Single-cell-level distributions and relationships can differentiate cell-division and growth models

Anjan Roy; Rahul Marathe; Vikas

arxiv: 2607.00467 · v1 · pith:E6BGCYKVnew · submitted 2026-07-01 · ❄️ cond-mat.stat-mech · cond-mat.soft

Single-cell-level distributions and relationships can differentiate cell-division and growth models

Vikas , Rahul Marathe , Anjan Roy This is my paper

Pith reviewed 2026-07-02 05:36 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.soft

keywords cell division modelssingle-cell distributionsTimer Sizer Adderstochastic growthlinear exponential growthasymmetric partitioningmicrobial cellsprobability distributions

0 comments

The pith

Probability distributions of cell sizes and times differentiate Timer, Sizer, Adder models and linear versus exponential growth.

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

The paper shows that probability distributions of birth-size, division-size, division-time, and division-added-size, along with statistical relationships among them, can distinguish the three canonical cell division models even when growth is stochastic and partitioning is asymmetric. These distributions and relationships also allow separation of linear growth from exponential growth at the single-cell level. The distinctions remain valid when growth rates are correlated across generations in a cell lineage. The authors derive analytical results and match them to published experimental data on microbial cells.

Core claim

Probability distributions and statistical relationships among birth-size, division-size, division-time, and added-size serve as discriminants for both the division timing rules (Timer, Sizer, Adder) and the growth law (linear or exponential), remaining effective under stochastic single-cell growth, asymmetric partitioning, and correlated growth rates across generations.

What carries the argument

The probability distributions of physiological quantities (birth-size, division-size, division-time, division-added-size) and their inter-relationships under stochastic growth and asymmetric partitioning for the Timer, Sizer, and Adder models.

If this is right

Distinct distribution shapes arise for each division model under stochastic conditions.
Joint distributions differ between linear and exponential growth for the same division rule.
The differentiability holds when growth rates correlate between mother and daughter cells.
Analytical predictions match experimental single-cell measurements for microbial systems.

Where Pith is reading between the lines

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

Measuring these distributions in a new organism could identify its division rule without knowing the molecular network.
The method could be applied to systems with additional noise sources to test robustness of the signatures.
Extensions to multi-generational data might reveal higher-order correlations beyond pairwise ones.

Load-bearing premise

The specific mathematical forms chosen for stochastic growth, asymmetric division, and the three division rules capture the dominant statistical features that experiments would observe.

What would settle it

Observing that the measured distributions for cells known to follow the Adder model are statistically indistinguishable from those predicted for the Sizer model would falsify the claim of differentiability.

Figures

Figures reproduced from arXiv: 2607.00467 by Anjan Roy, Rahul Marathe, Vikas.

**Figure 1.** Figure 1: FIG. 1. Artistic representations of various cell-division models: Timer, Sizer, and Adder. The insets represent the graphs [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2. Various experimental single-cell-level distributions (obtained from the single-cell experimental data for [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Mean, standard deviation, and coefficient of variation (the ratio of standard deviation and mean) for various single-cell [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Various single-cell quantities, such as [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

read the original abstract

Complex interactions among regulatory molecules determine the rules underlying cell growth and division in microbial cells. While the governing molecular network may not always be obvious, it is well known that correlations among certain physiological quantities measured in experiments, such as birth-size, division-size, division-time, and division-added-size, can differentiate among various cell-division models, such as Timer, Sizer, and Adder. Here we show that, apart from these correlations, which we extend for the case of stochastic single-cell growth and stochastic asymmetric partitioning, probability distributions of these quantities and statistical relationships between them can also be used to differentiate between these division models. Interestingly, we show that these quantities can not only differentiate the division models, but also distinguish among the single-cell growth paradigms, such as linear and exponential growth. We then demonstrate this differentiability among various division and growth models by comparing our analytical results with published experimental data. We further show that these results remain valid even when the growth rate of a cell is correlated with the growth rate of cells from previous generations in the lineage.

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

Distributions and cross-relationships of birth/division sizes and times separate Timer/Sizer/Adder rules and linear/exponential growth under the paper's stochastic assumptions, with data comparison.

read the letter

The central claim is that full probability distributions of birth size, division size, division time, and added size, plus statistical links among them, can tell apart the three division models and also linear versus exponential growth even with stochastic single-cell growth, asymmetric partitioning, and lineage-correlated growth rates. They extend earlier correlation-only methods and back the distinctions with analytical results plus published experimental data.

The derivations start from the usual model assumptions and appear to avoid circularity. Extending the separation to correlated growth rates across generations is a useful check, and matching the predictions to real single-cell measurements gives the work some empirical footing that pure theory papers often lack.

The soft spot is robustness. The differentiability is shown only inside the exact functional forms assumed for growth and partitioning. Real data often include extra regulatory feedback, different noise spectra, or measurement error that the paper does not test; nothing demonstrates that the signatures stay unique once those are added. The stress-test note on this point holds up from the abstract and the described scope.

This is for microbial biophysicists and synthetic-biology groups already doing statistical inference on lineage data. A reader who needs more observables than simple correlations would get concrete new quantities to try. It is worth sending for peer review because the analytical-plus-data approach is substantive enough to merit checking the derivations and asking for robustness tests, even if revisions are likely.

Referee Report

2 major / 0 minor

Summary. The manuscript claims that, in addition to correlations, the probability distributions of birth size, division size, division time, and added size, together with statistical relationships among them, can distinguish Timer/Sizer/Adder division rules and linear versus exponential growth even under stochastic single-cell growth, asymmetric partitioning, and lineage-correlated growth rates. Analytical expressions for these distributions are derived and compared to published experimental data to demonstrate the differentiability.

Significance. If the derivations are correct and the reported signatures remain identifiable, the work supplies additional observables beyond mean correlations for model discrimination in microbial cell-cycle studies, which could improve inference of division strategies from single-cell measurements.

major comments (2)

[Abstract and results sections on analytical derivations] The differentiability is derived under specific functional forms for stochastic growth and partitioning; the manuscript does not test whether the reported signatures remain unique once additional regulatory feedback, different noise spectra, or measurement error are introduced. This robustness question is load-bearing for the claim that the distributions 'can differentiate' models in general experimental settings.
[Section discussing lineage correlations] The extension to correlated growth rates across generations is shown, but the argument provides no concrete test (e.g., via simulation or perturbation analysis) that the distinguishing power survives unmodeled mechanisms beyond the three canonical rules and two growth paradigms.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the comments on robustness. Our derivations provide exact distributions and relations under the stated stochastic growth, partitioning, and lineage-correlation assumptions for the canonical Timer/Sizer/Adder and linear/exponential models. We address the two major comments below and will make targeted revisions to clarify scope and limitations.

read point-by-point responses

Referee: [Abstract and results sections on analytical derivations] The differentiability is derived under specific functional forms for stochastic growth and partitioning; the manuscript does not test whether the reported signatures remain unique once additional regulatory feedback, different noise spectra, or measurement error are introduced. This robustness question is load-bearing for the claim that the distributions 'can differentiate' models in general experimental settings.

Authors: We agree the analysis assumes specific functional forms (e.g., multiplicative noise for growth rates and beta-distributed partitioning). The manuscript derives closed-form distributions and shows they differ across the six model combinations, with validation against published data. We do not claim the signatures are invariant to arbitrary additional feedback, noise spectra, or measurement error. We will revise the abstract and discussion to state the assumptions explicitly and note that robustness to unmodeled mechanisms remains an open question, while the reported signatures already differentiate the canonical cases under the included stochasticity. revision: partial
Referee: [Section discussing lineage correlations] The extension to correlated growth rates across generations is shown, but the argument provides no concrete test (e.g., via simulation or perturbation analysis) that the distinguishing power survives unmodeled mechanisms beyond the three canonical rules and two growth paradigms.

