Exact Statistical Mechanical Investigation of a Finite Model Protein in its environment: A Small System Paradigm

TL;DR

This paper provides an exact statistical mechanical analysis of a finite model protein, exploring how size and energetics influence folding, entropy, and energy landscapes, with implications for understanding small protein thermodynamics.

Contribution

It introduces an exact enumeration method for finite proteins on a lattice, analyzing the effects of size, energetics, and conformational restrictions on thermodynamics and energy landscapes.

Findings

Canonical entropy exceeds microcanonical entropy in small systems.

Small proteins do not exhibit self-averaging properties.

The model shows no energy gap despite sharp folding transition expectations.

Abstract

We consider a general incompressible finite model protein of size M in its environment, which we represent by a semiflexible copolymer consisting of amino acid residues classified into only two species (H and P, see text) following Lau and Dill. We allow various interactions between chemically unbonded residues in a given sequence and the solvent (water), and exactly enumerate the number of conformations W(E) as a function of the energy E on an infinite lattice under two different conditions: (i) we allow conformations that are restricted to be compact (known as Hamilton walk conformations), and (ii) we allow unrestricted conformations that can also be non-compact. It is easily demonstrated using plausible arguments that our model does not possess any energy gap even though it is supposed to exhibit a sharp folding transition in the thermodynamic limit. The enumeration allows us to investigate exactly the effects of energetics on the native state(s), and the effect of small size on protein thermodynamics and, in particular, on the differences between the microcanonical and canonical ensembles. We find that the canonical entropy is much larger than the microcanonical entropy for finite systems. We investigate the property of self-averaging and conclude that small proteins do not self-average. We also present results that (i) provide some understanding of the energy landscape, and (ii) shed light on the free energy landscape at different temperatures.