On the inclusion of self regulating branching processes in the working paradigm of evolutionary and population genetics

Abstract

The principal goal of this methodological paper is to suggest to a general audience in the genetics community that the consideration of recent developments of self regulating branching processes may lead to the possibility of including this class of stochastic processes as part of working paradigm of evolutionary and population genetics. This class of branching processes is self regulating in the sense that an evolving population will grow only to a total population size that can be sustained by the environment. From the mathematical point of view the class processes under consideration belongs to a subfield of probability and statistics sometimes referred to as computational applied probability and stochastic processes. Computer intensive methods based on Monte Carlo simulation procedures have been used to empirically work out the predictions of a formulation by assigning numerical values to some point in the parameter space and computing replications of realizations of the process over thousands of generations of evolution. Statistical methods are then used on such samples of simulated data to produce informative summarizations of the data that provide insights into the evolutionary implications of computer experiments. Briefly, it is also possible to embed deterministic non-linear difference equations in the stochastic process by using a statistical procedure to estimate the sample functions of the process, which has interesting methodological implications as to whether stochastic or deterministic formulations may be applied separately or in combination in the study of evolution. It is recognized that the literature on population genetics contains a substantial number of papers in which Monte Carlo simulation methods have been used. But, this extensive literature is beyond the scope of this paper, which is focused on potential applications of self regulating branching processes in evolutionary and population genetics.

Stochastic Modeling and Simulation of Viral Evolution

“…However, as seen before, the computational implementation of the phenotypic model required the introduction a cut off K in order to bound the growth of the population. If the cut off is taken as basic constituent of the phenotypic model, and not merely a convenient device, then it can play a role similar to a carrying capacity and the model may no longer be considered a "pure" branching process, but a self-regulating branching process [73,74].…”

Modeling Y-Linked Pedigrees through Branching Processes

“…Branching Processes (BPs) (see [1]) are stochastic mathematical models widely used in population genetics (see [2]) as an alternative approach to the Wright-Fisher model when the classical assumption of constant population size is dropped (see [3]). Not only the simplest model, the Bienaymé-Galton-Watson BP, has been applied (see, for example, the pioneering work [4], or more recently [5,6], or [7]), but also more complex BPs have been used to better describe newly arisen genetic issues (see, for example, logistic BP [8], infinite-allele BP [9], or multi-type self-regulating BP [10]. )…”

Mutation in populations governed by a Galton–Watson branching process

“…7]. However, as argued by Mode et al [28], the influence of models based on GW branching processes has in general been overshadowed, at least within the text book literature, by that of Wright-Fisher (WF) based models. Much of the WF model's dominance can be attributed to the intuitive appeal of the coalescent [21], which is a natural consequence of WF models but mathematically formidable for a GW process [24], and to the WF model's well-known diffusion limit via the forward Kolmogorov equation, as championed by Kimura [18,19,20].…”