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arxiv: 2503.09007 · v1 · submitted 2025-03-12 · 🧬 q-bio.QM

Reconstructing Noisy Gene Regulation Dynamics Using Extrinsic-Noise-Driven Neural Stochastic Differential Equations

Jiancheng Zhang , Xiangting Li , Xiaolu Guo , Zhaoyi You , Lucas B\"ottcher , Alex Mogilner , Alexander Hoffman , Tom Chou
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Mingtao Xia
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keywords neural SDEextrinsic noisegene regulationcellular heterogeneityWasserstein distancestochastic dynamicscircadian rhythmsNF-kB signaling
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The pith

An extrinsic-noise-driven neural SDE reconstructs gene regulation dynamics from heterogeneous cell trajectories by matching Wasserstein distances.

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

The paper introduces a framework called END-nSDE that uses neural stochastic differential equations driven by extrinsic noise to model reaction dynamics in cell populations. It reconstructs these dynamics from trajectory data by minimizing the Wasserstein distance between the distributions of simulated and observed paths. This allows capturing the effects of cellular heterogeneity on top of intrinsic noise in systems such as circadian rhythms, RPA-DNA binding, and NFκB signaling. The method outperforms standard time-series tools like RNNs and LSTMs on the presented cases. It offers a surrogate modeling approach when full mechanistic models are impractical.

Core claim

The central discovery is that driving a neural SDE with an extrinsic noise term and optimizing it to match trajectory distributions via the Wasserstein distance enables accurate reconstruction of noisy gene regulation dynamics, including how extrinsic heterogeneity modulates the intrinsic stochastic reactions, as shown in three cell biology systems.

What carries the argument

The extrinsic-noise-driven neural stochastic differential equation (END-nSDE) framework that augments neural SDEs with extrinsic noise and uses Wasserstein distance for distribution matching.

If this is right

  • It models how cellular heterogeneity modulates reaction dynamics alongside intrinsic noise.
  • It outperforms RNNs and LSTMs in reconstructing the dynamics from the tested systems.
  • By inferring heterogeneities, it reproduces the noisy experimental dynamics.
  • It provides a surrogate model for complex biophysical processes where mechanistic models are hard to build.

Where Pith is reading between the lines

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

  • Applying this to other noisy biological systems could help predict how environmental changes affect population-level behavior through altered heterogeneity.
  • Combining it with single-cell data might reveal which molecular factors drive the inferred extrinsic noise.
  • Validating against systems with independently measured noise sources would test if the separation of intrinsic and extrinsic effects holds.

Load-bearing premise

Trajectory data from heterogeneous cells can be reconstructed accurately by a neural SDE driven by extrinsic noise when distributions are matched using Wasserstein distance, without needing system-specific mechanistic details.

What would settle it

A new experiment on one of the systems where cell heterogeneity is deliberately reduced or increased, showing that the reconstructed model does not adjust its predictions accordingly.

Figures

Figures reproduced from arXiv: 2503.09007 by Alexander Hoffman, Alex Mogilner, Jiancheng Zhang, Lucas B\"ottcher, Mingtao Xia, Tom Chou, Xiangting Li, Xiaolu Guo, Zhaoyi You.

Figure 1
Figure 1. Figure 1: Workflow of our proposed END-nSDE prediction on parameters altering stochastic dynamics. A. The predicted trajectories are generated through the reconstructed SDE dXˆ = fˆ(Xˆ ; ω)dt+σˆ(Xˆ ; ω)dBt. B. The drift and diffusion functions, fˆ and σˆ, are approximated using parameterized neural networks. The parameterized neural-network-based drift function fˆ(Xˆ ; ω) and diffusion function σˆ(Xˆ ; ω) take the s… view at source ↗
Figure 2
Figure 2. Figure 2: A continuous-time discrete Markov chain model for multiple RPA molecules binding to long ssDNA. The possible steps in the biomolecular kinetics of RPA on ssDNA create a complex scenario involving multiple RPA molecules binding. The RPA in the free solution can bind to ssDNA with rate k1 provided there are at least 20 nu￾cleotides (nt) of consecutive unoccupied sites. This bound “20-nt mode” RPA unbinds wit… view at source ↗
Figure 3
Figure 3. Figure 3: Simplified schematic of the NFκB Signaling Network. TNF binds its receptor, activating IKK, which degrades IκBα and releases NFκB. The free NFκB translo￾cates to the nucleus and promotes IκBα transcription. Newly synthesized IκBα then binds NFκB and exports it back to the cytoplasm. Red arrows indicate noise that we consider in the corresponding SDE system. To incorporate the intrinsic noise within the NFκ… view at source ↗
Figure 4
Figure 4. Figure 4: Reconstructing the circadian model using END-nSDE. Temporally decoupled squared W2 losses Eq. (3) and errors in the reconstructed drift and diffusion functions for different types of the diffusion function and different values of (σ0, c). A-C. The temporally decoupled squared W2 loss between the ground truth trajectories and the trajectories generated by the reconstructed nSDEs for the constant-type diffus… view at source ↗
Figure 5
Figure 5. Figure 5: Reconstructed trajectories of the RPA-DNA binding problem. A. Sample ground truth and reconstructed trajectories evaluated at lg k2 = −4, where we use the convention that lg = log10. B. Sample ground truth and reconstructed parameters evaluated at lg k2 = −1.5. C. Temporally decoupled squared W2 distances (see Eq. (8)) between the ground truth and reconstructed trajectories evaluated at different lg k2 val… view at source ↗
Figure 6
Figure 6. Figure 6: Reconstruction of NFκB signaling dynamics. A. Sample trajectories of nuclear NFκB concentration as a function of time with σ1 = 10−3.2 , σ2 = 10−2.5 . B. Sample trajectories of nuclear NFκB concentration as a function of time with σ1 = 10−2.2 , σ2 = 10−1.5 . C. Reconstructed nuclear NFκB trajectories generated by the trained neural SDE versus the ground truth nuclear NFκB trajectories under noise intensiti… view at source ↗
Figure 7
Figure 7. Figure 7: Workflow of reconstructing experimental data via END-nSDE. Workflow for reconstructing experimental data using the trained parameterized nSDE and the parameter-inference neural network (NN). The boxes on the left outline the steps of the experimental data reconstruction process, while the boxes on the right illustrate the corresponding results at each step. IV. DISCUSSION In this work, we used a W2-distanc… view at source ↗
Figure 8
Figure 8. Figure 8: Inferring intrinsic noise intensities and reconstructing experimental data via END-nSDE. A. Plots showing the mean (solid circles) and variance (error bars) of the relative error in the reconstructed noise intensities (ˆσ1, σˆ2) predicted by the parameter-inference NN for the testing dataset, as a function of the group size of input trajectories. B. Heatmaps showing the relative error in the reconstructed … view at source ↗
read the original abstract

Proper regulation of cell signaling and gene expression is crucial for maintaining cellular function, development, and adaptation to environmental changes. Reaction dynamics in cell populations is often noisy because of (i) inherent stochasticity of intracellular biochemical reactions (``intrinsic noise'') and (ii) heterogeneity of cellular states across different cells that are influenced by external factors (``extrinsic noise''). In this work, we introduce an extrinsic-noise-driven neural stochastic differential equation (END-nSDE) framework that utilizes the Wasserstein distance to accurately reconstruct SDEs from trajectory data from a heterogeneous population of cells (extrinsic noise). We demonstrate the effectiveness of our approach using both simulated and experimental data from three different systems in cell biology: (i) circadian rhythms, (ii) RPA-DNA binding dynamics, and (iii) NF$\kappa$B signaling process. Our END-nSDE reconstruction method can model how cellular heterogeneity (extrinsic noise) modulates reaction dynamics in the presence of intrinsic noise. It also outperforms existing time-series analysis methods such as recurrent neural networks (RNNs) and long short-term memory networks (LSTMs). By inferring cellular heterogeneities from data, our END-nSDE reconstruction method can reproduce noisy dynamics observed in experiments. In summary, the reconstruction method we propose offers a useful surrogate modeling approach for complex biophysical processes, where high-fidelity mechanistic models may be impractical.

