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Volume 9 | Number 37 | 7 October 2007 | Pages 5057–5172

Soft Matter is really taking off.

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I can see from the quality of the authors publishing that it is already attracting some of the best people in the field

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Soft Matter www.softmatter.org

Volume 3 | Number 2 | 7 February 2007 | Pages 125–248

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Soft Matter

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Pages 125–248

ISSN 1463-9076

COVER ARTICLE Astumian Design principles for Brownian molecular machines: how to swim in molasses and walk in a hurricane

COMMUNICATION Lei Jiang et al. Directional adhesion of superhydrophobic butterfly wings

REVIEW Jovica D. Badjić et al. Prospects in controlling morphology, dynamics and responsiveness of supramolecular polymers

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www.softmatter.org COMMUNICATION Marston et al. Organic acid formation in the gas-phase ozonolysis of !-pinene

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PERSPECTIVE

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Design principles for Brownian molecular machines: how to swim in molasses and walk in a hurricane R. Dean Astumian* Received 14th June 2007, Accepted 13th July 2007 First published as an Advance Article on the web 28th August 2007 DOI: 10.1039/b708995c Protein molecular motors—perfected over the course of millions of years of evolution—play an essential role in moving and assembling biological structures. Recently chemists have been able to synthesize molecules that emulate in part the remarkable capabilities of these biomolecular motors (for extensive reviews see the recent papers: E. R. Kay, D. A. Leigh and F. Zerbetto, Angew. Chem., Int. Ed., 2006, 46, 72–191; W. R. Browne and B. L. Feringa, Nat. Nanotechnol., 2006, 1, 25–35; M. N. Chatterjee, E. R. Kay and D. A. Leigh, J. Am. Chem. Soc., 2006, 128, 4058–4073; G. S. Kottas, L. I. Clarke, D. Horinek and J. Michl, Chem. Rev., 2005, 105, 1281–1376; M. A. GarciaGaribay, Proc. Natl. Acad. Sci., U. S. A., 2005, 102, 10771–10776)). Like their biological counterparts, many of these synthetic machines function in an environment where viscous forces dominate inertia—to move they must ‘‘swim in molasses’’. Further, the thermal noise power exchanged reversibly between the motor and its environment is many orders of magnitude greater than the power provided by the chemical fuel to drive directed motion. One might think that moving in a specific direction would be as difficult as walking in a hurricane. Yet biomolecular motors (and increasingly, synthetic motors) move and accomplish their function with almost deterministic precision. In this Perspective we will investigate the physical principles that govern nanoscale systems at the single molecule level and how these principles can be useful in designing synthetic molecular machines.

Dean Astumian received his PhD in Mathematical Sciences/ Chemistry from the University of Texas at Arlington in 1984. After a post-doc at NIH and four years on the staff at NIST he moved to the University of Chicago in the Departments of Surgery and in Biochemistry and Molecular Biology where he spent the next nine years. He moved to his current position, professor of Physics at the University of R. Dean Astumian Maine in 2001. He is a fellow of the American Physical Society (APS), and currently serves as the Chair of the Division of Biological Physics of the APS.

1. Introduction Recently it has become possible to follow the stretching and unfolding of a single protein pulled by an atomic force microscope (AFM)1 and to monitor the stepping of an individual molecular motor driven by a chemical fuel and influenced by an external force exerted by an optical trap.2 These experiments (see Fig. 1) are in many ways analogous to experiments devised by Galileo to test the fundamental laws of Department of Physics, University of Maine, Orono, Maine, 044695709, USA. E-mail: [email protected]

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macroscopic mechanics. Just as in Galileo’s experiments the position of an object is followed as a function of time and the motion is influenced with externally applied forces. However, unlike large spheres falling through air or rolling down inclined planes, small particles and molecules in water are subject to very significant thermal fluctuations, and to viscous drag so large that in almost all cases inertia (the acceleration term in Newton’s equation) is negligible.3 A theory to describe single molecule experiments is of great importance, not only for nanotechnology, but also for fundamental understanding of nanoscale processes in solution. Because acceleration is negligible for nanometer scale motions of small objects in solution a very simple description of the dynamical behavior of nanoscale systems interacting with externally applied time dependent fields, with thermal gradients or with a far from thermodynamic equilibrium chemical fuel is possible. The perspective I will develop is that, in a liquid, individual nanoscale particles or macromolecules, including molecular machines, are mechanically equilibrated systems that can serve as a conduit for the flow of energy between a non-equilibrium source (say an external field or a chemical reaction where the chemical potential of reactant is greater than that of the product) and the bath. The energy flow is modulated by the equilibrium fluctuations of the force on the particle or macromolecule due to the solvent. The equilibrium-like behavior can be understood in the context of the generalized fluctuation dissipation theorem (GFDT)4 _ X" P½a; ¼ eDE=kB T _ Xy" P½#a;

ð1:1Þ

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Fig. 1 Two types of single molecule experiments. (a) A protein is attached at one end to the cantilever of an atomic force microscope and at the other to a substrate on a translatable stage. A force is applied through the cantilever tip and maintained constant via feedback. After a waiting period the molecule undergoes an unfolding transition thereby relieving its internal tension. A plot of the extension versus time obtained by simulation for a two-state process is shown, with the trajectory near the transition illustrated in detail in the exploded view. The external force was taken to be constant in time. (b) Cartoon of an optical trap used to apply a force to a bead attached to a single molecular motor (kinesin) that uses chemical energy from ATP hydrolysis to drive directed motion. The experimentally obtained position of the bead vs. time is shown.

This theorem states that even under strongly thermodynamically non-equilibrium conditions the ratio of the probability of a transition to the probability of the time reverse of that transition is the exponential of the change in the internal energy of the system—the dissipation—due to the transition. In eqn (1.1) a is a generalized position (and hence a_ is a generalized velocity), X = X(a, t) is a generalized force (and Xw = X(a, #t) is its time reverse) that can depend on both position and time due to an _ external source that feeds energy into the system, P[a,X] is the probability density for a trajectory or sequence of values a_ given the sequence X, and R R the change in internal energy of the system _ = Xda is the integral of the generalized force DE = Xadt times generalized displacement. The striking relation he#DE=kB T i ¼ 1

ð1:2Þ

for the change in internal energy averaged over many trajectories follows immediately from eqn (1.1).5 The original derivation of eqn (1.1) and (1.2) for the case that the system is connected to an external source that continually feeds energy into the system was carried out using Hamiltonian mechanics. As we will show in section 5 the theorem can be very simply derived for over-damped systems such as those of interest here by using the Onsager–Machlup6 thermodynamic action theory. The fluctuation–dissipation form of eqn (1.1) is obvious from the physical perspective—those trajectories in which the internal energy of the bath increases at the expense 5068 | Phys. Chem. Chem. Phys., 2007, 9, 5067–5083

of the source are relatively more likely than those trajectories in which the internal energy of the bath decreases.5 However, on short time scales there is an appreciable chance that the fluctuations reverse the net flow of energy such that the work done by the source in the interval is negative, i.e., that the bath does work on the source.7 When coupled to two external sources, molecules can function as Brownian machines8–11 by transmitting energy from one source to the other to perform some specific function. The flow of energy from the stronger source can rectify the occasional fluctuation driven reversal of the flow of energy between the bath and the weaker source, allowing energy to be pumped from the bath to do work on the weaker source. The overall energy for the reversal is provided by the stronger source, but the mechanism takes advantage of the omnipresent fluctuations in the energy flows due to thermal or other noise. When structural asymmetry of the molecule is combined with Brownian motion to allow non-directional energy input to drive directed motion in space, the device has come to be known as a Brownian motor. Brownian motors and machines carry out their functions and can do work on the environment while remaining in mechanical equilibrium throughout a work cycle. These Brownian machines8–12 share far more in common with the coupled transport processes described by Onsager6,13 than they do with the mechanisms of macroscopic machines. Notably, there has been significant recent progress in harnessing Brownian motion in constructive role in synthetic molecular motors (for reviews see ref. 14–18). Here we will explore the design principles and constraints that apply to nanoscale machines in solution. The plan of the paper is as follows. In section 2 we will briefly discuss the meaning of equilibrium for single molecules in solution, in section 3 we present a simple derivation of an equilibrium result from a single particle perspective. In section 4 we describe a simple Brownian motor that uses the ineluctable thermal fluctuations coupled with an energy input and structural asymmetry to drive directed motion. In section 5 we give symmetry relations for trajectories of a Brownian motor. The generalized fluctuation dissipation relation is extended to include a relation between forward trajectory and a backward trajectory in forward time. In section 6 we discuss these concepts in the context of discrete lattice kinetic models for Brownian machines and in section 7 extend the discussion to motors and machines driven by a thermodynamically nonequilibrium chemical reaction. In section 8 the approach is generalized by using the fact that for systems without inertia—this includes all single molecules in water—time appears explicitly in the dynamics of the system only through the dissipation. In section 9 the applicability of Brownian motors to understanding the mechanisms of biomolecular motors is discussed. Finally, in section 10 the discussion is concluded with a comparison of macroscopic machine inspired mechanical approaches to controlling motion and synthesis at the nanoscale to Brownian machine inspired chemical approaches.

2. What is equilibrium? Two scientists, a chemist and a physicist are each asked to use whatever experimental technique they can to arrive at the This journal is

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answer to a ‘‘simple’’ question: Is a small spherical object (e.g., a pearl) in equilibrium or not as it falls through a very viscous medium (e.g., a thick, rich shampoo)? After a week or so, both scientists present their findings. The chemist announces that by using the most sensitive calorimetric techniques available it was shown that as the sphere falls a tiny but measurable increase in the temperature of the solution was observed. Since the flow of heat is inconsistent with equilibrium it can be safely concluded that the falling pearl is not in equilibrium. The physicist responds that, with all due respect, his colleague is mistaken. Using the latest optical techniques the position of the falling pearl was measured at a sequence of equally spaced times. The distance traveled in any one interval of time was, to within experimental precision, the same as the distance traveled in both earlier and in later intervals. Since the particle was not accelerating the net force acting must have been zero. Clearly, the pearl, even though falling, undergoes uniform linear motion and is in equilibrium. Both scientists are of course correct in the context of the definition of equilibrium used most commonly in their respective disciplines. When the two worlds meet however, as in single molecule experiments, we are bound to get much confusion unless great effort is taken to clearly define terms. The definition we learn in freshman physics is that if the vector sum of the forces and torques on an object is zero, X! X !! F ¼ 0 and T ¼ 0; ð2:3Þ

there is no acceleration and the object is in mechanical equilibrium. Thus a book resting on a table is in mechanical equilibrium—the normal force exerted by the table exactly counterbalances the force of gravity and there is no acceleration. A parachutist falling at terminal velocity through air, or a small sphere falling through a very viscous medium is also in mechanical equilibrium. The drag force due to the air (or to the viscous medium) exactly counterbalances the force of gravity and there is no acceleration. In this case the velocity relative to the ground is constant but certainly not zero. We learn a second definition of equilibrium in freshman chemistry—a system is in statistical equilibrium or thermodynamic equilibrium if the ratio of the probabilities to find the system in particular states ‘‘i’’ and ‘‘j’’ is given by a Boltzmann distribution Pj gj #DU=kB T ¼ e all i and j Pi gi

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3. The unreasonable effectiveness of equilibrium theory for describing thermodynamically non-equilibrium systems The statistical equilibrium distribution for ideal colloidal particles in dilute aqueous suspension follows the familiar barometric or exponential law as in eqn (2.4)20–22 ceq ðhj Þ ¼ e#mgDh=kB T ; ceq ðhi Þ

ð3:5Þ

where Dh ( hj # hi, ceq (hj) and ceq (hi) are the equilibrium concentrations of particles at heights hj and hi, respectively, g is the acceleration due to gravity, kB is Boltzmann’s constant, and T is the absolute temperature. For spherical particles of radius r the effective mass is m = 4pr3(rp # rw)/3, where rp and rw are the mass densities of the particle and of water, respectively. Eqn (3.5) is a statistical equilibrium result that implicitly involves many particles so that the concentrations (particle densities) are well defined. We can look at colloids from the very different perspective of a single particle falling through solution23 (see Fig. 2). _ The forces acting are gravity, mg, the viscous drag force, Rh, where R B Zr is the coefficient of viscous drag with Z the viscosity of the solution, and a random thermal noise force e(t) due to the molecular movement of the water molecules. The

ð2:4Þ

where gj and gi are the degeneracies and DU ( Uj # Ui is the difference in the potential energies of the two states ‘‘j’’ and ‘‘i’’. Another way of saying this is that at thermodynamic equilibrium the chemical potentials of all components of a system are equal and that all thermodynamic flows of heat, energy, entropy, etc. vanish. In the past there has been little chance of confusion resulting from these two different concepts of equilibrium—it is pretty obvious whether one is dealing with a single baseball or with a graduate cylinder of 108 colloidal particles or with a test tube of 1016 protein molecules. Recent advances in experimental techniques however have blurred this boundary between mechanical systems and statistical ensembles by allowing the This journal is

direct observation and manipulation of individual micronand sub-micron sized particles19 and even single macro-molecules.1,2 On experimentally relevant time scales these systems are in mechanical equilibrium—the viscous force is equal and opposite any mechanical forces due to external manipulation or internal strains in the molecules—but NOT generally in thermodynamic equilibrium with the environment since there can be a net flow of energy into the bath. To sharpen our understanding of the difference between statistical and mechanical equilibrium, and how the rapid mechanical equilibration of macromolecules in solution can lead to a simple theory for describing molecular machines operating under far from thermodynamic equilibrium conditions, let us first consider a very simple system—colloidal particles in water.

Fig. 2 A single colloidal particle falling in solution due to gravity. The size of the particle as drawn and the probability density curve P(h) for the center of mass height is approximately to scale for a 1 mm sphere with time measured in units of m/R.

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equation of motion can be written mh€ þ Rh_ ¼ mg þ eðtÞ

ð3:6Þ

where e(t) is modeled as Gaussian white noise with statistical properties he(t)i = 0 and he(t1)e(t2)i = 2RKBTd(t2 # t1) where d(t2 # t1) is the Dirac delta function. After a sufficiently long time (about m/R = 10#6 s for a micron sized particle) the particle reaches terminal velocity vterm = mg/R where the force of gravity is balanced by the viscous drag force and there is no further acceleration. When viewed on time scales long compared to m/R the Langevin equation for the motion of the particle becomes Rh_ * mg þ eðtÞ

ð3:7Þ

An equivalent partial differential equation for the conditional probability density P(hj, t|hi, 0) ( P(Dh, t) to find the particle in the interval hj + dh at time t given that it was in the interval hi + dh at time 0 is Fick’s equation for diffusion with drift,24 2

@PðDh; tÞ @ PðDh; tÞ mg @PðDh; tÞ þ ¼D ; @t @Dh2 R @Dh

ð3:8Þ

where D = kBT/R is the diffusion coefficient. The solution to eqn (3.8) is

PðDh; tÞ ¼

n

e

# ½Dhþðmg=RÞt"2 4Dt

pffiffiffiffiffiffiffiffiffiffi 4pDt

o

;

ð3:9Þ

Eqn (3.9) describes a Gaussian distribution with mean position m = #(mg/R)t and variance s2 = hs2 = Dh(t)2i–hDh(t)2 = 2Dti. Although it is more likely that the particle moves downward, at short times there is a reasonable chance that thermal noise will cause the particle to be found slightly higher than where it started. The relationship between the occasional upward motion and the second law of thermodynamics has been discussed by Smoluchowski in a paper, the title of which translates into English as ‘‘Limits on the validity of the second law of thermodynamics’’.25 Eqn (3.9) for the evolution of the probability density for any single particle depends on time. However, by transposing hi and hj in the probability density to obtain P(hi,t|hj,0) ( P(#Dh,t) we find the probability density for a particle to be at hi at t given that it started at hj at t = 0. The ratio P(Dh,t)/ P(#Dh,t) is PðDh; tÞ ¼ e#mgDh=kB T : Pð#Dh; tÞ

ð3:10Þ

where we used Einstein’s relation26 RD = kBT. Remarkably, time has disappeared altogether in eqn (3.10), and we have regained the equilibrium barometric law, eqn (3.5), which now relates the conditional probability to be at hj at t given that it was at hi at t = 0 to the conditional probability to be at hi at t given that it was at hj at t = 0. An analogous relation for an over-damped particle in an arbitrary potential was derived in a more general context by Bier et al.27 using the Onsager– Machlup6 thermodynamic action approach. 5070 | Phys. Chem. Chem. Phys., 2007, 9, 5067–5083

Because the probability R +N R +N density function is normalized, #N P(Dh,t)dDh = #N P(#Dh,t)dDh = 1, we also have Z þ1 hemgDh=kB T i ¼ emgDh=kB T PðhÞdD h ¼ 1: ð3:11Þ #1

The quantity mgDh is the energy lost when a particle falls a distance Dh. Note that the analysis resulting in derivation of eqn (3.10) and eqn (3.11) is independent of whether the individual particle on which we focus is drawn from a statistically equilibrium ensemble, or from an ensemble arbitrarily far from statistical equilibrium. We can understand the origin of eqn (3.10) in terms of the principle of microscopic reversibility27—the idea that at equilibrium every process is as likely as the exact (microscopic) reverse of that process. The equilibrium probability to observe a particle move from some position hi at time 0 to a position hj at time t by any trajectory is ceq (hi)P(hj, t|, , ,|hi, 0), which at equilibrium must equal the probability to observe the exact reverse process, ceq ðhi ÞPðhj ; tj , , , jhi ; 0Þ ¼ ceq ðhj ÞPy ðhi ; tj , , , jhj ; 0Þ:

ð3:12Þ

27

From eqn (3.5) we thus have

Pðhj ; tj , , , jhi ; 0Þ ¼ e#mgDh=kB T Py ðhi ; tj , , , jhj ; 0Þ

ð3:13Þ

for any trajectory, where the w indicates the microscopic reverse trajectory. Although eqn (3.13) was derived using knowledge of the behavior of the system at equilibrium, it is valid irrespective of how different the actual concentration distribution is from the statistical equilibrium distribution. The conditional probabilities P(hi, t|, , ,|hj, 0) and Pw(hj, t|, , ,|hi, 0) are defined independent of any reference to the concentrations themselves. The derivation of eqn (3.13) is analogous to the use of detailed balance13 (a corollary of microscopic reversibility) to derive that the ratio of rate constants for a chemical reaction is equal to the exponential of the free energy difference over the product of the gas constant and temperature, exp(DG0/RT). Consider a simple reaction kf

AÐB kr

ð3:14Þ

where kf and kr are the forward and reverse rate coefficients, respectively. Detailed balance dictates that kfceq,A = krceq,B where ceq,A and ceq,B are the equilibrium concentrations of species A and B, respectively. Because ceq,B/ceq,A = exp(DG0/ RT) we also have kf/kr = exp(DG0/RT), a relation that holds irrespective of how different the actual concentration ratio cA(t)/cB(t) at any time is from the equilibrium ratio ceq,B/ceq,A.

4. A simple Brownian motor The occasional upward trajectories seen in Fig. 2 can be exploited in a non-isotropic system to allow the random input of energy to drive directed motion by a Brownian motor mechanism.9,11,12,28–30 In Fig. 3 a particle is trapped at the bottom of a potential well created, e.g., by the interdigitated electrodes in Fig. 4. When the potential is turned off the particle begins to move to the left on average at a velocity This journal is

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Fig. 3 Basic model for a flashing ratchet. A particle trapped at ‘‘0’’ begins to drift to the left due to the force F, and to diffuse symmetrically because of thermal noise when the pinning potential is turned off. When the potential is turned on, the particle is again trapped in one of the wells at positions nL. Because of the asymmetry of the potential it is more likely at short times that the particle is trapped at +L to the right than at #L despite the homogeneous force acting to move the particle to the left. Turning the asymmetric potential on traps the occasional fluctuations uphill to the right, allowing net directed motion against an applied force. The energy comes NOT from the thermal noise, but from the external source when the potential is turned back on. The energy supplied by the source however is directionless, and it is the combination of the asymmetry and the thermal noise that provides the mechanism for the directionless input energy to drive directed motion against an external force.

