Pith. sign in

REVIEW 3 major objections 6 minor 12 references

Muscle Crossbridge Theory With Internal Crossbridge Dynamics

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A variable rest length lets muscle crossbridges pull at constant force.

desk verdict Fresh internal-variable crossbridge model worth refereeing, but the quick-release validation needs sensitivity analysis on kse and a quantitative misfit before it carries the weight. read the letter →

arxiv 2505.20198 v1 pith:PQZNO56J submitted 2025-05-26 physics.bio-ph

Mengjian Hua , Charles S. Peskin This is my paper · ORCID
classification physics.bio-ph MSC 92C1092C40
keywords crossbridgetheorymusclecontractionvariablerestlengthinternaldynamicsforce-velocityrelationquickreleaseevent-drivensimulationcatch-bonddetachment
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to resolve a standing contradiction in muscle mechanics: attached crossbridges behave as linear springs with stiffness 3.3 pN/nm, yet during active shortening the force they bear stays roughly constant as strain is relieved. The proposed resolution is to give each attached crossbridge an internal dynamical variable, its rest length $r$, which evolves according to $dr/dt = v_{\max}(1 - p/p_{\infty})$, so that force relaxes toward a load-dependent plateau. In the limit of infinite crossbridge stiffness, the model exactly reproduces A.V. Hill's force-velocity curve and heat of shortening, and this fixes a detachment rate $\beta(p)$ that decreases linearly with crossbridge force. With parameters fit to Piazzesi's steady-state data, the model's stochastic quick-release simulations fall within the scatter of the experimental records. If correct, the theory replaces the Huxley picture of pushing and pulling crossbridges with one in which all attached crossbridges pull, while a variable rest length absorbs the shortening.

What carries the argument

The central object is the variable rest length $r(t)$ of an attached crossbridge, with force given by $p = k(r - x)$; the internal dynamics $dr/dt = v_{\max}(1 - p/p_{\infty})$ closes the system into $dp/dt = k(v_{\max}(1 - p/p_{\infty}) - v)$, giving the exponential force rise $p(t) = p_{\infty}(1 - v/v_{\max})(1 - e^{-kv_{\max}(t-t_0)/p_{\infty}})$ that carries the steady-state analysis. In the $k \to \infty$ limit, matching Hill's data forces the detachment rate to be the linearly decreasing function $\beta(p) = (\alpha/4)(1 + 20(1 - p/p_{\infty}))$, and this same form is retained at finite $k$. The steady-state machinery couples attachment-detachment balance to the force distribution, leading to the exact relation $P(v) = p_{\infty}(5U(v)/4 - 1/5)$ and to an integrable ODE for the force density $u(v,p)$; the event-driven simulation then propagates the system using exact exponential waiting times for attachment and Newton-solved waiting times for force-dependent detachment.

What would settle it

Measure the detachment rate of single attached crossbridges as a function of force in an optical-trap or half-sarcomere preparation: the model requires $\beta(p) = (\alpha/4)(1 + 20(1 - p/p_{\infty}))$ to fall linearly with $p$, with fitted values $\alpha \approx 68$ s$^{-1}$ and $p_{\infty} \approx 10$ pN. If the measured detachment rate rises or is flat with force, the derivation from Hill's laws cannot be transplanted to finite stiffness and the quick-release agreement would not follow. A second, cheaper check is the steady-state identity $P = p_{\infty}(5U/4 - 1/5)$: plotting mean force per crossbridge against attachment probability across shortening velocities should fall on a straight line that extrapolates to force $p_{\infty}/5$ at zero attachment probability.

