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REVIEW 3 major objections 5 minor 80 references

Ultrasensitivity without conformational spread: A mechanical origin for non-equilibrium cooperativity in the bacterial flagellar motor

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Mechanical torque, not nearest-neighbor coupling, can drive the flagellar motor's ultrasensitive switching, with cooperativity that grows with stator number.

desk verdict A clean, testable mechanism for non-equilibrium cooperativity in the flagellar motor, with preliminary experimental support that is not yet decisive; worth careful review. read the letter →

arxiv 2502.03290 v1 pith:KQ3HPHHS submitted 2025-02-05 physics.bio-ph cond-mat.stat-mechq-bio.SC

Henry H. Mattingly , Yuhai Tu This is my paper · ORCID
classification physics.bio-phcond-mat.stat-mechq-bio.SC
keywords flagellarmotorultrasensitivitycooperativitynon-equilibriumswitchingglobalmechanicalcouplingtorquebalanceHillcoefficientconformationalspread
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 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.

What carries the argument

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 $H \approx (4/M)\,\mathrm{Var}(N_e^+) \leq M$.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

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.

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 (3)
  1. [Minimal model of motor mechanics (Eqs. 5-6)] 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.
  2. [Experimental evidence (Fig. 4)] 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.
  3. [GMC eases a speed-sensitivity trade-off (Fig. 5)] 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.
minor comments (5)
  1. [Abstract/Discussion] The phrase "an unique starting point" should be "a unique starting point."
  2. [Introduction] 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."
  3. [Fig. 1 caption] In the caption, "Fig. 1AB" should be written as "Fig. 1A,B" or "panels A and B" for clarity.
  4. [Eq. (8)] The derivative notation "∂∆F ⟨N_e^+⟩" is unconventional; consider writing ∂⟨N_e^+⟩/∂(ΔF) evaluated at ΔF = 0.
  5. [Experimental section] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the GMC derivation is self-contained, and the experimental comparison is an openly labeled tentative fit, not a fitted-input prediction.

full rationale

The paper's central derivation is self-contained. The torque-balance equations (5)-(6) follow explicitly from the stated no-slip, rigid-rotor assumptions; Eq. (7) is a derived stationary distribution; Eq. (8) is a fluctuation-response identity for a binary collective variable, and the bound H ≤ M is a variance bound, not an input. The prediction that H increases with M is a model consequence, not an assumed result. The experimental support fits a Hill function to published CW-bias data from Ref. [22]; although H(19%) and K(19%) are fixed from that reference, H(0%) is freely fitted, so the comparison is a genuine test of the qualitative trend rather than a tautology. The paper explicitly states that 'Simultaneous measurements of M, CheYp, and CW bias are needed to fully test this prediction' and calls the evidence 'tentative.' The self-citation of Tu 2008 (for peaked interval-time distributions implying non-equilibrium) is an external mathematical result used for motivation, not a load-bearing step in deriving GMC. The no-slip/C-ring-rigidity assumptions are physical limitations acknowledged in the Discussion, not circular reasoning. Overall, no step reduces by construction to its own input.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The model rests on several dimensionless parameters (gamma, M, beta, N, J, k0) and on explicit structural assumptions about stator-rotor mechanics. No new physical entities such as new particles or forces are postulated. The most fragile assumptions are no-slip gear coupling, rigid rotor, and even torque distribution, which together produce the global coupling.

