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Astumian 1

Stochastic Conformational Pumping: a Mechanism

for Free-Energy Transduction by Molecules

R. D. ASTUMIAN

Department of Physics, University of Maine, Orono, ME 04469 USA

Key Words Free-energy transduction, Fluctuating enzymes, Molecular motors, Molec-

ular pumps, Microscopic reversibility, fluctuation-dissipation theorem

Abstract Proteins and other macromolecules can act as molecular machines that convert energy from one

form to another through cycles of conformational transitions. In a macroscopically fluctuating environment

or at the single molecule level the probability for a molecule to be in any state j fluctuates, and the

probability current from any other state i to state j is given as the sum of a steady state current and

a pumped current, Iij = Issij + FijdPjdt where Fij is the fraction of the fluctuating current into/out of

state j coming directly from state i, and dPjdt is the rate of change of the probability for the molecule

to be in state j. If the fluctuations arise from an equilibrium source microscopic reversibility guarantees

that the time average of the pumped current is zero. If however the fluctuations arise due to the action of a

non-equilibrium source the time average of the pumped current is not in general zero, and can be opposite

in sign to the steady state current. The pumped current provides a mechanism by which fluctuations arising

either from an internal non-equilibrium chemical reaction or from an external driving can do electrical,

mechanical, or chemical work on a system by coupling into the equilibrium conformational transitions of a

protein. In this paper I will review work done on elaborating the mechanism of stochastic pumping and also

discuss a thermodynamically consistent approach for modeling the effects of dynamic disorder on enzymes

and other proteins.

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CONTENTS

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

AN ADIABATICALLY PUMPED MOLECULAR MACHINE . . . . . . . . . . . . . . . . 4

3-Catenance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Adiabatic pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Preventing backward motion - why a 2-Catenane doesnt work . . . . . . . . . . . . . . . . . . . 8

STOCHASTIC PUMPING ACROSS MEMBRANES . . . . . . . . . . . . . . . . . . . . 9

Kinetic mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

pumped currents in the small perturbation limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

FLUCTUATING PROTEINS AND DYNAMIC DISORDER . . . . . . . . . . . . . . . . . 14

Stochastic Pumping and ATP driven pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Maxwells and Smoluchowskis demons: Engineering with bi-lability and bi-stability . . . . . . 16

Developing a thermodynamically consistent model for molecular fluctuations and dynamic disorder 17

MICROSCOPIC REVERSIBILITY AND CONFORMATIONAL TRANSITIONS . . . . . 19

Myoglobin binding and dissociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Cycles of molecular machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Molecular machines in a thermal environment. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Catalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

SUMMARY POINTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1 INTRODUCTION

The ability to convert free-energy from one form to another is essential for life. Our cells

store energy in chemical form (often Adenosine Triphosphate, ATP, or Guanosine Triphos-

2

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Astumian 3

phate, GTP) and then use the energy released by catalytic breakdown of these molecules

to do various types of work, e.g., to pump ions from low to high electrochemical potential,

to polymerize and synthesize necessary macro-molecules, or to power motion of molecu-

lar motors to move material from one place to another (Fig. 1). Despite the fundamental

similarity between these different energy conversion processes, the function of a molecu-

lar motor is typically described in a very different way than is the function of a molecular

pump or a synthase or polymerase.

Models for molecular motors (39,64), e.g., have focussed on an ATP driven mechani-

cal power stroke - a viscoelastic relaxation process where the protein starts from a non-

equilibrium, strained conformation due to the action of ATP at the active site. The sub-

sequent relaxation following product (ADP or Pi) release can be visualized much as the

contraction of a stretched rubber band.

In contrast, molecular pumps are most often modeled in terms of chemical kinetics (46),

where ATP energy is used to change the relative affinities of and barrier heights between

binding sites by sequentially favoring different conformational states of the protein as ATP

is bound, hydrolyzed, and the products released. The conformational relaxation and molec-

ular transport across the membrane are thermally activated steps.

The perspective I will develop in this review is that at the single molecule level molec-

ularmachines are mechanically equilibrated systems that serve as conduits for the flow of

energy between a source such as an external field or a non-equilibrium chemical reaction

and the environment on which work is to be done (9). Consider the process

o / B o / okAi

/ A okBA

kAB/ B okBj / o / A o /

where A and B are states in the conformational cycle of one of the free-energy transduction

processes shown in Fig. 1. We normally focus on the steady state probability current

IssAB = kABPssA kBAP

ssB to assess the average direction of cycling - whether the net flow

is to the left or to the right. In a fluctuating environment or at the single molecule level,

however, we must add a term that reflects the correlations between the fluctuating rates and

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4 Stochastic Pumping

fluctuating probabilities

IAB(t) = IssAB(t) + FAB(t)dPB(t)dt . (1)

where FAB, is the fraction of the transient change in the probability to be in state B,

dPBdt, coming directly from (to) state A. (5, 8, 38). If the fluctuations in FAB and inPB are uncorrelated the long time average of the second term, the pumped current, is

zero. If, however, the system is pumped, e.g. by simultaneously raising and lowering the

free energy of state B and the energy barrier between B and A (and hence modulating kAB

and kBj, but with the product kABkBj constant) the average pumped flux is positive since

FAB

(t

)is greater than 1/2 when PB is increasing and FAB

(t

)is less than 1/2 when PB is

decreasing (17).

The physical motion by which a single molecule of protein in state A is converted to B is

the same irrespective of how different the probability ratio PAPB is from the equilibriumconstant KAB = kABkBA or whether the system is pumped or not. The conformationaltransitions in the cycles by which these machines carry out their function are intrinsically

equilibrium processes between states that are very close to thermal equilibrium. A time-

dependent external energy source or a non-equilibrium chemical reaction modulates the

relative stabilities of the states in the cycle and the rate constants for the conformational

transitions between them in a correlated way thereby driving net flux through the cycle and

the performance of work by a mechanism known as stochastic pumping (69).

2 AN ADIABATICALLY PUMPED MOLECULAR MACHINE

A particularly simple example of stochastic pumping involves a recently synthesized cate-

nane based molecular motor (47). Catenanes are molecules with two or more interlocked

rings. Fig. 2a shows a 3-catenane (three interlocked rings) and Fig. 2b illustrates a 2-

catenane (two interlocked rings). The salient feature of these two molecules is that binding

sites or bases (the blue, red, and green boxes labelled 1,2, and 3, respectively) for the

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Astumian 5

small purple rings can be designed and located on the large orange ring. The bases can

be designed such that their interaction energies with the purple rings can be independently

externally controlled, e.g. by protonation/de-protonation or by oxidation/reduction (37). A

sequence of external cyclical changes to the interaction energies of the sites can squeeze

the purple rings to undergo directional rotation by a mechanism similar to peristalsis (6)

but where the transitions occur by thermal noise so the system operates as a Brownian

motor (3, 15).

2.1 3-Catenance

Let us first consider the three ring catenane shown in Fig. 2 a). The larger yellow ring has

three distinct recognition stations, labeled 1,2, and 3, for the two identical purple rings.