Watch

Extended reading notes

Core claim

The central claim is that an attached crossbridge is a Hookean spring whose rest length is not fixed but is an internal dynamical variable satisfying $dr/dt = v_{\max}(1 - p/p_{\infty})$. Since strain is $r - x$ and force is $p = k(r - x)$, the spring responds stiffly to sudden length changes on a fast time scale, while on a slower time scale the rest length itself runs, keeping the force nearly constant during shortening. Attachment occurs with zero strain, and the maximum shortening velocity is reached when the internal rest-length rate equals the shortening rate, so no attached crossbridge is ever pushed into compression. The detachment rate $\beta(p) = (\alpha/4)(1 + 20(1 - p/p_{\infty}))$ is derived uniquely from the requirement that the $k \to \infty$ limit match Hill's 1938 force-velocity and heat equations; the paper then uses the same $\beta(p)$ for the measured finite stiffness $k = 3.3$ pN/nm, noting that the dimensionless parameter $\epsilon = \alpha p_{\infty}/(k v_{\max}) = 0.08$ is small. Fitted to Piazzesi's steady-state data, the model yields about 116 cycling crossbridges per half-sarcomere and reproduces the quick-release length traces for loads from 0.88 down to 0.14 times the isometric force.

Load-bearing premise

The whole construction leans on the assumption that the detachment rate $\beta(p)$ derived in the infinite-stiffness limit stays valid at the measured finite stiffness $k = 3.3$ pN/nm; the paper justifies this only by plausibility and the smallness of $\epsilon = 0.08$.

Editorial extensions

If this is right

  • In normal shortening all attached crossbridges pull and none are compressed, unlike Huxley's theory where zero net force at maximum velocity comes from a balance of tension and compression.
  • Hill's empirical force-velocity curve and heat of shortening emerge from a single detachment-rate hypothesis in the $k \to \infty$ limit, providing a mechanistic explanation of Hill's constants and of the coincidental equality between the optimal shortening velocity and the velocity at which maintenance heat equals shortening heat.
  • The predicted steady-state identity $P(v) = p_{\infty}(5U(v)/4 - 1/5)$ between mean crossbridge force and attachment probability is directly testable against existing data without invoking the simulation machinery.
  • The quick-release simulations reproduce the overall shortening trajectory and the back-extrapolated intercept of the steady-velocity line, though the simulated instantaneous length drop at release is smaller and the subsequent damped oscillations are largely absent, suggesting that series compliance or inertia not included in the model shapes the transient.
  • The model predicts a fundamental distinction between an isometric state and an isotonic state with zero mean velocity: series elasticity couples force fluctuations to filament motion only in the isometric case, which may explain the anomalous isometric force-per-crossbridge data point.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • One implication the paper leaves implicit: the decreasing detachment rate is not an added biological hypothesis but is forced by matching Hill's energetics, so any crossbridge model aiming to reproduce both the force-velocity curve and the heat of shortening may need a catch-bond-like detachment law or an equivalent internal variable.
  • The variable rest length makes each attached crossbridge behave like a Maxwell-type viscoelastic element, a spring in series with a dashpot driven by $v_{\max}(1 - p/p_{\infty})$; connecting this to macroscopic muscle constitutive laws could give a parameter-free bridge between the molecular model and continuum descriptions.
  • A direct experimental discriminator would be to measure single-motor dwell times under controlled force: the model predicts that mean attached duration increases linearly with force because $\beta(p)$ decreases linearly with $p$, whereas a conventional slip-bond motor would show the opposite trend.
  • A model variant worth testing would let the detachment rate itself depend on $\epsilon$, the ratio of attachment stiffness to internal dynamics rate; the paper leaves open whether a finite-$k$ correction to $\beta(p)$ would improve the quick-release transient while preserving the steady-state fits.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper proposes a crossbridge model in which an attached crossbridge is a linear spring of stiffness k = 3.3 pN/nm whose rest length r(t) obeys an internal dynamical law dr/dt = vmax(1 - p/p∞), so that the crossbridge is stiff on fast timescales but develops and relaxes force on slower timescales. The detachment rate β(p) is not postulated freely but is derived in Section 3 from the k → ∞ limit by requiring the model to reproduce Hill's 1938 force-velocity and heat-of-shortening relations, yielding β(p) = (α/4)(1 + 20(1 - p/p∞)). The paper then derives exact steady-state distributions for finite k (Section 4), fits p∞, vmax, α, and Nc to Piazzesi's steady-state data (Section 5), introduces series elasticity with an arbitrarily chosen stiffness kse = 100 pN/nm (Section 6), and develops an event-driven stochastic quick-release simulation (Sections 7-8) whose results are compared visually with Piazzesi's quick-release records. The central claim is that the model explains how a crossbridge can behave as a linear spring on fast timescales yet maintain roughly constant force during shortening.

