Revisiting bacterial division control with likelihood inference: limited identifiability of simple models
Abstract
Cell-division control in bacteria has been studied for many years, but gaps in understanding its logic still remain. Simple candidate models of cell division control have been studied, but are assessed with heuristic analysis of single-cell data that does not provide a quantitative scale for comparison. We recast division control mechanism identification as a likelihood-based inference problem, defining an explicit metric for model comparison: among candidate mechanisms, the model with higher likelihood provides the better explanation of the data. We demonstrate that within a broad class of models, discrimination depends only on the conditional distribution of cell-cycle durations. Under mild independence assumptions, variability in growth rates and division asymmetry do not contribute to the likelihood-based comparison, effectively separating the problem of understanding growth from that of identifying division control. Applying the likelihood framework to simulations and experimentally measured long-term single-cell lineages of Escherichia coli, we find that candidate simple models are statistically identifiable in principle, but only weakly separated in practice. Sizer, adder, and related mechanisms all achieve comparable and generally high likelihoods, but no single one consistently outperforms others: their relative likelihood depends on the dataset and growth condition. In contrast, a flexible model trained directly on the data always achieves significantly higher likelihood, revealing structure not captured by any one of the simplified rules; surprisingly, it can be well fit by a low-dimensional variable combination. More broadly, this work establishes a general likelihood-based framework for identifying stochastic regulatory mechanisms and for comparing future mechanistic models on a quantitative scale.
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In-depth summary
Claude Opus 5.5 · xhigh effort · 8 Nov 2026 · from the full text of version 2 (9 Sept 2026) · CC BY 4.0
Likelihood comparison on six E. coli mother-machine datasets finds sizer, adder, and log-foldchange division rules only weakly separable, while a neural conditional cycle-duration model fits best.
Sizer and adder division-control rules have mostly been assigned from correlation heuristics (such as birth size versus added size), which lack a quantitative comparison scale and break down when the division threshold fluctuates with temporal memory. Casting mechanism identification as maximum-likelihood model comparison supplies that scale and separates division control from growth-rate and asymmetry variability.
Key results
ShownAcross six E. coli mother-machine datasets, the sizer, adder and log-foldchange (LFC) models all fit about equally well and no single one wins everywhere. The preferred rule changes with temperature, medium and dataset, even for the same strain in the same lab.
Each dataset was fit separately per mechanism, with 20 repeated optimizations and scoring on held-out cycles. Tanouchi MC4100 at 37°C and 27°C favored sizer, while 25°C favored LFC. Sassi rich medium slightly favored sizer and poor medium favored adder. Bar heights were similar and pooled per-cycle likelihood CDFs overlapped, even though most datasets had a significant winner. · B. Mechanism comparison on experimental data; B. Additional experimental data results; Figure legends
ShownA neural network predicts generalized-gamma parameters from current and previous-cycle cell state. It reaches the highest likelihood on every dataset. Its most likely (modal) cycle duration closely follows an affine function of log growth rate and log birth size.
The data-driven model (red bars) beat sizer, adder and LFC in all six datasets and over most of the pooled per-cycle likelihood range. The combination of log(α) and log(xb) correlated with the network's modal prediction at Pearson 0.864, versus 0.019 for the sizer combination. · C. A data-driven division control rule learned…; D. A data-driven generalized-gamma model learned…; Figure legends
ShownIf growth-rate and division-asymmetry parameters are independent of the control parameters, the generalized likelihood ratio between mechanisms depends only on conditional cycle-duration densities. This separates growth modeling from identifying the division-control rule.
Proposition A.1 splits the lineage likelihood into a mechanism-dependent cycle-duration term and a mechanism-independent growth and asymmetry term. Lemma A.2 proves that the nuisance parameters cancel from the ratio, including for history-dependent models. Sizer, adder and LFC are shown to be special cases of this family. · II. From heuristic classifications to a…; Likelihood-based hypothesis testing; B. Proofs; c)Log-fold-change (LFC)
ShownOn simulated lineages with a known mechanism, maximum-likelihood fits land near the true threshold timescale and noise level and rank the true mechanism highest. Wrong mechanisms still come close, so the rule is identifiable in principle but weakly separated in practice.
