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arxiv: 1907.05108 · v1 · pith:N7FNCQCNnew · submitted 2019-07-11 · 🧮 math.ST · math.AP· math.PR· stat.TH

Estimating the division rate from indirect measurements of single cells

Marie Doumic (MAMBA) , Ad\'ela\"ide Olivier (UP11 UFR Sciences) , Lydia Robert (MICALIS) This is my paper

Pith reviewed 2026-05-24 22:57 UTC · model grok-4.3

classification 🧮 math.ST math.APmath.PRstat.TH
keywords division rate estimationadder modelbacterial growthinverse problemsize distributionreconstruction formulasingle-cell dataill-posed problem
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The pith

Division rates in the adder model of bacterial growth can be reconstructed from cell size measurements via an explicit formula.

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

The paper establishes that division rate estimation is possible when the rate depends on size increment but only size itself is observed. It provides a reconstruction formula first in a deterministic setting and then in a statistical one, then implements the method numerically on both simulated and experimental data. A sympathetic reader would care because direct observation of division triggers is experimentally difficult while size tracking is routine, so the formula turns common measurements into estimates of growth control. The inverse problem is severely ill-posed yet the numerical recoveries remain satisfactory. The work centers on proving recoverability and delivering the practical mapping from size data to the division rate function.

Core claim

In the incremental model the division rate depends on the size increment between birth and division; this rate can be recovered from the observable size distribution by a reconstruction formula that is first stated deterministically and then extended to a statistical setting, after which numerical solution on simulated and experimental single-cell size data is shown to be feasible.

What carries the argument

The reconstruction formula that inverts the size distribution to recover the division rate function under the adder assumption.

If this is right

  • Division rate functions are recoverable from exact size measurements in the deterministic case.
  • The statistical version of the formula handles population-level or noisy size data.
  • Numerical implementation produces usable estimates on both simulated trajectories and real bacterial measurements.
  • The method works despite the inverse problem being severely ill-posed.

Where Pith is reading between the lines

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

  • The same size data could be used to test whether the adder assumption itself holds by checking consistency of the recovered rate.
  • The approach suggests a general strategy for other models in which the controlling variable is not directly measured but a proxy is.
  • Experimental designs that track only size could now be re-analyzed to infer division strategies without new measurements.
  • The reconstruction might be combined with time-lapse imaging to separate growth from division effects in mixed populations.

Load-bearing premise

The population obeys the incremental adder model in which division rate depends specifically on the size increment between birth and division.

What would settle it

Applying the reconstruction to size data from a population known to follow a different division rule (for example size-dependent rather than increment-dependent) and obtaining negative rates or rates inconsistent with direct increment measurements would show the claim fails.

Figures

Figures reproduced from arXiv: 1907.05108 by Ad\'ela\"ide Olivier (UP11 UFR Sciences), Lydia Robert (MICALIS), Marie Doumic (MAMBA).

Figure 1
Figure 1. Figure 1: Protocol 1 – Reconstruction of B when both UB,x and LB are (almost) exactly known. The oracle choice for h gives us the value 1/4.75. Reconstruction of B B Numerical [0;2] [0;2.5 ] sampling ∆a = 0.01 ∆a = 0.01 Protocol 1 0.0730 0.2065 Protocol 2 0.0849 0.1321 [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Protocol 2 – Reconstruction of B when UB,x is (almost) exactly known but not LB. The oracle choice for h gives us the value 1/5. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Results of Protocols 1 and 2. (x stands for size, ξ for frequency and a for increment of size) 21 [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Protocol 3 – Reconstruction of B when UB,x is reconstructed from X1, . . . , Xn i.i.d. ∼ UB,x but LB is (almost) exactly known. The oracle choice for h3 gives us values that range between 1/3.25 for n = 500 and 1/4.75 for n = 50 000. We set $n = 1/n. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Results of Protocol 3 for n = 2000 and M = 100 Monte Carlo samples. (x stands for size, ξ for frequency and a for increment of size) 23 [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Protocol 4 – Reconstruction of B when both UB,x and LB are reconstructed from X1, . . . , Xn i.i.d. ∼ UB,x. The parameter h1 is automatically chosen by the kernel smoothing func￾tion ksdensity; h2 is deduced from h1. The oracle choice for h3 gives us values that range between 1/3.25 for n = 500 and 1/4.5 for n = 50 000. We set $n = 1/n. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Results of Protocol 4 for n = 2000 and M = 100 Monte Carlo samples. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Results of Protocols 3 and 4 – Estimation of the division rate [PITH_FULL_IMAGE:figures/full_fig_p027_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Results of Protocols 3 and 4 – Reduction of the mean error over [PITH_FULL_IMAGE:figures/full_fig_p029_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Testing the procedure on experimental data. Initial step: estimation of the size distribution (a) Estimation of fB(a) (b) Estimation of the division rate B(a) [PITH_FULL_IMAGE:figures/full_fig_p030_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Testing the procedure on experimental data. Final step: estimation of the increment-structured division rate 30 [PITH_FULL_IMAGE:figures/full_fig_p030_12.png] view at source ↗
read the original abstract

