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REVIEW 3 major objections 5 minor 84 references

Landscapes and nonequilibrium fluctuations of eukaryotic gene regulation

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A single ratio of chromatin-switch rate to protein turnover rate shapes gene-regulation landscapes and fluctuations.

desk verdict A useful multi-gene extension of the eddy/landscape model, with a compelling omega-based explanation for Nanog heterogeneity, but the central entropy-production claim rests on an unvalidated continuous approximation of discrete chromatin states. read the letter →

arxiv 2502.10067 v2 pith:R34P57NB submitted 2025-02-14 physics.bio-ph q-bio.MN

Masaki Sasai , Bhaswati Bhattacharyya , Shin Fujishiro , Yoshiaki Horiike This is my paper · ORCID
classification physics.bio-phq-bio.MN
keywords generegulationchromatin-statetransitionsnonequilibriumfluctuationsstochasticexpressionadiabaticityprobabilitycurrententropyproductionpluripotency
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 argues that eukaryotic gene regulation cannot be understood from circuit wiring alone: the ratio of the chromatin-transition rate to the protein-turnover rate, encoded in the adiabaticity parameter $\omega$, reshapes the regulatory landscape and controls fluctuation behavior. Extending a previously developed stochastic model from a single gene to multi-gene circuits, it shows that in the mildly nonadiabatic regime $0.1 \lesssim \omega \lesssim 1$ the probability current develops a global circular flow, entropy production rises, and gene switching becomes hysteretic with chromatin-state change preceding protein-concentration change. In the intensely nonadiabatic regime $0.01 \lesssim \omega \lesssim 0.1$, entropy production per chromatin turnover time peaks. Applied to the three core pluripotency genes of mouse embryonic stem cells, the model reproduces the observed wide fluctuations of Nanog coexisting with narrow single-peak distributions of Oct4 and Sox2 when the three genes have different $\omega$ values. A deeper two-layer model ties this heterogeneity to domain-wide versus local epigenetic regulation and predicts measurable probability currents.

What carries the argument

The load-bearing object is the extended landscape $U(p,y)=-\log P(p,y)$ built from the stationary distribution of coupled Langevin equations for continuous protein concentration $p_i$ and continuous chromatin state $y_i$ (and later the enhancer-promoter state $z_i$). The equations are derived by writing the discrete-state master equation in path-integral form and applying the saddle-point approximation, which truncates fluctuations at second order and yields Gaussian noise with diffusion constants set by the volume $\Omega$ and the adiabaticity $\omega_i$. The circular, divergence-free component of the probability current $J$ signals broken detailed balance; entropy production is computed from $\dot{S}=\int\int dp\,dy\,(J_p^2/(P D_p)+J_y^2/(P D_y))$; and time ordering is quantified by the cross-correlation $A(t)$, whose peak at $t\approx 1/\omega$ marks the chromatin turnover time. These objects carry the argument: the current drives hysteresis, $A(t)$ detects it, and $\dot{S}$ quantifies dissipation.

What would settle it

Simulate the original discrete-state jump master equation (for example, by the Gillespie algorithm) with the same transition rates and compare the cross-correlation $A(1/\omega)$ and entropy production $\dot{S}/\omega$: if the positive time-ordering and the dissipation peak near $0.01 \lesssim \omega \lesssim 0.1$ disappear or move outside the eddy regime, the continuous saddle-point approximation is the source of the claimed nonequilibrium effects.

