- occursuncoupled to rotation.

ChemPhysChem 2016, 17,1719 –1741 www.chemphyschem.org 1722 2016 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Reviews sitions—red for those involving the energy barrier (saddle 3. Non-EquilibriumThermodynamics of point) e*,green for those involving the energy barrier e,and Molecular Machines orange for those involving the energy barrier e**—are shown on the right. The motion of the motor can be described as In every completion of arandom walk from (q þ iDq; x þ jDxÞ [28] arandom walk on this lattice of states. Downwardtransi- to ½q þ ðÞi þ 1 Dq; x þ ðÞj þ 1 Dx ,the F1 ATPase undergoes a tions are strongly favored when ATPisinexcess and ADP is in +Dq (clockwise) rotation and hydrolyses three ATPs (i.e.,com- deficit of their equilibrium amounts, but the most probable pletes astep on the chemical axis of Dx). This process is la- downward transition is determined by the relative barrier beled f.Inthe reverse process, (q þ iDq; x þ jDxÞ to heights. As with the reaction–diffusionapproach, applying the ½q þ ðÞi À 1 Dq; x þ ðÞj À 1 Dx ,termed fR,the F1 ATPase un- constraints of microscopicreversibility assuresconsistency with dergoes a ÀDq (counterclockwise) rotation and synthesizes the underlying model governed by Equation (1). three ATPs. In every completion of awalk from

It is not useful to describe the effect of achemical gradient (q þ iDq; x þ jDxÞ to ½q þ ðÞi À 1 Dq; x þ ðÞj þ 1 Dx ,the F1 in terms of atilt along the vertical axis in Figure 1a when one ATPase undergoes a ÀDq (counterclockwise) rotation and hy- attempts to understand the microscopicpicture of the bind- drolyses three ATPs. This process is labeled backward, b.Inthe ing, release, and chemical processes and how they relate to reverse(completionofawalk from (q þ iDq; x þ jDxÞ to the conformationalchanges,althoughsuch phenomenological ½q þ ðÞi þ 1 Dq; x þ ðÞj À 1 Dx ), termed bR,the F1 ATPase un- modelscould offer heuristic insight in comprehending the dergoes a +Dq (clockwise) rotationand synthesizes three mechanism. Bindingofaligand such as ATPispurely local. The ATPs. There are also processes involving uncoupled ATPhydrol- chemicalpotentials of ATP, ADP,and Pi in the bulk do not influ- ysis (futile cycling, c/cR)and processes involving rotation with- ence in any way the internal energies of the protein (although out ATPhydrolysis/synthesis (slip, s/sR). These possibilities they influence the free energy of the overall state). In reac- are illustrated explicitly in the kinetic lattice model of Fig- tion–diffusion or kineticmodels, the effect of concentration is ure 1c. handled consistently by allowing the rates of processes in The ratio of the probability for any process and its micro- which ATP, ADP,orPiare boundtobeproportionaltothe con- scopic reverse is athermodynamic identity. For the complete centrations(or more correctly,activities) of [ATP],[ADP],or[Pi], cycles shown in Figure 1b these identities can be written as respectively.The best description of the effect of the chemical Equation (3):[37] hiÀÁ ½ATP 0 potentialdifference, Dm ¼ RT ln ½½ADP Pi þ ln K ATP ,between Pf Pb Ps Pc ¼ eXxÀXq ; ¼ eXxþXq ; ¼ eÀXq ; ¼ eXx ð3Þ reactantand product is the phrase “mass action”asused by P P P P fR bR sR cR Guldberg and Waage[32] in the second half of the 19th century. It is perhaps abit more sensible to describe the effect of an where the “generalized thermodynamic forces” for the F1 appliedtorque as atilt between the initial andfinal point of ATPase, X ¼TDq and X ¼ 3Dm,are the mechanical work the mechanical coordinate, q,once the chemical potentialis q x and chemical work done in moving aperiod Dq and Dx,re- [33,43] added but the torque will influenceboth the net tilt, and spectively,and where T represents the applied torque. Under also distort the local features (e.g.,well and saddle point ener- physiological conditions for hydrolysis of ATP Dm70 kJmolÀ1. gies) of the energy surface. The relative likelihood for ATPtobind and for ADP and Pi to It is possible to get useful insightinto the applied torque or dissociate versus the likelihood for ADP and Pi to bind and ATP the appliedforce against which the motor can do work by to dissociate—that is,the effect of Dm on the process—is in- adding the chemistry (free energy in the standard state plus corporated in the factor eXx appearing in both ratios P =P the concentration effect, which is equivalent to the chemical b bR and P =P .The net probability for completion of aclockwise potential) to the conformational free energy profile to model f fR rotation, P ,and for ATPhydrolysis, P ,are given by the action of the motor[30,33–35] in the presence of ATPinexcess Dq;net ATP;net Equations (4a) and (4b): of its equilibrium amount. In this representation, the effect of concentration appears as areduction of the state energies and ÂÃÀÁÀÁÂÃÀÁ P ¼ P þ P À P þ P þ P À P ð4aÞ also of the effective activation barrier when going from an un- Dq;net f bR fR b s sR bound state to abound state along the functional pathway ÂÃÀÁÂÃÀÁ P ¼ ðÞÀP þ P P þ P þ P À P ð4bÞ (see Figure 4). This approachallows forMonte Carlo or Lange- ATP;net f b fR bR c cR vin dynamics simulations on the simplified functional surface, thus eliminating the need for simulations with many parti- Using Equations (3), and recognizing that the currents [36] cles. (fluxes)can be writtenasthe products of the net probabilities We can also circumvent the necessity of considering many and acommon inverse time constant, tÀ1,weget Equation- of the local detailsofenergy coupling by focusingonhow the s(5a)and (5b): energy surface relates to complete cycles of the machine[37] in which some number of ATPs are hydrolyzed/synthesized, and 8 9 > > the motor makes some number of complete rotations in the < = À1 ÀXx þXq Xq ÀXx Xq Jq ¼ t ½ðÞÀ1 À e qeðÞÀ e þ LqðÞ1 À e ð5aÞ clockwise/counterclockwise direction. :>|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl } |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} ;> coupled transport uncoupled transport

ChemPhysChem 2016, 17,1719 –1741 www.chemphyschem.org 1723 2016 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Reviews

8 9 The term r eXq is the ratio of backward/counterclockwise <> => 0 À1 ÀXxþXq Xq ÀXx ÀXx (Pb þ Pf þ Ps )toforward/clockwise (Pf þ Pb þ Ps)steps/ Jx ¼ t ½ðÞþ1 À e qeðÞÀ e þ LxðÞ1 À e R R R >|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} > À1 : ; rotations and C ¼ t ðPf þ Pb þ PsÞ is akinetic pre-factor. coupled transport uncoupled transport R Taking asimple Boltzmann expression, C ¼ C eaXq ,Equa- ð5bÞ 0 tion (6) can be cast in dimensionless form as Equation (8): 2 3 ~ where Jq and Jx are the currents in units of steps of angle Dq ðÞ1À Xq ~ 1 À r0 per unit time and stoichiometricconversions (ATPs hydrolyzed) J~ ¼ ra Xq 4 5 ð8Þ q 0 1 À r per unit time, respectively.The controlling factor for the cou- 0

Xq De=kBT pled transport is the ratio Pb=Pf ¼ qe ,where q e is governed solely by the difference in activation barriers forthe ~ where Xq is the generalized mechanical “force” normalized by b and f paths. The four paths Pb; Pb ; Pf; and Pf are related Xq ~ R R the stopping force (the force at which r0e ¼ 1), and Jq is the by symmetry[38] and determine completely the terms labeled flux normalized by the flux evaluated with Xq ¼ 0. The normal- “coupled transport”. Plots of the fluxes normalized by the flux ~ ~ ized flux, Jq,isshown as afunction of Xq for severalvalues of at zero applied torque are shown in Figure 2. “a”. As apoint of comparison, the hyperbolic expression ~ ~ ~ [41] Jq ¼ Kð1 À XqÞ=ðK þ XqÞ proposed by Hill to model the shortening force versus velocitycurves for muscle is shownas adashed curve in Figure 3.

Figure 2. Plots of the coupled transport termsinEquation(5a)(solid blue line) and Equation(5b)(dashed lines) are shownfor q ¼ eÀ7 and eÀXx ¼ eÀ5 (dashed green curve) and q ¼ eÀ5 and eÀXx ¼ eÀ7 (dashed orange curve)

À1 À1 Xq =2 with t ¼ t0 e .

When eÀXx q,the system is under thermodynamic control.

Forcing the motor backwards by an applied torque causesthe À13 Figure 3. Plots of Equation(7) (solid curves) with r0 ¼ e for three different ~ ~ ~ motor to move over the low barriers, but in the reversedirec- values of a.The hyperboliccurve Jq ¼ Kð1 À XqÞ=ðK þ XqÞ proposed by Hill, tion, resulting in synthesis of ATP. This regime, also known as with K=0.2 as used by Hill to fit experimental data obtained for the force tight coupling, is the regime that is often considered in litera- versus velocity curve of muscle, is shown as the dashedblue curve. ture on the thermodynamicsofmolecular machines, and seems to be the case for the F1 ATPase. Asimple mathematical analysisofAndrewHuxley’searly When eÀXx q,the system is under kinetic control. In this mechanistic model of muscle[42] can be cast into the hyperbolic case, forcing the motor backward by an appliedtorque causes form proposed as aphenomenological fit to the data. Howev- atransition over the higherbarriers to carry out increased ATP er,when we analyze Huxley’s model,[42] taking rate constants hydrolysis. This possibilitywas pointed out by Bier and Astu- that are consistentwith microscopic reversibility,the result is mian[28] and given experimental support by Nishiyama, Higuchi, of the form of Equation (8)[43] and not of the hyperbolic form and Yanagida[39] and by Carter and Cross[40] for kinesin in partic- of Hill.[41] ular.

3.1. Response of aMolecular Machine to External Load 3.2. Rethinkingthe Terms “Torque Generation” and “Force Generation” Equation (5a) can be rewritten in avery simple form as Equa- tion (6): When we say that JosephPriestly was the first chemist to generate oxygen we mean that he could bottle it, stopper the J ¼ CðÞ1 À r eXq ð6Þ q 0 bottle,and send the bottled oxygen to other labs. Theuse of the termenergy generation is similarly meaningful, the storage where r0 is given by Equation (7): form of energy being potential energy.There is, however,no ðÞþeÀXx þ q L r ¼ q ð7Þ equivalent potentialtorque or force. It is neither possible to 0 ÀX ðÞþ1 þ qe x Lq store, nor to produce and consume torque, and it is certainly

ChemPhysChem 2016, 17,1719 –1741 www.chemphyschem.org 1724 2016 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Reviews not correcttodescribe torque or force as a“product of areac- pathways, that is, by constructing adetailed free-energy sur- tion”.[44] face forthe most relevant mechanical and chemical degrees of

In arecent paper on the FOF1 ATPsynthase, Mukherjee and freedom, one can proceedtowards aholisticunderstanding. [45] Warshel referred to the “experimentally observed torque”, Using the example of F1 ATPaseand Figure 1a,wesee that which essentially implies the torque inferred from the experi- the direction of motion is independent of the energies of the mentally observed rotational motion,and not torque as states (minima on the energy landscape);sowhat feature of adirectobservable of the single molecule experiments.The the energy landscape actually controlsthe direction of imprecision of the phrase highlightsanimportantdifference motion?Using our mind’s eye, we see that the transformation between the macroscopicworld of our experience and the e* $ e,which switchesbetween red and geen in the kinetic molecular world in which even large macromolecular com- lattice picture, does, in avisually clearway,change the sense plexes such as the flagellarmotor,actinomyosin (muscle), and of rotation—the minimum energy zigzag valley runs between

