In protein structure prediction, statistical potentials or knowledge-based potentials are scoring functions derived from an analysis of known protein structures in the Protein Data Bank (PDB). The original method to obtain such potentials is the quasi-chemical approximation, due to Miyazawa and Jernigan. It was later followed by the potential of mean force (statistical PMF ), developed by Sippl. Although the obtained scores are often considered as approximations of the free energy—thus referred to as pseudo-energies—this physical interpretation is incorrect. Nonetheless, they are applied with success in many cases, because they frequently correlate with actual Gibbs free energy differences. Overview Possible features to which a pseudo-energy can be assigned include: interatomic distances, torsion angles, solvent exposure, or hydrogen bond geometry. The classic application is, however, based on pairwise amino acid contacts or distances, thus producing statistical interatomic potentials. For pairwise amino acid contacts, a statistical potential is formulated as an interaction matrix that assigns a weight or energy value to each possible pair of standard amino acids. The energy of a particular structural model is then the combined energy of all pairwise contacts (defined as two amino acids within a certain distance of each other) in the structure. The energies are determined using statistics on amino acid contacts in a database of known protein structures (obtained from the PDB). History Initial development Many textbooks present the statistical PMFs as proposed by Sippl as a simple consequence of the Boltzmann distribution, as applied to pairwise distances between amino acids. This is incorrect, but a useful start to introduce the construction of the potential in practice. The Boltzmann distribution applied to a specific pair of amino acids, is given by: where is the distance, is the Boltzmann constant, is the temperature and is the partition function, with The quantity is the free energy assigned to the pairwise system. Simple rearrangement results in the inverse Boltzmann formula, which expresses the free energy as a function of : To construct a PMF, one then introduces a so-called reference state with a corresponding distribution and partition function , and calculates the following free energy difference: The reference state typically results from a hypothetical system in which the specific interactions between the amino acids are absent. The second term involving and can be ignored, as it is a constant. In practice, is estimated from the database of known protein structures, while typically results from calculations or simulations.