{"id":"0bfae7be-13e3-4515-9955-dad53a9b311f","arxiv_id":"2505.20198","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"A crossbridge model with a time-varying rest length derives a force-decreasing detachment rate and matches steady-state and quick-release muscle data.","lead":"This paper proposes a new model of muscle contraction in which each myosin crossbridge is a spring whose rest length changes over time, allowing it to act stiff on fast timescales yet keep force steady during slow shortening. The model reproduces Hill's classic force-velocity law and roughly matches Piazzesi's quick-release experiments, though with several hand-chosen assumptions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Quick-release validation rests on an arbitrary series elasticity kse=100 pN/nm with no sensitivity analysis or quantitative misfit metric; the claimed reproduction of Piazzesi's data is therefore not yet established.","rationale":"The reader's weakest assumption is the infinite-stiffness derivation of β(p) used at finite k, a concern the authors themselves flag as 'somewhat questionable' in Section 3. I agree that this is a real issue, but it is primarily a model-identification issue: if β(p) at finite k differs, the parameter fit and the Hill link change, yet the qualitative claim of stiff-fast/constant-force-slow behavior could survive with a different β. By contrast, the quick-release simulation is the paper's only out-of-sample test of that dynamic behavior, and it depends on an unconstrained series elasticity. The steady-state comparisons in Section 5 fit the same data used to set parameters, so they do not independently confirm the model. The paper itself notes the 'arbitrary choice' of kse in Section 6, reports no sensitivity study of that choice, and offers only qualitative visual comparison in Section 8. A simple sweep of kse would settle whether the agreement is meaningful. Until that is done, the central claim should remain conditional rather than accepted. This is consistent with the reader's CONDITIONAL verdict, but for a reason different from the specific weakest assumption named.","tokens_in":20673,"tokens_out":10243,"duration_ms":110473,"concrete_test":"Rerun the event-driven quick-release simulation of Section 7 for kse = 30, 100, and 300 pN/nm, keeping all other parameters fixed at the Table 1 values, and evaluate the t=0 intercept and the steady shortening velocity for each of the five loads against Piazzesi's experimental records. Report the shift in the intercept; if it exceeds the experimental scatter (order 1 nm per half-sarcomere) across this plausible range, the agreement is not robust. As a second check, fit kse to the measured instantaneous compliance from the release records; if the best-fit value is far from 100 pN/nm, the displayed match is parameter-dependent rather than a test of the internal dynamics.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central validation of the internal crossbridge dynamics is the quick-release simulation of Sections 7-8. This is the only out-of-sample check: the steady-state fits in Section 5 optimize p∞, vmax, α, and Nc to the same Piazzesi data, and the paper itself notes that ϵ is poorly determined by those data, so those fits are in-sample. The quick-release comparison, however, contains an unconstrained parameter: kse is set to 100 pN/nm by an explicit 'arbitrary choice' in Section 6 and is never varied or fitted. The transition-phase length jump at release is proportional to (1/kse + 1/(kNA)) via Eq. 93 of Section 7, so kse directly controls the t=0 intercept, which is one of the two features the authors claim to match in Section 8. Without a sensitivity analysis, the displayed agreement could be specific to this value. The comparison is also only visual: no RMS error or confidence interval is reported, and the text concedes that the simulated traces miss the damped oscillations seen experimentally. Thus the quick-release evidence does not yet pin down the claimed mechanism.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":21012,"tokens_out":2711,"duration_ms":32248,"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":[{"comment":"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.","section":"Section 3, Eq. (28)"},{"comment":"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.","section":"Section 6 and Section 8, kse = 100 pN/nm"},{"comment":"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.","section":"Section 5, parameter fitting and Table 1"}],"minor_comments":[{"comment":"The manuscript contains numerous typographical errors, including 'atachement', 'detachement', 'corssbrdge', 'paramter', 'simnulation', 'ocurs', 'advantate', 'dimensionkess', 'approprite', and 'shoten'. A careful proofreading pass is needed.","section":"Throughout"},{"comment":"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.","section":"Section 2, Eq. (3)"},{"comment":"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.","section":"Section 6, Eq. (66) and following"},{"comment":"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.","section":"References [1] and [2]"},{"comment":"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.","section":"Figure 1 caption"},{"comment":"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.","section":"Appendix A, Eq. (105)"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Sam,\n\nThe headline: this is a genuinely new crossbridge model with an internal rest-length variable, and the steady-state math is solid; the quick-release claim is the part that doesn't fully land yet. Worth sending to review, not desk-rejecting.\n\nWhat's new: Eq. (3) dr/dt = vmax(1 - p/p∞) gives a spring that is stiff on fast timescales and roughly constant-force on slow timescales, which is exactly the Piazzesi paradox. The detachment rate β(p) = (α/4)(1 + 20(1 - p/p∞)) falling linearly with force (a catch bond) is derived from Hill's 1938 curves in the k→∞ limit, and the exact finite-k steady-state formulas in Section 4 reduce properly back to that limit. The event-driven simulation is clean, and the authors are admirably transparent about the by-construction nature of the Hill agreement.\n\nThe soft spots, in order of real weight:\n\n1. The β(p) identification is circular in the k→∞ limit: they solve β out of the very force-velocity and heat curves they then \"reproduce.