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Frequency dependence of catalyzed reactions in a weak oscillating fieldBaldwin Robertson and R. Dean AstumianNational Institute of Standards and Technology, Gaithersburg, Maryland 20899

(Received 14 January 199 1; accepted 11 February 199 1)

The frequency dependence of the average rates of reactions catalyzed by one or more catalystsin a weak oscillating field is derived. The average rates are sums of Lorentzian curves whose

characteristic frequencies are the inverse relaxation times of the normal modes of the kineticsystem and whose amplitudes are quadratic in the field. The signs of the Lorentz amplitudescan be either positive or negative, so the rates versus frequency can have a variety of shapes,including frequency windows. One can get relaxation times and amplitudes by measuringsteady-state rates as a function of the frequency of the field. The theory is applied to determinethe Lorentz amplitudes and characteristic frequencies of ion transport rates catalyzed by Na+-Kf-ATPase in erythrocytes.

INTRODUCTION

Relaxation kineticsle5 is concerned with the response ofa system of chemical reactions to an external disturbance.This perturbation of the chemical system from a steady statecan be a change in concentration of one or more reactants orproducts, or it can be a change in a thermodynamic param-eter, which affects one or more rate coefficients. The linearresponse to a step function change is a sum of exponentialswhose amplitudes and inverse time constants are those ofthenormal modes of the system. An equivalent approach is toapply an oscillating perturbation and measure one or moreoscillating concentrations as a function of frequency.6 Theamplitude of the oscillation is a sum of Lorentzian curves

whose amplitudes and frequencies are those of the normalmodes. This oscillatory response is the Laplace transform ofthe transient response.7y8Another alternative is to measurethe power absorbed by the system. This also is a sum ofLorentzians, one for each normal mode, but the Lorentzamplitudes are quadratic in the perturbation.

In this paper we focus on the steady-state rate of a cata-lyzed reaction in an oscillating field.‘,” We show that theaverage rate as a function of frequency is also a sum of Lor-entz curves, whose characteristic frequencies are the inversedecay times of the normal modes, and whose amplitudes canbe calculated from the kinetic parameters of the system.Thus information about the elementary kinetic parametersof a catalytic process can be obtained by measuring thesteady-state rate versus frequency.

We apply this theory to the ac-electric-field-inducedtransport of ions through the plasma membrane of erythro-cytes by Na+-K +-ATPase.“*‘2 The electric field effectarises because the enzyme has electric charges that movewhen it undergoes a conformational change during its cata-lytic cycle. The effect is large for transmembrane proteinsbecause the applied electric field results in a lo3 or lo4 timeslarger field in the membrane, because the membrane pre-vents the protein from rotating and thereby evading the ef-fect of the field, and because the conformational changes of

many membrane proteins involve large displacementcharge.13

CHEMICAL SYSTEM IN AN OSCILLATING FIELD

Consider a system of chemical reactions catalyzed byone or more enzymes. The kinetic equations for this systemare

dE,/dt = - i (k,Ei - k,iEj), i = l,...,n, (1)j= 1

where the E, are enzyme-state probabilities, n is the numberof enzyme states, and the k, are rate coefficients (withk,, ~0). Substrate and product concentrations are assumedto vary slowly. They are not written explicitly here but havebeen absorbed into the rate coefficients for brevity. If there ismore than one enzyme, the E, are to be interpreted as prod-

ucts of the individual enzyme probabilities. Since theEi areprobabilities,

&El = , (2)

which can be used to eliminate E,, in favor of the other E,.When this is done, Eqs. ( 1) may be written as

dE/dt+PE=Q, (3) where E is an (n - 1 -dimensional column vector whose ithelement is E,, where Q is an (n - 1 )-dimensional columnvector whose ith element is kni, and where P is the

(n - 1) X (n - 1) relaxation matrix, whose ith diagonalelement is

k,i + i kij,= I

(4)

and whose off-diagonal element in the ith row andjth col-umn is

kni - kj, . (5)An explicit example is given in Appendix A for a four-statesystem.

When an electric potential $ is applied, the equilibriumconstant K, in the absence of the potential must be replaced

byK,> = K, exp(q,$/RT), (6)

7414 J.Chem.Phys.Q4(11),1 June1991 0021-9606/91/117414-06$03.00 @ 1991 Americanlnstituteof Physics

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B. Robertson and R. D. Astumian: Catalyzed reactions in an oscillating field

REACTION RATEhere q,, ( = - q,i ) is the electric charge that moves acrossthe potential Jt during the i toj transition. Alternatively, theexponent could be the scalar product of the electric fieldtimes the change in electric dipole moment during the i tojtransition, or the pressure times a volume change, etc.

The equilibrium constant K I; is the ratio k ;/kb, so theeffect of the field must be apportioned between the two rate

coefficients. When this is done and an oscillating potential$4= Ij, cos(wt) (7)

is applied, the rate coefficients k, in P and Q are replacedwith

k; = k, exp[zU cos(wt)], (8)

where z,, = ai,qti@, /RT and ati ( = 1 - aii> is the appor-tionment constant.

SOLUTION

Since the rate coefficients vary periodically, the matrix

P can be expanded in a Fourier seriesP = PO + 2P’ cos(wt) + *. * (9)

with

The rate of the transition from state i to statej is

Jo = k ;Ei - k;,E,, (16)where i andj are not equal and take the values 1,2,...,n. Thiscan be written

Jg = R,E + SO, (17)where R, is an (n - 1 -dimensional row vector and S, is anumber defined as follows. There are three cases. If neither inorj equals n, then the elements of R, are all zero, except theith element is k ; and thejth element is - kk, and

S, = 0, i, j n. (18)If either i orj equals n, use Eq. (2) to eliminate E, from Eq.( 16). The elements of R, all equal k Ai, except the ith ele-mentisk:,, +kLi,and

Sin = -k;,,. (19)The elements of R, all equal - k nj except thejth element is

- kj,, - k L,, and

P, =F, P, = Pcos(wt) ,...) (10)where the overbar denotes an average over one period of theapplied field. Similar equations apply for the column vectorQ. These quantities are easily calculable in terms of modifiedBessel functions.14 The result is that the matrix PO is thesame as P except that k J is replaced by

S,,j = k :, . (20)An explicit example is given in Appendix A for a four-statesystem.

kj,o = kijI0 (Zg 1, (11)

and P, is the same as P except that k > is replaced by

The average of Eq. ( 17) over a cycle is

$ = R,oA, + R,, A, + S,,, (21)where the subscripts 0 and 1 mean that R, and S, are com-posed using expressions ( 11) and ( 12)) respectively, in placeof k ;. When Eqs. (15) and ( 14) are substituted into this, itreduces to

k,,., = &I, (zti 1. (12)Similar replacements apply to Q,, and Q, . In this paper weexpand in powers of z and keep terms to second order. Thusthe modified Bessel function in Eq. ( 11) may be approxi-mated by lo(z) =: 1 +,?/4, and that in Eq. (12) byI, (z) -,z/2.

3, =I; +xq piPi +a2

Y, (22)

where

Since P and Q are periodic, the solution E, after anyinitial transients have decayed, will also be periodic with thesame period and so can be expanded in a Fourier series

E=& +A, cosot+B, sin&+--., (13)

where the coefficients are to be calculated. The value of thein-phase first-harmonic coefficient is”

57 = R,,P, ‘Q. + S,,o (23)

is the rate of the i to j transition at infinite frequency,

xii =R,, - R,,P;- ‘P, (24)is an (n - 1 )-dimensional row vector, and

Y =2Po-‘(Q, -P,P,‘Qo) (25)is an (n - 1 -dimensional column vector, which equals A,

at zero frequency. Special cases of these equations are dis-cussed in Appendix B.

A, = 2(P: + w2) - ‘P, (Q, - P, PO ‘Q, 1, (14)correct to first order in $, . It is proportional to $, and is thein-phase linear response from relaxation kinetics.14 Theaverage of the A, term in Eq. ( 13) over a cycle is of coursezero. The average of the state probability isI

NORMAL MODES

Equation (22) leads immediately to the principal resultof this paper. In general the eigenvalues of PO are distinct, soPO can be diagonalized. I6 Then the i to j transition rate re-duces to

A, =P;'(Qo -P,A,), (15)correct to second order in 3,. The oscillating field causes achange in the average state probability, which could be mea-sured, e.g., by binding a fluorescent indicator to the enzyme.The effect on the average state probability contributes to theaverage rate of the catalyzed reaction, as shown in the nextsection.

where l/?-k is the k th eigenvalue of PO, and where the numer-ator is the product of the two numbers

Xij,k = XiiUk (27)

and

7415

(26)

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Yk = VkY. (28)

Here rk and yk are the relaxation time and the forcing func-tion of the k th normal mode of the system, the uk are thecolumn eigenvectors of the matrix P,, , and the vk ar e the roweigenvectors.

