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Exponential Growth of Bifurcating Processes with Ancestral Dependence

Part of: Markov processes

Published online by Cambridge University Press:  22 February 2016

Sana Louhichi  and
Bernard Ycart
Sana Louhichi*
Affiliation:
Université Grenoble Alpes
Bernard Ycart*
Affiliation:
Université Grenoble Alpes
*
∗ Postal address: Laboratoire Jean Kuntzmann, Université Grenoble Alpes, 51 rue des Mathématiques, 38041 Grenoble cedex 9, France.
∗ Postal address: Laboratoire Jean Kuntzmann, Université Grenoble Alpes, 51 rue des Mathématiques, 38041 Grenoble cedex 9, France.
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Abstract

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Branching processes are classical growth models in cell kinetics. In their construction, it is usually assumed that cell lifetimes are independent random variables, which has been proved false in experiments. Models of dependent lifetimes are considered here, in particular bifurcating Markov chains. Under the hypotheses of stationarity and multiplicative ergodicity, the corresponding branching process is proved to have the same type of asymptotics as its classic counterpart in the independent and identically distributed supercritical case: the cell population grows exponentially, the growth rate being related to the exponent of multiplicative ergodicity, in a similar way as to the Laplace transform of lifetimes in the i.i.d. case. An identifiable model for which the multiplicative ergodicity coefficients and the growth rate can be explicitly computed is proposed.

Keywords

Branching processbifurcating Markov chainmultiplicative ergodicitycell kinetics

MSC classification

Primary: 60J85: Applications of branching processes
Secondary: 92D25: Population dynamics (general)
Type
Research Article
Information
Advances in Applied Probability , Volume 47 , Issue 2 , June 2015 , pp. 545 - 564
Copyright
© Applied Probability Trust 

References

Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.CrossRefGoogle Scholar
Bellman, R. and Harris, T. (1952). On age-dependent binary branching processes. Ann. Math. (2) 55, 280–295.Google Scholar
Benjamini, I. and Peres, Y. (1994). Markov chains indexed by trees. Ann. Prob. 22, 219–243.Google Scholar
Bitseki Penda, S. V., Djellout, H. and Guillin, A. (2014). Deviation inequelities, moderate deviations and some limit theorems for bifurcating Markov chains with application. Ann. Appl. Prob. 24, 235–291.CrossRefGoogle Scholar
Cowan, R. and Staudte, R. (1986). The bifurcating autoregression model in cell lineage studies. Biometrics 42, 769–783.CrossRefGoogle ScholarPubMed
Crump, K. S. and Mode, C. J. (1969). An age-dependent branching process with correlations among sister cells. J. Appl. Prob. 6, 205–210.Google Scholar
De la Peña, V. H. and Lai, T. L. (2001). Theory and applications of decoupling. In Probability and Statistical Models with Applications, Chapman & Hall/CRC, Boca Raton, FL, pp. 117–145.Google Scholar
De Saporta, B., Gégout-Petit, A. and Marsalle, L. (2011). Parameters estimation for asymmetric bifurcating autoregressive processes with missing data. Electron. J. Statist. 5, 1313–1353.Google Scholar
Delmas, J.-F. and Marsalle, L. (2010). Detection of cellular aging in a Galton–Watson process. Stoch. Process. Appl. 120, 2495–2519.Google Scholar
Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd edn. Springer, New York.Google Scholar
Esary, J. D., Proschan, F. and Walkup, D. W. (1967). Association of random variables, with applications. Ann. Math. Statist. 38, 1466–1474.CrossRefGoogle Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
Glynn, P. W. and Whitt, W. (1994). Large deviations behavior of counting processes and their inverses. Queueing Systems Theory Appl. 17, 107–128.CrossRefGoogle Scholar
Guyon, J. (2007). Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging. Ann. Appl. Prob. 17, 1538–1569.Google Scholar
Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
Harvey, J. D. (1972). Synchronous growth of cells and the generation time distribution. J. General Microbiol. 70, 99–107.Google Scholar
Joag-Dev, K. and Proschan, F. (1983). Negative association of random variables, with applications. Ann. Statist. 11, 286–295.Google Scholar
John, P. C. L. (ed.) (1981). The Cell Cycle. Cambridge University Press.Google Scholar
Kendall, D. G. (1952). On the choice of a mathematical model to represent normal bacterial growth. J. R. Statist. Soc. B 14, 41–44.Google Scholar
Kleptsyna, M. L., Le Breton, A. and Viot, M. (2002). New formulas concerning Laplace transforms of quadratic forms for general Gaussian sequences. J. Appl. Math. Stoch. Anal. 15, 323–339.CrossRefGoogle Scholar
Kleptsyna, M., Le Breton, A. and Ycart, B. (2014). Exponential transform of quadratic functional and multiplicative ergodicity of a Gauss–Markov process. Statist. Prob. Lett. 87, 70–75.CrossRefGoogle Scholar
Kontoyiannis, I. and Meyn, S. P. (2003). Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Prob. 13, 304–362.CrossRefGoogle Scholar
Korevaar, J. (2004). Tauberian Theory. A Century of Developments. Springer, Berlin.Google Scholar
Markham, J. F. et al. (2010). A minimum of two distinct heritable factors are required to explain correlation structures in proliferating lymphocytes. J. R. Soc. Interface 7, 1049–1059.Google Scholar
Meyn, S. and Tweedie, R. L. (2009). Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press.Google Scholar
Nordon, R. E., Ko, K.-H., Odell, R. and Schroeder, T. (2011). Multi-type branching models to describe cell differentiation programs. J. Theoret. Biol. 277, 7–18.Google Scholar
Pemantle, R. (1992). Automorphism invariant measures on trees. Ann. Prob. 20, 1549–1566.Google Scholar
Pemantle, R. (1995). Tree-indexed processes. Statist. Sci. 10, 200–213.Google Scholar
Pitt, M. K., Chatfield, C. and Walker, S. G. (2002). Constructing first order stationary autoregressive models via latent processes. Scand. J. Statist. 29, 657–663.Google Scholar
Powell, E. O. (1956). Growth rate and generation time of bacteria with special reference to continuous culture. Microbiol. 15, 492–511.Google ScholarPubMed
Rahn, O. (1932). A chemical explanation of the variability of the growth rate. J. Gen. Physiol. 15, 257–277.Google Scholar
Shao, Q.-M. (2000). A comparison theorem on moment inequalities between negatively associated and independent random variables. J. Theoret. Prob. 13, 343–356.CrossRefGoogle Scholar
Spitzer, F. (1975). Markov random fields on an infinite tree. Ann. Prob. 3, 387–398.Google Scholar
Wang, P. et al. (2010). Robust growth of Escherichia coli . Curr. Biol. 20, 1099–1103.CrossRefGoogle ScholarPubMed