Cell State Transitions Beyond the Small-Noise Limit

Jianzhe Wei; Jingwen Zhu; Liang Luo; Pan Chu; Xiongfei Fu

arxiv: 2510.07797 · v3 · pith:TMQRNPZ5new · submitted 2025-10-09 · ❄️ cond-mat.stat-mech · q-bio.CB· q-bio.MN

Cell State Transitions Beyond the Small-Noise Limit

Jianzhe Wei , Jingwen Zhu , Pan Chu , Liang Luo , Xiongfei Fu This is my paper

Pith reviewed 2026-05-21 21:40 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech q-bio.CBq-bio.MN

keywords cell state transitionssmall-noise regimeKramers theorysynthetic genetic circuitmother machinemultiplicative noisefirst-passage analysisbiological noise

0 comments

The pith

Single-cell tracking shows bacterial state transitions occur outside the small-noise regime with no single characteristic rate.

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

The paper reports the first single-cell observations of state transitions in a synthetic bacterial genetic circuit by tracking more than 1000 cells in a mother machine for 27 hours. First-passage analysis and dynamical reconstruction demonstrate that these transitions take place beyond the small-noise limit where classical Kramers theory applies. The process exhibits no single characteristic rate, which questions the assumption of discrete cell states. Significant multiplicative noise distorts the effective potential landscape while increasing the time required for transitions to complete. These results indicate that new theoretical frameworks are required to describe biological state transitions without relying on the small-noise assumption.

Core claim

State transitions in a synthetic bacterial genetic circuit, directly observed at single-cell resolution, occur outside the small-noise regime. First-passage statistics lack a single characteristic rate and dynamical reconstruction shows that multiplicative noise distorts the effective potential yet lengthens transition times, challenging the direct applicability of Kramers theory to biological systems.

What carries the argument

First-passage time statistics and dynamical reconstruction applied to single-cell trajectories recorded in a mother machine device.

If this is right

Cell state transitions cannot be modeled with a single exponential rate constant.
Multiplicative noise lengthens rather than shortens transition times despite distorting the potential landscape.
Classical Kramers escape formulas do not apply directly to these observed biological transitions.
Theoretical descriptions of cell fate must incorporate large multiplicative noise effects.

Where Pith is reading between the lines

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

Natural gene regulatory networks may routinely operate in this large-noise regime, producing more variable switching statistics than small-noise models predict.
Engineering synthetic switches should account for noise-induced slowing when designing circuits for reliable state changes.
The findings may extend to understanding how phenotypic variability arises in bacterial populations facing fluctuating environments.

Load-bearing premise

The observed first-passage statistics and reconstructed dynamics are taken to reflect the intrinsic transition process of the genetic circuit without major confounding from the mother-machine environment or circuit-specific artifacts.

What would settle it

Finding an exponential distribution of first-passage times with a clear single rate constant, or transition times that shorten with added noise strength as expected in the small-noise limit, would contradict the central claim.

Figures

Figures reproduced from arXiv: 2510.07797 by Jianzhe Wei, Jingwen Zhu, Liang Luo, Pan Chu, Xiongfei Fu.

**Figure 2.** Figure 2: FIG. 2. (a) Reconstructed landscape [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Mean first passage time (MFPT) [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

read the original abstract

State transitions are fundamental in biological systems but challenging to observe directly. Here, we present the first single-cell observation of state transitions in a synthetic bacterial genetic circuit. Using a mother machine, we tracked over 1007 cells for 27 hours. First-passage analysis and dynamical reconstruction reveal that transitions occur outside the small-noise regime, challenging the applicability of classical Kramers' theory. The process lacks a single characteristic rate, questioning the paradigm of transitions between discrete cell states. We observe significant multiplicative noise that distorts the effective potential landscape yet increases transition times. These findings necessitate theoretical frameworks for biological state transitions beyond the small-noise assumption.

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 tracks over 1000 cells in a mother machine and reports that state transitions in a synthetic circuit fall outside the small-noise Kramers regime, with multiplicative noise lengthening times and no single rate, but device artifacts remain a live concern.

read the letter

The main thing to know is that this work uses long-term single-cell tracking to argue that transitions in a synthetic bacterial circuit do not fit the small-noise limit that Kramers theory assumes. They see no single characteristic rate and find that multiplicative noise distorts the effective landscape while actually increasing transition times. That is the central experimental claim from over 1007 cells tracked for 27 hours with first-passage analysis and dynamical reconstruction. It is the first such direct observation in a synthetic circuit, which gives the result some weight if the data hold up. The scale of the experiment and the attempt to reconstruct dynamics from real trajectories are the parts that stand out as useful contributions. The authors engage the classical theory by testing where it breaks rather than just simulating alternatives. The soft spot is the mother-machine environment. Physical confinement, division events, and flushing couple directly to growth and local conditions, which can generate apparent multiplicative noise or stretch escape times without any change to the underlying circuit. The abstract gives no sign that divisions were masked, growth-rate fluctuations regressed out, or circuit-free controls run in parallel. Until those checks appear, the departure from small-noise behavior stays open to device-specific explanations. This paper is for people who model stochastic gene circuits and need to know when standard approximations stop working. Readers who care about experimental tests of noise effects in synthetic systems would find it worth reading. It deserves a serious referee because the claim is important if the methods check out, even though the current write-up leaves the controls underspecified. I would send it to review with a request for explicit details on post-processing and environmental controls.

Referee Report

2 major / 2 minor

Summary. The manuscript reports the first single-cell experimental observation of state transitions in a synthetic bacterial genetic circuit. Using a mother-machine microfluidic device, the authors track fluorescence trajectories in over 1007 cells across 27 hours. First-passage time analysis and dynamical reconstruction are used to argue that the transitions occur outside the small-noise regime (challenging Kramers' theory), that the process lacks a single characteristic rate (questioning discrete-state paradigms), and that significant multiplicative noise distorts the effective potential landscape while increasing transition times, thereby calling for new theoretical frameworks beyond the small-noise limit.

Significance. If the central claims hold after addressing potential experimental confounds, the work would be significant for stochastic dynamics in biology. It supplies direct, large-scale single-cell data on non-classical escape behavior in a controlled genetic circuit, which could motivate extensions to Kramers-type theories for gene regulation and cell-state switching. The scale of the dataset (>1000 cells) is a clear strength for statistical power in first-passage statistics.

major comments (2)

[Abstract and Experimental Methods] The central claim that transitions lie outside the small-noise regime and that multiplicative noise increases transition times rests on first-passage statistics and landscape reconstruction (Abstract). The manuscript provides no explicit description of how division events were masked, whether growth-rate fluctuations were regressed from the fluorescence trajectories, or whether parallel control strains without the circuit were measured. This is load-bearing: mother-machine confinement couples division timing, local nutrient depletion, and mechanical stress to the observed dynamics, which could generate apparent multiplicative noise and lengthened escape times without altering the intrinsic circuit potential.
[Results (dynamical reconstruction subsection)] The dynamical reconstruction is presented as evidence that multiplicative noise distorts the effective potential yet prolongs transitions. No quantitative error bars, bootstrap uncertainties, or direct comparison to small-noise Kramers predictions (e.g., escape-time scaling with barrier height) are reported for the reconstructed quantities. Without these, the departure from classical theory cannot be rigorously assessed and remains vulnerable to post-processing choices.

minor comments (2)

[Abstract] The abstract states 'over 1007 cells' but supplies no exact count, no mention of error bars, and no statistical tests supporting the key claims about rates and noise effects.
[Results] Notation for the reconstructed potential and noise terms should be defined more clearly when first introduced to aid readers unfamiliar with the specific circuit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the importance of addressing potential experimental confounds and providing quantitative support for the claims. We respond to each major comment below and indicate the revisions that will be incorporated.

read point-by-point responses

Referee: [Abstract and Experimental Methods] The central claim that transitions lie outside the small-noise regime and that multiplicative noise increases transition times rests on first-passage statistics and landscape reconstruction (Abstract). The manuscript provides no explicit description of how division events were masked, whether growth-rate fluctuations were regressed from the fluorescence trajectories, or whether parallel control strains without the circuit were measured. This is load-bearing: mother-machine confinement couples division timing, local nutrient depletion, and mechanical stress to the observed dynamics, which could generate apparent multiplicative noise and lengthened escape times without altering the intrinsic circuit potential.

Authors: We agree that explicit details on data processing steps are necessary to rule out confounds and support the central claims. In the revised manuscript we will expand the Experimental Methods section to describe the procedure used to mask division events in the fluorescence trajectories, the regression of growth-rate fluctuations, and the decision not to include parallel control strains (the experimental design focused on the synthetic circuit under standardized mother-machine conditions). We will also add a brief analysis showing that growth-rate variations do not account for the observed multiplicative noise structure, thereby addressing the possibility that confinement effects dominate the reported dynamics. revision: yes
Referee: [Results (dynamical reconstruction subsection)] The dynamical reconstruction is presented as evidence that multiplicative noise distorts the effective potential yet prolongs transitions. No quantitative error bars, bootstrap uncertainties, or direct comparison to small-noise Kramers predictions (e.g., escape-time scaling with barrier height) are reported for the reconstructed quantities. Without these, the departure from classical theory cannot be rigorously assessed and remains vulnerable to post-processing choices.