The purple rings cannot pass one another, nor can they occupy the same station, as they

make thermally activated transitions from one station to another. Thus there are a total of

three distinguishable states, labeled A,B, and C. The interaction between a purple ring with

a station is characterized by an interaction energy E1, E2, E3 0. Each transition involves

breaking the interaction between one station and one ring. For example the transitions

from state A to state B requires breaking the interaction of the ring on site 1, as does the

transition from state C to B. Using this analysis the rate constants for the transitions are

kAB = kCB =Ae(E1E

)

kBC = kAC =Ae(E3E

) (2)

kCA = kBA =Ae(E2E

)

where A is a frequency factor, = (kBT)1 is the inverse of the thermal energy, andE > 0 is the energy of the barrier between the stations, which we assume to be the same

for all transitions.

Periodic modulation of the energies E1, E2, and E3, even modulation carried out so

slowly that the state probabilities are given by an equilibrium relation (PiPj = kjikij for

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i,j = A,B,C) at every instant, can drive directional rotation of the small rings about the

larger ring. Beginning with E1, E3

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Astumian 7

where is the frequency of the modulation. Under adiabatic conditions (very slow modu-

lation) the state probabilities are at equilibrium at every instant and the fraction FadAB of the

change in PeqB directly to/from A is independent of whether P

eqB is increasing or decreas-

ing,

FadAB =kBA

kBA + kBC=

kABPeqA

kABPeqA + kCBP

eqC

(4)

We find FadAB = (1 + e(E3E2))1 and PeqB = 1 + e(E1E2) + e(E1E3)1 for the rateconstants in Eq. (1). The adiabatic integrated current per cycle FadABdPeqB - the area

enclosed in a plot of FadAB vs. PeqB parametrized by time - is purely geometric and does

not depend on frequency. Further, since both FadAB and PeqB can vary at most between 0

and 1, the limit is one cycle through the states per cycle of modulation, and 1 turn for each

ring per 2 cycles of the modulation (8, 30). The rate constants Eq. (1) obey the simple

relation kABkBCkCA = kACkCBkBA irrespective of the values of the energies E1, E2, E3.

This relation is a necessary and sufficient condition for the steady state component of

the current to be zero, IssAB = 0 but, as we have seen, in a fluctuating environment this

relation between the rate constants, despite assertions in the literature to the contrary, is

not a sufficient condition to assure that the average net cyclic flux, IAB is zero (19,61, 80).

The stochastic pumping of the 3-catenane can be implemented experimentally by out-of-

phase modulation of the pH and reduction/oxidation (redox) potential as discussed in (8)

(Fig. 2c), where the energy E1 is controlled by the redox potential and the energy E2 is

controlled by the pH (Fig. 2c).

The geometric effect by which pumping is achieved for a 3-catenane is what Michael

Berry has termed the geometric phase (24). A similar picture emerges for a wide variety of

physical phenomena including dissipationless pumping of electrons (14,72, 73), a mecha-

nism for bio-molecular ion pumps (5, 70), phase control in oscillating chemical reactions

(42), and swimming at low Reynolds number (59, 67).

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2.3 Preventing backward motion - why a 2-Catenane doesnt work

The pumping mechanism for the 3-catenane based molecular motor illustrates an impor-

tant principle for molecular machines that is very different than the mechanism by which

macroscopic machines function. In a macroscopic machine input energy is used to cause

the desired motion, and without the energy there would be no motion at all. For molecular

machines, however, the parts of the machine are constantly moving about even at thermal

equilibrium. The design of the 3-catenane motor focuses on restricting or preventing the

undesired parts of the thermal motion (4,10). In each transition, one of the small rings acts

as an immobile obstacle, thereby setting the direction of motion of the other ring when,

by thermal noise, the mobile ring moves from an unstable site to a stable site. By restrict-

ing the backward motion, only the desired motion remains - a concept often described as

biased brownian motion.

We can better understand this concept by contrasting the case of the 3-catenane with

that of a 2-catenane which cannot be induced to undergo directional cycling by adiabatic

modulation of the interaction energies between the yellow ring and the bases. For the

2-catenane in Fig. 2b the rate constants out of any state are the same for clockwise and

counterclockwise transition

kAB = kAC =Ae(E1E

)

kBC = kBA =Ae(E3E

) (5)

kCA = kCB =Ae(E2E

)

With these rate constants we have FadAB = 12, a constant, and thus there is no possibilityfor directional adiabatic pumping. When, e.g., E1 is increased the probability for transi-

tion out of state A to state C is exactly the same as the probability for transition to state B.

Non-adiabatic pumping (14) at higher frequencies is possible for the 2-catenane (30, 60),

however, since the state probabilities appearing in FAB are no longer given by their equi-

librium values and are frequency dependent. A very general and easily implemented com-

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Astumian 9

putational scheme for calculation of the instantaneous state probabilities and currents for

a cyclic system with arbitrarily large amplitude and frequency modulation was given by

Robertson and Astumian (61). The 3-Catenane three state model is perhaps the simplest

example of adiabatic stochastic pumping where only binding site energies are modulated.

The directionality requires the interaction between the two purple rings. If in addition to a

binding site energy a barrier energy can be directly modulated a two-state model for pump-

ing is possible (16,56) where both adiabatic and non-adiabatic components of the pumping

can be analytically evaluated. In the next section we will focus on such a two-state model

for a membrane pump which is also analogous to the Michaelis-Menten mechanism for

enzyme catalysis.

3 STOCHASTIC PUMPING ACROSS MEMBRANES

ATP driven pumps are proteins that span a cell or organelle membrane and use energy

from adenosine triphosphate (ATP) hydrolysis to pump ligand (often ions such as Na+,

K+, H+, or Ca++) across the membrane, thus generating and maintaining the ion electro-

chemical gradients essential for life (46). In a simple picture of a membrane pump (figure

1b), the protein structure presents energy barriers (gates) for ligand permeation at the two

entrances, one on either side of the membrane, surrounding an energy well (binding site)

in the middle. Fig. 2a shows an energy diagram for this two-barrier, one-site model of an

ion transporter. The differential barrier height u and well energy are internal parameters

(27) controlled by the conformation of the protein and do not influence the overall trans-

port equilibrium = 1 2. In the absence of input energy, ions flow from high to

low electrochemical potential. Conformational fluctuations of the pump protein cause the

relative energies of the two gates, as well as the binding energy (well depth) for ligand, to

fluctuate - i.e., both (t) and u(t) depend on time through the protein conformation. Un-driven fluctuations, however, are not correlated and so cannot drive uphill pumping. The

absence of correlations in the fluctuations of (t) and u(t) is reflected in the principle of

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microscopic reversibility (74).

When ATP is bound to the protein, hydrolyzed, and product released in chemically

driven pumping the protein undergoes shape changes in which the relative gate and bind-

ing energies for the ion fluctuate in a correlated way, causing transport of ions across the

membrane from low to high electrochemical potential.

In general, different conformations of a protein have different dipole moments. Thus an

external oscillating electric field can also drive structural changes of a pump protein (45)

and can cause non-equilibrium correlated modulation of the relative barrier height u(t)and the well energy (t), thereby driving uphill pumping. This was shown experimen-

tally by Tsong and colleagues (49, 65, 81, 82) who applied a fluctuating external electric

field to suspensions of red blood cells. The zero average applied fields were able to drive

thermodynamically uphill transport via the ion pump sodium, potassium ATPase (Na,K

ATPase) even under conditions where ATP hydrolysis could not occur. Interpretation of

these experiments led to the development of the theory of electro-conformational coupling

(19, 7678, 80) that explains how an external oscillating or fluctuating perturbation can

drive, by coupling into intrinsic conformational degrees of freedom of a protein, pumping

of ligand from low to high electrochemical potential (19), catalysis of a chemical reac-

tion away from equilibrium (17), or performance of mechanical work on the environment

(11). It seems likely that the same conformational motions are exploited in ATP hydrolysis

driven pumping.