F/R, but there is also symmetric diffusion left and right. After some time t the probability distribution for the position of the particle is given by the Gaussian function shown, with pffimean ffiffiffiffiffiffiffiffiffi position m = #(F/R)t and full width at half-height ¼ 4pDt. If the potential is turned back on at time t most particles will return to the electrode at x = 0, but some will be caught in the well at x = +L. For the case shown in Fig. 3 and small t, there is almost no chance that a particle will be caught in the well at h = #L even though the average velocity when the potential is to the left. Thus by turning the potential cyclically on and off, either by a periodic or stochastic modulation scheme, the symmetric diffusion when the potential is off is asymmetrically biased when the potential is turned back on thus leading to net motion against the applied force, and to the performance of work. An excellent simulation of a prototypical Brownian motor can be found at http://monet.unibas.ch/Belmer/bm/. An electrophoretic separation technique based on this idea (shown schematically in Fig. 4) has been constructed and used to separate DNA strands of different length by Bader and colleagues.32 The basic ratchet mechanism has been extended to a two dimensional technique33–35 where particles are driven through a maze of asymmetric obstacles by an electric field and separation in the axes parallel and perpendicular to the field scale differently with the radius of the particle.

5. Symmetry relations for trajectories of a Brownian motor A Brownian motor such as that shown in Fig. 3 is a nanoscale or molecular device that combines the effects of thermal noise, spatial or temporal asymmetry, and directionless input energy This journal is

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Fig. 4 Schematic illustration of how a flashing ratchet can be used for separation of particles. Interdigitated electrodes create an asymmetric potential energy profile for charged particles (e.g., latex spheres) of different sizes. When the potential is turned off, the particles begin to diffuse. Since the diffusion coefficient is inversely proportional to the radius of the particle, the small particles diffuse more rapidly than the larger particles, while the larger particles feel the effect of gravity more strongly. When the potential is turned back on, the smaller particles have a greater chance of being trapped in a well to the right of where they started, while the larger particles are more likely to be trapped in a well to the left due to gravity. By cycling the potential on and off at the right frequency it is possible to give the small particles a net velocity to the right and the large particle a net velocity to the left. More complicated pulsing protocols can be used to achieve the same goal of having particles of different sizes move in different directions without a net dc force (gravity in the case shown).31

to drive directed motion. In solution, viscous drag and thermal noise dominate the inertial forces that drive macroscopic machines. Because of the strong viscous drag, the motion of such a Brownian motor is over-damped and in one dimension can be described by the simple equation6 Ra_ # X ¼ eðtÞ

ð5:15Þ

where e(t) is Gaussian noise with mean m = 0 and variance s2 = 2RkBT/dt, and R is the coefficient of viscous friction. In the following (and in the rest of the paper) we use units where the thermal energy kBT = 1. The generalized force X = X(a, c(t)) can be written as the gradient of a scalar potential X = #qH/qa where Hða; cðtÞÞ ¼ UðaÞ þ cðtÞzðaÞ

ð5:16Þ

is the sum of an intrinsic potential due to chemical interactions and any external load and an external time dependent forcing term that is the product of canonically conjugate intensive and extensive thermodynamic parameters z(a) and c(t), respectively.8 The conjugate parameters include, e.g., molecular volume and pressure, entropy and temperature, or dipole moment and field. The underlying system is typically spatially periodic (possibly with a homogeneous force or load F) so that U(a + L) = U(a) + DU, where DU = FL, and z(a + L) = z(a). For any fixed value of c detailed balance requires Pðai þ L; Tj , , , jai ; 0Þ ¼ e#DU : Py ðai ; Tj , , , jai þ L; 0Þ

ð5:17Þ

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where P(ai + L, T|, , ,|ai, 0) is the conditional probability density that a particle starting at position ai at time 0 goes to position ai + L at time T by the specific trajectory (sequence of positions and times) denoted by , , ,, and Pw(ai, T|, , ,|ai + L, 0) is the conditional probability to follow the reverse of that process. The ratio depends only on the difference in energy between the initial and final points. It further holds that PðL; Tj0; 0Þ ¼ e#DU Pð0; TjL; 0Þ

ð5:18Þ

R R where the net probability P(L,T|0, 0) = 0, , , LP(ai + L, T|, , ,|ai, 0) is the integral over all trajectories from (0, 0) to (L, T). A time dependent modulation, c(t), causes dissipation and breaks detailed balance, in which case eqn (5.17) and (5.18) do not hold. As shown in Fig. 3 it is even possible to have PðL; Tj0; 0Þ 41; e#DU o1 Pð0; TjL; 0Þ

ð5:19Þ

where the external stimulus c(t) provides energy to drive uphill motion.29,30 The generalized fluctuation–dissipation theorem4,5 states that even under strongly thermodynamically non-equilibrium conditions the ratio of the probability of a forward (F) transition to the probability of the time-reverse (FR) of that transition is the exponential of the change in the internal energy of the system due to the transition PF ðL; Tj , , , j0; 0Þ ¼ eW#DU : PFR ð0; #Tj , , , jL; 0Þ

ð5:20Þ

where W is the work supplied to the system by the external modulation in the forward trajectory. We can derive an extension of the generalized fluctuation dissipation theorem for a Brownian motor to obtain the ratio between the probability for the motor to take a forward step and the probability to take a backward step in forward time.36 First, we write eqn (5.15) as a more rigorous finite difference or update equation and convert to unit normal Gaussian noise N(0, 1)37 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aiþ1 # ai # R#1 Xiþ1 Dt ¼ 2R#1 DtNð0; 1Þ ð5:21Þ

The time interval Dt is chosen to be sufficiently short that the change in position Da is very small. We used the relation N(m, s2) = m + sN(0, 1) where N(0, 1) is a Gaussian random variable with zero mean and unit variance pffiffiffiffiffi n, of # the$ values, which occur with probability, PðnÞ ¼ exp #n2 =2 =ð 2pÞ. Any two values of n are uncorrelated hninki = di,k. Broken symmetry is an essential feature of a Brownian motor, so we split each of the position dependent terms into even and odd components U(a) = Ue(a) + Uo(a) and z(a) = ze(a) + zo(a), where for any function f e(#a) = f e(a) and f o(#a) = #f o(a). Finite difference expressions for the even and odd components of the generalized force, Xi+1 = Xei+1 + Xoi+1, are ðe;oÞ

Xiþ1 ¼ #

ðo;eÞ DUiþ1

ðo;eÞ ciþ1 Dziþ1

þ aiþ1 # ai

5072 | Phys. Chem. Chem. Phys., 2007, 9, 5067–5083

ð5:22Þ

Fig. 5 Depiction of symmetry related trajectories of a Brownian particle in a periodic ratchet potential. (a) Snapshot of the potential described in eqn (6.37) with a particle at a = 0. For any constant c, PF/PB = e#DU and the net motion of the particle is to the left. External modulation c(t) breaks detailed balance and can drive net motion to the right against the load. (b) With an external forcing c(t) all four trajectories—F, FR, B, BR—are distinct. As described in the text, an extension of the generalized fluctuation dissipation relation can be used to derive the ratios of the probability densities for these symmetry related trajectories.

where Df ki+1 = f k(ai+1)–f k(ai) for f = U, z and k = e, o. For every forward trajectory {a(t), c(t)} (see Fig. 5) with probability PF, defined by c1

c2

cm#1

cm

F ( 0 #! a1 #! , , , #! am#1 #! L; PF ¼

M#1 Y i¼0

ð5:23Þ

Pðaiþ1 jai ; ciþ1 Þ;

there are three symmetry related trajectories. One is a time reverse trajectory38 {a(#t), c(#t)} obtained by switching the sign of time. For a time periodic system reversing time is equivalent to the transformation t - (T–t), cm

cm#1

c2

c1

FR ( L #! am#1 #! , , , #! a1 #! 0; PFR ¼

M#1 Y i¼0

ð5:24Þ

Pðai jaiþ1 ; ciþ1 Þ;

Another is a backward trajectory {#a(t), c(t)} obtained by switching the sign of the position variable. For a space periodic system this is equivalent to the transformation a (L # a), c1

c2

cm#1

cm

B ( 0 #! #a1 #! , , , #! #am#1 #! #L; PB ¼

M#1 Y i¼0

Pð#aiþ1 j # ai ; ciþ1 Þ:

ð5:25Þ

The third is a backward reverse trajectory {a(#t), c(#t)} obtained by switching the sign of both time and of the position This journal is

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variable. For a time and space periodic system this is equivalent to the transformation a - (L # a) and t # (T # t), cm

cm #1

c2

c1

BR ( #L #! #am#1 #! , , , #! #a1 #! 0; PBR ¼

M#1 Y i¼0

Pð#ai j # aiþ1 ; ciþ1 Þ;

ð5:26Þ

Viewing eqn (5.21) as a mapping between the ‘‘noise’’ space and ‘‘position’’ space,27,39 the conditional probability density given that the system is at position ai after the ith step, and that the value of the field is ci+1 for the (i + 1)st step is seen to be Pðaiþ1 jai ; ciþ1 Þ ¼

#ðDa#R#1 Xiþ1 DtÞ2 4R#1 Dt

e

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4pR#1 Dt

ð5:27Þ

where Da = (ai+1 # ai). The ratio of the probability density for the forward and time reverse step is Pðaiþ1 jai ; ciþ1 Þ ¼ eXiþ1 Da Pðai jaiþ1 ; ciþ1 Þ

ð5:28Þ

and the ratio between the forward and time reverse trajectory is ! M#1 X PF ¼ exp Xiþ1 Da ¼ eWF #DU ð5:29Þ P FR i¼0 where WF ¼

M#1 X i¼0

ciþ1 ðDze ðaÞ þ Dzo ðaÞÞ

ð5:30Þ

is the total external work done in the forward trajectory. Eqn (5.29) is the generalized fluctuation dissipation relation4,5 and the change in the internal energy of the system, DE = DU # WF, is the dissipated work. The ratio of the probability density for a backward and backward time reverse is similarly obtained, PB ¼ eWB þDU PBR

ð5:31Þ

where WB ¼

M #1 X i¼0

ciþ1 ðDze ðaÞ # Dzo ðaÞÞ

ð5:32Þ

is the total external work done in the backward trajectory. Finally, the ratio between a forward and backward step is e e o #1 Pðaiþ1 jai ; ciþ1 Þ ¼ eXiþ1 Da#Xiþ1 Xiþ1 R Dt Pð#aiþ1 j # ai ; ciþ1 Þ

