Graphical models have become powerful frameworks for protein structure prediction, protein–protein interaction, and free energy calculations for protein structures. Using a graphical model to represent the protein structure allows the solution of many problems including secondary structure prediction, protein-protein interactions, protein-drug interaction, and free energy calculations. There are two main approaches to using graphical models in protein structure modeling. The first approach uses discrete variables for representing the coordinates or the dihedral angles of the protein structure. The variables are originally all continuous values and, to transform them into discrete values, a discretization process is typically applied. The second approach uses continuous variables for the coordinates or dihedral angles.
Discrete graphical models for protein structure Markov random fields, also known as undirected graphical models are common representations for this problem. Given an undirected graph G = (V, E), a set of random variables X = (Xv)v ∈ V indexed by V, form a Markov random field with respect to G if they satisfy the pairwise Markov property:
any two non-adjacent variables are conditionally independent given all other variables:
X u ⊥ ⊥ X v | X V ∖ { u , v } if { u , v } ∉ E . {\displaystyle X_{u}\perp \!\!\!\perp X_{v}|X_{V\setminus \{u,v\}}\quad {\text{if }}\{u,v\}\notin E.}
In the discrete model, the continuous variables are discretized into a set of favorable discrete values. If the variables of choice are dihedral angles, the discretization is typically done by mapping each value to the corresponding rotamer conformation.
Model Let X = {Xb, Xs} be the random variables representing the entire protein structure. Xb can be represented by a set of 3-d coordinates of the backbone atoms, or equivalently, by a sequence of bond lengths and dihedral angles. The probability of a particular conformation x can then be written as:
p ( X = x | Θ ) = p ( X b = x b ) p ( X s = x s | X b , Θ ) , {\displaystyle p(X=x|\Theta )=p(X_{b}=x_{b})p(X_{s}=x_{s}|X_{b},\Theta ),\,}
where Θ {\displaystyle \Theta } represents any parameters used to describe this model, including sequence information, temperature etc. Frequently the backbone is assumed to be rigid with a known conformation, and the problem is then transformed to a side-chain placement problem. The structure of the graph is also encoded in Θ {\displaystyle \Theta } . This structure shows which two variables are conditionally independent. As an example, side chain angles of two residues far apart can be independent given all other angles in the protein. To extract this structure, researchers use a distance threshold, and only a pair of residues which are within that threshold are considered connected (i.e. have an edge between them). Given this representation, the probability of a particular side chain conformation xs given the backbone conformation xb can be expressed as
p ( X s = x s | X b = x b ) = 1 Z ∏ c ∈ C ( G ) Φ c ( x s c , x b c ) {\displaystyle p(X_{s}=x_{s}|X_{b}=x_{b})={\frac {1}{Z}}\prod _{c\in C(G)}\Phi _{c}(x_{s}^{c},x_{b}^{c})}
where C(G) is the set of all cliques in G, Φ {\displaystyle \Phi } is a potential function defined over the variables, and Z is the partition function. To completely characterize the MRF, it is necessary to define the potential function Φ {\displaystyle \Phi } . To simplify, the cliques of a graph are usually restricted to only the cliques of size 2, which means the potential function is only defined over pairs of variables. In Goblin System, these pairwise functions are defined as
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