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

2 major / 2 minor

Summary. The manuscript introduces an extrinsic-noise-driven neural stochastic differential equation (END-nSDE) framework that trains a neural SDE by driving it with an extrinsic noise process and minimizing Wasserstein distance to observed single-cell trajectories. The central claim is that this reconstructs how cellular heterogeneity modulates intrinsic reaction dynamics in gene regulation, demonstrated on simulated and experimental data from circadian rhythms, RPA-DNA binding, and NFκB signaling, while outperforming RNNs and LSTMs as a surrogate model.

Significance. If the separation of extrinsic modulation from intrinsic noise holds and the learned dynamics are identifiable, the approach could offer a practical data-driven surrogate for noisy biophysical processes where mechanistic models are intractable, extending neural SDE methods to heterogeneous cell populations.

major comments (2)
  1. [Abstract and §3] Abstract and §3 (Methods): the claim that END-nSDE 'models how cellular heterogeneity modulates reaction dynamics' requires that the extrinsic driver plus learned drift/diffusion isolate modulation effects rather than merely matching marginal trajectory statistics via Wasserstein distance. No identifiability argument, recovery of known modulation parameters, or out-of-distribution test is described for the three cases, leaving the separation assumption untested.
  2. [§4] §4 (Results, circadian/RPA/NFκB cases): the reported outperformance over RNNs/LSTMs is stated without quantitative metrics, error bars, or ablation on the extrinsic-noise driver component, so it is unclear whether gains arise from the extrinsic driver or from the neural SDE architecture alone.
minor comments (2)
  1. [§2] Notation for the extrinsic noise process and its coupling to the neural SDE should be defined explicitly with an equation in §2 or §3.
  2. [Abstract] The abstract states 'reproduce noisy dynamics observed in experiments' but does not specify which experimental observables (e.g., period, amplitude distributions) are matched.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (Methods): the claim that END-nSDE 'models how cellular heterogeneity modulates reaction dynamics' requires that the extrinsic driver plus learned drift/diffusion isolate modulation effects rather than merely matching marginal trajectory statistics via Wasserstein distance. No identifiability argument, recovery of known modulation parameters, or out-of-distribution test is described for the three cases, leaving the separation assumption untested.

    Authors: The END-nSDE framework is constructed precisely to achieve this separation: an explicit extrinsic noise process (sampled to reflect observed cellular heterogeneity) is provided as a time-varying driver to the neural SDE, while the neural network learns only the drift and diffusion that govern the underlying reaction dynamics. Minimizing Wasserstein distance then aligns the generated trajectory distributions with those observed in heterogeneous populations. This design ensures the learned components capture modulation effects rather than merely reproducing marginal statistics. Although the manuscript does not contain a formal identifiability proof or out-of-distribution parameter-recovery tests, the consistent performance on both simulated data (where ground-truth modulation is known) and three experimental systems supports the validity of the approach for data-driven reconstruction. We therefore maintain the claim as stated and do not plan to alter it. revision: no

  2. Referee: [§4] §4 (Results, circadian/RPA/NFκB cases): the reported outperformance over RNNs/LSTMs is stated without quantitative metrics, error bars, or ablation on the extrinsic-noise driver component, so it is unclear whether gains arise from the extrinsic driver or from the neural SDE architecture alone.

    Authors: We agree that §4 would benefit from additional quantitative detail. In the revised manuscript we will report explicit performance metrics (e.g., Wasserstein distances and trajectory prediction errors) with error bars across multiple random seeds, together with an ablation that removes the extrinsic-noise driver while retaining the neural SDE architecture. revision: yes

Circularity Check

0 steps flagged

No circularity: data-driven neural SDE reconstruction is self-contained

full rationale

The paper presents END-nSDE as a neural SDE framework trained on trajectory data via Wasserstein distance matching to reconstruct dynamics under extrinsic noise. No derivation chain reduces a claimed result to its inputs by construction, no self-citations are load-bearing for the central method, and no fitted parameters are renamed as independent predictions. The approach is a standard surrogate modeling procedure validated on three separate biological cases, remaining self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities.

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Reference graph

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