Stochastic modeling of chemical–mechanical coupling in striated muscles

Abstract

We propose a chemical-mechanical model of myosin heads in sarcomeres, within the classical description of rigid sliding filaments. In our case, myosin heads have two mechanical degrees-of-freedom (dofs)one of which associated with the so-called power stroke-and two possible chemical states, i.e. bound to an actin site or not. Our major motivations are twofold: (1) to derive a multiscale coupled chemical-mechanical model, and (2) to thus account-at the macroscopic scale-for mechanical phenomena that are out of reach for classical muscle models. This model is first written in the form of Langevin stochastic equations, and we are then able to obtain the corresponding Fokker-Planck partial differential equations governing the probability density functions associated with the mechanical dofs and chemical states. This second form is important, as it allows to monitor muscle energetics, and also to compare our model with classical ones, such as the Huxley'57 model to which our equations are shown to reduce under two different types of simplifying assumptions. This provides insight, and gives a Langevin form for Huxley'57. We then show how we can calibrate our model based on experimental data-taken here for skeletal muscles-and numerical simulations demonstrate the adequacy of the model to represent complex physiological phenomena, in particular the fast isomet

Thermodynamic properties of muscle contraction models and associated discrete-time principles

“…Physically, the property a(s − , t) = a(s + , t) = 0 appears when the attachment rates k 1 (s) and k −2 (s) vanish, while the detachment rates k −1 (s) and k 2 (s) go to infinity on the boundaries of the interval [s − , s + ]. Note that the energy levels and the transition rates are linked by the detailed balance (1), which implies that the energy of the attached level goes to infinity on the boundaries of the interval [s − , s + ]. In a nutshell, the parameter functions must satisfy…”

“…Note that a nonlinear hyperelastic behavior could be considered, at the price of having to deal with the dimensionless extension e s as an additional internal variable. Nevertheless, in physiological conditions the extension e s remains small and a linear behavior is adequate [1]. As regards viscosity, we here incorporate a simple component of viscous modulus ν in parallel with the active part in the sarcomere branch, and we recall that viscosity is also present in the parallel branch as provided by the term ∂Ψ v ∂ė = ηė.…”

“…Moreover, we present how these microscopic models can be incorporated into a macroscopic model of muscle fibers in the spirit of [2] with the aim of following these thermodynamic balances at the macroscopic level for the continuous-time dynamics but also after adequate time discretization. This last part is general with respect to the chemical microscopic model of interest and could also be extended to similar types of models [3,19], or those mixing mechanical and chemical modeling elements, for instance [1,17,22].…”

Biophysically detailed mathematical models of multiscale cardiac active mechanics

“…In order to capture the separation between the fastest time scales (i.e. between the first two phases following a fast step either in force or in length), an explicit representation of the power-stroke must be included in the model, by introducing a multistable discrete [15,75] or continuous [20,28,29,69,76] degree of freedom, representing the angular position of the rotating MH.…”

“…In the past decades, several efforts have been dedicated to the construction of mathematical models describing the complex dynamics of the processes taking place in sarcomeres [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]. However, because of the intrinsic complexity of the phenomenon of force generation, huge computational costs are associated with the numerical approximation of such models, thus limiting their application within multiscale EM simulations.…”