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Using stochastic thermodynamics with internal variables to capture orientational spreading in cell populations undergoing cyclic stretch

T0 review · 2 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Cells that start sharply aligned with an energy maximum first spread out and then refocus at the energy minimum, according to a two-variable reduction of the Fokker-Planck model for cyclic-stretch reorientation.

desk verdict A von-Mises reduction of the cell reorientation Fokker-Planck equation that is worth reading, but the proposed switch experiment starts on an invariant line where the reduced model does not produce the two-stage prediction. read the letter →

arxiv 2507.15694 v1 pith:PI5XCUQG submitted 2025-07-21 cond-mat.stat-mech cond-mat.softmath-phmath.MPmath.PR

Rohan Abeyaratne , Sanjay Dharmaravan , Giuseppe Saccomandi , Giuseppe Tomassetti This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.softmath-phmath.MPmath.PR MSC 60H1035Q9292C10
keywords cellreorientationcyclicstretchFokker-PlanckequationvonMisesdistributionstochasticthermodynamicswithinternalvariablesorderparametertwo-stageorientationspreading
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

Populations of cells on a cyclically stretched substrate need not reorient as a coherent block. This paper claims that when the population starts sharply aligned with a direction of maximum energy, it first broadens into a more disordered spread, rapidly moves its mean orientation to the energy minimum, and only then sharpens again around that minimum. The claim comes from reducing the Fokker-Planck equation for orientation to a two-dimensional autonomous system for a mean orientation and an order parameter, using the stochastic-thermodynamics closure with internal variables of [15] and a pi-periodic von Mises ansatz. The reduced system reproduces the full equation's steady-state spreading and its transient evolution, and it matches experimental transient curves once the characteristic cell time is taken to depend on strain amplitude. If the prediction is right, a simple switch experiment should show a brief broadening of the orientation histogram immediately after the stretch direction is changed by 90 degrees.

What carries the argument

The load-bearing object is the $\pi$-periodic rescaled von Mises family $\tilde p(\theta|\mu,\kappa)$, the circular analogue of a Gaussian wrapped onto $(0,\pi)$, with internal variables $\mu$ (mean orientation) and $\kappa$ (order parameter). The projection machinery of [15] turns the Fokker-Planck operator into evolution equations for these internal variables; because the cross-metric term $\langle \partial \tilde s/\partial \mu, \partial \tilde s/\partial \kappa\rangle_{\tilde p}$ vanishes, the resulting system (13) is diagonal and autonomous, with drift and diffusion coefficients expressed through modified Bessel functions. Phase-plane analysis of this two-dimensional system is what reveals the non-monotone $\kappa(t)$ path underlying two-stage reorientation.

What would settle it

Run the switch experiment described in Remark 8: stretch cells cyclically along one axis until they reach steady alignment, then rotate the stretch axis by 90 degrees and follow the orientation histogram. Two-stage reorientation predicts an immediate transient broadening, meaning the order parameter must dip before recovering; its absence would falsify the central claim. Alternatively, repeat the protocol of [21] at decreasing stretch frequencies: if the spreading of orientations keeps increasing as frequency drops, the period-averaged constant-noise assumption fails, as the paper concedes.

Watch

Extended reading notes

Core claim

The central claim is that the full Fokker-Planck dynamics (6) can be faithfully compressed into the two-dimensional ODE system (13) for the mean orientation $\mu(t)$ and the order parameter $\kappa(t)$ of a $\pi$-periodic von Mises distribution. On that reduced landscape, a population that begins narrowly peaked near an energy maximum does not move directly toward the energy minimum. Instead, in the first stage the distribution broadens, so $\kappa(t)$ decreases while $\mu(t)$ rapidly converges to the minimum at $\pi/2$; in the second stage $\kappa(t)$ grows again as the distribution refocuses around that minimum. This two-stage reorientation is a property of the reduced system and, as the numerical Kullback-Leibler projection tests show, also of the original Fokker-Planck equation. The same reduced model yields a unique stationary concentration $\kappa^*$ that decreases with $\tau/\varepsilon$, matching the empirical rule that larger strains give sharper orientation peaks, and with a strain-dependent characteristic time $\eta/K$ it reproduces the transient order-parameter curves measured in [21].

