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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 acceler- ation) 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 amechani- cal 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 struc- ture 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.
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