You ain't seen nothing, and yet: Future biochemical concentrations can be predicted with surprisingly high accuracy

Ilya Nemenman; Ketevan Danelia; Sean A. Ridout

REVIEW 3 major objections 4 minor 51 references

You ain't seen nothing, and yet: Future biochemical concentrations can be predicted with surprisingly high accuracy

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read A cell that knows the shape of a signal can read its final strength early, with accuracy better than simple molecule counting.

desk verdict A clean, genuinely new result for sensing with known structured profiles; the super-Poisson scaling depends on exact knowledge of a and k, and the paper needs to face that limitation before publication. read the letter →

arxiv 2602.00261 v2 pith:L4F3QZQF submitted 2026-01-30 physics.bio-ph q-bio.MN

Ketevan Danelia , Sean A. Ridout , Ilya Nemenman This is my paper · ORCID

classification physics.bio-phq-bio.MN

keywords concentrationsensingBayesianinferencemaximumaposterioriestimationBerg-Purcelllimitshotnoisemorphogengradientpositionalinformationdevelopmentalbiology

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 when a cell knows the family of possible concentration profiles (as in embryogenesis), it can treat concentration sensing as Bayesian inference over that family instead of merely counting molecules. For profiles whose early rate of rise correlates strongly with the eventual plateau level — c(t) ≈ k c_∞^a t before saturation — the MAP estimator achieves fractional error 1/(a√N) for N binding events. For a > 1 this beats the standard Poisson counting limit 1/√N, so the cell can estimate the future steady-state concentration (and hence its position in the embryo) before the gradient has settled. The authors show the required computations can be implemented by simple biochemical reaction networks, and connect the result to the speed–precision puzzle in Drosophila development.

What carries the argument

The key object is the maximum a posteriori (MAP) estimator over a parameterized family of concentration profiles c_θ(t), where each binding time is drawn from an inhomogeneous Poisson process with rate r c_θ(t). The workhorse identity is the Cramér–Rao bound applied to the log-posterior: ε² ≈ [∫₀ᵀ (r/c̄)(∂c̄/∂θ)² dt]⁻¹. For the piecewise-linear saturating profiles, this integral evaluates to a²N, directly yielding the super-Poisson error scaling. The analysis also derives a stochastic differential equation (Eq. 17) for how the estimate updates between and at binding events, and shows that for a=2 the update rule reduces to operations (ratios of clock-measured time squared and binding count)

What would settle it

Fit a measured mutant morphogen profile where the early rise and the eventual plateau are decoupled (so the assumed c(t)=k c_∞^a t form fails). If the precision of downstream boundary placement does not degrade from the predicted 1/(a√N) scaling, or if a sensor using only post-saturation events matches the full-trajectory predictor, the central claim would be contradicted. A cleaner test: in an embryo with a manipulated early profile (e.g., shifted production onset), measure whether the MAP-predicted boundary position — computed from binding events in the first minutes — anticipates the eventu

Watch

Extended reading notes

Core claim

The central claim is that prior knowledge of a restricted class of spatiotemporal concentration profiles converts the sensing problem from estimating an unknown concentration to identifying which member of a known family produced the observed binding times. In the piecewise-linear model c(c∞, t) = k c_∞^a t for t < c_∞^{1-a}/k and c∞ afterward, the MAP estimator has variance ε²/c∞² ≈ 1/(a²N) during the growth phase. Since N is the number of binding events, a > 1 means the relative error is smaller than 1/√N, the Poisson limit that bounds classical maximum-likelihood and Berg–Purcell sensing. The same scaling holds for positional inference, with the improvement mirrored through the profile's

Load-bearing premise

The sensor must know the exact parametric family of profiles (including the exponent a), the prior over the parameter, and the time origin measured from the arrival of the first molecules; if the real profile deviates from the assumed family, or the cell cannot establish t=0 and the prior, the 1/(a√N) accuracy is not guaranteed.

Editorial extensions

If this is right

A nucleus in a developing embryo can infer its position before the morphogen gradient reaches steady state, using binding events from the very beginning of the profile.
The effective sensing period extends back to the arrival of the first molecules, so N counts events from the start of the process, not just from stabilization — multiplying accuracy gains.
For profiles with a≈1.8 (as fitted to a diffusion–degradation model of Bicoid), the error prefactor is roughly 3.2 times smaller (1/a² ≈ 0.31), a substantial improvement over Poisson counting.
The super-Poisson advantage disappears (a=1) or reverses (a<1) when early dynamics do not amplify differences in the eventual saturation value, so the effect is specific to strongly coupled profiles.
The same MAP formalism applies to any reproducible, pre-programmed spatiotemporal signal, not just development, with the appropriate prior over profiles.