Significance. If the central claim is correct, the paper offers a qualitatively new resolution of a long-standing tension in muscle mechanics: the linear-spring behavior seen in quick stretches/releases and the nearly constant per-crossbridge force during steady shortening are reconciled by an internal rest-length degree of freedom. The paper is unusually transparent about its own limitations, explicitly flagging the questionable transfer of β(p) from the k → ∞ derivation to finite k, the arbitrary choice of kse, and the model's failure to reproduce the damped oscillations in the quick-release data. The mathematical work is a genuine strength: the steady-state ODE solution in Section 4, the asymptotic reduction to the k → ∞ limit, and the event-driven simulation are internally consistent, and the simulation is described as exact realizations of the stochastic process. The paper also gives a concrete mechanistic interpretation of the catch-bond-like detachment rate. However, the out-of-sample validation is weakened by the arbitrary series elasticity and by the absence of quantitative misfit or sensitivity analysis, so the empirical support for the mechanism is not yet conclusive.

major comments (3)
  1. [Section 3, Eq. (28)] The derivation of β(p) from the k → ∞ limit is a fitting step, not an independent confirmation: Eqs. (14)-(15) are Hill's empirical curves, and Eq. (28) is solved from those very curves. The paper acknowledges this in the discussion after Eq. (28), but the finite-k agreement with Piazzesi's steady-state data is then presented as support for the model. Since the same β(p) is used at k = 3.3 pN/nm, the steady-state fit in Section 5 cannot independently validate the force-velocity and heat predictions. A concrete test would be to allow a two-parameter generalization of β(p), e.g., β(p) = (α/4)(A + B(1 - p/p∞)), fit A and B directly to the finite-k Piazzesi steady-state data without imposing the Hill-derived values, and report whether A = 1 and B = 20 lie within the confidence region. Without such a test, the claim that the finite-k model reproduces Hill's laws is a consistency check rather than a validation.
  2. [Section 6 and Section 8, kse = 100 pN/nm] The quick-release comparison is the only out-of-sample test of the internal crossbridge dynamics, but it depends on an unconstrained parameter: kse is set to 100 pN/nm by an explicit 'arbitrary choice' in Section 6 and is never varied or fitted. Equation (93) shows that the transition-phase length jump scales as (1/kse + 1/(kNA)), so kse directly controls the t = 0 intercept, which is one of the two features claimed to match in Section 8. The displayed agreement could therefore be specific to this value. The authors should either fit kse to the observed intercept and report the resulting value with uncertainty, or perform a sensitivity analysis over a plausible range of kse and show that the conclusions are unchanged. In addition, the comparison in Figure 2 is visual only; a quantitative misfit metric, such as RMS deviation between the simulated and experimental length traces over a defined time window, would allow an assessment of whether the discrepancies (including the missing damped oscillations) are acceptable.
  3. [Section 5, parameter fitting and Table 1] The paper states that the fit is 'not sensitive to ϵ' and that any ϵ ∈ [0, 0.15] gives an almost equally good fit, yet the final parameter set in Table 1 is reported as a single point with no uncertainty intervals. This matters because α is determined from Eq. (58) after eliminating ϵ using Eq. (48); if ϵ is poorly constrained, then α and the derived quantities (including the cycling rate and step length) inherit a substantial uncertainty. The authors should report confidence intervals or a profile-likelihood analysis for the fitted parameters, and should state how the quick-release simulation results depend on the spread of plausible parameter values.
minor comments (6)
  1. [Throughout] The manuscript contains numerous typographical errors, including 'atachement', 'detachement', 'corssbrdge', 'paramter', 'simnulation', 'ocurs', 'advantate', 'dimensionkess', 'approprite', and 'shoten'. A careful proofreading pass is needed.
  2. [Section 2, Eq. (3)] The notation p∞ is introduced as a limiting force but it is also a fitted parameter; the text would benefit from an explicit statement that p∞ is finite and that the t → ∞ approach in Eq. (5) is cut short by detachment, since the paper later uses p∞ as the isometric-force scale.
  3. [Section 6, Eq. (66) and following] The same symbol P is used for the total half-sarcomere force and for the expected force per crossbridge (e.g., 'P = NcP' in Section 6). This overloading is confusing; a distinct symbol such as F for total force would improve readability.
  4. [References [1] and [2]] References [1] and [2] appear to be the same paper by Alcazar, Csapo, Ara, and Alegre, duplicated with different entry formatting. One duplicate should be removed.
  5. [Figure 1 caption] The caption says the red curves are the fit of the 'initial guess' and the blue curves are the best fits, but the text in Section 5 describes the red curves as the k → ∞ fit and the blue curves as the k = 3.3 pN/nm fit; the caption should state this more explicitly.
  6. [Appendix A, Eq. (105)] The text says the left-hand side of Eq. (105) is a 'strictly increasing function from [0,1]', but the left-hand side is exp(-∫β dτ), which is decreasing; the intended statement is that the function whose root is sought is strictly increasing, or that the CDF is increasing. Please clarify.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Hill-based detachment rate is explicitly calibrated, and the central test is an out-of-sample quick-release comparison.

