Overcoming free energy barriers using

JOURNAL OF CHEMICAL PHYSICS

VOLUME 121, NUMBER 7

15 AUGUST 2004

Overcoming free energy barriers using unconstrained molecular dynamics simulations Je´roˆme Heńin and Christophe Chipota) Equipe de dynamique des assemblages membranaires, UMR CNRS/UHP 7565, Institut nanceíen de chimie molećulaire, Universite´ Henri Poincare´, BP 239, 54506 Vandœuvre-le`s-Nancy cedex, France

共Received 1 March 2004; accepted 21 May 2004兲 Association of unconstrained molecular dynamics 共MD兲 and the formalisms of thermodynamic integration and average force 关Darve and Pohorille, J. Chem. Phys. 115, 9169 共2001兲兴 have been employed to determine potentials of mean force. When implemented in a general MD code, the additional computational effort, compared to other standard, unconstrained simulations, is marginal. The force acting along a chosen reaction coordinate ␰ is estimated from the individual forces exerted on the chemical system and accumulated as the simulation progresses. The estimated free energy derivative computed for small intervals of ␰ is canceled by an adaptive bias to overcome the barriers of the free energy landscape. Evolution of the system along the reaction coordinate is, thus, limited by its sole self-diffusion properties. The illustrative examples of the reversible unfolding of deca-L-alanine, the association of acetate and guanidinium ions in water, the dimerization of methane in water, and its transfer across the water liquid-vapor interface are examined to probe the efficiency of the method. © 2004 American Institute of Physics. 关DOI: 10.1063/1.1773132兴

for coordinate ␰, obtained by including in the statistical simulation an external potential Ubias( ␰ ) that virtually rubs out the rugosity of the free energy landscape, thereby guaranteeing a uniform sampling along ␰. Ideally, the best choice for Ubias( ␰ ) is the opposite of the free energy, i.e., ⫺A( ␰ ), which implies the a priori knowledge of the PMF. Departing from the target biasing potential, ⫺A( ␰ ), when setting up the simulation usually results in inefficient to poor sampling of those regions of the configurational space that contribute significantly to the free energy change. Furthermore, for qualitatively new problems, determining a reasonable estimate of Ubias( ␰ ) is rarely possible. Ever since the US approach was devised, attempts to improve its efficiency have been put forward, in particular by optimizing continuously the initial guess of the biasing potential as the simulation progresses.4,5 Aside from the US approach and its variants, which, in essence, restrain the sampling of the phase space to areas of interest, a number of algorithms utilize constrained MD to freeze the reaction coordinate at discrete values of ␰. In this spirit, free energy perturbation 共FEP兲 and thermodynamic integration 共TI兲,6 in conjunction with appropriate algorithms for enforcing holonomic constraints,7,8 have been employed extensively to characterize the association of complex molecular systems. In most circumstances, it was assumed that the reaction coordinate ␰ can be constrained to a fixed value without affecting the sampled statistical distribution, which is likely to be wrong. Several years ago, it was hypothesized that the first derivative of the free energy dA( ␰ )/d ␰ is intimately related to the constraint force acting along the reaction coordinate.9 Only recently, however, was a formal demonstration of this relationship established, taking into account the influence of the constraint on the sampling of the phase space.10 To what

I. INTRODUCTION

The overwhelming number of degrees of freedom that are described explicitly in statistical simulations of complex macromolecular systems, in particular those of both chemical and biological relevances, rationalizes the need for a synthetic description of thermodynamic properties. Potentials of mean force 共PMFs兲 fulfill this requirement by providing the evolution of the free energy for chosen degrees of freedom. Determination of PMFs, whenever a reaction coordinate can be defined, however, still constitutes a challenging task from the perspective of numerical simulations. Using conventional Boltzmann weighted sampling of the phase space, overcoming the high free energy barriers that separate the thermodynamic states of interest constitute sufficiently rare events, which are unlikely to occur on the time scales amenable to molecular dynamics 共MD兲. To circumvent the inherent difficulties of Boltzmann sampling, an arsenal of approaches has been devised over the years, able to explore high-energy regions of the phase space in statistical simulations of reasonable lengths 共for instance, see Ref. 1 and citations therein兲. Among these methods, umbrella sampling 共US兲共Refs. 2 and 3兲 has proven to be a powerful tool for determining free energy profiles along a chosen reaction coordinate ␰, based on the probability to find the system in the thermodynamic state characterized by a particular value of ␰: A 共 ␰ 兲 ⫽⫺

1 ln P␰ ⫺Ubias共 ␰ 兲 ⫹A 0 . ␤

共1兲

Here, ␤ ⫽1/k B T, where k B is the Boltzmann constant and T is the temperature. P␰ denotes the biased probability density a兲

Author to whom correspondence should be addressed. Electronic mail: [email protected]

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J. Chem. Phys., Vol. 121, No. 7, 15 August 2004

Overcoming free energy barriers

extent will the constraint of the reaction coordinate affect the sampling of other degrees of freedom remains yet to be demonstrated. The inherent slow-converging nature of free energy calculations prompted the development of alternative methods aimed at reducing the cost of the simulation without sacrificing its accuracy. Recently, application of the relationship between the equilibrium PMF and the ensemble average over nonequilibrium free energy differences, usually associated with a cumulant expansion, yielded interesting results that open up new vistas for the computation of PMFs.11–13 By and large, free energy calculations, in particular those targeted at the estimation of PMFs, remain computationally intensive, thus limiting their range of applicability to molecular systems of moderate size.14 –18 It should be pointed out, however, that the amount of sampling necessary to reach the desired accuracy will necessarily depend upon the methodology employed, thus calling for a careful characterization of its efficiency. An important step forward was made recently to sample in an effective manner the phase space along a chosen reaction coordinate. In this context, the concept of effectiveness embraces both the homogeneous sampling over ranges of ␰ and, indirectly, the relaxation of all other degrees of freedom. In essence, the method relies on the continuous application of a biasing force that compensates the current estimate of the PMF, thereby eliminating the roughness of the free energy landscape as the system progresses along the reaction coordinate. In this spirit, the algorithm devised by Darve and Pohorille can be viewed as an elegant approach for the uniform sampling of the phase space when considering selected degrees of freedom.19,20 In this contribution, thermodynamic integration in association with unconstrained MD and an adaptive biasing force are employed to determine the free energy profile along a chosen reaction coordinate. After outlining in the following section the theoretical framework of the method together with its implementation in a general MD code, the proposed approach will be illustrated through four well-documented examples that may serve as proofs of concept—viz. 共a兲 the reversible unfolding of deca-L-alanine in the gas phase, where the reaction coordinate ␰ is an intramolecular distance, 共b兲 the association of acetate and guanidinium ions, 共c兲 the dimerization of methane in an aqueous solution, where, in both instances, ␰ is an intermolecular distance, and last, 共d兲 the transfer of methane across a water liquid-vapor interface, where ␰ is the abscissa along a given direction of Cartesian space.