FoF1 ATPsynthase carry out their functions. If amacroscopic, or the lower left and upper right hand corners when e*< e.Thus, even amesoscopic, object is observedtoundergo persistent we conclude that the directionality is controlled by the relative rotation, the inference that there is an underlying mechanical energies of the barriers and not by the energies of the states. torque causingthe rotation is absolutely secure. On the other Another recent study has revealed the molecular origins of hand, in the microscopic world, gating, as in the original pre- e and e*, showing that mutations of certain parts of the F1 sentationofaninformation ratchet,[46,47] can lead to persistent ATPase rotary subunit can lead to the destruction of the zigzag rotational motion even if the mechanical torquealong the re- path by effectively putting ee*. This leads to systemswhere @U action path, À @q,iszero or negative. the mechanical rotation is uncoupled from the chemical steps, It is tempting to characterize molecular rotors according to thereby generating afutile molecular motor incapable of whether there is or is not alocal mechanical torqueasthe showingdirectional motions poweredbyATP hydrolysis.[45] rotor moves, but this distinction leads to confusion. On the energy scale of the coarse-grained molecular landscape of F 1 3.3. Mechano-ChemicalCoupling—An Overview ATPase shown in Figure 1a, UA/C is comparable to UB,but the directionofmotion does not depend on whether UA/C >UB or The single parameter describing mechano-chemical coupling is De=kB T UB > UA/C (or in general,onthe relative energies of any of the q e where De is the differenceinactivation energy for states—the energy minima—of the system). With UA/C >UB,we a“functional” forwardprocess, f,inwhich substrate is con- could perhaps seemingly justify words such as “deposition of verted to product concomitantwith aforward step, and aback- chemicalenergy results in lifting the system energetically from ward process, b,inwhich substrate is converted to product Btothe pre-power stroke state A/C from which the system ex- concomitant with abackward step. Amore precise expression ecutes the powerstroke A/C!B, thereby generating torque as the exponential of the differenceofthe Onsager–Machlup and causing rotation”. However,when UB > UA/C,the mecha- thermodynamic actionscan be calculated from the equilibrium nism remains fundamentally the same, but transitions along energy landscape, with arbitrary stoichiometry,[38] but in the the mechanical coordinate are predominately energetically simplest case (such as that showninFigure 1a)the processes uphill. Irrespective of whether UA/C > UB or UB > UA/C the mecha- are controlled by single rate-limiting barriers so De is very nism by which rotational motion occurs is that of an informa- easily defined. At equilibrium,ofcourse,completionofafor- tion ratchet[45,46] where ATPbinding/release is fast, and ADP ward cycle f is exactly as likely as completion of its micro- binding/release is slow at some values of q and ATPbinding/ scopic reverse fR,and completionofabackward cycle b is as releaseisslow,and ADP binding/release is fast at other values likely as completion of its microscopicreverse bR so there is of q.This gating, combined with ever-present thermalnoise, is no net transport, in consistency with the second law of ther- sufficient to drive the experimentally observed rotation. For modynamics. Away from chemical equilibrium, however, f is chemicaldriving,the mechanical torque on the equilibrium more likely than fR,and b is more likely than bR.Inthis case, potentialissimply irrelevant for determining the direction, the structural bias by which f is more likely than b comes to stoppingforce, andmaximal efficiency of the motor.Itshould the forefront and we have net clockwise rotationpowered by be additionally notedthat gaining athermodynamically robust hydrolysis of ATP. understanding of the directionality or gating mechanism of As an example of this perspective, considerthe recent com- molecular motors is not possible through the application of ex- putational modelfor the overall functional cycle of myosin V.[34] ternal forces or torque, as adopted by many as apreferred The authors show that the free energy of the pre-“power route to perform forced-molecular dynamics simulations of stroke”state is 11 kCal molÀ1 lower than the post-“power biological systems. Investigations of the relaxationsofadeter- stroke” state for the forward functional cycle (f)ofmyosin V ministicelastic network model following sudden changes of in whichanATP is hydrolyzed and myosin movesone step the constraints applied to atoms at the active site or to those toward the “plus” end of actin—energy is absorbed, not re- involved in allosteric conformational changes of aprotein are leased in the process termed the “power stroke”. In the back- similarly unhelpful with regard to understanding the thermo- ward, non-functional cycle (b)inwhichanATP is hydrolyzed dynamics of mechano-chemical coupling by amolecular ma- and the myosin movesone step towardthe “minus” end of chine. Conversely,our analysisshows that by revealing the un- actin, the free energy of the pre-“power stroke” state is 15 kCal derlyingnature of the relative energies of the variouspossible molÀ1 (or by adifferent pathway 3kCal molÀ1)higherinenergy

ChemPhysChem 2016, 17,1719 –1741 www.chemphyschem.org 1725 2016 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Reviews than the post-“power stroke”—energy is released in the step causal language is doomed to failure.[23] The equilibrium free- labelled“power stroke” in the backward non-functional cycle. energy surfacedefines q,and the flux is given by

À1 ÀXx À1 Thus, according to apowerstroke model, myosin Vshould Jq ¼ t ðÞ1 À q ðÞ1 À e .The time constant, t ,does not move toward the “minus” end of actin, but in factexperimen- depend thermodynamically on Xx,but it does depend kineti- tal observation shows that myosin Vmovestoward the “plus” cally on the absolute concentrations of substrate and product end of actin. Mukherjeeand Warshel[34] show that the energy as asaturable functionofthe concentration. barrierinthe forwardfunctional direction is lower than the Both of the positive definite coefficients q and tÀ1 depend energy barrier in the backward direction, and that it is this ki- on the applied torque (or force), T ,kinetically since not only netic difference in activation barriersthat governs the direc- does T influence the net tilt along the mechanical axis, Xq,but tion—that is, myosin Vfunctions as an information ratchet. In it may also distortthe energy landscape, and hence ef; eb, Figure 4, schematic energy diagramsinspired by those of Mu- and De may all depend on T .The dependenceofq and tÀ1 kherjeeand Warshel[34] are shown to illustrate that the direc- on T is important for fitting kinetic data, but the thermody- tion of motion is kineticallyselected based on the relative namic dependenceofthe coupled system on T is captured in heights of the maximum energy barriers rather than by the the terms that involve Xq. energy released (absorbed) by the power stroke. Equations (6) and (8) are exact rewritten formsofEqua- When the chemical reaction is away from thermodynamic tion (5a) and are convenient for examiningthe velocity versus equilibrium(Xx6¼0), q is unchanged,but there is net mechani- torque response of amolecular machine. Equation (8) in partic- ÀXx cal flux Jq becauseðÞ 1 À e 6¼0, and the flux is proportional ular highlights the factthat the thermodynamic theory based to atime constant, tÀ1.The flux arises because of mass on trajectories[37] fits the data that had previously been de- action,[32] and any attempt to describe coupling in terms of scribed by using Hill’s hyperbolic expression for the force–flux relation.[41]

4. Irrelevanceofthe Power Stroke for Chemically Driven Motors From the above analysiswesee that the internal energy released in any single transition between states is irrelevant for determining the intrinsic directionality,the stopping force, and the optimal efficiency of any chemically driven motor.[37,43,48, 49] Even so, the concept of the “power stroke” for chemically driven molecular machines stubbornly persists in the literature. There are perhaps two reasons for this persistence. First, the powerstroke is unarguably an important determinant of the directionality of light-drivenmotors,[50] the rate constantsof which do not obey microscopic reversibility,and of motors driven by externalmodulation of thermodynamic parameters such as electric field strength, pH, and redox potential.[7] Second, the diagrams by which the powerstroke models are explained are very persuasivefrom the perspective of macro- scopicmechanical intuition.[51,52] First, let us consider aheuristic model as shown in Figure 5, which illustrates how random fluctuations can be used in synergywith externally supplied energy or information to drivedirected motion. Figure 4. Energy profiles for the a) forward functional (plus end directed) pathway (green curve) and b) the backward (minus end directed) pathway (red or blue curves) for two-headed myosinVwalking on actin are shown 4.1. Simple Model for Using aRatchet to Harness Random (adapted from ref. [35]).The computational results suggest that the “power Energy to Drive Directed Motion stroke” in the forward pathway is endergonic (DG >0), whereas that in the ps Imagineasmall car subjecttoaviolent hail storm.[8b] The backward pathway is exergonic (DGps <0). The forward pathway nevertheless is strongly preferred over the backwardpathwaybecause the highest random pushes resulting from the hail can be exploited to activation barrier in the backward path is much higher than the highest bar- move the car uphill by use of aspecially designed ratchet rier in the forward pathway (greencompared with red curve), De=eÀe*. brake as shown in Figure 5a.When the brake is engaged, the Even when one considersthat the energetically costlyconformational change in one legofmyosin Viscompensated completely by the downhill car is locked in place at the notch of the ratchet as shown. conformationalchangeinthe other leg, still the system goesthroughthe When the brake is released, the car on average rolls backward blue curve where e*>e,although De is much lower than that for the red (and the ratchet gear rotates counterclockwise), but because curve. It should be notedthat acomplete and simultaneous compensation of the hail it sometimesmovesforward, rotatingthe ratchet of the conformational changes in the two legs is unlikelytooccur and myosin Vmost likely adopts amuch high energy pathwayfor backstepping (i.e., gear clockwise. The asymmetry of the ratchet brake is such red curve). that the distance to move forwardtowhere the brake, when

ChemPhysChem 2016, 17,1719 –1741 www.chemphyschem.org 1726 2016 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Reviews

If the driver observes the position of the car relative to ape- riodic array of fire hydrants that serve as fiduciary points and appliesthe brake only when the car is in such apositionthat the brake will catch on the next clockwise notch of the ratchet, the car can be moved very reliably uphill. Remarkably,the design of the brake can be simplified to where the cogs teeth are simple rectangular cutouts.When the pawl is engaged, the car remains free to diffuse but only within alimited range. If the driver releases the brake only when in the forward most part of the cog’s constrained diffusive range and appliesthe brake after the cog has rotated forward past the next tooth, the car inexorably movesforward even thoughatnotime Figure 5. Schematic illustration of how random energyfrom ahail storm can does the driver perform any force times distance work. In this make it possible for avery small car to drive uphill given an appropriate case, the energy for the uphill motion comes from the hail, but brake design (adapted from ref. [8b]). There are two possible mechanisms shown, a) an energyratchetand b) an informationratchet. The car is mod- this is allowed by the information used by the driver in decid- eled as asmall green sphere in each,where the fire hydrants act as fiduciary ing when to apply and release the brake. The driver playsthe markers. a) In the energy ratchet, the car is equipped with aspecial brake role of Maxwell’s demon.[9,23, 31] modelled after amechanical ratchet shown in the upper left hand corner. To understand the relevance of these models to real molecu- When the brake is on, the car is forced into the notch of the ratchetjust to the rear of the fire hydrant. When the brake is released, the cartends to roll lar machines, it is necessary to incorporate physically consis- backwardbut because of the hail the caralso jitters back and forth. Owing tent descriptions of amolecule rather than of acar,ofthermal to the asymmetry of the ratchet teeth it is more likely for the car to initially noise insteadofahail storm, and of allostericinteractions move forward past the hydrant to its front than backward past the hydrant rather than of an observant driver,and to explicitly describe to its rear,althougheventually the car will move downhillifthe brake is kept off for along time. Reapplying the brake, however,atintermediate how the binding substrateand release of product effectively times when the car is morelikely to have moved the short distanceforward appliesand then releases the brake. In other words, one must past the hydrant to the front than the long distancebackward past the hy- have in hand aquantitative free energy description of the drant to the rear,forces the car on average forward to the nextnotch to the chemicalstate dependentconformational changes calculated front. This process canberepeated,resulting in net uphill motion of the car. Note that the energy comesnot from the hail itself, but from the effort ex- from the atomistic 3D structure of biological motors and then pended by the driver in applying the brake—that is, from apower stroke. scrutinize the nature of the motor throughthe lens of micro- This mechanism does not requirethe driver to observethe position of the scopicreversibility.When this is done, one sees that the car in determining whether to apply or release the brake, but only to make energy ratchet mechanism cannotoperate when powered by sure the brake is not keptoff for too long.b)Analternate method involves the driver observing the position of the car relative to the hydrants. If the an autonomous exergonic chemical reaction, but that the in- driver releases the brake only whenthe car is near the hydrant in front, and formation ratchet can, and that the presence or absence of applies the brake whenever the car is near the hydranttothe rear,the car apower stroke is irrelevant for determiningthe direction and inexorably moves uphill, even with avery simple brake that simply prevents thermodynamics of the molecular machine. slippageand where no force needs be exerted when applying the brake. Here, the car moves uphill by virtue of the information obtained to deter- Now,with this in mind, let us examine asupposed power mine whentoapply and release the brake. stroke mechanism from the literature that has been used to describe the flagellar rotary motor,[52] and that is very similar to an earlier mechanism for muscle.[51] depressed, will catch on the next tooth in the clockwise direction is much smaller than the distance to move backward to 4.2. PowerStroke Models in the Literature where the brake will catch on the next tooth in the counterclockwise direction. If the hill is not too steep and the ratchet Adiagram inspired by work on the flagellar rotor[52a] is shown is designed correctly,onaveragedepressing the brake after in Figure 6, where in the original illustration the authors includ- ashort period of having released it, will apply atorque that ed only the mechanism indicated by the solid arrows in the moves the car forward, uphill. This cycle can be repeated to shadedwindows.The slopeofthe potential is a trompe l’oeil move the car uphill when the driver applies and releases the that fools the naïve reader into accepting the implicitidea that brake at set intervals withoutthe need to use any knowledge the transitions labelled “power strokes” are important for de- of the cars positiontodetermine what to do. termining the direction of the motor.[48] The mechanism, The work of moving the car uphill is done by the driver de- known as ashift ratchet, seems to have been inspired by anal- pressing the brake pedal. The hail, withoutwhich the mecha- ogy with amechanical escapement mechanism rather than nism fails, simply serves to allow the car to move randomly asimple brake as discussed in Figure 5. Ahuge difference be- uphill or downhill when the brake is released. This mechanism tween amechanical escapement mechanism and abraking has been termed an energy ratchet, and the action of the mechanism is that, seemingly,nodiffusion is required fordi- driver in pressing the brake can be very reasonablydescribed rected motionbythe escapement. Unfortunately,this descrip- as apowerstroke. An alternate mechanism termedaninforma- tion, when appliedtomolecules, leads to afalsely mechanical tion ratchet(Figure 5b)isinmany ways even simpler than the perspective of motion in which biological rotary motors have energy ratchet. been described as the “world’s smallest wind-up toy”.[52b] Blunt-

