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Understanding protein structure is of crucial importance in science, medicineand biotechnology. For about two decades, knowledge based potentials based onpairwise distances -- so-called "potentials of mean force" (PMFs) -- have beencenter stage in the prediction and design of protein structure and thesimulation of protein folding. However, the validity, scope and limitations ofthese potentials are still vigorously debated and disputed, and the optimalchoice of the reference state -- a necessary component of these potentials --is an unsolved problem. PMFs are loosely justified by analogy to the reversiblework theorem in statistical physics, or by a statistical argument based on alikelihood function. Both justifications are insightful but leave manyquestions unanswered. Here, we show for the first time that PMFs can be seen asapproximations to quantities that do have a rigorous probabilisticjustification: they naturally arise when probability distributions overdifferent features of proteins need to be combined. We call these quantitiesreference ratio distributions deriving from the application of the referenceratio method. This new view is not only of theoretical relevance, but leads tomany insights that are of direct practical use: the reference state is uniquelydefined and does not require external physical insights; the approach can begeneralized beyond pairwise distances to arbitrary features of proteinstructure; and it becomes clear for which purposes the use of these quantitiesis justified. We illustrate these insights with two applications, involving theradius of gyration and hydrogen bonding. In the latter case, we also show howthe reference ratio method can be iteratively applied to sculpt an energyfunnel. Our results considerably increase the understanding and scope of energyfunctions derived from known biomolecular structures.

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