Full Statistics of Conjugated Thermodynamic Ensembles in Chains of Bistable Units

Manon Benedito,Stefano Giordano,Fabio Manca

doi:10.3390/inventions4010019

Abstract

The statistical mechanics and the thermodynamics of small systems are characterized by the non-equivalence of the statistical ensembles. When concerning a polymer chain or an arbitrary chain of independent units, this concept leads to different force-extension responses for the isotensional (Gibbs) and the isometric (Helmholtz) thermodynamic ensembles for a limited number of units (far from the thermodynamic limit). While the average force-extension response has been largely investigated in both Gibbs and Helmholtz ensembles, the full statistical characterization of this thermo-mechanical behavior has not been approached by evaluating the corresponding probability densities. Therefore, we elaborate in this paper a technique for obtaining the probability density of the extension when force is applied (Gibbs ensemble) and the probability density of the force when the extension is prescribed (Helmholtz ensemble). This methodology, here developed at thermodynamic equilibrium, is applied to a specific chain composed of units characterized by a bistable potential energy, which is able to mimic the folding and unfolding of several macromolecules of biological origin.

Highlights

The recent developments of thermodynamics and statistical mechanics concern the thermodynamics of small systems, kept far from the thermodynamic limit, and the stochastic thermodynamics, which is based on Langevin or stochastic differential equations
When we adopt the approximation of the energy wells with two quadratic functions, we lose the information about the energy barrier between the wells and we can not use this version of our model to deal with out-of-equilibrium regimes [55]. This approach has been recently used to investigate the properties of several two-state systems and macromolecular chains [74,75,76,77,78]. Both the Gibbs and the Helmholtz ensembles can be studied by the spin variables methodology, permitting to draw direct comparisons between isotensional and isometric conditions, provided that we work at thermodynamic equilibrium
In this work we considered the comparison of Gibbs and Helmholtz ensembles of the statistical mechanics in the context of the stretching of chains of bistable units

Summary

Introduction

The recent developments of thermodynamics and statistical mechanics concern the thermodynamics of small systems, kept far from the thermodynamic limit, and the stochastic thermodynamics, which is based on Langevin or stochastic differential equations. In the second theoretical approach, the out-of-equilibrium statistical mechanics is introduced by means of the Langevin and Fokker-Planck equations, which represent the stochastic evolution of the phase-space variables and of their probability density, respectively [7,8,9,10] In this context, the first and the second principles of the thermodynamics can be re-demonstrated [11,12,13,14] and other important fluctuation-dissipation theorems have been elaborated [15,16,17,18,19,20,21]. This approach has been recently used to investigate the properties of several two-state systems and macromolecular chains [74,75,76,77,78] Both the Gibbs and the Helmholtz ensembles can be studied by the spin variables methodology, permitting to draw direct comparisons between isotensional and isometric conditions, provided that we work at thermodynamic equilibrium. These two possible mechanisms of stretching generate different stochastic mechanical behaviors of the system, which can be studied by calculating the corresponding configurational partition functions

The Gibbs Ensemble

The Helmholtz Ensemble

Complete Probability Densities in the Gibbs and Helmholtz Ensembles

Discussion and Conclusions