Reading between the lines

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

A direct implication the authors leave implicit: the improvement is equivalent to an effective count of a²N, so a downstream readout that behaves like a counter would need to count correlated events to mimic the Bayesian estimator.
One could test the theory experimentally by perturbing the early phase of a morphogen profile (e.g., shifting production rate or start time) and measuring whether downstream gene expression boundaries track the MAP prediction rather than the instantaneous concentration.
The requirement that the sensor know t=0 (arrival of first molecules) suggests a testable prediction: if a cell cannot set time origin reliably, the predicted 1/(a√N) precision should degrade; this could be probed by artificially varying the delay between fertilization and the first detectable binding.
The biochemical implementation shown for a=2 suggests a design principle: cells may use constitutive clocks (protein accumulation) to normalize event counts by time factors, which could be sought in real regulatory networks.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The headline: this paper adds something real to the biochemical sensing literature. It shows that if a cell has a prior over a known class of monotonically saturating concentration profiles, MAP estimation can predict the eventual steady-state concentration with error ε²/c∞² ≈ 1/(a²N) — better than the Poisson counting 1/N when the profile exponent a>1. The Fisher-information calculation is straightforward and clean, the constant-concentration limit recovers the known Endres–Wingreen/ML result, and the piecewise-linear model is explicitly presented as an analytically tractable surrogate, not as a literal morphogen profile. The biochemical reaction-network sketch is a plausible existence proof, and the numerics line up.

The soft spots are real. The most important is the parameter-identifiability issue: in the pre-saturation regime the likelihood depends on c∞, a, and k only through k c∞^a. With a and k unknown, the Fisher information matrix for the triple is rank 1, so no amount of data separates c∞ from a or k. Any continuous prior on a or k leaves a posterior width in log c∞ that does not vanish as N→∞. So the 1/(a²N) scaling is only as good as the sensor's exact knowledge of a and k. The reader's note flagged this indirectly; the stress-test makes it precise. The paper should own this limitation and either argue that a and k are effectively hardwired constants the cell knows, or analyze the error with priors over them. The start-time assumption is similarly idealized: time measured from arrival of first molecules may be physically accessible, but the paper gives it only a passing mention.

These are caveats, not fatal flaws. The core mathematical claim holds as a conditional limit: given the profile family and parameters, the bound is exactly what the paper says. The paper is honest about the idealized setup, and the Drosophila discussion is appropriately speculative. I'd take it as a solid theoretical contribution.

Recommendation: send to peer review. The authors need to add a section on robustness to unknown profile parameters and start-time uncertainty; otherwise a sharp referee will raise the identifiability point, and it should be addressed before publication. I would cite it.

Referee Report

3 major / 4 minor

Summary. The paper studies the precision with which a cell can estimate the future steady-state concentration (or its spatial position in an embryo) from the stochastic binding times of ligand molecules to a receptor, when the concentration profile is known to belong to a parametric family. The authors formulate the problem as Bayesian MAP estimation for an inhomogeneous Poisson process. For a piecewise-linear saturating profile c(t) = k c_∞^a t for t < t1 and c∞ thereafter, they derive ε²/c∞² ≈ 1/(a²N) when the measurement is made during the growth phase, where N is the total number of binding events. Since a≈1.8 for a simple Bicoid model, this beats the Poisson counting limit 1/N. They further show that a simple biochemical network can approximately implement the MAP estimator. The constant-concentration limit recovers the standard 1/N result, and simulations in Fig. 3(b) support the error scaling.

