Statistical potential

In protein structure prediction, a statistical potential or knowledge-based potential is an energy function derived from an analysis of known protein structures in the Protein Data Bank.the quantities, referred to as `statistical potentials,' `structurebased potentials,' or `pair potentials of mean force', as derived fromthe protein data bank, are neither `potentials' nor `potentials ofmean force,' in the ordinary sense as used in the literature onliquids and solutions. In protein structure prediction, a statistical potential or knowledge-based potential is an energy function derived from an analysis of known protein structures in the Protein Data Bank. Many methods exist to obtain such potentials; two notable methods are the quasi-chemical approximation (due to Miyazawa and Jernigan) and the potential of mean force (due to Sippl ). Although the obtained energies are often considered as approximations of the free energy, this physical interpretation is incorrect. Nonetheless, they have been applied with a limited success in many cases because they frequently correlate with actual (physical) free energy differences. Possible features to which an energy can be assigned include torsion angles (such as the ϕ , ψ {displaystyle phi ,psi } angles of the Ramachandran plot), solvent exposure or hydrogen bond geometry. The classic application of such potentials is however pairwise amino acid contacts or distances. 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 Protein Data Bank). Many textbooks present the potentials of mean force (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 r {displaystyle r} is the distance, k {displaystyle k} is the Boltzmann constant, T {displaystyle T} isthe temperature and Z {displaystyle Z} is the partition function, with The quantity F ( r ) {displaystyle F(r)} is the free energy assigned to the pairwise system.Simple rearrangement results in the inverse Boltzmann formula,which expresses the free energy F ( r ) {displaystyle F(r)} as a function of P ( r ) {displaystyle P(r)} : To construct a PMF, one then introduces a so-called referencestate with a corresponding distribution Q R {displaystyle Q_{R}} and partition function Z R {displaystyle Z_{R}} , and calculates the following free energy difference: The reference state typically results from a hypotheticalsystem in which the specific interactions between the amino acidsare absent. The second term involving Z {displaystyle Z} and Z R {displaystyle Z_{R}} can be ignored, as it is a constant. In practice, P ( r ) {displaystyle P(r)} is estimated from the database of known proteinstructures, while Q R ( r ) {displaystyle Q_{R}(r)} typically results from calculationsor simulations. For example, P ( r ) {displaystyle P(r)} could be the conditional probabilityof finding the C β {displaystyle Ceta } atoms of a valine and a serine at a givendistance r {displaystyle r} from each other, giving rise to the free energy difference Δ F {displaystyle Delta F} . The total free energy difference of a protein, Δ F T {displaystyle Delta F_{ extrm {T}}} , is then claimed to be the sumof all the pairwise free energies: