Estimating the division kernel of a size-structured population

Hoang, Van Ha

ESAIM: Probability and Statistics, Tome 21 (2017), pp. 275-302.

We consider a size-structured model describing a population of cells proliferating by division. Each cell contain a quantity of toxicity which grows linearly according to a constant growth rate $\alpha$. At division, the cells divide at a constant rate $R$ and share their content between the two daughter cells into fractions $\Gamma$ and $1-\Gamma$ where $\Gamma$ has a symmetric density $h$ on $[0,1]$, since the daughter cells are exchangeable. We describe the cell population by a random measure and observe the cells on the time interval $[0,T]$ with fixed $T$. We address here the problem of estimating the division kernel $h$ (or fragmentation kernel) when the division tree is completely observed. An adaptive estimator of $h$ is constructed based on a kernel function $K$ with a fully data-driven bandwidth selection method. We obtain an oracle inequality and an exponential convergence rate, for which optimality is considered.

Classification : 60J80, 60J85, 62G05, 62G07, 92D25
Mots clés : Random size-structured population, division kernel, nonparametric estimation, Goldenshluger-Lepski’s method, adaptive estimator, penalization

H. Aguilaniu, L. Gustafsson, M. Rigoulet and T. Nyström, Asymmetric Inheritance of Oxidatively Damaged Proteins During Cytokinesis. Sci. 299 (2003) 1751. | DOI

M. Ackermann, S.C. Stearns and U. Jenal, Senescence in a Bacterium with Asymmetric Division. Sci. 300 (2003) 1920. | DOI

K.B. Athreya and P.E. Ney, Branching Processes. Springer edition (1970). | MR | Zbl

V. Bansaye, Proliferating parasites in dividing cells: Kimmels branching model revisited. Ann. Appl. Probab. (2008) 967–996. | MR | Zbl

V. Bansaye, J.-F. Delmas, L. Marsalle and V.C. Tran, Limit theorems for Markov processes indexed by continuous time Galton - Watson trees. Ann. Appl. Probab. 21 (2011). | DOI | MR | Zbl

V. Bansaye and V.C. Tran, Branching Feller diffusion for cell division with parasite infection. ALEA, Lat. Am. J. Probab. Math. Stat. 8 (2011) 95–127. | MR | Zbl

V. Bansaye, J.C. Pardo and C. Smadi, On the extinction of continuous state branching processes with catastrophes. Electron. J. Probab. 18 (2013) 1–31. | DOI | MR | Zbl

B. Bercu, B. De Saporta and A. Gégout-Petit, Asymptotic analysis for bifurcating autoregressive processes via a martingale approach. Electron. J. Probab. 14 (2009) 2492–2526. | DOI | MR | Zbl

K. Bertin, C. Lacour and V. Rivoirard, Adaptive pointwise estimation of conditional density function. Ann. Inst. Henri Poincaré Probab. Statist. (2015). | MR

S. Valère Bitseki Penda, Deviation inequalities for bifurcating Markov chains on Galton – Watson tree. ESAIM: PS 19 (2015) 689–724. | DOI | Numdam | MR | Zbl

B. Cloez, Limit theorems for some branching measure-valued processes. Adv. Appl. Probab. 49 (2017) 549–580. | DOI | MR | Zbl

J.-F. Delmas and L. Marsalle, Detection of cellular aging in a Galton-Watson process. Stochastic Processes and their Application 120 (2010) 2495–2519. | DOI | MR | Zbl

M. Doumic, M. Hoffmann, N. Krell and L. Robert, Statistical estimation of a growth-fragmentation model observed on a genealogical tree. Bernoulli 21 1760–1799. | MR

M. Doumic, M. Hoffmann, P. Reynaud-Bouret and V. Rivoirard, Nonparametric estimation of the division rate of a size-structured population. SIAM J. Numer. Anal. 50 (2012). | DOI | MR | Zbl

S.N. Evans and D. Steinsaltz, Damage segregation at fissioning may increase growth rates: A superprocess model. Theoret. Popul. Biol. 71 (2007) 473–490. | DOI | Zbl

