Failure of the work-hamiltonian connection for free energy calculations. Abstract

Transcription

1 Failure of he work-hamilonian connecion for free energy calculaions Jose M. G. Vilar 1 and J. Miguel Rubi 1 Compuaional Biology Program, Memorial Sloan-Keering Cancer Cener, 175 York Avenue, New York, NY 11 Deparamen de Fisica Fonamenal, Universia de Barcelona, Diagonal 647, 88 Barcelona, Spain Absrac Exensions of saisical mechanics are rouinely being used o infer free energies from he work performed over single-molecule nonuilibrium rajecories. A key elemen of his approach is he ubiquious expression dw / d = H ( x, ) /, which connecs he microscopic work W performed by a ime-dependen force on he coordinae x wih he corresponding Hamilonian H( x, ) a ime. Here we show ha his connecion, as pivoal as i is, canno be used o esimae free energy changes. We discuss he implicaions of his resul for single-molecule experimens and aomisic molecular simulaions and poin ou possible avenues o overcome hese limiaions. PACS numbers: 5.4.-a, 5..-y, 5.7.Ln 1

2 Hamilonians provide wo key ingrediens o bridge he microscopic srucure of naure wih macroscopic hermodynamic properies: hey compleely specify he underlying dynamics and hey can be idenified wih he energy of he sysem [1]. A uilibrium, he link wih he hermodynamic properies is esablished hrough he pariion funcion = x, which here uses he Hamilonian H( x) in he coordinae space β H( x) Z e d x as he energy of he sysem []. In paricular, he free energy is given by 1 G = ln Z, where β β 1/ kt B is he inverse of he emperaure T imes he Bolzmann s consan k B. Thermodynamic properies play an imporan role because hey provide informaion ha is no readily available from he microscopic properies, such as wheher or no a given process happens sponaneously. The connecion beween work and Hamilonian expressed hrough he relaion d W d = H ( x, ), or uivalenly hrough is inegral represenaion W = H( x( '), ') d ', is ypically used o exend saisical mechanics o far-from- ' uilibrium siuaions [3-5]. These relaions are mean o imply ha he work W performed on a sysem is used o change is energy. The poenial advanage of his ype of approach is ha i would allow one o infer hermodynamic properies even when he relevan deails of he Hamilonian are no known or when hey are oo complex for a direc analysis. Experimens and compuer simulaions can hus be performed o probe he microscopic mechanical properies from which o obain hermodynamic properies. Time-dependen Hamilonians, however, provide he energy up o an arbirary facor ha ypically depends on ime and on he microscopic hisory of he sysem. Such

3 dependence, as we show below, prevens his approach from being generally applicable o compue hermodynamic properies. To illusrae how work and Hamilonian fail o be generally conneced, we consider a sysem described by he Hamilonian H ( x) under he effecs of a ime-dependen force f (). The oal Hamilonian is given by H( x, ) = H ( x) f( ) x+ g( ), where g () is an arbirary funcion of ime, which leads o a oal force F = H / x+ f( ). The funcion g () does no affec he oal force bu i changes he Hamilonian. Therefore, g () has o be chosen so ha he Hamilonian can be idenified wih he energy of he sysem. In general, he arbirary ime dependence of he Hamilonian,, canno be chosen so ha he Hamilonian gives a consisen energy. Consider, for insance, ha he sysem, g () being iniially a x, is subjeced o a sudden perurbaion f () f Θ (), where f is a consan and Θ() is he Heaviside sep funcion. The work performed on he sysem, W = f ( x x ), where x x ( ) represens he value of he coordinae x a ime, is in general differen from H( x ', ') d' = fx + g( ) g(), irrespecive of he explici ' form of he funcion g (). To illusrae he consuences of he lack of connecion beween work and changes in he Hamilonian, we focus on he domain of validiy of nonuilibrium work relaions [3] of he ype e βδge = e βw, 3

4 which have been widely used recenly o obain esimaes Δ GE of free energy changes from single-molecule pulling experimens [6] and aomisic compuer simulaions [7]. The promise of his ype of relaions is ha hey provide he values of he free energy from irreversible rajecories and herefore do no ruire uilibraion of he sysem. Ye, in almos all insances in which his approach has been applied, he agreemen wih he canonical hermodynamic resuls has no been complee and in some cases he discrepancies have been large. These discrepancies have been aribued o he presence of saisical errors in he esimaion of he exponenial average e βw [8]. Currenly, he mahemaical validiy of hese ype of nonuilibrium work relaions appears o be well esablished: hey have been derived using approximaions [3] and rigorously for sysems described by Langevin uaions [4, 5]. However, all hese derivaions rely in differen ways on he work-hamilonian connecion, which as we show below prevens hem from giving general esimaes of hermodynamic free energies. The free energy difference beween wo saes is defined as Δ G = W, where rev W rev is he work ruired o bring he sysem from he iniial o he final sae in a reversible manner []. Noe ha, if he sysem is no macroscopic, W rev is in general a flucuaing quaniy. A quasi-uilibrium, he exernal force f () balances wih he sysem force H( x) / x. Afer inegraion by he displacemen, he reversible work done on he sysem is given by Wrev = H( x ) H( x). Therefore, he free energy follows from Δ G = W P ( x, ) P ( x,) dxdx, rev 4