full rationale

The derivation chain is self-contained and does not disguise fits as predictions. Section 3 derives beta(p) by explicitly requiring the k to infinity steady-state model to match Hill's empirical force-velocity and heat equations: the text says 'we require (23) to agree with (15), and (22) to agree with (14)' and later calls the use of Eq. (28) for finite k 'somewhat questionable.' This is transparent calibration, not a claimed prediction; no step equates a fitted parameter with an independent output. The model's free parameters (p_infinity, v_max, alpha, N_c) are fitted to Piazzesi steady-state data in Section 5, and the quick-release traces in Section 8 are a separate protocol compared visually; the paper does not claim to have predicted the steady-state data it used for fitting. The one free non-fitted quantity, k_se, is explicitly declared arbitrary ('we just make the arbitrary choice of kse = 100pN/nm'), so its role in the transition-phase jump (Eq. 93) is a limitation or correctness risk rather than a circular reduction. The self-citation to Lacker and Peskin [9] is contextual and not load-bearing. The paper's own caveats: the finite-k use of beta(p), the approximate nature of Hill's 1938 summary, and the unexplained damped oscillations, are acknowledged limitations, not evidence that the central internal-dynamics mechanism is equivalent to its inputs by definition.

Assumptions & free parameters 6 free parameters · 7 assumptions · 1 invented entities

The model rests on a new constitutive law for the internal rest length, on a detachment rate identified by fitting the k→∞ limit to Hill's empirical curves and then assumed valid at finite k, and on standard field assumptions (dense binding sites, one ATP per cycle, Markov detachment). The fitted parameters are α, p∞, vmax, Nc, ϵ, and the hand-chosen kse.

