{"id":"2aa447bc-8385-4111-97b9-7a94077e39ae","arxiv_id":"2502.03290","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"A torque-balance model shows that stator forces can couple FliG subunits globally, producing non-equilibrium ultrasensitive switching whose Hill coefficient grows with stator number.","lead":"This paper proposes that mechanical torques from the motor's stator units act back on the switch proteins to align them, creating a non-equilibrium 'tug of war' that makes the motor switch direction very sensitively. The proposed mechanism predicts that switching sensitivity grows with the number of stator units, and the authors find tentative support in published motor data.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The no-slip rigid-gear assumption is load-bearing for the GMC prediction; if stator–FliG coupling slips, local torque is no longer set by the global majority, and the predicted H∝M scaling can fail.","rationale":"The paper is careful and honest: it labels the experimental evidence 'tentative' and explicitly asks for simultaneous M, CheYp, and CW-bias measurements. The theoretical construction is internally coherent, and Eq. (8) is a genuine identity for binary variables. My stress-test pass therefore does not find an internal inconsistency or an obvious hidden mathematical error. The most load-bearing vulnerability is exactly what the reader identified: the torque-balance step (Eqs. (5)-(6)) assumes no-slip stator-FliG coupling, a rigid C-ring, and uniform torque distribution. Under that assumption, a minority subunit feels the full differential torque that drives the positive feedback; without it, the all-to-all effective coupling and the H∝M prediction may be quantitatively weakened or lost. Because the paper's novel claim is precisely that mechanical torque can replace nearest-neighbor conformational spread, this physical assumption is not a peripheral detail. A finite-stiffness simulation is a concrete way to decide whether the mechanism survives realistic compliance. If it survives, the model is an important contribution; if it does not, the prediction would need substantial revision. Neither outcome is established by the current manuscript, so the appropriate verdict remains CONDITIONAL, unchanged from the reader's assessment. A secondary concern is that the experimental support in Fig. 4 depends on fixing H(19%)=10.3 and K=3.1 from Ref. [22] rather than fitting all parameters freely; a likelihood-ratio test against a single shared H would sharpen this, but this is a validation weakness, not the theoretical crux.","tokens_in":13044,"tokens_out":7026,"duration_ms":70239,"concrete_test":"Extend the mechanical part of the model from no-slip to finite stiffness: couple each stator to FliG through a harmonic spring of stiffness κ (and optionally add C-ring bending elasticity), recover the reported no-slip results in the limit κ→∞, and run Gillespie simulations for M=3,4,...,11 at γ=3.5 to compute H(M). If H(M) remains approximately linear for κ values calibrated to observed motor compliance (e.g., torque-speed measurements or stator-resurrection fluctuations), the concern is refuted. If H(M) flattens or the bimodal distribution disappears at realistic κ, the central prediction fails because local torques are no longer globally determined.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equations (5)-(6) are the mechanism's engine: the no-slip, rigid-rotor torque balance makes each engaged FliG subunit's local torque a deterministic function of the global engaged fraction N_e^+/M, producing the all-to-all effective coupling in Eq. (7) and the bound H≤M. If stator-FliG contacts are compliant or the C-ring deforms, the torque felt by a lone minority subunit is set by local elastic strain, not by the full majority imbalance, so the minority may not experience τ±>1 and the exponential amplification f(τ)=exp(γτ) loses its cooperative punch. The paper's own Discussion lists 'strong MotA-FliG interactions and C-ring rigidity' as assumptions that 'more detailed modeling could relax,' but no test of robustness to finite stiffness is provided. Since every novel result—bimodal switching, broken detailed balance, H∝M—flows from this torque-balance step, this is the load-bearing physical assumption rather than a fitting detail.