Exactness of the cluster variation method and factorization of the equilibrium probability for the Wako-Saito-Munoz-Eaton model of protein folding

TL;DR

This paper proves that the cluster variation method provides an exact solution for the equilibrium probability distribution in a specific protein folding model, by showing it factors into local cluster probabilities.

Contribution

It demonstrates the exactness of the cluster variation method for a long-range interaction protein folding model through factorization of the equilibrium probability.

Findings

Cluster variation method is exact for the model.

Equilibrium probability factors into local cluster probabilities.

Clusters include single sites, pairs, and square plaquettes.

Abstract

I study the properties of the equilibrium probability distribution of a protein folding model originally introduced by Wako and Saito, and later reconsidered by Munoz and Eaton. The model is a one-dimensional model with binary variables and many-body, long-range interactions, which has been solved exactly through a mapping to a two-dimensional model of binary variables with local constraints. Here I show that the equilibrium probability of this two-dimensional model factors into the product of local cluster probabilities, each raised to a suitable exponent. The clusters involved are single sites, nearest-neighbour pairs and square plaquettes, and the exponents are the coefficients of the entropy expansion of the cluster variation method. As a consequence, the cluster variation method is exact for this model.