Adder datasets had 50 lineages of 25 cycles. Estimates of τu and σu clustered near the true values (σu shows Hurwicz finite-sample bias). The adder had the highest mean log-likelihood and the highest per-cycle log-likelihood over nearly all quantiles. Sizer and LFC ground truths behaved similarly, and for sizer data part of the adder parameter space reached nearly the same likelihood. · A. Mechanism comparison on simulation data; A. Additional simulation results; Figure legends
SuggestedCorrelation heuristics can misclassify mechanisms when the threshold is correlated over time. A sizer with a correlated threshold can produce an adder-like linear-map slope, which may explain earlier adder-leaning readings of the same data.
The derivation gives slope a ≃ 2ρu for a sizer, so ρu ≃ 1/2 yields a ≃ 1. Tanouchi et al. reported a ≃ 0.871, 1.02 and 1.08 across three temperatures. Simulated birth-size versus added-size plots identified the mechanism correctly only with τu = 2 min, far below the mean cycle time. · II. From heuristic classifications to a…; B. Mechanism comparison on experimental data; A. Misclassification as adder by the noisy…; Figure legends
Methods, data and reproducibility
Division is modeled as an indicator first crossing an Ornstein-Uhlenbeck threshold (a cyclewise-constant threshold for the log-foldchange, LFC, model). Sizer and adder cycle-duration densities use neural-network generalized-gamma surrogates. Mechanisms are ranked by trimmed maximum likelihood against a flexible neural model fit to lineages.
- Tanouchi et al. MC4100 mother-machine lineages in LB at three temperatures, recorded at one-minute resolution · Escherichia coli (strain MC4100) · n = 160, 54 and 65 mother-cell lineages (37°C, 27°C, 25°C)
- Sassi et al. ProVenus mother-machine lineages at 30°C in rich (MOPS EZ) and poor (supplemented M9) media · Escherichia coli (ProVenus strains)
- Salman MG1655: wild-type long-trace mother-machine lineages at 30°C in LB, from Susman et al. · Escherichia coli (MG1655)
- Simulated adder-mechanism lineage datasets (plus sizer and LFC sets in the appendix) used for identifiability and parameter-recovery tests · n = 50 simulated lineages of 25 cycles per adder dataset
- Simulated first-passage-time samples over independently sampled (u0, xb, α, θu) configurations, used to select and train the generalized-gamma surrogates · n = 1000 FPTs per configuration per fine-tuning batch
- Methods & tools
- Ornstein-Uhlenbeck stochastic threshold, first-passage-time model, Generalized-gamma surrogate density, Multilayer perceptron surrogates (shared trunk, three output heads), Trimmed maximum likelihood (ρ = 0.15), Generalized likelihood ratio model comparison, Wasserstein-1 distance for parametric family selection, Lasso affine baseline with one-standard-error rule, Adam optimizer with cosine-annealing schedule, Subsampling bootstrap for likelihood variability, Mother-machine microfluidics with time-lapse microscopy (source data), Noisy linear-map regression heuristic (Tanouchi et al.)
- Reproducibility gaps
- The experimental lineages are reused from earlier studies (Tanouchi, Sassi, Susman), and no accession numbers or data URLs are given. Software versions and hardware are not reported.
Limitations (6)
- The authors do not interpret the low-dimensional log(α), log(xb) approximation to the learned timing rule biologically. They leave this for future work with more extensive data analysis. · stated by the authors
- For non-nested, misspecified candidates, the generalized likelihood ratio is only asymptotically guaranteed to pick the model closest in Kullback-Leibler divergence. Because there is no finite-sample guarantee, robustness is checked by subsampling bootstrap. · stated by the authors
- The growth and asymmetry terms cancel only if their parameters are independent of the division-control parameters. Richer growth-rate models can be swapped in without breaking a fair comparison only while this independence holds. · stated by the authors
- Estimates of the threshold standard deviation carry a systematic finite-sample (Hurwicz) bias. In sizer simulations the threshold mean cannot be estimated because of median-based normalization. · stated by the authors
- The flexible network takes a 14-dimensional input that includes previous-cycle variables, while the mechanistic models have few parameters. Likelihoods are compared without a complexity penalty, so part of the network's advantage may come from richer covariates rather than a distinct control rule. · noted in this summary
- Sizer and adder likelihoods rely on neural generalized-gamma surrogates of first-passage densities. Because the gaps between mechanisms are small, surrogate error could change the rankings, and its size relative to those gaps is not reported. · noted in this summary
Who should read:Read this if you study bacterial cell-size homeostasis or need a principled likelihood criterion for comparing stochastic threshold models of division timing in single-cell lineage data.
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