Synchronization of two bacterial flagella as a stochastic process

Jing Qin; Nariya Uchida

REVIEW 4 major objections 4 minor 41 references

Synchronization of two bacterial flagella as a stochastic process

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper derives explicit formulas for the phase difference and heat dissipation along individual trajectories of two synchronizing flagella under white noise.

desk verdict Genuinely new single-trajectory heat formula for a two-rotor flagellar model, but the printed derivation has sign and period errors that must be fixed before the central claim is verifiable. read the letter →

arxiv 2411.18103 v4 pith:5SYDH2DZ submitted 2024-11-27 nlin.AO physics.bio-ph

Jing Qin , Nariya Uchida This is my paper · ORCID

classification nlin.AOphysics.bio-ph MSC 60H1082C31

keywords flagellarsynchronizationstochasticthermodynamicsphaseoscillatorsheatdissipationLangevinequationOrnstein-Uhlenbeckprocesshydrodynamiccouplingsingle-trajectoryenergetics

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

This paper asks whether the energetics of synchronizing flagella can be described trajectory by trajectory, without averaging over many noise realizations. For two hydrodynamically coupled rotors driven by white noise, the authors derive explicit stochastic-process formulas for the phase difference and the total heat dissipation under two approximations: the slow-variable approximation and the small-phase-difference approximation. They show that the phase difference is an Ornstein-Uhlenbeck process whose variance approaches $2D/(K\cos\beta)$, and that heat dissipation contains a linear drift, a periodic term, and a Brownian term. A sympathetic reader would care because it makes single-trajectory heat fluctuations analytically tractable in a biologically motivated setting, connecting synchronization to stochastic thermodynamics.

What carries the argument

The central object is the pair of stochastic phase equations for two rotors, analyzed through the slow variable $\Phi_i=\varphi_i-\omega_\varphi t$: separating the fast intrinsic rotation leaves a slow, noise-driven phase dynamics. The second load-bearing element is the small-phase-difference linearization $\sin\delta\approx\delta$, which turns the phase-difference equation into an exactly solvable Ornstein-Uhlenbeck process; the same linearization, inserted into the Stratonovich definition of heat, yields the closed-form formulas for phase and heat. The slow-variable approximation is what allows the rapidly oscillating coupling term to be averaged away.

What would settle it

Track two bead-rotor flagella (or simulate the full Langevin equations) with parameters such that $K\cos\beta$ is comparable to or smaller than $D$, for example $\beta$ near $\pi/2$ with weak coupling, and measure the single-trajectory heat variance at long times: if it grows approximately as $t^3$ rather than linearly, the closed-form formula for $Q'(t)$ is not a valid description in that regime.

Watch

Extended reading notes

Core claim

Starting from a Langevin description of two bead rotors modeling tethered flagella coupled by Blake-tensor hydrodynamic interactions, the paper claims the following explicit solution. When $2D/(K\cos\beta)\ll 1$ and the slow variable approximation holds, the phase difference $\delta(t)=\varphi_1-\varphi_2$ is a zero-mean Ornstein-Uhlenbeck process with variance $V(t)=\frac{2D}{K\cos\beta}\left(1-e^{-2K\omega\cos\beta\, t}\right)$; the individual phases are $\varphi_i(t)=\omega_\varphi\left(1+\frac{K}{2}\right)t+\frac{K\omega}{4\omega_\varphi}[\cos(2\omega_\varphi t+\beta)-\cos\beta]+\sqrt{2D\omega}\,W_i(t)$ up to first order in $K$ and $\sqrt{D}$; and the normalized total heat is $Q'(t)=2\omega_\varphi^2(1+K)t-2K\omega\sin(\omega_\varphi t+\beta)\sin(\omega_\varphi t)+2\omega_\varphi\sqrt{D\omega}\,W_t$. These formulas are validated by numerical simulations when both approximations hold; for $\beta=0.49\pi$, where the small-phase-difference assumption fails at long times, the heat variance grows as $t^3$ rather than linearly.

Load-bearing premise

The load-bearing premise is that the phase difference between the two rotors stays small enough that $\sin\delta$ can be replaced by $\delta$; if noise is strong relative to the synchronization tendency, this fails and the explicit phase and heat formulas no longer describe the late-time behavior.

