Simulating the Emergence and Survival of Mutations Using a Self Regulating Multitype Branching Processes

Abstract

It is difficult for an experimenter to study the emergence and survival of mutations, because mutations are rare events so that large experimental population must be maintained to ensure a reasonable chance that a mutation will be observed. In his famous book, The Genetical Theory of Natural Selection, Sir R. A. Fisher introduced branching processes into evolutionary genetics as a framework for studying the emergence and survival of mutations in an evolving population. During the lifespan of Fisher, computer technology had not advanced to a point at which it became an effective tool for simulating the phenomenon of the emergence and survival of mutations, but given the wide availability of personal desktop and laptop computers, it is now possible and financially feasible for investigators to perform Monte Carlo Simulation experiments. In this paper all computer simulation experiments were carried out within a framework of self regulating multitype branching processes, which are part of a stochastic working paradigm. Emergence and survival of mutations could also be studied within a deterministic paradigm, which raises the issue as to what sense are predictions based on the stochastic and deterministic models are consistent. To come to grips with this issue, a technique was used such that a deterministic model could be embedded in a branching process so that the predictions of both the stochastic and deterministic compared based on the same assigned values of parameters.

Simulating the Emergence of Mutations and Their Subsequent Evolution in an Age-Structured Stochastic Self-Regulating Process with Two Sexes

“…As demonstrated in recent publications, Mode and Sleeman [9], Mode et al (2011), [12,13], by deriving formulas for the conditional expectation of any random variable at time , given the evolution of the process prior to , it is possible to derive a set of recursive nonlinear difference equations, such that, given the initial values of the random functions at time = 0, estimates of the sample functions of the process can be derived for all > 1. In previous publications dating back at least a decade, this derived set of equations has been called the deterministic model embedded in the stochastic process under consideration.…”

“…More precisely, the components of the vector are Z ( ; , ) = ( ( ; , , 1) , ( ; , , 2) , ( ; , , 3)) . (13) Observe for each = 1, 2, 3 the elements in the vector ( ; , , ) for = 1, 2, 3 are the random number of females of genotype and age that have sexual contacts with a males of genotype . Altogether there are 9 types of sexual contacts when there are 3 genotypes of each sex as was shown in the previous section.…”

“…Moreover, the risk of power outage, resulting in the loss of simulated data, during such a time period was significant for the home computer used to carry out the experiments reported in this paper. If a reader is interested in a more in depth discussion of the issues that arise in choosing the number of replications for a Monte Carlo simulation experiment, the paper by Mode et al [13] may be consulted.…”

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

“…For if more than three types are considered in a computer simulation experiment, then it is more difficult statistically to provide informative summarizations of samples of simulated data than if only three types are considered in an experiment. If a reader is interested in other illustrative experiments using the three type model of chapter 10, it is suggested that Mode et al (2011b) be consulted.…”

“…While carrying out the computer experiments reported in the book, it was not observed that a rare beneficial mutation would fail appear in the population, using the self regulating process three type branching process under consideration. But if a reader is interested in experiments using this model such that an advantageous mutant genotype did not appear in the population in experiment completed after work on the book was finished, the recent paper of Mode et al (2011b) may also be consulted, where other experiments in which the predictions of the embedded deterministic model are not consistent with those of the process.…”