The frequency-dependent part of the rate (26) is a sumof Lorentz cur ves with amplitudes xijkyk. Since both x~,~and yk are proportional to $, , the amplitudes of the Lorentzcurves are proportional to 6.

Equations (ll), (12), (24), (25), (27),and (28) deter-mine the Lorentz amplitudes from the kinetic parameters k,and zii. Numerical routines are available” that calculate thereal and/or complex eigenvalues l/r, and column eigenvec-tors uk of a real general matrix such as P,, . The row eigenvec-tors are obtained b y inverting the matrix whose k th columnis uk. The k th row of the inverse matrix is the row vector vk .

The normalization of the eigenvectors does not affectthe Lorentz amplitudes. Proof: Compute the inverse of theproduct of the matrix of column eigenvectors times a diag-

onal matrix whose diagonal elements are not zero. Thusmultiplying a uk by a nonzero constant results in the corre-sponding vk being divided by the same constant, and theLorentz amplitude is not changed.

If the eigenvalues are order ed and l/r, ( l/r, + , forsome m, then in the frequency interval l/r,,, 40 Q l/r, + ,the rate (26) is constant. If one eigenvalue is small and allthe rest are large, then, in the intermediate frequency inter-val, the rate (26) reaches a plateau whose height is given by aMichaelis-Menten expression.”

COMPARISON WITH EXPERIMENT

Equation (26) says that the average rates as a functionof frequency are sums of Lorentz curves. Since the individualLorentz amplitudes can be either positive or negative, thecurves can combine to form a peak as shown in Fig. 1. Asimilar frequency dependence has been observed experimen-tally.

Tsong and co-workers’ ‘,I2have used radiotracer meth-ods to measure unidirectional ion transport catalyzed byNa +-K + -ATPase in erythrocytes as a function of an ap-plied oscillating electric field. They show that a 20 V/cm (O-pk) external field (8 mV oscillating membrane potential)causes net influx of Rb + (an analog of K + ) and net efflux

of Na + . Their data as a function of frequency” can be fit byour theory, ther eby determining the amplitudes and relaxa-tion times of the Lorentz curves.

At first, we fit a sum of two Lorentzi ans to the Na +efflux data and independently fit a sum of three Lorentziansto the Rb + influx data and found that one of the characteris-tic frequencies from the first fit coincided within 1% withone of the frequencies from the second fit. We can not saywhether the closeness of the two characteristic frequenciesoccurs because a normal mode is shared by the Na+ andRb+ transport processes or is only a coincidence. It doessuggest that just four characteristic frequencies with two

nonzero amplitudes for the Na + efflux and three nonzeroamplitudes for the Rb+ influx could equal ly well be fit si-

7416 B. Robertso n and Ft. D. Astumian: Catalyzed reactions in an oscillating field

j==;;;---------._

‘\ ‘\\ \\ \\ \\ \\ \ \

a \ \

2

\\\

P/

\\\\\.\\ \

'\I '-.

10 IO2 IO3Frequency

FIG. 1. The solid curve is a typical curve of reaction rate vs frequency. It i sthe difference between two Lorentzians. which are the dashed curves.

multaneously to both sets of data. The results of this simulta-neous fit ar e reported in Table I and Fig. 2.

In relaxation kinetics, determining relaxation times andamplitudes unambiguously is difficult when the relaxationtimes are close to each other. The same is true here. Forexample, the first two modes of Rb + transport are poorlydetermined, while the highest mode for Rb + is well deter-mined because t is isolated.

A kinetic scheme can be compared with experiment byfirst fitting the Lorentz curves to the experimental data andthen fitting the kinetic parameters to the Lorentz parametersusing the formulas of the previous four sections. We apply

this technique to Na + -K + -ATPase, simplifying the prob-lem by considering the Rb + influx separately from the Na +efflux. The simplest model capable of fitting the Rb + data isa four-state model, which is described in Appendix A.

