Modelling the variability of lag times and the first generation times of single cells of (original) (raw)

Kinetics of Single Cells: Observation and Modeling of a Stochastic Process

Applied and Environmental Microbiology, 2006

Appl. Environ. Microbiol. 70:675-678, 2004), and the histograms they generated were used as empirical distributions to simulate growth of the population as the result of the multiplication of its single cells. This way, a stochastic birth model in which the underlying distributions were measured experimentally was simulated. To validate the model, analogous bacterial growth curves were generated by the use of different inoculum levels. The agreement with the simulation was very good, proving that the growth of the population can be predicted accurately if the distribution of the first few division times for the single cells within that population is known. Two questions were investigated by the simulation. (i) To what extent can we say that the distribution of the detection time, i.e., the time by which a single-cell-generated subpopulation reaches a detectable level, can be identified with that of the lag time of the original single cell? (ii) For low inocula, how does the inoculum size affect the lag time of the population?

Modeling Bacterial Population Growth from Stochastic Single-Cell Dynamics

Applied and Environmental Microbiology, 2014

A few bacterial cells may be sufficient to produce a food-borne illness outbreak, provided that they are capable of adapting and proliferating on a food matrix. This is why any quantitative health risk assessment policy must incorporate methods to accurately predict the growth of bacterial populations from a small number of pathogens. In this aim, mathematical models have become a powerful tool. Unfortunately, at low cell concentrations, standard deterministic models fail to predict the fate of the population, essentially because the heterogeneity between individuals becomes relevant. In this work, a stochastic differential equation (SDE) model is proposed to describe variability within single-cell growth and division and to simulate population growth from a given initial number of individuals. We provide evidence of the model ability to explain the observed distributions of times to division, including the lag time produced by the adaptation to the environment, by comparing model predictions with experiments from the literature for Escherichia coli, Listeria innocua, and Salmonella enterica. The model is shown to accurately predict experimental growth population dynamics for both small and large microbial populations. The use of stochastic models for the estimation of parameters to successfully fit experimental data is a particularly challenging problem. For instance, if Monte Carlo methods are employed to model the required distributions of times to division, the parameter estimation problem can become numerically intractable. We overcame this limitation by converting the stochastic description to a partial differential equation (backward Kolmogorov) instead, which relates to the distribution of division times. Contrary to previous stochastic formulations based on random parameters, the present model is capable of explaining the variability observed in populations that result from the growth of a small number of initial cells as well as the lack of it compared to populations initiated by a larger number of individuals, where the random effects become negligible.

Cell division theory and individual-based modeling of microbial lag

International Journal of Food Microbiology, 2005

This paper is the second in a series of two, and studies microbial lag in cell number and/or biomass measurements caused by temperature changes with an individual-based modeling approach. For this purpose, the theory of cell division, as discussed in the first part of this series of research papers, was implemented in the individual-based modeling framework BacSim. Simulations of this model are compared with experimental data of Escherichia coli, growing in an aerated, glucose-rich medium and subjected to sudden temperature shifts. The premise of a constant cell volume under changing temperature conditions predicts no lag in cell numbers after the shift, in contrast to the experimental observations. Based on literature research, two biological mechanisms that could be responsible for the observed lag phenomena are proposed. The first assumes that the average cell volume depends on temperature while the second assumes that a lag in biomass growth occurs after the temperature shift. For a lag in cell number caused by an increased average cell volume, the cell biomass always increases at the maximal rate. Therefore, cells are evidently not stressed and do not have to adapt to the new conditions, as opposed to a lag in biomass growth. Implementation and simulation of both mechanisms are found to describe the experimental observations equally well. Therefore, further research is needed to distinguish between the two mechanisms. This can be done by observing, in addition to cell numbers, a measure for the average cell volumes. In conclusion, the individual-based modeling approach is a good methodology to investigate and test biological theories and assumptions. Also, based on the simulations, suggestions for further experimental observations can be made. D

Cell growth and division: a deterministic/probabilistic model of the cell cycle

Journal of Mathematical Biology, 1986

A model of the cell cycle, incorporating a deterministic cell-size monitor and a probabilistic component, is investigated. Steady-state distributions for cell size and generation time are calculated and shown to be globally asymptotically stable. These distributions are used to calculate various statistical quantities, which are then compared to known experimental data. Finally, the results are compared to distributions calculated from a Monte-Carlo simulation of the model.