Significance. If correct, the result is significant because it identifies a mechanism—using prior knowledge of concentration-profile structure—by which cells could extract more than Poisson-level information from a fixed number of binding events and predict future concentrations before steady state is reached. This could contribute to explaining the speed-precision paradox in developmental decisions. The Fisher-information calculation is transparent and internally consistent; the authors are explicit that the piecewise-linear profile is an analytically tractable representative rather than a full biophysical model. The paper also provides a concrete biochemical network implementation, which is a strength. The main weakness is that the advertised super-Poisson scaling requires the sensor to know the exact profile family, including parameters a and k and the start time, and the paper does not analyze the consequences of relaxing these assumptions.

major comments (3)

[Section III, Eq. (26)] The likelihood for T ≤ t1 depends on (c∞, a, k) only through μ = r k c∞^a. The 3×3 Fisher information matrix for (c∞, a, k) is thereby rank 1, so if a or k are not known exactly, the posterior variance of c∞ does not vanish as N→∞; the ε²/c∞² ≈ 1/(a²N) scaling is a property of the perfectly calibrated model. The manuscript does not state this assumption explicitly or discuss its implications. Please add an explicit statement that a and k are assumed known, and analyze the robustness of the result, e.g., by placing priors on log a and log k and showing how the error floor scales with their widths. The biochemical network in Eqs. (29)–(33) similarly requires rate constants tuned to 1/(rk), which is a strong requirement.
[Section III, before Eq. (23)] The derivation sets t=0 at the arrival of the first molecules, but the sensor cannot observe this time until the first binding event occurs. The paper acknowledges that 'a sensor cannot know that development has already started' but then proceeds as if the start time were known. An unknown start time t0 adds a parameter; even if it is identifiable in principle, it will increase the estimation error and may weaken the improvement over the Poisson limit. Please analyze this case or add a caveat to the claims of prediction accuracy.
[Section V] The Discussion and abstract present the super-Poisson result as a general property of structured concentration profiles, but the main derivation is for the specific piecewise-linear family in Eq. (23). The only link to actual Bicoid profiles is the scaling of the maximum rate of change, v ~ c∞^1.8 (Fig. 1(c)). The paper should state more explicitly that the ε²/c∞² ≈ 1/(a²N) result is proved only for this representative family, and that the Fisher information for realistic saturating profiles may differ.

minor comments (4)

[Section II, after Eq. (3)] Typo: 'renrmalize' should be 'renormalize'.
[Section IV, after Eq. (33)] The text refers to 'D ss', but no variable D has been defined; this should likely read 'Z ss' or the appropriate readout variable.
[Section II, after Eq. (19)] The statement 'Numerical averages over n=100 repetitions produced estimator means and variances consistent with our calculations here in the N≫1 limit (not shown)' is unsupported because no results are shown. Please either include the data or cite the simulation in Fig. 3(b) as the verification.
[Section III, Eq. (24)] The estimator is implicit because t1 depends on c∞. The paper notes this, but a short demonstration of how the self-consistency is resolved in practice would improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central Fisher-information derivation is self-contained and does not reduce to fitted outputs or self-citations.

full rationale

The paper's central claim is derived from an explicitly stated inhomogeneous Poisson likelihood (Eq. 3) and an explicitly parameterized concentration profile family (Eq. 23). Equation (26) is obtained by computing the expected Fisher information for c_infty in that model; the relation N(T)=r k c_infty^a T^2/2 and the error epsilon^2/c_infty^2 = 1/(a^2 N) follow from the same likelihood, not from fitting the predicted error to sensing data. The value a ~ 1.8 is imported from a separate diffusion-degradation simulation (Fig. 1c), so the super-Poisson prefactor is not a re-fit of the quantity the paper claims to predict. No load-bearing step reduces to a self-citation. Prior papers by the authors, e.g. [17], are used for background, for a standard log-normal fluctuation model in Appendix A, and for biochemical implementation examples, but the central concentration-estimation result does not depend on those citations for its mathematical content. The paper itself acknowledges that practical implementation remains an open question and that the piecewise-linear profile is an analytically tractable representative rather than the full biological profile; these are limitations, not circular steps. The possible unidentifiability of c_infty when a and k are unknown is a robustness/assumption concern about the model class, not a case where the derivation is equivalent to its inputs by construction.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The derivation is a Fisher information calculation for an assumed model. It has one structural input (a) that controls the claimed advantage, plus standard domain assumptions about Poisson binding, known priors, and exact knowledge of the profile class and start time. The exponent a is the main external input; it is estimated from a separate biophysical simulation, not from the sensing error itself.