ChemPhysChem 2016, 17,1719 –1741 www.chemphyschem.org 1727 2016 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Reviews

Figure 7. a) Illustrationofacatalytically driven shift ratchet. The clear impli- cationisthat the slope of the potential dictatesthe direction of motion.For asystem inwhich the flipping between the two potentials is accomplished externally by,for example, an applied electric field, this is in fact the case. b) However,whenthe flipping between the two potentialsismediated by the binding of substrate and release of product in acatalyticprocess, the rate constantsare constrained by microscopic reversibility and we see that the direction of motion is governed not by the slopes of the potentials but by the q dependence of the chemical specificities.[28]

Figure 6. Aratchet mechanism inspired by apaper on the bacterial flagellar ple, af ðÞ¼q arðÞ¼q bfðÞ¼q brðÞ¼q G for all q,the directionali- motor[52a] is formally similar to amechanical escapement (upper right hand ty is indeedcontrolled by the slope of the potentials. corner). Similar pictures have beengiven for many other molecular machines,including myosin moving on actin.[51] Only the solid arrows are shown for the mechanism in the figure by Xing et al.[52] where k and k on off 4.3. Enforcing the Constraints of Microscopic Reversibility are described as rates for composite conformational transition and proton association from the periplasm or dissociation to the cytoplasm, respectively. The situation is very,very different when the driving is mediat- The figureisatrompe l’oeil that leads the naïve readertothe conclusion that the slopeofthe potential dictates the directionofmotion and other ed by catalysis of achemical reaction such as hydrolysis of ATP thermodynamic properties. In fact, the slope does not dictate the direction (as in the F1 ATPase), or by transport of protons across amem- of rotation—thedirectionofmotion is determined by selection between the brane from high to low chemical potential (as in the bacterial pathway to the rightshownbythe solid arrows and the pathway to the left flagellar motor and the F portion of the ATPsynthase).[28] In indicatedbythe dottedarrows, aselection dictatedbythe q dependence of o the specificities for binding/release of protontothe cytoplasm/peri- these cases,the rate constants are constrained by the principle plasm.[42, 48] of microscopicreversibility.One constraint of microscopic reversibility is immediatelyevident from Equation (9): ly put, attempts to analyze the behavior of molecular machines af ðÞq bf ðÞq ¼ eXx ð9Þ in terms of mechanical devices without consideration of funda- arðÞq brðÞq mental thermodynamic principles is futile. Fortunately,the principle of microscopic reversibilityprovides asolid platform for any q.Byuse of Equation (2), asecond constraint can be from which to launch adetailed analysis of the mechanisms by derived from the picture involving two arbitrary values of which molecular machines carry out their functions. q—q0 and q00—shown in the expanded view in Figure 7b.Irre- When we carefully analyze the dynamics of the system in spectiveofwhether the system is at or away from thermody- light of microscopic reversibility,wefind that if the specificities namic equilibrium, the forwardand microscopically reversetra- 0 for the reactions are the same at q ¼ q and at q ¼ 0, the mac- jectories that begin and end at the exact same point must roscopically implausible, and deterministically impossible, path- have equal probabilities. way shownbythe dotted arrows is just as likelyasthe process "#ÀÁ"#ÀÁÀÁÀÁ shown by the solid arrows, and hence the machine does not, P q0 ! q00 P q0 q00 a q00 a q0 [28] 1 2 f r on average, rotate at all. Further,when the specificity for 0 00 0 00 0 00 ¼ 1 ð10Þ P ðÞq q P ðÞq ! q a ðÞq a ðÞq proton to/fromperiplasm is greater at q ¼ q0 than at q ¼ 0, 1 2 f r and the specificityfor protonto/from cytoplasm is greater at q ¼ 0than at q ¼ q0,the system undergoes, on average, coun- with an analogousequationholding for the b coefficients. terclockwise rotation! From Equation (10) and Equation (2) it is straightforwardto To better understand what governs the directionality of derive Equation (11): amolecular machine, consider the model shown in Figure 7a ÀÁ0 ÀÁ00 ÀÁ0 ÀÁ00 where the system undergoes switchingbetween two poten- af q ar q bf q br q 0 00 0 00 ¼ ¼ e½U1 ðÞq ÀU1ðÞq À½U2ðÞq ÀU2 ðÞq [28] a ðÞq0 a ðÞq00 b ðÞq0 b ðÞq00 tials, U1ðÞq and U2ðÞq . If the transitions between U1ðÞq and r f r f

U2ðÞq are caused by some external modulation, with, for exam- ð11Þ

ChemPhysChem 2016, 17,1719 –1741 www.chemphyschem.org 1728 2016 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Reviews for any pair q0 ; q00.From Equations (9) and (11) we can derive shownvertically as in Figure 7b rather than as acycleasin another relationship that will prove useful [Eq. (12)]:[49] Figure7a. Potential energy surfaces obtained by specifyingthe

ÂÃÀÁ ÀÁÂÃÀÁ ÀÁ energies UiðÞÆjDq=2 ; UiðÞÆjDq ; i ¼ 1; 2and transition-state a q0 þ b q0 a q00 þ b q00 f r r f energies eiðÞÆjDq=2 ; eiðÞÆj Dq ; i ¼ a; b,for integer j,with ½a ðÞþq0 b ðÞq0 ½a ðÞþq00 b ðÞq00 linear extrapolation between the specified energies, are shown r f f r ð12Þ ÂÃÀÁ0 ÂÃÀÁ00 1 þ s q eÀXx 1 þ s q in Figure 8b and 8c.For simplicity,wetake DU1þDU2 ¼ 0 00 e e ðÞ¼ÆjDq e ðÞ¼ÆjDq=2 e and e ðÞ¼ÆjDq e ðÞ¼ÆjDq=2 e*. ½1 þ sðÞq ½1 þ sðÞq eÀXx a b b a Let us comparetwo processes:one in whichthe system un- where we have introduced the chemical specificities dergoes chemical reaction from U1 to U2 at q=0, moves from bf ðÞq sðÞq ,whichare independentofthe energies U ðÞq and q= 0toq=+Dq/2 on U2 (powerstroke 1), undergoes chemi- arðÞq 1 brðÞq ÀX U ðÞq .Using Equation (9), we have ¼ sðÞq e x . cal reactionfrom U2 to U1 at + Dq/2, and then moves from + 2 af ðÞq Imposing the constraints of microscopic reversibility,Equa- Dq/2 to +Dq while on U1 (power stroke 2), completingafor- tions (9) and (11), on the rate constants ensures that any reac- ward step;and another in which the systemmovesfrom q =0 tion diffusion or kinetic model involving states and transitions to q =ÀDq/2 on U1 (un-power stroke 2?), undergoes chemical reactionfrom U to U at q= ÀDq/2, movesfrom ÀDq/2 to between them is consistent with an underlying model in 1 2 which over-damped motion occurs on asingle time-independ- ÀDq while on U2 (un-power stroke 1?), and undergoes chemi- ent energy surface.The constraint given in Equation (10) is the cal reaction from U2 to U1 at ÀDq,thus completing abackward discrete equivalent of the requirementthat the force field de- step. The ratio of the probability for abackward step and afor- rived as the gradient of the potentialbecurl free, that is, that ward step is given by Equation (13): rÂrUðÞ¼x; q 0. The transitions between the states occur P ðÞDq=2 Dq P ðÞ0 Dq=2 because of thermal noise as describedbyEquation (1) and r ¼ 1 2 Â P ðÞDq=2 ! Dq P ðÞ0 ! Dq=2 hence the machine functions as a“Brownian motor”. To better 1 2 ð13Þ illustrate this point, the ratchet mechanism shown in Figure 7a ½af ðÞþDq=2 brðDq=2Þ ½arðÞþ0 bfðÞ0 eXq is redrawn in Figure 8a where the a and b transitions are ½arðÞþDq=2 bf ðDq=2Þ ½afðÞþ0 brðÞ0

where we have used periodicity to deduce the identities

P1ðÞ¼ÀDq=2 0 P1ðÞDq=2 Dq and

P2ðÞÀDq ÀDq=2 ¼P2ðÞ0 Dq=2 .Using Equations (2), (9), (11), and (12) we find:[49]

ÀXx ÀXx ½1 þ sðÞDq=2 e ½1 þ sðÞ0 ðÞþe þ q vuc r ¼ eXq ¼ eXq ½1 þ sðÞ0 eÀXx ½1 þ sðÞDq=2 ðÞþ1 þ qeÀXx v |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} uc

r0 ð14Þ

where we identify q ¼ sðÞ0 sÀ1ðÞDq=2 ,and where

À1 ÀXx vuc ¼ ½s ðÞþDq=2 sðÞ0 e .The salient point of Equation (14) is that the ratio of forward to backward steps depends only on the chemical specificities andthe generalized thermodynamic

“forces”, Xx and Xq,and not on the energies of the states.

