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 and the potential of mean force. 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 free energy differences.
Assigning an energy
Possible features to which an energy can be assigned include torsion angles, 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 in the structure. The energies are determined using statistics on amino acid contacts in a database of known protein structures.
Sippl's potential of mean force
Overview
Many textbooks present the potentials of mean force 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. For example, could be the conditional probability of finding the atoms of a valine and a serine at a given distance from each other, giving rise to the free energy difference . The total free energy difference of a protein, , is then claimed to be the sum of all the pairwise free energies: where the sum runs over all amino acid pairs and is their corresponding distance. In many studies does not depend on the amino acid sequence. Intuitively, it is clear that a low value for indicates that the set of distances in a structure is more likely in proteins than in the reference state. However, the physical meaning of these PMFs have been widely disputed since their introduction. The main issues are the interpretation of this "potential" as a true, physically valid potential of mean force, the nature of the reference state and its optimal formulation, and the validity of generalizations beyond pairwise distances.
Justification
Analogy with liquid systems
The first, qualitative justification of PMFs is due to Sippl, and based on an analogy with the statistical physics of liquids. For liquids, the potential of mean force is related to the radial distribution function, which is given by: where and are the respective probabilities of finding two particles at a distance from each other in the liquid and in the reference state. For liquids, the reference state is clearly defined; it corresponds to the ideal gas, consisting of non-interacting particles. The two-particle potential of mean force is related to by: According to the reversible work theorem, the two-particle potential of mean force is the reversible work required to bring two particles in the liquid from infinite separation to a distance from each other. Sippl justified the use of PMFs - a few years after he introduced them for use in protein structure prediction - by appealing to the analogy with the reversible work theorem for liquids. For liquids, can be experimentally measured using small angle X-ray scattering; for proteins, is obtained from the set of known protein structures, as explained in the previous section. However, as Ben-Naim writes in a publication on the subject:
the quantities, referred to as `statistical potentials,' `structure based potentials,' or `pair potentials of mean force', as derived from the protein data bank, are neither `potentials' nor `potentials of mean force,' in the ordinary sense as used in the literature on liquids and solutions.
Another issue is that the analogy does not specify a suitable reference state for proteins.
Analogy with likelihood
Baker and co-workers justified PMFs from a Bayesian point of view and used these insights in the construction of the coarse grained ROSETTA energy function. According to Bayesian probability calculus, the conditional probability of a structure, given the amino acid sequence, can be written as: is proportional to the product of the likelihood times the prior . By assuming that the likelihood can be approximated as a product of pairwise probabilities, and applying Bayes' theorem, the likelihood can be written as: where the product runs over all amino acid pairs , and is the distance between amino acids and. Obviously, the negative of the logarithm of the expression has the same functional form as the classic pairwise distance PMFs, with the denominator playing the role of the reference state. This explanation has two shortcomings: it is purely qualitative, and relies on the unfounded assumption the likelihood can be expressed as a product of pairwise probabilities.
Applications
Statistical potentials are used as energy functions in the assessment of an ensemble of structural models produced by homology modeling or protein threading - predictions for the tertiary structure assumed by a particular amino acid sequence made on the basis of comparisons to one or more homologous proteins with known structure. 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.
