Full Statistics of Conjugated Thermodynamic Ensembles in Chains of Bistable Units

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.

Stochastic thermodynamics of holonomic systems

“…In this regards, Eqs. (52) and (53) would be helpful to quantitatively establish if the entropy production is negligible or not depending on the extent of the perturbation.…”

“…A classical example deals with the thermal and elastic behavior of DNA [43,44] and other polymer chains [45][46][47], which have been largely investigated with statistical methodologies. In addition, the conformational transitions observed in several macromolecules of biological origin (nucleic acids, proteins and so on) have been modeled with chains of bistable units [48][49][50][51][52]. In this context, the equivalence of the ensembles in the thermodynamic limit has been discussed to give a correct interpretation of isotensional (Gibbs) and isometric (Helmholtz) ensembles [53,54].…”

Rate-dependent force–extension models for single-molecule force spectroscopy experiments

“…Before showing the results of the integration of the Langevin equations stated in previous Section 2, we introduce here the spin variables approach, which is a mathematical method useful to obtain the force extension relations for very low traction speeds v 0 (ideally v 0 → 0) [36][37][38][39][40][41][42][43][44]. It means that in this Section we consider rate-independent processes.…”

“…This methodology has been specialized to deal with bistable units through the so-called spin variables approach. While this idea has been originally introduced to study the mechanics of muscles [34,35], it is currently used to analyse many different two-state systems [36][37][38][39][40][41][42][43][44]. The bistable potential energy of each unit is approximated by two quadratic functions representing the folded and unfolded states.…”

Unfolding pathway and its identifiability in heterogeneous chains of bistable units

“…Moreover, the intermediate cases, in-between the Gibbs and the Helmholtz ensembles, have been recently studied by introducing the real 55 stiffness of the adopted devices [8]. These results can be obtained with the method of the spin variables, which introduces a discrete variable (spin-like) for each unit, able to define the potential well explored by the unit itself (folded or unfolded state) [33][34][35][36]. This approach, originally introduced to develop 60 a chemo-mechanical model of the muscle behavior [37,38], has Figure 1: Folding and unfolding processes in homogeneous and heterogeneous chains.…”