II. METHODS

where g(r) is the pair correlation function of the two particles. It should be clarified that the vocabulary ‘‘PMF’’ has been extended to a plethora of reaction coordinates that go far beyond the simple distance between atoms or molecules—e.g., torsional angles.22 In this perspective, generalization of the above expression is not straightforward. For this reason, it has been often chosen to adopt a definition suitable for any type of reaction coordinate, ␰: A 共 ␰ 兲 ⫽⫺

w 共 r 兲 ⫽⫺

1 ln g 共 r 兲 , ␤

共2兲

1 ln P␰ ⫹A 0 , ␤

共3兲

where A( ␰ ) is the free energy of the state defined by a particular value of ␰, which corresponds to an iso-␰ surface of the phase space. Here, A 0 is a constant and P␰ is the probability density to find the chemical system at ␰: P␰ ⬀

冕␦ ␰

关 ⫺ ␰ 共 x兲兴 exp关 ⫺ ␤ H共 x,px 兲兴 dx dpx ,

共4兲

which is equal to an integral over an iso-␰ surface. In many instances, PMFs have been determined using TI in conjunction with constrained MD:23

冓冔

dA 共 ␰ 兲 ⳵ V共 x兲 ⫽ d␰ ⳵␰

共5兲

, ␰

where V共x兲 is the potential energy. This approach is, however, arguable and somewhat deceiving. First, Eq. 共5兲 neglects an important contribution to the free energy, as will be shown in the following paragraphs. The second, questionable assumption that was made in constrained TI calculations lies in the fact that constraints enforced along ␰ have virtually no impact on the sampled phase space at constant ␰. In reality, the latter has two consequences: A ‘‘thermodynamic’’ effect, which modifies the sampled distribution,10 and a ‘‘kinetic’’ effect, which puts certain regions of phase space out of reach within reasonable simulation time scales—viz. the so-called quasi-nonergodicity problem. As is, Eq. 共4兲 cannot be used in a straightforward fashion for obvious practical reasons. One possible route is to turn to generalized coordinates, separating configurational and kinetic contributions: P␰ ⬀

冕

exp关 ⫺ ␤ V共 q, ␰ 兲兴兩 J 兩 dq

冕

exp关 ⫺ ␤ T共 px 兲兴 dpx , 共6兲

where V共q,␰兲 is the potential energy, function of the generalized coordinates 兵q其, and the reaction coordinate ␰. T(px ) is the kinetic energy. 兩 J 兩 is the determinant of the Jacobian for the inverse transformation from generalized to Cartesian coordinates:

A. Theoretical background

Strictly speaking, a PMF is the reversible work supplied to bring two solvated particles from infinite separation to a contact distance:21

2905

J⫽

冉

⳵ x 1 / ⳵␰

⳵x1 /⳵q1

⳵ x 2 / ⳵␰

⳵x2 /⳵q1

]

]

⳵ x 3N / ⳵␰

⳵ x 3N / ⳵ q 1

¯

⳵ x 1 / ⳵ q 3N⫺1

]

¯

⳵ x 2 / ⳵ q 3N⫺1

¯

⳵ x 3N / ⳵ q 3N⫺1

冊

.

共7兲

In the first derivative of the free energy defined in Eq. 共3兲,


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J. Heńin and C. Chipot


dA 共 ␰ 兲 1 d ln P␰ 1 1 dP␰ ⫽⫺ ⫽⫺ , d␰ ␤ d␰ ␤ P␰ d ␰ dP␰ /d ␰ can be evaluated using Eq. 共6兲, dP␰ ⬀ d␰

冕

⫻

再

exp关 ⫺ ␤ V共 q, ␰ 兲兴 ⫺ ␤ 兩 J 兩

冕

共8兲

冎

⳵ V共 q, ␰ 兲 ⳵ 兩 J 兩 ⫹ dq ⳵␰ ⳵␰

exp关 ⫺ ␤ T共 px 兲兴 dpx .

共9兲

Combining Eqs. 共8兲 and 共9兲, it can be seen that the kinetic contribution vanishes. Introducing next an integral over ␰* and a Dirac function, it follows that: 1 1 dA 共 ␰ 兲 ⫽⫺ d␰ ␤ P␰

再

冕

⫻ ⫺␤兩J兩

exp关 ⫺ ␤ V共 q, ␰ * 兲兴 ␦ 共 ␰ * ⫺ ␰ 兲

冎

⳵ V共 q, ␰ * 兲 ⳵ 兩 J 兩 ⫹ dq d ␰ * . ⳵␰ ⳵␰

共10兲

Back transformation into Cartesian coordinates yields 1 1 dA 共 ␰ 兲 ⫽⫺ d␰ ␤ P␰

再

⫻ ⫺␤

冕

exp关 ⫺ ␤ V共 x兲兴 ␦ 关 ␰ 共 x 兲 ⫺ ␰ 兴

冎

⳵ V共 x兲 1 ⳵兩J兩 ⫹ dx, ⳵␰ 兩 J 兩 ⳵␰

共11兲

which can then be expressed in terms of configurational averages at constant ␰:

冓

⳵ V共 x兲 1 ⳵ ln兩 J 兩 dA 共 ␰ 兲 ⫽ ⫺ d␰ ⳵␰ ␤ ⳵␰

冔

␰

⫽⫺ 具 F ␰ 典 ␰ .

