Applications of the Galton-Watson process to human DNA evolution and demography

作者：Neves AGM^*; Moreira CHC

来源：Physica A: Statistical Mechanics and Its Applications , 2006, 368(1): 132-146.

DOI：10.1016/j.physa.2005.11.055

• branching processes
• mitochondrial eve
• mitochondrial DNA
• Y chromosome
• evolutionary genetics

We show that the problem of existence of a mitochondrial Eve can be understood as an application of the Galton-Watson process and presents interesting analogies with critical phenomena in Statistical Mechanics. In the approximation of small survival probability, and assuming limited progeny, we are able to find for a genealogic tree the maximum and minimum survival probabilities over all probability distributions for the number of children per woman constrained to a given mean. As a consequence, we can relate existence of a mitochondrial Eve to quantitative demographic data of early mankind. In particular, we show that a mitochondrial Eve may exist even in an exponentially growing population, provided that the mean number of children per woman N is constrained to a small range depending on the probability p that a child is a female. Assuming that the value p approximate to 0.488 valid nowadays has remained fixed for thousands of generations, the range where a mitochondrial Eve occurs with sizeable probability is 2.0492 < N < 2.0510. We also consider the problem of joint existence of a mitochondrial Eve and a Y chromosome Adam. We remark why this problem may not be treated by two independent Galton-Watson processes and present some simulation results suggesting that joint existence of Eve and Adam occurs with sizeable probability in the same TV range. Finally, we show that the Galton-Watson process may be a useful approximation in treating biparental population models, allowing us to reproduce some results previously obtained by Chang and Derrida et al.

• 出版日期2006-8-1