Editorial extensions

If this is right

For parameters satisfying both approximations, single-trajectory heat can be characterized without ensemble averaging; the variance of heat grows linearly in time in the regime where the formulas apply.
The steady-state phase-difference distribution is a von Mises distribution, so the tendency to synchronize is controlled by the ratio $K/(2D\cos\beta)$, and no synchronization tendency exists at $\beta=\pi/2$.
The explicit phase formula gives a parameter-free prediction for bead-assay trajectories, allowing the work done by a bead on the surrounding fluid to be compared with measured time series.
Because the model is of a generic form, the same approximation scheme can be carried over to other mesoscale synchronizing systems driven by white noise, and extended to additive active noise.

Reading between the lines

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

Implicit in the results is that the $t^3$ growth of heat variance for $\beta$ near $\pi/2$ is a regime marker: when phase diffusion is no longer confined by coupling, heat fluctuations acquire superdiffusive character, so the linear variance formula should be read as conditional on synchronization strength.
One could test the method's scope by measuring, in the same setup, the distribution of single-trajectory heat at fixed time; the theory predicts a Gaussian heat fluctuation driven by the Brownian term, whereas beyond the small-angle regime the distribution should broaden and skew.
The slow-variable scheme suggests a direct extension to $N$ rotors: each pair contributes an Ornstein-Uhlenbeck phase difference, and total heat would be a sum of independent such contributions only in the weak-coupling limit; otherwise pair correlations enter.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The new thing here is Eq. (26), a closed-form single-trajectory expression for heat dissipation in a two-rotor flagellar model, plus the Ornstein-Uhlenbeck phase-difference process Eq. (20). That goes beyond the ensemble-averaged energetics in Refs. [27,28] and is a legitimate extension. The steady-state distribution is a standard von Mises result and the model itself comes from Ref. [34], so the novelty is concentrated in the trajectory-level formulas.

The paper does a fair job of checking itself. The derivation is parameter-free: K, D, and beta are set by the earlier model, with no fitting to the simulation. That is a real strength. The Euler-Maruyama checks in the valid regime (beta = 0.25 pi) match the corrected formulas, and the authors honestly show the failure at beta = 0.49 pi, where the small-phase approximation breaks down and the heat variance grows as t^3 rather than linearly. That honesty counts.

But the derivation as printed is not reproducible. The stress-test note is right: Eq. (23) has the wrong sign. Integrating what is printed gives Sigma(t) with -K omega_phi t and the opposite periodic sign, not Eq. (24). Equations (25) and (26) are built on Eq. (24), so a reader cannot verify the central claim without silently fixing the sign. There is also a period inconsistency in Section III: the fast term is said to average out over the period pi/omega, but the actual fast frequency is 2 omega_phi = 2 omega sin beta, so the stated period does not zero the term. The reader's note about Eq. (10) putting cos beta in the wrong place and Eq. (22)/Eq. (23) notation inconsistencies is also accurate. These look like fixable typos rather than conceptual errors, but as printed they block verification.

The title overstates scope. This is a specific simplified rotor model with white noise and no experimental data; the validity window is narrow, and the paper itself demonstrates the small-phase approximation failing at long times for beta close to pi/2. The Discussion is appropriately cautious about the 'slow variable and strong coupling approximations,' but the title and abstract suggest broader applicability than the model delivers.

Who is this for? Stochastic thermodynamics practitioners and bead-assay experimentalists who want single-trajectory predictions. They will find Eq. (26) useful, and the paper is written accessibly. The citation pattern is fine; leaning on Ref. [34] is legitimate self-citation because the model comes from there.

I would send this to a serious referee. The core result is checkable and probably correct, and the errors are correctable. The referee should verify the sign corrections and ask the authors to state the approximation windows explicitly. With those fixes, this is a solid extension worth publishing.

Referee Report

4 major / 4 minor

Summary. The paper studies two hydrodynamically coupled bacterial flagella modeled as active rotors driven by white noise. Using slow-variable and small-phase-difference approximations, it derives approximate stochastic-process descriptions: the phase difference becomes an Ornstein-Uhlenbeck process, and the individual phases and total heat dissipation are expressed as explicit functions of time and Wiener processes. The analytical formulas are compared with Euler-Maruyama simulations for three values of the force-tilt angle beta. The central claim is that single-trajectory heat fluctuations can be characterized without ensemble averaging.