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Extended reading notes

Core claim

The central claim is that $\omega$, the ratio of the chromatin-state transition rate to the protein concentration change rate, is a control parameter for nonequilibrium fluctuations in eukaryotic gene circuits, not just a correction to adiabatic gene-network theory. As $\omega$ is lowered below unity, landscapes $U=-\log P$ develop additional basins; in the eddy regime $0.1 \lesssim \omega \lesssim 1$ the stationary probability current circulates, detailed balance is broken, and the cross-correlation $A(t)$ between chromatin state $y$ and protein concentration $p$ becomes positive at positive lag, meaning $y$ changes first and $p$ follows in both activation and inactivation. The entropy production rate $\dot{S}$ per chromatin turnover time is low in the adiabatic limit, rises through the eddy regime, peaks for $0.01 \lesssim \omega \lesssim 0.1$, and decreases at smaller $\omega$. In the mutually repressing two-gene circuit the circular fluxes appear even before the basin bifurcation, acting as precursors. For the Oct4\textendash Sox2\textendash Nanog circuit, heterogeneous adiabaticity\textemdash slow chromatin transitions for Nanog ($\omega_3 \approx 0.2$\textendash $0.5$) and fast transitions for Oct4 and Sox2 ($\omega_1 = \omega_2 \approx 10$)\textemdash produces exactly the experimentally observed heterogeneous fluctuation pattern without invoking Nanog self-activation or mutual repression.

Load-bearing premise

The entire calculation depends on treating the discrete chromatin state $y$ (and, in the deeper model, the enhancer-promoter state $z$) as continuous variables with Gaussian noise through the saddle-point approximation; if real chromatin transitions are genuinely discrete and rare, the continuous landscape, the circular probability current, and the entropy-production formula may describe an artifact of that approximation rather than the cell.

Editorial extensions

If this is right

  • If the central claim holds, eukaryotes with $\omega \approx 0.1$ are generically in the nonadiabatic regime, so slow histone modifications should produce extra landscape basins and larger expression fluctuations than adiabatic models predict.
  • The circular probability flux appears before the emergence of the off-diagonal basins in the two-gene toggle, so circular currents are a precursor or early-warning signal of switching; the paper suggests RNA-velocity-style measurements could detect them.
  • For mouse embryonic stem cells, the observed wide Nanog fluctuations can be explained by slow chromatin transitions at the Nanog locus alone, without Nanog self-activation or mutual repression among the three core genes.
  • Because $\omega$ can be tuned by chromatin-modifying enzyme activity, cells could pass from stable nonadiabatic basins through a fluctuating eddy regime to different stabilized states, providing a physical route for cell-type transitions.
  • In the deep-epigenetic model, the predicted signature is a diagonal circular current between Nanog's enhancer-promoter state $z_3$ and protein concentration $p_3$, plus a projected current flowing from high- to low-Nanog states; the paper argues this is experimentally testable with combined Hi-C, ChIP-seq, and RNA-seq data.

Reading between the lines

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

  • The same qualitative phenomenology\textemdash basin proliferation, positive $A(t)$, and a dissipation peak\textemdash should survive in the original discrete jump master equation, but the location of the peak may shift; a Gillespie simulation of the discrete model with the same parameters would calibrate the continuous approximation.
  • The per-gene $\omega$ mechanism suggests a general design principle for synthetic gene circuits: heterogeneity can be programmed by choosing chromatin modification rates rather than promoter strengths, which might be cheaper or more reversible in engineered cells.
  • If the entropy-production peak is confirmed, it implies that regulation operating at timescales comparable to protein turnover is thermodynamically costly; this could reflect a trade-off between stability (slow switching) and flexibility (fast response) that natural selection tunes.
  • The projected divergence of the probability current in the Oct4-Nanog plane is a falsifiable signature: time-resolved single-cell data that show no directional flow from high- to low-Nanog would count against the heterogeneous-$\omega$ hypothesis.
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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 manuscript extends a stochastic landscape model of eukaryotic gene regulation from a single gene to multi-gene circuits. The authors define an adiabaticity parameter omega as the ratio of the chromatin-state transition rate to the protein-concentration relaxation rate, replace discrete chromatin states by continuous variables through a saddle-point approximation, and integrate coupled Langevin equations for protein concentration and chromatin state (Eq. 4). For a self-activating single-gene circuit and a mutually repressing two-gene circuit, they compute stationary landscapes, probability currents, the cross-correlation A(t), and an entropy-production measure (Eq. 9). They report that decreasing omega increases the number of landscape basins, that the eddy regime 0.1-1 hosts circular probability currents and hysteresis, and that the entropy production per chromatin turnover time peaks in the intensely nonadiabatic regime. They then apply the model to a three-gene Oct4-Sox2-Nanog circuit in mouse embryonic stem cells, proposing that heterogeneous per-gene omega values, extended by a 'deep epigenetic' two-layer chromatin model, can explain the wide Nanog fluctuations and narrow Oct4/Sox2 distributions observed experimentally.

Significance. If its central claims survive validation, the paper offers a physically suggestive control parameter, omega, for nonequilibrium fluctuations in gene circuits, and it makes concrete, testable predictions about circular probability currents and hysteresis in chromatin-protein dynamics. The manuscript is transparent in its parameter choices, tabulates all rates, and includes useful robustness checks (monomer versus dimer transcription factors, two-state versus three-state chromatin, alternative parameterizations in Appendices C-E). The main strength is the breadth of the conceptual framework: it connects a single quantitative parameter to basin structure, current circulation, entropy production, and cell-to-cell heterogeneity in pluripotency genes. The main limitations are that the central quantitative claims rest on an uncontrolled continuous approximation of discrete chromatin variables, and the mES cell application currently selects parameters after the fact rather than validating them against independent data.

major comments (3)
  1. [II.B, Eq. (4), Appendix A] The continuous Gaussian description of the discrete chromatin variables is an uncontrolled approximation. The underlying process is a three-state Markov chain for y_i with rates given by Eqs. 2-3. The saddle-point/truncation at second order has no small parameter for y_i, and the noise amplitude D_yi = (G_yi + F_yi)/(2 omega_i) actually grows as omega_i decreases, i.e., in the regime where the paper's central peak in Fig. 9D appears. The reflecting-wall treatment in Appendix A, which excludes 5% of the domain from the Eq. 9 integral, demonstrates that the diffusion dynamics generate spurious boundary currents; without a direct simulation of the original jump master equation (or a Gillespie simulation of the discrete-state process) there is no way to know whether the circular currents, the hysteresis measure A(t), and the entropy-production peak are physical or artifacts. I request an explicit comparison of P(p,y), J, and Sdot between Eq. 4 and the discrete-state master equation at representative omega values, including omega = 0.01, 0.1, and 1.
  2. [III, Eq. (9), Fig. 9D] Equation 9 is the diffusion-form entropy production rate, integral of J^2/(2DP), but for the original jump process the correct entropy production rate is the sum over transitions of (r_forward P_backward_state - r_backward P_forward_state) times ln(r_forward/r_backward). These two expressions need not have the same dependence on omega. Moreover, Fig. 9D plots Sdot/omega, the entropy produced per chromatin turnover time, not the absolute entropy production rate, so the abstract's phrase 'maximum entropy production' is not directly supported by the figure. The authors should either compute the discrete-state entropy production rate and show that it peaks in the same regime, or revise the abstract to say 'maximum entropy production per chromatin turnover time' and clearly state that this is the plotted quantity.
  3. [IV, Figs. 10-11, Appendix E] The mES cell result is obtained with adiabaticity parameters that appear to be chosen after the fact: omega1 = omega2 = 10 and omega3 = 0.5 are set so that the desired wide-Nanog/narrow-Oct4 pattern emerges, and Appendix E shows that neighboring parameterizations destroy the match. As presented, this is a proof of principle that heterogeneous omega values can tune fluctuations in a circuit, not a validation of the proposed explanation for the experimental data. To make the explanatory claim load-bearing, the authors should state this status explicitly and provide a falsifiable prediction that can be tested against independent data, such as measured chromatin-state lifetimes at the three loci, single-cell time-series cross-correlations, or the projected current divergence pattern of Fig. 14D.
minor comments (5)
  1. [Eq. (13)] In the second line of Eq. 13 the variable is written as v_i, while the text and the rest of the equation use y_i; the notation should be made consistent.
  2. [Appendix A] The numerical section does not state the stochastic integration convention. The Euler discretization implicitly uses the Ito convention; this should be stated explicitly, since the noise amplitudes depend on p and y and the convention affects boundary behavior and current statistics.
  3. [Fig. 9] The caption lists panel (C) twice and omits a caption for panel (F); the labels should be corrected, and the definition of Sdot/omega should be repeated in the caption for clarity.
  4. [Throughout] There are several typographical errors, including 'dacetylated', 'calcutated', 'adiavaticiy', 'landscapt', and 'swiching' in Sections V and Appendix E; these should be corrected in a final proofreading pass.
  5. [References] References [27] and [29] appear to share the same title 'Enhancer dynamics: Unraveling the mechanism of transcriptional bursting' but cite different venues; please verify that both entries are correct and distinct.

Circularity Check

1 steps flagged · score 6.0 of 10

The mES-cell fluctuation 'prediction' is wired in via chosen omega values; the core landscape and entropy-production results are independent.

  1. fitted input called prediction [Section IV.B, Fig. 11, and Eq. 6]
    "Consequently, our hypothesis suggests that Nanog exhibits slow transitions in its chromatin state, leading to an adiabaticity of omega3 = 0.2 ~ 0.5. In contrast, Oct4 and Sox2 demonstrate rapid transitions, resulting in greater adiabaticity of omega1 and omega2. ... Fig. 11 shows the calculated landscape with heterogeneous adiabaticity parameters: omega1 = omega2 = 10 for Oct4 and Sox2 and omega3 = 0.5 for Nanog. ..."

    Eq. 6 defines Dyi = (Gyi + Fyi)/(2 omega_i), so omega directly controls the noise amplitude of the chromatin variable. Choosing omega3 = 0.5 and omega1 = omega2 = 10 therefore preselects the wide-Nanog, narrow-Oct4/Sox2 fluctuation pattern that is then presented as consistent with the experimental data. No independent estimate of omega3 versus omega1/omega2 is supplied; the values are chosen after the fact. Appendix E confirms that other parameter choices eliminate the match, showing that the agreement is a selected input rather than an emergent model prediction. The paper itself notes at Eq. 6 that 'the volume Omega and the adiabaticity omega determine the fluctuation amplitude of p and y, respectively.'

full rationale

The paper's main theoretical results—the increase in landscape basin number as omega decreases, the eddy-regime circular probability current, the entropy-production peak, and the hysteresis/cross-correlation time-ordering—are generated by explicit simulation of Eq. 4 and are not fitted to the phenomena they explain; they are compared with independent experimental observations such as lambda/tet repressor switching or chromatin timescale estimates. The genuinely circular element is the Circuit C application: the heterogeneous mES fluctuation pattern is imposed by selecting low omega3 and high omega1/omega2, and Eq. 6 makes omega the fluctuation amplitude by construction. The authors are transparent that this is a hypothesis rather than an independently measured parameterization, but the resulting 'consistency' and the Fig. 14 probability currents are presented as predictions even though the key output (wide Nanog, narrow Oct4/Sox2) is encoded in the chosen inputs. This partial circularity does not invalidate the Circuits A and B results or the general entropy-production/hysteresis analysis.

Assumptions & free parameters 6 free parameters · 6 assumptions · 1 invented entities

The central claims load on several modeling choices: the continuous Langevin treatment of discrete chromatin states, the assumed entropy production formula, and, for the mES application, timescale parameters chosen to match the data. These are the costs the reader pays upstream of the results.

free parameters (6)
  • Adiabaticity parameter omega_i = Circuits A and B: scanned 10^-3 to 10^2; Circuit C: omega1=omega2=10, omega3=0.5
    Controls the ratio of chromatin-state transition rate to protein degradation rate. In the mES application, the omega values are chosen to reproduce the observed heterogeneity, making them effectively fitted parameters (Sec. IV.B, Fig. 11).
  • Typical protein concentration pbar = 1.0 (Circuit A), 1.2 or 3 (Circuit B), 2 (Circuit C); scanned 0 to 10
    Sets the scale of protein synthesis rates and affects basin structure. The phase diagrams in Figs. 6 and 7 depend on it; it is scanned, not fitted, but chosen by hand.
  • Chromatin transition coefficients mu and gamma (Table II) = e.g., mu_{0,-1}=k, gamma_{0,-1}=0.6k in Circuit A; signs and magnitudes in Table II
    Set to plausible values 'of order k' based on estimated histone modification timescales. They determine the transition rates that shape the eddy dynamics.
  • TF binding parameter ratios h0/f and h1/f = h0/f=10 in all circuits; h1/f=200 in Circuit C
    Chosen to make the TF binding/unbinding time (1/f ~ 1-10 s) much faster than protein turnover (1/k ~ 2-10 h), so that x_i can be adiabatically eliminated.
  • Normalized protein synthesis rates xi_xy (Table II) = e.g., xi11=pbar, xi01=0.2 pbar, xi10=0.2, xi00=0.2, others 0
    Chosen as simple parameterizations of activation/repression; Appendix D shows that reducing the intermediate state changes basin structure, so these choices affect the central results.
  • Deep epigenetic parameters (Table III) = nu, sigma, gamma_x, gamma_z coefficients in k units
    Introduced for the z variable in Sec. V; they are plausible constants with no direct empirical constraint, and the qualitative result depends on their signs following the consistency assumption.
assumptions (6)
  • domain assumption The saddle-point approximation (truncation of fluctuations at second order) maps the discrete-state master equation to continuous Langevin equations.
    Invoked in Sec. II.B to obtain Eq. 4; validity for rare chromatin-state transitions is not checked against the original jump process.
  • domain assumption Chromatin state y and EP state z can be treated as continuous variables in [-1,1] with reflecting walls.
    Discrete two- or three-state chromatin transitions are replaced by diffusion on a continuous line; this underpins all landscape and current calculations (Sec. II.B, Appendix A).
  • domain assumption TF binding equilibrates rapidly (adiabatic approximation), so x_i = h/(f+h).
    Used throughout to eliminate x_i dynamics (Sec. II.A.2), based on 1/f = 1-10 s versus 1/k = 2-10 h.
  • ad hoc to paper The entropy production rate of Eq. 9 is the true physical entropy production of the underlying gene-switching process.
    No derivation connects the continuous-current formula to the underlying chemical master equation; the peak claim rests on this identification.
  • ad hoc to paper In Circuit C, Nanog's transcription initiation complex spans the domain, linking its EP state to the collective histone state, giving a low omega_z3.
    This hypothesis is introduced in Sec. V to justify the parameter choice that reproduces the observed heterogeneity; no direct experimental evidence is cited.
  • domain assumption The noise in Eq. 4 is interpreted in the Ito sense for numerical integration.
    Not stated in the text; the Euler discretization implies Ito, but the path-integral derivation's convention is not specified.
invented entities (1)
  • EP state variable z_i (enhancer-promoter state) independent evidence
    purpose: Adds a local chromatin layer between TF binding and collective histone state in the deep epigenetic regulation model, to explain different timescales and generate testable predictions.
    The variable itself is a modeling abstraction with no direct observable, but the model produces falsifiable predictions for probability currents (Fig. 14D) and for heterogeneous fluctuation patterns that can be tested with single-cell multi-omics experiments.