Load-bearing premise

The prediction rests on cells sensing only the period-averaged strain, with a constant angular noise level, because the reorientation time is assumed to be much longer than the stretch period; the paper itself notes that frequency-dependent spreading observed in [21] lies outside this picture.

Editorial extensions

If this is right

  • Steady-state spreading is tied to the ratio $\tau/\varepsilon$: lower effective strain gives a smaller stationary order parameter $\kappa^*$, so orientation histograms are broader, as observed in experiments.
  • Switching the stretch axis should produce a measurable transient broadening of the orientation distribution before cells refocus along the new minimum; Remark 8 turns this into a testable protocol.
  • The reduced ODE model can be used in place of the full Fokker-Planck equation for qualitative phase-plane studies, including asymptotic estimates of the equilibrium concentration.
  • Matching the transient data of [21] requires the characteristic time $\eta/K$ to increase with strain amplitude, indicating that constant-viscosity descriptions relax too quickly at large strains.

Reading between the lines

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

  • Going beyond the paper: if the switch experiment confirms transient broadening, tissue-engineering protocols that alternate stretch directions may need to budget for a temporary loss of alignment after each switch.
  • Going beyond the paper: the finding that $\eta/K$ must grow with strain amplitude suggests a concrete test, namely measuring reorientation completion time versus strain amplitude and checking whether it increases, as a strain-dependent viscosity would predict.
  • Going beyond the paper: the same internal-variable closure could be applied to a bimodal von Mises ansatz for two-minimum energies; even if closed-form coefficients are unavailable, a numerical projection would show whether two-stage reorientation persists when two peaks compete.
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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

2 major / 6 minor

Summary. The paper revisits the Loy–Preziosi Fokker–Planck model for cell orientation under cyclic stretch and reformulates it using the stochastic-thermodynamics-with-internal-variables framework of Leadbetter et al. The orientation distribution is projected onto a π-periodic von Mises family, yielding a two-dimensional ODE system (13)/(50) for the mean orientation μ(t) and concentration κ(t). The authors compute the coefficients in closed form in terms of modified Bessel functions, perform a phase-plane analysis for single-well energies, identify a two-stage reorientation scenario for initial conditions near an energy maximum, compare the reduced ODE with numerical solutions of the full Fokker–Planck equation and with the experimental data of Mao et al., and propose a switch experiment in Remark 8 as a falsifiable test.

Significance. If the central claims hold, the paper provides a tractable low-dimensional reduction of a Fokker–Planck equation with explicit coefficients, a transparent phase-plane analysis, and a falsifiable transient prediction. The derivation from the Leadbetter et al. scheme is structurally sound, the closed-form coefficients (33)–(36) are a genuine technical asset, and the two-stage reorientation is independently observed in the full Fokker–Planck numerics for off-maximum initial conditions (Fig. 11b). The authors are also candid about limitations: the constant-viscosity assumption, the frequency-dependence of spreading, and the interpretive nature of the numerical comparison in Remark 11. However, the proposed switch experiment is not actually reproduced by the reduced ODE for the protocol as stated, and the uniform-initial-condition numerical test contains an internal inconsistency, so the manuscript needs substantive revision before the central predictive claim can be accepted.