5 where he uilibrium probabiliies P are obained, in he usual way, from he Bolzmann disribuion P ( x, ) e β 1 H ( x) =, Z(). To be explici, le us consider a harmonic sysem described by H x kx 1 ( ) = and () g =, wih k a consan. In his case, we can compue exacly he free energy change: 1 Δ G = kx, where x f() / k, which leads o a posiive value as ruired for non-sponaneous processes. One migh have been emped o use he pariion funcion o esimae changes in free energy according o he expression Δ G = 1 ln( Z( ) / Z()), where Z β H( x ) Z() e β, d = x is he ime-dependen quasi-uilibrium pariion funcion [3, 4]. However, his relaion is no valid when changes in he Hamilonian canno be associaed wih changes in energy. In he case of he harmonic poenial, he use of he ime-dependen pariion funcion leads o Δ G = kx, a negaive value inconsisen wih a process ha is no Z 1 sponaneous. More generally, he Hamilonian H x, = kx f x γ ), where γ is a 1 ( ) ( )( consan parameer ha does no affec he dynamics of he sysem, leads o Δ G = kx ( γ x ), which can be posiive or negaive depending on he value of γ. 1 Z Therefore, he esimaes Δ GZ are no suiable o predic ypical hermodynamic properies, such as wheher or no a process happens sponaneously. To wha exen does he failure of he work-hamilonian connecion impac nonuilibrium work ualiies? In he case of a sudden perurbaion and a harmonic poenial discussed previously, he following resul follows sraighforwardly: 5

6 , W ( ) e β e β f x x P x P x dxdx = (, ) (,) = 1 e β which is differen from Δ G. An inriguing quesion hen arises: why do experimens and compuer simulaions someimes lead o resuls ha agree wih nonuilibrium work ualiies? Le us consider a siuaion closer o he experimenal and compuaional seups, wih a harmonic imedependen force ha consrains he moion on he coordinae x : H( x ) H ( x) K( x X ) 1, = +. Here K is a consan and X is he ime-dependen uilibrium posiion for he consraining force. In his case, wih H x kx 1 ( ) = and X =, we also have 1 Δ G = Wrev = kx, where now x X. K k+ K For quasi-uilibrium displacemens of X, so ha he work performed is ual o he reversible work, W = Wrev = H( x ) H( x), we have = dx, β W rev β( H ( x ) ( )) e e H x P( x, ) P( x,) dx which leads o e k( k+ K) β x k+ K βw e k + rev = ( ) ( K). K( k + K) This resul indicaes ha quasi-uilibrium does no guaranee he accuracy of he exponenial esimae of he free energy from nonuilibrium work relaions. The free energy change ΔG and is exponenial esimae Δ GE agree wih each oher only for large values of K. The reason is ha, in his case, work and Hamilonian are conneced o each 6

7 oher when boh quasi-uilibrium and large- K condiions are fulfilled simulaneously. Under such condiions, he work-hamilonian connecion is valid because x x X implies ha he rae of change of he Hamilonian, H( x, ) / = K( x X ) dx / d, uals he power associaed wih he exernal force, dw / d = K( x X ) dx / d. Ineresingly, large values of K suppress flucuaions and lead o quasi-deerminisic dynamics. Indeed, he experimenal daa [6] and compuer simulaions [7] indicae ha he agreemen beween he free energy change Δ G and is exponenial esimae Δ GE occurs mainly for relaively slow perurbaions ha lead o quasi-deerminisic rajecories. Bringing hermodynamics o nonuilibrium microscopic processes [9] is becoming increasingly imporan wih he adven of new experimenal and compuaional echniques able o probe he properies of single molecules [6, 7]. Our resuls show ha he classical connecion beween work and changes in he Hamilonian canno be applied sraighforwardly o ime-dependen sysems. As a resul, quaniies ha are based on he work-hamilonian connecion, such as hose obained from nonuilibrium work relaions and ime-dependen pariion funcions, canno generally be used o esimae hermodynamically consisen free energy changes. A possible avenue o overcome hese limiaions, as we have shown here, is o idenify he paricular condiions for which work and changes in he Hamilonian are conneced o each oher. References [1] H. Goldsein, Classical mechanics (Addison-Wesley Pub. Co., Reading, Mass., 198). 7

8 [] R. C. Tolman, The principles of saisical mechanics (Oxford Universiy Press, London, 1955). [3] C. Jarzynski, Physical Review Leers 78, 69 (1997). [4] G. Hummer, and A. Szabo, Proc Nal Acad Sci USA 98, 3658 (1). [5] A. Imparao, and L. Pelii, Physical Review E 7, (5). [6] J. Liphard e al., Science 96, 183 (). [7] S. Park e al., Journal of Chemical Physics 119, 3559 (3). [8] J. Gore, F. Rior, and C. Busamane, Proc Nal Acad Sci USA 1, 1564 (3). [9] D. Reguera, J. M. Rubi, and J. M. G. Vilar, Journal of Physical Chemisry B 19, 15 (5). 8