Immortal branching Markov processes: Averaging properties and PCR applications

Abstract

The immortal branching Markov process (iBMP) is a modification of the usual branching model, in which each particle of generation n is counted, in addition to its offspring, as a member of generation $n+1$, its state being unchanged. When the number of offspring is Bernoulli, iBMP accounts, for instance, for the variability of the biological sequences that are produced by polymerase chain reactions (PCRs). This variability is due to the mutations and to the incomplete replications that affect the PCR. Estimators of PCR mutation rate and efficiency have been proposed that are based, in particular, on the mean empirical law $\eta_n$ of the mutations of a sequence. Unfortunately, $\eta_n$ is not analytically tractable. However, the infinite-population limit $\eta^*_n$ of $\eta_n$ is easily characterized in the two following, biologically relevant, cases. The Markovian kernel describes a homogeneous random walk, either on the integers or on some finite Cartesian product of a finite set. In the PCR context, this corresponds to infinite or finite targets, respectively. In this paper, we provide bounds of the discrepancy between $\eta_n$ and $\eta^*_n$ in these two cases. As a consequence, iBMP exhibits a strong averaging effect, even for surprisingly small starting populations. The bounds are explicit functions of the offspring law, the Markovian kernel, the number of steps n, the size of the initial population and, in the finite-target case, the size of the target. They concern every moment and, what might be less expected, the histogram itself. In the finite-target case, some of the bounds undergo a phase transition at an explicit value of the mutation rate per site and per cycle. We use precise estimates of the harmonic means of classical nondecreasing branching processes, whose proofs are included in the Appendix.