major comments (2)
  1. [Remark 8 and §4 (Eqs. (50), (66), (78))] The proposed switch experiment starts from the old steady state, whose symmetry places the initial condition exactly on the invariant line μ = π/2. Since f(π/2, κ) = 0 for all κ by Eq. (66), the reduced ODE (50) keeps μ(t) = π/2 for all t, and on this line Eq. (78) gives κ̇ < 0 for large κ, followed by approach to the fixed point (π/2, κ*) located at the energy maximum. Thus the ODE never produces the announced 'brief broadening then refocusing around the new minimum' of Remark 8; the two-stage trajectory in Fig. 6 uses μ0/π = 2·10⁻², not the actual post-switch state. The full Fokker–Planck solution preserves symmetry and its KL projection jumps from μ = π/2 to μ = 0 with κ passing through 0, a scenario that the unimodal reduced ODE cannot represent. This breaks the link between the reduced model and the proposed experiment. The authors should either revise the protocol to include a characterized symmetry-breaking perturbation, or analyze the KL projection of the symmetric full solution and explain how the reduced system handles the jump.
  2. [§5, uniform initial condition (Eqs. (42), (66), (78))] The text states that for a uniform initial condition 'we have chosen μ0 = 0', but Fig. 9 shows μ(t) instantaneously converging to π/2 and κ(t) increasing to κ*. Both features are impossible with μ0 = 0: by Eq. (66) f(0, κ) = 0, so μ(t) = 0 for all t, and by Eq. (78) κ̇ < 0 on μ = 0. The reported numerical test is therefore internally inconsistent as written. The authors should correct the stated initial condition (probably μ0 = π/2) and discuss the degeneracy of the KL projection at κ = 0, where μ is undetermined.
minor comments (6)
  1. [§7, first sentence] The phrase 'an order parameter ω(t)' should read κ(t); the symbol ω is not defined in the paper.
  2. [§6, Eq. (59) and surrounding text] The order parameter S(t) is defined as ⟨cos 2θ⟩, but two sentences state 'expected value of cos θ'; these should refer to cos 2θ.
  3. [§5, after Eq. (57)] The sentence 'These are the same parameters that defermine κ∗' contains a typo ('defermine' should be 'determine'), and the same symbol κ∗ is used for the KL-optimal stationary value and for the ODE fixed point; these should be distinguished.
  4. [§2.1 and §4] There are wording slips: 'spreading is more pronounced for small small strains' has a duplicated word, and 'the mean concentration µ(t) evolves monotonically' in the two-stage reorientation paragraph should be 'the mean orientation µ(t)'.
  5. [§5, numerical validation] The comparison in Figs. 9–11 shows agreement between the ODE parameters and the KL-optimal parameters of the full solution, but it does not quantify the actual error between p(θ,t) and p̃(θ|μ(t),κ(t)); reporting a direct distance (e.g., L¹ or KL divergence) would strengthen the validation.
  6. [Remark 9 and §5] Minor textual errors: 'is suffices' should be 'it suffices' in Remark 9, and 'ration τ/ε' should be 'ratio τ/ε' in Section 5.

Circularity Check

1 steps flagged · score 2.0 of 10

The reduction is self-contained and the two-stage prediction is not fitted, but the numerical validation compares the ODE against the same KL-optimal projection used to build the ODE, a mild self-referential check.

  1. self definitional [Section 5 (Numerical tests), eq. (58) and Remark 11]
    "At each time t, we define μ̄(t) and κ̄(t) by solving the minimization problem (μ̄(t), κ̄(t)) = argmin_{μ,κ} D_KL(p(·,t)∥p̃(·;μ,κ)) ... Their agreement with μ(t) and κ(t) serves as our metric for assessing the quality of our approximation. ... In exact terms, our test does not aim to quantify the error between the true solution p(θ,t) and the approximation p̃(θ|μ(t),κ(t)). Instead, it seeks to demonstrate that the solution (μ(t),κ(t)) of the dynamical system (13) closely matches the optimal parameters μ̄(t) and κ̄(t) obtained via an information-theoretic criterion."

