Immortal branching Markov processes: Averaging properties and PCR applications

• 成果类型: Article
• 署名作者: Piau, D
• 署名单位: Universite Claude Bernard Lyon 1
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1078415838
• 发表日期: 2004
• 页码: 337-364
• 关键词: polymerase-chain-reaction; replication errors; CONVERGENCE; dna

摘要

The immortal branching Markov process (iBMP) is a modification of the usual branching model, in which each particle of generation 17 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* 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* 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.