ð5:33Þ

The ratio for the probability densities for a forward and backward trajectory, follows immediately R T XeXo PF dt # ¼ e#DU e 0 R ð5:34Þ PB where we have taken the limit Dt - 0 to get the integral form. Unlike the symmetry relations for the forward and reverse (and backward and backward reverse) trajectories, the ratio for the forward and backward trajectories involves the whole path. Eqn (5.34) highlights the importance of broken symmeThis journal is

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Fig. 6 (a) Toy model of an external time-dependent field driven Brownian rotor. An ac or stochastic field c(t) drives directional rotation because the dipole is not oriented either parallel or perpendicular to the chemically preferred axis. With the orientation shown the rotation is clockwise. (b) A two-state model appropriate for analysis of the behavior of the system when the chemical interaction is much stronger than the interaction between the dipole and the field.

try—if either Xo or Xe is zero, the ratio of the probability for a forward step to a backward step is governed solely by the homogeneous force acting on the system and is independent of the work pumped in by the time dependent modulation. The results can be summarized using the Onsager–Machlup thermodynamic action6,23 Z 1 T _ þ Xi =RÞ2 dt; i ¼ F; B; FR ; BR Si ¼ ðaðtÞ ð5:35Þ 4 0 where XF,FR = Xe + Xo and XB,BR = Xe#Xo. Then Pi ¼ eRðSj #Si Þ ; i; j ¼ F; B; FR ; BR Pj

ð5:36Þ

The least action (optimal) trajectory is that for which the _ = (a(t) _ Lagrangian L(a,a) + Xi/R)2 solves the Euler– _ _ = 0. Optimizing Lagrange equation [@/@a–(d/dt)@/@ a]L(a, a) a Brownian motor requires that we design a system and modulation scheme that maximizes the difference between the least action in the forward and backward directions.40

6. Discrete state models Much of the early work on what has come to be known as Brownian motors was done in the context of chemical kinetic models with external time dependent perturbations.8,41–44 It was shown that even zero average perturbation can drive directional motion through a cycle of states. As a concrete example consider the example in Fig. 6 that depicts a ‘‘Brownian’’ machine consisting of a rotor inside a fixed stator. There are two chemical sites on the rotor that interact with two sites on the stator such that the chemical potential energy is minimized at angles a = 0 and a = p. There is a permanent dipole where the component of the dipole moment parallel to the field is z(a) = z0 cos(a # sp) and #1 r s r 1. An applied torque produces a load that tends to turn the rotor counterPhys. Chem. Chem. Phys., 2007, 9, 5067–5083 | 5073

clockwise such that the energy of the system decreases by DU each turn. Because of the asymmetric arrangement of the permanent dipole relative to two chemical bonding sites, an oscillating or fluctuating electric field applied via the waveform generator to the capacitor plates above and below the machine causes a net tendency to turn clockwise. A sufficiently large amplitude and high frequency field can overcome the effect of the load and cause clockwise rotation. We can approximate the angle dependent terms in the potential experienced by the rotor as45 UðaÞ ¼ U0 cosð2aÞ zðaÞ ¼ z0 ½cosðspÞ cosðaÞ þ sinðspÞ sinðaÞ"

ð6:37Þ

Similar dipolar molecular systems have been synthesized46 and computationally studied47 under both large driving where thermal noise is not required, and smaller driving where thermal noise is required for operation.48 For U0 4 c0z0, (i.e., where the chemical interaction is stronger than the effect of the field c(t) on the dipole), a single spatial period of the potential has two clearly defined energy wells, say 0 and 1. The sign of C determines which of the two wells has a lower energy, and which of the two barriers has a greater height. A two-state representation is shown at the bottom of the figure. We can then describe the motion as a random walk on a lattice !

!

!

k0

k0

k1

!

k1

. 0Ð1Ð0Ð1Ð0 . k1

k0

k1

k0

ð6:38Þ

The transition constants are !

k0 ¼ k ½e#DU ðfe fo Þ"

1=4

!

k1 ¼ k ½e#DU =ðfe fo Þ"

holds at every instant. Nevertheless, any time dependent modulation c(t) drives motion to the right when DU = 0 and can do work against a small non-zero DU 4 0. The infinitely extended lattice model can more conveniently be written as a cycle as in Fig. 6b, where a clockwise transition indicates a half-step to the right, and a counterclockwise transition indicates a half-step to the left on the lattice. The net motion can be solved analytically for small amplitude (c0z0 o 1),44 and in special cases, such as square wave perturbation, where c oscillates between +C and #C, for larger amplitude (U0 4 c0z0 4 1).8 6.1 A minimal Brownian machine For the specific case that c(t) is externally generated dichoto% & g mic noise þC #! # # C in which c(t) switches between +C g

and #C with a Poisson distributed random lifetime (average 1/g) the combined stepping/switching process can be described by a single diagram8

ð6:43Þ

This case is particularly relevant for Brownian motors that are driven, e.g., by the stochastic binding of chemical fuel molecule and release of product and emphasizes the idea of a minimal Brownian motor as two coupled two-state processes. One process is& the externally driven dichotomic modulation % g þC #! # # C and the other is the thermally activated stepping. g

1=4

k0 ¼ k ½eDU ðfe =fo Þ"

1=4

k1 ¼ k ½eDU ðfo =fe Þ"

1=4

ð6:39Þ

where fe = e4z0c(t)cos(sp) and fo = e4z0c(t)sin(sp). The even part of the time dependent perturbation cos(sp) influences the relative energies of the two states 0 and 1, and hence the equilibrium constants

The overall diagram can be broken into six cycles8—two cycles for the uncoupled stepping, one with fixed +C (I) and the other with fixed #C (II), two cycles for the dissipative back and forth motion with no net stepping (III and IV), and two cycles describing net stepping coupled to the external fluctuation (V and VI).

!

Ki ¼

ki

k1#i

; i ¼ 0; 1:

ð6:40Þ

The odd part of the time dependent perturbation sin (sp) influences the relative heights of the two distinguishable barriers and hence the splitting probabilities

ð6:44Þ

!

Ski ¼

ki ki

; i ¼ 0; 1:

ð6:41Þ

Irrespective of the value of s or of the form of c(t) a corollary of detailed balance for rate processes, ! !

k0 k1 k0 k1

¼ e#DU

5074 | Phys. Chem. Chem. Phys., 2007, 9, 5067–5083

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The last two, coupled, cycles are of particular interest. The forward, reverse, backward, and backward reverse paths are

ð6:45Þ

The probability for completion of a cycle is proportional to the product of the transition constants in the cycle. The proportionality constants involve rate constants for back and forth transitions, lifetimes of the states within the cycle, etc. Importantly, since F, FR, B, and BR directional cycles (eqn (6.51)) involve the same states, the proportionality constants are the same for all of these symmetry related cycles. Thus, with ki+ = ki(c(t) = +C), it is easy to derive !

!

k0 k1 PF ¼ þ # ¼ eW#DU P FR k k 1þ 0# k0 k1 PB ¼ þ ! # ¼ eWþDU P BR ! k1þ k0#

ð6:46Þ

where W = ln(fo+) is the work done in the forward cycle when the energy is increased by 2z0Csin (sp) in going from 0,# - 0, + and again from 1, + - 1, #. Here and below f(e,o) = + f(e,o)(c(t) = +C). The ratio of the probabilities for a forward and backward cycle is !

!

PF k0þ k1# ¼ ¼ e#DU feþ PB k k 0þ 1#

ð6:47Þ

and the ratio of the net forward to backward steps is % e o& P F þ P BR #DU 1 þ fþ fþ ¼e ð6:48Þ P B þ P FR foþ þ feþ The expansion of the coefficient in eqn (6.53) involves only even powers of the amplitude c0 of the external driving—the Brownian motor mechanism is a fundamentally non-linear effect of the external driving.44,49 It does not necessarily require a large amplitude driving to observe experimentally50–52 however since the linear term is, by symmetry, identically zero. The non-monotonic frequency response observed experimentally50–52 however since the linear term is, by symmetry, identically zero. The nonmonotonic frequency response observed experimentally50 and explained theoretically53 arises from the uncoupled trajectories which take on greater or lesser importance depending on the frequency of the applied signal. The symmetry relations derived here are evidently frequency independent. Note that although we have introduced a specific example of a rotor (Fig. 6) the predominate trajectory by which the system cycles through its possible This journal is

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states is entirely specified by the energies of the states and of the barriers between them, and the interaction energies between the field and the system. The diagram (6.43) could just as well represent a molecular motor moving on a track, where the 0 and 1 states denote left-head bound and right-head bound and the two direction for each transition forward and backward, or a molecular pump where the 0 and 1 states represent binding site unoccupied and binding site occupied, respectively and the two directions to/from left reservoir and to/from right reservoir.54 Unlike the case of inertial motion, there is no significant fundamental difference between overdamped linear and rotary motion, or for that matter motion of an ion or of a macromolecule.55 Directed motion is achieved by coupling two functions very familiar to chemists, switching and gating, and can be understood in terms of the diagram below. The relatively more stable states are shown in bold and between brackets [ij], and the higher energy barrier (‘‘closed’’ gate) is indicate by hash marks (J) through the transition.