free parameters (6)
  • gamma = varied 0-4; estimate gamma ~ 3.2
    Dimensionless torque sensitivity controlling the strength of torque-dependent FliG switching. It is not directly measured; the estimate uses an assumed projection factor alpha ~ 0.1. Central to all GMC predictions.
  • M = 3-10 in simulations; ~6 and ~11 inferred for the two load conditions in the data
    Number of stator units driving rotation. The central prediction is that Hill coefficient grows with M. Treated as an input, inferred from load in the experimental comparison.
  • beta = beta ~ 2.5 (used in limit M/beta >> 1)
    Dimensionless rotational drag on the flagellum. Only the ratio M/beta enters, and the theory takes it large. Taken from prior experiments, not fit here.
  • N = N = 30 (experimental C-ring values near 34)
    Number of FliG subunits in the ring. Used in simulations; the coarse-grained theory's main results depend more strongly on M than on N.
  • J = varied 0-3
    Nearest-neighbor coupling strength, used in the speed-sensitivity trade-off section. Not needed for the pure GMC mechanism but included for the synergistic model.
  • k0 = set to 1
    Base switching rate, sets the time scale. Times are rescaled, so the absolute value is immaterial.
assumptions (6)
  • domain assumption Strong no-slip interaction between stators and FliG subunits, so they rotate together like gears.
    Introduced in 'Minimal model of motor mechanics'. Required for Eq. (5)-(6) to make local torque a function of the global motor state.
  • domain assumption The rotor (C-ring) rotates as a rigid object.
    Stated in the same section. Needed for the torque balance that yields the global coupling; if the ring deforms, the simple relation tau± = 1 ∓ Omega breaks down.
  • domain assumption Overdamped limit, fast stator refueling and ion release, and negligible torque fluctuations.
    Stated in 'Minimal model of motor mechanics'. These choices simplify the torque balance but are not directly verified for the motor.
  • ad hoc to paper Torque is distributed evenly among stator-engaged FliG subunits.
    Used in 'Local torques can drive FliG conformation cycles' to argue that minority subunits experience larger torque than majority subunits. This is a modeling choice that makes tau- != tau+ and drives the non-equilibrium cycle.
  • domain assumption Force-dependent switching rates follow f(tau) = exp(gamma tau), a slip-bond-like form.
    Introduced in Eq. (3), citing slip-bond models (Refs. 38-39). This exponential dependence is plausible but is not derived for FliG conformation changes.
  • domain assumption Fast-switching limit k0 >> 1 for the stationary distribution of N+e.
    Used to derive Eq. (7). It separates conformational switching from stator engagement dynamics; the authors state it when presenting P(N+e).

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Cite this review

Pith. "Pith review of Ultrasensitivity without conformational spread: A mechanical origin for non-equilibrium cooperativity in the bacterial flagellar motor." pith.science (2026). https://pith.science/paper/KQ3HPHHS

@misc{pith2026250203290,
  author       = {Pith},
  title        = {Pith review of: Ultrasensitivity without conformational spread: A mechanical origin for non-equilibrium cooperativity in the bacterial flagellar motor},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KQ3HPHHS}},
  note         = {Machine review of arXiv:2502.03290}
}
read the original abstract

Flagellar motors enable bacteria to navigate their environments by switching rotation direction in response to external cues with high sensitivity. Previous work suggested that ultrasensitivity of the flagellar motor originates from conformational spread, in which subunits of the switching complex are strongly coupled to their neighbors as in an equilibrium Ising model. However, dynamic single-motor measurements indicated that rotation switching is driven out of equilibrium, and the mechanism for this dissipative driving remains unknown. Here, based on recent cryo-EM structures, we propose that local mechanical torques on motor subunits can affect their conformation dynamics. This gives rise to a tug of war between stator-associated subunits, which produces cooperative, non-equilibrium switching responses without requiring nearest-neighbor interactions. Since subunits are effectively coupled at a distance, we call this mechanism ``Global Mechanical Coupling." Our model makes a qualitatively new prediction that the motor response cooperativity grows with the number of stators driving rotation. Re-analyzing published motor dose-response curves in varying load conditions, we find tentative experimental evidence for this prediction. Finally, we show that operating out of equilibrium enables motors to achieve high cooperativity with faster responses compared to equilibrium motors. Our results suggest a general role for mechanics in sensitive chemical regulation.

Figures

Figures reproduced from arXiv: 2502.03290 by the authors.

Figure 1
Figure 1. FIG. 1. A) Schematic top view of the motor, showing FliG [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Microscopic Gillespie simulations of the full FliG [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Tentative evidence for GMC. Measured CW bias [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: FIG. 5. GMC eases speed-sensitivity trade-off of equilib [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.