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

Yiwei Wang, Chun Liu

TL;DR

Problem: bridging microscopic cross-bridge energetics with macroscopic muscle mechanics under a thermodynamically closed framework. Approach: formulate a three-state cross-bridge cycle within the Energetic Variational Approach ($\\text{EnVarA}$) to yield a strain-resolved Fokker–Planck–jump model with state-dependent landscapes and local detailed balance for ATP hydrolysis; derive reduced two-state FP models under chemostatted fast-equilibration, and recover a $Huxley$-type transport–reaction equation in a singular limit. Key results: simulations reproduce a Hill-type force–velocity curve and show how ATP availability reshapes the curve while preserving Hill-like form. Contributions: energy-dissipation balance partitions chemical input, mechanical power, and irreversible dissipation, and links Hill's cycle view with Huxley's mechanics within a thermodynamically consistent framework. Significance: establishes a scalable micro–macro framework for muscle contraction, enabling ATP-limited regime analysis and fatigue-like behavior.

Abstract

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.

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

TL;DR

Problem: bridging microscopic cross-bridge energetics with macroscopic muscle mechanics under a thermodynamically closed framework. Approach: formulate a three-state cross-bridge cycle within the Energetic Variational Approach () to yield a strain-resolved Fokker–Planck–jump model with state-dependent landscapes and local detailed balance for ATP hydrolysis; derive reduced two-state FP models under chemostatted fast-equilibration, and recover a -type transport–reaction equation in a singular limit. Key results: simulations reproduce a Hill-type force–velocity curve and show how ATP availability reshapes the curve while preserving Hill-like form. Contributions: energy-dissipation balance partitions chemical input, mechanical power, and irreversible dissipation, and links Hill's cycle view with Huxley's mechanics within a thermodynamically consistent framework. Significance: establishes a scalable micro–macro framework for muscle contraction, enabling ATP-limited regime analysis and fatigue-like behavior.

Abstract

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.
Paper Structure (3 sections, 53 equations, 2 figures)

This paper contains 3 sections, 53 equations, 2 figures.

Table of Contents

  1. Mechanical power output.
  2. Chemical power input.
  3. Efficiency and power budget.

Figures (2)

  • Figure 1: From the reduced Huxley-type model to a Hill-type F--V curve. (a): normalized cross-bridge force $F_{\rm cb}(V)/F_0$, where $F_{\rm cb}(V)$ is computed by \ref{['eq:Fcb_reduced']} using $n(x;V)$ and $F_0:=F_{\rm cb}(0)$. (b): steady-state solutions $n(x;V)$ of \ref{['eq:steady_huxley_reduced']} for representative shortening velocities.
  • Figure 2: The effect of ATP concentration on the force-velocity relation. Forces are normalized by the isometric tension at the reference concentration ($c_{\mathrm{ATP}}=3c_0$).