3.1 Kinetic mechanism

We can understand conformational pumping in terms of the simple kinetic model in Fig.

3a. The instantaneous net current between reservoir 1 and the well is

I1(t) = I1(t)I1(t) + I2(t)

dPEL(t)dt

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Astumian 11

which, by decomposing the currents Ii(t) = Issi (t) + Ii(t) and observing that Iss1 (t) +Iss2 (t) = 0 at every instant, can be written

I1 = Iss1 +F1

dPssEL

dt

+dpEL

dt , (6)

where we suppress the explicit denotation of the time dependence of the quantities, where

the probability for the well to be occupied, PEL = PssEL+pEL, is split into the instantaneous

steady state value and a deviation from that value, and where F1 = I1(I1 + I2) isthe ratio of the fluctuating current between reservoir 1 and the well to the total fluctuating

current into/out of the well. The term F1dPssELdt is the adiabatic, and the term F1dpELdt

is the non-adiabatic contribution to the pump current. The derivation can be generalized

to continuous systems using Gauss law for converting between the rate of change of a

density within a volume and the total current into the volume (see Fig. 3 d).

From elementary energetic considerations we have the simple relations between pairs of

rate constants and the internal parameters, and u, and the external parameters 1 and 2

k1

k1= e(1);

k2

k2= e(2);

k1

k2= eu ;

k1

k2= e(u). (7)

where = 1 2. From Eq. (7) we easily derive another relation between all four rate

constants in which the internal parameters and u disappear,

k1k2

k1k2= e. (8)

The kinetic mechanism for ligand transport shown in Fig. 3a is identical to the Michaelis-

Menten (MM) mechanism for catalysis of the chemical reaction L1 L2 where k1 and

k2 are effective first order rate constants into which the concentrations [L1] and [L2],respectively, have been subsumed. In this model 1 and 2 are the chemical potentials of

substrate, L1, and product, L2, respectively. The relative gate height u has the interpreta-

tion of the chemical specificity (relative lability) of the enzyme - when u < 0 the enzyme

is specific for L1 (i.e., the binding/dissociation ofL1 is faster than the binding/dissociation

ofL2) and when u > 0 the enzyme is specific for L2 (i.e., the binding/dissociation ofL2 is

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faster than the binding/dissociation ofL1) . The well depth specifies the binding affinity

(stability) of the enzyme.

The rate of change of the binding probability is

dPEL

dt= 1PEL + (k1 + k2) (9)

where = (k1 + k1 + k2 + k2)1 is the relaxation time for ligand binding in the well.The instantaneous steady state probability for the well to be occupied is obtained by set-

ting dPELdt = 0 in Eq. (9) and solving for PEL to find PssEL = (k1 + k2). Thefraction of fluctuating well occupancy coming from reservoir 1 is F1 = (k1 + k1),and the instantaneous steady state current Iss1 = k1

(1 PssEL

) k1P

ssEL can be written

Iss1 = (k1k2 k1k2) = (1 e)k2k1. If the internal parameters u and fluctuate,the rate constants and hence, PssEL, F1, and I

ss1 , all vary in time. Nevertheless, irrespec-

tive of the instantaneous values of u(t) and (t), the sign of Iss1 is determined solely by = 2 1. This is not true however of the pumped current. If we follow the sequence

(u < 0) ( < 1) (u > 0) ( > 2) (u < 0), etc. the pumped current willbe positive (from reservoir 1 to reservoir 2) even though 2 > 1. The maximum proba-

bility for an ion to be pumped in one cycle is achieved in the limit u . In this casethe steady state current is nearly zero because one of the gates is very high at every instant.

The probability to pump an ion then is the difference in occupancy between the state where

( < 1) and that where ( > 2). This probability can be written tanh [( )2]since the well equilibrates with the reservoir to which it has finite access. Thus the max-

imum average output energy per cycle is Emax, out = tanh [( )2], the min-

imum input energy per cycle is Ein,min = , with the limiting thermodynamic efficiency

(5,6)

max =

tanh [( )2]. (10)

For an input energy of = 201 (i.e., the energy provided by ATP hydrolysis under

physiological conditions) this maximum efficiency is about 75%.

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Astumian 13

3.2 pumped currents in the small perturbation limit

In order to compare the theory for stochastic pumping with experimental results of Tsong

and colleagues (49) consider a situation in which the internal parameters change periodi-

cally in time with frequency . For small amplitude oscillations u(t) = u0 + u cos(t)and (t) = 0 + cos(t ), Eq. 6 can be evaluated in the small perturbation limit (16)to yield the simple equation for the pumped flux (14)

I1 Iss1 = K

sin() + cos ()21 + 2

(11)

where K is a constant that is proportional to the product u. In Fig 3c the data of Liu,

Astumian, and Tsong (49) for ac field induced pumping of both Rb+

(an analogue of K+

)

and Na+ by the Na,K ATPase as functions of the reduced frequency are fit to Eq. (17),

with = 106s for sodium and = 103s for rubidium based on the optimal pumping

frequencies observed in the experiments. The fit parameter = 2.1 is very close to 2suggesting that the non-adiabatic contribution is essentially negligible except at very high

frequencies >> 1. Thus we conclude that the Na,K ATPase may work in many respects

like an adiabatic pump, where two internal parameters are caused by the applied field to

oscillate out of phase with one another.

Fig. 3b illustrates a simple two-state mechanism by which a single external parameter,

the oscillating external field (t), can cause two internal parameters to oscillate out ofphase with one another. There are two major conformational states, EA and EB. State

EA has high affinity for ligand (A > 0 and Pss,AEL > 12) and easy access between the

well and the reservoir 1 (uA < 0, and F1,A > 12) and the other, state EB, has low affinityfor ligand (B < 0 and Pss,BEL < 12) and easy access between the well and the reservoir 2(uB > 0, and F1,B < 12). IfEA and EB have different dipole moments, an external acfield will alternately favor first one, and then the other state, causing the average values of

F1 and ofPssEL to oscillate. Let the conformational transition be governed by two relaxation

times, one fast relaxation time that govens u and the other slow relaxation time that

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governs . As a result of the different relaxation times F1, PssEL, and pEL will oscillate out

of phase with one another. The phase lag between F1 and PssEL is caused by an internal

conformational degree of freedom being out-of-equilibrium with the applied modulation.

Even at low frequency the system is not in global equilibrium, but only in equilibrium with

respect to the degree of freedom corresponding to ion transport. Non-adiabatic flux, in

which pEL fluctuates out of phase with F1 has also been discussed (16, 18).

In the experiment of Tsong et al. (49) the conformational oscillation was driven by an

applied oscillating electric field. In chemically driven pumping, where e.g. ATP hydrolysis

drives transport, the stochastic binding of reactants and release of products cause transitions

between states of the protein. In this case, after phosphorylation or dephosphorylation, the

differential barrier height that controls the parameter F1 rapidly approaches its final value,

followed by a slower relaxation of the well energy (i.e., PssEL) to its new value. In this way,

a stochastic input (ATP hydrolysis) is converted into two on average phase shifted outputs.