free parameters (2)

a (profile exponent) = ≈1.8 from a minimal Bicoid diffusion-degradation simulation (Fig. 1c); the model assumes a>1 for the claimed improvement
Appears in c(t)=k c_∞^a t (Eq. 23); the central error formula ε²/c_∞²=1/(a²N) inherits this exponent. It is not fit to sensing-error data, but it is an input estimated from a separate biophysical model.
k (profile rate constant) = not assigned in the paper (model input)
Parameter of the assumed profile family (Eq. 23). It cancels or appears only through N in the final error formula, so it is not a hidden constraint on the main scaling result.

assumptions (6)

standard math Cramér-Rao bound / inverse Fisher information approximates estimator variance in the large-N limit
Eq. (7) is used to derive all error formulas; standard asymptotic estimation theory.
domain assumption Concentration profile is deterministic given θ: P(c_θ(t)|θ)=δ(c_θ(t)−c̄_θ(t))
Eq. (4) removes concentration noise in the main derivation; Appendix A adds small fluctuations but does not change the leading-order estimator.
domain assumption Binding events form an inhomogeneous Poisson process with rate r c(t); bound times are ignored
Eq. (3) adopts the standard Berg-Purcell-type idealization, stated explicitly in Section II.
domain assumption The sensor knows the prior P(θ) and the profile family
Section II: 'we assume [the prior] is also known to the sensor.' The whole MAP advantage depends on this knowledge.
domain assumption Time is measured from arrival of the first molecules; the sensor has a clock that knows T
Section III: 'time is measured from the arrival of the first molecules to the sensor'; Section IV introduces clock species X,Y. The paper does not explain how a cell knows this start time.
ad hoc to paper Piecewise-linear saturating profile c(t)=k c_∞^a t for t<t_1 is representative of developmental profiles
Eq. (23) is chosen for analytic tractability and motivated by the simulation in Fig. 1(c); it is not derived from first principles.

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Pith. "Pith review of You ain't seen nothing, and yet: Future biochemical concentrations can be predicted with surprisingly high accuracy." pith.science (2026). https://pith.science/paper/L4F3QZQF

@misc{pith2026260200261,
  author       = {Pith},
  title        = {Pith review of: You ain't seen nothing, and yet: Future biochemical concentrations can be predicted with surprisingly high accuracy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L4F3QZQF}},
  note         = {Machine review of arXiv:2602.00261}
}

abstract

Accurate sensing of chemical concentrations is essential for numerous biological processes. The accuracy of this sensing, for small numbers of molecules, is limited by shot noise. Corresponding theoretical limits on sensing precision, as a function of sensing duration, have been well-studied in the context of quasi-static and randomly fluctuating concentrations. However, during development and in many other cases, concentration profiles are not random but exhibit predictable spatiotemporal patterns. We propose that leveraging prior knowledge of these structured profiles can improve and accelerate concentration sensing by utilizing information from current molecular binding events to predict future concentrations. By framing the constrained sensing problem as Bayesian inference over an allowed class of spatiotemporal profiles, we derive new theoretical limits on sensing accuracy. Our analysis reveals that maximum a posteriori (MAP) estimation can outperform the classical Berg-Purcell and maximum-likelihood (Poisson counting) limits, achieving a sensing precision of $\delta c/c = 1/\sqrt{a^2N}$, where $N$ is the number of binding events, and $a > 1$ in certain cases. Thus knowledge of the statistical structure of concentration profiles enhances sensing precision, providing a potential explanation for the rapid yet highly accurate cell fate decisions observed during development.

Figures

Figures reproduced from arXiv: 2602.00261 by the authors.

**Figure 2.** FIG. 2. Model piecewise-linear concentration profiles defined in Eq. (23) for different values of the parameter [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3. Concentration estimation by a biochemical net [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

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Works this paper leans on

51 extracted references · 1 linked inside Pith

[1]

Tkaˇ cik and W

G. Tkaˇ cik and W. Bialek, Annual Review of Condensed Matter Physics7, 89 (2016)

2016
[2]

I. M. De la Fuente, L. Mart ´ ınez, J. Carrasco-Pujante, M. Fedetz, J. I. L´ opez, and I. Malaina, Frontiers in ge- netics12, 644615 (2021)

2021
[3]

Levchenko and I

A. Levchenko and I. Nemenman, Current Opin. Biotechn. 28, 156 (2014)

2014
[4]

Driever and C

W. Driever and C. N¨ usslein-Volhard, Cell54, 83 (1988)

1988
[5]

M. J. Berridge, P. Cobbold, and K. Cuthbertson, Philo- sophical Transactions of the Royal Society of London. B, Biological Sciences320, 325 (1988)

