BackgroundMany organisms coordinate cell growth and division through size control mechanisms: cells must reach a critical size to trigger a cell cycle event. Bacterial division is often assumed to be controlled in this way, but experimental evidence to support this assumption is still lacking. Theoretical arguments show that size control is required to maintain size homeostasis in the case of exponential growth of individual cells. Nevertheless, if the growth law deviates slightly from exponential for very small cells, homeostasis can be maintained with a simple ‘timer’ triggering division. Therefore, deciding whether division control in bacteria relies on a ‘timer’ or ‘sizer’ mechanism requires quantitative comparisons between models and data.ResultsThe timer and sizer hypotheses find a natural expression in models based on partial differential equations. Here we test these models with recent data on single-cell growth of Escherichia coli. We demonstrate that a size-independent timer mechanism for division control, though theoretically possible, is quantitatively incompatible with the data and extremely sensitive to slight variations in the growth law. In contrast, a sizer model is robust and fits the data well. In addition, we tested the effect of variability in individual growth rates and noise in septum positioning and found that size control is robust to this phenotypic noise.ConclusionsConfrontations between cell cycle models and data usually suffer from a lack of high-quality data and suitable statistical estimation techniques. Here we overcome these limitations by using high precision measurements of tens of thousands of single bacterial cells combined with recent statistical inference methods to estimate the division rate within the models. We therefore provide the first precise quantitative assessment of different cell cycle models.

We raise the issue of estimating the division rate for a growing and dividing population modelled by a piecewise deterministic Markov branching tree. Such models have broad applications, ranging from TCP/IP window size protocol to bacterial growth. Here, the individuals split into two offsprings at a division rate B(x) that depends on their size x, whereas their size grow exponentially in time, at a rate that exhibits variability. The mean empirical measure of the model satisfies a growth-fragmentation type equation, and we bridge the deterministic and probabilistic viewpoints. We then construct a nonparametric estimator of the division rate B(x) based on the observation of the population over different sampling schemes of size n on the genealogical tree. Our estimator nearly achieves the rate n −s/(2s+1) in squared-loss error asymptotically, generalizing and improving on the rate n −s/(2s+3) obtained in [13,15] through indirect observation schemes. Our method is consistently tested numerically and implemented on Escherichia coli data, which demonstrates its major interest for practical applications.

Abstract.We give a short overview of recent results on a specific class of Markov process: the Piecewise Deterministic Markov Processes (PDMPs). We first recall the definition of these processes and give some general results. On more specific cases such as the TCP model or a model of switched vector fields, better results can be proved, especially as regards long time behaviour. We continue our review with an infinite dimensional example of neuronal activity. From the statistical point of view, these models provide specific challenges: we illustrate this point with the example of the estimation of the distribution of the inter-jumping times. We conclude with a short overview on numerical methods used for simulating PDMPs. General introductionThe piecewise deterministic Markov processes (denoted PDMPs) were first introduced in the literature by Davis ( [Dav84,Dav93]). Already at this time, the theory of diffusions had such powerful tools as the theory of Itō calculus and stochastic differential equations at its disposal. Davis's goal was to endow the PDMP with rather general tools. The main reason for that was to provide a general framework, since up to then only very particular cases had been dealt with, which turned out not to be easily generalizable.PDMPs form a family of càdlàg Markov processes involving a deterministic motion punctuated by random jumps. The motion of the PDMP {X(t)} t≥0 depends on three local characteristics, namely the jump rate λ, the flow φ and the transition measure Q according to which the location of the process at the jump time is chosen. The process starts from x and follows the flow φ(x, t) until the first jump time T 1 which occurs either spontaneously in a Poisson-like fashion with rate λ (φ(x, t)) or when the flow φ(x, t) hits the boundary of the state-space. In both cases, the location of the process at the jump time T 1 , denoted by Z 1 = X(T 1 ), is selected by the transition measure Q(φ(x, T 1 ), ·) and the motion restarts from this new point as before. This fully describes a piecewise continuous trajectory for {X(t)} with jump times {T k } and post jump locations {Z k }, and which evolves according to the flow φ between two jumps. Article published online by EDP Sciences and available at http://www.esaim-proc.org or http://dx

We explore statistical inference in self-similar conservative fragmentation chains when only approximate observations of the sizes of the fragments below a given threshold are available. This framework, introduced by Bertoin and Martinez [Adv. Appl. Probab. 37 (2005) 553--570], is motivated by mineral crushing in the mining industry. The underlying object that can be identified from the data is the step distribution of the random walk associated with a randomly tagged fragment that evolves along the genealogical tree representation of the fragmentation process. We compute upper and lower rates of estimation in a parametric framework and show that in the nonparametric case, the difficulty of the estimation is comparable to ill-posed linear inverse problems of order 1 in signal denoising.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ274 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm