Modeling actin-myosin interaction: beyond the Huxley-Hill framework (Q6060140)

The present paper deals with a jump-diffusion stochastic model as an alternative to the Huxley--Hill (HH) model of the actin-myosin interaction. The modeling framework of the actin-myosin interaction is described and, in particular, the HH model is formulated. The well-posedness of the HH model is proved in the space \(C([0,T]\times [-d/2, d/2],[0,1])\cap C^1(]0,T[\times ]-d/2, d/2[,[0,1])\), for some \(T>0\), with \(d\) denoting the distance between two consecutive actin sites. Moreover, its stochastic formulation of the jump process is described to the full real line. The authors propose a parametrized myosin head by the position \(X_t\) (of its head) with respect to its anchor point on the myosin filament at time \(t\), and a relative position \(h_t\) with respect to its closest binding site on the actin filament. The \(h\)-model is formulated for a \(\{0,1\}\times\mathbb{R}\times \mathbb{T}_d\)-valued Markov process \((\alpha_t, X_t, h_t)_{t\geq 0}\) as the right-continuous with left limits (càdlàg) solution to a stochastic system, involving a continuous overdamped Langevin dynamics for the position of the head, and a discrete Poisson dynamics for the state \(\alpha_t\) (attached (\(\alpha_t = 1\)) or detached (\(\alpha_t = 0\))). The one-dimensional torus \(\mathbb{T}_d\) of width \(d\) is centered at \(X_t\). The deterministic approach is based on the concise PDE: \(\partial_t p =Q(\cdot,p)+\eta^{-1}k_BT_a(\partial^2_{xx}+\partial^2_{ss})p\) in \(]0,T[\times \mathbb{R}\times \mathbb{T}_d\), via the change of variable \(s=\pi(x)+h\) in \(\mathbb{T}_d\), where \(\pi:\mathbb{R}\rightarrow \mathbb{T}_d\) stands for the canonical projection. Here, \(k_B\) denotes the Boltzmann constant, \(T_a\) is the ambient temperature, \(\eta\) is a constant damping coefficient, and \(Q\) is a nonlocal operator which depends on \(p\) in a bounded linear way. To relate the \(h\)-model with the HH model, a \(h\)-reduced model is derived as a heuristic limit of the \(h\)-model via an adiabatic elimination of fast variables. For all models (HH, \(h\)-model and \(h\)-reduced model), the compatibility with the thermodynamic principles is studied. Also, numerical illustrations are provided for the calibration of the \(h\)-model and its reduced version, and for the reproduction of key indicators of the cardiac muscle contraction physiology.