pairwise differences, and (iii) the reference ratio is determined by the prior distribution. Reference ratio Expressions that resemble statistical PMFs naturally result from the application of probability theory to solve a fundamental problem that arises in protein structure prediction: how to improve an imperfect probability distribution over a first variable using a probability distribution over a second variable , with . Typically, and are fine and coarse grained variables, respectively. For example, could concern the local structure of the protein, while could concern the pairwise distances between the amino acids. In that case, could for example be a vector of dihedral angles that specifies all atom positions (assuming ideal bond lengths and angles). In order to combine the two distributions, such that the local structure will be distributed according to , while the pairwise distances will be distributed according to , the following expression is needed: where is the distribution over implied by . The ratio in the expression corresponds to the PMF. Typically, is brought in by sampling (typically from a fragment library), and not explicitly evaluated; the ratio, which in contrast is explicitly evaluated, corresponds to Sippl's PMF. This explanation is quantitive, and allows the generalization of statistical PMFs from pairwise distances to arbitrary coarse grained variables. It also provides a rigorous definition of the reference state, which is implied by . Conventional applications of pairwise distance statistical PMFs usually lack two necessary features to make them fully rigorous: the use of a proper probability distribution over pairwise distances in proteins, and the recognition that the reference state is rigorously defined by . Applications Statistical potentials are used as energy functions in the assessment of an ensemble of structural models produced by homology modeling or protein threading. Many differently parameterized statistical potentials have been shown to successfully identify the native state structure from an ensemble of decoy or non-native structures. Statistical potentials are not only used for protein structure prediction, but also for modelling the protein folding pathway. See also Scoring functions for docking Discrete optimized protein energy CASP CAMEO3D Lennard-Jones potential Bond order potential Notes References Potentials Computational biology Bioinformatics Molecular modelling Protein structure