[Paper Review] Ising Models of Cooperativity in Muscle Contraction

Elaheh Saadat, Matthieu Caruel+7 more|arXiv (Cornell University)|Mar 4, 2026

Cardiomyopathy and Myosin Studies0 citations

TL;DR

The paper develops a one-dimensional Ising model to explain cooperativity in thin filament activation during muscle contraction, linking Ca2+ concentration and myosin motor force to a Hill-like response, and tests predictions against experiments including Omecamtiv Mecarbil (OM).

ABSTRACT

Regulation of contraction in striated muscle is controlled by a dual mechanism involving both thin filaments containing actin and thick filaments containing myosin. The thin filament is activated by calcium ions binding to troponin, leading to tropomyosin azimuthal displacement which allows the activation of a regulatory unit (composed of one troponin, one tropomyosin and seven actin monomers) that exposes the actin sites for interaction with the myosin motors. Motor attachment to actin contributes to spreading activation within and beyond a regulatory unit along the thin filament through a cooperative mechanism. We introduce a one-dimensional Ising model to elucidate the mechanism of cooperativity in thin filament activation in relation to the force generated by the attached myosin motor. The model characterizes thin filament activation and cooperativity using only two parameters: one related to calcium concentration and the other to the force exerted by the attached myosin motor, which is modulated by temperature. At any force, the model is able to determine the extent of actin-myosin interactions on a correlation length ranging from two to seven actin monomers in addition to the seven actin monomers of the regulatory unit. Our theoretical predictions are successfully tested on experimental data, and our tests also include the condition of hindered filament activation by the use of the specific drug Omecamtiv Mecarbil (OM). According to our model, the effect of OM results in an anti-cooperativity mechanism accounting for the experimental data.

Motivation & Objective

• Explain how thin filament activation exhibits cooperativity in response to Ca2+ and motor-generated force.
• Provide a minimal, mechanistic Ising-model framework that maps to the Hill coefficient and force-pCa relation.
• Quantify how motor force and temperature modulate cooperativity, and assess the impact of Omecamtiv Mecarbil (OM).
• Offer a quantitative comparison with existing multi-state models and derive correlation-length predictions.
• Calibrate the model against experimental force-pCa data and extract interpretable parameters (J and [Ca2+]50).

Proposed method

• Model the thin filament as a 1D chain of N two-state units (spin s_i = ±1) with nearest-neighbor coupling J and external field h capturing Ca2+ concentration, via H1 = -sum_i (J s_i s_{i+1} + h s_i).
• Map the Ising model to a two-state Markov process with detailed-balance transition rates k_±^i, yielding k_+^i/k_-^i = exp{2[J(s_{i+1}+s_{i-1})+h]}.
• Relate h and J to Ca2+ concentration and Hill coefficient via h = (1/2) log(c) and J = (1/2) log(n_H).
• Compute force f = (1+<s>)/2 using transfer-matrix methods, obtaining f = 1/2 {1 + (c-1)/sqrt[(c-1)^2 + 4 n_H^{-2} c]}.
• Calibrate the model to experimental force-pCa data to extract [Ca2+]50 and n_H; report J ≈ 0.57 (n_H ≈ 3.14) at 25°C without OM.
• Analyze correlation length xi = [log((λ_+)/(λ_-))]^{-1} with xi = [log((n_H+1)/(n_H-1))]^{-1} at c=1.
• Compare with Rice et al.’s four-state model, reformulating it to a two-parameter analog f = 1/2 {1 + (x-1)/sqrt[(x-1)^2 + 4 n_H^{-2} x]}.
• Investigate OM effects by showing OM reduces n_H (often <1) and induces anti-cooperativity (J<0) in the tested regime.

Experimental results

Research questions

• RQ1 How does Ca2+ concentration and attached-motor force modulate cooperative activation along the thin filament?
• RQ2 Can a single-layer Ising (two-state) model reproduce the experimentally observed force-pCa curves and their temperature/OM dependence?
• RQ3 What is the relationship between the Hill coefficient n_H and the Ising coupling J, and how does this relate to correlation length?
• RQ4 How does Omecamtiv Mecarbil (OM) influence cooperativity and force generation according to the model?
• RQ5 How does the single-layer Ising model compare to Rice et al.’s four-state model in explaining force-pCa data?

Key findings

• A single-layer Ising model reproduces the force-pCa relation with f = 1/2{1 + (c-1)/sqrt[(c-1)^2 + 4 n_H^{-2} c]} and a bijection n_H = exp(2J).
• Calibrated parameters yield [Ca2+]50 ≈ 10^(-6.5) M and n_H ≈ 3.14 (J ≈ 0.57) at 25°C without OM, with similar fits across 12–35°C (without OM).
• Hill coefficient n_H increases with motor force F0, i.e., higher cooperativity at larger F0; with OM, n_H falls below 1, indicating anti-cooperativity (J<0).
• Correlation length xi grows with F0, ranging from about 2 to about 6–7 units as temperature varies, indicating longer-range cooperative spread at higher force.
• The model’s predictions for force-pCa are consistent with experimental data and OM effects, supporting a cooperative mechanism driven by motor-force-assisted spreading of activation.
• Compared to Rice et al.’s four-state model, the two-state model is not over-determined for force-pCa data and can be reframed to a similar two-parameter form; the four-state model is more complex and less able to fit high- and low-Ca2+ data with a single parameter set.