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a new mechanism, 'Global Mechanical Coupling' (GMC), for the ultrasensitive switching of the bacterial flagellar motor. The model treats stator-engaged FliG subunits as mechanically coupled through torque balance on a rigid C-ring, so that the local torque on a subunit depends on the global fraction of engaged subunits in the +1 conformation. This effective all-to-all coupling produces bimodal switching without nearest-neighbor interactions, breaks detailed balance by coupling to the ion-motive force, and yields a Hill coefficient H that is bounded by the number of engaged stators M. The authors derive a coarse-grained stationary distribution (Eq. 7), verify it with Gillespie simulations, and show that the fluctuation-dissipation relation H = (4/M)Var(N_e^+) (Eq. 8) implies H ≤ M. They re-analyze published dose-response data from Zhu et al. and report H(0%) ≈ 6.0 versus H(19%) ≈ 10.3 as tentative evidence that H grows with M. They also demonstrate that combining GMC with nearest-neighbor couplings can produce the same Hill coefficient as equilibrium conformational spread with roughly 10-fold faster response times.","tokens_in":13224,"tokens_out":5799,"duration_ms":53611,"significance":"If the mechanism survives further scrutiny, it offers a new physical origin for non-equilibrium cooperativity in the flagellar motor, distinct from the equilibrium conformational-spread picture, and it makes a concrete, falsifiable prediction that response cooperativity increases with the number of torque-generating stators. The theoretical core is internally consistent: the torque-balance derivation is self-contained, the Gillespie simulations corroborate the mean-field results, and the Hill-coefficient bound H ≤ M follows directly from the statistics of M binary variables. The experimental support is appropriately labeled as tentative, and the authors explicitly call for simultaneous measurements of M, CheYp, and CW bias. The paper also opens a potentially general principle that mechanical force can mediate long-range coupling in other biological tug-of-war systems, such as bidirectional cargo transport.","major_comments":[{"comment":"The central prediction H∝M rests on the assumption that stator-rotor coupling is perfectly rigid and the C-ring rotates as a rigid body, so that the local torque on each engaged FliG subunit is a deterministic function of the global fraction N_e^+/M. If stator-FliG contacts are compliant or the C-ring deforms, the torque on a lone minority subunit is set by local elastic strain rather than the full majority imbalance, and the exponential amplification in Eq. (3) may not generate the same cooperative enhancement. The Discussion acknowledges this limitation (\"strong MotA-FliG interactions and C-ring rigidity\" are assumptions that \"more detailed modeling could relax\"), but the manuscript provides no test of robustness to finite stiffness. Because every novel result (bimodality, broken detailed balance, H≤M, H∝M) flows from this torque-balance step, the authors should either add a stiffness parameter and show how H versus M degrades as coupling becomes compliant, or explicitly delineate the parameter regime in which the mechanism is expected to operate.","section":"Minimal model of motor mechanics (Eqs. 5-6)"},{"comment":"The comparison in Fig. 4 is not parameter-free and is partly circular. The authors fix H(19%) = 10.3 and K(19%) = 3.1 μM from Ref. [22] and then fit H(0%) and K(0%) to the same published dataset; the adapted stator numbers M(0%) ≈ 6 and M(19%) ≈ 11 are not measured in the same cells but are taken from separate population-level studies. Thus the reported increase from H(0%) ≈ 6.0 to H(19%) ≈ 10.3 is consistent with H∝M only under additional assumptions about unchanged CheYp concentration and load-dependent stator adaptation. The authors acknowledge this (\"Simultaneous measurements of M, CheYp, and CW bias are needed to fully test this prediction\"), but as it stands the experimental section is suggestive rather than a quantitative test. The authors should present the prediction as a ratio H(0%)/H(19%) ≈ M(0%)/M(19%) with propagated uncertainties, or fit the two datasets with M as a shared parameter and report the resulting confidence intervals.","section":"Experimental evidence (Fig. 4)"},{"comment":"The claim that non-equilibrium GMC motors achieve the same Hill coefficient as equilibrium motors with approximately 10-fold faster responses is based on a single definition of response speed (mean time to reach N_e^+ = 0 after a step in ΔF from 1/2 to -1/2) and specific values of the bias before and after the stimulus. The authors state that they \"expect similar qualitative behavior regardless of the specific values,\" but this expectation is not demonstrated. Since this speed advantage is a key advertised benefit of GMC, the authors should provide a parameter scan (e.g., varying the magnitude of the bias step or the definition of the response time) to show that the ~10-fold advantage is not an artifact of the chosen protocol.","section":"GMC eases a speed-sensitivity trade-off (Fig. 5)"}],"minor_comments":[{"comment":"The phrase \"an unique starting point\" should be \"a unique starting point.\"","section":"Abstract/Discussion"},{"comment":"The text \"added non-equilibrium effects ad hocto the conformational spread model\" appears to be a typo: it should read \"added non-equilibrium effects ad hoc to the conformational spread model.