Given the Lorentz parameters for just the one experi-ment,12 there is not enough information to determineuniquely the kinetic parameters for Rb + influx catalyzed byNa + -K + -ATPase. Additional measurements are neces-sary, e.g., for different substrate and product concentrat ions,etc. However, one example of kinetic parameters that givethe Lorentz parameters of Table I appears in Table II. Otherkinetic parameters exist that would do just as well. Thus onlyguarded conclusions about Na + -K + -ATPase can bedrawn from Table II. The values are presented only to illus-trate what kind of information can be obtained using ourmethod.

TABLE I. Lorentz parameters for Na+ and Rb+ transport by Nat-K+-ATPase in erythrocytes.

k 1 2 3 4

1/27-n-, ms-‘) 0.564 0.601 186 429071 (p(s) 282 265 0.855 0.0371Rb+: (amol/RBC/hr)IyI - 278.2 272.4 7.24 0

Na + : (amol/RBC/hr)I yI 0 0 - 35.80 32.05

J. Chem. Phys., Vol. 94, No. 11,l June 199 1


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PAZ32.bE2.3=c+e

fJz

iiCT\5ii5E+2

B. Robertsonand R.D. Astumian:Catalyzed eactions n an oscillating ield 7417

30,25-20-

15-

IO-

0' 1 I I I I I I I012345678

Log (Freq. in Hz)

60

5040

30

20

1003

012345678Log (Freq. in Hz)

FIG. 2. Theoretical frequency dependence of membrane transport fluxcompared with experiment. The upper curve is a sum of three Lorentziansand the lower curve s a sum of two Lorentzians,with one characteristicfrequency common to the two curves. These curves are fitted simultaneous-ly to Rb + and Na + flux measur ements from Ref. 12. The best-fit par am-eters are given in Table I. The bars on the data points represent one standarddeviation uncertainty. There is only one experimental point that deviatesmuch f rom the theoretical curve, and it is within 1.7 standard deviations.

POWER

The reaction rate can change sign as a function of fre-

quency,‘8*‘9 even for a nonelectrogenic catalyzed reaction.When the flux is uphill, the energy for this must come fromthefield.6,‘0.‘3.‘0In this section, we derive expressions for thefrequency dependence of the chemical and electrical power.

TABLE II. Example ofkinetic parameters that give the Lorentz parameter sfor Rb ’ in Table I.

ij 12 2,3 394 4,1

% 0 - 0.0339 0 0.05395 0 0.0339 0 - 0.0539k,,(ms-‘) 2.45 568 1.36 0.616k,,(ms-‘) 1.26 600 2.27 0.6814 1.94 0.947 0.600 0.905

The power supplied by the applied field is the time aver-age of the voltage times the electric current

9 e,ec &+ cosW>Jg, (29)

where we have normalized by dividing by RT. Insert Eq:( 17) into this and do the average over a cycle to get

9 elec=+@(Ri& ++R,,A, +&I.

This can be evaluated as before to yield

(30)

where

9:” elec =fp(R,J,‘P, +s,,,,

the x,k are numbers given by

Xe.k = + &?jRty,o%, (33)

(31)

(32)

and the yk are the same numbers as before. Equation (3 1)for the electrical power has the same sum-of-Lorentziansform as Eq. (26) for the rate. We expect the Lorentz ampli-tudes for the electric power all to be negative so the powerincreases monotonically with frequency.

The chemical power output is

9 them pJP (34)

wherepu,,, is the chemical potential of the ligand for thei toj

transition. This expression can be evaluated using Eq. (26)to get

(35)

where

9’”hem = c lug; (36)ifi

is the chemical power at infinite frequency, the x,,~ arenumbers satisfying

xcvk = ;/hvxij,k, (37)

and the yk are the same as before. Equation (35) for thechemical power output has the same sum-of-Lorentziansform as Eqs. (26) and ( 3 1) for rate and electrical power.

SUMMARY

This paper shows that the average rates of a catalyzedreaction versus the frequency of an applied oscillating fieldare sums of Lorentz curves, and it gives a systematic proce-dure for calculating the Lorentz amplitudes and frequenciesfrom the kinetic parameters of the chemical system. Thisprovides a general method for determining kinetic param-eters by measuring the steady-state rate of a catalyzed reac-tion as a function of the frequency.

