We review recent results of ours concerning branching processes with general lifetimes and neutral mutations, under the infinitely many alleles model, where mutations can occur either at the birth of particles or at a constant rate during their lives. In both models, we study the allelic partition of the population at time . We give closed-form formulae for the expected frequency spectrum at and prove a pathwise convergence to an explicit limit, as , of the relative numbers of types younger than some given age and carried by a given number of particles (small families). We also provide the convergences in distribution of the sizes or ages of the largest families and of the oldest families. In the case of exponential lifetimes, population dynamics are given by linear birth and death processes, and we can most of the time provide general formulations of our results unifying both models. 1. Introduction We consider a general branching model, where particles have i.i.d. (not necessarily exponential) life lengths and give birth at a constant rate during their lives to independent copies of themselves. The genealogical tree thus produced is called splitting tree [1–3]. The process that counts the number of the alive particles through time is a Crump-Mode-Jagers process (or general branching process) [4] which is binary (births occur singly) and homogeneous (constant birth rate). We enrich this genealogical model with mutations. In Model I, each child is a clone of her mother with probability and a mutant with probability . In Model II, independently of other particles, each particle undergoes mutations during her life at constant rate (and births are always clonal). For both models, we are working under the infinitely many alleles model; that is, a mutation yields a type, also called allele, which was never encountered before. Moreover, mutations are supposed to be neutral; that is, they do not modify the way particles die and reproduce. For any type and any time , we call family the set of all particles that share this type at time . Branching processes (and especially birth and death processes) with mutations have many applications in biology. In carcinogenesis [5–10], they can model the evolution of cancerous cells. In [11], Kendall modeled carcinogenesis by a birth and death process where mutations occur during life according to an inhomogeneous Poisson process. In [8, 10], cancerous cells are modeled by a multitype branching process where a cell is of type if it has undergone mutations and where the more a cell has undergone mutations, the faster it
J. Geiger and G. Kersting, “Depth-first search of random trees, and Poisson point processes,” in Classical and Modern Branching Processes (Minneapolis, MN, 1994), vol. 84 of IMA Volumes in Mathematics and its Applications, pp. 111–126, Springer, New York, NY, USA, 1997.
P. Jagers, Branching Processes with Biological Applications, Wiley-Interscience, London, UK, 1975, Wiley Series in Probability and Mathematical Statistics—Applied Probability and Statistic.
M. A. Nowak, F. Michor, and Y. Iwasa, “The linear process of somatic evolution,” Proceedings of the National Academy of Sciences, vol. 100, no. 25, pp. 14966–14969, 2003.
S. Sagitov and M. C. Serra, “Multitype Bienaymé-Galton-Watson processes escaping extinction,” Advances in Applied Probability, vol. 41, no. 1, pp. 225–246, 2009.
R. Durrett and S. Moseley, “Evolution of resistance and progression to disease during clonal expansion of cancer,” Theoretical Population Biology, vol. 77, no. 1, pp. 42–48, 2010.
K. Danesh, R. Durrett, L. J. Havrilesky, and E. Myers, “A branching process model of ovarian cancer,” Journal of Theoretical Biology, vol. 314, pp. 10–15, 2012.
A. Lambert and P. Trapman, “Splitting trees stopped when the first clock rings and Vervaat's transformation,” Journal of Applied Probability. In press.
B. Haegeman and R. S. Etienne, “Relaxing the zero-sum assumption in neutral biodiversity theory,” Journal of Theoretical Biology, vol. 252, no. 2, pp. 288–294, 2008.
A. Lambert, “Species abundance distributions in neutral models with immigration or mutation and general lifetimes,” Journal of Mathematical Biology, vol. 63, no. 1, pp. 57–72, 2011.
W. J. Ewens, “The sampling theory of selectively neutral alleles,” Theoretical Population Biology, vol. 3, pp. 87–112, 1972, erratum, ibid. vol. 3, p. 240, 1972; erratum, ibid. vol. 3, p. 376, 1972.
R. C. Griffiths and A. G. Pakes, “An infinite-alleles version of the simple branching process,” Advances in Applied Probability, vol. 20, no. 3, pp. 489–524, 1988.
J. Bertoin, “The structure of the allelic partition of the total population for Galton-Watson processes with neutral mutations,” The Annals of Probability, vol. 37, no. 4, pp. 1502–1523, 2009.
J. Bertoin, “A limit theorem for trees of alleles in branching processes with rare neutral mutations,” Stochastic Processes and their Applications, vol. 120, no. 5, pp. 678–697, 2010.
Y. E. Maruvka, N. M. Shnerb, and D. A. Kessler, “Universal features of surname distribution in a subsample of a growing population,” Journal of Theoretical Biology, vol. 262, no. 2, pp. 245–256, 2010.
Y. E. Maruvka, D. A. Kessler, and N. M. Shnerb, “The birth-death-mutation process: a new paradigm for fat tailed distributions,” PLoS ONE, vol. 6, no. 11, article e26480, 2011.
N. Champagnat and A. Lambert, “Splitting trees with neutral Poissonian mutations I: small families,” Stochastic Processes and their Applications, vol. 122, no. 3, pp. 1003–1033, 2012.
N. Champagnat and A. Lambert, “Splitting trees with neutral Poissonian mutations II:,” Largest and Oldest families. In press, http://arxiv.org/abs/1108.4812.
O. Nerman, “On the convergence of supercritical general (C-M-J) branching processes,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 57, no. 3, pp. 365–395, 1981.
M. Richard, “Limit theorems for supercritical age-dependent branching processes with neutral immigration,” Advances in Applied Probability, vol. 43, no. 1, pp. 276–300, 2011.
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, vol. 55 of National Bureau of Standards Applied Mathematics Series, Superintendent of Documents, U.S. Government Printing Office, Washington, DC, USA, 1964.
P. Jagers and O. Nerman, “Limit theorems for sums determined by branching and other exponentially growing processes,” Stochastic Processes and their Applications, vol. 17, no. 1, pp. 47–71, 1984.