[Paper Review] A statistical mechanics perspective for protein folding from $q$-state Potts model

Theja N. De Silva, Vattika Sivised|arXiv (Cornell University)|Sep 14, 2017

Spectroscopy and Quantum Chemical Studies3 references3 citations

TL;DR

This paper models protein folding using a q-state Potts model on a lattice to study thermodynamic transitions via statistical mechanics. Using variational mean-field theory, it identifies a strongly first-order phase transition with a critical temperature of $ k_B T_C = 0.58J_0 $, validated by exact transfer matrix and large-q expansion methods, showing mean-field overestimates $ T_C $ by 9% in 2D square lattices.

ABSTRACT

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.

Motivation & Objective

• To understand how protein folding emerges from collective interactions among amino acids using statistical mechanics.
• To model the folding process as a phase transition driven by competition between entropy and energy.
• To evaluate the accuracy of mean-field approximations against exact and large-q expansion methods in one and two dimensions.
• To investigate the role of local environment and external fields on correlation length and folding transition.
• To provide a qualitative but insightful framework linking statistical physics to protein folding thermodynamics.

Proposed method

• Model the protein as a q-state Potts spin system on a lattice, where each site represents an amino acid with q possible states corresponding to residue types.
• Apply variational mean-field theory to map the interacting Potts Hamiltonian into an effectively non-interacting system for analytical tractability.
• Derive the Helmholtz free energy (HFE) as a function of the order parameter to analyze the folding transition.
• Use the transfer matrix method to compute exact correlation lengths and thermodynamic properties in one-dimensional systems with an external field.
• Perform high- and low-temperature expansions of the partition function on a 2D square lattice, then apply duality and large-q expansion to construct the free energy series.
• Compare mean-field results with exact 1D transfer matrix and 2D large-q expansion results to assess critical temperature accuracy.

Experimental results

Research questions

• RQ1 Does the q-state Potts model capture the thermodynamic features of protein folding as a phase transition?
• RQ2 How does the mean-field approximation describe the folding transition, and how accurate is it compared to exact methods?
• RQ3 What is the nature of the phase transition (first-order, second-order) in the Potts model representation of protein folding?
• RQ4 How does an external field influence the correlation length and onset of folding in one-dimensional systems?
• RQ5 To what extent does the large-q expansion in two dimensions correct the mean-field prediction of the critical temperature?

Key findings

• The protein folding transition in the q-state Potts model is strongly first-order, as indicated by a discontinuity in the free energy at the critical point.
• The critical temperature is found to be $ k_B T_C = 0.58J_0 $, derived from the large-q expansion in two dimensions.
• Mean-field theory overestimates the critical temperature by 9% compared to the large-q expansion result in the 2D square lattice.
• In one dimension, the correlation length exhibits a sharp peak at the folding transition onset under a small external field, signaling a critical behavior.
• The high- and low-temperature expansions of the partition function are combined via duality to construct a series expansion of the free energy up to seventh order in $ v = (e^K - 1)/\sqrt{q} $.
• The model successfully captures the competition between entropy (unfolded state) and energy (folded state), with a clear transition at $ T_C $, supporting a thermodynamic view of folding.