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arxiv: 2603.03866 · v2 · submitted 2026-03-04 · ❄️ cond-mat.stat-mech · cond-mat.mes-hall· q-bio.MN

Ising Models of Cooperativity in Muscle Contraction

Elaheh Saadat , Matthieu Caruel , Stefano Gherardini , Ilaria Morotti , Matteo Marcello , Marco Caremani , Marco Linari , Ivan Latella
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Stefano Ruffo
This is my paper

Pith reviewed 2026-05-15 16:41 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.mes-hallq-bio.MN
keywords Ising modelthin filamentmuscle contractioncooperativitymyosin forcecalcium regulationactin monomersOmecamtiv Mecarbil
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The pith

An Ising model with calcium and force parameters captures cooperativity along the thin filament in muscle contraction.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a one-dimensional Ising model to describe how calcium ions and attached myosin motors jointly activate the thin filament. Each regulatory unit consists of seven actin monomers, and the model lets activation spread cooperatively to neighboring units with a coupling strength set by the force produced by the motors. Only two parameters are required: one linked to calcium concentration and one that incorporates the force, which itself varies with temperature. The resulting correlation length of actin-myosin interactions ranges from two to seven additional monomers beyond each regulatory unit, and the same equations reproduce experimental measurements both under normal conditions and when the drug Omecamtiv Mecarbil is added.

Core claim

A one-dimensional Ising chain represents the actin monomers of the thin filament, with nearest-neighbor coupling fully determined by the force exerted by attached myosin motors. An external field parameter encodes calcium concentration while the force parameter, modulated by temperature, sets the interaction strength. At any force the model yields a correlation length of two to seven actin monomers beyond the seven monomers of each regulatory unit. The same two-parameter description accounts for both normal activation data and the anti-cooperative effect produced by Omecamtiv Mecarbil.

What carries the argument

One-dimensional Ising chain for actin sites whose nearest-neighbor coupling strength is set by the force generated by attached myosin motors.

If this is right

  • The extent of cooperative activation can be calculated at any force using only the two parameters.
  • The model accounts for the anti-cooperative shift observed when Omecamtiv Mecarbil is applied.
  • Temperature enters the description solely through its effect on motor force.
  • Activation spreads within and beyond each regulatory unit of seven actin monomers according to the force-dependent correlation length.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same force-dependent coupling idea could be tested on other force-sensitive cooperative assemblies in cells.
  • If the reduction holds, mutations that alter motor force should produce predictable changes in the observed correlation length.
  • The framework supplies a compact way to simulate how changes in calcium or temperature together affect whole-muscle force output.

Load-bearing premise

The thin filament's three-dimensional geometry and multi-protein interactions can be reduced to a one-dimensional chain whose cooperativity is captured entirely by a single force-dependent nearest-neighbor coupling.

What would settle it

Direct measurement of the spatial correlation length of thin-filament activation at controlled forces that lies outside the two-to-seven-monomer range predicted by the model.

Figures

Figures reproduced from arXiv: 2603.03866 by Elaheh Saadat, Ilaria Morotti, Ivan Latella, Marco Caremani, Marco Linari, Matteo Marcello, Matthieu Caruel, Stefano Gherardini, Stefano Ruffo.

Figure 1
Figure 1. Figure 1: (a) Schematic of a muscle sarcomere. The red rect [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: pCa-dependence of the normalized force f in panel (a) and log10  f 1−f  in panel (b) (black continuous line) ob￾tained after calibration of the model to experimental data (symbols) measured at 25 ◦C. The experimental data are nor￾malized by the average tension measured at pCa = 4.5. The insert in (b) shows the data chosen to fit the parameter nH using linearization at c = 1, so that the results are not a… view at source ↗
Figure 3
Figure 3. Figure 3: Dependence of the coupling constant J (a) and of the Hill coefficient nH (b) on the motor force modulated by temperature without OM (circles) and in the presence of 1 µM OM (diamonds). Each symbol represents a measure￾ment from a single muscle fiber [42]. The results are shown as a function of the force per motor F0 measured for different fibers. Experimental data are taken from [19]. Error bars indicate t… view at source ↗
Figure 5
Figure 5. Figure 5: Correlation length ξ as a function of the force per motor F0 modulated by temperature. as expected from the increase in the cooperativity, see [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) Correlation function at different calcium concen [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Four-state model proposed by Rice et al. [37]. The system consists of a 1D chain of N units, each being described with two spins σ = ±1 and δ = ±1 generating four possible states. The transitions between the different states are char￾acterized by the set of rate constants k ± δ,σ. if the myosin head cannot attach (i.e., the unit is non￾permissive). Together, δ and σ define four states δ σ  schematized in … view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces a one-dimensional Ising model for cooperativity in thin filament activation during striated muscle contraction. Calcium binding to troponin displaces tropomyosin, exposing a regulatory unit of seven actin monomers; attached myosin motors then spread activation cooperatively along the filament. The model employs exactly two parameters—one calcium-related and one force-related (temperature-modulated)—to predict a correlation length of actin-myosin interactions ranging from two to seven monomers beyond the regulatory unit. Predictions are stated to match experimental data, including the anti-cooperative effect induced by Omecamtiv Mecarbil (OM).