4.4. ACautionary Tale It is easy to become confused, however.Ifwetake simple, plausible, expressions for the rate coefficients

Xx=4 ÀXx=4 af ðÞ¼q bf ðÞ¼q e and arðÞ¼q brðÞ¼q e forall q,anas- signment that is clearly in agreement with constraint Equa- Figure 8. a) Schematic picture of how achemical process can drivedirected mechanical motion.The mechanism illustrated with the solid arrows in- tion (9), andwhere we blithelyassert that Xx parametrizes the volves two “powerstrokes”, downhill “slides” on the slopes of U1 and U2. deviation from equilibrium, we find mathematically that in the The mechanism with the dottedblue arrowslooks impossible from the limit Xx 0, we seem to have Equation (15): pointofview of macroscopic physics, but whenthe activationenergies for ÀX ÀðÞDU1 þDU2 x the chemicalprocesses are equal, e ¼ e*,the mechanism with dottedblue rincorrect ¼ e e ð15Þ arrows in whichthe motorsteps left is just as likely as the mechanism in- volvingthe solid blue arrows in which the motorstepsright. b) and c) Po- where tential energysurfaces for the ratchet mechanism in Figure7afor the cases e* < e and e < e*,respectively, where the most probable trajectories are DU1 þ DU2 ½U1ðÞÀDq=2 U1ðÞDq þ½U2ðÞÀ0 U2ðÞDq=2 .This shownbythe white dashed and solidcurves. result seemingly supportsthe power stroke modelsince

ChemPhysChem 2016, 17,1719 –1741 www.chemphyschem.org 1729 2016 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Reviews

DU1 þ DU2 is the energy dissipated in the two “power strokes” Acommonobjection to the use of Onsager–Machlup ther- in the mechanism of Figure 7a.Hill, Eisenberg, and Chen,[51] modynamic action theory for molecular machines is the claim based on asimilar picture, asserted that the maximum efficien- that Onsager’stheory is only valid near equilibrium in the cy for amolecular machine is h ¼ ðÞDU1þDU2 .However,the q linear regime. Indeed, Onsager and Machlup explicitly state max Xx independentassignment of the rate constants violates the that “The essential physicalassumption about the irreversible constraint imposed by Equation (11) and this assignmentisnot processes is that they are linear;that is, that the fluxes depend consistentwith amechanism for autonomous chemical driving. linearly on the forces that ”cause“ them”.[17] This situation, how- The efficiency for chemically drivenmotors is not controlled by ever,isthe case with all over-dampedsystems obeying Equa- [49] d~r ðÞDU1 þ DU2 butbythe chemical specificities, and in the tion (1) in which the linearity between the velocity, dt,and the limit that q eÀXx the efficiency approaches unity when force that causes it, ÀrUðÞ~r ,ismanifest. The required equilibri- [48] Xq ! Xx irrespectiveofthe value of ðÞDU1 þ DU2 . um is the mechanical equilibrium by which acceleration can be It is perhaps worthwhile to step back and recognizefrom ignoredfor trajectoriesofsingle molecules. Thereisnore-

Equation (13) possible limiting cases. When Xx is very large the quirementthat the overall system be near thermodynamic transition constants for the dashed red and green arrows can equilibrium. The results of our analysis, such as Equations (5a) be ignoredinfavor of those for the solidred and green and (5b), remain valid for large generalized thermodynamic arrows. It is also reasonable to take the approximationwhere “forces” Xq and Xx because the underlying process, described the transitions for the red arrows are kinetically blockedsuch at the single-molecule level by Equation (1), is in mechanical that the transition constantsfor the red arrows can be ignored equilibrium. in favor of those for the green arrows. It is not, however,rea- Although the powerstroke is irrelevant for the thermody- sonable to take both approximations, as is done implicitly in namics of chemically driven molecular machines, apower Figure 4b of Xing et al.,[52] since this leads to division by zero stroke can be, and often is, essential for light-driven molecular in Equation (13) for r. machines.Wecan understand this fundamental difference be- There is an unfortunate tendency in the literature for au- tween light-driven and thermally activated processes in the thors to incant the words“far from equilibrium” severaltimes context of binding of CO to myoglobin, an extremely well- and then proceed to assign rate constants ad libatum with no studied process. attentionwhatsoever to the fundamental physical principles such as microscopicreversibility that governthe dynamics of the system. It is not clear what, if any,contribution is made by 5. Microscopic Reversibility and the Physics of such unconstrained“theoretical” analysis. Ligand Binding to Proteins Asimilar point has been made with regard to hypothetical mechanisms for evolutionofhomochirality (mechanismfor the Molecular dynamics simulations have shownthat aCOmole- establishment of apreponderance of l or d isomersfrom arac- cule can arrive at the binding site of myoglobin by diffusing emic mixture) in biological systems where hypotheses un- through the bulk of the protein.[54] Any specific realization of grounded by microscopic reversibility were described, accu- this process is of course very unlikely.The microscopic reverse rately,ifunflatteringly,as“if pigs could fly” chemistry.[53] The of the process where CO leaves the binding site by diffusing problemwith regardtoflying pig proposalssuch as the power through the protein is also very unlikely,but the ratio of the stroke is that from amacroscopic perspective the power stroke probability forthe forwardand microscopic reverse of these seems to follow from common sense,and indeed the power rare trajectoriesisexactly the same as the ratio of the probabil- stroke is important for externally and optically driven motors. ities for any other,possibly far more likely,binding/unbinding Nevertheless, the assertion of the importance of apower pathway.This is the essence of microscopicreversibility reflect- stroke for molecular machines driven by catalysis of achemical ed in Equation (2), and which is stated by IUPAC as:[55] reactionisjust plain wrong. “In areversible reaction, the mechanism in one direction is exactly the reverseofthe mechanism in the otherdirection. This does not apply to reactions that begin with aphotochemi- 4.5. Motion on an Energy Landscape cal excitation.” The 2D energy landscapes derived from the model in Fig- Early experiments on dissociation of CO from myoglobin ure 7a are shown in Figure 7b for e*< e,and in Figure 7c for seemedtochallenge this fundamental principle. In the mecha- e < e*.The white dashedand solid lines are the pathsofleast nism for association,COdiffuses to the iron-containing binding thermodynamic action as calculated[48] from Onsager and site of the deoxy form, deoxyMb, of the protein and there in- Machlup’s theory.[17] In Onsager and Machlup’s derivationof teracts with the iron in the high-spin, out-of-plane state. The the thermodynamic action, the probability for any trajectory iron undergoes alocal configurationchange to the in-plane involves arather complicated pre-factor in addition to the ex- low-spin form, followed by aglobal conformational change to ponential of the energy differencebetween the initial and final the bound form MbCO. An essential feature is that the myo- points on the trajectory.The key to obtaining Equation (2) is globin conformational change starts locally and propagates recognizing that this pre-factorisdirection independentand globally,consistent with our experience in the macroscopic hence cancels in the ratio of the probability for any trajectory world in which waves propagate away from asource of excita- and its microscopic reverse.[24] tion as in water,see Figure 9.

ChemPhysChem 2016, 17,1719 –1741 www.chemphyschem.org 1730 2016 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Reviews

preceded by aglobal to local “unquake” followed by dissociation of CO. The critical point is that the experimentsofAnsari et al.[56] werecarriedout by flash photolysis with photochemi- cally induced desorption, adistinction that changes everything (Figure 10). The appearance of anon-reciprocal cycle—a situation where Figure 9. When aneutraldensity ball falls on aviscousliquid (a) waves prop- the dissociation of CO is not the microscopicreverseoftheas- agate outward from the ball (b,c,d) until finallythe liquid, with the ballrest- ing on it, is quiescent. If the ball is removed from the surface by some exter- sociation of CO—isofparamount importance for understand- nal means (a’), wavesonce againpropagate outward from where the ball ing the operation of amolecular machine. Such acycle is nec- had been (b’,c’,d’), until finallythe liquid is again quiescent. This backward essary for the generation of directed motion,and is at the mechanism is very different than the microscopic reverse mechanism (gray heart of the mechanism for all molecular machines. There are dashed arrows) for removal of the ball from the surfaceinwhich waves several excellent reviewsthat discussthe hydrodynamicsby spontaneously propagateinward toward the ball (dR,cR,bR)until finallythe energyofthe wave coalesces at the ball, propelling the ball awayfrom the which non-reciprocal shapechangeslead to directed surface. The external removal of the ball corresponds molecularly to photo- motion.[59,60] Such anon-reciprocal cycle can result either from [56] [57] dissociation, external changes to thermodynamic parameters (electric opticalorexternal driving, or from driving by catalysis of field strength, pressure, pH, or redox potential),ortocomputational disap- parition.[58] Fromthe macroscopic or even mesoscopic perspective, the pro- anon-equilibrium chemical reaction, but the designprinciples cess dR ! cR ! bR ! aR seems remarkablyunlikely,requiring as it does by which amolecular motor can use these two types of energytospontaneously concentrate from many degreesoffreedom to the energy are totally different. single degree of freedom of the ball. Even so, this is the most likely mechanism for thermally activateddissociation as requiredbythe principle of microscopic reversibility. 5.1. Seeingthe Light versus Feeling the Heat Unfortunately,there is astrong tendency in the literature to Frauenfelderand colleagues[56] studied the dissociation of analogize chemical driving with optical driving. Indeed, in pop- CO, findingthe mechanism to first involve dissociation of CO ular animationsofkinesin and myosin, which are widely avail- from the in-planeMbCO. Then, the iron undergoes alocal con- able on the web, the ATPhydrolysis step is denoted by aflash figuration change to the out-of-plane high-spin iron state, fol- of light. This analogy, however,isseriously misleading and lowed by aglobalchange to the deoxyMbform. Once again, tends to gloss over avery important symmetry difference be- the local to global progression is consistent with experience tween thermally activated transitions and optically induced based on removing an object from quiescent water.These re- transitions. sults led Frauenfelderand colleagues to argue that “Bindingor The immediately apparent difference between optically and dissociation of aligand at the heme iron causes aprotein chemically driven processes is quantitative—hydrolysis of quake.”However,although the local to global progression for amolecule of ATPunder physiological conditions can provide both bindingand dissociation is consistentwith macroscopic around20kBT energy whereas asingle photon of green light experience, it is not consistentwith microscopic reversibility, can provide around 100 kBT of energy.Aseemingly natural as- according to which the thermal dissociation of CO mustbe sumption is that achemically driven molecular machine can approachthe behavior of asimilaroptically driven molecular machine in the limit that the reaction being catalyzed is very, very far from equilibrium. This natural assumption, however,is wrong. The essential difference between light-driven andthermally activated processesisbased on afundamentalsymmetry difference between the Bose–Einsteinrelations[61] and microscopic reversibility.According to theBose–Einstein relations, the transition coefficient for absorption is identicaltothe transition coefficient for stimulated emissionifthe degeneracies of the ground and excited states are the same. In contrast, the microscopic reversibility [Equation (2)] that holds for all thermally activated processes requires that the ratio of forward and reverse transition coefficients be proportional to the exponential of the energy differencebetween starting and endingstates. Thus, the description of achemicalprocess as driving areaction between a“groundstate” N-dimensional energy surfaceand an “excited state” N-dimensional energy surface[62] is problem- Figure 10. Illustration of the difference between a) equilibrium, thermally ac- atic and can lead to seriously incorrect conclusions. Instead, tivated, dissociationinwhich the most probable path for dissociation of CO the effect of substrate bindingand catalysis of achemical reac- is the microscopic reverse of that for association of CO and b) photolysisin- duced dissociation of CO in which the dissociation is not the microscopic re- tion must be described in terms of motion on asingle (N+ 1)- verse of association. dimensional energy surface (see Figure 8a–c). The conclusions

