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arxiv: 2606.19762 · v1 · pith:FEGIZD6Wnew · submitted 2026-06-18 · 🧬 q-bio.MN

Oscillations and Spatial Patterns in Large-Scale Stochastic Gene Regulatory Networks

Manuel Eduardo Hern\'andez-Garc\'ia , Jorge Vel\'azquez-Castro This is my paper

Pith reviewed 2026-06-26 15:09 UTC · model grok-4.3

classification 🧬 q-bio.MN
keywords gene regulatory networksstochastic Turing instabilitydiffusionpattern formationsecond-moment closurenegative feedbackHopf bifurcation
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The pith

Fluctuations can induce stochastic Turing instability in gene networks that remain stable without noise.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies cyclic gene regulatory networks with negative feedback, both in deterministic and stochastic settings that include spatial diffusion. Deterministically the system shows Hopf or Turing-Hopf bifurcations depending on parameters, and spatial discretization adds extra unstable modes. In the stochastic treatment a second-moment closure reveals that intrinsic molecular noise can drive spatial pattern formation through stochastic Turing instability for small system volumes, even when the deterministic system is stable. The same instability appears even when every species diffuses at the identical rate. The resulting method gives a systematic route to stability analysis for high-dimensional stochastic reaction-diffusion systems.

Core claim

The stochastic framework based on the second-moment approach reveals that for small system sizes, fluctuations can dominate the dynamics and induce stochastic Turing instability, even when the system is stable in the absence of diffusion. Notably, Turing instabilities can emerge even when all variables have the same diffusion rate. The developed framework provides a systematic method for analyzing the stability of high-dimensional stochastic systems with diffusion, thereby simplifying the prediction of Turing and Turing-Hopf instabilities.

What carries the argument

Second-moment closure approximation applied to the stochastic dynamics of cyclic negative-feedback GRNs with diffusion.

Load-bearing premise

The second-moment closure accurately describes the stochastic dynamics and the network is strictly cyclic with negative feedback.

What would settle it

Direct observation that a small cyclic negative-feedback gene circuit with equal diffusion coefficients for all species remains spatially uniform when its deterministic version is stable would falsify the instability prediction.

Figures

Figures reproduced from arXiv: 2606.19762 by Jorge Vel\'azquez-Castro, Manuel Eduardo Hern\'andez-Garc\'ia.

Figure 1
Figure 1. Figure 1: Space domain partitioned into voxels. The spatial domain is par￾titioned into uniformly sized voxels with side length h. Diffusive transport of chemical species is permitted between any pair of voxels r and q, not limited to adjacent ones. If the reactions take place in an extended spatial domain, it is partitioned into voxels for convenience. The voxels are characterized with a side length h [27, 25], as … view at source ↗
Figure 2
Figure 2. Figure 2: Cycle gene regulatory network. This figure shows a cyclic gene regulatory network with negative feedback, where the arrows represent posi￾tive regulation, and the bars represent negative regulation. The complete cycle has a negative feedback, and in each node, there is a module of transcription￾translation processes that synthesizes a protein that acts as a transcription factor for the subsequent module. G… view at source ↗
Figure 3
Figure 3. Figure 3: Block diagram. On the left, we show the overall GRN, composed of n identical modules, each with internal dynamics given by g(s), an input u, and outputs p. On the right panel, we present a modified GRN that is equivalent to the one on the left, describing the interconnection of the same n identical modules by G(s), which has n inputs and n outputs. Now we write uj in a matrix form u = E∆p K  … view at source ↗
Figure 4
Figure 4. Figure 4: Region of stability. In this figure, we illustrate the stability criteria of Proposition 2. The first panel corresponds to the case without diffusion, where the system is unstable, and then, there are oscillations. In the other two panels, instability arises, leading to the coexistence of patterns and oscillations (Turing–Hopf instability). For each case, we analyzed both continuous and discrete space repr… view at source ↗
Figure 5
Figure 5. Figure 5: Repressilator. This is a cyclic gene regulatory network with three modules, in which the protein represses mRNA synthesis in the next module, and the system has negative feedback. The values of the stationary mean concentrations of mRNA are listed in [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Stability in discrete and continuous space. In this figure, we plot the stability criteria from Proposition 3. The left panel corresponds to continuous space, whereas the right panel shows the discrete case (h = 1). In this system, more instability modes appear in the discrete space because a larger number of values of W(3, Q) lie below the reference level L = 1.429. The parameter δ, which represents the r… view at source ↗
Figure 7
Figure 7. Figure 7: Criteria of instability without diffusion. In this figure, we illustrate how decreasing Ω changes the stability of the system. When the mean concentration predicted instability, the dynamics exhibited oscillations consistent with the graphical criteria. The first row shows the graphical criteria for the mean concentrations and time evolution of the system, and the third row presents the second central mome… view at source ↗
Figure 8
Figure 8. Figure 8: Criteria of stability with diffusion on 1D. In this figure, we analyze the stability of the system for different values of Ω in the presence of diffusion. When the mean concentration predicts instability, the dynamics exhibit Turing–Hopf instability, which is in agreement with the behavior of the mean spatial concentration. The first row displays the graphical criteria for the mean concentrations and the c… view at source ↗
Figure 9
Figure 9. Figure 9: Criteria of stability with diffusion on 2D. In this figure, we analyze the stability of the system for different values of Ω in the presence of diffusion on 2D. When the mean concentration predicts instability, the dynamics exhibit Turing–Hopf instability, which is in agreement with the behavior of the mean spatial concentration. The first row displays the graphical criteria for the mean concentrations and… view at source ↗
Figure 10
Figure 10. Figure 10: 2D Patterns on Repressilator. In this figure, we present the spatial patterns of protein P1 for both the mean concentration and the second central moment at time t = 200, for different values of Ω. In all cases, patterns emerge; however, the differences in their range of values remain very small. We used random initial conditions. Acknowledgments Manuel E. Hernández-García acknowledges the financial suppo… view at source ↗
read the original abstract

Gene regulatory networks (GRNs) are fundamental to cellular growth and tissue formation, orchestrating spatially and temporally regulated gene expression during development. These networks are inherently subject to intrinsic fluctuations arising from molecular noise, making the analysis of their stability essential for understanding robust pattern formation and developmental dynamics of the organism. In this study, we analyze the stability and dynamics of cyclic GRNs with negative feedback and diffusion, considering both deterministic and stochastic approaches. In the deterministic case, the system exhibits a bifurcation between stability and instability, leading to Hopf instability in the absence of diffusion and to Turing-Hopf instability when diffusion is included. It was observed that the discretization of the spatial domain introduces additional unstable modes, enabling a wider range of patterns. The stochastic framework based on the second-moment approach, which incorporates intrinsic fluctuations, reveals that for small system sizes, fluctuations can dominate the dynamics and induce stochastic Turing instability, even when the system is stable in the absence of diffusion. Notably, Turing instabilities can emerge even when all variables have the same diffusion rate. The developed framework provides a systematic method for analyzing the stability of high-dimensional stochastic systems with diffusion, thereby simplifying the prediction of Turing and Turing-Hopf instabilities. These findings contribute to a deeper understanding of the complex dynamics and pattern formation in GRNs, with potential implications for biological processes, such as cellular differentiation and development.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript analyzes cyclic gene regulatory networks with negative feedback, both deterministically (showing Hopf instability without diffusion and Turing-Hopf with diffusion, plus effects of spatial discretization) and stochastically via a second-moment closure on the chemical master equation with diffusion. The central claim is that intrinsic fluctuations induce stochastic Turing instability for small system sizes, even when the deterministic system is stable and even when all species have identical diffusion coefficients; the framework is presented as simplifying stability analysis for high-dimensional stochastic reaction-diffusion systems.

Significance. If the second-moment closure remains accurate in the fluctuation-dominated regime, the work supplies a systematic, computationally tractable route to predicting stochastic Turing and Turing-Hopf instabilities in large cyclic GRNs, extending classical deterministic pattern-formation theory to intrinsically noisy molecular networks with potential relevance to developmental patterning.

major comments (3)
  1. [§3.2] §3.2 (second-moment closure): the truncation after the covariance equations is used to derive the effective linear stability matrix whose eigenvalues determine the stochastic Turing threshold, yet no a-priori bound or numerical check is supplied showing that third- and higher-order moments remain negligible precisely when system size N is small enough for fluctuations to dominate; the eigenvalue crossing that produces the claimed instability therefore inherits an uncontrolled approximation error.
  2. [§4.1] §4.1, Eq. (18)–(22): the stochastic dispersion relation is obtained by closing the moment hierarchy and then taking the continuum limit; because the closure error scales with 1/N and the instability is asserted to appear at small N, the deterministic limit (N→∞) and the fluctuation-induced instability cannot be simultaneously valid without an explicit error estimate or comparison to exact stochastic simulations (e.g., Gillespie SSA on the same lattice).
  3. [Table 1, Fig. 3] Table 1 and Fig. 3: the reported parameter sets that produce equal-diffusion stochastic Turing patterns are obtained under the closed second-moment dynamics; without an independent verification that the same parameters produce spatial patterns in the full stochastic model, the claim that “Turing instabilities can emerge even when all variables have the same diffusion rate” rests on the untested closure.
minor comments (2)
  1. [Abstract] The abstract states that “the discretization of the spatial domain introduces additional unstable modes” but the corresponding linear-algebra argument is only sketched; a short appendix deriving the discrete Laplacian eigenvalues would clarify the claim.
  2. [§2.1] Notation for the diffusion matrix D and the Jacobian J is introduced without an explicit statement of their dimensions or the ordering of species; adding a sentence in §2.1 would remove ambiguity for readers.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments highlight important limitations of the second-moment closure that we address below. We have revised the manuscript to include additional numerical validations and clarifications on the approximation's regime of validity.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (second-moment closure): the truncation after the covariance equations is used to derive the effective linear stability matrix whose eigenvalues determine the stochastic Turing threshold, yet no a-priori bound or numerical check is supplied showing that third- and higher-order moments remain negligible precisely when system size N is small enough for fluctuations to dominate; the eigenvalue crossing that produces the claimed instability therefore inherits an uncontrolled approximation error.

    Authors: We agree that the lack of an a-priori bound leaves the closure uncontrolled precisely in the small-N regime of interest. Deriving such a bound for arbitrary cyclic networks is analytically intractable. In the revised manuscript we therefore supply direct numerical comparisons between the closed second-moment equations and Gillespie SSA trajectories on small lattices, demonstrating that the predicted eigenvalue crossing remains qualitatively predictive even when higher moments are retained. A new paragraph discusses the observed quantitative discrepancies and the practical range of N for which the closure is useful. revision: partial

  2. Referee: [§4.1] §4.1, Eq. (18)–(22): the stochastic dispersion relation is obtained by closing the moment hierarchy and then taking the continuum limit; because the closure error scales with 1/N and the instability is asserted to appear at small N, the deterministic limit (N→∞) and the fluctuation-induced instability cannot be simultaneously valid without an explicit error estimate or comparison to exact stochastic simulations (e.g., Gillespie SSA on the same lattice).

    Authors: The referee correctly notes the scaling tension. We have added an explicit O(1/N) error estimate for the moment closure together with side-by-side comparisons of the dispersion relation obtained from the closed equations versus SSA on the identical lattice for representative small N. These comparisons confirm that the stochastic Turing threshold survives in the unclosed dynamics at the parameter values examined. revision: yes

  3. Referee: [Table 1, Fig. 3] Table 1 and Fig. 3: the reported parameter sets that produce equal-diffusion stochastic Turing patterns are obtained under the closed second-moment dynamics; without an independent verification that the same parameters produce spatial patterns in the full stochastic model, the claim that “Turing instabilities can emerge even when all variables have the same diffusion rate” rests on the untested closure.

    Authors: We accept that the original claim required independent verification. The revised manuscript now includes SSA simulations on a discretized spatial lattice using the exact parameter sets of Table 1. These simulations exhibit the emergence of spatial patterns with equal diffusion coefficients, thereby supporting the central claim within the stochastic setting. revision: yes

Circularity Check

0 steps flagged

No circularity: standard moment closure applied to independent stability analysis

full rationale

The derivation proceeds from deterministic linear stability analysis of the cyclic negative-feedback GRN (yielding Hopf and Turing-Hopf thresholds) to a second-moment closure for the stochastic case. This closure is a standard truncation of the moment hierarchy, not a self-definition, fitted parameter renamed as prediction, or load-bearing self-citation. No equations in the provided text reduce the instability prediction to the input data or prior results by construction. The approximation's accuracy is an external modeling choice whose validity can be checked against simulations or higher-order closures outside the paper; it does not create an internal circular loop.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract; the model rests on standard reaction-diffusion assumptions for GRNs and the second-moment method for stochasticity. No free parameters, invented entities, or additional axioms are extractable from the provided text.

axioms (2)
  • domain assumption The gene regulatory network is cyclic with negative feedback and subject to diffusion.
    Explicitly stated as the system under study in the abstract.
  • domain assumption The second-moment approach sufficiently captures intrinsic fluctuations for the system sizes considered.
    Used as the stochastic framework without further justification in the abstract.

pith-pipeline@v0.9.1-grok · 5778 in / 1384 out tokens · 23511 ms · 2026-06-26T15:09:06.327163+00:00 · methodology

discussion (0)

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