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ð6:49Þ The rotor is bi-stable and the relative stability of the two states is switched by changing the sign of the field. The rotor is also bi-labile (there are two paths, the upper and lower in Fig. 6b) by which the rotor can undergo a transition between its two states. The relative lability or gating is also controlled by the field. When the asymmetry parameter ‘‘s’’ is 0, the stability of the two states is switched by the field, but not the lability— both paths are equally open at all values of the field—and there is no directed motion. When the asymmetry parameter is +1, the relative lability or gating depends on the field, but not the stability, and once again there is no directed motion. At intermediate values of s both the relative lability and stability depend on the field and directed motion is achieved by the correlated changing of the stability of the two states and lability of the two pathways. When the field is ‘‘+’’, state 1+ is more stable than state 0+ so the probability to be in 1 is higher than to be in 0. When the field is switched from ‘‘+’’ to ‘‘#’’ the relative stabilities switch—0# is more stable than 1#. Thus the system makes a transition to 0, predominately in the clockwise (forward) direction since the counterclockwise transition, which would require the positive end of the dipole to rotate toward the positive electrode, is kinetically blocked. Then, when the field is again switched back to the ‘‘+’’ state, the dipole rotates to the 1 orientation, again in the clockwise direction, thus completing a full clockwise turn. Through a cycle, the time average of the dipole moment is zero, and the time average of the field is zero. However, the time average of the product of the dipole moment and the field is not zero. It is the cross correlation between the field and the dipole—it is more likely for the rotor to be in the 0 state when the field switches from ‘‘+’’ to ‘‘#’’, and it more likely for the rotor to be in state 1 Phys. Chem. Chem. Phys., 2007, 9, 5067–5083 | 5075

when the field switches from ‘‘#’’ to ‘‘+’’—that pumps energy into the system and drives the directional motion. 6.2 Fluctuations and detailed balance The net motion described above comes from the imposition of external dichotomous noise on the system. There is Johnson–Nyquist noise at the electrode pair in Fig. 6 even when the electrodes are not powered. However, in this case we have internal or endogenous noise and the transition constants are constrained by a corollary of detailed balance. Consider the kinetic scheme in eqn (6.50) where the rate constants are shown explicitly

systems) switches the relative stability of the two states or/and the relative lability of the two transitions. Consider the diagram below

ð7:54Þ

where the lateral cycles describe a catalytic process S + E# ! E+S ! E# + P which can be written

ð7:55Þ ð6:50Þ

Detailed balance places the constraint !

!

k0þ k1þ k0þ k1þ

!

¼

!

k0# k1# k0# k1#

¼ e#DU

ð6:51Þ

for stepping at a fixed value of the field. Further, the product of the clockwise rate coefficients divided by the product of the counterclockwise rate coefficients for each of the two cycles in mechanism (6.50) must equal unity !

!

a0# k0þ a1þ k1# !

!

!

!

!

!

k0# a1# k1þ a0þ

¼1

ð6:52Þ

¼1

ð6:53Þ

When the chemical potential of substrate is higher than that of product the directional flow of S - P breaks detailed balance that at equilibrium would preclude directional motion.56 By incorporating bi-lability (a kinetically ‘‘blocked’’ transition is indicated by hash marks J) in the transition rates and bistability (a relatively stable state is shown in bold and in brackets [ij]) in the energies of the states 0 and 1 it is possible to design a motor that undergoes directional (horizontal) transport when the vertical cycle is powered by the chemical fuel.

and a0# k0þ a1þ k1# k0# a1# k1þ a0þ

With rate constants that obey the relations eqn (6.51), eqn (6.52), and eqn (6.53) we see that the net motion is zero when DU = 0, to the left when DU 4 1 and to the right when DU o 1. ! ! The mechanism in (6.50), with a0# ¼ a0þ ¼ a1# ¼ a1þ ¼ g was first proposed as a model for the effects of external fluctuations driven by an external generator on an enzyme or free-energy transducing protein such as a motor or pump.8 This choice is consistent with the constraint of eqn (6.52) but not with the constraint of eqn (6.53). The external fluctuations break detailed balance and can drive net motion to the right or left (horizontal) if symmetry is broken even though the perturbation directly influences only the up and down (vertical) transitions.

7. Chemical Brownian motors

ð7:56Þ I call this an energy ratchet55 because the substrate binding switches the relative energies of the two states. The preferred forward process (F) when mS 4 mP couples conversion of S to P to drive forward motion

The kinetically blocked transitions are left out of the diagram for clarity. The chemical transitions unused in the path are shown as a dashed line. A sufficiently large external load (DU) acts to push the motor to the left and favors (relative to the forward path) the reverse (R) process in which the motor moves backward and P is converted to S against the chemical potential gradient

The mechanism shown in eqn (6.49) can be adapted to illustrate how a motor or rotor can be driven by a catalyzed chemical reaction. The idea is that binding of substrate S or product P (ATP and ADP in the case of many biomolecular 5076 | Phys. Chem. Chem. Phys., 2007, 9, 5067–5083

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optimal information ratchet is described by a single completely coupled path.

where PF ¼ eDm#DU P FR

ð7:57Þ

However, there is an alternative, backward, pathway in which the motor moves backwards as S is converted to P, but the binding of S and release of P occur in the opposite state as compared to the forward (F) process

where PB ¼ Sa0þ Sa1# PFR

ð7:58Þ

!

where Saij ¼ aij =aij : The ‘‘splitting’’ factors reflect the two transitions leading from states 1# and 0+ in the paths B and FR. In general both splitting factors are greater than one when Dm 4 1 so the backward path is favored over the forward reverse path under a large load. There is a similar dichotomy for the forward path F and the backward reverse path BR. This energy ratchet mechanism makes a striking prediction, already noted in 1996.56 Under the action of a strong enough load to cause back stepping, the backward motion is, in this model, enhanced by increased concentration of substrate (ATP for kinesin). In a completely coupled model, in contrast, backward stepping would ineluctably be accompanied by conversion of product to substrate and would be hindered by addition of substrate. The prediction of the energy ratchet model was recently confirmed experimentally by Carter and Cross57 for load induced backward stepping of kinesin. The energy ratchet model also explains why the ‘‘stopping’’ force observed for kinesin is significantly less than would be predicted based on a completely coupled kinetic cycle model where forward stepping would not stop until the energy released by the conversion of substrate to product is equal to the energy gain by moving a step forward against the load. These observations lend strong support to our hypothesis that kinesin functions as an energy ratchet.58 An alternative to the energy ratchet is an ‘‘information ratchet’’55,59 in which only the gating is controlled. This requires an allosteric interaction by which one state is specific for binding/release of substrate, and the other state is specific for binding/release of product. Effectively, in this mechanism information about what mechanical state the motor is in determines the kinetic properties with respect to substrate and product.

The paths B and BR are eliminated by the kinetic blocks. At equilibrium the forward motion on this path is as likely as reverse motion, but when S is in excess and P in deficit of the equilibrium amount, net motion to the left is enforced by simple mass action. Exactly one molecule of S is converted to P for each step taken, and the stopping force Fstop = Dm/L. If Dm 4 DU the motor steps right and converts S to P, and if DU 4 Dm the motor steps left and converts P to S. It may well be that molecules such as the mitochondrial FoF1 ATPase (a protein in mitochondria that, under normal conditions uses energy from a proton electrochemical gradient to synthesize ATP) works as an information ratchet since mechanically turning the rotor backwards does in fact synthesize ATP.60 Chemical implementations of both the energy61,62 and information ratchet63 have been synthesized using catenanes. These make use of tightly controlled external manipulation of the chemical environment to change the relative stabilities of states and heights of barriers. Typically molecules are switched and gated by, e.g., reduction/oxidation or protonation/deprotonation cycles enforced by manually changing the conditions (adding base or acid, or reducing or oxidizing agents, etc.) Similar systems have been coupled to drive macroscopic transport,64 giving great optimism for future applications. A major goal of research remains incorporating a catalytic function into a switchable and gateable molecule to achieve autonomous motion.

8. A general approach for coupled transport The trajectory for a system without inertia depends on time only through dissipation. Consider the chemically driven twostate motor eqn (7.54). This system can be described in terms of a random walk on a periodic 2-D lattice where the vertical coordinate describes the chemical transitions and the horizontal coordinate describes the spatial stepping transitions.

ð8:60Þ

ð7:59Þ

By removing the kinetically blocked transitions from the diagram we see that, in contrast to the energy ratchet, the This journal is

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The states labelled FR, u, BR, l, r, B, d and F all represent the state 0# but reached from the central initial state 0# by paths Phys. Chem. Chem. Phys., 2007, 9, 5067–5083 | 5077

involving different changes in the environment (i.e., conversion between substrate and product S 2 P and stepping to the right and left), and hence different amounts of dissipation. For example, the paths to the right hand lower corner are the coupled ‘‘forward’’ paths in which the motor steps to the right by one period while converting one substrate molecule to one product molecule. The random walk (i.e., the path probabilities and lifetimes) on this lattice is specified by the energies of the states Eij and of the barriers between them E+ijz and Ei+jz, where we take +i to indicate the barrier to the right of state ij, #i to indicate the barrier to the left of state ij, +j to indicate the barrier below of state ij, and #j to indicate the barrier above state ij of state ij. The transition constants can be written in terms of these energies

!

k ij ¼ A k e

z

Eþi

j

z

E#i

; kij ¼ Ak e

j

;

z z E E ! aij ¼ Aa e iþj ; aij ¼ Aa e i#j

ð8:61Þ

where Ak and Aa are frequency factors. Each state lifetime tij is the inverse of the sum of the rate constants out of the state !

tij ¼ ðkij þ kij þ aij þ aij Þ#1 !

ð8:62Þ

and the transition probabilities for the individual steps are !

pðði þ 1Þj jij Þ ¼ kij tij ; pðði # 1Þj jij Þ ¼ kij tij ; !

pðiðjþ1Þ jij Þ ¼ aij tij ; pðiðj#1Þ jij Þ ¼ aij tij

ð8:63Þ

The probability of any path is the product of the probabilities of the transitions on the path, and the average time to complete a path is the sum of the lifetime of the states on the path. The net probability to reach the ‘‘m’’th site, where m = F, B, FR, BR, u, d, l, r, is the sum of all paths that lead from the central 0# state to the ‘‘m’’th state before arriving at any of the other 0# states, P

pðmjkÞ , , , pðjjiÞpðij0# Þ Z XX pðmjkÞ , , , pðjjiÞpðij0# Þ; Z¼ Pm ¼

S

m

ð8:64Þ

S

for m = F, B, FR, BR, u, d, l, r, and where SS indicates a sum over all paths from the center 0# to m, i.e., a sum over ‘‘i’’, the nearest neighbors to 0#, ‘‘j’’, the nearest neighbors to i, ‘‘k’’, the nearest neighbors to ‘‘m’’, and all states in between, but NOT including F, B, FR, BR, u, d, l, r. Generalizing this lattice approach to ever finer graining in the spatial coordinate, and to inclusion of more chemical states, presents computational but not conceptual hurdles. The general picture can be thought of in terms of the 5078 | Phys. Chem. Chem. Phys., 2007, 9, 5067–5083

diagram (8.65)

