Enhancing sampling with free-energy calculations

Many important phenomena of biological relevance occur on timescales largely exceeding milliseconds (ms). Following, for instance, by means of atomistic simulations broad movements between protein domains connected allosterically would supply a comprehensive, dynamic picture inaccessible to experiment, and help meet challenges hitherto insuperable, like establishing the chronology of conformational transitions triggered by external stimuli [1]. In spite of significant advances, the timescales amenable to molecular dynamics (MD) of very large objects, commonly from tens of nanoseconds (ns) per day on supercomputers to more than a microsecond on special-purpose computers [2], still remain orders of magnitude less than those spanned by biological processes. Computational investigation of ms–to–s rare events, therefore requires more than just brute-force MD on fast computers.

In computer simulations, a process assumed to be ergodic may appear somewhat non-ergodic as a consequence of incomplete sampling rooted in the slow diffusion of the molecular process. In many biological processes, this shortcoming arises from high free-energy barriers between distinct volumes of phase space. Transitions between these regions, thus, appear like rare events, difficult to capture over the common brute-force MD timescales. Free-energy calculations with enhanced sampling techniques represent a cogent alternative to act on the relevant slow degrees of freedom (DOFs), and encourage sampling along them.

Many methods aimed at the determination of free-energy differences share the common concept of progress variable [3], or reaction coordinate (RC), describing the transition between metastabilities in configurational space. Assuming timescale separation, the RC is ordinarily modeled with a few collective variables (CVs) [4,5], supposed to embrace all the DOFs controlling the transition. This leap of faith rests on our ability to identify all the slow DOFs, essentially ignoring the fast ones. An exaggeratedly reductionist view of the CV space invariably results in an inadequate description of the RC, and, hence, of the dynamics of the transition. Although the free-energy difference does not depend on the path followed for its determination, and, hence, on the CV space wherein this path lies, efficiency and convergence of the simulation stringently depends on the choice of the CVs. Since their identification cannot simply rely on the human eye—or on our intuition, a host of methods has been devised to extract the relevant DOFs associated to transitions in configurational space, including by machine learning [6].

Accelerating sampling to investigate rare events is commonly achieved by encouraging exploration of important regions of configurational space along the surrogate RC, ξ(x). Here, we focus on free-energy methods, whereby an external bias is applied along ξ(x). Owing to the ever-expanding palette of algorithms, and the difficulty to choose the best-suited one for the problem at hand, an attempt to classify the available methods [7] and show how they are interconnected is necessary (see Figure 1).

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