\"","section":"Introduction"},{"comment":"In the caption, \"Fig. 1AB\" should be written as \"Fig. 1A,B\" or \"panels A and B\" for clarity.","section":"Fig. 1 caption"},{"comment":"The derivative notation \"∂∆F ⟨N_e^+⟩\" is unconventional; consider writing ∂⟨N_e^+⟩/∂(ΔF) evaluated at ΔF = 0.","section":"Eq. (8)"},{"comment":"The term \"CW bias\" is used without a definition in the main text; a brief definition (e.g., fraction of time the motor rotates clockwise) would help readers not specialized in the field.","section":"Experimental section"}],"recommendation":"major_revision","confidential_remarks":"The theoretical mechanism is novel and internally consistent, and the paper is well written. The main risk is the rigidity/no-slip assumption underlying Eqs. (5)-(6), which the authors themselves flag as an idealization. I believe the paper can be made publishable by adding a robustness analysis with compliant coupling and by making the experimental comparison more transparent regarding the fixing of parameters and the lack of simultaneous M measurements. The experimental section is already appropriately cautious, but the fitting procedure needs clarification."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe key idea here is worth your time: the flagellar switch may owe its cooperativity to a global mechanical tug-of-war rather than to nearest-neighbor conformational spread. The authors show that if each stator transmits a local torque that depends on the global conformational state of the C-ring (via reaction torque), then subunits are effectively all-to-all coupled. This is a genuine mechanistic addition over earlier torque-dependent switching models, which either treated torque as a fixed parameter or ignored the reaction torque on stators. The derivation from torque balance to the effective Hamiltonian is clean, the bound H ≤ M is a nice result, and the Gillespie simulations match the theory. The additional observation that non-equilibrium motors can achieve the same Hill coefficient about ten times faster than equilibrium ones is a useful contribution.\n\nThe soft spots are real but not fatal. The most load-bearing assumption is the no-slip rigid-gear engagement between stators and FliG. If that coupling slips or the C-ring deforms, the local torque on a minority subunit will be set by local strain rather than the global majority, and the cooperative amplification weakens. The paper lists this assumption in the Discussion but does not test robustness to finite compliance, which a referee should ask for. Similarly, the exponential torque-dependent rate f(τ)=exp(γτ) is plausible but the value γ≈3.2 rests on an order-of-magnitude guess for α; a sensitivity analysis would help.\n\nThe experimental support is tentative, as the authors concede. They fix H(19%) = 10.3 and K(19%) = 3.1 from the published data, then fit the low-load curve and infer CheYp per cell. That is post-hoc, not a parameter-free test. The prediction H∝M is specific enough to be tested with simultaneous measurements of stator number, CheYp, and CW bias; the present data are suggestive, not conclusive. I also had trouble reproducing the simulations from the text alone; no code or full SI is provided.\n\nOverall, this is a serious paper with a clear, testable mechanism. For anyone working on the flagellar motor or on general mechanisms of non-equilibrium cooperativity, it deserves a rigorous peer review and should be published if the robustness analysis and data-access issues are addressed. I would bring it to the reading group and would cite it in related work.\n\nRecommendation: send to peer review.","headline":"A clean, testable mechanism for non-equilibrium cooperativity in the flagellar motor, with preliminary experimental support that is not yet decisive; worth careful review.","tokens_in":13766,"tokens_out":3618,"would_cite":true,"duration_ms":30635,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Mechanical torque, not nearest-neighbor coupling, can drive the flagellar motor's ultrasensitive switching, with cooperativity that grows with stator number.","keywords":["flagellar motor","ultrasensitivity","cooperativity","non-equilibrium switching","global mechanical coupling","torque balance","Hill coefficient","conformational spread"],"falsifier":"A direct test would be to measure the Hill coefficient $H$ of the CW-bias response while experimentally varying the number of engaged stators $M$ (for example by load changes or stator resurrection) in cells with controlled CheYp levels; if $H$ does not rise roughly proportionally to $M$ and saturate near $M$ for large $\\gamma$, the GMC prediction fails.","tokens_in":12796,"feed_emoji":"⚙️","tokens_out":5954,"duration_ms":51632,"temperature":0.7,"pith_summary":"The paper proposes that the bacterial flagellar motor's ultrasensitive