# Stochastic modeling of chemical–mechanical coupling in striated muscles

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## 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 isometric transients in which the power stroke is known to have a crucial role, thus circumventing a limitation of many classical models.

## Keywords

Muscle modeling Sarcomere Sliding filament Cross-bridge Power stroke Langevin equations Fokker–Planck equations## Notes

### Acknowledgements

We would like to warmly thank our colleagues from the Laboratory of Physiology of Firenze University—Vincenzo Lombardi and Marco Linari, in particular—for their invaluable feedback on this work, François Kimmig (Ecole Polytechnique and Inria) for insightful discussions on the thermal equilibrium model and Lev Truskinovsky (ESPCI) for stimulating exchanges on the stochastic model.

### Compliance with ethical standards

### Conflicts of interest

The authors declare that they have no conflict of interest.

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