ChemPhysChem 2016, 17,1719 –1741 www.chemphyschem.org 1731 2016 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Reviews based on these two very different pictures can be tremendous- ly different, and correspond to the differencebetween light-activated and thermally activated processes.[42,48] In aphotochemi- cal process, light absorption and emission is followed by anon-equilibrium relaxation of the conformational state of aprotein and controlled by the exothermicity,the reorganiza- tion energy, and leakage to the originalground state.[63] The resulting transitions are described by Ansarietal.[56] as “function- ally important motions”. The relaxation of the protein on either the ground- or excited-state surfaces can, in the case of flash photolysis, be well described as a“power stroke” that can result in mechanicalmotion. In contrast, the internal degrees of freedom of the protein remain in equilibrium at every instant during catalysis of even avery strongly exergonicreaction.[23] For example, in ATPhy- drolysis an ATPdiffusestothe active site of the enzyme (an ATPase), binds, and undergoes conversion to ADP+Pi at the active site, and then ADP and Pi dissociate and diffuse away. The reverse, in which ADP andPidiffuse to the activesite, bind, and undergo in situ conversion to ATP, which then disso- ciates anddiffuses away,results in ATPsynthesis. As far as the individual protein is concerned, bothofthese are equilibrium processes.The character of the motion of the protein is independentofthe chemical potentials of ATP, ADP,and Pi in the bulk. Any thermodynamicdisequilibrium in the bulk is mani- Figure 11. Schematic figure used to illustrate how input energy canbeused fest only in achange of the relative frequencies of trajectories to maintain anon-equilibrium steady state in which the relative concentra- leadingtohydrolysis versus those leadingtosynthesis, and tions in states 2and 0are not given by aBoltzmann expression. not to achange in the character of the protein motion. The chemicalpotential difference acts to impose apreferred direction of reactionbymass action.[32] It is very misleading to char- steady-state levels. If we take the barrier between states 0and acterizethe effect of ATPasdelivering “violent kicks”[64a] to the 1tobevery high, the approximate ratio of the steady-state enzyme that catalyzes its hydrolysis, and proposal of chemo- levels of states 2and 0can be written down by inspection as acousticwaves[64b] (in analogy with the photo-acoustic effect) Equation(17): as amechanism for enhanced diffusion of enzymes is ill-found- ed. P2 y0k0 !1 1k1!2 y01 * * ¼ eDG12 eDG0* 1* ð17Þ P k y k y 0 ss 2!1 1 1* !0* 0 1 0 5.2. Why the PowerStroke Model is Correct for Light- jjDG00* Driven Processes and Wrong for Chemically Driven In light that is very bright at both the frequency n0 ¼ h jjDG11* Processes and n1 ¼ h ,weobtain from the Bose–Einstein relations that y01 1ifthe degeneraciesofstates 0and 0* and 1and Considerasimple model for the light-driven pumpingofpro- y10 1* are the same. In this case, the optically driven steady-state tons by bacterio-rhodopsin.[65] ratio between states 2and 0isgiven by Equation (18): When used to describe light-driven processes, adrawing such as Figure 11 represents aprocess in which external P2 eDG12þDG0* 1* ð18Þ energy from light is used to cause transitions between P a“ground-state”energy surface(shown in blue) and an “excit- 0 ss; opt ed-state” energy surface(shown in red). The transitions on either the ground or excited energy surfacesare thermally acti- The transitions 0* ! 1* and1! 2can be very reasonably vated processes where the ratios of the forward and backward described as “powerstrokes” and theenergy released in these rate constants have the standard interpretations [Eq. (16)]: powerstrokes is necessary to allow light energy to maintain anon-equilibrium steadystate. k0!1 k1!2 k0 !1 We can contrast this with the equilibrium case where there ¼ eDG01 ; ¼ eDG12 ;and * * ¼ eDG0* 1* ð16Þ k1!0 k2!1 k1* !0* is no light and where all transitions occur on the ground-state surface, in which case Equation (19) applies: It is asimple matter to calculate the concentrations(or prob- * * abilities) of the five “states” (0,1,2,0 ,1 )asafunctionoftime P2 k0!1k1!2 DG þDG DG ¼ e 12 01 ¼ e 02 ð19Þ given arbitrary initial conditions, and simpler still to calculate P0 eq k2!1k1!0

ChemPhysChem 2016, 17,1719 –1741 www.chemphyschem.org 1732 2016 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Reviews

The ratio at steady state in bright light is changed from that When these relations are inserted into the expression for the at equilibrium by afactor eDG0* 1* ÀDG01.This condition holds also steady-state ratio between state 2and 0, we find Equa- if the optical densities for ablack-bodyradiator at the same tion (24): temperature as the molecular machine are used in the Bose– ÀÁ [61] [48] ÀDmATP Einstein relations, as shown by Astumian. P2 y0k0 !1 1k1!2 1 þ s0e ðÞ1 þ s1 * * ¼ eDG02 P k y k ÀDmATP What happens, though, if the transitions are caused, not by 0 ss;ATP 2!1 1 1* !0* 0 ðÞ1 þ s1e ðÞ1 þ s0 light, but by the binding of aligand,L.Inthis case, the vertical ð24Þ transitions must be interpreted as thermally activated processes, where we can take the red energy surfacetobethat for bf;0 bf;1 where s0 ¼ and s1 ¼ . the ligand bound protein and the blue surface for the un- ar;0 ar;1 bound form, and where Equation (20) applies: Both ATPhydrolysis and light absorption can support anon- equilibrium steady state, but there is ahuge difference in the y0 cLk0!0 y1 cLk1!1 designprinciple. For alight-driven process, it is necessary that ¼ * ¼ c eDG00* ; ¼ * ¼ c eDG11* ð20Þ L L DG DG ,whereasfor achemically driven process this is 0 k0* !0 1 k1* !1 0* 1* 01 not necessary at all. What is necessary is that s1 s0,that is, the kinetic specificities be suchthat state 0isspecific for ATP(i.e., It is important to note that although the figure does not ATPbinding/release is fast and ADP and Pi binding/release is change depending on whether the transitionsbetween the slow in state 0) and state 1isspecific for ADP and Pi (i.e.,ATP surfaces are caused by light or by binding of aligand,the in- binding/release is slow and ADP and Pi binding/release is fast terpretation must change entirely.This understanding is very in state 1). important when interpreting diagrams such as those used to describe, for example, Marcus theory for electron transfer reactions.[66] As noted, an optically driven system must be inter- 5.3. Spectroscopy and Dynamical Contributions to Enzyme preted as aprocess in which transitions are driven between Catalysis two separate energy surfaces. However,for asystem involving Because of the fundamental differenceinthe symmetry con- ligand binding/dissociation(or any system driven by internal straintsfor optical versus thermaltransitions, it is imperative to processes),the transitions between two effectively 1D energy be especially careful in interpreting results based on spectro- surfaces are mediated by thermally activated binding, and dis- scopic investigations of proteins.Inrecent work, several sociationcan be described as diffusive motion on asingle 2D groups have shown that light can induce slightly under- energy surfacebyEquation (1). damped collective motion in aprotein,[67,68] lysozyme, where Now,when we calculate the steady state in the presence of the frequency of vibration is influenced by whether the protein ligand,wefind Equation (21): is or is not bound by aligand. These very interesting results, unfortunately,have led to over-the-top descriptions in press re- P2 y0k0 !1 1k1!2 * * ¼ eDG00* ÀDG11* þDG12þDG0* 1* ¼ eDG02 leases from the two groups’respective home institutions in P k y k 0 ss;L 2!1 1 1* !0* 0 which proteins are described as “ringinglike bells” while “play- ð21Þ ing the symphony of life”. High frequency (1–10 THz) modes of motion certainly exist in proteins and these may well involve In other words, the steady state when the transitions be- collective dynamics of many degrees of freedom, and may be tween the two surfacesare mediatedbyasingle thermally ac- somewhat under-damped when excited by light. In the experi- tivated process (binding/dissociation of aligand)isjust the ments,the damping frequency wasfound to be about half the equilibrium state. This should not be asurprise, butitiscom- value of the oscillator frequency,around 2THz. These results forting that the math works out correctly. seem to support earlier proposalsthat there may exist “pro- This being the case, how can catalysis of achemical reaction tected”degreesoffreedom that store energy and that are im- such as ATPhydrolysis support anon-equilibrium steady state? portant for catalysis and for chemo-mechanical energy cou- Without specifying the detailsofthe chemical mechanism, let pling.[69] To betterunderstandwhether this proposal is reasonable, us consider y0 ¼ðaf;0 þ br;0Þ, y1 ¼ðaf;1 þ br;1Þ, let us consider the implications of the recentspectroscopic re- 0 ¼ðbf;0 þ ar;0Þ,and 1 ¼ðbf;1 þ ar;1Þ where,bymicroscopic reversibility Equations (22) and (23) apply: sults on proteins in the context of amolecular dynamicsinves- tigation of asmall molecule molecular machine, amolecular rotor (diethyl sulfide) on agold surfacedrivenbyaterahertz af;0bf;0 af;1bf;1 Dm ¼ ¼ e ATP ð22Þ oscillating electric field applied normal to the surface.[70] This ar;0br;0 ar;1br;1 work wasinspired by experimental studies of thioether molecular rotors.[71,72] The THz ac field causes rapid (gigahertz) direc- and tional rotation of the diethyl sulfide as assessed by molecular dynamic simulations, where the directionality is governed by af;0ar;1 br;0bf;1 ¼ ¼ eDG01þDG0* 1* ð23Þ the chirality of the diethyl sulfide. The frequency response was af;1ar;0 bf;0br;1 fit with asimple Brownian oscillator model coupled with

ChemPhysChem 2016, 17,1719 –1741 www.chemphyschem.org 1733 2016 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Reviews aparametric resonator.Alibrational frequency[70] of 2.4 THz, with adamping frequency of 1.2 THzwas used to fit the molecular dynamics simulationofthe high-frequency rotationin- duced by the driving. These values are very close to the oscillator and damping frequencies found for lysozyme.While there is certainly some inertia, as evident from afew rotations (<2) being completed after the field was turned off, the rotation was completely damped within apicosecond (see also ref.[73] for asimilar discussion on the possible role of dynamics on enzymecatalysis). The most appropriate onomatopoeia for the tintinnabulation of this particularbell would be, perhaps, “thunk”, not to be confused with the past tenseofthink, an action with which the description of aprotein as aringing bell has no relationship. The source of the damping for diethyl sulfide on the gold surfaceisdielectric friction resulting from the interaction between the electrons of the diethyl sulfide with those of the gold atoms that make up the surface. Since electrons and other charged particles are ubiquitous in molecules, the dielec- tric friction probably sets an insurmountable fundamentallimit for the damping coefficient. Further,asnoted above, the catalysis of even ahighly exergonic reaction is, from the perspective of an individual protein molecule, amechanical equilibrium process. Olsson,Parson, and Warshel[74] and Kamerlin and Warshel[75] have reviewed the literature searching for evidencefor “dy- namicalcontributionstoenzyme catalysis” and found that within all of the reasonably defined dynamicaleffects proposed by various groups,none contributes to catalysis. Further, there is afundamental reason—microscopic reversibility—that no enzyme carrying out its functioninthe ground state can exploit dynamical effects to store and harnessenergy.This recent conclusion is consistent with that of Stackhouse et al.[76] who summarize the resultsoftheir experiments in the abstract to their 1985 paper,“no evidence was found to support asig- Figure 12. Trianglereaction with an external load Xq and:a)noexternal nificant contribution to the rate of catalysis by dynamic funnel- driving (ND); b) optical driving (OD);and c) chemical driving (CD).The ratio, ing of vibrational energy within the protein molecule”. r,isthe probability of acounterclockwise cycle divided by the probability of aclockwise cycle, andiscalculated as the ratio of the product of the counterclockwise rates divided by the product of the clockwise rates. In very 6. Microscopic Reversibility and Motor Cycles bright light, the optical transition coefficients obeythe simple relation wAE wEA and wBE wEB.Incontrast, the ratio of each forward and reverse Many modelsfor molecular machines involvekinetic cycles. rate constantfor each pair of processes that are microscopic reverses of one One of the simplest non-trivialexamples is the three-state tri- another must be proportional to the exponentialofthe free-energy difference of the states they connect[see Eq. (2)]. angle reaction discussed by Onsager in his paper on reciprocal relations in chemistry.[14] This “triangle reaction” is shown in Figure 12 for three cases, a) no driving;b)opticaldriving;and Onsager.[16] For an optically driven molecular machine, Lehn[77] c) chemical driving, where an external load tends to drive proposed atriangle mechanism for optical conversion between counterclockwise cycling when Xq 0. The ratio of the probabil- the anti and syn forms of imines. In the ground state, the ities for completion of acounterclockwise and clockwise cycle imine is aromatic and hence planar, but in the excited state is equal and the ratio of the products of the net clockwise the molecule cannot be planar.Ifthe free energies of the anti rates and net counterclockwise rates are shown forthe three and syn are different (DGAB6¼0) andthe molecule is on asur- cases of no driving(ND), optical driving (OD), and chemical face, constant illumination will result in steady-state cycling. In

DGAB Xq driving (CD). this case, rOD ¼ e e and the optically driven transitions An equilibrium, no driving,realization of the trianglereac- AQEand BQEsustain anon-equilibrium steady state where tion is the isomerization reactionbetween 1-butene, cis-2- clockwise or counterclockwise cycling occurs when Xq ¼ 0, de- [23] Xq butene, and trans-2-butene. In this case, r0;ND ¼ e (with pending on whether DGAB is positive or negative.