1988
[6]

Niethammer, C

P. Niethammer, C. Grabher, A. T. Look, and T. J. 9 Mitchison, Nature459, 996 (2009)

2009
[7]

Atay and J

O. Atay and J. M. Skotheim, Journal of Cell Biology216, 317 (2017)

2017
[8]

H. C. Berg and E. M. Purcell, Biophysical journal20, 193 (1977)

1977

Show all 51 references

[9]

R. G. Endres and N. S. Wingreen, Physical review letters 103, 158101 (2009)

2009
[10]

Bialek and S

W. Bialek and S. Setayeshgar, Proceedings of the Na- tional Academy of Sciences102, 10040 (2005)

2005
[11]

Kaizu, W

K. Kaizu, W. De Ronde, J. Paijmans, K. Takahashi, F. Tostevin, and P. R. Ten Wolde, Biophysical journal 106, 976 (2014)

2014
[12]

R. G. Endres and N. S. Wingreen, Progress in biophysics and molecular biology100, 33 (2009)

2009
[13]

B. Hu, W. Chen, W.-J. Rappel, and H. Levine, Physical review letters105, 048104 (2010)

2010
[14]

Ellison, A

D. Ellison, A. Mugler, M. D. Brennan, S. H. Lee, R. J. Huebner, E. R. Shamir, L. A. Woo, J. Kim, P. Amar, I. Nemenman,et al., Proceedings of the Na- tional Academy of Sciences113, E679 (2016)

2016
[15]

Mugler, A

A. Mugler, A. Levchenko, and I. Nemenman, Proceedings of the National Academy of Sciences113, E689 (2016)

2016
[16]

Mora and N

T. Mora and N. S. Wingreen, Physical review letters104, 248101 (2010)

2010
[17]

Mora and I

T. Mora and I. Nemenman, Physical review letters123, 198101 (2019)

2019
[18]

Singh, M

V. Singh, M. Tchernookov, and I. Nemenman, Physical Review E94, 022425 (2016)

2016
[19]

E. D. Siggia and M. Vergassola, Proceedings of the Na- tional Academy of Sciences110, E3704 (2013)

2013
[20]

Desponds, M

J. Desponds, M. Vergassola, and A. M. Walczak, Elife9, e49758 (2020)

2020
[21]

Malaguti and P

G. Malaguti and P. R. Ten Wolde, Elife10, e62574 (2021)

2021
[22]

Sartori and Y

P. Sartori and Y. Tu, Journal of statistical physics142, 1206 (2011)

2011
[23]

Smith, S

T. Smith, S. Fancher, A. Levchenko, I. Nemenman, and A. Mugler, Physical biology13, 035004 (2016)

2016
[24]

Singh and I

V. Singh and I. Nemenman, PLoS computational biology 13, e1005490 (2017)

2017
[25]

Carballo-Pacheco, J

M. Carballo-Pacheco, J. Desponds, T. Gavrilchenko, A. Mayer, R. Prizak, G. Reddy, I. Nemenman, and T. Mora, Physical Review E99, 022423 (2019)

2019
[26]

Paulsson, Nature427, 415 (2004)

J. Paulsson, Nature427, 415 (2004)

2004
[27]

T. E. Saunders and M. Howard, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics80, 041902 (2009)

2009
[28]

Tostevin and P

F. Tostevin and P. R. Ten Wolde, Physical review letters 102, 218101 (2009)

2009
[29]

Struhl, K

G. Struhl, K. Struhl, and P. M. Macdonald, Cell57, 1259 (1989)

1989
[30]

T. J. Rink and R. Jacob, Trends in Neurosciences12, 43 (1989)

1989
[31]

Wolpert, Journal of theoretical biology25, 1 (1969)

L. Wolpert, Journal of theoretical biology25, 1 (1969)

1969
[32]

J. C. Smith, Cold Spring Harbor perspectives in biology 1, a002477 (2009)

2009
[33]

Gregor, E

T. Gregor, E. F. Wieschaus, A. P. McGregor, W. Bialek, and D. W. Tank, Cell130, 141 (2007)

2007
[34]

K. W. Rogers and A. F. Schier, Annual review of cell and developmental biology27, 377 (2011)

2011
[35]

Wieschaus, Current topics in developmental biology 117, 567 (2016)