Such hysteretic behavior is very general in proteins or for that matter for any relatively

complex molecule (16,83).

4 FLUCTUATING PROTEINS AND DYNAMIC DISORDER

4.1 Stochastic Pumping and ATP driven pumping

A simple model (18) for stochastic pumping by a single external parameter (t) is shownin Fig. 4a alongside a kinetic model for ATP driven pumping involving the same protein

states in Fig. 4b. The model in Fig. 4a can also be written in the form

L1 +EAO

BA AB

ok1A

k1A / EALO

BA AB

ok2A

k2A / EA + L2O

BA AB

L1 +EB o

k1B

k1B / EBL ok2B

k2B / EB +L2

(12)

The sets of rate constants kiA and kiB (i = 1,2) separately satisfy Eqs. (7) and (8),

with uA and A, and with uB and B , respectively. Additionally there is a detailed balance

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Astumian 15

condition (77)

k2Bk2A

k2Bk2A=

ABBA

BAAB=

k1Bk1A

k1Bk1A(13)

that constrains how the rate constants for fluctuation between the two states can be as-

signed. Considering the case in Fig. 4, neither EA nor EB are very good catalysts - there

is one large barrier preventing free transport of ligand for each form. The possibility of

fluctuations between the two forms can have a significant beneficial effect on the catal-

ysis. If the s and s are large (meaning fast fluctuations), the transporter is a much

better catalyst than either of its two conformational states alone! By using the pathway

EA+L1 EAL EBL

L2 EB EA ligand can pass between reservoir 1 and 2

without surmounting a large activation barrier. The flow of ligand however is from high

to low chemical potential - the protein is just a catalyst, but a better catalyst because of

the conformational fluctuations. Surprisingly, if we drive the conformational fluctuation

with some external forcing (t) such that AB = 0ABe(t)2, BA = 0BAe(t)2,AB =

0ABe

(t)2, BA = 0BAe

(t)2, where (t) is any autonomous function of timewhatsoever (19), there is net flow of ligand from reservoir 1 to reservoir 2 when = 0.

Note that drops out of the product ABBA

(BAAB

)so the detailed balance condi-

tions are satisfied at every instant.

The protein is a poor catalyst in state EA and a poor catalyst in state EB. Equilibrium

conformational fluctuation between the states turns the protein into a better catalyst. These

same conformational transitions in the presence of external driving or involved in the catal-

ysis of a non-equilibrium chemical reaction such as ATP hydrolysis allow the protein to

function as a free-energy transducer that harvests energy from the external driving or ATP

hydrolysis to pump ligand from low to high electrochemical potential.

Sunney Xie and colleagues have recently provided compelling experimental evidence at

the single molecule level that conformational fluctuations are important for enzymes (50).

This is consistent with much work on the general importance of conformational flexibility

in enzyme catalysis reviewed in (36) and with a general model for enzyme conformational

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flexibility given in (23) and role of dynamics in protein function (22). How can we theoret-

ically model the effects of internal fluctuations, present even at equilibrium, on enzymes?

4.2 Maxwells and Smoluchowskis demons: Engineering with bi-

lability and bi-stability

Consider a simple two-state system based on a rotaxane, a mechanically interlinked molecule

formed when a long rod shaped molecule is threaded through a macrocylic ring compound

and then stopper groups added to the ends of the rod to prevent the macrocycle from

escaping (see Fig. 5). As with the catenane discussed in Fig. 2, separate binding sites for

the macrocycle can be chemically incorporated on the rod. We consider two such sites in

the model in Fig. 5, where there is a steric barrier between the two sites hindering but not

preventing exchange of the ring between sites 1 and 2. At equilibrium the occupancy of the

macrocycle at the two sites is determined by the relative interaction energies of the sites.

There are two ways in which the relative occupancy can be shifted away from the equi-

librium value. These are illustrated here by a Maxwells demon, an intelligent being

that uses information about the location of the macrocycle to determine when to open and

close a gate, and a Smoluchowskis demon drawn as a blindfolded being that randomly

raises and lowers the interaction energy between the macrocycle and one of the binding

sites (29).

In the Maxwells demon case, if the demon, spotting the location of the ring, lowers

the barrier when the ring is on station 1, and raises the barrier when ring is on station 2,

the ring will obviously spend more time on station 2 than on station 1 despite the fact that

the two station have identical interaction energies at every instant. Just this scenario has

been implemented experimentally by using both a photo-activated (66) and a chemically

activated barrier (1), where the sensitivity of the trigger mechanism depended on the loca-

tion of the macrocyclic ring. The raising and lowering of the gate illustrates one of the key

design principles of a brownian motor or stochastic pump of bilability (9) which has been

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Astumian 17

investigated experimentally by (68).

A second important design principle is bistability, illustrated by the Smoluchowski de-

mon who randomly raises and lowers the interaction energy for one of the sites, thereby

switching the relative stabilities of sites 1 and 2 back and forth. When site 1 is less sta-

ble escape to site 2 is rapid, whereas when site 2 is less stable , escape to site 1 is less

rapid. As a result of the speed of escape the average occupancy of site 2 is greater than the

average occupancy of site 1 under the fluctuating conditions even though on average, the

interaction energies of sites 1 and 2 are the same. Leigh et al. have used this principle of

bistability (coupled with bilability) to design a catenane based rotary molecular motor (?).

4.3 Developing a thermodynamically consistent model for molecular

fluctuations and dynamic disorder

The two principles of bistability and bilability are the cornerstones of a minimal brown-

ian motor or stochastic pump (9). Roughly speaking, the lability of a kinetic pathway is

reflected in the term Fij in Eq. (1), and the switching of the stability is reflected in the

term dPj

dt in Eq. (1). The fact that a single term

(t

)can drive a system away from

equilibrium even if fluctuates randomly forces us to ask how we can describe, in a single

theory, both the assuredly present equilibrium fluctuation in and also non-equilibrium

driving of.

Consider the Michaelis-Menten scheme with rate constants that depend on some con-

trol parameter (t) (17)

L1 +E o

k1eb(t)

k1ea(t)

/ EL o

k2eb(t)

k2ea(t)

/ E + L2

Note that in the ratio Eq. (8), cancels in the numerator and denominator. When a = b the

system is set up for a Maxwells demon who, by increasing when PEL is greater than

average, and decreasing when PEL is less than average drives pumped current from L1 to

L2 even when < 0. When a 0 and b = 0 the system is set up for a Smoluchowskis

demon who by causing random fluctuations also drives pumped current from L1 to L2

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(1619). How can we model both equilibrium and non-equilibrium fluctuations in (t)?The equation for the rate at which the bound state probability changes can be written

(see Eq. (9))

dPEL

dt= 1() [PEL PssEL()] (14)

It is tempting to consider the model proposed by Zwanzig (84) and subsequently adopted

by Wang and Wolynes (79), Schenter, Lu, and Xie (63), and Lerch, Rigler, and Mikhailov

(48), among others, to describe the effects of dynamic disorder on proteins, including

enzymes. Zwanzigs model assumes that it is reasonable to have an internal stochastic

control variable (e.g., (t)) that influences the dynamics of the rate process dPELdtbut which is not influenced by the value of PEL, so that the equation of motion for (t) isgiven by the autonomous Langevin equation ddt = +(t), where (t) is taken to bewhite noise. This picture, however, is not thermodynamically consistent for endogenous