Xq ¼ 0) and the probability for aclockwise cycle is the same as There are many realizationsofchemically driventriangle re- the probability for acounterclockwise cycle as emphasized by actions including enzyme-catalyzedisomerization,[78] ion chan-

ChemPhysChem 2016, 17,1719 –1741 www.chemphyschem.org 1734 2016 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Reviews nels,[79] muscle contraction,[80] and the bacterial flagellar fit parameters are very interesting with regard to understand-

ÀXx ing how the structure of the motor determines its function. ðÞ1þsAe ðÞ1þsB [81] Xq motor. For all of these cases, rCD ¼ ÀX e where ðÞ1þsBe x ðÞ1þsA The assertion that this result suggeststhat the motor operates s ¼ bf;i ; i ¼ A; B. Energyfrom the catalyzed non-equilibrium by a“power stroke”mechanism is, however, a non sequitor. i ar;i The fraction of the step, Dq,taken in any transition has noth- (X 6¼0) chemical reaction sustains anon-equilibrium steady x ing to do with whether apower stroke is important for the state if, and only if, s 6¼s .The direction of cycling, and the ex- A B mechanism. The further assertion that the fact that most of ternal force necessary to cause the cyclingrate to be zero, is

br;A br;B the mechanical motion occurs in the transition AQBindicates determined by the chemical specificities, sA ¼ and sB ¼ , af;A af;B that this transition “dissipates most of the availablefree but is independent of DG .This recognition is the basis of the AB energy” is wrong, as can be recognized from the fact that the assertion that the power stroke is irrelevant for chemically experimentally observed angular velocityversus applied driven molecular machines.[48] torque can be fit whether DGAB is positive,negative, or zero. It is not necessary that the load against which achemically driven machine works be less than DG as claimed by 6.1. Deconstructing the “Power Stroke” AB Howard.[82] Further,the concept of identifying afraction of the The rate of cycling for achemically driven motor is given by dissipation with asingle transition is misguided as pointed out by Hill andEisenberg who conclusively demonstrated that for Equation (6), Jq ¼ CðÞ1 À rCD .The kinetic pre-factor C is acom- plicated function of all of the rate constants, but it is positive chemically driven motors,asopposed to light-driven motors, and definite and so the stopping force and direction of cycling free energy dissipation is aproperty of the cycle as awhole and cannot be assignedtoone or afew transitions within the are determinedsolely by rCD.The pre-factor C depends kinetically on the appliedtorque, T,and is important for fitting cycle.[83] torque versusangular velocity data, but has no bearingonthe relevance of apower stroke forthe mechanism. Berry and Berg[81] showedthat the shape of the torque–angular velocitycurve observed for the bacterial flagellar rotor (similartothe shape in Figure3with a= 0) is bestfit with rate constantsinwhich mostofthe physical rotation occurs in the 7. Dissipation-Driven Assembly of Non- transition AQB. An energy diagram for this transition is shown Equilibrium Structures in Figure 13 7.1. Enzymes Use Input EnergytoDrive aReaction Away The resultsfrom Berryand Berg[81] are consistent with d 1 from Equilibrium and a 0, that is, with asituation in which most of the motion occurs in the BQAtransition, and where most of the It is often claimed in the literature that having the product of counterclockwise rate constantsequal the product of clock- torque dependence is absorbed in the rate constant kAB.These wise rate constants(kAEkEBkBA ¼ kBEkEAkAB for the cycle in Fig-

ure 12a) is sufficient to guarantee that the flux Jq is zero. This is not true in the presence of external fluctuations that causethe rate constants to depend on time. If, for 0 z yðÞt 0 Àz yðÞt example, kAE ¼ kAEe and kEB ¼ kEBe the condition

kAEkEBkBA ¼ kBEkEAkAB holds at every instant and yet the system undergoes clockwise flux. The rate of cycling is dependent on the detailsofyðÞt .Indeed, many molecular systems, both biological and non-biological, can harvest energy from oscillating,[84] or even noisy,[85a,b] externalperturbations to generate order and to do work, therebydriving asystem away from equilibrium.[86] Asimple illustration is shown in Figure 14 where oscillation of the free energy of an intermediate state and akinetic barrier for aMichaelis–Menten enzyme can causethe enzyme to drive the catalyzedreactionaway from equilibrium.[86] Figure 13. Anatomyofa“powerstroke”. In clockwise cycling, the motor The rate constants are given by the expression moves through the transition B ! A. Seemingly, the energy DG provides AB k ¼ k0ezi yðÞt ; i ¼Æ1; 2, where z are parameters of the enzyme the power for the power strokeand according to Howard[82] allows the i i i motortodowork against an applied force or applied torqueprovided such as activation volumewhen the externalperturbation, [51] DGAB Xq,with amaximum efficiency of hmax ¼ DGAB=Xx.This analysis is yðÞt ,ispressure;ordipole momentofactivation when the ex- correct for an optically driven machine, but is totally wrongfor achemically ternalperturbation, yðÞt is electric field strength, etc. driven motor.Inthe thermodynamic control limit,irrespective of whether For high-frequency externally imposed oscillations, the DG is positive, negative, or zero, achemicallydrivenmotor can do work AB “steady-state” concentration gradientisgiven by Equa- against an applied forceorapplied torqueprovided Xx Xq,and the maximum efficiency is bounded only by unity. tion (25):[86]

ChemPhysChem 2016, 17,1719 –1741 www.chemphyschem.org 1735 2016 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Reviews

Figure 15. Kinetic mechanism for an enzyme, describing the effect of externally driven Poisson fluctuations between two states, Æy.The fluctuations can drive the reaction S Ð Paway from equilibrium.[57]

These and similar ideas on the effects of externalperturba- Figure 14. Kinetic barrier model for an enzyme.Under the influence of an tions[87,88] have led to recent advances in the design and syn- external driving, yðÞt ,which changes the energy of the intermediate ES and thesis of molecular machines, including those that use dissipa- the kinetic barriers (transition states) the enzyme can drivethe reaction SQP tion of energy to achieve self-assembly.[89,90] One system,in awayfrom rather than towards equilibrium. particular, on which we shall focus, has been termed asynthetic molecular pump, where externally driven oscillations of the redox potential leads to formation and maintenance of ½P ez1 yðÞt Á ez2yðÞt ¼ K ð25Þ astrongly disfavored non-equilibrium structure.[10,91] ½S eq SS ezÀ1 yðÞt Á ezÀ2yðÞt

7.2. The Demon is in the Details where the over-barindicates atime average, and where K eq is In Figure 16 we consider aring molecule [cyclobis(paraquat-p- the time-independent equilibrium constant for conversion of phenylene) (CBPQT)] interactingwith adumbbell-shaped rod substrate Stoproduct P. In Figure 14, z1 ¼Àz2 ¼ z,and molecule, abistable [2]pseudorotaxane with abipydidinium zÀ1 ¼ zÀ2 ¼ 0, so the steady-state concentration ratio is differ- recognition site, a2-isopropyphenyl (IPP), shown in green, as 2 ent from the equilibrium ratio by afactor ðÞezyðÞt .For the case asteric barrier, with one end capped by charged 3,5-dimethyl + shown (and in generalwhen z1 þ z2 þ zÀ1 þ zÀ2 ¼ 0), the over- pyridinium (PY )and the other end capped by abulky stopper all equilibrium constant for the catalyzed reaction, K eq,isinde- group (showninblack on the right hand side of the dumbbell) pendentofthe external driving parameter yðÞt .The input elec- that provides an insurmountable barriertothe CBPQTring. tric energy is Pin ¼ yðÞt d½ES =dt and for large, high-frequency The IPP is connected through atriazoletoaring collecting oli- perturbation, the efficiency can approach 100%.[86] gomethylene chain that is terminated by the bulky stopper An important point to note is that the enzyme itself pro- group. The ring collecting chain can, in principle, be occupied vides amechanismfor pumping energy into the system—the by several of the CBPQT rings, but thiswould be avery high equilibriumconstantofthe catalyzed reaction itself, K ¼ k1k2 energyand lowentropy structure since even one ring has eq kÀ1kÀ2 does not depend on y.The conclusioncan be dramatically ahigher energy (by DGout)onthe ring than in the bulk. emphasized by considering asituation with mS ¼ mP in the Under either reducing or oxidizing conditions, only avery presence of ahigh-frequency oscillating or fluctuating pertur- small fraction of the collecting chains will be occupiedbyeven bation yðÞt but without the enzyme. Because there is no asingle ring, and the ratio of occupied(assembled, [A]) to un- mechanism to absorb energy from yðÞt ,the system is in over- occupied(disassem bled [D]) forms is given by the equilibrium ½A all thermodynamic equilibrium. Then, when we add asmall expression ¼ eÀDGout .However,repetitively changing be- ½D eq amount of the enzyme—a catalyst—the reaction spontaneous- tween oxidizing and reducing conditions pumps aring onto ly moves away from equilibrium until aconcentration ratio the collecting chain with almostunity certainty,and asecond given by Equation (25) is reached because the enzymeabsorbs cycle can even place asecond CBPQT ring on the collecting and harnesses energy from yðÞt ,asource of energy that could chain to form ahighly non-equilibrium structure.[91] Energyis not be utilized in the absence of the enzyme, to attain and dissipated each time the redox potential is changed, and some maintain anon-equilibrium steady state. of this energy can be used for assembling non-equilibrium We can better understand the effect of an external fluctuat- structures. The mechanism canbeconsidered in analogy with ing perturbation in the context of asimple kinetic modelin the action of amolecular “demon”,[31,47,92,93] which selects which yðÞt is taken to fluctuate between its maximal (þy) apath that leads to an ordered structure over another path value given by the dashedorange line and minimal (Ày)value that leads to adisordered structure, as illustrated in Figure 17. given by the solid orange line (Figure 14) with aPoisson dis- External modulation of the redox potential allows the tributed lifetime, G,ateach value,[57] + y !G Ày. system to act as aSmoluchowski’senergy demon or energy G For sufficiently large G and y,the dominant mechanism is ratchet (Figure 18a) whereas catalysis of anon-equilibrium À À þ + S+ E QES QES QE +Pwithaneffective “equilibrium” con- chemicalreaction can poweraMaxwell’s information demon stant of k1k2 ðÞezy 2 (Figure15). or information ratchet (Figure 18b). kÀ1kÀ2

ChemPhysChem 2016, 17,1719 –1741 www.chemphyschem.org 1736 2016 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Reviews

Figure 18. Driving by a) an externalfluctuation that results in an energy ratchetmechanism;and by b) an energy-releasingcatalyzed reaction that results in an informationratchetmechanism.

be very slow.For the energy ratchet,Figure 18 athis can be calculated as the product of the “equilibrium”constantsalong Figure 16. Illustration of asynthetic molecular pump. The energy profiles for the path DQD*QI*QIQAtogive Equation (26): aring underboth reducing(top) and oxidizing(bottom) conditionsare shown, where the overall free-energy difference between having aring on the collection site is the sameirrespectiveofwhether the ring is oxidized ½A DG þDG ÀDG ¼ K* K ¼ e * e out ð26Þ 4+ 2(C +) DI IA (CBPQT )orreduced (CBPQT ). ½D ss;en: ratchet

The transitions D*!I* in the reduced state and I ! Ainthe oxidizedstate are both power strokes with equilibrium constants greater than unity,sothe externalmodulation between reducing and oxidizing conditions supports anon-equilibrium steady state in which it is more likely for the collecting chain

to be occupied than not despite the unfavorable DGout. The power strokesfor this pump are absolutely essential. Absent the increaseinthe energy of the complex between the ring and recognition site in the oxidized state relative to the reduced state the mechanism fails. Similar ideas have been used in the design of light-driven motors,[89,94,95] where the power stroke is also essential. When consideringfrom atheo- retical standpoint how to turn these mechanismsinto autono- Figure 17. Illustration of amoleculardemon that selectively shepherds mous chemically driven motors,itsoon became apparent that, aring in an intermediate statetoassemble rather than disassembleeven thoughthe assembled state is higher in energy.The demon can operate because the principle of microscopic reversibility constrains either as ablind energy ratchet(Smoluchowski demon), raising and lowering the transition constants of the exergonic catalyzed reaction, the energies of states and barriers irrespectiveofthe state of the molecule, apower stroke is not sufficient to allow the energy released by or as asighted informationratchet(Maxwell demon), raising and lowering the reaction to drive pumping.[10] Even more surprisingly,per- barriersdepending on the state of the molecule. The design principlesnec- essary for synthetic implementation of the two types of “demons” are very haps, the powerstroke is not even necessary.This can be seen different.[10,49] by calculating the product of the ratios of the forward and reverse rates for the path DQD*QI*QIQAinthe mechanism shown in Figure 18btoobtain Equation (27):