共12兲

具 F ␰ 典 ␰ is the average force acting along reaction coordinate ␰, the expression of which was published in different forms by Carter et al.,24 Ruiz-Montero et al.,25 Sprik and Cicotti,26 and den Otter27. This formulation was subsequently generalized by den Otter and Briels to an ensemble of reaction coordinates.28 Compared to Eq. 共5兲, the present formula takes into account explicitly the ␰ dependence of the volume element in generalized coordinates, which may be shown easily to be proportional to 兩 J 兩 . It should be emphasized that, in most circumstances, selected fast-relaxing degrees freedom are constrained, in particular, to allow larger time steps to be utilized to integrate the equations of motion. Since covalent chemical bonds constitute sufficiently ‘‘hard’’ degrees of freedom, constraining them to their equilibrium geometry is commonly assumed to have a marginal impact on the sampled phase space. In the convention adopted here, constraint forces are not taken into account in the evaluation of F ␰ . It should, therefore, be ascertained that contributions due to holonomic constraints do not contaminate the measured force. This requirement is met under the sine qua non condition that a given Cartesian coordinate does not appear simultaneously in the expression of ⳵V共x兲/⳵␰ in Eq. 共12兲 and in a constrained degree of freedom. Whereas the average force 具 F ␰ 典 ␰ is physically meaningful, the same cannot be said for the instantaneous components F ␰ , from which the former is evaluated. Intuitively, it is tempting to assimilate F ␰ to the force exerted along ␰—i.e., ⫺ ⳵ H(x,px )/ ⳵␰ . It should be reminded, however,

that neither F ␰ nor ⫺ ⳵ H(x,px )/ ⳵␰ are fully defined by the sole choice of the reaction coordinate. The rigorous definition of these quantities implies that ␰ be associated with a complete set of generalized coordinates, the choice of which bears some arbitrariness. In the framework of the average force approach, F ␰ is accumulated in small windows or bins of finite size ␦␰, thereby providing an estimate of dA( ␰ )/d ␰ defined in Eq. 共12兲. The adaptive biasing force19,29 ABF applied along the reaction coordinate, thus, writes FABF⫽“ x Ã ⫽⫺ 具 F ␰ 典 ␰ “ x ␰ .

共13兲

Here, Ã denotes the current estimate of the free energy and 具 F ␰ 典 ␰ , the current average of F ␰ . As sampling of the phase space proceeds, the estimate “ x Ã is progressively refined. The biasing force FABF introduced in the equations of motion guarantees that in the bin centered about ␰, the force acting along the reaction coordinate averages to zero over time. Evolution of the system along ␰ is, therefore, governed mainly by its self-diffusion properties. It should be understood, however, that if sampling is facilitated along ␰, convergence of the ensemble average in Eq. 共12兲 still relies upon the relaxation properties of the degrees of freedom other than ␰. We note in passing that, instead of biasing the evolution of the system along ␰ by means of an average force, one might envision to decouple the reaction coordinate from other degrees of freedom employing modified equations of motion— either using Langevin dynamics20 or an associated high temperature and large mass.30 Compared to US, for which the guess of the correct biasing potential—optimally, ⫺A( ␰ ), may rapidly turn out to be a daunting challenge, the present approach ensures a uniform sampling along the reaction coordinate by constantly adapting the force required to overcome free energy barriers. In a similar spirit, adaptive US methods4,5 update the shape of the biasing potentials, based on the sampling behavior of short simulations. Whereas for idealistically fast relaxing degrees of freedom, the two alternative approaches should prove to be equally efficient, the present scheme is expected to perform somewhat better in quasi-non-ergodicity scenarios. In the latter, adaptive US is likely to yield an incomplete sampling along the reaction coordinate, because the preliminary simulations aimed at optimizing the biasing potentials are too short to render a faithful description of the free energy landscape. B. Implementation

1. Computing the instantaneous force component

In a nutshell, the implementation in a general MD code of the method described in the preceding section can be devised as an independent set of routines that collect the Cartesian coordinates and the forces exerted on those atoms involved in the reaction coordinate ␰, and feed back the biasing force to the core MD integrator. The general expression for the instantaneous force is given by F ␰ ⫽⫺

⳵ V共 x兲 1 ⳵ ln兩 J 兩 ⫹ . ⳵␰ ␤ ⳵␰

共14兲




The first contribution is related to the individual Cartesian forces exerted onto the particles: 3N

⫺

⳵ V共 x兲 ⳵ V共 x兲 ⳵ x k ⳵x ⫽⫺ ⫽F• . ⳵␰ ⳵␰ ⳵␰ k⫽1 ⳵ x k

兺

共15兲

To evaluate the latter, the Cartesian coordinates of the system will be transformed into generalized ones. As has been underlined previously, the definition of ⳵x/⳵␰ is intimately connected to the choice of the generalized coordinates. 2. Special case of a distance separating two particles

For this reaction coordinate, the coordinate transform is chosen, (x 1 ,y 1 ,z 1 ,x 2 ,y 2 ,z 2 )哫(x M ,y M ,z M , ␰ , ␪ , ␸ ) where (x M ,y M ,z M ) is the center of the segment joining particles 1 and 2, and 共␰,␪,␸兲 are the usual spherical coordinates for the vector borne by the segment connecting the two particles. The Jacobian of this transformation is given by expression 共7兲, and its determinant reduces to 兩 J 兩 ⫽ ␰ 2 sin ␪ .

