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From noisy cell size control to population growth: when variability can be beneficial

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Pith's one-line read Noise in single-cell growth rates can increase a population's growth rate when slow-growing cells divide at smaller sizes.

desk verdict A careful small-noise Euler-Lotka extension that finally couples growth-rate noise to size-control and partitioning noise; the math holds up, but the headline beneficial-variability regime rests on an uncorrelated-growth-rate kernel that the author explicitly flags. read the letter →

arxiv 2411.08512 v1 pith:OT4DKEOT submitted 2024-11-13 q-bio.PE cond-mat.stat-mechphysics.bio-ph

Arthur Genthon This is my paper · ORCID
classification q-bio.PEcond-mat.stat-mechphysics.bio-ph MSC 92D2592C37
keywords populationgrowthrateMalthusparametercellsizecontrolsingle-cellvariabilityEuler-Lotkaequationadder-sizermechanismsstochasticcoefficientofvariation
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

Single-cell experiments show that genetically identical cells differ in growth rate and in the sizes at which they are born and divide, and this paper asks how those fluctuations set the population growth rate—the Malthus parameter that serves as a fitness proxy. The author builds a stochastic population model whose inheritance law factorizes into a size-control kernel, a growth-rate distribution, and a volume-partitioning kernel, and derives closed-form small-noise expansions for the population growth rate and the population-level mean birth size. The headline result is that growth-rate variability can be beneficial: when the target division size depends on single-cell growth rate in the direction of slow-growing cells dividing at smaller sizes, the coefficient multiplying $\text{CV}_\alpha^2$ in $\lambda/\langle\alpha\rangle$ flips sign, so populations with more variable growth rates grow faster. The same formalism also shows that, for exponentially growing cells, noisier size control can partly offset the cost of growth-rate fluctuations and that the sizer mechanism outgrows the adder at fixed noise, while for linearly growing cells division-noise corrections are positive at first order. If correct, these formulas connect single-lineage measurements in mother machines to population-level fitness.

What carries the argument

The engine of the paper is the generalized Euler–Lotka equation (eq. 6), the population analogue of the renewal equation that relates the distribution of birth, division, and growth-rate states in the population tree to the population growth rate $\lambda$. The inheritance law entering it is assumed to factorize as $K(s_d,s_b,\alpha|s'_d,s'_b,\alpha') = \psi(s_d|s_b,\alpha)\,\nu(\alpha)\,\kappa(s_b|s'_d)$, with $\psi$ generated by the noisy linear map $s_d = a s_b + b + \eta$ (eq. 11). Factorization implies that, in the forward-lineage statistics, birth size and growth rate are independent and that the forward birth-size distribution obeys a self-consistent equation; the author expands both sides of the Euler–Lotka equation in powers of the coefficients of variation and matches coefficients to get explicit formulas such as eqs. (17), (22), and (30). The parameter $a$ indexes the size-control strategy—0 is a sizer, 1 is an adder, 2 would be a timer—and appears explicitly in every coefficient, which is how the paper compares mechanisms.

What would settle it

Measure, in a mother-machine experiment, the joint distribution of mother and daughter single-cell growth rates: if the child growth rate is not independent of the mother's growth rate and birth size, the factorized kernel fails and eq. (30) is not the correct leading-order prediction. A second direct check is to measure $S_a$ and $S_b$ from linear fits of division size versus birth size in growth-rate bins, then compare population growth rates across conditions with different $\text{CV}_\alpha$; if $\lambda$ does not rise once $\bar{a}S_a + (2-\bar{a})S_b$ exceeds $2\ln 2 - (\ln 2)^2$, the claimed beneficial effect is absent.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that the population growth rate $\lambda$ is, to leading order in the coefficients of variation, a sum of squares of single-cell noise: growth-rate noise $\text{CV}_\alpha^2$, size-control noise $\text{CV}_\phi^2$, volume-partitioning noise $\text{CV}_p^2$, and, in the $\alpha$-dependent case, the sensitivities $S_a$ and $S_b$ of the slope and intercept of the noisy linear map $s_d = a s_b + b + \eta$. For exponentially growing cells the central formula is $\lambda/\langle\alpha\rangle \approx 1 - \bigl(1 - \tfrac{\ln 2}{2} - \tfrac{\bar{a}S_a + (2-\bar{a})S_b}{2\ln 2}\bigr)\,\text{CV}_\alpha^2$ (eq. 30). When $\bar{a}S_a + (2-\bar{a})S_b$ is large enough, the correction is positive and fluctuations in single-cell growth rates increase population growth; the author argues this is the regime where slow-growing cells divide at smaller sizes than fast-growing cells. For the previously studied case where size control does not depend on growth rate, the paper recovers the known detrimental effect and adds new cross-terms: noise in size control is always beneficial, while noise in volume partitioning is beneficial only below $a \approx 0.97$, so the sizer is faster than the adder for fixed noise levels. The same Euler–Lotka machinery, applied to linear growth, yields first-order positive corrections from size-control and partitioning noise even when growth rates are identical across cells.