The steady-state ratio [A]/[D] for both mechanisms can be ÀÁÀÁ ½A a þ b a þ b easily calculated, and the task becomes particularly simple ¼ K* K ÀÁf;D r;D ÀÁr;I f;I ð27Þ ½D DI IA when the transitions denoted by the red arrows are taken to ss;inf: ratchet ar;D þ bf;D af;I þ br;I

ChemPhysChem 2016, 17,1719 –1741 www.chemphyschem.org 1737 2016 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Reviews

By applying the constraints of microscopic reversibility

af;ibf;i ¼ eXx ; i ¼ D; I; A ð28Þ ar;ibr;i and a a b b K a a b b K f;D r;I ¼ r;D f;I ¼ DI ; f;I r;A ¼ r;I f;A ¼ IA ð29Þ ar;Daf;I bf;Dbr;I K DI* ar;Iaf;A bf;Ibr;A K IA* to Equation (27), we obtain Equation (30): Figure 19. Akinetic lattice model to describe the mechanism of using ÀXx ½A ðÞ1 þ sDe ðÞ1 þ sI energyreleased by catalysis of achemical reaction to pump aring from the ¼ eÀDGout ð30Þ ÀXx bulk onto ahigh-energycollecting site. The greenzigzag path denotes the ½D ss;inf:ratchet ðÞ1 þ sIe ðÞ1 þ sD desired mechanism for coupling. where s ¼ br;D and s ¼ br;I . D af;D I af;I The essential difference between an energy ratchet, which The presenceorabsence of apowerstroke is, simply put, ir- describes the effect of external and opticaldriving, and an in- relevant in the information ratchet mechanism by which all formation ratchet, which describes the operation of an autono- chemically driven molecular motors function. The essential re- mous chemically driven molecular machine, is microscopicre- quirementisseen to be amechanism for gating the catalysis versibility.The constraint Equation (29) can be rewritten as: such that reaction with substrate is fast and reactionwith K a a K b b K a a K b b product slow in one state of the mechanical cycle, and reac- DI r;D f;I ¼ DI f;D r;I ¼ IA r;I f;A ¼ IA f;I r;A ¼ 1 ð31Þ tion with substrateslow and reactionwith product fast in adif- K DI* af;Dar;I K DI* br;Dbf;I K IA* af;Iar;A K IA* br;Ibf;A ferent state of the mechanical cycle. This is achieved in Fig- ure 18bwhen sD 1 sI.This gatedspecificityassures that Equation (31) represents the discrete equivalent of the curl- the mechanicaland chemical steps are interleaved with one free condition rÂrUðÞ¼q; x 0for the gradient of the under- another,acondition that, combined with Xx 0for the cata- lying potential. From the point of view of asingle molecular lyzed reaction, is both necessary and sufficient to assure direc- machine, chemo-mechanical coupling is an equilibrium process tional pumping of the rings onto the collecting chain. in which the internal degrees of freedom of the protein or The energy ratchet is apower stroke mechanism—if DG* macromolecule remaininequilibriumatevery instant.The and DG are both zero, there is no enhancementofthe assem- coupled processes can be described as motion on asingle bled structure versus the disassembled structure. In contrast, time-independent multi-dimensional energy landscape. Ther- althoughthere may be powerstrokes,aninformation ratchet modynamic disequilibriuminthe bulk is manifestbychanges is not apower stroke mechanism.The enhancement of the as- in the frequency of carrying out forward and reversetrajecto- sembled structure versus the disassembled structure is inde- ries on this landscape (i.e.,bymassaction), but not by pendentofDG* and DG and is governedsolely by the specific- achange in the character of the motion of the machine from ities. We can better understand the constraints of microscopic the equilibrium case. The constraints of microscopic reversibili- reversibility by considering akinetic lattice model of the infor- ty ensure that kinetic and reaction diffusion modelsfor molec- mation ratchet as shown in Figure 19. ular motors and pumps are consistentwith an underlying pic- This mechanism is very similartothat described by Jencks ture of diffusive motion on asingle potential energy surfaceas for aCaATPase ion pump,[96] and was proposed explicitly by described by Equation(1). Astumian and Derenyi for understandingATP-driven ion pump- [45] ing by membrane ATPases. Asimilar picture was given very 8. Conclusion recentlybyAlhadeff and Warshel[97] for asodium/proton anti- porter.The mechanism in Figure 19 is an information ratchet In their 1981discussion of why “free energy transduction because the essential requirement is that the specificity of the cannotbelocalized at some crucial part of the enzymatic catalytic active site is controlled by the mechanical state of the cycle”,[83] Eisenberg and Hill observed that the disagreement in ring. Achemically driven information ratchet has recently been the fieldregardingthis point was only conceptual. Neverthe- synthesized by Leigh and colleagues.[98] The key design criteri- less, they argued, resolutionofthe controversy was important on for the autonomous chemically driven information ratchet because of the great amount of intellectual and experimental is that the transitions leading off of the coupled pathway be effort devoted to finding the crucial step (powerstroke?) and slow,indicated here by the red arrows. Given this, mass action energized state (pre-power stroke state?) in the overall mecha- resultingfrom the non-equilibrium chemical reaction driving nism. Hill and Eisenberg point out that the basic idea—that the system from top to bottom leads to asignificant enhance- there is acrucial step—is wrong and that free energy transduc- ment in the amount of the assembled product at the steady tion by chemically driven molecular machines must be under- state as shown in Equation (30). stood in terms of the cycle as awhole.

ChemPhysChem 2016, 17,1719 –1741 www.chemphyschem.org 1738 2016 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Reviews

Science has progressed to the point where it is possible to and product concentrationsisgoverned by barrierheights study biomolecular machines at the single-molecule level,[1–6] (transition states), not by welldepths(state free energies). and even to synthesize molecular machines.[7–13] The controver- 3. Thermal noise is essential to the operation of chemically sy highlighted by Hill and Eisenberg 35 years ago now has driven molecular machines, although the free energy barri- very practical implications—the chemical requirements for syn- ers of the relevant coupled allosteric changes dictate di- thetic design of an energized state, the escape from which is rectionality. driven by apower stroke, are very different than the require- 4. Enzymes, including chemically driven molecular machines, ments necessary to achieve gating of acatalyzed reaction de- operate at mechanical equilibrium. pending on the state of the catalyst. 5. Microscopic reversibility provides afundamental thermo- The necessity of the powerstroke for light-driven and exter- dynamic grounding for development of atheory of chemi- nally driven processes may explain why experiments often find cally driven molecular machines. that there are in fact strongly exergonic conformational 6. Light-driven processes are fundamentally different to ther- changes in the mechanisms of molecular machines. In the ear- mally activated processes, and analogybetween the two liest stages of evolution, the machines necessarytocarry out can lead to seriously incorrect conclusions. transport, pumping, and other essential tasks may well have 7. The presence or absence of apower stroke (an exergonic been driven by light, or by periodic changes in the thermody- conformational change) is irrelevant for determining the namic parameters of the system, for example, by circulation in direction and thermodynamics of chemically driven molec- atemperature gradientnear athermal vent. Subsequentevo- ular machines. lution developed allosteric gating mechanisms to allow for uti- 8. In contrast, optical and externally driven molecular ma- lization of energy from acatalyzed chemical reaction, but chines can (and often do) operate by apowerstroke there was never pressure to eliminate the vestigial “power mechanism. stroke” from the operation of the precursor molecular ma- 9. The best descriptionofthe overall mechanism by which chines that evolvedtobecome muscle, flagella, kinesin, an exergonic reactioniscoupled to drive mechanical dynein,FOF1 ATPsynthase, etc. Thus, experimentalists, when motion is “mass action”. seekingthe structuralorigins of the power stroke, do indeed 10. Acomplete understanding of the mechanism of amolecu- often find strongly exergonic conformational changes in the lar motorcan be gained only by viewing the chemo-me- mechanisms of many biomolecular motors to which they can chanical cycles holistically.Experiments on parts of the point. The pseudo-deterministic modelsinvoking thesepower cyclecan fill in the details, but over-interpretation often strokes are very attractive and persuasive from amacroscopic leads to erroneous conclusions, especially when analogies perspective. It is only by examining the mechanisms through are drawn with macroscopic elements. the lens of microscopicreversibility that we are abletosee that in fact the naive conclusions regarding the importance of the powerstroke for chemically driven motors and pumps are Acknowledgments not correct. The directionality and thermodynamicproperties are instead governed solelybychemical gating—theability of This work was supported by the National Science Foundation the molecular complextodiscriminate different mechanical (grantMCB-1243719), National Institute of Health (R01- states of the motor.One of the most important agendas for AI055926), and NationalCancer Institute (grant 1U19CA105010). development of autonomous chemically driven molecular ma- We also thank the University of Southern California High Per- chines is achieving better control of allosteric mechanismsby formance Computing and Communication Center (HPCC)for which gating can be achieved.[99–102] It can be expected that computational resources. the ongoing efforts devoted to the design and constructionof synthetic molecular pumps and motors will result both in better understanding of biomolecular machines, and inthe de- Keywords: Brownian motors · energy landscapes · information velopment of remarkable tools for organizing complex ratchet · microscopic reversibility · molecular machines matter,[103] for developing networked nanoswitches for catalysis,[104] and for harnessing the power of molecule-by-molecule [1] M. Schliwa, G. Woehlke, Nature 2003, 422,759–765. assembly.[105] [2] R. D. Vale, R. A. Milligan, Science 2000, 288,88–95. [3] P. D. Boyer, Annu.Rev.Biochem. 1997, 66,717 –749. [4] H. C. Berg, Annu.Rev.Biochem. 2003, 72,19–54. [5] P. B. Moore, Annu.Rev.Biophys. 2012, 41,1–19. [6] C. Bustamante, W. Cheng,Y.X.Mejia, Cell 2011, 144,480 –497. Physical Perspectives on Molecular Machines [7] J. M. Abendroth, O. S. Bushuyev, P. S. Weiss, C. J. Barrett, ACS Nano 2015, 9,7746–7768. [8] a) A. Coskun, M. Banaszak, R. D. Astumian, J. F. Stoddart, B. A. Grzybow- 1. Molecular machines can be understood in terms of their ski, Chem.Soc. Rev. 2012, 41,19–30;b)R.D.Astumian, Sci. Am. 2001, free-energy landscape, not by analogy with macroscopic 285,56–64. mechanical devices. [9] a) S. Erbas-Cakmak, D. A. Leigh, C. T. McTernan, A. L. Nussbaumer, Chem. Rev. 2015, 115,10081–10206; b) E. R. Kay,D.A.Leigh, F. Zerbet- 2. Directionality of chemically driven molecular machines op- to, Angew.Chem. Int. Ed. 2007, 46,72–191; Angew.Chem. 2007, 119, erating under the thermodynamic constraints of reactant 72–196.

