Exact results for the probability and stochastic dynamics of fixation in the Wright-Fisher model. (7th October 2017)

• Record Type: Journal Article
• Authors: Shafiey, Hassan; Waxman, David
• Is Part Of: Journal of theoretical biology. Volume 430(2017)
• Page Start: 64
• Page End: 77
• Publication Date: 2017-10-07
• Subjects: Random genetic drift -- Frequency trajectories -- Stochastic process -- Frequency-dependent selection -- Strong selection; Biology -- Periodicals; Biological Science Disciplines -- Periodicals; Biology -- Periodicals; Biologie -- Périodiques; Theoretische biologie; Biology; Periodicals; 571.05
• DOI: 10.1016/j.jtbi.2017.06.026
• Languages: English
• ISSNs: 0022-5193

Abstract

Highlights: We provide a set of exact results for a fundamental model of population genetics. We present exact results for the fixation probability, for the Wright-Fisher model, for some schemes of selection. The exact results hold for selection that can be strong and frequency dependent. Knowledge of the fixation probability, for all frequencies, fully determines the Wright-Fisher model. We present the exact stochastic rule that generates purely fixing trajectories. Abstract: In this work we consider fixation of an allele in a population. Fixation is key to understanding the way long-term evolutionary change occurs at the gene and molecular levels. Two basic aspects of fixation are: (i) the chance it occurs and (ii) the way the gene frequency progresses to fixation. We present exact results for both aspects of fixation for the Wright–Fisher model. We give the exact fixation probability for some different schemes of frequency-dependent selection. We also give the corresponding exact stochastic difference equation that generates frequency trajectories which ultimately fix. Exactness of the results means selection need not be weak. There are possible applications of this work to data analysis, modelling, and tests of approximations. The methodology employed illustrates that knowledge of the fixation probability, for all initial frequencies, fully characterises the dynamics of the Wright–Fisher model. The stochastic equations for fixing trajectories allow insight into the way fixation occurs. They provide the alternative picture that fixation is driven by the injection of one carrier of the fixing allele into the population each generation. The stochastic equations allow explicit calculation of some properties of fixing trajectories and their efficient simulation. The results are illustrated and tested with simulations.