Structural Fluctuation of Protein in Water around Its Native State: A New Statistical Mechanics Formulation

TL;DR

This paper introduces a novel statistical mechanics framework that models protein structural fluctuations in water, linking them through coupled equations derived from the generalized Langevin equation and 3D-RISM theory, with applications to molecular recognition.

Contribution

It develops a new formulation combining Langevin dynamics and RISM theory to describe protein-water fluctuations at the atomic level, including a method to evaluate the free energy Hessian.

Findings

Derived coupled equations for protein and water fluctuations.

Identified the Hessian matrix as the second derivative of free energy.

Proposed a method to compute the Hessian using 3D-RISM theory.

Abstract

A new statistical mechanics formulation of characterizing the structural fluctuation of protein correlated with that of water is presented based on the generalized Langevin equation and the 3D-RISM/RISM theory of molecular liquids. The displacement vector of atom positions and their conjugated momentum, are chosen for the dynamic variables for protein, while the density fields of atoms and their momentum fields are chosen for water. Projection of other degrees of freedom onto those dynamic variables using the standard projection operator method produces essentially two equations which describe the time evolution of fluctuation concerning the density field of solvent and the conformation of protein around an equilibrium state, which are coupled with each other. The equation concerning the protein dynamics is formally akin to that of the coupled Langevin oscillators, and is a generalization of the latter, to atomic level. The most intriguing feature of the new equation is that it contains the variance-covariance matrix as the "Hessian" term describing the "force" restoring an equilibrium conformation, which is the second moment of the fluctuation of atom positions. The "Hessian" matrix is naturally identified as the second derivative of the free energy surface around the equilibrium. A method to evaluate the Hessian matrix based on the 3D-RISM/RISM theory is proposed. Proposed also is an application of the present formulation to the molecular recognition, in which the conformational fluctuation of protein around its native state becomes an important factor as exemplified by so called "induced fitting".