2012/10/11 by Marie Doumic, Doumic, Marie, Marc Hoffmann +5
Biochemistry, Genetics and Molecular Biology · Mathematics · #35A05 #35B40 #45C05 #45K05 #62G05 #62G20 #82D60 #92D25 #FOS: Biological sciences #FOS: Mathematics #Probability (math.PR) #Quantitative Methods (q-bio.QM) #Statistics Theory (math.ST) #math.PR #math.ST #msc:35A05 #msc:35B40 #msc:45C05 #msc:45K05 #msc:62G05 #msc:62G20 #msc:82D60 #msc:92D25 #q-bio.QM #stat.TH
paper · pdf · doi:10.48550/arxiv.1210.3240
46 pages, 4 figures
arxiv created 2015/05/28 · arxiv updated 2015/05/29
We model the growth of a cell population by a piecewise deterministic Markov branching tree. Each cell splits into two offsprings at a division rate B(x) that depends on its size x. The size of each cell grows exponentially in time, at a rate that varies for each individual. We show that the mean empirical measure of the model satisfies a growth-fragmentation type equation if structured in both size and growth rate as state variables. We 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. When the growth rate is assumed to be identical for every cell, we retrieve the classical growth-fragmentation model and our estimator improves on the rate n-s/(2s+3) obtained in \citeDHRR, DPZ through indirect observation schemes. Our method is consistently tested numerically and implemented on \it Escherichia coli data.