ð8:65Þ

The vertical path from the origin (0 - d) describes the chemical transformation S - P uncoupled to stepping, and the horizontal path (0 - r) describes stepping right uncoupled to the chemical process. The diagonal paths (0 - F) and (0 B) describe stepping right or left, respectively, coupled to chemistry S - P. Each path has a reverse (u, l, FR and BR) involving the reverse function. The overall function (the velocity in space and the rate of the chemical reaction) is obtained by summing the probabilities of paths and cycle times for each type of cycle. The weighting for each path is proportional to the product of the probabilities for the steps in that path so short paths are a priori more likely than longer paths. The cycle time is the sum of the lifetimes of the states on the path. The symmetry relations between forward and reverse paths are governed by thermodynamics. hPF i hPB i ¼ eðDm#DUÞ ; ¼ eðDmþDUÞ ; hPFR i hPBR i hPu i hPr i ¼ eDm ; ¼ e#DU hPd i hPl i

ð8:66Þ

There are also symmetry relations for the ‘‘last touch first touch’’ times between the starting point 0 and the endpoints.27,65 tF ¼ tFR ; tB ¼ tBR ; tu ¼ td ; tr ¼ tl

ð8:67Þ

In the periodic setting discussed here these are the cycle completion times—the average time of those cycles that start in the center and first reach the indicated periodically identical state F, B, etc. The symmetry relation for the forward and reverse last touch first touch times27,65 has been investigated recently in a number of contexts including membrane transport,66 single molecule enzymology,67 and molecular dynamics.68 The relations between the forward (F) and backward (B) paths are governed by the symmetry of the system and can be described in terms of kinetic splitting factors and the like. Designing a ‘‘coupling’’ machine is a matter of engineering the stabilities of the states and labilities of the transitions to maximize the probability for a diagonal transition and to minimize the probability for a vertical or horizontal transition. When DU = Dm = 0 the probability for completion of any forward path is exactly as likely as the probability for completion of the reverse of that path but the system still explores its energy landscape by thermal noise. When Dm, DU a 0 the two directions are not equally likely, and the relative likelihood is the exponential of Dm # DU, PF/PFR = eDm # DU. The nonzero values for Dm and DU also change the relative weightings This journal is

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of the other paths and so can lead to complicated and rich coupling. The function of a Brownian machine is specified entirely by the energies of the states (stabilities) and of the barriers between them (labilities). Structure is important only insofar as it determines the energies. The stochastic transitions of the resulting diffusive exploration of the energy landscape can give rise to mechanisms that rival deterministic processes in terms of their predictability, a feature discussed a quarter of a century ago by Charles Bennet in comparing Brownian computers with computers based on deterministic processes.69 Once the most probable path on the energy landscape has been determined experimentally it is very tempting to ‘‘project’’ this path onto a single dimension and call it ‘‘the mechanism’’. The transition constants for the mechanism are then used as fit parameters to capture the observed experimental behavior of the system. This is a very successful approach in many instances. It must always be remembered, however, that changing the chemical potential gradient Dm and the load DU can change the most probable path. This behavior is seen for the energy ratchet particularly. When DU is large enough that backstepping is induced, the ‘‘mechanism’’ changes. A striking prediction results—load induced backstepping should be stimulated by addition of substrate. This is in contrast to the prediction made based on a completely coupled mechanism established at smaller load where the backstepping would be retarded by addition of substrate. This prediction56 was recently experimentally confirmed for load induced backstepping of kinesin.57 Terrell Hill70 has given a very elegant and general approach for analysis of biochemical cycles such as those contained implicitly in diagram (8.65). He showed that the net number, nm, of completions for any cycle ‘‘i’’, no matter how complex the overall diagram in which it is embedded, can be described by a simple partial differential equation71 @Pðnm ; tÞ @ 2 Pðnm ; tÞ ! ¼ðom þ om Þ @t 2@n2m @Pðnm ; tÞ þ ðom # om Þ ; @nm !

ð8:68Þ

where P(nm, t) is the probability of having completed nm cycles of type ‘‘m’’ in a time t c tm, where tm is the average cycle completion time and m = F, B, d, or r. Eqn (8.68) (compare eqn (3.8)) has the solution Pðnm Þ ¼ with mean

2 1 2 pffiffiffiffiffi e#ðnm #mm Þ =2sm ; sm 2p

!

om om

¼ eðDGm Þ

ð8:72Þ

We have for the ratio of the probability to complete nm forward cycles to the probability to complete nm reverse cycles (i.e., #nm the time independent relation Pðnm Þ ¼ e#2nm tanh ðDGm =2Þ : Pð#nm Þ

ð8:73Þ

Eqn (8.68) is identical to Fick’s equation for diffusion with drift eqn (3.8). This equivalence suggests that a good physical model for understanding chemo-mechanical coupling by Brownian motors is given by diffusion of a charged particle in a microporous material under the action of gravity.72–75 An external load can be applied by an electric field orthogonal gravity as shown in Fig. 7.33,49 In moving one step to the right (left) the particle gains (loses) DUel = qELx electrical energy, where Lx is the spatial period in the horizontal direction and q is the charge on the particle. In moving one step up (down) the particle gains (loses) DUgr = mgLz electrical energy, where Lz is the spatial period in the vertical direction. The symmetry relations for the forward and reverse processes, and for the last touch first touch times, are the same as in eqn (8.64) and (8.67) where we substitute DUel for DU and DUgr for Dm. The ratio PF/PB on the other hand clearly depends on the shapes and positions of the obstacles. In the absence of inertia, dissipation plays the role of time. In Fig. 7 ‘‘down’’ is the direction of decreasing potential energy and the ‘‘natural’’ direction of motion. Note that this is opposite to the typical representation of ‘‘space–time’’ diagrams (Fig. 5) where forward time is, by convention, drawn in the upward direction. I have drawn paths corresponding to one forward path and the three symmetry related reverse, backward, and backward reverse paths. Gravity breaks the symmetry between the forward and forward reverse paths such that PF/PFR = e(mgLz # qELx) and between the backward and backward reverse paths such that PB/PBR = e(mgLz + qELx). As seen in the figure, symmetry is also broken between the forward and backward path by the asymmetric obstacles. The backward path corresponding to the forward path drawn is simply physically blocked—the particle cannot penetrate the obstacle. The ratio

ð8:69Þ

!

ð8:70Þ

!

ð8:71Þ

mm ¼ ðom # om Þt

change DGm due to one forward step

and variance s2m ¼ ðom þ om Þt

The constants (~ om) and (om ) are effective net rate constants for the forward and reverse cycles, respectively, and are generally rather complicated functions of all individual rate constants of the overall diagram. The ratio of the two however is a simple function—the exponential of the net free energy This journal is

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Fig. 7 Illustration of a microporous sieve that acts as a 2-dimensional ratchet. Particles falling due to gravity move, on average, to the right because of the asymmetry of the obstacles. It is possible to use this effect to couple gravity to do work against a weak applied field.

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PF/PB depends on the shape and spacing of the obstacles, and the diffusivity and electrical mobility of the particle. As the particle diffuses downward due to gravity, the triangular obstacles tend to shepherd it to the right by blocking backward paths even though the field tends to move the particle left. If DUgr 4 DUel, and the obstacles sufficiently asymmetric, it is possible for gravity to drive the positively charged particle against an applied voltage. The mechanism works to convert gravitational energy to electrical energy not by using the force of gravity to increase the probability for a trajectory to the right (to F) but by decreasing the probability for a trajectory to the left (to B). This process can be understood in terms of Onsager’s theory of coupled flows and forces,13 but we need to go beyond the regime linear in the applied forces.49 In Fig. 7, the linear coupling terms are identically zero by symmetry. Let us return to the original question addressed in section 2, is the particle in equilibrium as it wends its way through the maze of obstacles in Fig. 7. Once again, the answer depends on how we view the system. Certainly the particle loses gravitational potential energy as it falls, and when it moves right it gains electrical energy. The difference appears as heat in the environment, so there is net dissipation and there are flows of energy, heat and matter. Further, there is no guarantee that the fluctuations of the position of the particle, or of the energy dissipated are Gaussian about the expected average values. Neither are the velocities in the horizontal and vertical direc~ tions generally linear functions of the external forces g and E, respectively, i.e., the system is not necessarily in the linear response regime. Clearly the system is out of, and depending on the magnitudes of the forces, may be far from, thermodynamic equilibrium. On the other hand, because of viscous drag there is NO acceleration and the fluctuations of the force acting between the particle and medium (e.g. water) due to thermal noise (e in eqn (5.15)) are assured to be Gaussian by the central limit theorem. There are 1021 collisions between the particle and solvent molecules each second!76 Further, since there is no inertia, the system fulfils the requirement given by Onsager for application of his thermodynamic action theory, ‘‘the velocity of the particle is at every instant a linear function of the local force that causes it’’. We conclude that the particle is at every instant arbitrarily close to mechanical equilibrium.

9. Brownian machines and biomolecular motors Many protein motors (e.g., kinesin or myosin) look structurally like mechanical devices. It is very tempting to view the chemical kinetic lattice models, with rate and equilibrium constants but no direct structural connection, as temporary, phenomenological models awaiting further structural studies that will surely facilitate development of a ‘‘real’’ mechanical theory. However, bio-motors and other biomolecular machines undergo an enormous number of collisions with water molecules each second.76 Despite the impression left by the beautiful and elegant static structures that so often decorate the covers of Nature, Science, Proceedings of the National Academy of Science, USA and many other journals, many 5080 | Phys. Chem. Chem. Phys., 2007, 9, 5067–5083

Fig. 8 On the left, a representation of a deterministic macroscopic machine, and on the right an ineluctably probabilistic quantum mechanical process, reflection of a photon. Between these is a cartoon of kinesin, a biomolecular motor. Is it best described by macroscopic mechanics, or inherently stochastic laws of chemical kinetics.