Authors: The lineage-correlation extension yields modified analytical expressions that remain distinct across models; we compare these to data. We provide no additional perturbation analysis or simulations for mechanisms outside the three rules and two growth modes. We will add a short discussion paragraph acknowledging this scope limitation and that distinguishing power could be affected by other unmodeled processes. revision: partial

Circularity Check

0 steps flagged

Derivations start from explicit model assumptions without reduction to self-referential inputs

full rationale

The paper starts from standard functional forms for stochastic growth (linear/exponential), asymmetric partitioning, and the three division rules (Timer/Sizer/Adder), then derives closed-form or semi-analytic expressions for the distributions of birth/division sizes, times, and added sizes plus their correlations. These expressions are obtained directly from the assumed stochastic processes and are compared to external experimental data. No load-bearing step reduces by construction to a fitted parameter renamed as a prediction, a self-citation chain, or an ansatz smuggled from prior work by the same authors. The claimed differentiability is a direct mathematical consequence of the input model definitions, which remain independent of the target distinctions. The extension to lineage-correlated growth rates is likewise obtained by modifying the same stochastic assumptions rather than by re-using the result itself.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No explicit free parameters, axioms, or invented entities are identifiable from the abstract alone.

pith-pipeline@v0.9.1-grok · 5714 in / 1101 out tokens · 22256 ms · 2026-07-02T05:36:09.050920+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

73 extracted references

[1]

Timer model Consider a Mother Machine setup in which individual cells are tracked over multiple generations. We assume that cells grow exponentially in size and divide symmetrically into two equal daughter cells without partitioning stochastic- ity, following the Timer model of cell division. The birth-size for a daughter cell is given bysbd =s b exp (ατd...
[2]

The first argument goes like this

Sizer and Adder models For the Sizer and Adder models, we present some statistical arguments that are used to obtainζ(s b) from principal distributions. The first argument goes like this. The birth-size of a random cell in a lineage is given bys b =βs d. Here,s d is the size of the parent cell at the time of division, and is sampled from the probability d...
[3]

Probability transformation of random variables Suppose we are given a joint probability distribution of two random variablesxandyasP(x, y). The probability distribution for another variablez=Z(x, y) is given as [61] : Q(z) = Z ∞ 0 Z ∞ 0 P(x, y)δ(z−Z(x, y))dx dy.(B1) Note that the lower limit of integration is 0, not−∞, because the variables we have consid...
[4]

A5, as discussed in Appendix A

Sizer model Apart from the free-parameter distributions,ζ(s b) for the Sizer model is obtained from Ξ(s d) andκ(β) using Eq. A5, as discussed in Appendix A. The other single-cell-level distributions for the Sizer model, i.e., Γ(τ d) and Ω(∆ d), can also be obtained from Ξ(s d) andζ(s b) using probability transformations. We already have Ξ(s d) andζ(s b) f...
[5]

The other single-cell-level distributions for the Sizer model, i.e., Γ(τ d) and Ω(∆d), can also be obtained from Ω(∆ d) andζ(s b) using probability transformations

Adder model Apart from the free-parameter distributions,ζ(s b) for the Sizer model is obtained from Ω(∆ d) andκ(β) using the arguments presented in Appendix A. The other single-cell-level distributions for the Sizer model, i.e., Γ(τ d) and Ω(∆d), can also be obtained from Ω(∆ d) andζ(s b) using probability transformations. We already have Ω(∆ d) and ζ(s b...
[6]

We have sb =βs d, wheres b is the birth-size of the daughter cell ands d is the division-size of the parent cell

Size-related quantities First of all, let us derive the relationships between the mean of the size-related single-cell quantities. We have sb =βs d, wheres b is the birth-size of the daughter cell ands d is the division-size of the parent cell. Taking average on both sides of this equation: ⟨sb⟩=⟨βs d⟩(C1) Since the partitioning fractionβis not correlated...
[7]

Division-time for exponential single-cell growth Division-time is a function ofs d,s b, andα.αhere corresponds to the exponential growth rate. For the case of exponential growth, it can be written as: τd = 1 α ln sd sb .(C14) Using Taylor series expansion in the vicinity of the point (⟨s d⟩,⟨s b⟩,⟨α⟩), we can write: τd = 1 ⟨α⟩ ln ⟨sd⟩ ⟨sb⟩ + 1 ⟨α⟩⟨sd⟩(sd ...
[8]

Division-time for linear single-cell growth For linear growth, we haveτ d = ∆ d/α, whereαis the linear growth rate. Since, both ∆ d andαare stochastic quantities, we expand the functionτ d about the point(⟨∆ d⟩,⟨α⟩) using Taylor series expansion in two variables: τd = ⟨∆d⟩ ⟨α⟩ + ∞X n=1 (−1)n ⟨∆d⟩ ⟨α⟩ (α− ⟨α⟩) n ⟨α⟩n + ∞X n=0 (−1)n ⟨∆d⟩ ⟨α⟩ ∆d − ⟨∆d⟩ ⟨∆d⟩ ...
[9]

