Statistical mechanics of the Huxley-Simmons model

Abstract

The chemomechanical model of Huxley and Simmons (HS) [A. F. Huxley and R. M. Simmons, Nature 233, 533 (1971)NATUAS0028-083610.1038/233533a0] provides a paradigmatic description of mechanically induced collective conformational changes relevant in a variety of biological contexts, from muscles power stroke and hair cell gating to integrin binding and hairpin unzipping. We develop a statistical mechanical perspective on the HS model by exploiting a formal analogy with a paramagnetic Ising model. We first study the equilibrium HS model with a finite number of elements and compute explicitly its mechanical and thermal properties. To model kinetics, we derive a master equation and solve it for several loading protocols. The developed formalism is applicable to a broad range of allosteric systems with mean-field interactions.

Bi-stability resistant to fluctuations

“…In the hard device case, the stiffness was shown to be sign indefinite (Caruel and Truskinovsky, 2016), while here K is always positive which confirms that the two ensembles are not equivalent. If we denote by κ = K/N, the stiffness per element, we obtain κ(σ, β) = 1 + χ(σ, β) −1 ,…”

“…In the three Appendices A, B and C we briefly discuss the equilibrium susceptibility, the thermal behavior of the system and the limits of bi-stability. In a companion paper (Caruel and Truskinovsky, 2016) we show that although the hard device version of the HS model does not exhibit a cooperative snap-through behavior, its mechanical response is characterized by an intriguing negative stiffness.…”

“…which has the Cauchy-Born and the fluctuations induced contributions; the only difference with the behavior in a hard device (Caruel and Truskinovsky, 2016) is that the fluctuation part comes with plus sign, which makes the stiffness always positive. The equilibrium stiffness κ is represented as function of the external force and the temperature in Fig.…”

“…For us, of particular interest in this respect are the zero dimensional systems with infinitely long range (mean-field) interactions (Kometani and Shimizu, 1975;Desai and Zwanzig, 1978). In the context of fast force recovery in skeletal muscles, a mean-field model was first proposed in the seminal paper by Huxley and Simmons (HS) (Huxley and Simmons, 1971;Caruel and Truskinovsky, 2016), where the authors considered a parallel bundle of bi-stable elements subjected to white noise. However, since the study of HS was performed in a hard device (length clamp) prohibiting collective effects, the cooperative behavior was not found.…”

Full Statistics of Conjugated Thermodynamic Ensembles in Chains of Bistable Units

“…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.…”