ChemPhysChem 2016, 17,1719 –1741 www.chemphyschem.org 1739 2016 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Reviews

[10] C. Cheng,P.R.McGonigal, J. F. Stoddart, R. D. Astumian, ACS Nano [59] E. Lauga, Soft Matter 2011, 7,3060 –3065. 2015, 9,8672–8688. [60] T. Sakaue, R. Kapral, A. S. Mikhailov, Eur.Phys. J. B 2010, 75,381 –387. [11] S. Sengupta, M. E. Ibele, A. Sen, Angew.Chem.Int. Ed. 2012, 51,8434 – [61] A. Einstein,P.Ehrenfest, Z. Phys. 1923, 19,301 –306. 8445; Angew.Chem. 2012, 124,8560 –8571. [62] X. Bai, P. G. Wolynes, J. Chem. Phys. 2015, 143,165101. [12] B. L. Feringa, J. Org. Chem. 2007, 72,6635 –6652. [63] A. Warshel,D.W.Schlosser, Proc. Natl. Acad.Sci. USA 1981, 78,5564 – [13] G. S. Kottas, L. I. Clarke, A. Dominik Horinek, J. Michl, Chem. Rev. 2005, 5568. 105,1281 –1376. [64] a) J. Liphardt, Nat. Phys. 2012, 8,638 –639;b)C.Riedel, R. Gabizon, [14] F. Julicher, A. Ajdari, J. Prost, Rev.Mod.Phys. 1997, 69,1269 –1281. C. A. M. Wilson,K.Hamadani, K. Tsekouras, S. Marqusee, S. PressØ, C. [15] R. D. Astumian, Phys. Chem. Chem. Phys. 2007, 9,5067 –5083. Bustamante, Nature 2015, 517,227 –230. [16] L. Onsager, Phys. Rev. 1931, 37,405 –426. [65] A. Warshel, Methods Enzymol. 1986, 127,578–587. [17] L. Onsager,S.Machlup, Phys. Rev. 1953, 91,1505 –1512. [66] R. A. Marcus, Rev.Mod. Phys. 1993, 65,599–610. [18] E. M. Purcell, Am. J. Phys. 1977, 45,3–11. [67] D. A. Turton, H. M. Senn, T. Harwood, A. J. Lapthorn, E. M. Ellis, K. [19] R. D. Astumian, Science 1997, 276,917 –922. Wynne, Nat. Commun. 2014, 5,3999. [20] R. D. Astumian, P. Hänggi, Phys. Today 2002, 55,33–39. [68] G. Acbas, K. A. Niessen, E. H. Snell, A. G. Markelz, Nat. Commun. 2014, [21] P. Reimann, Phys. Rep. 2002, 361,57–265. 5,3076. [22] S. Chandrasekhar, Rev.Mod.Phys. 1949, 3,114–126. [69] J. A. McCammon, P. G. Wolynes, M. Karplus, Biochemistry 1979, 18, [23] R. D. Astumian, Nat. Nanotechnol. 2012, 7,684–688. 927 –942. [24] M. Bier,I.DerØnyi, M. Kostur,R.D.Astumian, Phys. Rev.E1999, 59, [70] J. Neumann, K. E. Gottschalk, R. D. Astumian, ACS Nano 2012, 6,5242 – 6422 –6432. 5248. [25] R. D. Astumian, Am. J. Phys. 2006, 74,683–688. [71] H. L. Tierney,C.J.Murphy, A. D. Jewell, A. E. Baber,E.V.Iski, H. Y. Kho- [26] R. D. Astumian, M. Bier, Phys. Rev.Lett. 1994, 72,1766 –1769. daverdian, A. F. McGuire, N. Klebanov,E.C.H.Sykes, Nat. Nanotechnol. [27] M. Magnasco, Phys. Rev.Lett. 1994, 72,2656 –2659. 2011, 6,625–629. [28] R. D. Astumian, M. Bier, Biophys. J. 1996, 70,637–653. [72] J. Michl, E. C. H. Sykes, ACS Nano 2009, 3,1042 –1048. [29] D. Keller,C.Bustamante, Biophys. J. 2000, 78,541 –556. [73] A. V. Pisliakov,J.Cao, S. C. L. Kamerlin, A. Warshel, Proc. Natl. Acad. Sci. [30] S. Mukherjee, A. Warshel, Proc. Natl.Acad. Sci. USA 2011, 108,20550 – USA 2009, 106,17359 –17364. 20555. [74] M. H. M. Olsson, A. William, W. Parson, A. Warshel, Chem.Rev. 2006, [31] R. D. Astumian, I. DerØnyi, Eur.Biophys. J. 1998, 27,474 –489. 106,1737 –1756. [32] E. O. Voit, H. A. Martens, S. W. Omholt, PLoS Comput. Biol. 2015, 11, [75] S. C. L. Kamerlin, A. Warshel, Proteins Struct. Funct. Bioinf. 2010, 78, e1004012. 1339 –1375. [33] S. Mukherjee, R. P. Bora,A.Warshel, Q. Rev.Biophys. 2015, 48,395 – [76] J. Stackhouse,K.P.Nambiar, J. J. Burbaum,D.M.Stauffer,S.A.Benner, 403. J. Am. Chem. Soc. 1985, 107,2757–2763. [34] S. Mukherjee, A. Warshel, Proc. Natl.Acad.Sci.USA 2012, 109,14876 – [77] J.-M. Lehn, Chem. Eur.J.2006, 12,5910–5915. 14881. [78] E. Z. Eisenmesser, D. A. Bosco, M. Akke, D. Kern, Science 2002, 295, [35] S. Mukherjee,A.Warshel, Proc. Natl. Acad.Sci. USA 2013, 110,17326 – 1520 –1523. 17331. [79] E. A. Richard, C. Miller, Science 1990, 247,1208 –1210. [36] A. Burykin,M.Kato,A.Warshel, Proteins Struct. Funct. Bioinf. 2003, 52, [80] C. Y. Seow, J. Gen. Physiol. 2013, 142,561–573. 412 –426. [81] R. M. Berry,H.C.Berg, Biophys. J. 1999, 76,580–587. [37] R. D. Astumian, Biophys. J. 2010, 98,2401 –2409. [82] J. Howard, Curr.Biol. 2006, 16,R517 –R519. [38] a) R. D. Astumian, Phys. Rev.E2007, 76,020102; b) R. D. Astumian, [83] T. L. Hill, E. Eisenberg, Q. Rev.Biophys. 1981, 14,463 –511. Phys. Rev.E2009, 79,021119. [84] D. S. Liu, R. D. Astumian, T. Y. Tsong, J. Biol. Chem. 1990, 265,7260 – [39] M. Nishiyama, H. Higuchi, T. Yanagida, Nat. Cell Biol. 2002, 4,790 –797. 7267. [40] N. J. Carter,R.A.Cross, Nature 2005, 435,308–312. [85] a) R. D. Astumian, P. B. Chock,T.Y.Tsong, Y. D. Chen,H.V.Westerhoff, [41] A. V. Hill, Proc. R.Soc. London Ser.B1938, 126,136 –195. Proc. Natl. Acad. Sci. USA 1987, 84,434 –438;b)T.D.Xie, P. Marszalek, [42] A. F. Huxley, Prog. Biophys.Biophys. Chem. 1957, 7,255–318. [43] R. D. Astumian, Top. Curr.Chem. 2015, 369,285–316. Y. D. Chen,T.Y.Tsong, Biophys.J.1994, 67,1247 –1251. [44] C. Bustamante, Y. R. Chemla, N. R. Forde, D. Izhaky, Annu. Rev.Biochem. [86] R. D. Astumian, B. Robertson, J. Am. Chem. Soc. 1993, 115,11063 – 2004, 73,705 –748. 11068. [45] S. Mukherjee, A. Warshel, Proc. Natl. Acad.Sci.USA 2015, 112,2746 – [87] R. D. Astumian, Phys. Rev.Lett. 2003, 91,118102. 2751. [88] R. D. Astumian, Proc. Natl. Acad.Sci. USA 2007, 104,19715 –19718. [46] R. D. Astumian, Appl. Phys. A 2002, 75,193–206. [89] G. Ragazzon, M. Baroncini, S. Silvi, M. Venturi, A. Credi, Nat. Nanotech- [47] I. DerØnyi, M. Bier,R.D.Astumian, Phys. Rev.Lett. 1999, 83,903 –906. nol. 2015, 10,70–75. [48] R. D. Astumian, Biophys. J. 2015, 108,291–303. [90] K. V. Tretiakov, I. Szleifer,B.A.Grzybowski, Angew.Chem. Int. Ed. 2013, [49] R. D. Astumian, Annu.Rev.Biophys. 2011, 40,289 –313. 52,10304–10308; Angew.Chem. 2013, 125,10494–10498. [50] J. Conyard, K. Addison, I. A. Heisler,A.Cnossen, W. R. Browne,B.L.Fer- [91] C. Cheng, P. R. McGonigal, S. T. Schneebeli, H. Li,N.A.Vermeulen, C. inga,S.R.Meech, Nat. Chem. 2012, 4,547 –551. Ke, J. F. Stoddart, Nat. Nanotechnol. 2015, 10,547–553. [51] T. L. Hill,E.Eisenberg, Y. D. Chen, R. J. Podolsky, Biophys. J. 1975, 15, [92] M. N. Chatterjee, E. R. Kay,D.A.Leigh, J. Am. Chem.Soc. 2006, 128, 335 –372. 4058 –4073. [52] a) J. Xing,F.Bai, R. Berry,G.Oster, Proc. Natl. Acad. Sci. USA 2006, 103, [93] G. K. Feld,M.J.Brown, B. A. Krantz, Protein Sci. 2012, 21,606–624. 1260 –1265;b)R.M.Berry, Curr. Biol. 2005, 15,R385 –R387. [94] J. Bauer, L. Hou, J. C.M.Kistemaker,B.L.Feringa, J. Org. Chem. 2014, [53] D. G. Blackmond, Angew.Chem. Int. Ed. 2009, 48,2648 –2654; Angew. 79,4446 –4455. Chem. 2009, 121,2686 –2693. [95] H. Li, C. Cheng, P. R. McGonigal, A. C. Fahrenbach, M. Frasconi, W.-G. [54] D. A. Case, M. Karplus, J. Mol. Biol. 1979, 132,343 –368. Liu, Z. Zhu, Y. Zhao, C. Ke, J. Lei, R. M. Young, S. M. Dyar,D.T.Co, Y. -W. [55] InternationalUnion of Pure and Applied Chemists (IUPAC) Compendi- Yang, Y. Y. Botros, W. A. Goddard, III, M. R. Wasielewski, R. D. Astumian, um of Chemical Terminology;http://goldbook.iupac.org. J. F. Stoddart, J. Am. Chem.Soc. 2013, 135,18609 –18620. [56] A. Ansari, J. Berendzen,S.F.Bowne, H. Frauenfelder,I.E.Iben, T. B. [96] W. P. Jencks, Annu. Rev.Biochem. 1997, 66,1–18. Sauke, E. Shyamsunder, R. D. Young, Proc. Natl. Acad.Sci. USA 1985, [97] R. Alhadeff, A. Warshel, Proc. Natl. Acad. Sci. USA 2015, 112,12378 – 82,5000 –5004. 12383. [57] R. D. Astumian, P. B. Chock, T. Y. Tsong, H. V. Westerhoff, Phys. Rev.A [98] M. Alvarez-PØrez, S. M. Goldup, D. A. Leigh, A. M. Z. Slawin, J. Am. 1989, 39,6416 –6435. Chem. Soc. 2008, 130,1836 –1838. [58] A. Cressman,Y.Togashi, A. S. Mikhailov,R.Kapral, Phys. Rev.E2008, 77, [99] J. J. Henkelis, A. K. Blackburn, E. J. Dale, N. A. Vermeulen, M. S. Nassar, 050901. J. F. Stoddart, J. Am. Chem.Soc. 2015, 137,13252 –13255.

ChemPhysChem 2016, 17,1719 –1741 www.chemphyschem.org 1740 2016 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Reviews

[100] A. I. Share, K. Parimal, A. H. Flood, J. Am. Chem. Soc. 2010, 132,1665 – [104] M. Schmittel, Chem. Commun. 2015, 51,14956 –14968. 1675. [105] S. Kassem, A. Lee, D. A. Leigh, A. Markevicius,J.Sola, Nat. Chem. 2016, [101] O. V. Makhlynets, E. A. Raymond, I. V. Korendovych, Biochemistry 2015, 8,138–143. 54,1444 –1456. [102] K. Deckert, S. J. Budiardjo, L. C. Brunner,S.Lovell, J. Karanicolas, J. Am. Chem. Soc. 2012, 134,10055 –10060. Manuscript received:February 21, 2016 [103] J.-M. Lehn, Angew.Chem.Int. Ed. 2013, 52,2836–2850; Angew.Chem. AcceptedArticle published:May 6, 2016 2013, 125,2906 –2921. Final Article published:June 14, 2016

ChemPhysChem 2016, 17,1719 –1741 www.chemphyschem.org 1741 2016 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim