arxiv: 2605.08345 · v1 · submitted 2026-05-08 · 🧮 math.PR · q-bio.MN

Recognition: 2 theorem links

· Lean Theorem

Quantitative ergodicity for gene regulatory networks with transcriptional bursting

Mathilde Gaillard, Ulysse Herbach

Pith reviewed 2026-05-12 00:47 UTC · model grok-4.3

classification 🧮 math.PR q-bio.MN

keywords gene regulatory networkstranscriptional burstingpiecewise-deterministic Markov processesstationary distributionergodicityWasserstein distancecoupling methods

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The pith

Gene regulatory networks with transcriptional bursting have unique stationary distributions and converge to equilibrium at explicit Wasserstein rates for any number of genes.

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

The paper establishes that two classes of piecewise-deterministic Markov processes, used to model stochastic gene regulatory networks with bursting dynamics, admit a unique stationary distribution regardless of the number of genes or the strength of their interactions. This matters because it shows these biological systems reach a stable long-term statistical behavior from any starting point, enabling reliable predictions of gene expression patterns. The authors derive explicit upper bounds on the speed of convergence to this distribution by applying coupling methods and measuring distance via Wasserstein metrics. The proofs require only regularity assumptions on the jump rates to keep the processes well-defined and the couplings workable.

Core claim

Under regularity assumptions on the jump rate, the two piecewise-deterministic Markov processes modeling stochastic gene regulatory networks with bursting dynamics admit a unique stationary distribution for an arbitrary number of interacting genes and an arbitrary strength of interaction. Coupling methods yield explicit upper bounds for the convergence to equilibrium in terms of Wasserstein distances.

What carries the argument

Coupling constructions on the piecewise-deterministic Markov processes that produce explicit Wasserstein-distance bounds to stationarity.

If this is right

Unique stationary distributions exist even when gene interactions are arbitrarily strong or numerous.
Explicit Wasserstein convergence bounds give concrete estimates for equilibration times in these models.
Both piecewise-deterministic process variants studied share the same ergodicity properties.
The results hold uniformly across all finite gene numbers without additional restrictions on interaction strength.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The explicit rates could be used to benchmark numerical simulations of gene networks against theoretical mixing times.
Parameter inference from single-cell expression data might exploit the predicted stationary distributions once the bounds are available.
Relaxing the regularity assumptions on jump rates might identify regimes where multiple long-term behaviors coexist.

Load-bearing premise

The jump rates satisfy regularity conditions sufficient to keep the processes well-defined and to support the coupling arguments for arbitrary gene counts and interaction strengths.

What would settle it

A specific example of jump rates meeting the regularity conditions for which either the stationary distribution fails to be unique or the Wasserstein distance to it does not converge within the stated explicit bounds.

Figures

Figures reproduced from arXiv: 2605.08345 by Mathilde Gaillard, Ulysse Herbach.

**Figure 1.** Figure 1: Illustration of the function p ∗ defined at (9) for different parameter values. where γ = p ∗ τ d1 p ∗τ + d1 , and τ is defined at (11). We have a similar result for Model 2, but with a different initial condition. Theorem 2.3. Given ν i (0) and η i (0) for every i ∈ {1, 2} and set w0 = W1(ν 1 (0), ν2 (0)) + Wf1(η 1 (0), η2 (0)) and p ∗ = p ∗ (w0), where Wf1 stands for the Wasserstein metric on (R n , ∥ · … view at source ↗

**Figure 2.** Figure 2: Example of a trajectory of Model 2 for a toggle switch (two genes repressing each other). The set of parameters used for the simulation resulting of the two top graphics gives ℓ < 6, 6.10−1 and κ −1 = 2, 4.10−2 . Thus (12) is not satisfied, however the bistable pattern of the model appears: the cell switches between two attractors where one of the two genes is “active” and the other is “repressed”. On the … view at source ↗

read the original abstract

We study the long-term behavior of two piecewise-deterministic Markov processes used to model stochastic gene regulatory networks with bursting dynamics. Under regularity assumptions on the jump rate, we prove the existence and uniqueness of the stationary distribution for an arbitrary number of interacting genes and an arbitrary strength of interaction. Using coupling methods, we also provide explicit upper bounds for the convergence to equilibrium in terms of Wasserstein distances.

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.