switching between rotation directions can arise from mechanical torque balance rather than from nearest-neighbor conformational coupling among switch proteins. In the proposed mechanism, each stator-engaged FliG subunit feels a local torque set by the global state of the rotor, so a subunit in the minority conformation is pushed harder and flips faster into the majority, creating a cooperative, energy-dissipating response. This 'Global Mechanical Coupling' predicts that the motor's Hill coefficient grows with the number of active stators, and the authors find tentative support by re-analyzing published dose-response data under different loads. If correct, the result offers a concrete physical origin for non-equilibrium switching and suggests that dissipative mechanics can serve as a general route to sensitive chemical regulation.","feed_headline":"Flagellar switch steepness grows with stator count, model predicts","feed_subtitle":"A torque tug-of-war replaces conformational spread and buys faster, equally sharp switching at the cost of dissipation.","key_machinery":"The central object is the Global Mechanical Coupling (GMC) mechanism, defined by the torque-balance equations of the minimal stator–C-ring model: the dimensionless motor speed $\\Omega(t) = \\frac{M/\\beta}{1+M/\\beta}\\left(2N_e^+/M - 1\\right)$ and the local torques $\\tau_\\pm(t) = 1 \\mp \\Omega(t)$. These equations make the local torque on each stator-engaged FliG subunit a deterministic function of the global fraction of engaged $+1$ subunits, so all engaged subunits are effectively coupled at a distance. Switching rates of engaged subunits are torque-dependent via $f(\\tau)=\\exp(\\gamma\\tau)$, which breaks detailed balance whenever $\\gamma\\neq0$ and $\\tau_- \\neq \\tau_+$; the free energy dissipated per cycle is $\\gamma(\\tau_- - \\tau_+)$. In the large-$\\gamma$ limit the effective Hamiltonian in the stationary distribution reduces to a non-equilibrium Monod–Wyman–Changeux form, and the Hill coefficient satisfies $H \\approx (4/M)\\,\\mathrm{Var}(N_e^+) \\leq M$.","core_discovery":"On its own terms, the paper claims that local mechanical torque on FliG subunits is the coordinator of the flagellar switch. Because the C-ring is treated as a rigid gear engaged with several stators, the torque on any one engaged subunit depends on how many engaged subunits are in the +1 conformation: $\\tau_\\pm = 1 \\mp \\Omega$, with the rotation speed $\\Omega$ itself a linear function of the global alignment. A subunit that differs from the majority is therefore driven against its stator's torque, and via force-dependent switching rates $f(\\tau)=e^{\\gamma\\tau}$ it flips faster into the majority, producing a positive feedback. The stationary distribution of the coarse-grained model is bimodal for large $\\gamma$, and the Hill coefficient of the response to the chemical bias $\\Delta F$ approaches $M$, the number of stators. The paper further shows that this non-equilibrium mechanism coexists synergistically with nearest-neighbor coupling, giving the same cooperativity at roughly tenfold faster response than equilibrium conformational-spread models.","pith_inferences":["The GMC prediction could be tested more directly by measuring the Hill coefficient of the same motor while changing stator number $M$ via controlled load or stator resurrection, while holding the CheYp distribution fixed; if $H$ does not track $M$, the mechanism would need revision.","If GMC operates in other macromolecular machines, force-dependent detachment in bidirectional cargo transport along microtubules is a natural analogue, where a tug-of-war between opposing motors could produce chemical sensitivity without explicit coupling between motor copies.","The model's speed advantage suggests that any dissipative mechanical coupling—not just flagellar torque—could be a generic design principle for sensitive switches that must respond quickly, though the paper itself only gestures at this generalization.","The re-analysis of [22] leaves the CheYp concentration and stator number inferred rather than directly measured, so a combined measurement of $M$, CheYp, and CW bias in single cells would separate GMC's contribution from load-dependent changes in other parameters."],"forward_implications":["The motor's steep CW-bias response to CheYp does not require strong nearest-neighbor coupling among C-ring subunits; mechanical torque balance alone can supply the cooperativity.","The Hill coefficient of the switching response should increase with the number of engaged stators $M$, saturating near $M$, so motors under higher load should appear more ultrasensitive.","Non-equilibrium operation through GMC can achieve the same Hill coefficient as equilibrium conformational-spread models with roughly tenfold faster response, easing the speed-sensitivity trade-off.","The large-$\\gamma$ limit of GMC reproduces a non-equilibrium MWC-like model, connecting the mechanical picture to classic allosteric descriptions.","Because dissipation underlies the cooperativity, the ion-motive force that drives rotation also pays for the sharpness of the chemotactic switch."],"supporting_citations":[{"why":"Supplies the experimental CW-bias dose-response data at two loads re-analyzed here, giving tentative evidence that the Hill coefficient grows with stator number.","marker":"[22]"},{"why":"Supplies the conformational-spread model that GMC is proposed to replace, with the cooperative switching curve used as the comparison baseline.","marker":"[13]"},{"why":"Establishes that equilibrium processes cannot produce peaked switching-interval distributions, motivating the non-equilibrium GMC mechanism.","marker":"[18]"},{"why":"Cryo-EM structure of the stator complex supporting the gear-like torque transmission from stators to FliG.","marker":"[23]"},{"why":"Structures showing FliG changes pose between CCW and CW states, the conformational transition that local torque is proposed to bias.","marker":"[26]"},{"why":"Structural basis of the C-ring as a gear and of directional switching, used to justify rigid coupling at a distance.","marker":"[25]"},{"why":"Load-dependent assembly of stators, giving the relation between load and stator number used in the re-analysis.","marker":"[30]"},{"why":"The Monod–Wyman–Changeux model that the large-$\\gamma$ limit of GMC reduces to, connecting to classic allostery.","marker":"[45]"}],"fun_headline_variants":["Mechanical tug-of-war explains flagellar switch ultrasensitivity","Stator-driven torque sum sets flagellar cooperativity","Non-equilibrium torque coupling sharpens flagellar switch","Mechanical coupling, not neighbor spread, boosts flagellar switch","Stator number dictates flagellar switch steepness"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the C-ring and stators engage like rigid gears with no slip, so the local torque on a lone minority FliG subunit is a deterministic function of the global fraction of engaged +1 subunits; any slip or deformation that weakens this link would reduce or destroy the predicted cooperativity.","fun_headline_variants_meta":{"raw":{"variants":["Mechanical tug-of-war explains flagellar switch ultrasensitivity","Stator-driven torque sum sets flagellar cooperativity","Non-equilibrium torque coupling sharpens flagellar switch","Mechanical coupling, not neighbor spread, boosts flagellar switch","Stator number dictates flagellar switch steepness"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000483,"raw_usage":{"total_tokens":2401,"prompt_tokens":978,"completion_tokens":1423,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":594,"completion_tokens_details":{"reasoning_tokens":1345}},"tokens_in":594,"tokens_out":1423,"duration_ms":10159,"temperature":1.0,"reasoning_tokens":1345,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T05:14:52.596358+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct test would be to measure the Hill coefficient $H$ of the CW-bias response while experimentally varying the number of engaged stators $M$ (for example by load changes or stator resurrection) in cells with controlled CheYp levels; if $H$ does not rise roughly proportionally to $M$ and saturate near $M$ for large $\\gamma$, the GMC prediction fails.","supporting_citations":[{"cited_title":"stimulus,","cited_arxiv_id":null,"evidence_quote":"Supplies the experimental CW-bias dose-response data at two loads re-analyzed here, giving tentative evidence that the Hill coefficient grows with stator number."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the conformational-spread model that GMC is proposed to replace, with the cooperative switching curve used as the comparison baseline."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes that equilibrium processes cannot produce peaked switching-interval distributions, motivating the non-equilibrium GMC mechanism."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Cryo-EM structure of the stator complex supporting the gear-like torque transmission from stators to FliG."},{"cited_title":"Santiveri, A","cited_arxiv_id":null,"evidence_quote":"Structures showing FliG changes pose between CCW and CW states, the conformational transition that local torque is proposed to bias."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Structural basis of the C-ring as a gear and of directional switching, used to justify rigid coupling at a distance."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Load-dependent assembly of stators, giving the relation between load and stator number used in the re-analysis."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The Monod–Wyman–Changeux model that the large-$\\gamma$ limit of GMC reduces to, connecting to classic allostery."}],"review_version":1}