共16兲

Adopting the above coordinate transform, ⳵x/⳵␰ may be explicitly restated as

⳵x 1 ⫽ 关共 x 1 ⫺x 2 兲 , 共 y 1 ⫺y 2 兲 , 共 z 1 ⫺z 2 兲 , 共 x 2 ⫺x 1 兲 , 共 y 2 ⳵␰ 2 ␰ ⫺y 1 兲 , 共 z 2 ⫺z 1 兲 ,0,...兴 ,

共17兲

where the components associated with those atoms other than 1 and 2 is zero. Equation 共15兲 can then be rewritten as F•

⳵x 1 ⫽ 共 F ⫺F1 兲 •uˆ12 , ⳵␰ 2 2

共18兲

where uˆ12 is a unit vector. Combination of expressions 共18兲, 共16兲, and 共14兲 provides the expression of the force: F ␰ ⫽ 12 共 F2 ⫺F1 兲 •uˆ12⫹

2 . ␤␰

共19兲

As has been noted previously, introduction of a Jacobian correction into the expression of the force F ␰ guarantees a uniform sampling along the reaction coordinate. The difference in the availability of phase space for the two particles in contact and at large separations, however, results in the appearance of a repulsive pseudoforce acting between them. Normalization of the PMF could, therefore, be performed a posteriori, to ensure that the free energy tends towards zero when ␰→⫹⬁. With this convention, the PMF complies with the original definition of Eq. 共2兲, rather than the more general one in Eq. 共3兲. This is achieved by subtracting from the free energy profile the term arising from the Jacobian correction. 3. Application of the adaptive bias

One noteworthy characteristic of F ␰ is that it fluctuates very strongly. Consequently, in the beginning of an ABF simulation, the accumulated average in each bin is likely to take large, inaccurate and physically meaningless values. Under these circumstances, applying a biasing force along ␰ according to Eq. 共13兲 may severely perturb the dynamics of

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the system, biasing artificially the accrued average, and, thus, impede convergence. In some extreme cases, the latter is expected to be virtually unreachable. To avoid such undesirable effects, no biasing force is applied in a bin centered about ␰ until a reasonable number of force samples has been collected. When the user-defined minimum number of samples is reached, the biasing force is introduced progressively in the form of a linear ramp. Not too surprisingly, this minimal number of samples accrued prior to applying FABF is intrinsically system dependent, and should, therefore, be chosen carefully, as will be discussed amply in the following sections. Furthermore, to alleviate the deleterious effects due to abrupt variations of the force, the corresponding fluctuations are smoothed out, using a weighted running average over a present number of adjacent bins, in lieu of the average on the current bin. It is, however, crucial to ascertain that the free energy profile varies regularly in the ␰ interval, over which the average is performed. With these two precautions in mind, the adaptive force is anticipated to be a continuous function of both ␰ and time. C. Computational details

All the simulations reported herein were performed with the MD program NAMD,31,32 in which a specific module for modifying the sampling by means of an adaptive biasing force has been implemented. The CHARMM27 force field has been used to describe all molecular systems,33 with the TIP3P model to represent water molecules.34 For all simulations, the temperature was kept at 300 K, employing moderately damped Langevin dynamics. Long-range electrostatic forces were taken into account by means of the particle-mesh Ewald 共PME兲 algorithm,35 and the equations of motion were integrated using the r-RESPA multiple time step scheme36,37 for updating short-range, van der Waals, and long-range, electrostatic, contributions every two and four steps, respectively. Unless specified otherwise, a time step equal to 1.0 fs was used. In all instances, a reference simulation of 10.0 ns was carried out, supplemented by additional, shorter simulations of 5.0, 2.5, and 1.0 ns, to probe the efficiency of the method. Moreover, barring a few special cases, one hundred samples were accrued by default over four contiguous bins on each side of ␰, before the force was applied progressively to the system. Generally speaking, estimation of statistical errors in free energy calculations is a difficult task, implying a number of stringent hypotheses. Recently, extending the earlier work of Straatsma et al.38 to ABF simulations, Rodriguez-Gomez et al. have developed the relevant formulas to obtain standard deviations in PMF calculations.29 As will be seen in the first example chosen to illustrate the method, the underlying assumption that the time series of the force 兵 F ␰ (t) 其 is stationary may not necessarily always be satisfied, thereby making the convergence of the corresponding autocorrelation function difficult. Alternatively, the so-called ‘‘sampling ratio,’’ 1⫹2␬—i.e., the minimum number of correlated samples between two independent data points38—can be estimated from the variance of block averages over N samples:


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␴ 2 共 F ␰ ,N 兲 ⫽

␴ 2 共 F ␰ ,i 兲共 1⫹2 ␬ 兲 N

共20兲

in conjunction with a linear fit as a function of 1/N. Here N F ␰ ,N stands for the average of 兵 F ␰ ,i 其 , i.e., 1/N 兺 i⫽1 F ␰ ,i , and 2 the variance, ␴ (F ␰ ,N )⫽Var(F ␰ ,N ) is defined by

␴ 2 共 F ␰ ,N 兲 ⫽E 兵关 F ␰ ,N ⫺E 共 F ␰ ,N 兲兴 2 其 ,

共21兲

where E(F ␰ ,N ) denotes the expectation value of F ␰ ,N . Next, the standard deviation of the free energy difference ⌬A( ␰ ) is evaluated, using the formula proposed by Rodriguez-Gomez et al.:29

␴ 关 ⌬A 共 ␰ 兲兴 ⯝⌬ ␰

␴ 共 F ␰ ,i 兲共 1⫹2 ␬ 兲 1/2 N 1/2

.

共22兲

As expected, ␴ 关 ⌬A( ␰ ) 兴 decreases as 1/N 1/2 and increases linearly with ␴ (F ␰ ,i ). It is worth stressing that Eq. 共22兲 only supplies a coarse, upper-bound estimate of the actual error. 1. Reversible unfolding of deca-L-alanine

To estimate the free energy for unfolding deca-L-alanine in vacuo, the reaction coordinate was chosen as the distance separating the first and the last ␣-carbon atoms of the peptide chain. In order to sample the full range of conformations spanning from the unique, compact ␣-helical form to the ensemble of extended structures, ␰ varied between 12.0 and 32.0 Å. The average forces were accumulated in 0.1 Å wide bins. A time step of 0.5 fs was used to integrate the equations of motion. Electrostatic and van der Waals interactions were truncated using a 12 Å spherical cutoff in conjunction with a switching function. No multiple time stepping was utilized in this particular application. 2. Acetate-guanidinium association in water