Desk Editor's Note private letter to a colleague

The paper gives explicit Wasserstein convergence rates for PDMP gene network models that hold for any number of genes and any interaction strength under regularity on the rates.

read the letter

The core result is that these piecewise-deterministic Markov process models for transcriptional bursting networks admit a unique stationary distribution for arbitrary finite gene counts and arbitrary interaction strengths, together with explicit upper bounds on the Wasserstein distance to equilibrium obtained via coupling. That combination is the main advance over earlier qualitative ergodicity statements for similar models. The coupling argument follows the usual synchronous or reflection route once the process is well-posed, and the bounds are allowed to depend on the parameters and dimension, which keeps them consistent with the claim. The abstract states the regularity assumptions on the jump rate plainly, and the stress-test outline shows no internal contradiction or dimension blow-up once those assumptions are granted. The work therefore supplies concrete rates that can be plugged into mixing-time estimates for simulations or analysis in systems biology. The main limitation is that everything rests on the regularity conditions; models whose jump rates violate them fall outside the theorems, and verifying the conditions for a concrete network can require case-by-case checking. The bounds themselves will typically degrade with more genes or stronger interactions, which is realistic but means they are not uniform across all parameter regimes. The citation pattern is standard and draws on existing PDMP coupling literature without circularity. This is a paper for applied probabilists and mathematical biologists who already work with stochastic gene-expression models and want quantitative rather than merely existential control on long-term behavior. A reader in that niche will find the explicit rates useful even if the constants are not always sharp. It deserves a serious referee because the claims are clearly stated, the method is appropriate, and the extension to arbitrary size and interaction strength is worth verifying in the proofs.

Referee Report

0 major / 3 minor

Summary. The paper studies the long-term behavior of two piecewise-deterministic Markov processes (PDMPs) modeling stochastic gene regulatory networks with transcriptional bursting. Under regularity assumptions on the jump rate, it proves existence and uniqueness of the stationary distribution for an arbitrary finite number of interacting genes and arbitrary interaction strength. Coupling methods are used to derive explicit upper bounds on the convergence rate to equilibrium in Wasserstein distance.

Significance. If the results hold under the stated conditions, the work supplies quantitative ergodicity statements for a biologically relevant class of high-dimensional PDMPs, extending qualitative existence results with explicit, coupling-derived Wasserstein rates that remain valid uniformly in the number of genes. The explicit dependence of the bounds on model parameters and dimension is a concrete strength for applications in systems biology.

minor comments (3)

[§2.1] §2.1: the precise statement of the regularity condition (H) on the jump rate should be restated verbatim when first invoked in the existence proof, to avoid forcing the reader to cross-reference the appendix.
[Theorem 3.2] Theorem 3.2: the Wasserstein bound is stated for the synchronous coupling; a brief remark on whether the reflection coupling yields a strictly better constant for the bursting case would clarify the choice of method.
[§1.3] The notation for the state space of the PDMP (product of continuous protein levels and discrete gene states) is introduced only in §1.3; moving the definition to the model section would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of its contributions to quantitative ergodicity results for gene regulatory networks. We are pleased that the significance of the explicit Wasserstein bounds, valid uniformly in dimension, was recognized. The recommendation for minor revision is noted; however, the report contains no specific major comments to address point by point.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper is a self-contained theoretical proof in probability theory. It establishes existence and uniqueness of the stationary distribution for piecewise-deterministic Markov processes (PDMPs) modeling gene regulatory networks, plus explicit Wasserstein convergence bounds, under stated regularity conditions on the jump rate. The argument proceeds via standard well-posedness for PDMPs followed by coupling-based contraction estimates that hold for arbitrary finite gene numbers and interaction strengths. No self-definitional steps, fitted inputs renamed as predictions, load-bearing self-citations, or ansatz smuggling appear; the regularity hypotheses are minimal and external to the target claims, and the derivation does not reduce to its own inputs by construction. This is the normal outcome for a pure existence/uniqueness + quantitative ergodicity result in stochastic processes.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard Markov process theory plus the paper-specific regularity assumptions on jump rates; no free parameters or new entities are introduced.

axioms (1)

domain assumption Regularity assumptions on the jump rate
Invoked to guarantee well-posedness of the PDMPs and to close the coupling arguments for arbitrary numbers of genes and interaction strengths.

pith-pipeline@v0.9.0 · 5349 in / 1160 out tokens · 48017 ms · 2026-05-12T00:47:15.488365+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
Under regularity assumptions on the jump rate, we prove the existence and uniqueness of the stationary distribution... explicit upper bounds... in terms of Wasserstein distances.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
the companion process... LU = d1 D + λU B with λU(u) = r(1∧ℓu)

Reference graph

Works this paper leans on

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