Cox Markov models for estimating single cell growth

Recent experimental techniques produce thousands of data of single cell growth, consequently stochastic models of growth can be validated on true data and used to understand the main mechanisms that control the cell cycle. A sequence of growing cells is usually modeled by a suitable Markov chain. In this framework, the most interesting goal is to infer the distribution of the doubling time (or of the added size) of a cell given its initial size and its elongation rate. In the literature, these distributions are described in terms of the corresponding conditional hazard function, referred as division hazard rate. In this work we propose a simple but effective way to estimate the division hazard by using extended Cox modeling. We investigate the convergence to the stationary distribution of the Markov chain describing the sequence of growing cells and we prove that, under reasonable conditions, the proposed estimators of the division hazard rates are asymptotically consistent. Finally, we apply our model to study some published datasets of E-Coli cells.