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Cite this review

Pith. "Pith review of Landscapes and nonequilibrium fluctuations of eukaryotic gene regulation." pith.science (2026). https://pith.science/paper/R34P57NB

@misc{pith2026250210067,
  author       = {Pith},
  title        = {Pith review of: Landscapes and nonequilibrium fluctuations of eukaryotic gene regulation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R34P57NB}},
  note         = {Machine review of arXiv:2502.10067}
}
read the original abstract

Understanding the interplay among processes that occur over different timescales is a challenging issue in the physics of systems regulation. In gene regulation, the timescales for changes in chromatin states can differ from those for changes in the concentration of product protein, raising questions about how to understand their coupled dynamics. In this study, we examine the effects of these different timescales on eukaryotic gene regulation using a stochastic model that describes the landscapes and probability currents of nonequilibrium fluctuations.This model shows that slow, nonadiabatic transitions of chromatin states significantly impact gene-regulation dynamics. The simulated circular flow of the probability currents indicates a maximum entropy production when the rates of chromatin-state transitions are low in the intensely nonadiabatic regime. In the mildly nonadiabatic regime, this circular flow fosters hysteresis, suggesting that changes in chromatin states precede changes in transcription activity. Furthermore, calculations using a model of a circuit involving three core genes in mouse embryonic stem cells illustrate how the timescale difference can tune fluctuations in individual genes. These findings highlight the rich effects of nonadiabatic chromatin-state transitions on gene regulation in eukaryotic cells.

Figures

Figures reproduced from arXiv: 2502.10067 by the authors.

Figure 1
Figure 1. FIG. 1. Landscapes explaining the gene expression dynamics. [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. A model of eukaryotic gene regulation. The enhancer [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Gene circuits examined in this study: ( [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (16 more)
Figure 4
Figure 4. Figure 4: shows the landscape, U(p, y) = − log P(p, y), which was obtained from the stationary distribution, P(p, y), in the self-activating single-gene circuit (Circuit FIG. 4. The landscape, U(p, y), of the self-activating single￾gene circuit, Circuit A, derived from the numer…
Figure 6
Figure 6. Figure 6: FIG. 6. Phase diagram of the number of basins on the land [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Phase diagram of the number of basins on the [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: shows the probability current in the self-activating single-gene circuit (Circuit A). When ω ≫ 1, the current is not very noticeable in the steady state, suggesting that the state can be effectively approximated by an equilib￾rium. Conversely, in the nonadiabatic case …
Figure 9
Figure 9. Figure 9: FIG. 9. Prominent nonequilibrium features in the eddy regime of 0 [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. The landscape of the circuit of the three core genes [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. The landscape of the circuit of the three core genes [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]
Figure 12
Figure 12. Figure 12: We define zi = −1 when histones in the EP re￾gion are dacetylated making the EP region condensed, zi = 0 when they are acetylated and the EP region has an extended structure, and zi = 1 when the extended EP region is forming a transcription initiation complex. We cons…
Figure 14
Figure 14. Figure 14: FIG. 14 [PITH_FULL_IMAGE:figures/full_fig_p013_14.png]
Figure 13
Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_13.png]
Figure 15
Figure 15. Figure 15: FIG. 15. The probability current [PITH_FULL_IMAGE:figures/full_fig_p015_15.png]
Figure 16
Figure 16. Figure 16: compares the monomer and dimer models for the self-activating single-gene circuit (Circuit A) by vary￾ing the chromatin adiabaticity ω. In both models, when the circuit is in the adiabatic regime at ω = 10, a single basin is observed at the active state of the chromat…
Figure 17
Figure 17. Figure 17: compares the monomer and dimer models for the mutually repressing two-gene circuit (Circuit B) at a relatively high protein production level of ¯p = 3. At this high level of protein production, the dimer model displays two coexisting basins even at ω = 1. In con￾trast…
Figure 18
Figure 18. Figure 18: FIG. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_18.png]
Figure 20
Figure 20. Figure 20: FIG. 20 [PITH_FULL_IMAGE:figures/full_fig_p018_20.png]
Figure 21
Figure 21. Figure 21: A in the present Appendix. However, this feature disappears under specific parameter settings as shown in Figs. 21B–21D. Fig. 21B illustrates the landscapes generated by the lower adiabaticity for Oct4 and Sox2 (with ω1 = ω2 = 1). In this parameter setting, the adiaba…

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.