Load-bearing premise

The whole calculation rests on the assumption that a newborn cell's growth rate is drawn independently of its mother's growth rate and of the size it was born with, encoded in the factorized inheritance kernel of eq. (8); if mother–daughter growth-rate correlations are present, the predicted size and sign of the growth-rate effect would need to change.

Editorial extensions

If this is right

  • For exponentially growing cells with $\alpha$-independent size control, the population growth penalty from growth-rate noise is reduced by size-control noise; at fixed $\text{CV}_\phi$, $\text{CV}_p$, and $\text{CV}_\alpha$, the sizer yields a larger $\lambda$ than the adder or timer-like mechanisms.
  • Noise in division-size control is never harmful at leading order for exponential growth, while noise in volume partitioning helps only for $a \lesssim 0.97$ and is essentially neutral for the adder.
  • For linearly growing cells, noise in size control and partitioning raise $\lambda$ even when all cells share the same growth rate, with the adder being the slowest strategy.
  • When $a(\alpha)$ and $b(\alpha)$ are extracted from single-cell data, the model predicts that growth-rate variability can push $\lambda$ above the value for a uniform population, and the bound $\lambda \le \langle\alpha\rangle$ can fail while $\lambda \ge \ln 2/\langle\tau_d\rangle_{\mathrm{fw}}$ survives.
  • Mother-machine estimates of $\text{CV}_\alpha$, $\text{CV}_\phi$, and $\text{CV}_p$ are directly usable in these expansions to predict population-level fitness from lineage data.

Reading between the lines

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

  • A natural next test is to measure the joint distribution of mother and daughter growth rates in the same experiments; the factorized-kernel assumption (eq. 8) is the main place where real cells could depart from the model, and the paper itself notes that mother–daughter correlations can already make fluctuations beneficial.
  • If eq. (30) survives data, it reframes the evolution of cell size control: variability in growth rate would not be a nuisance to be suppressed but a trait that can be selected for, provided slow growers divide small—an argument the author does not develop into an evolutionary claim.
  • The linear-growth result suggests that for organisms with linear single-cell growth, measurement of size-control noise $\text{CV}_\phi$ alone is enough to predict a fitness advantage of noisy division, without requiring growth-rate noise; this is a sharper, testable prediction than the exponential case.
  • The same expansion method could be applied to growth laws with more than two phases or to time-varying growth rates; the supplementary treatment of bi-linear fission yeast growth indicates the machinery extends, but the paper leaves those extensions for future work.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 7 minor

Summary. This paper develops a unified framework, based on a generalized Euler-Lotka equation (Eq. 6), to compute the steady-state population growth rate λ and the mean birth size in the tree statistics for cell populations with stochastic single-cell growth rates, noisy size control (noisy linear map, Eq. 11), and noisy volume partitioning (Eq. 9). The authors derive small-noise expansions for exponentially growing cells (Eqs. 16-17) and linearly growing cells (Eqs. 21-22), and they extend the model in Section III.C to allow the slope and intercept of the size-control map to depend on the growth rate (Eqs. 23-24). The central result is Eq. (30): λ/⟨α⟩ ≈ 1 − (1 − ln2/2 − (āSa + (2−ā)Sb)/(2 ln2)) CVα², so that growth-rate fluctuations can increase λ when slow-growing cells tend to divide at smaller sizes. The theoretical predictions are compared with agent-based simulations and show good agreement. An analysis of E. coli mother-machine data indicates that the size-control parameters do depend on growth rate, motivating the α-dependent model.

Significance. The paper provides a systematic and careful treatment of how different sources of single-cell noise affect population growth, extending earlier work by Lin and Amir and by Thomas. The result that size-control noise always increases λ for exponential growers, and that partitioning noise can be beneficial or detrimental depending on the control strategy, is new. The α-dependent size-control mechanism offers a distinct route for beneficial variability, separate from mother-daughter growth-rate correlations. The simulations are extensive and the expansions are tested beyond leading order; the recovery of known results as special cases (e.g., the asymmetric-division result of Barber et al.) lends confidence. The framework is general enough to cover linear and bi-linear growth laws, which broadens its applicability to organisms such as M. tuberculosis and S. pombe. If the identified limitations are addressed, the paper should be a useful contribution to quantitative cell biology.