Authors: We acknowledge that the absence of uncertainty quantification and explicit comparisons limits the strength of the evidence for departure from the small-noise regime. In the revision we will add bootstrap-derived error bars and uncertainty estimates to all reconstructed quantities (potential, noise amplitude, and first-passage statistics). We will also include a direct comparison of the measured escape times against the scaling predicted by small-noise Kramers theory as a function of barrier height, allowing a quantitative assessment of the observed lengthening of transition times. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical measurements and reconstruction stand independently

full rationale

The paper reports direct single-cell tracking in a mother-machine device (>1007 cells, 27 h) followed by first-passage analysis and dynamical reconstruction. These yield empirical statements about escape times, absence of a single rate, and the effect of multiplicative noise on an effective landscape. No derivation chain is presented that reduces a claimed prediction or first-principles result back to a fitted parameter, self-citation, or definitional identity; the central claims are statistical summaries of observed trajectories rather than closed mathematical loops. External controls or artifact checks are not addressed in the provided text, but that is a question of experimental validity, not circularity of derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract does not introduce new free parameters, axioms, or invented entities; claims rest on experimental data and standard first-passage analysis.

pith-pipeline@v0.9.0 · 5644 in / 1009 out tokens · 24233 ms · 2026-05-21T21:40:21.767311+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

transitions occur outside the small-noise regime... multiplicative noise that distorts the effective potential landscape yet increases transition times
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean costAlphaLog_high_calibrated_iff unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

reconstructed potential landscape U(r) ... rescaled landscape Ursc(r)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages

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[1] [1]

M. B. Elowitz and S. Leibler, A synthetic oscillatory net- work of transcriptional regulators, Nature403, 335 (2000)

work page 2000

[2] [2]

T. S. Gardner, C. R. Cantor, and J. J. Collins, Construc- tion of a genetic toggle switch in Escherichia coli, Nature 403, 339 (2000)

work page 2000

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E. M. Ozbudak, M. Thattai, H. N. Lim, B. I. Shraiman, and A. van Oudenaarden, Multistability in the lactose uti- lization network of, 427 (2004)

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Zhang, H

R. Zhang, H. Goetz, J. Melendez-Alvarez, J. Li, T. Ding, X. Wang, and X.-J. Tian, Winner-takes-all resource com- petition redirects cascading cell fate transitions, Nature Communications 12, 853 (2021)

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Weinreb, A

C. Weinreb, A. Rodriguez-Fraticelli, F. D. Camargo, and A. M. Klein, Lineage tracing on transcriptional landscapes links state to fate during differentiation, Science 367, eaaw3381 (2020)

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H. A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, physica 7, 284 (1940)

work page 1940

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H¨ anggi, P

P. H¨ anggi, P. Talkner, and M. Borkovec, Reaction-rate theory: Fifty years after Kramers, Reviews of Modern Physics 62, 251 (1990)

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work page 1991

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Ao, Potential in stochastic differential equations: Novel construction, Journal of Physics A: Mathematical and General 37, L25 (2004)

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work page 2004

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Y. Tang, R. Yuan, G. Wang, X. Zhu, and P. Ao, Po- tential landscape of high dimensional nonlinear stochas- tic dynamics with large noise, Scientific Reports 7, 15762 (2017)

work page 2017

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X. Fang, K. Kruse, T. Lu, and J. Wang, Nonequilibrium physics in biology, Reviews of Modern Physics 91, 045004 (2019)

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Stolovitzky and E

G. Stolovitzky and E. S. Ching, Characterization of stationary distributions using conditional expectations, Physics Letters A 255, 11 (1999)

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R. Friedrich, J. Peinke, M. Sahimi, and M. R. Rahimi Tabar, Approaching complexity by stochastic methods: From biological systems to turbulence, Physics Reports (2011)

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Rosas, I

A. Rosas, I. L. D. Pinto, and K. Lindenberg, Kramers’ rate for systems with multiplicative noise, Physical Review E 94, 012101 (2016)