biochemists have recognized that a protein is a ‘‘kicking and screaming stochastic molecule’’.77 In Fig. 8 we see a macroscopic object, representing a machine, on the right, and on the left a quantum mechanical process, the interaction of a photon with a partially reflecting mirror. In order to understand the working of a macroscopic machine comprised of cogs, levers, pulleys, and the like it is necessary to know how the components are connected to one another—i.e., to know the structure of the machine. The effect of an applied force can then be calculated deterministically. In contrast, quantum mechanical processes are fundamentally probabilistic. We can give only the probability that a single photon will reflect in any given experiment. On average, one out of four photons is reflected, but ‘‘which one’’, and the outcome of any single experiment with a single photon, is simply a matter of chance. In between these two extremes is shown a representation of kinesin, a biomolecular motor that uses energy from ATP hydrolysis to move preferentially toward the ‘‘+’’ end of its bio-polymeric track. How are we to think about such a nanoscale machine? Is it best modeled in analogy with a macroscopic machine, where evolution has somehow found a way to win the fight against thermal noise and viscous drag, or is it best modeled as a stochastic system, undergoing a random walk that harnesses thermal noise constructively in the mechanism? The current paradigm in biophysics and structural biology is that biomolecular motors can be treated as miniaturized macroscopic machines, best understood in terms of the arrangement and connections of the component atoms. The expectation is that mechanical forces arising from chemical conversion of ATP to ADP under ‘‘non-equilibrium’’ conditions are propagated and amplified by structural elements in the protein to give rise to directed motion along the microtubule track. Thus, it is argued that by obtaining the structures of the motor in all chemical states to sufficient resolution, a mechanism similar in detail to the one by which we understand the workings of a wristwatch should be attainable. The unavoidable, but unwanted, effects of thermal noise are manifest in this picture by occasional backward steps, and by occasional hydrolysis of ATP without a forward step, both processes that decrease the efficiency of the mechanism. In contrast we have given here a picture of a molecular motor as an object in mechanical equilibrium undergoing a random walk on a lattice of states that are distinguished from This journal is

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one another by differing spatial positions within a period (mechanical coordinate) and by differing chemical substituents (chemical coordinate). The equilibrium energies of the states and of the barriers between the states constrain the motion on the multi-dimensional lattice to one or a few paths in which completion of a chemical cycle is coupled to moving forward by a spatial period. The function of a Brownian machine is specified entirely by the equilibrium energies of the states (stabilities) and of the barriers between them (labilities). Structure is important only insofar as it determines the energies. Addition of the chemical substrate such that the free energy released is greater than the increase in the potential energy due to motion against an applied load causes the system to undergo directed motion by mass action. We can gain insight into this question of whether a mechanical or a statistical picture is most appropriate for describing biomolecular motors by comparing a very small macroscopic motor with a biomolecular motor. In his now famous after dinner talk, ‘‘Plenty of Room at the Bottom’’ Richard Feynman issued a challenge to build a motor that, not counting the power supply and connecting wires, would fit into a cube 1/64th of an inch (a bit less than 1/2 a millimeter) on a side. This challenge was successfully accomplished by an engineer, William McClellan, only a year later. It is reported that Feynman was disappointed that no new principles were applied, McClellan’s motor was simply a tour de force of miniaturization. When viewed under a microscope without any source of external energy the motor does absolutely nothing—it simply sits there, totally still. Imagine, now, that we look at a single kinesin molecule 10 nm on a side at chemical equilibrium (mATP = mADP). We see a very different picture. The molecule is vigorously moving about on its microtubule track because of thermal noise, sometimes stepping left, sometimes stepping right, sometimes binding ATP, sometimes binding ADP, sometimes hydrolyzing ATP, sometimes making ATP from ADP. The chemical equilibrium is maintained not by a static opposition of equal magnitude forces, but by dynamic processes in which every forward motion is exactly as likely as the microscopic reverse of that motion. What changes when we add more ATP so that the system is no longer at chemical equilibrium? Are the accessible states of the protein different? Certainly not, there is no way for a protein to know what the chemical potential of ATP in the bulk solution is, the protein senses only whether it is bound or it is not bound to ATP. Do the transitions between the states of the protein have a different character when the chemical potential of ATP is higher than that of ADP and Pi in the bulk solution? For the same reason—the protein cannot be directly influenced by the chemical potentials of reactants in the bulk—the answer is also certainly no, the character of a transition between two states of the protein when the chemical reaction is away from equilibrium is exactly the same as the character of that transition when ATP is at equilibrium with ADP and Pi. Logically, the only difference in the presence of excess ATP is that a motor molecule that is not bound to ATP has a greater chance of binding to ATP. How does this increase in the rate of binding ATP translate into biasing the random walk of the motor protein so strongly that the motor This journal is

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Fig. 9 Energy diagram illustrating how binding ATP changes the relative energies of two states. Thermal noise provides the mechanism by which the system escapes the ATP destabilized state, and the direction of escape is controlled by the relative barrier heights.

can do work on the environment with almost unit efficiency and where the motor molecule takes almost one step for each ATP that is hydrolyzed? The answer can be found only in a statistical theory such as that outlined in the preceding sections. The energies of some barriers and states (Fig. 9) depend on whether ATP is or is not bound, but when we ask what causes the system to leave a high energy state and go to a lower energy state there is only one answer—thermal noise. Certainly it is important to understand structurally the allosteric mechanisms by which equilibrium energies of states and barriers depend on whether ATP is present at the active site. We should not, however, expect any mechanical description of how ATP hydrolysis causes a force at the active site that propagates through the molecule thus driving a ‘‘power stroke’’. We are left with the perhaps unsatisfying, but nonetheless accurate statement, that ATP hydrolysis drives directed motion by mass action. The energetic picture in Fig. 9 used to describe how, in a Brownian motor mechanism, ATP hydrolysis drives directed motion is precisely the same as common descriptions of the mechanism by which ion pumps use ATP hydrolysis to drive uphill transport of ions from low to high chemical potential reservoirs.78–80 ATP induces a conformational transition (i.e., changes the relative stabilities of the states) in which a protein goes from a high affinity state with access between the ion binding site and the low chemical potential reservoir to a low affinity state with access between the ion binding site and the high chemical potential reservoir.81,82 As Peter Lauger pointed out: ‘‘Ion pumps do not function by a power stroke mechanism; instead, pump operation involves transitions between molecular states, each of which is very close to thermal equilibrium with respect to its internal degrees of freedom, even at very large overall driving force.’’83 Finally the energetic picture is also essentially the same as that describing how chemical manipulation can drive synthetic molecular motors in a specific direction.14,18,62 Ion pumps and biomolecular motors do not share any particular structural homology, and the synthetic motors are not even made of amino acids. The fundamental aspects of how a macromolecule can mediate inter-conversion of chemical and electrical or mechanical energy is not based on any specific structural features, but rather on common energetic Phys. Chem. Chem. Phys., 2007, 9, 5067–5083 | 5081

motifs of ligand induced changes in state energies (stabilities) and barrier energies (labilities).

10. Conclusions—design principles for machines at the nanoscale There are two main approaches to the design and operation of nanoscale machines.14 One approach can be considered an adaptation of the principles of macroscopic machines, in which a molecule is engineered to be rigid in all degrees of freedom except that for the desired motion. External energy is used to ‘‘push’’ the machine through the sequence of configurations that defines the function. Examples of this approach include crystalline molecular ‘‘gyroscopes’’,17,84 nano-cars,85 and molecular rotors.16 An analogous macroscopically inspired mechanical approach to synthesis of nanoscale molecular complexes has been proposed and called a molecular assembler.86 The mechanical approach is similar to the way a fine wristwatch is made—a highly skilled watchmaker painstakingly snaps together each individual component to create the final working watch. I am sure that every watchmaker on a particularly clumsy day has felt cursed with fat sticky fingers, a beautifully evocative analogy used by Smalley87 to illustrate some problems associated with controlled mechanical assembly at the nanoscale. In designing synthetic approaches and nano-machines along the lines of macroscopic mechanics, it is desirable to minimize friction, specific and non-specific chemical interactions, and van der Waals forces, and to reduce insofar as is possible the effects of thermal noise. Consequently, most examples of nanoscale machines based on mechanical principles are carried out in the solid state, or at the gas surface interface. Brownian molecular machines, on the other hand, are first and foremost molecules and are governed by the laws of chemistry rather than of mechanics. Examples of such synthetic soft-matter systems include surface liquid crystal rotors88 solution phase organic rotors,89,90 DNA walkers,91–94 and catenanes.18,62,63 The dynamical behavior of machines based on chemical principles can be described as a random walk on a network of states. In contrast to macroscopic machines whose function is determined predominately by the connections between the elements of the machine, the function of a Brownian machine in response to an external stimulus is completely specified by the equilibrium energies of the states and of the heights of the barriers between them. Chemists have long experience with approaches for controlling stabilities of molecules, and for designing systems sterically or energetically hindered pathways allowing for kinetic rather than thermodynamic control of mechanisms. This experience will be crucial in the next steps of interfacing synthetic molecular machines with the macroscopic world. A key difference between Brownian and mechanical approaches is that, due to thermal noise, a nanoscale system explores all possible motions and configurations. This feature admits a uniquely chemical approach to controlling motion at the nanocale. By using chemical design and input energy to constrain Brownian motion, and to prevent motion that we do not want, what is left behind is the motion that we do want.95 5082 | Phys. Chem. Chem. Phys., 2007, 9, 5067–5083

