Publication: Modelos matemáticos en genética de poblaciones
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2022-06-22
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Abstract
En este trabajo, se estudia matemáticamente el comportamiento en función del tiempo de una frecuencia alélica en individuos diploides de una población de tamaño fijo y finito, con generaciones no superpuestas y bajo la única influencia de la deriva genética aleatoria. Para ello, se sigue el artículo de referencia [13]. En primer lugar, se plantea el modelo probabilístico de Wright-Fisher y, a partir de las características del modelo, se deriva una aproximación de difusión con la ecuación en derivadas parciales de Fokker-Planck. A continuación, se resuelve dicha ecuación estudiando existencia y unicidad de una nueva solución propuesta en dicho artículo. Finalmente, esta nueva solución permite analizar algunas propiedades del modelo inicial de una forma más sencilla que en el trabajo anteriormente desarrollado por Kimura en [6].
In this project, the behaviour of an allele frequency through time at diploid individuals in a finite population of fixed size, with non-overlapping generations and under the sole influence of the random genetic drift is studied mathematically. For this purpose, reference article [13] is followed. Firstly, the probabilistic WrightFisher model is set out and, a diffusion approximation with the Fokker-Planck partial differential equation is derived from the characteristics of the model. Then, this equation is solved by studying existence and uniqueness of a new solution proposed in that article. Finally, this new solution allows analyzing some properties of the initial model more simply than in the work developed by Kimura in [6].
In this project, the behaviour of an allele frequency through time at diploid individuals in a finite population of fixed size, with non-overlapping generations and under the sole influence of the random genetic drift is studied mathematically. For this purpose, reference article [13] is followed. Firstly, the probabilistic WrightFisher model is set out and, a diffusion approximation with the Fokker-Planck partial differential equation is derived from the characteristics of the model. Then, this equation is solved by studying existence and uniqueness of a new solution proposed in that article. Finally, this new solution allows analyzing some properties of the initial model more simply than in the work developed by Kimura in [6].