Potentials of Mean Force for Protein Structure Prediction Vindicated, Formalized and Generalized

Abstract

Understanding protein structure is of crucial importance in science, medicine and biotechnology. For about two decades, knowledge-based potentials based on pairwise distances – so-called “potentials of mean force” (PMFs) – have been center stage in the prediction and design of protein structure and the simulation of protein folding. However, the validity, scope and limitations of these potentials are still vigorously debated and disputed, and the optimal choice of the reference state – a necessary component of these potentials – is an unsolved problem. PMFs are loosely justified by analogy to the reversible work theorem in statistical physics, or by a statistical argument based on a likelihood function. Both justifications are insightful but leave many questions unanswered. Here, we show for the first time that PMFs can be seen as approximations to quantities that do have a rigorous probabilistic justification: they naturally arise when probability distributions over different features of proteins need to be combined. We call these quantities “reference ratio distributions” deriving from the application of the “reference ratio method.” This new view is not only of theoretical relevance but leads to many insights that are of direct practical use: the reference state is uniquely defined and does not require external physical insights; the approach can be generalized beyond pairwise distances to arbitrary features of protein structure; and it becomes clear for which purposes the use of these quantities is justified. We illustrate these insights with two applications, involving the radius of gyration and hydrogen bonding. In the latter case, we also show how the reference ratio method can be iteratively applied to sculpt an energy funnel. Our results considerably increase the understanding and scope of energy functions derived from known biomolecular structures.

Predicting RNA 3D structure using a coarse-grain helix-centered model

“…The knowledge-based potentials are based on coarse-grained measures whose distributions differ between native structures (target distribution) and structures sampled from the proposal distribution (reference distribution). For each of these coarse-grained measures, we will present examples of the target distribution and the reference distribution (as calculated from a decoy) as well as the associated energy calculated by the reference ratio method (Hamelryck et al 2010;Valentin et al 2014). The energy associated with a value x of the measure is calculated as the log of the ratio of the target distribution [p t (x)] divided by the reference distribution [p r (x)] and multiplied by a factor c which serves as a parameter for tuning how closely the target distribution should match the sampled values (see Supplemental Section A.6.2):…”

On statistical energy functions for biomolecular modeling and design

“…For example, if a potential of mean force of the radius of gyration is to be applied in conjunction with a set of local interaction terms, either the reference distribution for the local interaction terms should be sampled with a given potential of mean force for the radius of gyration, or the reference distribution for the radius of gyration should be estimated with conformations sampled with given local interaction terms [31].…”

Simplification of complexity in protein molecular systems by grouping amino acids: a view from physics

“…On the residue level, the strength of pairwise interactions could be determined statistically based on their occurrences in native structures, and are stored in the form of matrix [72][73][74][80][81][82]. The features of proteins have already been implicitly included.…”