    The reduced ODE (13) is derived by applying the Leadbetter-Purohit-Reina variational closure to the Fokker-Planck equation using the same information-theoretic (KL) metric that defines the optimal approximating parameters μ̄,κ̄. The numerical test then evaluates (13) by comparing it to those same KL-optimal parameters of the exact solution. Thus the agreement is a consistency check with the closure's own construction criterion: if the von Mises family were exactly invariant under the FP flow, the ODE trajectory and the KL projection would coincide by definition. The paper's Remark 11 explicitly narrows the test to matching the information-theoretic criterion, acknowledging that it does not measure error to the true solution.

full rationale

The central derivation of the reduced ODE system (13) is a self-contained projection of the Fokker-Planck equation (6) onto the von Mises family, with coefficients computed in closed form from Bessel functions; the fixed-point equation (55) is derived, not fitted. The two-stage reorientation prediction is a genuine phase-plane consequence of (13), and the paper independently verifies (in Fig. 11b) that the same phenomenon appears in the full Fokker-Planck solution's optimal projection, so the main claim does not reduce to an input. The experimental comparison in Section 6 is a transparent parameter fit (τ, k, η/K tuned to Mao et al. data), not a prediction from the model, so it is not a circular step. The only noteworthy circularity is the numerical validation of Section 5: the ODE is constructed from the KL/information-theoretic projection, and the test judges it by comparison to the KL-optimal projection, so part of the agreement is by construction. This is acknowledged in Remark 11 and does not undermine the independent content of the phase-plane analysis. A separate correctness concern, not circularity, arises in Remark 8's proposed switch experiment: the post-switch steady state lies exactly on the invariant line μ=π/2 of the reduced ODE (f(π/2,κ)=0), so the reduced ODE predicts only monotonic broadening from that exact initial condition, whereas the two-stage prediction requires an initial condition slightly off the maximum; the full Fokker-Planck solution, by contrast, preserves symmetry and the optimal μ jumps, so the experimental protocol as stated may not test the reduced model's two-stage prediction without a symmetry-breaking perturbation. This does not affect the circularity score.

Assumptions & free parameters 4 free parameters · 6 assumptions · 2 invented entities

The central reduction sits on: two inherited domain assumptions about the cell noise and time-averaged strain response; the von Mises closure ansatz introduced here; the external variational framework of [15]; the single-well energy restriction; and standard Bessel identities. The experimental section introduces five fitted numbers (τ, k, and three viscosity values) in addition to the scenario initial conditions used to illustrate the prediction. No new physical entities are introduced except a strain-dependent effective viscosity, which is inferred from the fit and has no independent falsifiable handle.