major comments (2)
  1. [Section III.C, Eqs. (23)-(24), Fig. 5] The α-dependent parameterization a = ā(1 + Sa x) and b = b̄(1 + Sb x) is introduced without restricting the range of x. In the original linear map (11), the constraints b ≥ 0 and 0 ≤ a ≤ 2 are assumed. For the parameter values used to illustrate the beneficial effect in Fig. 5 (e.g., Sb = 3 with CVα up to 0.3), the added size b becomes negative for slow-growing cells (those with x < −1/Sb). A negative b implies sd < sb for the adder, so the generation time τd = ln(sd/sb)/α is not well-defined and the Euler-Lotka equation (6) no longer describes a meaningful cell cycle. The simulation results in Fig. 5 may therefore lie outside the model's domain of validity. The authors should either restrict the sensitivities so that b(α) > 0 and a(α) ∈ [0, 2) for all α in the support of ν (for example, use Sb values below the positivity bound 1/(2CVα) and choose Sb = 1.5 rather than Sb = 3 in the figure), or explicitly discuss the interpretation and consequences of negative b. This is directly relevant to the central claim because the beneficial regime is realized only for large positive S, where constraint violations are most likely.
  2. [Section II.A, Eq. (8), and Section III.C, Eq. (30)] The factorized inheritance kernel K = ψ(sd|sb,α)ν(α)κ(sb|s'd) is the central structural assumption. It makes the tree distribution of growth rates equal to ν(α) and underlies the derivation of Eq. (30). The paper correctly notes in the Introduction that mother-daughter correlations in α are neglected and that such correlations can by themselves make fluctuations beneficial (ref. [14]). However, because real mother-machine data, including the E. coli data analyzed in Section C, exhibit such correlations, the quantitative threshold S > 2 ln2 − (ln2)² for beneficial fluctuations is not directly transferable to those data. I recommend that the authors (i) state this limitation explicitly in the abstract and conclusion, and (ii) ideally provide a short analysis or simulation for a correlated kernel ν(α|α′) to indicate how the coefficient cα changes. At minimum, the discussion should make clear that Eq. (30) applies to populations with uncorrelated single-cell growth rates.
minor comments (7)
  1. [Abstract and Discussion] The phrase 'fluctuations in single-cell growth rates can be beneficial' in the abstract should be qualified with 'within our model, which neglects mother-daughter growth-rate correlations' to avoid overstatement, since the discussion already notes this limitation.
  2. [Section III.A] There is a typo: 'mircro-organisms' should be 'micro-organisms'.
  3. [Section III.A] The sentence 'on can also consider the situation' should read 'one can also consider the situation'.
  4. [Section III.B] The phrase 'allows use to identify' should be 'allows us to identify'.
  5. [Supplementary Material, Section III] In the text introducing the derivation, 'sensitivies' is a typo for 'sensitivities'.
  6. [Section III.C] The sentence 'This analysis also shows that the bounds on the population growth rate, ⟨α⁻¹⟩⁻¹ ≤ λ ≤ ⟨α⟩, derived in [10] when ψ(sd|sb,α)=ψ(sd|sb) are no longer valid in this case' could be clarified by adding 'in general' and explicitly noting that the lower bound ln2/⟨τd⟩fw still holds, which the paper does later.
  7. [Figure 5] For panels (a) and (b), the authors should state the order of the truncation used for the theoretical curves (first order in CVα²) and indicate which parameter combinations keep b(α) > 0 for all α in the plotted range.

Circularity Check

0 steps flagged · score 0.0 of 10

No material circularity: the Euler-Lotka expansions are derived from the model equations, and the E. coli sensitivities enter only as fitted inputs, not as fits to lambda.

full rationale

The derivation chain is self-contained. The generalized Euler-Lotka equation (eq. 6) follows from the population balance, and the small-noise coefficients in eqs. (16), (17), (21), (22), (29) and (30) are obtained by expanding that equation, not by fitting the population growth rate. The forward-lineage relation (eq. 13) fixes the birth-size distribution independently of the growth-rate distribution, so the expansion coefficients are analytic functions of the model parameters and noise levels. In Section C, the sensitivities S_a and S_b are fitted to sd-vs-sb mother-machine regressions from external data [7] and then enter as inputs into the derived formula for lambda; no fitted quantity is renamed as a prediction. The factorized kernel (eq. 8) and the restriction to uncorrelated growth rates are explicit modeling assumptions, and the paper itself states in Section II.A that correlated kernels nu(alpha|alpha') can make fluctuations beneficial [14], which is an honest limitation rather than a circular step. The self-citations [39,40] are used only in the Supplementary Material as a cross-check that the lower bound ln2/<tau_d>_fw holds, and the main results do not depend on them. Known limits (eq. 7 and the checks against [10,12,15]) are reproduced, supporting the independence of the calculation. No step was found in which an Eq. X reduces to Eq. Y by construction, and no prediction is statistically forced by a fit to the target quantity.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

No free constants are fitted to the target population growth rate; the derivation is analytic in a and the CVs. The only fitted numbers are the linear-map sensitivities a_bar, Sa, b_bar, Sb for E. coli, used to illustrate the alpha-dependent control scenario. The structural assumptions are the factorized kernel (no correlations in alpha), the noisy linear map, small noise, constant single-cell growth rate, and linear alpha-dependence of a and b in the final section.

free parameters (4)
  • a_bar (mean size-control slope) = 0.90 +/- 0.01 (glucose)
    Fit of sd vs sb across 8 alpha-bins in E. coli mother-machine data from [7]; enters eqs (26)-(30).
  • Sa (slope sensitivity) = -1.12 +/- 0.25 (glucose)
    Linear fit of binned a versus standardized alpha; sign and magnitude control whether variability is beneficial in eq (30).
  • b_bar (mean intercept) = 2.42 +/- 0.03 um (glucose)
    Fit of sd vs sb; sets the mean birth size scale.
  • Sb (intercept sensitivity) = 0.89 +/- 0.20 (glucose)
    Linear fit of binned b versus standardized alpha; positive Sb means slow growers divide at smaller sizes.
assumptions (6)
  • domain assumption The single-cell growth rate alpha of a newborn is drawn independently from a fixed distribution nu(alpha), with no mother-daughter correlations and no correlation with birth size.
    Introduced in eq (8) as the factorized kernel K=psi nu kappa; the main results and the beneficial-variability condition depend on this independence. The paper notes that relaxing it changes the outcome (ref [14]).
  • domain assumption Cell division follows the noisy linear map sd = a sb + b + eta with 0 <= a < 2, b > 0, <eta>=0, and eta independent of alpha.
    Phenomenological model from [8,31], used throughout; Section II C.
  • domain assumption Fluctuations are small, allowing truncation of the Euler-Lotka expansion at second order in the CVs.
    The small-noise expansions eqs (12),(15),(19),(20),(27),(28) require peaked distributions; later terms are kept only for figure comparisons.
  • ad hoc to paper In Section C, the slope and intercept of the linear map depend linearly on the standardized growth rate: a = a_bar(1 + Sa x), b = b_bar(1 + Sb x).
    Motivated by binned linear fits to E. coli data [7], but linearity is assumed rather than derived; nonlinear dependence would alter eqs (26),(29),(30).
  • standard math Volume is conserved at division, with daughter fraction p having mean 1/2.
    Eq (9) and <p>=1/2; standard biological constraint.
  • domain assumption Each cell grows at a constant rate alpha during its cycle, under exponential, linear, or bi-linear growth laws.
    Stated at the start of Section II A; time-varying growth rates are excluded.

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Pith. "Pith review of From noisy cell size control to population growth: when variability can be beneficial." pith.science (2026). https://pith.science/paper/OT4DKEOT

@misc{pith2026241108512,
  author       = {Pith},
  title        = {Pith review of: From noisy cell size control to population growth: when variability can be beneficial},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OT4DKEOT}},
  note         = {Machine review of arXiv:2411.08512}
}
abstract

Single-cell experiments revealed substantial variability in generation times, growth rates but also in birth and division sizes between genetically identical cells. Understanding how these fluctuations determine the fitness of the population, i.e. its growth rate, is necessary in any quantitative theory of evolution. Here, we develop a biologically-relevant model which accounts for the stochasticity in single-cell growth rates, birth sizes and division sizes. We derive expressions for the population growth rate and for the mean birth size in the population in terms of the single-cell fluctuations. Allowing division sizes to fluctuate reveals how the mechanism of cell size control (timer, sizer, adder) influences population growth. Surprisingly, we find that fluctuations in single-cell growth rates can be beneficial for population growth when slow-growing cells tend to divide at smaller sizes than fast-growing cells. Our framework is not limited to exponentially-growing cells like $\textit{Escherichia coli}$, and we derive similar expressions for cells with linear and bi-linear growth laws, such as $\textit{Mycobacterium tuberculosis}$ and fission yeast $\textit{Schizosaccharomyces pombe}$, respectively.

Figures

Figures reproduced from arXiv: 2411.08512 by the authors.

Figure 1
Figure 1. FIG. 1. Schematics of the population tree and the mechanisms of cell size control. (a) Population tree starting with one [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Impact of single-cell variability on population growth and mean birth size for exponentially-growing cells. Noises in size [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Impact of single-cell variability on population growth and mean birth size for linearly-growing cells. Noises in size [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Analysis of mother machine data from Ref. [ [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Impact of single-cell variability on population growth and mean birth size with [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

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Pith tools

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