Modeling Neutral Evolution Using an Infinite-Allele Markov Branching Process

We consider an infinite-allele Markov branching process (IAMBP). Our main focus is the frequency spectrum of this process, that is, the proportion of alleles having a given number of copies at a specified time point. We derive the variance of the frequency spectrum, which is useful for interval estimation and hypothesis testing for process parameters. In addition, for a class of special IAMBP with birth and death offspring distribution, we show that the mean of its limiting frequency spectrum has an explicit form in terms of the hypergeometric function. We also derive an asymptotic expression for convergence rate to the limit. Simulations are used to illustrate the results for the birth and death process. 1. Introduction The infinite-allele branching process was first introduced by Griffiths and Pakes [1]. As a special type of branching process, this process allows individuals to mutate into infinitely many allelic variants, each of which is “new” in the sense of being different from all previously existing variants. This idealization is approximately correct for rare point mutations in long DNA sequences. Fundamental results for the discrete-time case (simple branching process) and for the continuous-time case (Markov branching process) have been obtained in [1, 2]. These include the number of alleles at a given generation or time, the generation number or time of the last mutation, and the limiting frequency spectrum. There exists an analogy between the results for the discrete-time and the continuous-time cases; however, the characteristics in the continuous case are relatively easier to derive [2]. Many evolutionary processes may be considered time continuous, and frequently we assume Markov property in modeling. A classical example is the discrete-time Wright-Fisher model, which is typically either approximated by a continuous-time diffusion or replaced by a continuous-time Markov chain, the so-called continuous-time Moran process [3]. Therefore, the time-continuous infinite-allele Markov branching process (TCIAMBP, or simply IAMBP) seems to be appropriate for modeling evolution in population genetics. Consider a Markov branching process with neutral mutations. Suppose that the process starts from a group of individuals carrying the same allele, and individuals can mutate into new allelic variants. We assume that the mutation is independent of the previous history of the process, and the offspring distribution is independent of the allelic type, that is, the selection is neutral for all alleles. The process can be described as an