Single-Sweep Methods for Free Energy Calculations

The free energy (or potential of mean force) is the thermodynamic force driving structural processes such as conformational changes of macromolecules in aqueous solution, ligand binding at the active site of an enzyme, protein-protein association, etc. The free energy gives information about both the rate at which these processes occur and the mechanism by which they occur. This makes free energy calculations a central issue in biophysics. Molecular dynamics (MD) simulations provide a tool for performing such calculations on a computer in a way which is potentially both precise and inexpensive (e.g. [1,2,3]). Since a free energy is in essence the logarithm of a probability density function (see (1) below for a precise definition) it can in principle be calculated by histogram methods based on the binning of an MD trajectory. This direct approach, however, turns out to be unpractical in general because the time scale required for the trajectory to explore all the relevant regions of configuration space is prohibitively long. Probably the best known and most widely used technique to get around this difficulty is the weighted histogram analysis method (WHAM) [4]. Following [5], WHAM adds artificial biasing potentials to maintain the MD system in certain umbrella sampling windows. WHAM then recombines in an optimal way the histograms from all the biased simulations to compute the free energy. WHAM is much more efficient than the direct sampling approach, and generalizations such as [6] alleviate somewhat the problem of where to put the umbrella windows (usually, this requires some a priori knowledge of the free energy landscape). In practice, however, WHAM remains computationally demanding and it only works to compute the free energy in 2 or 3 variables. An interesting alternative to WHAM is metadynamics [7,8]. In essence metadynamics is a way to use an MD trajectory to place inverted umbrella sam-pling windows on-the-fly and use these windows both to bias the MD simulation and as histogram bins to sample the free energy directly (thereby bypassing the need of further histogram analysis in each window).

Both WHAM and metadynamics compute the free energy directly by histogram methods, but an alternative approach is possible. Unlike the free energy which is a global quantity, its negative gradient (known as the mean force) can be expressed in terms of a local expectation and thereby computed at a given point in the free energy landscape. This is the essence of the blue moon sampling strategy [9] and it offers the possibility to calculate first the mean force at a given set of locations, then use this information to reconstruct the free energy globally. In one dimension, this approach is known as thermodynamic integration and it goes back to Kirkwood [10]. In higher dimensions, however, this way to compute free energies has been impeded by two issues. The first is where to place the points at which to compute the mean force, and the second is how to reconstruct the free energy from these data

In this paper, we propose a method, termed singlesweep method, which addresses both of these issues in two complementary but independent steps. In a first step, we use the temperature-accelerated molecular dynamics (TAMD) proposed in [11] (see also [12,13]) to quickly sweep through the important regions of the free energy landscape and identify points in these regions where to compute the mean force. In the second step we then reconstruct the free energy globally from the mean force by representing the free energy using radial-basis functions, and adjusting the parameters in this representation via minimization of an objective function.

The single-sweep method is easy to use and implement, does not require a priori knowledge of the free energy landscape, and can be applied to map free energies in several variables (up to four, as demonstrated here, and probably more). The single-sweep method is also very efficient, especially since the mean force calculations can be performed using independent calculations on distributed processors (i.e. using grid computing facilities [14,15]).

The reminder of this paper is organized as follows. In Sec. II, we describe the two steps of the single-sweep method in detail, starting with the second one for convenience. In Sec. III we illustrate the method on a simple two-dimensional example. This example is then used for comparison with metadynamics in Sec. IV. In Sec. V we use the single-sweep method to compute the free energy of alanine dipeptide (AD) in solution in two and in four of its dihedral angles. Finally, concluding remarks are made in Sec. VI and the details of the MD calculation on AD are given in Appendix A

We shall consider a molecular system with n degrees of freedom whose position in configuration space Ω ⊆ R n will be denoted by x. We also introduce a set of N collective variables θ(x) = (θ 1 (x), . . . , θ N (x)) which are functions of x such as torsion angles, interatomi

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