Evolutionary genetics in extended populations

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Testing genetic evolutionary models is challenging because of the many parameters and the computational burden of large and multi-faceted datasets. Innovative statistical methods are necessary to take full advantage of existing and future datasets. My group develops and applies mathematical models of evolution that integrate biological, historical, and demographic information to better understand genomic diversity. ******One aim of this proposal is to develop new approaches to simulate the evolution of allele frequencies across populations that are faster, more stable, and more accurate than the state-of-the art. We propose to develop moment-based approaches to compute the distribution of allele frequencies under different evolutionary models. Using our expertise in partial differential equations and the development of open-source software, we therefore propose:***1) To develop efficient simulation and inference software that improves upon the state-of-the-art approaches to solving the diffusion equation***2) To generalize these solvers to cases where the diffusion approximation fails***3) To apply these models to genomic diversity datasets***4) To investigate the application of such moment approaches to the solution of partial differential equations beyond genetics. *********Our second aim is to develop genetic diversity models that take into account the continuous nature of extended populations. Most models of population genetics assume that populations are either spatially homogeneous, or subdivided in a small number of homogeneous sub-populations, or “demes”. The reason for such assumptions is that homogeneity and random mating simplifies numerical computation, and reduces the number of free parameters. Unfortunately, these assumptions are often inaccurate in realistic populations. Our goal is to improve our understanding of the distribution of genetic diversity in spatially extended populations. We propose to do this in three ways:******1) By developing empirical measures of population structure in extended populations***2) By developing stochastic models to efficiently simulate such systems***3) By applying these methods to better understand genetic diversity in extended populations.******The third aim of this proposal is to model spatial heterogeneity at the cellular level in tumors and metastases. This aim is related to the previous one, in the sense that it jointly models genetic evolution and spatial structure. However, the peculiarities of cancer evolution and cancer data means that the mathematical models that we will use are completely different. We will develop multi-scale models of diversity by combining partial differential equation models of growth, which are already used for imaging-based tumor modelling, to fluid dynamics models for stochastic fluctuations at the microscopic level. The goal here is to model heterogeneity at the cellular scale in large-scale tumors. ***********************