Isotensional and isometric force-extension response of chains with bistable units and Ising interactions

Abstract

The combination of bistability and cooperativity plays a crucial role in several biological and artificial micro-and nano-systems. In particular, the exhaustive understanding of the mechanical response of such systems under the effect of thermal fluctuations is essential to elucidate a rich variety of phenomena. Here, a linear chain composed of elastic units, which are bistable (folded or unfolded) and coupled through an Ising-like interaction, is selected as a case study. We assess the macroscopic thermoelastic response of this chain in terms of its microscopic description. For small systems, far from the thermodynamic limit, this response depends on the applied isometric or isotensional boundary conditions, which correspond to the Helmholtz or Gibbs ensembles of the statistical mechanics, respectively. The theoretical analysis is conducted through the spin variables approach, based on a set of discrete quantities able to identify the folded or unfolded state of the chain units. Eventually, this technique yields closed form expressions for the force-extension curves and the average number of unfolded units, as function of the applied fields. In addition, it allows to unveil a critical behavior of such systems, characterizing the operating regions with negative differential stiffness (spinoidal phase).

Full Statistics of Conjugated Thermodynamic Ensembles in Chains of Bistable Units

“…The mathematical details about this idea can be found in Refs. [74][75][76]. In the present analysis, this approach leads to closed form expressions for the probability densities defined above, and the final results can be interpreted by introducing a form of duality between the two ensembles, useful to better understand the specific features of the isotensional and isometric conditions.…”

“…Of course, 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 therefore 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.…”

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.…”

Unfolding pathway and its identifiability in heterogeneous chains of bistable units

“…The paradigmatic minimal model for describing all these physical situations is constituted of a chain of bistable units that may assume two states, classically named folded and unfolded configurations. The bimodal energy potential of each 35 unit is therefore composed of two energy wells with different basal energies, separated by a given energy barrier. This chain is typically considered at a given temperature T in order to study the effect of the thermal fluctuations on the transitions statistics between the states.…”

“…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.…”