## Bernoulli

• Bernoulli
• Volume 21, Number 3 (2015), 1760-1799.

### Statistical estimation of a growth-fragmentation model observed on a genealogical tree

#### Abstract

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 ( SIAM J. Numer. Anal. 50 (2012) 925–950, Inverse Problems 25 (2009) 1–22) through indirect observation schemes. Our method is consistently tested numerically and implemented on Escherichia coli data, which demonstrates its major interest for practical applications.

#### Article information

Source
Bernoulli, Volume 21, Number 3 (2015), 1760-1799.

Dates
Revised: March 2014
First available in Project Euclid: 27 May 2015

https://projecteuclid.org/euclid.bj/1432732036

Digital Object Identifier
doi:10.3150/14-BEJ623

Mathematical Reviews number (MathSciNet)
MR3352060

Zentralblatt MATH identifier
06470456

#### Citation

Doumic, Marie; Hoffmann, Marc; Krell, Nathalie; Robert, Lydia. Statistical estimation of a growth-fragmentation model observed on a genealogical tree. Bernoulli 21 (2015), no. 3, 1760--1799. doi:10.3150/14-BEJ623. https://projecteuclid.org/euclid.bj/1432732036

#### References

• [1] Baccelli, F., McDonald, D.R. and Reynier, J. (2002). A mean-field model for multiple TCP connections through a buffer implementing RED. Performance Evaluation 49 77–97.
• [2] Balagué, D., Canizo, J. and Gabriel, P. (2013). Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates. Kinetic and Related Models 6 219–243.
• [3] Banks, H.T., Sutton, K.L., Thompson, W.C., Bocharov, G., Roosec, D., Schenkeld, T. and Meyerhanse, A. (2011). Estimation of cell proliferation dynamics using CFSE data. Bull. Math. Biol. 73 116–150.
• [4] Bansaye, V. (2008). Proliferating parasites in dividing cells: Kimmel’s branching model revisited. Ann. Appl. Probab. 18 967–996.
• [5] Bansaye, V., Delmas, J.-F., Marsalle, L. and Tran, V.C. (2011). Limit theorems for Markov processes indexed by continuous time Galton–Watson trees. Ann. Appl. Probab. 21 2263–2314.
• [6] Baxendale, P.H. (2005). Renewal theory and computable convergence rates for geometrically ergodic Markov chains. Ann. Appl. Probab. 15 700–738.
• [7] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge: Cambridge Univ. Press.
• [8] Cáceres, M.J., Cañizo, J.A. and Mischler, S. (2011). Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations. J. Math. Pures Appl. (9) 96 334–362.
• [9] Chauvin, B., Rouault, A. and Wakolbinger, A. (1991). Growing conditioned trees. Stochastic Process. Appl. 39 117–130.
• [10] Cloez, B. (2011). Limit theorems for some branching measure-valued processes. Available at arXiv:1106.0660v2.
• [11] Douc, R., Moulines, E. and Rosenthal, J.S. (2004). Quantitative bounds on convergence of time-inhomogeneous Markov chains. Ann. Appl. Probab. 14 1643–1665.
• [12] Doumic, M., Hoffmann, M., Reynaud-Bouret, P. and Rivoirard, V. (2012). Nonparametric estimation of the division rate of a size-structured population. SIAM J. Numer. Anal. 50 925–950.
• [13] Doumic, M., Maia, P. and Zubelli, J.P. (2010). On the calibration of a size-structured population model from experimental data. Acta Biotheor. 58 405–413.
• [14] Doumic, M., Perthame, B. and Zubelli, J.P. (2009). Numerical solution of an inverse problem in size-structured population dynamics. Inverse Problems 25 1–22.
• [15] Doumic, M. and Tine, L.M. (2012). Estimating the division rate for the growth-fragmentation equation. J. Math. Biol. 67 69–103.
• [16] Doumic, M. and Gabriel, P. (2010). Eigenelements of a general aggregation–fragmentation model. Math. Models Methods Appl. Sci. 20 757–783.
• [17] Engler, H., Prüss, J. and Webb, G.F. (2006). Analysis of a model for the dynamics of prions. II. J. Math. Anal. Appl. 324 98–117.
• [18] Fort, G., Moulines, E. and Priouret, P. (2011). Convergence of adaptive and interacting Markov chain Monte Carlo algorithms. Ann. Statist. 39 3262–3289.
• [19] Gobet, E., Hoffmann, M. and Reiß, M. (2004). Nonparametric estimation of scalar diffusions based on low frequency data. Ann. Statist. 32 2223–2253.
• [20] Goldenshluger, A. and Lepski, O. (2011). Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality. Ann. Statist. 39 1608–1632.
• [21] Haas, B. (2003). Loss of mass in deterministic and random fragmentations. Stochastic Process. Appl. 106 245–277.
• [22] Harris, S.C. and Roberts, M.I. (2012). The many-to-few lemma and multiple spines. Available at arXiv:1106.4761v3.
• [23] Kaern, M., Elston, T.C., Blake, W.J. and Collins, J.J. (2005). Stochasticity in gene expression: From theories to phenotypes. Nat. Rev. Genet. 6 451–464.
• [24] Kubitschek, H.E. (1969). Growth during the bacterial cell cycle: Analysis of cell size distribution. Biophys. J. 9 792–809.
• [25] Laurençot, P. and Perthame, B. (2009). Exponential decay for the growth-fragmentation/cell-division equation. Commun. Math. Sci. 7 503–510.
• [26] Metz, J.A.J. and Diekmann, O., eds. (1986). The Dynamics of Physiologically Structured Populations. Lecture Notes in Biomathematics 68. Berlin: Springer. Papers from the colloquium held in Amsterdam, 1983.
• [27] Meyn, S. and Tweedie, R. (1993). Markov Chains and Stochastic Stability. Berlin: Springer.
• [28] Michel, P. (2006). Existence of a solution to the cell division eigenproblem. Math. Models Methods Appl. Sci. 16 1125–1153.
• [29] Michel, P., Mischler, S. and Perthame, B. (2005). General relative entropy inequality: An illustration on growth models. J. Math. Pures Appl. (9) 84 1235–1260.
• [30] Niethammer, B. and Pego, R.L. (1999). Non-self-similar behavior in the LSW theory of Ostwald ripening. J. Stat. Phys. 95 867–902.
• [31] Pakdaman, K., Perthame, B. and Salort, D. (2012). Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation. Available at http://hal.upmc.fr/hal-00794841.
• [32] Perthame, B. (2007). Transport Equations in Biology. Frontiers in Mathematics. Basel: Birkhäuser.
• [33] Perthame, B. and Ryzhik, L. (2005). Exponential decay for the fragmentation or cell-division equation. J. Differential Equations 210 155–177.
• [34] Perthame, B. and Zubelli, J.P. (2007). On the inverse problem for a size-structured population model. Inverse Problems 23 1037–1052.
• [35] Stewart, E.J., Madden, R., Paul, G. and Taddei, F. (2005). Aging and death in an organism that reproduces by morphologically symmetric division. PLoS Comput. Biol. 3 e45.
• [36] Sturm, A., Heinemann, M., Arnoldini, M., Benecke, A., Ackermann, M., Benz, M., Dormann, J. and Hardt, W.-D. (2011). The cost of virulence: Retarded growth of Salmonella typhimurium cells expressing type III secretion system 1. PLoS Pathog. 7 e1002143.
• [37] Tan, C., Marguet, P. and You, L. (2009). Emergent bistability by a growth-modulating positive feedback circuit. Nat. Chem. Biol. 5 842–848.
• [38] Wang, P., Robert, L., Pelletier, J., Dang, W.L., Taddei, F., Wright, A. and Jun, S. (2010). Robust growth of Escherichia coli. Curr. Biol. 20 1099–1103.