Significance. If the derivation is corrected, the paper provides an analytically tractable stochastic-thermodynamic description of synchronization in a concrete flagellar model, going beyond ensemble-averaged quantities. The formulas are parameter-free in the sense that K, D, and beta are inputs from the model, and the numerical verification for beta = 0.25 pi is a genuine test. The method is model-specific but potentially transferable to other mesoscale synchronizing systems. The main weakness is that several printed equations are internally inconsistent as written, so the central derivation is not reproducible without silent corrections.

major comments (4)

[Section II, Eqs. (1) and (3)] Equations (1) and (3) are inconsistent as printed. With the stated definitions F_phi = -F sin(beta) and F_r = F cos(beta), the hydrodynamic term in Eq. (1), after division by zeta b, becomes -K omega cos(phi_2 - beta) sin(phi_1), whereas Eq. (3) contains -K omega cos(phi_2 + beta) sin(phi_1). Since Eq. (3) is the starting point for all subsequent analytic and numerical work, Eq. (1) must be corrected or the sign convention for F_r must be stated consistently.
[Section III, Eq. (7)] The fast oscillatory term in Eq. (7) is printed as sin(Phi_i + Phi_j + 2 omega t + beta), and the text states that it averages out over the period pi/omega. From the definition Phi_i = phi_i - omega_phi t, the fast phase is actually 2 omega_phi t, so the averaging period should be pi/omega_phi, not pi/omega (unless sin(beta)=1). As printed, the averaging step is not justified and should be corrected; the final stationary distribution may survive the correction, but the derivation must be internally consistent.
[Section III, Eqs. (10)-(12)] The stationary solution of Eq. (9) is P proportional to exp[(K cos(beta)/(2D)) cos(Phi_2 - Phi_1)], with normalization 16 pi^2 I0(K cos(beta)/(2D)). The printed argument K/(2D cos(beta)) has the cosine in the wrong place and is inconsistent with the variance 2D/(K cos(beta)) quoted later in the same section. This must be corrected, as it also affects the stochastic entropy expression in Eq. (13).
[Section IV, Eqs. (23)-(24)] Equation (23) is printed with the wrong signs relative to the correct reduction of Eq. (22) to the sum variable. Using sigma = Sigma + 2 omega_phi t, the correct equation is dSigma/dt = -K omega sin(Sigma + beta + 2 omega_phi t) + K omega sin(beta) cos(delta) + 2 sqrt(D omega) xi_+, not the printed +K omega sin(...) - K omega cos(delta) sin(beta). Direct integration of the printed equation would produce the opposite signs in Eq. (24), and since Eqs. (25) and (26) are built on Eq. (24), the central formulas cannot be derived from the printed text without a silent sign correction. Please fix the sign and also clarify the K omega_phi notation in Eq. (22).

minor comments (4)

[Section IV and Fig. 3] The sign convention for delta is inconsistent: the text before Eq. (19) defines delta = phi_1 - phi_2, while the caption of Fig. 3 defines V_delta = <delta^2> = <(phi_2 - phi_1)^2>. Please make the convention uniform, since the sign matters for expressions such as Eq. (19).
[Section IV, Eq. (22)] The term printed as 'K omega phi cos delta' is ambiguous. If it means K omega_phi cos(delta), then because omega_phi = omega sin(beta) it equals K omega sin(beta) cos(delta); please write this explicitly to avoid dimensional confusion.
[Section V] Only the beta = 0.25 pi case provides a full quantitative test of Eqs. (20) and (26); for beta = 0.01 pi the slow-variable approximation is not valid, and for beta = 0.49 pi the small-phase-difference approximation breaks down at long times. The paper should state this limitation explicitly rather than implying uniform numerical verification across the three cases.
[General] There are several typographical and wording issues: 'under the the slow variable' in the Discussion, 'the coefficient before that is also from the addition of two noise' near Eq. (25), and the term 'dimensionless heat' for Q' = Q/(zeta b^2), which is not dimensionless as written. These should be cleaned up.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the analytical formulas are derived from the stated Langevin model, not fitted to the simulation; self-citations supply the model, not the target results.