free parameters (6)
  • α (attachment rate) = 68.2 s^-1
    Attachment rate per unattached crossbridge, fitted to Piazzesi Fig 4C using Eq. (58).
  • p∞ (limiting crossbridge force) = 9.98 pN
    Limiting force of an attached crossbridge as t→∞ under isometric conditions, fitted to Piazzesi data.
  • vmax (maximum shortening velocity) = 2.75 × 10^3 nm/s
    Maximum velocity at zero load, fitted to Piazzesi data.
  • Nc (number of cycling crossbridges) = 116
    Number of myosin crossbridges participating in the cycle in a half-sarcomere, fitted to Piazzesi Fig 4A.
  • ϵ = α p∞/(k vmax) = 0.08
    Dimensionless parameter; the fit is reported as insensitive over [0,0.15], and ϵ is effectively determined in the p∞, ϵ, Nc, vmax fitting step.
  • kse (series elasticity stiffness) = 100 pN/nm
    Chosen arbitrarily by the authors ('we just make the arbitrary choice of kse = 100 pN/nm'), not fitted; it directly affects the initial quick-release jump.
assumptions (7)
  • ad hoc to paper Internal dynamics dr/dt = vmax(1 - p/p∞) for the rest length of an attached crossbridge
    Eq. (3). This linear force-velocity relation for the rest length is the central new modeling assumption; it is not derived and is specific to this paper.
  • ad hoc to paper β(p) derived in the k→∞ limit (Eq. 28) remains valid for finite k = 3.3 pN/nm
    The authors call this 'somewhat questionable' in Section 3 and provide no rigorous justification beyond the smallness of ϵ=0.08 and plausibility arguments.
  • domain assumption Thin filament is a dense array of binding sites, so attachment occurs with zero strain at constant rate α
    Section 1 and Section 2; this continuum limit is standard in the field, cited to T.L. Hill [7], Lacker & Peskin [9], and Walcott et al. [12].
  • domain assumption Each crossbridge cycle hydrolyzes one ATP, so the energy consumption rate is proportional to the cycling rate 1/Tc(v)
    Used in Section 3 and Appendix B to convert Hill's heat-of-shortening Eq. (15) into a statement about cycling rate; standard in muscle energetics.
  • ad hoc to paper Hill's 1938 equations (14)-(15) are taken as the exact behavior of muscle in the k→∞ limit
    The authors assume their model's infinite-stiffness limit should reproduce Hill's 1938 summary, and use this to solve for β(p); they also acknowledge that Hill's 1938 description is approximate and later revised.
  • domain assumption Detachment probability per unit time depends only on instantaneous crossbridge force p
    Eq. (6) in Section 2; this Markov property is standard in crossbridge models.
  • domain assumption Series elasticity is a linear spring with stiffness kse and no rest length
    Section 6, Eqs. (66)-(67); a standard treatment of series compliance in muscle mechanics.
invented entities (1)
  • Variable rest length r(t) of the crossbridge spring independent evidence
    purpose: Introduces a slow internal degree of freedom so the spring can be stiff on fast timescales while force stays roughly constant on slow timescales.
    The model makes quantitative predictions (steady-state U(v), P(v), Sa(v), quick-release length traces) that are tested against Piazzesi's independent data; the rest length itself is not directly observable but its consequences are testable.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Muscle Crossbridge Theory With Internal Crossbridge Dynamics." pith.science (2026). https://pith.science/paper/PQZNO56J

@misc{pith2026250520198,
  author       = {Pith},
  title        = {Pith review of: Muscle Crossbridge Theory With Internal Crossbridge Dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PQZNO56J}},
  note         = {Machine review of arXiv:2505.20198}
}
read the original abstract

We describe in this paper a crossbridge model in which an attached crossbridge behaves like a linear spring with a variable rest length. We assume in particular that the rest length has a linear force-velocity relation, and that the force and rest length are both zero at the moment of crossbridge attachment. Crossbridges that are not attached in our model have a fixed probability per unit time of attachment, and attached crossbridges have a probability per unit time of detachment that is a function of the crossbridge force. This detachment rate is uniquely determined by the requirement that a limiting form of the model should reproduce the force-velocity curve and heat of shortening discovered by A.V.Hill~\cite{AVHILL}, and the detachment rate turns out to be a linearly decreasing function of the crossbridge force. The parameters of the model are determined by a fit to steady-state experimental data; and then an event-driven stochastic simulation methodology is introduced in order to study the behavior of the model in a simulated quick-release experiment. The model explains how the crossbridge can act like a linear spring on a fast time scale but have very different properties on a slower time scale.