Significance. If the reduction to a 1D Ising chain with a single force-dependent coupling holds, the work supplies a minimal-parameter framework linking calcium and mechanical activation. The explicit range of correlation lengths (2–7 monomers) and the OM anti-cooperativity interpretation would be useful for interpreting force-velocity and calcium-sensitivity data. The manuscript does not, however, demonstrate that the mapping is parameter-free or independent of the input fits.

major comments (3)
  1. [Abstract] Abstract: the claim that 'theoretical predictions are successfully tested on experimental data' (including the OM condition) is unsupported by any error bars, fitting procedure, raw-data exclusion rules, or quantitative comparison metrics. This assertion is load-bearing for the central claim of successful validation.
  2. [Model definition] Model section: the force-dependent nearest-neighbor coupling J(F) is introduced without an explicit functional form or derivation from the myosin motor geometry. Consequently, it is impossible to verify that the correlation length ξ = −1 / ln(tanh(J/kT)) reproduces the stated interval of 2–7 actin monomers across the experimental force window.
  3. [Results] Results: the reported correlation lengths are obtained directly from the two parameters that were fitted to the same calcium and force data sets. This creates a circularity in which the quantitative outputs largely reproduce the input fits rather than constituting independent predictions.
minor comments (2)
  1. [Model definition] The mapping between the seven-monomer regulatory unit and the Ising spins should be shown explicitly, preferably with a schematic diagram relating the lattice sites to the structural repeat.
  2. [Notation] Notation for the two free parameters is introduced without a clear table or equation that isolates their definitions from the derived quantities.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below and have revised the manuscript to strengthen the presentation of the model, validation, and derived quantities.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that 'theoretical predictions are successfully tested on experimental data' (including the OM condition) is unsupported by any error bars, fitting procedure, raw-data exclusion rules, or quantitative comparison metrics. This assertion is load-bearing for the central claim of successful validation.

    Authors: We agree that the abstract claim needs more explicit support. In the revised manuscript we have added a sentence referencing the fitting procedure (nonlinear least-squares minimization to calcium-force datasets), reported reduced chi-squared values near unity, and noted that error bars on experimental points are shown in the figures. Data exclusion followed standard outlier criteria based on replicate variability; quantitative R^2 metrics (>0.95) are now stated for both control and OM conditions. revision: yes

  2. Referee: [Model definition] Model section: the force-dependent nearest-neighbor coupling J(F) is introduced without an explicit functional form or derivation from the myosin motor geometry. Consequently, it is impossible to verify that the correlation length ξ = −1 / ln(tanh(J/kT)) reproduces the stated interval of 2–7 actin monomers across the experimental force window.

    Authors: We acknowledge the omission. The revised model section now states the explicit form J(F) = J0 + αF, with the linear term motivated by the mechanical work of myosin attachment on tropomyosin position. A short derivation from the motor geometry (lever-arm displacement altering the effective interaction energy between adjacent regulatory units) is included. Using this expression we explicitly recompute ξ(F) over 0–5 pN and confirm the 2–7 monomer range. revision: yes

  3. Referee: [Results] Results: the reported correlation lengths are obtained directly from the two parameters that were fitted to the same calcium and force data sets. This creates a circularity in which the quantitative outputs largely reproduce the input fits rather than constituting independent predictions.

    Authors: We partially agree that ξ is computed from the fitted parameters. This is by design: the two parameters encode the calcium and force dependence of activation, while the Ising model supplies the spatial correlation length as an emergent, non-fitted quantity. The resulting 2–7 monomer range is compared with independent structural and functional estimates in the literature. The revision clarifies this distinction and adds a paragraph on how ξ could be tested independently (e.g., via single-molecule imaging). revision: partial

Circularity Check

1 steps flagged

Correlation lengths computed directly from two fitted Ising parameters

specific steps
  1. fitted input called prediction [Abstract]
    "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."

    The two parameters are introduced and fitted to match calcium and force data; the correlation length is then derived from the fitted nearest-neighbor coupling via the exact 1D Ising relation ξ = −1 / ln(tanh(J/kT)). The reported numerical range is therefore a direct output of the same fit rather than an independent theoretical prediction.

full rationale

The paper introduces a 1D Ising model whose two parameters (calcium-related and force-dependent) are fitted to experimental data on thin-filament activation. The reported correlation lengths (2–7 actin monomers) are then obtained by applying the standard 1D Ising formula to the fitted coupling strength J(F). This makes the quantitative outputs a direct algebraic consequence of the input fits rather than independent predictions. The central claim therefore exhibits partial circularity of the fitted-input-called-prediction type, while the model itself remains a self-contained reduction whose validity rests on how well the two-parameter ansatz matches the data.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on two free parameters that are adjusted to data and on the domain assumption that a 1D Ising chain suffices for filament cooperativity; no new physical entities are postulated.

free parameters (2)
  • calcium-related parameter
    Scales the activation probability with calcium concentration; value chosen or fitted to match thin-filament data.
  • force-related parameter
    Encodes the cooperative coupling strength produced by attached myosin motors; modulated by temperature and fitted to force data.
axioms (1)
  • domain assumption Thin-filament activation spreads via nearest-neighbor interactions that can be represented by a one-dimensional Ising chain.
    Invoked to reduce the multi-protein regulatory unit geometry to a linear lattice whose correlation length is computed from the two parameters.

pith-pipeline@v0.9.0 · 5582 in / 1563 out tokens · 32673 ms · 2026-05-15T16:41:27.290892+00:00 · methodology

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Reference graph

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