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M. V. Moreno, D. G. Barci, and Z. G. Arenas, State- dependent diffusion in a bistable potential: Conditional probabilities and escape rates, Physical Review E 101, 062110 (2020)

work page 2020

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M. A. Coomer, L. Ham, and M. P. Stumpf, Noise dis- torts the epigenetic landscape and shapes cell-fate deci- sions, Cell Systems 13, 83 (2022)

work page 2022

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Risken, The Fokker-Planck Equation: Methods of So- lution and Applications , 2nd ed., Springer Series in Syner- getics No

H. Risken, The Fokker-Planck Equation: Methods of So- lution and Applications , 2nd ed., Springer Series in Syner- getics No. v. 18 (Springer-Verlag, New York, 1996)

work page 1996

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Van Kampen, Relative stability in nonuniform tem- perature, IBM Journal of Research and Development 32, 107 (1988)

N. Van Kampen, Relative stability in nonuniform tem- perature, IBM Journal of Research and Development 32, 107 (1988)

work page 1988

[26] [26]

J. Wang, Y. Zhang, and H. Zhao, Non-Gaussian normal diffusion induced by delocalization, Physical Review E93, 032144 (2016)

work page 2016

[27] [27]

Luo and M

L. Luo and M. Yi, Non-Gaussian diffusion in static dis- ordered media, Physical Review E 97, 042122 (2018)

work page 2018

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Pacheco-Pozo and I

A. Pacheco-Pozo and I. M. Sokolov, Convergence to a Gaussian by Narrowing of Central Peak in Brownian yet Non-Gaussian Diffusion in Disordered Environments, Physical Review Letters 127, 120601 (2021)

work page 2021

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Cell Type

What Is Your Conceptual Definition of “Cell Type” in the Context of a Mature Organism?, Cell Systems 4, 255 (2017)

work page 2017

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Mulas, A

C. Mulas, A. Chaigne, A. Smith, and K. J. Chalut, Cell state transitions: Definitions and challenges, Development 148, dev199950 (2021)

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Chahine, R

J. Chahine, R. J. Oliveira, V. B. P. Leite, and J. Wang, Configuration-dependent diffusion can shift the kinetic transition state and barrier height of protein folding, Pro- ceedings of the National Academy of Sciences 104, 14646 (2007)

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P. S. Stumpf, R. C. Smith, M. Lenz, A. Schuppert, F.-J. M¨ uller, A. Babtie, T. E. Chan, M. P. Stumpf, C. P. Please, S. D. Howison, F. Arai, and B. D. MacArthur, Stem Cell Differentiation as a Non-Markov Stochastic Process, Cell Systems 5, 268 (2017)

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Weinreb, S

C. Weinreb, S. Wolock, B. K. Tusi, M. Socolovsky, and A. M. Klein, Fundamental limits on dynamic infer- ence from single-cell snapshots, Proceedings of the Na- tional Academy of Sciences115, 10.1073/pnas.1714723115 (2018)

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Non-Kramers State Transitions in a Synthetic Toggle Switch Biosystem

X. Qiu, Y. Zhang, J. D. Martin-Rufino, C. Weng, S. Hos- seinzadeh, D. Yang, A. N. Pogson, M. Y. Hein, K. Hoi (Joseph) Min, L. Wang, E. I. Grody, M. J. Shurtleff, R. Yuan, S. Xu, Y. Ma, J. M. Replogle, E. S. Lander, S. Darmanis, I. Bahar, V. G. Sankaran, J. Xing, and J. S. Weissman, Mapping transcriptomic vector fields of single cells, Cell 185, 690 (2022)...

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A pre-trained model was used to identify and segment side channels within each FOV

Image Registration and Channel Detection: Time-lapse images from each field-of-view (FOV) were first regis- tered to correct for XY drift caused by stage movement. A pre-trained model was used to identify and segment side channels within each FOV

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Edge refinement was performed using Otsu’s thresholding algorithm to enhance cell boundary detection

Cell Segmentation: Individual cells were segmented using the re-trained segmentation model, which was trained by our own mother machine data to recognize bacterial morphology. Edge refinement was performed using Otsu’s thresholding algorithm to enhance cell boundary detection

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From these, cell parameters—including mask, length, width, and area—were extracted using a channel-aligned coordinate system

Cell Geometry Extraction: Cell midlines were calculated via interpolation to provide initial estimates of cell geometry. From these, cell parameters—including mask, length, width, and area—were extracted using a channel-aligned coordinate system

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Background fluorescence was estimated using the median pixel intensity of each channel

Fluorescence Quantification: Fluorescent protein expression levels were quantified by applying the segmented masks to fluorescence images. Background fluorescence was estimated using the median pixel intensity of each channel. The cellular fluorescence signal was computed by subtracting the background from the median intensity of pixels within each cell m...

work page 2010