On distinguishability among cell-division models based on population and single-cell-level distributions

TL;DR

This work assesses how Timer, Sizer, and Adder cell-division models can be distinguished from population- and single-cell-level data. It shows that, despite different division rules, population-level distributions such as age, size, and added-size are indistinguishable under both exponential and linear growth and even with stochastic growth rates, when analyzed through survival-probability and probability-transformation formalisms that connect principal and derived distributions. In contrast, correlations and relative fluctuations among single-cell quantities (e.g., $\langle s_d\rangle$, $\langle s_b\rangle$, $\langle Δ_d\rangle$, and their standard deviations) carry model-specific signatures that can distinguish Adder from Sizer, while Timer is equivalent to Adder under linear growth. The findings are supported by simulations and align with several experimental observations, though some data exhibit deviations that may reflect measurement noise or additional biological factors. Overall, the paper provides a robust framework to differentiate growth-division strategies and to compare linear vs exponential growth, with implications for interpreting single-cell data and optimizing bioprocesses.

Abstract

It is well known that the different cell-division models, such as Timer, Sizer, and Adder, can be distinguished based on the correlations between different single-cell-level quantities such as birth-size, division-time, division-size, and division-added-size. Here, we show that other statistical properties of these quantities can also be used to distinguish between them. Additionally, the statistical relationships and different correlation patterns can also differentiate between the different types of single-cell growth, such as linear and exponential. Further, we demonstrate that various population-level distributions, such as age, size, and added-size distributions, are indistinguishable across different models of cell division despite them having different division rules and correlation patterns. Moreover, this indistinguishability is robust to stochasticity in growth rate and holds for both exponential and linear growth. Finally, we show that our theoretical predictions are corroborated by simulations and supported by existing single-cell experimental data.

Figures (15)

• Figure 1: Artistic representations of various cell-division models: Timer, Sizer, and Adder. The insets represent the graph of birth-size of a random cell tracked for many generations in a simulated CSTR culture (containing 1 million cells) following the particular cell division model with stochasticity in growth rate and principle of cell division. IC1 and IC2 represent 'initial condition 1' (initial birth-size being 0.5) and 'initial condition 2' (initial birth-size being 1.5), respectively.
• Figure 2: Distribution of various quantities across the cells in CSTR like growth conditions for the three models of cell division, under exponential biomass growth: (A) age distribution, (B) size distribution, (C) added-size distribution, (D) birth-size distribution. The principal distribution for a division model, which is a free parameter based on which the population-level distributions are obtained, are chosen as follows - the division-time distribution $\Gamma(\tau_d)$ for the Timer model is Gaussian with a mean $\langle \tau_d \rangle =1$ hour and a standard deviation $\sigma_{\tau_d} = 0.1$ hour, the Sizer model's division-size distribution $\Xi(s_d)$ is Gaussian with a mean $\langle s_d \rangle =2$$\mu$m and standard deviation $\sigma_{s_d} = 0.1$$\mu$m, the division-added-size distribution $\Omega(\Delta_d)$ for the Adder model is taken to be Gaussian with a mean $\langle \Delta_d \rangle =1$$\mu$m and standard deviation $\sigma_{\Delta_d} = 0.1$$\mu$m. The lines indicate the analytical curves for the distributions (which are identical regardless of $\alpha$ being stochastic or deterministic), and the markers indicate the distributions obtained from the simulations. Timer, Sizer, and Adder in the legend indicate the results for the corresponding model when the growth rate is fixed. The growth rate for this case is taken to be $\ln(2)$ per hour. In contrast, stochastic Timer, stochastic Sizer, and stochastic Adder in the legend indicate the simulation results for the case when the growth rate is stochastic in time. The mean growth rate for this is taken to be $\ln(2)$ per hour, and the standard deviation in growth rate is taken to be $0.1$ per hour.
• Figure 3: Various correlations for the three models, obtained using simulations under exponential biomass growth: the first column is for the Timer model, the second is for the Sizer model, and the third is for the Adder model; the first row is division-time versus birth-size, the second is division-size versus birth-size, and the third is division-added-size versus birth-size. The slope of the linear fit to the data is shown in each figure, indicated by S, along with the Pearson Correlation Coefficient, indicated by P.C. The analytical curve corresponds to the correlations inferred analytically from the relations $s_d = s_b \exp{(\alpha \tau_d)}$ and $s_d = s_b + \Delta_d$. The parameters of the simulations are the same as mentioned in Figure \ref{['fig: allModelsExponential']}.
• Figure 4: Mean and standard deviations of various single-cell-level quantities plotted against each other. The symbols are for the experimental data on different bacterial species grown in different media, taken from the previously published studies taheriJun2015campos2014heerdenKempe2017fievetDucret2015tanouchi2017. (A) mean division-size versus mean birth-size, (B) standard deviation in division-size versus standard deviation in birth-size,(C) mean birth-size versus mean division-added-size, (D) standard deviation in birth-size versus standard deviation in division-added-size, (E) mean division-size versus mean division-added-size, (F) standard deviation in division-size versus standard deviation in division-added-size, (G) mean division-time versus $\log{(2)}$ times inverse of mean growth rate, (H) coefficient of variation of $\tau_d$ versus coefficient of variation of $s_d$.
• Figure 5: (A-D) Various distributions for the three models of cell division under linear biomass growth: (A) age distribution, (B) size distribution, (C) added-size distribution, (D) birth-size distribution; the division-time distribution $\Gamma(\tau_d)$ for the Timer model is taken to be Gaussian with a mean $\langle \tau_d \rangle =1$ hour and a standard deviation $\sigma_{\tau_d} = 0.1$ hour. The Sizer model's division-size distribution is Gaussian with a mean $\langle s_d \rangle =2$$\mu m$ and a standard deviation $\sigma_{s_d} = 0.1$$\mu m$. The Adder model's division-added-size distribution is Gaussian with a mean $\langle \Delta_d \rangle =1$$\mu m$ and a standard deviation $\sigma_{\Delta_d} = 0.1$$\mu m$. The growth rate for the case of non-stochastic growth is taken to be $1 \;\mu$m per hour. For the case of stochastic growth rate (in time), the mean growth rate is taken to be $1 \; \mu$m per hour, and the standard deviation in growth rate is taken to be $0.1 \; \mu$m per hour. (E-M) correlations between various single-cell-level quantities: (E, F, G) correlations between division-time and birth-size for Timer, Sizer, and Adder, respectively, (H, I, J) correlations between division-size and birth-size for Timer, Sizer, and Adder, (K, L, M) correlations between division-added-size and birth-size for Timer, Sizer, and Adder. The slope of the best-fit line is indicated by S, along with the Pearson Correlation Coefficient indicated by P.C. The parameters of the simulations are the same as mentioned for the plots (A-D). (N-U) Mean and standard deviations of various single-cell-level quantities plotted against each other for different bacterial species grown in different media: (N) mean division-size versus mean birth-size, (O) standard deviation in division-size versus standard deviation in birth-size, (P) mean birth-size versus mean division-added-size, (Q) standard deviation in birth-size versus standard deviation in division-added-size, (R) mean division-size versus mean division-added-size, (S) standard deviation in division-size versus standard deviation in division-added-size, (T) mean division-time versus $\sigma_{s_d} / 2 \langle \alpha \rangle$, (U) coefficient of variation of $\tau_d$ versus coefficient of variation of $s_d$. The experimental data have been taken from the previously published studies santiDhar2013chungKarAmir2024nobsMaerkl2014.