Therefore, we have: C(s di , sdj) =⟨s di sdj ⟩ − ⟨sd⟩2 =δ ijσ2 sd ,(D1) wheres di, ands dj are the division-sizes for theith andjth generation cells

Sizer For the Sizer model, division-size of a cell (s d) is selected independently of its birth-size (s b), and the division-sizes of cells from preceding generations in the lineage. Therefore, we have: C(s di , sdj) =⟨s di sdj ⟩ − ⟨sd⟩2 =δ ijσ2 sd ,(D1) wheres di, ands dj are the division-sizes for theith andjth generation cells. And, C(s d, sb) =⟨s dsb⟩...
[10]

Therefore, we have: C(∆ di ,∆ dj) =⟨∆ di∆dj ⟩ − ⟨∆d⟩2 =δ ijσ2 ∆d ,(D9) where ∆di, and ∆ dj are the division-added-sizes for theith andjth generation cells

Adder For the Adder model, division-added-size of a cell (∆ d) is selected independently of its birth-size (s b), and the division-added-sizes of cells from preceding generations in the lineage. Therefore, we have: C(∆ di ,∆ dj) =⟨∆ di∆dj ⟩ − ⟨∆d⟩2 =δ ijσ2 ∆d ,(D9) where ∆di, and ∆ dj are the division-added-sizes for theith andjth generation cells. And, C...
[11]

2, there are two separate histograms (forζ(s b), Ξ(sd), and Ω(∆d)) for the same experimental data

Explanation for two separate histograms for the same experimental data In Fig. 2, there are two separate histograms (forζ(s b), Ξ(sd), and Ω(∆d)) for the same experimental data. The only visible difference is the difference between single-cell growth types. One of them is for exponential growth, whereas the other is for linear growth. The same time-series...
[12]

Simulation correlatedα

Extra figures 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 sb (in m) 0.0 0.5 1.0 1.5 2.0 (sb) (A) 2.5 3.0 3.5 4.0 4.5 5.0 5.5 sd (in m) 0.0 0.2 0.4 0.6 0.8 1.0 (sd) (B) 1.0 1.5 2.0 2.5 3.0 3.5 d (in m) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 ( d) (C) 20 30 40 50 60 70 80 90 d (in min.) 0.00 0.01 0.02 0.03 0.04 0.05 ( d) (D) Experimental Exponential Experimental Linear Ana...

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Timer model Consider a Mother Machine setup in which individual cells are tracked over multiple generations. We assume that cells grow exponentially in size and divide symmetrically into two equal daughter cells without partitioning stochastic- ity, following the Timer model of cell division. The birth-size for a daughter cell is given bysbd =s b exp (ατd...

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The first argument goes like this

Sizer and Adder models For the Sizer and Adder models, we present some statistical arguments that are used to obtainζ(s b) from principal distributions. The first argument goes like this. The birth-size of a random cell in a lineage is given bys b =βs d. Here,s d is the size of the parent cell at the time of division, and is sampled from the probability d...

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Probability transformation of random variables Suppose we are given a joint probability distribution of two random variablesxandyasP(x, y). The probability distribution for another variablez=Z(x, y) is given as [61] : Q(z) = Z ∞ 0 Z ∞ 0 P(x, y)δ(z−Z(x, y))dx dy.(B1) Note that the lower limit of integration is 0, not−∞, because the variables we have consid...

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A5, as discussed in Appendix A

Sizer model Apart from the free-parameter distributions,ζ(s b) for the Sizer model is obtained from Ξ(s d) andκ(β) using Eq. A5, as discussed in Appendix A. The other single-cell-level distributions for the Sizer model, i.e., Γ(τ d) and Ω(∆ d), can also be obtained from Ξ(s d) andζ(s b) using probability transformations. We already have Ξ(s d) andζ(s b) f...

