Abstract
Considering a large class of muscle contraction models accounting for actin–myosin interaction, we present a mathematical setting in which solution properties can be established, including fundamental thermodynamic balances. Moreover, we propose a complete discretization strategy for which we are also able to obtain discrete versions of the thermodynamic balances and other properties. Our major objective is to show how the thermodynamics of such models can be tracked after discretization, including when they are coupled to a macroscopic muscle formulation in the realm of continuum mechanics. Our approach allows to carefully identify the sources of energy and entropy in the system, and to follow them up to the numerical applications.
Highlights
The modeling of the active mechanical behavior of muscles has been the object of intense research since the seminal work of Huxley [12] modeling the attachment-detachment process in the actin–myosin interaction responsible for sarcomere contraction
Numerous extensions—mostly based on refinements of the chemical process introduced by Huxley—of the previous model have been proposed in order to take into account different time scales of the actin–myosin interaction
Concluding remarks Considering a large class of muscle contraction models based on actin-myosin interaction— i.e. the Huxley’57 model and various extensions thereof, including the Piazzesi– Lombardi’95 model—we have presented a mathematical setting in which solution properties can be established, including fundamental thermodynamic balances
Summary
The modeling of the active mechanical behavior of muscles has been the object of intense research since the seminal work of Huxley [12] modeling the attachment-detachment process in the actin–myosin interaction responsible for sarcomere contraction. The question of the thermodynamic balances associated with the chemical machinery was intensively studied, notably with the fundamental contributions of Hill [10,11]. Note that these models are specific cases of molecular motors models without the natural diffusion introduced by the Fokker–Plank equation [2,3,14,18]. The first section presents the modeling ingredients of the microscopic models of actin–myosin interaction and we derive in a second section the fundamental properties of these models with the associated thermodynamic balances, up to the coupling with the macroscopic mechanical formulation. The last section illustrates our results with numerical investigations
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