References 1 M. Rief, M. Gautel, F. Oesterhelt, J. M. Fernandez and H. E. Gaub, Science, 1997, 276, 1109–1112. 2 H. Yin, M. D. Wang, K. Svoboda, R. Landick, S. M. Block and J. Gelles, Science, 1995, 270, 1653–1657. 3 E. Purcell, Am. J. Phys., 1977, 45, 3–11. 4 G. N. Bochkov and Yu. E. Kuzovlev, Sov. Phys. JETP, 1977, 45, 125–130. 5 G. N. Bochkov and Y. E. Kuzovlev, Physica A, 1981, 106, 443–479. 6 L. Onsager and S. Machlup, Phys. Rev., 1953, 91, 1505–1512. 7 G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles and D. J. Evans, Phys. Rev. Lett., 2002, 89, 050601-1–4. 8 R. D. Astumian, P. B. Chock, T. Y. Tsong and H. V. Westerhoff, Phys. Rev. A: At., Mol., Opt. Phys., 1989, 39, 6416. 9 R. D. Astumian, Science, 1997, 276, 917–922. 10 F. Julicher, A. Ajdari and J. Prost, Rev. Mod. Phys., 1997, 69, 1269–1281. 11 R. D. Astumian and P. Hanggi, Phys. Today, 2002, 55(11), 33–39. 12 P. Reimann, Phys. Rep., 2002, 361, 57–265. 13 L. Onsager, Phys. Rev., 1931, 37, 405–426. 14 E. R. Kay, D. A. Leigh and F. Zerbetto, Angew. Chem., Int. Ed., 2007, 46, 72–191. 15 W. R. Browne and B. L. Feringa, Nat. Nanotechnol., 2006, 1, 25–35. 16 G. S. Kottas, L. I. Clarke, D. Hornik and J. Michl, Chem. Rev., 2005, 105, 1281–1376. 17 M. A. Garcia-Garibay, Proc. Natl. Acad. Sci. U. S. A., 2005, 102, 10771–10776. 18 M. N. Chatterjee, E. R. Kay and D. A. Leigh, J. Am. Chem. Soc., 2006, 128, 4058–4073. 19 S.-H. Lee, K. Ladavac, M. Polin and D. G. Grier, Phys. Rev. Lett., 2005, 94, 110601. 20 J. Perrin, Ann. Chim. Phys., 1909, 18, 5–114. 21 K. Ito, T. Ieki and N. Ise, Langmuir, 1992, 8, 2952–2957. 22 M. D. Haw, J. Phys.: Condens. Matter, 2002, 14, 7769–7779. 23 R. D. Astumian, Am. J. Phys., 2006, 74, 683–688. 24 H. Berg, Random Walks in Biology, Princeton University Press, Princeton, NJ, 1983. 25 M. Smoluchowski, in Vortrage uber kinetische Theorie der Materie und der Elektrizitat (Mathematische Vorlesungen an der Universitat Gottingen, VI), ed., B. G. Teubner, Leipzig, 1914, pp. 87–105, translation in Usp. Fiz. Nauk, 1967, 93, 724–748. 26 A. Einstein, Ann. Phys. (Berlin), 1905, 17, 549–560. 27 M. Bier, I. Derenyi, M. Kostur and R. D. Astumian, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 1999, 59, 6422–6432. 28 A. Ajdari and J. Prost, C. R. Acad. Sci., Ser. II, 1992, 315, 1635. 29 R. D. Astumian and M. Bier, Phys. Rev. Lett., 1994, 72, 1766. 30 J. Prost, J. F. Chawin, L. Peliti and A. Ajdari, Phys. Rev. Lett., 1994, 72, 2652. 31 M. Bier and R. D. Astumian, Phys. Rev. Lett., 1996, 76, 4277–4280. 32 J. S. Bader, R. W. Hammond, S. A. Henck, M. W. Deem, G. A. McDermott, J. M. Bustillo, J. W. Simpson, G. T. Mulhern and J. M. Rothberg, Proc. Natl. Acad. Sci. U. S. A., 1999, 96, 13165–13169. 33 I. Derenyi, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 1998, 58, 7781–7784. 34 A. van Oudenaarden and S. G. Boxer, Science, 1999, 285, 1046–1048. 35 T. A. J. Duke and R. H. Austin, Phys. Rev. Lett., 1998, 80, 1552–1555. 36 R. D. Astumian, Phys. Rev. E, 2007, 6, 020102(R). 37 D. T. Gillespie, Am. J. Phys., 1995, 64, 225–239. 38 G. Crooks, J. Stat. Phys., 1998, 90, 1481–1493. 39 M. Bier and R. D. Astumian, Phys. Lett. A, 1998, 247, 385–390. 40 M. Tarlie and R. D. Astumian, Proc. Natl. Acad. Sci. U. S. A., 1998, 95, 2039–2043. 41 T. Y. Tsong and R. D. Astumian, Bioelectrochem. Bioenerg., 1986, 15, 457–476. 42 H. V. Westerhoff, T. Y. Tsong, P. B. Chock, Y.-der Chen and R. D. Astumian, Proc. Natl. Acad. Sci. U. S. A., 1986, 83, 4734–4738.

This journal is

' c

the Owner Societies 2007

43 R. D. Astumian, P. B. Chock, T. Y. Tsong, Y.-der Chen and H. V. Westerhoff, Proc. Natl. Acad. Sci. U. S. A., 1987, 84, 434–438. 44 R. D. Astumian and B. Robertson, J. Chem. Phys., 1989, 91, 4891–4901. 45 R. D. Astumian, Proc. Natl. Acad. Sci. U. S. A., 2005, 102, 1843–1847. 46 J. Vacek and J. Michl, Proc. Natl. Acad. Sci. U. S. A., 2001, 98, 5481–5486. 47 D. Horinek and J. Michl, J. Am. Chem. Soc., 2003, 125, 11900–11910. 48 D. Horinek and J. Michl, Proc. Natl. Acad. Sci. U. S. A., 2005, 102, 14175–14180. 49 M. Bier, M. Kostur, I. Derenyi and R. D. Astumian, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 2000, 265, 7184–7187. 50 D. S. Liu, R. D. Astumian and T. Y. Tsong, J. Biol. Chem., 1990, 265, 7260. 51 T. D. Xie, P. Marszalek, Y. D. Chen and T. Y. Tsong, Biophys. J., 1994, 67, 1247. 52 T. D. Xie, Y. D. Chen, P. Marszalek and T. Y. Tsong, Biophys. J., 1997, 76, 2496. 53 B. Robertson and R. D. Astumian, J. Chem. Phys., 1991, 94, 7414. 54 R. D. Astumian, J. Phys.: Condens. Matter, 2005, 17, S3753–S3766. 55 R. D. Astumian and I. Derenyi, Eur. Biophys. J., 1998, 27, 474–489. 56 R. D. Astumian and M. Bier, Biophys. J., 1996, 70, 637–653. 57 N. J. Carter and R. A. Cross, Nature, 2005, 435–439, 308–312. 58 R. D. Astumian and I. Derenyi, Biophys. J., 1999, 77, 993–1002. 59 I. Derenyi, M. Bier and R. D. Astumian, Phys. Rev. Lett., 1999, 83, 903. 60 H. Itoh, A. Takahashi, K. Adachi, H. Noji, R. Yasuda, M. Yoshida and K. Kinosita, Jr, Nature, 2003, 427, 465–468. 61 D. A. Leigh, J. K. Y. Wong, F. Dehez and F. Zerbetto, Nature, 2003, 424, 174–179. 62 J. V. Hernandez, E. R. Kay and D. A. Leigh, Science, 2004, 306, 1532–1537. 63 V. Serreli, C.-F. Lee, E. R. Kay and D. A. Leigh, Nature, 2007, 445, 523–527. 64 J. Bern, D. A. Leigh, M. Lubomska, S. M. Mendoza, E. M. P%rez, P. Rudolf, G. Teobaldi and F. Zerbetto, Nat. Mater., 2005, 4, 704–710. 65 I. Derenyi and R. D. Astumian, Phys. Rev. Lett., 1999, 82, 2623. 66 A. M. Berezhkovskii, G. Hummer and S. M. Bezrukov, Phys. Rev. Lett., 2006, 97, 020601. 67 H. Qian and S. X. Xie, Phys. Rev. E, 2006, 74, 010902.

This journal is

' c

the Owner Societies 2007

68 B. W. Zhang, D. Jasnow and D. W. Zuckerman, J. Chem. Phys., 2007, 126, 074504. 69 C. H. Bennet, Int. J. Theor. Phys., 1982, 21, 905–922. 70 T. L. Hill, Free Energy Transduction in Biology, Academic Press, New York, 1977. 71 T. L. Hill, Biochemistry, 1975, 14, 2127–2137. 72 S. Matthias and F. Muller, Nature, 2003, 424, 53–55. 73 J. E. Villegas, S. Savelev, F. Nori, E. M. Gonzalez, J. V. Anguita, R. Garcia and J. L. Vicent, Science, 2003, 302, 1188–1191. 74 J. F. Wambaugh, C. Reichhardt, C. J. Olson, F. Marchesoni and Franco Nori, Phys. Rev. Lett., 1999, 83, 5106–5109. 75 A. M. Song, Appl. Phys. A: Mater. Sci. Process., 2002, 75, 229–235. 76 S. Chandrasekhar, Rev. Mod. Phys., 1949, 21, 383. 77 G. Weber, Adv. Protein Chem., 1975, 29, 1–83. 78 T. Y. Tsong and R. D. Astumian, Annu. Rev. Physiol., 1988, 50, 273–290. 79 T. Y. Tsong and R. D. Astumian, Prog. Biophys. Mol. Biol., 1987, 50, 1–45. 80 R. D. Astumian, Phys. Rev. Lett., 2003, 91, 118102. 81 R. J. Naftalin, N. Green and P. Cunningham, Biophys. J., 2007, 92, 3474–3491. 82 W. P. Jencks, Methods Enzymol., 1989, 171, 145–173. 83 P. Lauger, Electrogenic Ion Pumps, Sinauer, Sunderland, MA, 1991. 84 T.-A. V. Khuong, H. Dang, P. D. Jarowski, E. F. Maverick and M. A. Garcia-Garibay, J. Am. Chem. Soc., 2007, 129, 839–845. 85 Y. Shirai, A. J. Osgood, Y. Zhao, K. F. Kelly and J. M. Tour, Nano Lett., 2005, 5, 2330–2334. 86 (a) K. E. Drexler, Engines of Creation: The Coming Era of Nanotechnology, Fourth Estate, London, 1990; (b) K. E. Drexler, Nanosystems: Molecular Machinery, Manufacturing and Computation, Wiley, Chichester, 1992. 87 R. E. Smalley, Sci. Am., 2001, 285(3), 68–69. 88 P. Palffy-Muhoray, T. Kosa and E. Weinan, Appl. Phys. A: Mater. Sci. Process., 2002, 293–300. 89 T. R. Kelly, H. De Silva and R. A. Silva, Nature, 1999, 150152. 90 B. L. Feringa, N. Koumura, R. A. van Delden and M. K. J. ter Wiel, Appl. Phys. A: Mater. Sci. Process., 2002, 75, 301–308. 91 W. B. Sherman and N. C. Seeman, Nano Lett., 2004, 4, 1203–1207. 92 J. S. Shin and N. A. Pierce, J. Am. Chem. Soc., 2004, 126, 10834–10835. 93 T. Ye and C. Mao, J. Am. Chem. Soc., 2004, 126, 11410–11411. 94 A. J. Turberfield, J. C. Mitchell, B. Yurke, A. P. Mills, M. I. Blakey and F. C. Simmel, Phys. Rev. Lett., 2003, 90, 118102. 95 R. D. Astumian, Sci. Am., 2001, 285(1), 56–64.

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