The Lorentz amplitudes of the reaction rates can be ei-

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7418 B. Robertson and R. D. Astumian: Catalyzed reactions in an oscillating field

ther positive or negative. This makes frequency windowspossible, in agreement with experiment.” The theory is notlimited to the catalyzed reaction being near equilibrium. Atsteady states very far from equil ibrium, the Lorentz ampli-tudes and frequencies may be complex, although they alwaysappear in complex-conjugate pairs so that the sum is real.

ACKNOWLEDGMENT

We thank Dr. Jeffrey Sachs for reading the manuscript.

APPENDIX A: FOUR-STATE SYSTEM

We write some expressions explicitly for the four-statesystem

k23 \

2k

EIL32

jr J

k21

k k43 34

12k L

14

El, kE, 43

41

(Al)

The time-dependent relaxation matrix for this system is

[

k;, + k;, + k;, k;, -k;, k’

P= -k;, k;, +k;, -ii’;, ,

k’43 k;3 -k;, k;, +k;, +k;,

1A21

the time-dependent inhomogeneous term is

k’

Q= ;’ ,

[ I

(A31k’43

and the expressions used to calculate the reaction rates are

R,, = [k;,, -k;,, O-J, S,, =O,

R,, = [O, k;,, -k;,], s2, =O,

R,, = [k;,, k;,, k;, +k;,], S,, = -k;j, (A41

R,, = [ -k;,-k;,, -k;,, -k;,], S,, =k;,.

The time-dependent rate coefficients in all of these expres-sions are given by Eq. (8). The matrices PO and P, , thecolumn vectors Q. and Q, , the row vectors R, and R, , andthe numbers S, and S, are obtained from Eqs. (A2)-( A4)by replacing the k I; in these equations with k,, and k,,using Eqs. ( 11) and ( 12), respectively.

Because the four-state scheme is a single cycle, the aver-age reaction rates for the four transitions must equal each

other and equal the rate of the catalyzed reaction. This ratecan be calculated using Eqs. (23)-(28).

APPENDIX B: SPECIAL CASES

Substantial simplification occurs for perturbationsabout equilibrium. For nonelectrogenic catalyzed reactions,y depends only on equilibrium constants, the chemical pow-er is zero, and at zero frequency the electrical power and allrates are zero. On the other hand, the rates at infinite fre-quency are zero provided the apportionment constants equall/2.

Since y equals A, at zero frequency, it is straightforwardusing Eqs. ( 3)) (7)) and ( 13 ) to show that y is $, times thederivative of P - ‘Q with respect to $. At equilibrium,P - ‘Q depends only on the field-dependent equilibriumconstants. For nonelectrogenic catalyzed reactions, the sys-tem remains in equilibrium when perturbed at zero frequen-cy. Thus in this case y depends only on equilibrium con-stants.

The average rate at zero frequency can be calculated asfollows. For a nonelectrogenic catalyzed reaction at equilib-rium, the instantaneous rates remain zero throughout thecycle, and so the average rates at zero frequency are zero.This can be used to eliminate the rate at infinite frequencyfrom Eq. (26), which then reduces to

Xij,kYk . (Bl)

A similar result is obtained for the electric power.Equation (23) for the average rate at infinite frequency

can also be derived from Eq. (2 1) by observing that at infi-nite frequency A, and B, are zero so that E is just A,, whichis Pop ‘Q,, . Equation (23) is the same as the usual result forsteady-state rate in the absence of the oscillating field*’ ex-

cept that the rate coefficients k, are multiplied by I, (zti).When the unperturbed catalyzed reaction (electrogenic ornot) is at equilibrium, only ratios of these rate coefficientsappear in P; ‘Q, , and I0 cancels out of the ratios when theapportionment constants for nonzero q. are all l/2. Thenthe rates at infinite frequency are all zero.

APPENDIX C: NARROWEST PEAK

The characteristic frequencies and amplitudes of theLorentzians in Eq. (26) are real numbers except very farfrom equilibrium. Here we discuss the dependence on fre-quency.

The difference of two Lorentzians of equal amplitude

J= JO JO1 tf’/fi - 1 f’/f:

forms a peak with a maximum at frequency

(Cl

f = cfifi ,“*=AJ. (C2,IIn terms of the maximum rate

J max = JOG -f, l/v; +.A 1,the rate (Cl ) reduces to

cc31

J=u-i +f,n-’

C./y +f”,v: +f”, Jmax*

The full width at half amplitude of this peak is

(C4)

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B. Robertson and R. D. Astumian: Catalyzed reactions in an oscillating field 7419

cc51If we let the characteristic frequencies f, and f2 approacheach other while holding f. and J,,,,, constant, JO increases,and the peak narrows, except that Af remains larger than itslimiting value 2fo. The effect of bringing the two frequenciestogether while holding JO constant merely reduces the am-plitude (C3) of the peak.