E. Wieschaus, Current topics in developmental biology 117, 567 (2016)

2016
[36]

Crick, Nature225, 420 (1970)

F. Crick, Nature225, 420 (1970)

1970
[37]

Kicheva, T

A. Kicheva, T. Bollenbach, O. Wartlick, F. J¨ ulicher, and M. Gonzalez-Gaitan, Current opinion in genetics & de- velopment22, 527 (2012)

2012
[38]

S. M. Kay,Fundamentals of statistical signal processing: estimation theory(Prentice-Hall, Inc., 1993)

1993
[39]

Bergmann, O

S. Bergmann, O. Sandler, H. Sberro, S. Shnider, E. Schejter, B.-Z. Shilo, and N. Barkai, PLoS biology 5, e46 (2007)

2007
[40]

Abu-Arish, A

A. Abu-Arish, A. Porcher, A. Czerwonka, N. Dostatni, and C. Fradin, Biophysical journal99, L33 (2010)

2010
[41]

Mora, arXiv preprint arXiv:1504.07203 (2015)

T. Mora, arXiv preprint arXiv:1504.07203 (2015)

2015 arXiv
[42]

Gregor, D

T. Gregor, D. W. Tank, E. F. Wieschaus, and W. Bialek, Cell130, 153 (2007)

2007
[43]

Tkaˇ cik, C

G. Tkaˇ cik, C. G. Callan Jr, and W. Bialek, Proceedings of the National Academy of Sciences105, 12265 (2008)

2008
[44]

Surkova, A

Manu, S. Surkova, A. V. Spirov, V. V. Gursky, H. Janssens, A.-R. Kim, O. Radulescu, C. E. Vanario- Alonso, D. H. Sharp, M. Samsonova,et al., PLoS biology 7, e1000049 (2009)

2009
[45]

H. G. Garcia, M. Tikhonov, A. Lin, and T. Gregor, Cur- rent biology23, 2140 (2013)

2013
[46]

Ali-Murthy and T

Z. Ali-Murthy and T. B. Kornberg, Elife5, e13222 (2016)

2016
[47]

Huang, C

A. Huang, C. Amourda, S. Zhang, N. S. Tolwinski, and T. E. Saunders, Elife6, e26258 (2017)

2017
[48]

Hern´ andez-L´ opez, A

C. Hern´ andez-L´ opez, A. Puliafito, Y. Xu, Z. Lu, S. Di Talia, and M. Vergassola, Proceedings of the Na- tional Academy of Sciences120, e2302879120 (2023)

2023
[49]

T. R. Sokolowski, T. Gregor, W. Bialek, and G. Tkaˇ cik, Proceedings of the National Academy of Sciences122, e2402925121 (2025)

2025
[50]

M. B. Elowitz, A. J. Levine, E. D. Siggia, and P. S. Swain, Science297, 1183 (2002)

2002
[51]

J. M. Raser and E. K. O’shea, Science309, 2010 (2005). Appendix A: Effect of small stochastic fluctuations around the mean concentration In the main text, we considered the case where, given the parameterθ, the concentration trajectory was determinis- tic. However, in addition...

2010

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Tkaˇ cik and W

G. Tkaˇ cik and W. Bialek, Annual Review of Condensed Matter Physics7, 89 (2016)

2016

[2] [2]

I. M. De la Fuente, L. Mart ´ ınez, J. Carrasco-Pujante, M. Fedetz, J. I. L´ opez, and I. Malaina, Frontiers in ge- netics12, 644615 (2021)

2021

[3] [3]

Levchenko and I

A. Levchenko and I. Nemenman, Current Opin. Biotechn. 28, 156 (2014)

2014

[4] [4]

Driever and C

W. Driever and C. N¨ usslein-Volhard, Cell54, 83 (1988)

1988

[5] [5]

M. J. Berridge, P. Cobbold, and K. Cuthbertson, Philo- sophical Transactions of the Royal Society of London. B, Biological Sciences320, 325 (1988)

1988

[6] [6]

Niethammer, C

P. Niethammer, C. Grabher, A. T. Look, and T. J. 9 Mitchison, Nature459, 996 (2009)

2009

[7] [7]

Atay and J

O. Atay and J. M. Skotheim, Journal of Cell Biology216, 317 (2017)

2017

[8] [8]

H. C. Berg and E. M. Purcell, Biophysical journal20, 193 (1977)

1977