The PMF characterizing the association of acetate and guanidinium ions in water has been generated as a function of the distance separating the carboxyl carbon atom of the anion from the central carbon atom of the cation, from 3.0 to 12 Å. The two ions were solvated in a box of 803 TIP3P water molecules. The atomic charges borne by the ions were extracted from the CHARMM27 force field. The simulation was carried out in the isobaric-isothermal ensemble, using a Langevin piston to maintain the pressure at a nominal value of 1 atm.39 In an unrestrained simulation, averaging of the forces over all possible orientations of the ions is numerically inefficient on account of the overwhelming number of relative positions to be sampled—albeit, stricto sensu, only the orientationnally averaged PMF is the physically meaningful quantity.40 Yet, when considering all possible approaches of the solutes, the pertinent information about their interaction turns out to be somewhat diluted in the resulting free energy profile. It was, therefore, chosen to restrain our description to the most favorable, biologically relevant approach, which corresponds to a C2 v symmetry of dimer, wherein the two oxygen atoms of acetate interact with the hydrogen atoms borne by two distinct nitrogen atoms of guanidinium. This low-entropy approach was enforced by means of a set of

properly selected harmonic restraints on both valence and torsional angles. It should be underlined that these restraints are not coupled to the reaction coordinate through common atoms, so that the corresponding forces have no direct influence upon the calculation of F ␰ . Since the PMF is markedly steep in the region of the contact-pair minimum, the bin width has been reduced to 0.05 Å to achieve an accurate description of the free energy derivative, together with an homogeneous sampling along ␰. As a basis of comparison to estimate the accuracy of our free energy profile, umbrella sampling calculations were performed using the same force field and MD protocol. The system was confined in three sequentially overlapping windows, by means of harmonic potentials. In addition, to ensure a reasonably uniform distribution of the reaction coordinate ␰, a set of linear biases were introduced, based on the knowledge of the PMFs determined previously using the average force formalism. 5.0 ns of sampling were generated in each window, making a total of 15.0 ns to cover the complete reaction path—an initial 3.0 ns run consisting of three overlapping windows sampled over 1.0 ns each proved to be insufficient to provide a smooth and converged PMF. Ultimately, the weighted histogram analysis method 共WHAM兲 was employed to derive self-consistently the complete free energy profile from the individual probability distributions.41 3. Dimerization of methane in water

To investigate the reversible association of two methane molecules in an aqueous solution, the reaction coordinate ␰ was defined as the simple carbon-carbon interatomic distance. The complete system consisted of two methane molecules solvated by 560 water molecules, in a box of 24.0 ⫻24.0⫻24.0 Å3. The simulation was carried out in the isobaric-isothermal ensemble, with a Langevin piston to maintain the pressure at a nominal value of 1 atm. 4. Transfer of methane across the water liquid-vapor interface

The free energy delineating the translocation of a single methane molecule across the water liquid-vapor interface was computed in the canonical ensemble. The reaction coordinate is the distance between the projections of the centers of mass of methane and the lamella formed by 790 water molecules, respectively, along the z direction, normal to the water liquid-vapor interface. Although constrained degrees of freedom in water molecules are coupled to ␰ through the center of mass of the water lamella, the measured instantaneous force is not affected because these degrees of freedom do not contribute to ⳵V共x兲/⳵␰ in Eq. 共12兲. This is a direct consequence of our choice of generalized coordinates for this particular chemical system. The dimensions of the simulation cell were 30.0⫻30.0⫻100.0 Å3, which corresponds to a thickness of the water lamella roughly equal to 20 Å. Difference in the plateau end points of the PMF provides an estimate of the free energy of hydration, which can be confronted directly to the experimental value. Since the accurate, in silico determination of hydration free energies in-




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herently depends upon the ability of the force field to reproduce solvation properties,42,43 it has been chosen to compare the present method with the free energy perturbation scheme:44 N 1 ⌬A hydration⫽⫺ ln具 exp兵 ⫺ ␤ 关 V共 x;␭ k⫹1 兲 ␤ k⫽1

兺

⫺V共 x;␭ k 兲兴其典 ␭ k ,

共23兲

where V(x;␭ k ) stands for the interaction energy of the solute with its environment, which is a function of the coupling parameter ␭. Double annihilation of the solute,45 in vacuum and in a box of 343 TIP3P water molecules, yields the free energy of dehydration—i.e.,—⌬A hydration . In practice, the absence of intramolecular ‘‘1– 4’’ interactions in methane makes the free energy contribution in vacuum nil. The dual topology paradigm was employed,46 wherein the interaction of the solute with its environment is scaled progressively to zero over N contiguous windows. Here, 34 windows of uneven width were employed to annihilate the methane molecule, involving 10 ps of equilibration and 20 ps of data collection—viz. a total of 1.02 ns for either the direct, annihilation, or the reverse, creation run. The simulation in the aqueous medium was carried out in the isobaric-isothermal ensemble, in the conditions described previously for the dimerization of methane. III. RESULTS AND DISCUSSION A. Reversible unfolding of deca-L-alanine

The free energy profile for unfolding deca-L-alanine is depicted in Fig. 1. The minimum of the curve occurs at ca. 13 Å, which corresponds to the ␣-helical form of the peptide. Between 15 and 25 Å, a sharp increase of the free energy is witnessed, as the intramolecular hydrogen bonds forming the ␣ helix are progressively disrupted. The PMF appears to level off beyond 25 Å, when the conformation of the peptide chain is extended. As can be observed, even on a quantitative level, the curve derived from the 2.5 ns simulation compares very well to the reference run, four times as long. Interestingly enough, the free energy profiles presented here are virtually identical to those described in a recent study by Park et al.,12 where a reference umbrella sampling 共US兲 simulation was compared with a series of fast-growth transformations, using the Jarzynski equality.47 To ensure proper convergence, the US unfolding required 200 ns of MD sampling—although it is likely that a shorter run might have provided a satisfactory estimate. This is about eighty times longer than the simulation time reported here, illustrating the efficiency of the average force formalism for computing PMFs. Alternatively, irreversible unfolding was performed throughout two series of hundred individual short, 0.2 and 2 ns, simulations, corresponding, respectively, to pulling speeds of 100 and 10 Å ns. Averaging of each series of simulations, using cumulant expansions, yields the free energy profile of reversible folding, in accordance with theJarzynski equality. Not too unexpectedly, the best agreement with the reference US simulation is reached with the smaller pulling speed. Whereas any individual run participating to the average gives, per se, no insight on the equilibrium be-