(internal) noise as had been previously pointed out by Astumian et al. (19). The joint

trajectories [PEL(t), (t)] do not obey microscopic reversibility and when = 0 thereis still net pumped flux from reservoir 1 to reservoir 2 in violation of the second law of

thermodynamics. For a thermodynamically consistent picture we must consider the back

reaction of the enzyme state on the likelihood for the control parameter to adopt some

particular value (19) by augmenting the equation of motion for (t) to read (11,52)d

dt= g(PEL, ) + fSD(t) + fMD(PEL) + (t) (15)

where we require

g(PEL, )PEL

+1() [PEL PssEL()]

= 0 . (16)

The autonomous function fSD(t) describes the action of Smoluchowskis demon, andthe function fMD(PEL) describes the action of Maxwells demon. When fSD(t) =fMD(PEL) = 0 (when the demons are at rest) the trajectories [PEL(t), (t)] obey mi-croscopic reversibility, the direction of the enzyme reaction is given solely by the chemical

potential difference , and the net flux is zero when = 0. The choice of g(PEL, )

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Astumian 19

defined in Eq. (16) assures that the curl of the vector field of the two reciprocally coupled

rates Eqs. (14) and (15) is zero dPELdt

, ddt

= 0.With a time dependent forcing fSD(t), or when there is a mechanism by which (t)

changes depending on whether the active site is or is not occupied fMD(PEL) , the in-teraction between (t) and the protein conformational transitions on which it acts allowfree-energy to be transduced from the source of the fluctuation fSD(t) or fMD(PEL) to dowork on the system by breaking the microscopic reversibility present at equilibrium.

5 MICROSCOPIC REVERSIBILITY AND CONFORMATIONAL

TRANSITIONS

Microscopic reversibility and its corollary, detailed balance, are among the most important

fundamental principles necessary for understanding free-energy transduction at the single

molecule level, and yet there is great confusion in the literature concerning these principles

and their applicability to non-equilibrium systems (26). In this section I will examine

how microscopic reversibility constrains possible designs for molecular machines, first by

considering a simple example of ligand binding to a protein.

5.1 Myoglobin binding and dissociation

Myoglobin is one of the most well studied proteins (20, 58), and one of the first for which

the x-ray crystal structure was determined (44). When oxygen or carbon monoxide bind

to the heme-group of myoglobin the heme undergoes a transition from a configuration in

which the iron atom is out of the plane of the heme to a configuration in which the iron

is in-plane. The local configurational change is followed by a large scale conformation

change of the protein. The mechanism is schematically illustrated in Fig. (5a).

Thinking about the reverse of the binding process, it is tempting to imagine a scenario

in which oxygen dissociates followed by return of the heme group to its original out-of-

plane configuration, and subsequently a global rearrangement of the molecule to restore

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the initial equilibrium conformation for the protein where the binding site is unoccupied.

Indeed, this picture seems to be well supported by experiments on ligand dissociation from

myoglobin where at low temperature a ligand is caused to dissociate by a LASER pulse,

and the subsequent conformational relaxation is studied. These investigations reveal that

after light induced dissociation the myoglobin molecule undergoes a local rearrangement

followed by a global conformational change in what has been termed a protein quake (2).

The mechanism for photolytic dissociation is shown in Fig. 5 b). The photolytic mech-

anism is clearly not the microscopic reverse of the reaction by which binding of oxygen

occurs. For non-photochemically assisted dissociation, however, according to microscopic

reversibility we must have for the most probable pathway the microscopic reverse of the

binding reaction, i.e., for thermally activated dissociation we have the mechanism shown

in Fig. 5 c). These two different mechanisms for dissociation of ligand from the heme

group, depending on whether the reaction occurs by photolysis or by thermal activation,

are in conformity with application of the principle of microscopic reversibility as defined

in the International Union of Pure and Applied Chemists (IUPAC) Compendium of Chem-

ical Terminology (http://goldbook.iupac.org/), known informally as the Gold Book:

Microscopic Reversibility - In a reversible reaction, the mechanism in one direction is

exactly the reverse of the mechanism in the other direction. This does not apply to reac-

tions that begin with a photochemical excitation.

The idea that, following either binding or dissociation of ligand at the heme, the conforma-

tional rearrangement of the protein starts locally and propagates through the protein until

the global change to the new equilibrium conformational state has occurred, follows very

naturally from macroscopic analogy. When we insert our finger into water, ripples prop-

agate from our finger outward. When we remove our finger ripples once again propagate

outward from where our finger had been. This picture was explicitly suggested by Ansari

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Astumian 21

et al. (2) for ligand association/dissociation to myoglobin. These authors stated that bind-

ing or dissociation of a ligand at the heme iron causes a protein-quake, with the heme as

the focus of the quake. Such a picture is not consistent with microscopic reversibility and

is possible only in the case of photochemically induced dissociation. If thermally activated

binding causes a quake propagating outward from the focus, then, counter-intuitive though

it may be, thermally activated dissociation must arise by an inward propagating un-quake

that triggers release of the ligand.

Conformational transitions - shape changes - of a protein (or any other deformable body

(59,67)) cause the center of mass of the protein to move relative to the fluid in which the

protein is immersed. Combination of any set of transitions constrained by microscopic

reversibility such that the backward reaction (e.g., mechanism Fig. 5c)) is the microscopic

reverse of the forward reaction (e.g., Fig. 5 a)) give rise to a reciprocal process (cycle)

(Fig. 5d) that according to Purcells scallop theorem (59) cannot, in the absence of inertia,

cause net directed motion in a cycle of the forward and backward transitions. Whatever

is done in the forward process is undone in the backward process. Thus, the thermally

activated binding/release of oxygen or carbon monoxide to myoglobin does not provide,

even in principle, a mechanism for propulsion of the protein through solution.

On the other hand, although there are doubtless many practical reasons that it is not

biologically relevant for myoglobin, there is no fundamental reason that a cycle of ther-

mally activated binding and photochemically induced dissocation of a ligand could not

provide an effective mechanism for propulsion under the right circumstances. The combi-

nation of Figs. 5a and 5b is shown in Fig. 5e where it is apparent that the conformational

relaxation following photo-assisted dissociation is not the microscopic reverse of the con-

formational relaxation following thermally activated binding. As a matter of principle any

non-reciprocal conformational cycle of a protein or polymer (or anything else) in viscous

solution can, and in general will, lead to directed motion (67), whether it be of an ion

across a membrane, stepping along a polymeric track, or self-propulsion through the aque-

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ous solvent (62). A particularly easy to understand example called the pushmepullyou has

be proposed and discussed by Avron (21). Let us now consider how such non-reciprocal

cyclical processes can be driven without photochemical activation.

5.2 Cycles of molecular machines

In a recent paper Togashi and Mikhailov (75) proposed that a polymer, described as an

elastic network, could be constructed to operate as a cyclic machine powered by ligand

binding. The binding was modeled by forming elastic links between the ligand and nearby

nodes of the elastic network and allowing the network to relax to its new conformational

energy minimum. The ligand was then removed (the elastic interactions were deleted)

and the system again allowed to undergo conformational relaxation. The overall process

resulting from adding ligand, relaxation, removing ligand, relaxation, adding ligand, etc.

was described by a simple cycle shown in Fig. (6a). Thermal noise was not included in

the computational study, and the transitionsEAL EBL and EB EA were deterministic

over-damped elastic relaxation processes. The mechanism is robust. Trajectories begun off

of the relaxation pathway feed into the pathway.