full rationale

The paper's derivation chain is self-contained once the model equations (3)-(4) are adopted from Ref. [34]. The steady-state distribution, the Ornstein-Uhlenbeck approximation for the phase difference, and the heat formula (26) are obtained by explicit calculation from those SDEs and from the standard stochastic-thermodynamics definition of heat. No parameter is fitted to the quantities being predicted: K, D, and beta are fixed model inputs, and the numerical simulations integrate the same equations as a consistency check rather than as a source of fitted parameters. The self-citation to Ref. [34] supplies the starting model, not the new results, and it is an externally published physical model rather than an unverified uniqueness claim. The paper itself flags the breakdown of its small-phase-difference approximation for beta = 0.49pi in Fig. 4 and the Discussion, which is a limitation statement rather than circular reasoning. The sign inconsistency in Eq. (23), which prevents a reader from deriving Eqs. (24)-(26) from that printed equation alone, is a correctness/reproducibility defect and not a circularity; the intended derivation can be recovered from Eq. (22). No load-bearing step reduces to its own input by construction.

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

No free parameters are fitted to data; the dimensionless parameters K, D, and beta come from the physical setup. The central results rest on the far-field Blake tensor, white-noise Langevin dynamics, and two approximation regimes (slow variables and small phase difference), all of which are stated in the paper but not independently justified.

assumptions (6)

domain assumption The two rotors are driven by constant tangential and radial force components F_r and F_phi, with F_phi < 0, following Ref. [34].
This force specification is taken from the earlier flagellar model and is not derived here; it sets the sign and form of the coupling in Eqs. (1) to (4).
domain assumption Hydrodynamic interaction is approximated by the far-field Blake tensor G approximately (3h^2/(2 pi eta d^3)) e_x e_x for distances d much larger than h, b, a.
Used in Section II to write the Langevin equations; it ignores near-field and other tensor components.
domain assumption Thermal fluctuations are additive white Gaussian noise with intensity sqrt(2D omega), and the heat is defined with Stratonovich integration.
Standard stochastic thermodynamics setup from Refs. [36,37]; the choice of Stratonovich product is essential for the heat formulas in Section IV.
ad hoc to paper Slow-variable approximation: Phi_i stay approximately constant over the fast period pi/omega, so the oscillatory term in Eq. (7) averages to zero.
Used to obtain the stationary Fokker-Planck equation (9); valid only when sin beta is much larger than K and sqrt(D). The paper states this in Sections III and V.
ad hoc to paper Small-phase-difference approximation: sin delta is replaced by delta in Eq. (19), valid when 2D/(K cos beta) is much smaller than 1.
This linearization is needed for the explicit solutions (20), (25), and (26); the paper notes in Section V that it breaks down for beta close to pi/2 at long times.
standard math The two noise processes xi_1 and xi_2 are independent standard white noises; the transformed noises xi_plus and xi_minus are independent Wiener processes.
Required for the variance and heat-noise calculations in Section IV.

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Cite this review

Pith. "Pith review of Synchronization of two bacterial flagella as a stochastic process." pith.science (2026). https://pith.science/paper/5SYDH2DZ

@misc{pith2026241118103,
  author       = {Pith},
  title        = {Pith review of: Synchronization of two bacterial flagella as a stochastic process},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5SYDH2DZ}},
  note         = {Machine review of arXiv:2411.18103}
}

read the original abstract

Synchronization with noise is important for understanding biophysical processes at nano- and micro-meter scales, such as neuronal firing and flagellar rotations. To understand the energetics of these processes, stochastic thermodynamics approaches are useful. Due to large fluctuations in a small system, ensemble averages of thermodynamic quantities are not sufficient to characterize the energetics of an individual sample. In this paper, we use a model for synchronization of bacterial flagella as an example, and develop an approximation method for analyzing the phase and heat dissipation in trajectories for different noise realizations. We describe the {temporal evolution} of the phase difference and heat dissipation as stochastic processes, and verify the analytical results by numerical simulations.

Figures

Figures reproduced from arXiv: 2411.18103 by the authors.

**Figure 2.** FIG. 2. (Color online) The case [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. (Color online) Variation of (a) the phase difference [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. (Color online) Variation of the dimensionless heat [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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(20) Using this, we ﬁnd ⟨δ⟩ = ⟨φ 1 − φ 2⟩ = 0 since the product is taken in Itˆ o’s sense

Assuming that the phase dif- ference is small, we approximate sin δ by δ and solve the equation as δ(t) = 2 √ Dωe −Kω cos β ·t ∫ t 0 eKω cos β ·sdWs = 2 √ Dω ( Wt − Kω cos β ∫ t 0 eKω cos β ·(s−t)Wsds ) . (20) Using this, we ﬁnd ⟨δ⟩ = ⟨φ 1 − φ 2⟩ = 0 since the product is taken in Itˆ o’s sense. The variance reads V (t) = ⟨δ(t)2⟩ = 4Dωe −2Kω cos β ·t ∫ t 0...