FIG. 1. Reversible unfolding of deca-L-alanine. Top: Free energy profiles obtained from a 1.0 共dotted line兲, 2.5 共solid line兲, 5.0 共dashed line兲, and 10.0 ns 共thick, solid line兲 unconstrained MD simulation with an adaptive bias. Middle: Free energy profiles derived from a 2.5 ns constrained MD simulation, wherein ␰ is the distance separating the carbon atoms of the terminal carbonyl moieties 共solid line兲, a 2.5 ns unconstrained MD simulation using the same reaction coordinate 共dotted line兲 and the reference 10.0 ns unconstrained run 共thick, solid line兲. Bottom: Free energy profiles obtained from 5.0 ns constrained 共solid line兲 and unconstrained 共dotted line兲 MD simulations using ␦␰⫽0.05 Å. Influence of accruing average force samples prior to application: No sample 共short-dashed line兲 and 100 samples 共long-dashed line兲. Comparison with the reference 10.0 ns unconstrained run 共thick, solid line兲. Each graph features in its inset the sampling distribution characteristic of each MD simulation.


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FIG. 3. Reversible unfolding of deca-L-alanine. Probability distributions of the instantaneous force component F ␰ exerted on the system along the reaction coordinate: ␰⫽13.0 共a: solid line兲, 17.0 共b: dotted line兲, and 25.5 Å 共c: dashed line兲. Points a, b, and c correspond in the PMF to a negative slope, a positive slope, and a small plateau, characterized, respectively, by a positive, a negative, and a zero average force. Each distribution is supplemented by its Gaussian fit. The free energy profile obtained from the reference 10.0 ns unconstrained MD simulation is included in the inset, with arrows pointing to values of ␰ where the F ␰ was sampled.

FIG. 2. Reversible unfolding of deca-L-alanine. Top: Evolution of the distance separating the terminal ␣-carbon atoms of the peptide chain as a function of time—from the 5.0 ns unconstrained MD simulation. Bottom: Evolution of the six intramolecular hydrogen bonds forming the scaffold of the ␣ helix as a function of time.

havior of the system, the same cannot be said for the unconstrained MD simulation using average forces, which, in the same amount of time, converges towards the anticipated answer. An analysis of 共i兲 the hydrogen bonds forming the ␣ helix and 共ii兲 the distance separating the first and the last ␣-carbon atoms, as a function of time, reveals that the native structure is intermittently broken and formed again. This is suggestive that unfolding, indeed, occurs in the conditions of thermodynamic reversibility, in sharp contrast with fastgrowth simulations. As can be seen in Fig. 2, over the simulation time of the reference run, viz., 10 ns, the ␣-helical conformation of the oligopeptide is observed several times, accompanied by the cooperative formation of i→i⫹4 intramolecular hydrogen bonds. Within a simulated time of a nanosecond, the whole reaction coordinate range is sampled, which implies that the adaptive bias compensates the free energy barriers in a very effective way. After a longer, 10.0 ns run, the sampling is reasonably uniform, showing that the free energy landscape is accurately described by the accumulated force data. A perfectly uniform sampling would, in all cases, require longer trajectories for mere statistical reasons—even with a perfect bias or a completely flat free energy surface.

Since the present computations are intended to serve as a proof of concept, demonstrating the efficiency of the algorithm, it was purposely chosen to use a single, broad reaction coordinate window. In order to sample quickly, yet adequately large ranges of ␰ values, it is straightforward to break the simulation into sequential windows, and integrate the force over all bins concurrently. This approach obviates the requirement of an overlap between contiguous windows, as well as the need for a matching algorithm—e.g., WHAM, as is the case in US calculations. The average force in each bin is rather noisy, due to the very broad distribution from which it is sampled 共see Fig. 3兲. The effect of that noise can however be significantly reduced when the PMF varies smoothly, by using a running average over several bins to compute the adaptive force. The force distributions are found to have a very simple structure. For all systems studied here, they are accurately described by Gaussian functions. The width is large—viz., typically, the standard deviation, ␴ (F ␰ ,i ), reaches ca. 20 kcal/mol/Å, compared to relevant values of the average force—viz. often within 1 kcal/mol/Å. This explains why a minimum of several thousand force samples per bin is required to guarantee an acceptable accuracy. Employing Eq. 共22兲, the maximum standard deviation on the free energy difference, ␴ 关 ⌬A( ␰ ) 兴 , over the whole reaction coordinate—viz. ⌬␰⫽20 Å is equal to ca. 0.2 kcal/mol for a 10.0 ns simulation and ca. 0.6 kcal/ mol for a 1.0 ns simulation. The implemented formalism relies upon the assumption that the dynamics is not constrained. Holonomic constraints may, however, be introduced without interfering with the computation of the bias and, hence, the PMF, granted that some precautions are taken. Here, as an illustration, ␰ was chosen as the distance separating 共i兲 the first and the last ␣-carbon atoms, and 共ii兲 the carbon atoms of the first and the last carbonyl moieties of the peptide, freezing or not those



FIG. 4. Reversible association of acetate and guanidinium in water. Top: Free energy profiles obtained from a 1.0 共dotted line兲, 2.5 共solid line兲, 5.0 共dashed line兲, and 10.0 ns 共thick, solid line兲 unconstrained MD simulation with an adaptive bias. The sampling distribution characteristic of each MD simulation is included in the inset. Bottom: Comparison of two PMFs obtained from the reference 10.0 ns unconstrained MD run 共solid line兲 and a US simulation consisting of three overlapping windows 共circles兲.

chemical bonds involving hydrogen atoms. The present computations confirm that, with a consistent choice of the reaction coordinate and constrained degrees of freedom, the resulting free energy profiles are unaffected by the constraints, as evidenced from Fig. 1. This allows us to constrain the geometry of TIP3P water, when present, using the SETTLE algorithm,48 as will be seen in the following sections. B. Acetate-guanidinium association in water