When the eigenvalues and Lorentz amplitudes are com-plex, the rates are sums of complex-conjugate pairs andhence are real. If the real part of the eigenvalue were smallcompared with the imaginary part, the sum would take aresonance form22*23with a narrow width. But if, as we expectfor a linear system such as ( 1 ), the imaginary part of theeigenvalues and Lorentz amplitudes are not larger than theirreal parts,24 the sum is at least as broad a function of fre-quency as Eq. (C4). A conclusion is that the bandwidth isnot narrowed by driving a linear system far from equilibri-um. This is important in interpreting the minimum detect-able perturbation in the presence of thermal noise.25

’ M. Eigen and L. DeMaeyer, in Techniquesof Chemistry, Vol. VI, Part II,Investigation of Rates and Mechanisms of Reactions, edited by G. G.Hammes (Wiley-Interscience, New York, 1974), pp. 63-146.

‘G. W. Castellan, Ber. Bunsenges. Physik. Chem. 67, 898 (1963).‘G. G. Hammes and P. R. Schimmel, J. Phys. Chem. 71,917 (1967).‘G. Schwarz, Rev. Mod. Phys. 40,206 ( 1968).‘P. B. Chock, Biochimie 53, 161 (1971).‘R. D. Astumian and B. Robertson, J. Chem. Phys. 91,489l (1989).’ K. Tamura and Z. A. Schelly, J. Phys. Chem. 84,2996 ( 1984).

‘K. Tamura and Z. A. Schelly, J. Chem. Phys. 83,4534 (1985).9B. Robertson and R. D. Astumian, Biophys. J. 57,689 ( 1990).I0 B. Robertson and R. D. Astumian, Biophys. J. 58,969 ( 1990).” E. H. Serpersu and T. Y. Tsong, 3. Biol. Chem. 259,7155 (1984).‘ID -S. Liu, R. D. Astumian, and T. Y. Tsong, J. Biol. Chem. 265, 7260

(1990).13T. Y. Tsong and R. D. Astumian, Bioelectrochem. Bioenerg. 15, 457

(1986).‘4 M. Abramowitz and I. A. Stegun, Handbookof Mathematical Functions,

National Bureau of Standards Applied Mathematics Series 55 (NationalBureau of Standards, Washington, D.C., 1964), Eqs. (9.6.10), (9.6.19),and (9.6.34), Fig. 9.7,andTable9.8.

“The derivation is the same as that given in Ref. 6 for a two-state system.Equations (4)-(6), (9)-( 16), (19), and (20) of this reference can beused as is, where the P, are now interpreted as matrices, and E, Q,,, A,,,,and B, as column vectors. Equations (17), (18), and (21)-(32) can beused when corrected for the noncommutivity of these matrices and col-umn vectors. Since these corrections are simple, the derivation need notbe repeated here.

I6 G. Strang, Linear Algebra And It s Applications, 2nd ed. (Academic, NewYork, 1980), Chap. 5.

“B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe, V. C.Klema, and C. B. Moler, Matrix Eigen-System Routines: EISPACKGuide (Springer-Verlag, Berlin, 1976).

‘sH. V. Westerhoff, T. Y. Tsong, P. B. Chock, Y.-D. Chen, and R. D. Astu-mian, Proc. Natl. Acad. Sci. U.S.A. 83, 4734 (1986).19R. D. Astumian, P. B. Chock, T. Y. Tsong, and H. V. Westerhoff, Phys.

Rev. A 39,6416 (1989).*‘V. S. Markin, T. Y. Tsong, R. D. Astumian, and B. Robertson, J. Chem.

Phys. 93, 5062 ( 1990).” T. L. Hill, Free Energy Transduction In Biology (Academic, New York,

1977).22P. Richter and J. Ross, Science 211, 715 (1981).2.1 . G. Lazar and J. Ross, Science 247, 189 ( 1990).24A. Nitzan and J. Ross, J. Chem. Phys. 59,241 (1973).*‘J. C. Weaver and R. D. Astumian, Science 247,459 (1990).

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