Is it possible to estimate the dependence of a growing and dividing population on a given trait in the case where this trait is not directly accessible by experimental measurements, but making use of measurements of another variable? This article adresses this general question for a very recent and popular model describing bacterial growth, the so-called incremental or adder model. In this model, the division rate depends on the increment of size between birth and division, whereas the most accessible trait is the size itself. We prove that estimating the division 10 rate from size measurements is possible, we state a reconstruction formula in a deterministic and then in a statistical setting, and solve numerically the problem on simulated and experimental data. Though this represents a severely ill-posed inverse problem, our numerical results prove to be satisfactory.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that estimating the division rate from size measurements is possible in the incremental (adder) model, where division depends on size increment but size is the observable. It derives a deterministic reconstruction formula from the integral equation relating the observable size density to the division rate B via the adder kernel, extends this to a statistical estimator by replacing the density with its empirical counterpart, and reports satisfactory numerical results on both simulated and experimental data despite the problem being severely ill-posed.

Significance. If the reconstruction formulas are rigorously justified and the numerical method proves stable, the work would advance inverse problems in mathematical biology by enabling inference of division mechanisms from accessible size data in bacterial populations. The explicit handling of both deterministic and statistical settings, plus validation on experimental data, represents a practical contribution.

major comments (2)
  1. [Abstract and statistical setting] Abstract and the section deriving the statistical reconstruction formula: the central claim requires the statistical estimator to be usable on finite data, yet no convergence rates, stability constants, or regularization analysis are supplied for the estimator obtained by substituting the empirical measure into the deterministic inversion formula, even though the abstract explicitly flags the problem as severely ill-posed.
  2. [Deterministic reconstruction] Section presenting the deterministic inversion (the integral equation relating size density to B via the adder kernel): while a reconstruction formula is stated, the manuscript provides no analysis of uniqueness, injectivity, or stability of the inversion operator, which is load-bearing for the claim that estimation is possible.
minor comments (2)
  1. [Introduction] The introduction would benefit from a brief explicit statement of the adder kernel and how it enters the integral equation, to improve accessibility for readers outside the immediate subfield.
  2. [Numerical results] Figure captions for the numerical results should include quantitative error metrics (e.g., L2 distance to ground truth on simulated data) rather than qualitative statements of 'satisfactory' performance.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below, clarifying the scope of our contributions while acknowledging where additional discussion can be added.

read point-by-point responses

  1. Referee: [Abstract and statistical setting] Abstract and the section deriving the statistical reconstruction formula: the central claim requires the statistical estimator to be usable on finite data, yet no convergence rates, stability constants, or regularization analysis are supplied for the estimator obtained by substituting the empirical measure into the deterministic inversion formula, even though the abstract explicitly flags the problem as severely ill-posed.

    Authors: The manuscript's central claim is that a reconstruction formula exists which makes estimation possible in principle; this is established by deriving the explicit inversion of the integral equation. The statistical version is the direct plug-in estimator using the empirical size density, and its practical usability is demonstrated via numerical experiments on simulated and experimental data. We agree that no convergence rates or regularization analysis is supplied, as developing a full statistical theory for this severely ill-posed problem (including choice of regularization) lies beyond the paper's scope, which focuses on the formula derivation and numerical validation. We will revise the abstract and add a clarifying remark on the absence of rates and the need for regularization in future work. revision: partial

  2. Referee: [Deterministic reconstruction] Section presenting the deterministic inversion (the integral equation relating size density to B via the adder kernel): while a reconstruction formula is stated, the manuscript provides no analysis of uniqueness, injectivity, or stability of the inversion operator, which is load-bearing for the claim that estimation is possible.

    Authors: The proof that estimation is possible consists in exhibiting the explicit reconstruction formula obtained by inverting the integral equation that links the observable size density to the division rate via the adder kernel. This derivation establishes that the mapping from B to the size density is invertible, so uniqueness holds by construction whenever the size density is known exactly. We acknowledge that a separate functional-analytic study of injectivity, uniqueness in appropriate spaces, or stability bounds for the inversion operator is not included. Given the explicit note that the problem is severely ill-posed, such stability is expected to be limited. We will add a short paragraph discussing the inversion operator and its ill-posed character. revision: partial

Circularity Check

0 steps flagged

No circularity; reconstruction formula derived from model equations

full rationale

The paper sets up an inverse problem for the adder model, deriving a deterministic reconstruction formula relating the observable size density to the division rate B via the integral kernel, then replacing the density by its empirical counterpart in the statistical setting. No quoted step reduces the claimed prediction or formula to a fitted parameter, self-definition, or self-citation chain; the derivation remains independent of the target data and is presented as an explicit inversion under stated model assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the domain assumption of the adder model; no free parameters or invented entities are described in the abstract.

axioms (1)
  • domain assumption Bacterial growth and division follows the incremental (adder) model in which division rate depends on the size increment between birth and division.
    This is the specific model for which the estimation method is developed, as stated in the abstract.

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Reference graph

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