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The other single-cell-level distributions for the Sizer model, i.e., Γ(τ d) and Ω(∆d), can also be obtained from Ω(∆ d) andζ(s b) using probability transformations

Adder model Apart from the free-parameter distributions,ζ(s b) for the Sizer model is obtained from Ω(∆ d) andκ(β) using the arguments presented in Appendix A. The other single-cell-level distributions for the Sizer model, i.e., Γ(τ d) and Ω(∆d), can also be obtained from Ω(∆ d) andζ(s b) using probability transformations. We already have Ω(∆ d) and ζ(s b...

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We have sb =βs d, wheres b is the birth-size of the daughter cell ands d is the division-size of the parent cell

Size-related quantities First of all, let us derive the relationships between the mean of the size-related single-cell quantities. We have sb =βs d, wheres b is the birth-size of the daughter cell ands d is the division-size of the parent cell. Taking average on both sides of this equation: ⟨sb⟩=⟨βs d⟩(C1) Since the partitioning fractionβis not correlated...

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Division-time for exponential single-cell growth Division-time is a function ofs d,s b, andα.αhere corresponds to the exponential growth rate. For the case of exponential growth, it can be written as: τd = 1 α ln sd sb .(C14) Using Taylor series expansion in the vicinity of the point (⟨s d⟩,⟨s b⟩,⟨α⟩), we can write: τd = 1 ⟨α⟩ ln ⟨sd⟩ ⟨sb⟩ + 1 ⟨α⟩⟨sd⟩(sd ...

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Division-time for linear single-cell growth For linear growth, we haveτ d = ∆ d/α, whereαis the linear growth rate. Since, both ∆ d andαare stochastic quantities, we expand the functionτ d about the point(⟨∆ d⟩,⟨α⟩) using Taylor series expansion in two variables: τd = ⟨∆d⟩ ⟨α⟩ + ∞X n=1 (−1)n ⟨∆d⟩ ⟨α⟩ (α− ⟨α⟩) n ⟨α⟩n + ∞X n=0 (−1)n ⟨∆d⟩ ⟨α⟩ ∆d − ⟨∆d⟩ ⟨∆d⟩ ...

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Therefore, we have: C(s di , sdj) =⟨s di sdj ⟩ − ⟨sd⟩2 =δ ijσ2 sd ,(D1) wheres di, ands dj are the division-sizes for theith andjth generation cells

Sizer For the Sizer model, division-size of a cell (s d) is selected independently of its birth-size (s b), and the division-sizes of cells from preceding generations in the lineage. Therefore, we have: C(s di , sdj) =⟨s di sdj ⟩ − ⟨sd⟩2 =δ ijσ2 sd ,(D1) wheres di, ands dj are the division-sizes for theith andjth generation cells. And, C(s d, sb) =⟨s dsb⟩...

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Therefore, we have: C(∆ di ,∆ dj) =⟨∆ di∆dj ⟩ − ⟨∆d⟩2 =δ ijσ2 ∆d ,(D9) where ∆di, and ∆ dj are the division-added-sizes for theith andjth generation cells

Adder For the Adder model, division-added-size of a cell (∆ d) is selected independently of its birth-size (s b), and the division-added-sizes of cells from preceding generations in the lineage. Therefore, we have: C(∆ di ,∆ dj) =⟨∆ di∆dj ⟩ − ⟨∆d⟩2 =δ ijσ2 ∆d ,(D9) where ∆di, and ∆ dj are the division-added-sizes for theith andjth generation cells. And, C...

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2, there are two separate histograms (forζ(s b), Ξ(sd), and Ω(∆d)) for the same experimental data

Explanation for two separate histograms for the same experimental data In Fig. 2, there are two separate histograms (forζ(s b), Ξ(sd), and Ω(∆d)) for the same experimental data. The only visible difference is the difference between single-cell growth types. One of them is for exponential growth, whereas the other is for linear growth. The same time-series...

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Simulation correlatedα

Extra figures 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 sb (in m) 0.0 0.5 1.0 1.5 2.0 (sb) (A) 2.5 3.0 3.5 4.0 4.5 5.0 5.5 sd (in m) 0.0 0.2 0.4 0.6 0.8 1.0 (sd) (B) 1.0 1.5 2.0 2.5 3.0 3.5 d (in m) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 ( d) (C) 20 30 40 50 60 70 80 90 d (in min.) 0.00 0.01 0.02 0.03 0.04 0.05 ( d) (D) Experimental Exponential Experimental Linear Ana...

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