Figure 4 shows the PMFs for the association of acetate and guanidinium in water. As a convention, the curves have been anchored at the appropriate value of the electrostatic interaction energy in a dielectric medium of permittity equal to 78.3 and a separation ␰ of 12 Å, that is ca. ⫺0.4 kcal/mol. The 10.0 ns simulation provides a smooth, nicely resolved profile, which exhibits a narrow well representing the ion ‘‘contact’’ pair state, supplemented by a second, softer minimum associated to a ‘‘solvent-separated’’ structure. The contact pair occurs at ␰⫽4.0 Å with a free energy of ⫺7.5 kcal/ mol. Comparison of the curves obtained over different simulation times indicates that a 1.0 ns run is sufficient to obtain a reasonable estimate of the converged PMF. This can be ascribed to the highly restrained reaction path, wherein


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the two ions evolve in a C2 v arrangement. Quantitative agreement within 0.2 kcal/mol is reached in 2.5 ns. Convergence is faster at short distances for the obvious reason that it corresponds to a smaller, easily sampled volume of phase space. In contrast, the profile remains noisier at greater distances, on account of the larger entropy, which increases slightly the overall statistical error. PMFs for this system were previously reported by Rozanska and Chipot,49 as well as, more recently, by Masunov and Lazaridis.50 US simulations were performed, during which a C2 v geometry was enforced. Both studies yielded results qualitatively similar to those described here, albeit not quantitatively so. The source of the discrepancies is likely to be rooted in differences in both the potential energy functions and the simulation protocols utilized. Rozanska and Chipot used the AMBER 共Ref. 51兲 force field and an Ewald lattice sum, whereas Masunov and Lazaridis used the unitedatom CHARMM19 with a spherical cluster of 200 water molecules, a spherical solvent boundary potential, and an 11 Å cutoff to truncate nonbonded interactions. It is worth mentioning that the set of charges for amino-acid side chains in CHARMM19 and CHARMM27 are strikingly different. It should also be noted that the free energy profile necessarily bears a small dependence on the precise set of restraints or constraints that are applied to ensure the C2 v symmetry of the association. Due to the acute sensitivity of the free energy to force fields, the only relevant comparison to probe both the efficiency and accuracy of the present method should involve a unique potential energy function and MD protocol. The US simulations described in the methodological section yielded the free energy profile of Fig. 4. The agreement with the PMFs computed with ABF is remarkable, thereby confirming the numerical equivalence of US and ABF formalisms. Here, the need for an adapted bias was reinforced by the conjunction of the rough free energy landscape and the noteworthy height of its barriers. Association of acetate and guanidinium offers a glaring example of the challenge in guessing ab initio the biasing potentials appropriate for steep and narrow valleys. This evidently also limits the efficiency of a ‘‘naive’’ US approach, which only resorts to confinement biases in many overlapping windows. Generation of the US PMF of Fig. 4 was made possible by virtue of the a priori knowledge of the free energy landscape revealed by independent average force calculations. In all likelihood, stand alone US simulations without this educated guess would have required several preliminary runs before an efficient protocol could be designed. C. Dimerization of methane in water

The free energy profile delineating the reversible association of two methane molecules in bulk water is presented in Fig. 5. Perhaps the most striking feature of this computation is that, compared to the reference, 10.0 ns, simulation, a reasonably accurate estimate of the PMF can be obtained within 1.0 ns. Qualitatively, the different curves exibit the usual characteristics expected in most free energy profiles determined for small hydrophobic solutes immersed in an


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nitude of the interaction involved in methane dimerization, polarizability effects are anticipated to compare with statistical errors. In all fairness, the reversible association of methane does not constitute a cogent application of the average force formalism, considering that the reduced free energy barrier, viz. approximately 1 kcal/mol, is spontaneously crossed at 300 K. For instance, the free energy profiles derived from unbiased simulations, wherein the two methane molecules move freely, compare well with either the present ABF calculations, or the 120 ns TI simulation of Rick,58 using a polarizable potential energy function, and the significantly shorter simulations of Chipot et al.,57 which utilizes the pairwise additive AMBER force field.51 Yet, the example of this simple chemical system represents a limiting case for the use of the average force method, illustrating the effects of poorly chosen parameters on its efficiency—e.g., the size of the bins, in which F ␰ is accrued. Furthermore, it illuminates the role of the Jacobian correction in Eq. 共12兲, as evidenced in Fig. 5. Not too unexpectedly, the sample distribution of an unbiased simulation is uneven and increases with the distance separating the two molecules.

D. Transfer of methane across the water liquid-vapor interface

FIG. 5. Reversible dimerization of methane in water. Top: Free energy profiles obtained from a 1.0 共dotted line兲, 2.5 共solid line兲, 5.0 共dashed line兲, and 10.0 ns 共thick, solid line兲 unconstrained MD simulation with an adaptive bias. The sampling distribution characteristic of each MD simulation is included in the inset. Bottom: Effect of the adaptive biasing force on the sampling along ␰—1.0 共dotted line兲 and 10.0 ns 共dashed line兲 free, unbiased MD runs vs 10.0 ns unconstrained MD simulation, using ␦␰⫽0.1 共solid line兲 and 0.05 Å 共thick solid line兲. Each graph features in its inset the sampling distribution characteristic of each MD simulation.

aqueous environment—i.e., a global, contact minimum and a secondary, solvent-separated one located at a greater distance.52 The general features of the PMFs obtained here follow the general trend observed in earlier work53–58—viz. the more probable contact pair, likely to be stabilized by favorable hydration entropy. Quantitatively, a comparison with previous investigations reveals a good agreement in the relative depths of the two free energy minima. The contact pair occurs around 3.8 Å with a free energy of ca. ⫺0.8 kcal/mol, whereas the local, solvent-separated minimum emerges around 7.1 Å with a small free energy of ca. ⫺0.1 kcal/mol. It is worth noting that irrespective of the protocol utilized, the energetic hierarchy is conserved within about 0.1 kcal/ mol. Although less dramatic than was observed for the acetate-guanidinum ion pair, the choice of the force field has been shown to influence somewhat the overall features of the PMF. In particular, inclusion of explicit induction effects appear to reduce the free energy difference between the contact and the solvent-separated pair.56,58 Yet, considering the mag-