Probabilistic Model of Microbial Cell Growth, Division, and Mortality

Applied and Environmental Microbiology, 2010

After a short time interval of length δ t during microbial growth, an individual cell can be found to be divided with probability P d ( t )δ t , dead with probability P m ( t )δ t , or alive but undivided with the probability 1 − [ P d ( t ) + P m ( t )]δ t , where t is time, P d ( t ) expresses the probability of division for an individual cell per unit of time, and P m ( t ) expresses the probability of mortality per unit of time. These probabilities may change with the state of the population and the habitat's properties and are therefore functions of time. This scenario translates into a model that is presented in stochastic and deterministic versions. The first, a stochastic process model, monitors the fates of individual cells and determines cell numbers. It is particularly suitable for small populations such as those that may exist in the case of casual contamination of a food by a pathogen. The second, which can be regarded as a large-population limit of the stochastic m...

Scaling laws governing stochastic growth and division of single bacterial cells

Proceedings of the National Academy of Sciences, 2014

Uncovering the quantitative laws that govern the growth and division of single cells remains a major challenge. Using a unique combination of technologies that yields unprecedented statistical precision, we find that the sizes of individual Caulobacter crescentus cells increase exponentially in time. We also establish that they divide upon reaching a critical multiple (≈ 1.8) of their initial sizes, rather than an absolute size. We show that when the temperature is varied, the growth and division timescales scale proportionally with each other over the physiological temperature range. Strikingly, the cell-size and division-time distributions can both be rescaled by their mean values such that the condition-specific distributions collapse to universal curves. We account for these observations with a minimal stochastic model that is based on an autocatalytic cycle. It predicts the scalings, as well as specific functional forms for the universal curves. Our experimental and theoretical analysis reveals a simple physical principle governing these complex biological processes: a single temperature-dependent scale of cellular time governs the stochastic dynamics of growth and division in balanced growth conditions.

Continuous Rate Modelling of bacterial stochastic size dynamics

2020

Bacterial division is an inherently stochastic process. However, theoretical tools to simulate and study the stochastic transient dynamics of cell-size are scarce. Here, we present a general theoretical approach based on the Chapman-Kolmogorov formalism to describe these stochastic dynamics including continuous growth and division events as jump processes. Using this approach, we analyze the effect of different sources of noise on the dynamics of the size distribution. Oscillations in the distribution central moments were found as consequence of the discrete translation invariance of the system with period of one doubling time, these oscillations are found in both the central moments of the size distribution and the auto-correlation function and do not disappear including stochasticity on division times or size heterogeneity on the population but only after include noise in either growth rate or septum position.

Continuous rate modeling of bacterial stochastic size dynamics

Physical Review E, 2021

Bacterial division is an inherently stochastic process with effects on fluctuations of protein concentration and phenotype variability. Current modeling tools for the stochastic short-term cell-size dynamics are scarce and mainly phenomenological. Here we present a general theoretical approach based on the Chapman-Kolmogorov equation incorporating continuous growth and division events as jump processes. This approach allows us to include different division strategies, noisy growth, and noisy cell splitting. Considering bacteria synchronized from their last division, we predict oscillations in both the central moments of the size distribution and its autocorrelation function. These oscillations, barely discussed in past studies, can arise as a consequence of the discrete time displacement invariance of the system with a period of one doubling time, and they do not disappear when including stochasticity on either division times or size heterogeneity on the starting population but only after inclusion of noise in either growth rate or septum position. This result illustrates the usefulness of having a solid mathematical description that explicitly incorporates the inherent stochasticity in various biological processes, both to understand the process in detail and to evaluate the effect of various sources of variability when creating simplified descriptions.

Timing the Start of Division in E. coli: a Single-Cell Study

Biophysical …, 2009

We monitor the shape dynamics of individual E. coli cells using time-lapse microscopy together with accurate image analysis. This allows measuring the dynamics of single-cell parameters throughout the cell cycle. In previous work, we have used this approach to characterize the main features of single-cell morphogenesis between successive divisions. Here, we focus on the behavior of the parameters that are related to cell division and study their variation over a population of 30 cells. In particular, we show that the single-cell data for the constriction width dynamics collapse onto a unique curve following appropriate rescaling of the corresponding variables. This suggests the presence of an underlying time scale that determines the rate at which the cell cycle advances in each individual cell. For the case of cell length dynamics a similar rescaling of variables emphasizes the presence of a breakpoint in the growth rate at the time when division starts, τ c . We also find that the τ c of individual cells is correlated with their generation time, τ g , and inversely correlated with the corresponding length at birth, L 0 . Moreover, the extent of the T-period, τ g − τ c , is apparently independent of τ g . The relations between τ c , τ g and L 0 indicate possible compensation mechanisms that maintain cell length variability at about 10%. Similar behavior was observed for both fast-growing cells in a rich medium (LB) and for slower growth in a minimal medium (M9-glucose). To reveal the molecular mechanisms that lead to the observed organization of the cell cycle, we should further extend our approach to monitor the formation of the divisome.