\" They say so themselves, so the only independent check is the finite-k Piazzesi steady-state fit plus quick-release. That's fine as an identification strategy, but it means the Hill agreement buys nothing.\n\n2. Using that same β(p) at finite k=3.3 pN/nm is asserted, not justified. The ϵ=0.08 smallness helps, and the steady-state fits are good, but the paper could say more about how wrong β(p) would have to be to break the Piazzesi agreement.\n\n3. The quick-release comparison is the real test, and it has two weaknesses the stress-test note correctly flags. kse=100 pN/nm is explicitly arbitrary, and it controls the t=0 intercept through Eq. (93). No sensitivity analysis. And the agreement is visual: no RMS error, no confidence intervals, and the simulations miss the damped oscillations. I don't think this is fatal—the steady-state slope and intercept do look close—but \"approximately right\" plus one arbitrary stiffness doesn't yet pin the mechanism.\n\n4. Excluding the isometric data point is a defensible choice given the anomaly, but it removes the one point where the model makes a distinct prediction.\n\nWho it's for: muscle biophysicists and mathematical biologists who work on crossbridge models. It deserves a serious referee: the idea is new, the derivations are checkable, and the issues are addressable in revision. I'd suggest the editor ask for sensitivity on kse, a quantitative misfit measure, and a paragraph on how β(p) at finite k could be tested rather than just assumed.","headline":"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.","tokens_in":21498,"tokens_out":2106,"would_cite":true,"duration_ms":41944,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["92C10","92C40"],"pacs":[],"model":"deepseek-v4-flash","headline":"A variable rest length lets muscle crossbridges pull at constant force.","keywords":["crossbridge theory","muscle contraction","variable rest length","internal dynamics","force-velocity relation","quick release","event-driven simulation","catch-bond detachment"],"falsifier":"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.","tokens_in":2082,"feed_emoji":"💪","tokens_out":3131,"duration_ms":86965,"temperature":0.7,"pith_summary":"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.","feed_headline":"Variable rest length makes crossbridges stiff yet constant-force","feed_subtitle":"Internal variable reconciles linear-spring behavior with steady force during shortening.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies Hill's 1938 force-velocity curve and heat of shortening that uniquely determine the detachment rate $\\beta(p)$ in the infinite-stiffness limit.","marker":"[5]"},{"why":"Provides the Piazzesi steady-state data and quick-release experimental records used for parameter fitting and for validating the simulations.","marker":"[10]"},{"why":"Defines the Huxley linear-spring crossbridge theory that the paper revises, including the tension-compression balance at maximum velocity.","marker":"[8]"},{"why":"Establishes the method of using Hill's macroscopic data to determine crossbridge kinetic functions, and introduces the continuum thin-filament binding-site assumption used here.","marker":"[9]"},{"why":"Supplies the quick-release experimental context that motivates the need for series elasticity and the simulated protocol.","marker":"[11]"},{"why":"Provides an earlier stochastic simulation of a crossbridge model used as precedent for the event-driven simulation methodology.","marker":"[3]"}],"fun_headline_variants":["Dynamic rest length makes crossbridge spring stiff yet steady","Crossbridge rest length evolves to keep force constant during shortening","New model: crossbridge spring rest length is a running variable","Variable rest length gives crossbridge fast stiffness, slow steady force"],"cache_read_input_tokens":23552,"weakest_assumption_plain":"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 $\\epsilon = 0.08$.","fun_headline_variants_meta":{"raw":{"variants":["Dynamic rest length makes crossbridge spring stiff yet steady","Crossbridge rest length evolves to keep force constant during shortening","New model: crossbridge spring rest length is a running variable","Variable rest length gives crossbridge fast stiffness, slow steady force"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000286,"raw_usage":{"total_tokens":1724,"prompt_tokens":1029,"completion_tokens":695,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":645,"completion_tokens_details":{"reasoning_tokens":627}},"tokens_in":645,"tokens_out":695,"duration_ms":8426,"temperature":1.0,"reasoning_tokens":627,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:58:30.428206+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Hill's 1938 force-velocity curve and heat of shortening that uniquely determine the detachment rate $\\beta(p)$ in the infinite-stiffness limit."},{"cited_title":"Piazzesi, M","cited_arxiv_id":null,"evidence_quote":"Provides the Piazzesi steady-state data and quick-release experimental records used for parameter fitting and for validating the simulations."},{"cited_title":"Huxley, 6 - muscle structure and theories of contraction , Progress in Biophysics and Biophysical Chemistry, 7 (1957), pp","cited_arxiv_id":null,"evidence_quote":"Defines the Huxley linear-spring crossbridge theory that the paper revises, including the tension-compression balance at maximum velocity."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the method of using Hill's macroscopic data to determine crossbridge kinetic functions, and introduces the continuum thin-filament binding-site assumption used here."},{"cited_title":"Podolsky, Kinetics of muscular contraction: the approach to the steady state, Nature, 188 (1960), pp","cited_arxiv_id":null,"evidence_quote":"Supplies the quick-release experimental context that motivates the need for series elasticity and the simulated protocol."},{"cited_title":"Duke, Molecular model of muscle contraction , Proceedings of the Na- tional Academy of Sciences, 96 (1999), pp","cited_arxiv_id":null,"evidence_quote":"Provides an earlier stochastic simulation of a crossbridge model used as precedent for the event-driven simulation methodology."}],"review_version":1}