Derivation of a Markov state model of the dynamics of a protein-like chain immersed in an implicit solvent

TL;DR

This paper derives a Markov state model from first principles to describe the dynamics of a protein-like chain in an implicit solvent, analyzing folding behavior and relaxation modes across different chain lengths and temperatures.

Contribution

It introduces a microscopic, first-principles derivation of a Markov state model for protein-like chains in implicit solvent, linking rate constants to escape rates and analyzing folding pathways.

Findings

Short chains exhibit single-exponential relaxation at low temperatures.

Longer chains show multi-exponential relaxation at intermediate temperatures.

Folding becomes rapid and single exponential at low temperatures for longer chains.

Abstract

A Markov state model of the dynamics of a protein-like chain immersed in an implicit hard sphere solvent is derived from first principles for a system of monomers that interact via discontinuous potentials designed to account for local structure and bonding in a coarse-grained sense. The model is based on the assumption that the implicit solvent interacts on a fast time scale with the monomers of the chain compared to the time scale for structural rearrangements of the chain and provides sufficient friction so that the motion of monomers is governed by the Smoluchowski equation. A microscopic theory for the dynamics of the system is developed that reduces to a Markovian model of the kinetics under well-defined conditions. Microscopic expressions for the rate constants that appear in the Markov state model are analyzed and expressed in terms of a temperature-dependent linear combination of escape rates that themselves are independent of temperature. The Markov model is studied by analyzing the eigenvalues and eigenvectors of the matrix of transition rates, and the equilibration process for a simple helix-forming system from an ensemble of initially extended configurations to mainly folded configurations is investigated as a function of temperature for a number of different chain lengths. For short chains, the relaxation is primarily single-exponential and becomes independent of temperature in the low-temperature regime. The profile is more complicated for longer chains, where multi-exponential relaxation behavior is seen at intermediate temperatures followed by a low temperature regime in which the folding becomes rapid and single exponential. It is demonstrated that the behavior of the equilibration profile as the temperature is lowered can be understood in terms of the number of relaxation modes or ``folding pathways'' that contribute to the evolution of the state populations.