The computed PMFs for the transfer of methane across the water liquid-vapor interface are shown in Fig. 6. The profiles exhibit a shallow minimum of ⫺0.5 kcal/mol, corresponding to favorable dispersion interaction of methane with the interface, and a plateau in the bulk liquid water at ca. ⫹2.4 kcal/mol, with respect to the gas phase. This implies a free energy of hydration for a single methane molecule equal to ⫹2.4 kcal/mol, which is within chemical accuracy to the value of ⫹2.5⫾0.3 kcal/mol obtained using the FEP method—based on a forward and a reverse ‘‘alchemical’’ transformation.45 These results emphasize the consistency of two completely independent approaches for estimating a hydration free energy, namely FEP and ABF. Interestingly enough, Shirts et al.43 recently published free energies obtained employing TI with a variety of force fields, including CHARMM. Based on a 73.2 ns simulation, they report a hydration free energy for methane equal to ⫹2.44 kcal/mol, in nice agreement with the present results. In addition, these in silico free energies agree reasonably well with the experimentally determined value of ⫹2.0 kcal/ mol reported by Ben Naim and Marcus,59 indicating that CHARMM27 provides a globally correct—if not utterly accurate—description of the interaction of a small hydrophobic solute with water. Comparing the curves of Fig. 6, it can be seen that convergence of the PMF is rapidly reached for ␰⬍⫺10 Å, i.e., outside the bulk liquid water. Inside the water lamella, due to the strong fluctuations of the forces exerted onto the solute, the profile remains significantly noisier, even over a 10.0 ns run. The shortest, 1.0 ns simulation, nevertheless, is able to sample the full reaction path, so that a qualitatively correct PMF is generated.




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No system-dependent fine tuning of these parameters should be necessary for subsequent, routine calculations. IV. CONCLUSION

FIG. 6. Transfer of methane across the water liquid-vapor interface. Top: Free energy profiles obtained from a 1.0 共dotted line兲, 2.5 共solid line兲, 5.0 共dashed line兲, and 10.0 ns 共thick, solid line兲 unconstrained MD simulation with an adaptive bias. The sampling distribution characteristic of each MD simulation is included in the inset. Bottom: Effect of the width of the bins on the PMFs—10.0 ns MD runs with ␦␰⫽0.05 共thick solid line兲 and 0.1 Å 共thick dashed line兲. Effect of smoothing and accruing the force before application—5.0 ns unconstrained MD runs without smoothing 共solid line兲 and with the accumulation of 500 samples 共dashed line兲. For comparison purposes, the free energy profile of a 10.0 ns free, unbiased MD simulation is shown 共dotted line兲. Each graph features in its inset the sampling distribution characteristic of each MD simulation.

Figure 6 also shows the sampling distributions for the various simulations. It is worth noting that, for both the 5.0 and 10.0 ns runs, the amount of sampling is almost constant across the 25.0 Å ␰ range. Of particular interest, crossing of the Gibbs dividing surface hardly affects these distributions. This is in sharp contrast with the results of an unbiased run, wherein no adaptive force is applied to the solute. The other profiles of Fig. 6 were obtained by varying given control parameters utilized in our implementation— viz. the bin width, the smoothing of the biasing force, and the minimum number of samples required in a bin prior to applying the bias. Except for the test case, where no bias was applied, all curves are very close to that given by the reference run. It can, therefore, be safely assumed that the algorithm is sufficiently robust, and does not suffer from a high sensitivity to nonphysical parameters. In fact, for each parameter, a range of reasonable values may be found, that yields for all the chemical systems reported here accurate results and an acceptable, if not optimal numerical efficiency.

In this contribution, association of unconstrained dynamics and an adaptive biasing force was applied to the formalism of thermodynamic integration to determine PMFs along a well-defined reaction coordinate ␰. Implementation in MD codes essentially requires the evaluation at every time step of the force exerted on the system along ␰. The computational overhead, compared to other routine, unconstrained MD simulations is sufficiently marginal to be ignored. The major step forward that this approach proposes is to overcome free energy barriers without the prerequisite of knowing their height and shape to introduce the appropriate biasing potentials in the simulation—as is the case in the US method. In the course of an ABF simulation, the system effectively experiences a flat free energy landscape in the direction of the reaction coordinate. Progression along ␰ is, thus, limited by the sole self-diffusion properties of the chemical system in that particular direction. In this respect, convergence of the free energy is likely to be improved, because for all values of the reaction coordinate, the average force necessary for escaping the energetic penalty is updated and refined continuously as the simulation advances. Four case examples were chosen to illustrate the efficiency of the approach. Reversible unfolding of deca-Lalanine illuminates the power of the method, able to sample the entire reaction coordinate uniformly within 5.0 ns—i.e., providing a PMF both qualitatively and quantitatively comparable to that generated by a US simulation forty times longer, performed in equivalent conditions. Association of acetate and guanidinium or dimerization of methane in water highlights the key role played by the width of the bins and the number of bins over which fluctuations of the force are smoothed out in the sampling along ␰, suggesting that these parameters should be chosen carefully when setting the MD simulation up. Last, the quantitative agreement between the present method and FEP double annihilation to reproduce the hydration free energy of methane confirms the robustness of the average force formalism. It should be underlined that in all four instances, thermodynamic reversibility was observed. It should be kept in mind, however, that evolution along ␰ and the associated convergence of the simulation is only one element of the free energy problem. The computational bottleneck, and, therefore, the cost of the simulation, still remains rooted in the relaxation of the remaining degrees of freedom involved in the system. How fast the average force 具 F ␰ 典 ␰ converges is, therefore, intimately connected to the intrinsic relaxation time of those degrees of freedom participating in the ensemble average. ACKNOWLEDGMENTS

The authors would like to thank Dr. Andrew Pohorille and Dr. Eric Darve for fruitful discussions, and the developers of NAMD for their valuable help with the implementation. The LORIA-CRVHP, Vandœuvre-le`s-Nancy, France,


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