A. Goldenshluger and O. Lepski, Bandwidth selection in kernel density estimation: oracle inequalities and adaptive minimax optimality. Ann. Statist. 39 (2011) 1608–1632. | DOI | MR | Zbl

J. Guyon, Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging. Ann. Appl. Probab. 17 (2007) 1538–1569. | DOI | MR | Zbl

T.E. Harris, The Theory of Branching Processes. Springer, Berlin (1963). | MR | Zbl

V.H. Hoang, Adaptive estimation for inverse problems with applications to cell divisions. PhD dissertation, Université de Lille 1 Sciences et Technologies, https://tel.archives-ouvertes.fr/tel-01417780 (2016).

M. Hoffmann and A. Olivier, Nonparametric estimation of the division rate of an age dependent branching process. Stoch. Process. Appl. 126 (2016) 1433–1471. | DOI | MR | Zbl

N. Ikeda and S. Wanatabe, Stochastic differential equations and diffusion processes. Vol. 24 of North-Holland Mathematical Library. | MR | Zbl

J. Jacob and A.N. Shiryaev, Limit Theorems for Stochastic Processes. Springer Verlag, Berlin (1987). | MR | Zbl

C.-Y. Lai, E. Jaruga, C. Borghouts and S.M. Jazwinski1, A Mutation in the ATP2 Gene Abrogates the Age Asymmetry Between Mother and Daughter Cells of the Yeast Saccharomyces cerevisiae. Genetics 162 (2002) 73–87. | DOI

C. Lacour and P. Massart. Minimal penalty for Goldenshluger-Lepski method. Stoch. Process. Appl. 126 (2016) 3774–3789. | DOI | MR | Zbl

A.B. Lindner, R. Madden, A. Demarez, E.J. Stewart and F. Taddei, Asymmetric segregation of protein aggregates is associated with cellular aging and rejuvenation. PNAS 105 (2015) 3076–3081. | DOI

P. Massart, Concentration Inequalities and Model Selection. Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour, 6-23 (2003). Springer (2007). | MR | Zbl

S.P. Meyn and R.L. Tweedie, Stability of Markovian processes iii: Foster - Lyapunov criteria for continuous-time processes. Adv. Appl. Probab. 25 (1993). | MR | Zbl

J.B. Moseley, Cellular Aging: Symmetry Evades Senescence. Current Biology, Vol. 23, Issue 19, R871 − R873 (2013).

J. Peter, A general stochastic model for population development. Scandinavian Actuarial J. 1969 (1969) 84–103. | DOI | Zbl

L. Robert, M. Hoffmann, N. Krell, S. Aymerich, J. Robert and M. Doumic, Division in Escherichia coli is triggered by a size-sensing rather than a timing mechanism. BMC Biology 12 (2014). | DOI

Sh.M. Ross, Stochastic Processes. Wiley, 2nd edition (1995). | MR | Zbl

P. Reynaud-Bouret, V. Rivoirard, F. Grammont and C. Tuleau-Malot, Goodness-of-fit tests and nonparametric adaptive estimation for spike train analysis. J. Math. Neuroscience 4 (2014). | DOI | MR | Zbl

B.W. Silverman, Density estimation for statistics and data analysis, volume 26. CRC press (1986). | MR | Zbl

E.J. Stewart, R. Madden, G. Paul and F. Taddei, Aging and Death in an Organism That Reproduces by Morphologically Symmetric Division. PLOS Biology 3 (2005). | DOI

V.C. Tran, Modèles particulaires stochastiques pour des problèmes d’évolution adaptive et pour l’approximation de solutions statisques. Ph.D. dissertation, Université Paris X - Nanterre, http://tel.archives-ouvertes.fr/tel-00125100 (2006).

V.C. Tran, Large population limit and time behaviour of a stochastic particle model describing an age-structured population. ESAIM: Probab. Statis. 12 (2008). | Numdam | MR | Zbl

A.B. Tsybakov, Introduction to Nonparametric Estimation. Springer series in Statistics. Springer (2004). | MR | Zbl