The proposed mechanism for autonomous generation of non-reciprocal cyclic motion is

not consistent with microscopic reversibility if the ligand that binds to state EA is the same

molecule as the ligand that dissociates from state EBL . Clearly, the energy of stateEA is

less than that of state EB, and the energy of state EBL is less than that of state EAL. No

matter what fixed arrangement of the energies of the bound states relative to the energies

of the non-bound states the overall cycle of binding ligand to EA, relaxing to EBL, and

releasing ligand from EB and relaxing back to EA needs energy which cannot be provided

by the binding and release of the same ligand under the same conditions.

In Togashi and Mikhailovs work, the ligand dissociating from the polymer was im-

plicitly different than the ligand that had associated. The details of the ligand binding

EA EAL and dissociation EBL EB were not explicitly discussed, nor was the role

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Astumian 23

of the chemical potential of ligand. The cycling EA EAL EBL EB EA can be

used to do work, W, on the environment in a manner similar to a single molecule opto-

mechanical cycle (40) so long as the motion on each of the two potentials is downhill,

aW < Ufree and (1 a)W < Ubound, where Ufree = UEB UEA and Ubound =UEAL UEBL.

5.3 Molecular machines in a thermal environment

Our goal is to understand how molecular motors convert chemical energy into non-reciprocal

conformational cycling, and hence into directed motion and mechanical work, in solution

at room temperature where thermal noise is very strong and there is a continual, reversible

exchange of energy between each polymer molecule and its environment. This fact has

extremely important ramifications for how we should describe and think about molecular

motors.

When we look at the mechanism in Fig. 6a it is tempting to term the elastic relaxation

processes EAL EBL and EB EA as power strokes, and indeed they are - power is

dissipated as the system undergoes elastic relaxation. In a thermal environment, though,

we can compare the power dissipated during the power stroke with the power that is con-

tinually and reversibly exchanged between the polymer and the environment to gauge the

relative importance of mechanical vs. thermal effects. If at some point on its energy profile

the polymer experiences a very large force of 100 pN that at that instant moves the center of

mass of the polymer with the very large velocity of 1 m/s, the power instantaneously being

dissipated by the power stroke is 1010 J/s. In contrast, the power reversibly exchanged

with the environment at room temperature is kBT = 4 1021J every thermal relaxation

time 1012s or 4 109 J/s, 40 times greater than the maximum power dissipated dur-

ing even a very powerful molecular power stroke! (9). Further, as recognized by Andrew

Huxley (41), the un-power strokes EBL EAL and EA EB will also occur with ap-

preciable rates in a thermal environment. This was recently shown experimentally (57)

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for a simple rotaxane molecule. The ratio of the probability for an uphill fluctuation to

a downhill relaxation by the microscopic reverse process is given by the simple relation

(7,25)

P(EB EA)P(EA EB) = e

(UfreeaW);P(EAL EBL)P(EBL EAL) = e

(Ubound(1a)W). (17)

In contrast to the patently non-equilibrium functionally important motions following

photo-dissociation of oxygen or carbon monoxide from myoglobin (2), the conformational

changes by which chemically driven molecular motors move are equilibrium processes.

The only difference between equilibrium and non-equilibrium for a thermally activated

mechanism is that away from equilibrium the probability to bind ligand when the polymer

is in state EA is different than the probability to bind ligand when the molecule is in state

EB. The physical motions of the molecule that follow binding of ligand are exactly the

same at and away from equilibrium.

The fact that the energy difference between the bound states and the non-bound states

depends on the chemical potential of ligand immediately suggests two approaches for how

to use ligand binding and dissociation to drive non-reciprocal cycling of the polymer con-

formational states - pumping and catalysis.

5.4 Pumping

Non-reciprocal conformational cycling (53) can be pumped by externally driven oscilla-

tions or fluctuations of the concentration of the ligand between a large and small concen-

tration. The large concentration [L]high >Kd, favors binding of ligand to EA followed byelastic relaxation EAL EBL, i.e., binding by the induced fit pathway. The small concen-

tration [L]low

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Astumian 25

the ligand concentration, although in principle, if the ligand were some intermediate in an

oscillating chemical reaction such as the Belousov-Zhabotinsky reaction (33) the process

could be driven without direct experimental manipulation of the concentrations.

The pumped energy flux into the system that allows work to be done on the environment

comes from the fact that, on average, ligand is bound while the chemical potential is high,

and dissociates when the chemical potential is low. Through a cycle of oscillation of the

ligand concentration the energy available to drive flux through the conformational cycle is,

at most, equal the amplitude of the oscillation of the chemical potential, but for very large

oscillation between very low ([L] > Kd) levels the ratio of the

probability to complete a clockwise vs. counter-clockwise cycle is

P(EA EAL EBL EB EA)P(EA EB EBL EAL EA) = e

(Ufree+UboundW). (18)

It is necessary to have at least one of the conformational changes involve a viscoelastic

relaxation for conformational pumping by an external source. The essential mechanism

is that of an energy ratchet (13) or Smoluchowskis demon - as soon as L binds when

[L] >> Kd the polymer relaxes from state EAL to state EBL because Ubound > 0, and

as soon as L dissociates when [L] 0.

5.5 Catalysis

The second approach for driving directional cycling is catalysis. If a polymer can be de-

signed to catalyze a reaction L1 L2, then, when the chemical potentials ofL1 and L2

are not equal, L1

L2

, the polymer will most likely bind whichever of L1 and L2 has

the higher chemical potential and release whichever ofL1 and L2 has the lower chemical

potential. Thus, catalysis autonomously achieves the bind high release low by which

pumping drives directional cycling. Note that, since L1 and L2 must be related chemically

(i.e., they are interconvertible), either L1 or L2 can bind to either state EA or state EB , but

with possibly different rates. We can express the ratio of the probabilities for clockwise

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and counterclockwise conformational cycling in terms of only the ratios of off rate con-

stants for L1 and L2 from EA and EB, the chemical potential difference = L1 L2 ,

and the work, W, on the environment required for non-reciprocal conformational cycling

P(EA EAL EBL EB EA)P(EA EB EBL EAL EA) =

(sA + 1) sBe + 1(sB + 1) (sAe + 1)e

W (19)

where si = kL1off,i

kL2off,i, i = A,B. In contrast to the case of external pumping, for catalyt-ically driven conformational cycling this ratio is independent of the elastic energy differ-

ences Ufree and Ubound . We can think of the mechanism as a chemically driven in-

formation ratchet (1, 13) where the active site acts as Maxwells demon (29), selecting

for L1 in state EA, and for L2 in state EB. Note however that irrespective of how strongly

asymmetric the selectivity of the active site is, the mechanism fails to drive directed trans-

port or to do work in the environment if = 0, in consistency with the second law of

thermodynamics. The non-reciprocal conformational cycling induced by catalysis may

well be the mechanism for enhanced diffusion during active catalysis by an enzyme (55),

where the rapid rotational rearrangement prevents the appearance of net directed motion.

6 CONCLUSIONS

The transitions within the conformational cycles by which molecular machines function are

equilibrium processes. The physical motions of the individual molecules are exactly the

same at and away from statistical equilibrium. Conformational pumping occurs when an

external source or an energy releasing chemical reaction entrains these equilibrium motions

to occur preferentially in a particular sequence thereby breaking microscopic reversibility

and allowing for a non-reciprocal conformational cycle by which directed motion occurs.

Unsatisfying though it may be, the best description of the mechanism by which, e.g., ATP

hydrolysis drives a molecular motor to move in one direction along a biopolymeric track

is mass action. The fact that binding ATP, converting it to ADP and Pi at the active site,

and releasing ADP and Pi is more likely than the reverse when the ATP hydrolysis reaction

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Astumian 27

is away from equilibrium imposes a temporal ordering on the equilibrium fluctuations of

the protein giving rise to correlations that drive directed motion, pumping ligand across

a membrane, or synthesis of important biopolymer. This kinder and gentler mechanism

does not involve judo throws, nor does it bear any resemblance whatsoever to a toy steam

engine. Perhaps the best macroscopic analogy is with an Archimedes screw, a device for

pumping water. In a perfectly constructed Archimedean screw the amount of water pumped

per cycle of turning of the screw is a constant. This geometrical type mechanism can work

effectively only in the very slow limit. However, for molecules, the relevant relaxation

times for the conformational transitions are often a few microseconds. Consequently a

molecular machine that carries out its function several thousand times a second can still be

well within the adiabatic limit. Further, because in a kinetic cycle all states can equilibrate

with one another even if one transition is kinetically blocked, a molecular machine can

do work against significant loads by designing mechanisms that sequentially move kinetic

blockades through the cycle, thereby preventing slip even under the influence of load (8).

Great progress has been made in the design and synthesis of artificial molecular ma-

chines (34,43,54,71) and DNA (35) and small molecule (32) walkers. It is becoming clear

that these are first and foremost molecules, governed by the laws of chemistry rather than

of mechanics. 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 by the heights of the

barriers between the states. Chemists have long experience with approaches for control-

ling stabilities and labilities of molecules, and for designing systems with sterically or

energetically hindered pathways allowing for kinetic rather than thermo-dynamic control

of mechanisms. This experience will be crucial in the next steps of interfacing synthetic

molecular machines with the macroscopic world.

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7 SUMMARY POINTS

1. Irrespective of how far a system is from statistical equilibrium the instantaneous probabil-

ity flux between any two conformational states of a macromolecule is the sum of a steady

state current and a pumped current, Iij = Issij + Fij

dPjdt

. In the absence of driving the time

average of the pumped current is guaranteed to be zero by microscopic reversibility. In

the presence of non-equilibrium driving, however, the average of the pumped current is

not in general zero, and can be opposite in sign and larger in magnitude than the steady

state current.

2. Conformational fluctuations present at equilibrium can be exploited as a mechanism by

which an external modulation can do work on the environment. The same conformational

fluctuations are important in free-energy transduction from a non-equilibrium chemical

reaction such as ATP hydrolysis.

3. A thermodynamically consistent model for the effects of fluctuations on proteins requires

inclusion of a reciprocal reaction between the source of the fluctuation and the protein. A

key test for any model is that in the absence of external driving the joint fluctuations of

the protein and the noise must obey microscopic reversibility.

4. Far from being predominately mechanical devices slightly perturbed by thermal noise,

molecular motors are overwhelmingly dominated by thermal effects and are first and fore-

most molecules that operate based on the laws of chemistry rather than of macroscopic

mechanics. A key design principle is that molecular machines take advantage of om-

nipresent thermal noise and function by using input energy in part to prevent backward

motion rather than to cause forward motion.

ACKNOWLEDGMENTS

I am grateful to the German Humboldt foundation for facilitating this work through

conferment of a Humboldt Research Award.

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Astumian 29

Figure Legends

Figure 1: Schematic illustrations of molecular machines that use energy from ATP

hydrolysis to accomplish specific tasks. a) A Molecular pump that move some ligand

across a membrane, possible from low to high electrochemical potential. b) A coupled

enzyme that synthesizes some necessary substance. c) A molecular motor that walks

along a polymeric track.

Figure 2: a) A three ring catenane that can operate as a molecular machine that moves

directionally in response to external stimulus (e.g., pH and redox potential modula-

tions). b) A two ring catenane that can undergo a precise cycle of states in response

to an external stimulus, but where the motion is not directional. c) Plot of FadAB = (1 +e(E3E2))1 and Peq

B =1 + e(E1E2) + e(E1E3)1 parametrized by time, with E1 =

(2+cos (t))1, E2 = (2+cos (t + 2))1, and E3 = 21. d) A parametric plot ofthe equilibrium probability for state A vs. the fraction of the flux into/out of A that comes

from/to B. The red curve is based on the rate constants for the 2-catenane for which FadAB

is constant, and the green curve is based on the rate constants for the 3-catenance for

which FadAB is controlled by pH and PeqA

is controlled by the redox potential, where for

simplicty we follow the cycle (E1 = 0,E2 = ) (E1 = ,E2 = )(E1 = ,E2 = 0) (E1 = 0,E2 = 0) (E1 = 0,E2 = ).

Figure 3: a) Potential energy diagram for a membrane pump. The two external pa-

rameters, 1 and 2 determine the direction of thermodynamically spontaneous current,

while the two internal parameters, u and fluctuate in time due to conformational fluc-

tuations of the protein. If the fluctuations are driven by, e.g., an oscillating field, the

correlated fluctuations ofu and can lead to uphill pumping of ligand. Below the mem-

brane is a kinetic mechanism for the pumping that is analogous to the Michaelis-Menten

mechanism for enzyme catalysis. b) Schematic diagram showing how a single external

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parameter, an oscillating field, can cause the internal parameters u and to oscillate out

of phase with one another. The inset shows how the area enclosed by the parametric

plot ofF1 vs. PEL is maximized when the external frequency matches the system

characteristic frequency 1. c) Fit of data from (49) to the sum of the adiabatic and

non-adiabatic currents Eq. (17). The maximum current was normalized to unity, and the

characteristic frequency for Rb+ pumping (pink squares) was 103 Hz and the character-

istic frequency for Na+ pumping (purple triangles) was taken to be 106 Hz, both cor-

responding to the optimal frequencies for pumping in the experiment. The inset shows

the parametric plots for the adiabatic ( counterclockwise loop on the right) and the non-

adiabatic (clockwise loop on the left) pumping. d) Illustration and general derivation

of the fundamental equation of stochastic pumping based on Gauss law. The pumped

current can be written as the product of the fraction of the total fluctuating current that

moves through the specific channel of interest (red arrow) multiplied by the volume

integral of the rate of change of the density within the volume. Pumping by using ex-

ternal driving to impose correlations on the fluctuations of the two factors can cause the

sign of the total current to be opposite that of the steady-state current and thereby do

work on the system.

Figure 4: Kinetic diagram showing how an external fluctuating field can entrain the

equilibrium conformational fluctuations of the pump protein to bind ligand from reser-

voir 1 and release ligand to reservoir 2 by alternately favoring EA and EB. b) Kinetic

mechanism showing how, at the single molecule level, ATP hydrolysis can accomplish

the same pumping that the fluctuating electric field causes by alternately phosphorylating

the protein (favoring EB) and dephosphorylating the protein (favoring EA).