free parameters (4)
  • τ (pseudo-temperature, τ²=ηD/K) = 0.04 (Eq. (64)); τ=0.1 in Section 5 numerics
    Fixed by matching the stationary value of S(t) in the Mao et al. data; controls the equilibrium spreading.
  • k (energy landscape parameter) = 2 (Eq. (64)); k=1.0 in Section 5 via Ks=0.7, K⊥=0.28, r=0.11
    Fixed by matching the stationary S(t); in principle a function of material constants and biaxiality (19).
  • η/K (characteristic time, fitted per strain amplitude) = 2.0, 3.5, 30 for εmax=0.02, 0.05, 0.1
    Three separate values needed to match the three transient S(t) curves; the factor-of-15 variation is interpreted as a strain-dependent effective viscosity, a post-hoc model modification.
  • initial (μ0, κ0) in the two-stage illustrations = e.g., μ0=0.02π, κ0=5 (Fig. 6); μ0=π/100, κ0∈{1,10,50} (Fig. 11); uniform case κ0=0, μ0 arbitrary (Remark 7)
    Scenario parameters chosen to display the predicted two-stage behavior and to test the closure; they are not fit to data.
assumptions (6)
  • domain assumption The stochastic dynamics is a Langevin equation dθ=Ω(θ)dt+√(2D)dW on [0,π) with constant angular diffusivity D and the Loy-Preziosi energy (18).
    Section 2.1, Eqs. (5)-(6); the entire reduction approximates this specific Fokker-Planck equation. If the true noise is multiplicative or strain-dependent, the reduced dynamics inherit the error.
  • domain assumption Cells respond to the period-averaged strain ε², i.e., the reorientation time η/(ε²K) is much larger than the period 1/ω.
    Section 2.1, paragraph after Eq. (17); justifies replacing εxx(t)² by ⟨εxx²⟩. The authors flag (Section 7) that observed frequency dependence of spreading is outside the model.
  • ad hoc to paper The solution p(θ,t) of the Fokker-Planck equation is well represented at all times by a π-periodic von Mises family p̃(θ|μ(t),κ(t)).
    Section 3.1, Eq. (29); this closure ansatz is the load-bearing modeling choice. Its adequacy is tested in Section 5, with sup-norm errors admitted as possibly not negligible (Fig. 8).
  • domain assumption The Leadbetter-Purohit-Reina variational principle (Eq. (26) from [15]) gives the best evolution of the internal variables (μ,κ).
    Section 2.2; the ODE system (13) is derived from [15, Eq. 7]. The paper relies on this framework's optimality claim.
  • ad hoc to paper The analysis is restricted to single-well energies, k≥1 and ¯k=1 (case (c) of the energy landscape).
    Section 4, Eq. (51); excludes the two-minima case (a) relevant for near-equibiaxial stretch, which the authors defer to future work (Section 7).
  • standard math Standard asymptotic and monotonicity properties of the modified Bessel functions I0, I1, I2.
    Appendix, Eqs. (69)-(70); used for the κ→0 and κ→∞ limits and for proving uniqueness of κ*.
invented entities (2)
  • Strain-dependent effective viscosity η(εmax)
    purpose: Reconciles the model's transient time scale with Mao et al. data at three strain amplitudes; Section 7 proposes η increases with strain amplitude.
    No functional form and no prediction at other amplitudes; it is a post-hoc interpretation of the per-curve fits.
  • Two-stage reorientation phenomenon independent evidence
    purpose: Predicted transient broadening when a narrowly aligned population starts at an energy maximum; motivates the switch experiment of Remark 8.
    The switch experiment gives a falsifiable handle outside the fitted data: if no brief broadening is observed after changing the stretch axis, the prediction fails. The phenomenon is also present in the full Fokker-Planck equation (Fig. 11b), so it is not an artifact of the ansatz.

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Pith. "Pith review of Using stochastic thermodynamics with internal variables to capture orientational spreading in cell populations undergoing cyclic stretch." pith.science (2026). https://pith.science/paper/PI5XCUQG

@misc{pith2026250715694,
  author       = {Pith},
  title        = {Pith review of: Using stochastic thermodynamics with internal variables to capture orientational spreading in cell populations undergoing cyclic stretch},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PI5XCUQG}},
  note         = {Machine review of arXiv:2507.15694}
}
read the original abstract

We revisit the modeling framework introduced in [N. Loy and L. Preziosi: Bull. Math. Bio., 85, 2023] to describe the dynamics of cell orientation under cyclic stretch. We propose a reformulation based on the principles of Stochastic Thermodynamics with Internal Variables introduced in [T. Leadbetter, P. Purohit, and C. Reina: PNAS Nexus, 2, 2023]. This approach allows us to describe not only the evolution of the orientation distribution, but also the observed spreading phenomenon. The insight provided by our model reveals an interesting phenomenon, which we call two-stage reorientation: when cells begin aligned with an energy maximum, their orientations spread before concentrating at the energy minimum. This theoretical prediction suggests a new experiment to test this modeling framework.

Figures

Figures reproduced from arXiv: 2507.15694 by the authors.