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The results are in good agreement with our theoretical results; Vδ increases exponentially with time as in Eq.( 20), while VQ′ linearly increases with the slope 4 ω 2 φ Dω = 0

We choose the ending the time as 100s to see both initial and steady behavior. The results are in good agreement with our theoretical results; Vδ increases exponentially with time as in Eq.( 20), while VQ′ linearly increases with the slope 4 ω 2 φ Dω = 0. 2 as in Eq.( 26). For β = 0 . 49π , the slow variable approximation holds but the small phase diﬀeren...

work page
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From the ﬁgure, we 5 FIG. 3. (Color online) Variation of (a) the phase diﬀerence δ and (b) the dimensionless heat Q′ versus time for β = 0. 25π . The solid lines show the numerical results, and the dashed lines show the theoretical results [Eqs.(20),(26)] FIG. 4. (Color online) Variation of the dimensionless heat for β = 0. 49π . The solid line shows the ...

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [34]

Silverman and M

M. Silverman and M. Simon: Nature 249 (1974) 73

work page 1974

[2] [1]

Pikovsky, M

A. Pikovsky, M. Rosenblum, and J. Kurths: Synchroniza- tion: A Universal Concept in Nonlinear Sciences (Cam- bridge University Press, 2001)

work page 2001

[3] [2]

(20) Using this, we ﬁnd ⟨δ⟩ = ⟨φ 1 − φ 2⟩ = 0 since the product is taken in Itˆ o’s sense

Assuming that the phase dif- ference is small, we approximate sin δ by δ and solve the equation as δ(t) = 2 √ Dωe −Kω cos β ·t ∫ t 0 eKω cos β ·sdWs = 2 √ Dω ( Wt − Kω cos β ∫ t 0 eKω cos β ·(s−t)Wsds ) . (20) Using this, we ﬁnd ⟨δ⟩ = ⟨φ 1 − φ 2⟩ = 0 since the product is taken in Itˆ o’s sense. The variance reads V (t) = ⟨δ(t)2⟩ = 4Dωe −2Kω cos β ·t ∫ t 0...

work page

[4] [3]

(23) From this and Eq.( 19), the change of Φ i is proportional to the dimensionless parameters K and √ D

Using Φ i = φ i − ω φ t and Σ = Φ 1 + Φ 2 it can be rewritten as dΣ dt =Kω sin(Σ + β + 2ω φ t) − Kω cos δ sin β + 2 √ Dωξ +. (23) From this and Eq.( 19), the change of Φ i is proportional to the dimensionless parameters K and √ D. that are in the slow variable approximation where sin β ≫ K, √ D. We obtain Σ to the ﬁrst order in K and √ D as Σ( t) = Kω φ t...

work page

[5] [4]

The results are in good agreement with our theoretical results; Vδ increases exponentially with time as in Eq.( 20), while VQ′ linearly increases with the slope 4 ω 2 φ Dω = 0

We choose the ending the time as 100s to see both initial and steady behavior. The results are in good agreement with our theoretical results; Vδ increases exponentially with time as in Eq.( 20), while VQ′ linearly increases with the slope 4 ω 2 φ Dω = 0. 2 as in Eq.( 26). For β = 0 . 49π , the slow variable approximation holds but the small phase diﬀeren...

work page

[6] [5]

From the ﬁgure, we 5 FIG. 3. (Color online) Variation of (a) the phase diﬀerence δ and (b) the dimensionless heat Q′ versus time for β = 0. 25π . The solid lines show the numerical results, and the dashed lines show the theoretical results [Eqs.(20),(26)] FIG. 4. (Color online) Variation of the dimensionless heat for β = 0. 49π . The solid line shows the ...

work page

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Strogatz: Sync: The Emerging Science of Spontaneous 6 Order (Penguin UK, 2004)

S. Strogatz: Sync: The Emerging Science of Spontaneous 6 Order (Penguin UK, 2004)

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S. M. Reppert and D. R. Weaver: Nature 418 (2002) 935

work page 2002