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

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.

💡 Research Summary

The paper revisits the long‑standing use of distance‑based potentials of mean force (PMFs) in protein structure prediction and design, addressing the persistent controversy over their theoretical justification, the choice of reference state, and their limited applicability to pairwise distances. The authors introduce a rigorous probabilistic framework called the “reference ratio method,” which shows that PMFs are in fact approximations to well‑defined quantities they term reference ratio distributions.

In this framework two probability distributions are considered: Q(X), a prior distribution over fine‑grained structural variables (e.g., fragment‑library dihedral angles), and P(Y), a target distribution over coarse‑grained variables that are deterministic functions of X (e.g., radius of gyration, hydrogen‑bond network, or any global feature). Because Y = f(X), the prior Q(X) implicitly defines a reference distribution Q_R(Y) – the distribution of Y when structures are sampled solely from Q(X). The correct combined distribution that respects both sources of information is

  P̂(X) = Q(X)·