A Three-State Thermodynamically Consistent Cross-Bridge Model for Muscle Contraction

Muscle contraction is a prototypical multiscale chemomechanical process in which ATP hydrolysis at the molecular level drives force generation and mechanical work at larger scales. A long-standing challenge is to connect microscopic cross-bridge dynamics to macroscopic observables while retaining an explicit, thermodynamically consistent energetic budget for chemical-to-mechanical transduction. Here we use the Energetic Variational Approach (EnVarA) to unify Hill's cycle-affinity viewpoint with Huxley's sliding-filament mechanics within a single thermodynamically closed framework. We formulate a three-state Fokker--Planck-jump description for cross-bridge populations evolving on state-dependent free-energy landscapes, in which ATP hydrolysis enters through local detailed balance and biases the transition rates. Filament sliding velocity is incorporated as a convective transport mechanism in the Fokker--Planck dynamics, so that mechanical power exchange with the external motion emerges transparently from the resulting energy-dissipation law together with chemical input and irreversible dissipation. Under chemostatted conditions and a fast-equilibration closure for the attached substates, the model reduces to a closed two-state molecular motor description; in a further singular limit, this reduction recovers a Huxley-type transport-reaction equation. Proof-of-concept simulations of the reduced model reproduce a Hill-like force-velocity relation and show how ATP availability modulates the force-velocity curve while preserving its characteristic Hill-type shape.