Figure 5: Illustration of a Maxwells information demon and Smoluchowskis energy

demon for controlling a simple two-state rotaxane based switch. Maxwells demon uses

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Astumian 31

information about the position of the ring to raise a barrier when the ring is on base

2, and lower the barrier when the ring is on base 1. Even though the interaction energy

between the ring and the two bases is identical at every instant in time, the demon impose

correlation between the height of the barrier and the position of the ring causes the

ring to spend most of the time on base 2. Smoluchowskis energy demon, which is

blindfolded, raises and lowers the interaction energy between the ring and base 1 at

random, with equal likelihood to raise the energy when the ring is on base 1 and on base

2. At equilibrium of course it would be more likely for the interaction energy of base

1 to fluctuate to a high level (red line) when the ring is on base 2 than when it is on

base 1. Smoluchowskis demon destroys this correlation expected at equilibrium, also

causing the ring to spend more than half the time on base 2. When the demons are at

rest, i.e., when there is no pumping of the system by an external source, the value of

still fluctuates but in a way that is consistent with microscopic reversibility. Simple

kinetic diagrams illustrating the effects of pumping are shown for the three cases of a

Maxwells demon, the demons at rest (no pumping), and Smoluchowskis demon.

Figure 6: a) Schematic mechanism for thermally activated binding of oxygen to myo-

globin contrasted with b) photochemically activated dissociation of oxygen and c) ther-

mally activated dissociation of oxygen. Combination of thermally activated association

and dissociation gives a reciprocal cycle d) in which the forward and backward processes

are the microscopic reverses of each other and which cannot drive directed motion. On

the other hand, combination of thermally activated binding with photochemically acti-

vated dissociation gives rise to a non-reciprocal cycle e) which can in principle provide

a mechanism for directed motion. f) Energy level diagram for the states involved in the

two cycles. The conformational rearrangements following photochemically activated

dissociation are patently non-equilibrium process, involving dissipation of more than 50

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kBT (fifty times the thermal energy) per transition. These types of conformational tran-

sitions are what Frauenfelder and colleagues (2) call functionally important motions,

but they are relevant only for photochemical processes. For thermally activated transi-

tions the energy changes are much more modest (at most around 20 kBT) and occur at

and away from equilibrium.

Figure 7: a) Illustration of cycling induced in an elastic network by binding ligand

and allowing viscous relaxation on the Bound energy surface, followed by removal

of ligand and relaxation on the Free energy surface. Whenever ligand is added or re-

moved, energy is deposited in the system and dissipated during the relaxation. If the

system is set up to harness this energy by attaching the relaxing network to a load, some

of the energy deposited on addition and removal of ligand can by harnessed to do work

(W) in the environment. b) In a thermal environment both forward and backward tran-

sitions are possible. The ratio of the probability for an uphill fluctuation to a downhill

dissipation is a state function (25), Eq. (25). c) If the polymer can bind some substrate

L1 and catalytically convert it to a different molecule L2, then the energy in the chemical

potential difference ofL1 and L2 can drive non-reciprocal cycling of the polymer and

do work on the environment. d) Illustration of a pumping mechanism for driving non-

reciprocal cycling by oscillation of the ligand concentration between a low level, where

the polymer is most likely free, and a high level, where the polymer is most likely

bound. e) Illustration of a catalysis mechanism where the concentration ofL1 is greater

than the dissociation constant, and the concentration of L2 is less than the dissociation

constant, so on average the polymer will bind L1, undergo conformational relaxation,

release L2, undergo conformational relaxation, bind L1, etc. For both the pumping and

the catalysis mechanism only unidirectional arrows have been shown. In a thermal envi-

ronment, however, the molecule will occasionally carry out a cycle in reverse.

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Astumian 33

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+

S +

+

reservoir 2 reservoir 2reservoir 1 reservoir 1

conformational cycle

ADP+PiATP


ADP+PiATP


ADP+ PiATP

P +

- - ++

a). Molecular Pump

c). Molecular Moto r

b). Coupled Enzym e

L1 L2

i i+1

Figure 1

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Astumian 41

2

31

2

31

2

31

A

BC

E1,E3 > E3 E1 ,E3 >> E2

1

2

F

Peq

Peq

AB

FAB

B

reducing,acidic

reducing,basic

oxidizing,basic

oxidizing,acidic

xx

x1

1

ad

ad

0.7

0.6

0.5

0.4

0.3 0.4 0.5 0.6

a) b)

c) d)

Figure 2

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1

2

EE+ EL

reservoir 1 reservoir 2

+

k-1

k2

k-2

k1

u

I1 I2

Ll L2

Fast

Fast

SlowSlowPss()

F(u)

(t)

b)

c) d)

a)

- 3 - 2 - 1 0 1 2 3 4

0.2

0.4

0.6

0.8

1

log( )

normalizedcurrent

F

0.1 0.2 0.3 0.4 0.5 0.6

0.55

0.45

0.35

0.1 0

area = constantarea ~

F(u(t))=1

=10

=0.1

EL

Pss((t))EL

PssELpEL

EA

EB

V

d

dt

VS

IS

I

I

I,ss

I

SIS

V

d

dtV

I,ss

F V

d

dtV

=

Figure 3

L1 L1L2 L2

ADP

ADP

ATP

ATP

L1 L1L2 L2

EB EBEA EA

EBL EBLEAL EAL

(t)External Field Driven Pumping ATP Hydrolysis Driven Pumping

Figure 4

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Astumian 43

21

21

Maxwells Information Demon - correlated lability

Smoluchowskis Energy Demon- uncorrelated stability

21

21

1A 1A1A

1B 1B 1B2B 2B 2B

2A 2A 2Ak e

k ea

e2(b-a)

k ea

k eb

k ebk e

k e

k e

k e

k e

k

k

Maxwells Demon-

Introduces correlationsDemons at rest -

detailed balance

Smoluchowskis Demon-

Destroys correlations

Clockwise cycling Clockwise cyclingNo cycling

x

xU(x,

)

U(x,

)

When on 1

When on 2

A

A

A

A

B

B

B

B

[2] > [1] [2] > [1][2] = [1]

1

2

2

1

Figure 5

Fe

FeFe

Fe

Fe Fe FeO2

FeO2

FeO2 Fe

Fe

Fe

Fe

O2

FeO2

O2

-O2

+O2

FeO2

Fe Fe FeO2

O2

O2

+O2

h

h

a) Thermally activated binding

Fe Fe Fe FeO2

O2

O2

+O2

d) Reciprocal cycle

e) Non-reciprocal cycle

c) Thermally activated dissociation

b) Photo-dissociation

-O2

-O2

-O2

h, -O2

+O2

-O2

Fe

Fe

FeFeO

2

FeO2

FeO2

DeoxyMb

OxyMb

f)

Figure 6

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Free

Bound

L1 L1 L2L2

exp{[(UEAL-UEBL)-(1-a)W]}

exp{[(UEB-UEA)-a W]}

EAL

EAL

EBL

EBL

EA

EA

EA

EB

EB

EB

EBLEAL

EA

EB

[L] > Kd [L1] > Kd > [L2][L] < Kd

EALEBL

EA

EBEAL

EBL

EA

EB

Pumping Catalysis

+L1+L -L

-L2

a)

c)

d) e)

L

exp{[(UEAL-UEBL)-(1-a)W]}

exp{[(UEB-UEA)-a W]}

EA

EB

EBLE

AL

b)

L

a W

(1-a) W

Figure 7

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