Quantitative Examination of Five Stochastic Cell-Cycle and Cell-Size Control Models for Escherichia coli and Bacillus subtilis

Abstract

We examine five quantitative models of the cell-cycle and cell-size control in Escherichia coli and Bacillus subtilis that have been proposed over the last decade to explain single-cell experimental data generated with high-throughput methods. After presenting the statistical properties of these models, we test their predictions against experimental data. Based on simple calculations of the defining correlations in each model, we first dismiss the stochastic Helmstetter-Cooper model and the Initiation Adder model, and show that both the Replication Double Adder (RDA) and the Independent Double Adder (IDA) model are more consistent with the data than the other models. We then apply a recently proposed statistical analysis method and obtain that the IDA model is the most likely model of the cell cycle. By showing that the RDA model is fundamentally inconsistent with size convergence by the adder principle, we conclude that the IDA model is most consistent with the data and the biology of bacterial cell-cycle and cell-size control. Mechanistically, the Independent Adder Model is equivalent to two biological principles: (i) balanced biosynthesis of the cell-cycle proteins, and (ii) their accumulation to a respective threshold number to trigger initiation and division.

Keywords: adder; bacterial cell cycle; bacterial cell size control; bacterial physiology; quantitative microbial physiology.

FIGURE 1

Physiological parameters that can be measured from single-cell experiments. (A) Time-lapse images of a single Escherichia coli cell growing in a microfluidics channel. The cell boundaries are segmented from phase contrast images whereas the replication forks are visualized using a functional fluorescently labeled replisome protein (DnaN-YPet). (B) Multifork replication: in most growth conditions, several replication cycles overlap. The direction of the arrows is not the direction of time, but to illustrate that the HC model’s core idea is to trace replication initiation backward in time by C+D from division. (C) Four models of E. coli cell cycle and their control variables, which can be measured from single-cell experiments. The sHC model describes cell size and cell cycle using three parameters: elongation rate λ = dln(l)/dt, where l is the cell length (not shown), τ_cyc = C+D, and the initiation size per origin of replication s_i. The IA model uses λ, τ_cyc and the added size per origin of replication between consecutive replication initiation events δ_ii. The RDA model uses λ, δ_ii and the added size per origin of replication from initiation to division δ_id. The IDA model uses λ, δ_ii and the added size from birth to division Δ_d. Note that both δ_id and τ_cyc can span multiple generations. The prefactor before s_i, δ_ii, δ_id reflects multiple replication origins at initiation as depicted above.

FIGURE 2

The four correlations ρ(S_b, Δ_d), ρ(s_i, δ_id), ρ(s_i, δ_ii), and ρ(s_i, τ_cyc) are computed for the 4 experimental datasets by Witz et al. (2019), and 15 experimental datasets that we produced (see Supplementary Methods). While the first three vanish for most experimental data, the ρ(s_i, τ_cyc) displays a consistent negative correlation, inconsistent with the sHC and IA models. Variables in each dataset were normalized by their mean. Numerical values for the Pearson correlation coefficients are given in Supplementary Data 2 file. Slopes can be inferred from the Pearson correlation coefficients and CVs in the approximation of bivariate Gaussian variables.

FIGURE 3

We applied the I-value analysis proposed by Witz et al. (2019) to 4 models (RDA, IDA, IA, and sHC) using 15 experimental datasets that we produced (see Supplementary Methods), and the 4 datasets published by Witz et al. (2019). In the top bar graphs, the 4 variables are λ, Δ_d, δ_ii, S_b for both the IDA model and λ, δ_id, δ_ii, s_i for the RDA model (see the choice of variables in Supplementary Methods). The length of the arrows indicate how much IDA or RDA model is favored by the data.

FIGURE 4

Each model (A) is characterized by a set of two correlations (B). Note that these defining correlations require 1 additional parameter S_b for the sHC model, 1 additional parameter s_i for the RDA and the IA models, whereas the IDA model requires 2 additional parameters, s_i and S_b. (C) As a result, the covariance matrix for the I-value analysis of these models, according to Witz et al. (2019), would be 4x4 for the sHC, IA and RDA models and 5x5 for the IDA model. Therefore, the I-values of these models cannot be meaningfully compared.

FIGURE 5

The theory based on the initiation-centric model predicts a more sizer-like behavior. We used Eq. (1) and the experimental values of σ_id/σ_ii from Si et al. (2019), Witz et al. (2019).