Figure 1
Figure 1. Sketch of the experimental setup. Livne et al. [17] had the intuition that cells reorient themselves to minimize an energy that depends on their orientation [8,17]. According to this point of view, the evolution of the orientation angle θ(t) is governed by the differential equation dθ dt = Ω(θ), (3) where Ω(θ) = − Kε2 η U ′ (θ), (4) 2 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. below illustrates the energy landscape of the function for ¯k = 1 and for several represen￾tative values of k ≥ 0. The energy landscape for k < 0 can be recovered from the profile with [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Graph of the rescaled von Mises distribution for [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (18 more)
Figure 4
Figure 4. Figure 4: Phase portrait and structure of the solution space for the ODE system (50), with param [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: shows the plots of the root κ∗(τ /ε, k) of (55) for different values of k. These plots confirm the monotone dependence of κ∗ on the ratio τ /ε. We note that for k > 1 the slope of the log-log plot is constant For τ /ε ≳ 1 the three curves collapse almost onto one anoth…
Figure 6
Figure 6. Figure 6: Example of two-stage reorientation. Parameters are [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: shows the relative error between κ∗ and κ∗. The agreement improves as k increases, and the smallest relative error is for τ /ε close to 1 [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: Equilibrium distributions: von Mises approximation (thin) vs. exact (thick transparent) [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 9
Figure 9. Figure 9: Comparison between the mean orientations (a) and the order parameters (b) extrapolated [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: Same as Fig. 9, but with initial conditions [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: Same as Fig. 9, but with initial conditions [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]
Figure 12
Figure 12. Figure 12: Plot of the function Sˆ(κ) = −I1 [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]
Figure 13
Figure 13. Figure 13: Time evolution of the order parameter S˜(κ(t)), as predicted by (13) for different values of εmax and η/K. 7 Conclusions We have shown that the full Fokker-Planck description of cell-orientation under cyclic stretch can be approximated by a two-dimensional autonomous …
Figure 14
Figure 14. Figure 14: Log-log plots of F1(κ) and F2(κ). Since we are assuming k2 ≥ 1, and since F2(κ) is smaller than 1, we also have that f(µ, κ) has the same sign as sin 2µ for all κ. As a consequence, f(µ, κ) > 0 for 0 < µ < π/2, f(µ, κ) < 0 for π/2 < κ < π. (68) Therefore, any stable f…
Figure 15
Figure 15. Figure 15: (a) Plots of G(κ), h(κ), and the ratio I2(κ/2) I1(κ/2) . 25 [PITH_FULL_IMAGE:figures/full_fig_p025_15.png]
Figure 16
Figure 16. Figure 16: Properties of b(κ) and c(κ). The functions b(κ) and c(κ), whose graphs are given in [PITH_FULL_IMAGE:figures/full_fig_p027_16.png]
Figure 17
Figure 17. Figure 17: Illustrative plot of Pκ(x) for κ ≪ 1. Behaviour of the roots xα(κ) as κ increases. We now characterize more in detail how the locations of the roots xα(κ) change as κ increases from 0 to ∞. To begin with, we differentiate the equation Pκ(xα(κ)) = 0 with respect to κ t…
Figure 18
Figure 18. Figure 18: Zero-level sets of ˙µ and ˙κ in the case ρ 2 < 1. The root x2(κ) is nonmonotone. Accord￾ingly, the corresponding value µ2(κ) from (94) is nonmonotone as well. Remark 12 (Non-monotone behaviour of κ(t)). We observe that in [PITH_FULL_IMAGE:figures/full_fig_p030_18.png]
Figure 19
Figure 19. Figure 19: Example of two possible configurations when the polynomial Pκ(x) has a root at x = −1. In the first case (blue dashed curve), the root corresponds to the smallest solution x1(κ) and the derivative P ′ κ (−1) is negative. In the second case (red dashed-dotted curve), t…
Figure 20
Figure 20. Figure 20: Zero-level sets of ˙µ and ˙κ in the case P ′ κ∗ (−1) > 0 (dashed blue line in [PITH_FULL_IMAGE:figures/full_fig_p032_20.png]
Figure 21
Figure 21. Figure 21: Curve in the (k2, ρ2 )-plane corresponding to the condition Pκ∗ (−1) = 0. 33 [PITH_FULL_IMAGE:figures/full_fig_p033_21.png]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.