A statistical mechanics perspective for protein folding from $q$-state Potts model

The folding of a peptide chain into a three dimensional structure is a thermodynamically driven process such that the chain naturally evolves to form domains of similar amino acids. The formation of this domain occurs by curling the one dimensional amino acid sequence by moving similar amino acids proximity to each other. We model this formation of domains or ordering of amino acids using q-state Potts model and study the thermodynamic properties using a statistical mechanics approach. Converting the interacting amino acids into an effectively non-interacting model using a mean-field theory, we calculate the Helmholtz free energy (HFE). Then by investigating the HFE, we study the properties of protein folding transition qualitatively. We find that the protein folding phase transition is a strongly first order and the specific heat shows the experimental signatures of this phase transition. Further, we compare these mean-field results with exact transfer matrix results in one dimension and then large $q$ expansion results in two dimensions.