Mathematics Colloquium: Genetic drift and its mathematical models: computation, analysis and applications

Speaker:  Qi Wang, George Mason University

Title: Genetic drift and its mathematical models: computation, analysis and applications 

Abstract:   Genetic drift describes random fluctuations in the number of gene variants (alleles) over time.  Mathematical models of genetic drift, such as Wright-Fisher, use discrete stochastic processes to model the dynamics of finite populations at the individual level, such that each copy of the gene of the new generation is independently and randomly selected from the entire gene pool of the previous generation.  In the large population limit, these processes can be approximated by diffusion, which describes the probability of fixation of a mutant with frequency-independent fitness.  While the continuum framework allows a systematic qualitative and quantitative analysis of a discrete model thanks to tools from modern analysis, it also inherits degenerate diffusion from the discrete stochastic process, which develops Dirac-delta singularities, posing great challenges to analytical and numerical studies.  

In this talk, we first describe and analyze an optimal mass transport method for the random genetic drift problem formulated as a degenerate reaction-advection-diffusion equation known as the Kimura equation.  The proposed numerical method can, on the one hand, quantitatively capture the evolution of Dirac delta singularities in genetic fixation to the fullest extent possible and, on the other hand, preserve several sets of biologically relevant and computationally favorable properties of genetic drift.  The second part of the talk introduces the well-posedness theory of the Kimura equation, for which we reformulate the concept of its weak solution and then establish a series of results that include the existence, uniqueness, regularity, and stability of the weak solution.  Our arguments establish an intrinsic connection between the genetic fixation probability and a stochastic process with two absorbing barriers.  Finally, extensions and applications of the mathematical results to some social sciences will be discussed.  

The first and last parts of this talk are accessible to graduate and undergraduate students with knowledge of the finite difference method and a little bit of distribution theory.

Time:  Friday, March 22, 3:30pm – 4:30pm

Place:  Exploratory Hall, room 4106

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