Figures

Figures reproduced from arXiv: 2505.20198 by the authors.

Figure 1
Figure 1. Left panel: Fitting (30) and (39) to the data in Figure 1c of [10]. [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Quick release simulations compared to experimental results. The [PITH_FULL_IMAGE:figures/full_fig_p032_2.png] view at source ↗
Figure 3
Figure 3. Steady-state simulation results (red filled squares) compared to [PITH_FULL_IMAGE:figures/full_fig_p033_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

  1. [1]

    Alcazar, R

    J. Alcazar, R. Csapo, I. Ara, and L. M. Alegre, On the shape of the force-velocity relationship in skeletal muscles: The linear, the hyperbolic, and the double-hyperbolic , Frontiers in physiology, 10 (2019), p. 438208. 30

    work page 2019

  2. [2]

    Alcazar, R

    J. Alcazar, R. Csapo, I. Ara, and L. M. Alegre, On the shape of the force-velocity relationship in skeletal muscles: The linear, the hyperbolic, and the double-hyperbolic , Frontiers in Physiology, 10 (2019)

    work page 2019

  3. [3]

    Duke, Molecular model of muscle contraction , Proceedings of the Na- tional Academy of Sciences, 96 (1999), pp

    T. Duke, Molecular model of muscle contraction , Proceedings of the Na- tional Academy of Sciences, 96 (1999), pp. 2770–2775

    work page 1999

  4. [4]

    Eisenberg, T

    E. Eisenberg, T. Hill, and Y. Chen, Cross-bridge model of muscle contraction. quantitative analysis, Biophysical Journal, 29 (1980), pp. 195– 227

    work page 1980

  5. [5]

    A. V. Hill, The heat of shortening and the dynamic constants of muscle , Proceedings of the Royal Society of London. Series B - Biological Sciences, 126 (1938), pp. 136–195

    work page 1938

  6. [6]

    Series B

    , The effect of load on the heat of shortening of muscle , Proceedings of the Royal Society of London. Series B. Biological Sciences, 159 (1964), pp. 297–318

    work page 1964

  7. [7]

    T. L. Hill, Theoretical formalism for the sliding filament model of contrac- tion of striated muscle part i , Progress in biophysics and molecular biology, 28 (1974), pp. 267–340

    work page 1974

  8. [8]

    Huxley, 6 - muscle structure and theories of contraction , Progress in Biophysics and Biophysical Chemistry, 7 (1957), pp

    A. Huxley, 6 - muscle structure and theories of contraction , Progress in Biophysics and Biophysical Chemistry, 7 (1957), pp. 255–318

    work page 1957

Show all 12 references
  1. [9]

    H. M. Lacker and C. Peskin, A mathematical method for the unique determination of cross-bridge properties from steady-state mechanical and energetic experiments on macroscopic muscle , in Lectures on mathematics in the life sciences, AMS, 1986, pp. 121–153

    1986

  2. [10]

    Piazzesi, M

    G. Piazzesi, M. Reconditi, M. Linari, L. Lucii, P. Bianco, E. Brunello, V. Decostre, A. Stewart, D. B. Gore, T. C. Irving, M. Irving, and V. Lombardi, Skeletal muscle performance determined by modulation of number of myosin motors rather than motor force or stroke size, Cell, ...

    2007

  3. [11]

    Podolsky, Kinetics of muscular contraction: the approach to the steady state, Nature, 188 (1960), pp

    R. Podolsky, Kinetics of muscular contraction: the approach to the steady state, Nature, 188 (1960), pp. 666–668

    1960

  4. [12]

    W alcott, D

    S. W alcott, D. M. W arshaw, and E. P. Debold, Mechanical coupling between myosin molecules causes differences between ensemble and single- molecule measurements, Biophysical journal, 103 (2012), pp. 501–510. 31 Figure 2: Quick release simulations compared to experimental resu...

    2012

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.