#### Project Title

Examining correlation of Wright-Fisher Model and Coalescent Theory based on Fractional Calculus

#### Academic department under which the project should be listed

College of Science and Mathematics

#### Research Mentor Name

Dr. Somayeh Mashayekhi

#### Abstract (300 words maximum)

Theoretical population genetics bridges mathematics and evolutionary biology. Kingman's coalescent theory and Wright-Fisher's model play an important role in bridging this gap. Kingman's coalescent models how alleles from a specific population with no mechanism to introduce genetic variation to further generations may have originated from a common ancestor; thus, studying allele frequencies backwards in time. Whereas, the Wright-Fisher model assumes a finite but constant population size, random mating, non-overlapping generations, and no selection. The gene frequencies in subsequent populations are composed of 2N (diploid gamete) draws from the gene frequency of the current population (represented by haploid N); modeling allele frequencies forward in time. In this research, we study Wright-Fisher's model in depth and compare it to an extension of Kingman's coalescent. It was found that while the Wright-Fisher model utilizes binomial sampling of the current generations gamete pool to generate a new generation of alleles, the Coalescent theory utilizes the exponential distribution function to calculate the time between two generations until they coalesce-find a common ancestor-. Thus, for a more accurate and specific representation of certain population types, the Coalescent theory was expanded by utilizing the Mittag Leffler function in place of the exponential distribution function, thus being coined the fractional coalescent. By replacing exponential distribution with the Mittag Leffler function, thereby introducing fractional calculus into population genetic models, a more accurate model can be created to depict allele frequencies over a specified time in different genetic environments. For example, modeling the survival of viruses in a heterogenetic population would call for the fractional coalescent.

Keywords: Kingman's Coalescent; Wright-Fisher ;Allele frequencies; Fractional Calculus; Binomial sampling; Heterogenetic population;

#### Disciplines

Applied Mathematics | Genetic Phenomena

Examining correlation of Wright-Fisher Model and Coalescent Theory based on Fractional Calculus

Theoretical population genetics bridges mathematics and evolutionary biology. Kingman's coalescent theory and Wright-Fisher's model play an important role in bridging this gap. Kingman's coalescent models how alleles from a specific population with no mechanism to introduce genetic variation to further generations may have originated from a common ancestor; thus, studying allele frequencies backwards in time. Whereas, the Wright-Fisher model assumes a finite but constant population size, random mating, non-overlapping generations, and no selection. The gene frequencies in subsequent populations are composed of 2N (diploid gamete) draws from the gene frequency of the current population (represented by haploid N); modeling allele frequencies forward in time. In this research, we study Wright-Fisher's model in depth and compare it to an extension of Kingman's coalescent. It was found that while the Wright-Fisher model utilizes binomial sampling of the current generations gamete pool to generate a new generation of alleles, the Coalescent theory utilizes the exponential distribution function to calculate the time between two generations until they coalesce-find a common ancestor-. Thus, for a more accurate and specific representation of certain population types, the Coalescent theory was expanded by utilizing the Mittag Leffler function in place of the exponential distribution function, thus being coined the fractional coalescent. By replacing exponential distribution with the Mittag Leffler function, thereby introducing fractional calculus into population genetic models, a more accurate model can be created to depict allele frequencies over a specified time in different genetic environments. For example, modeling the survival of viruses in a heterogenetic population would call for the fractional coalescent.

Keywords: Kingman's Coalescent; Wright-Fisher ;Allele frequencies; Fractional Calculus; Binomial sampling; Heterogenetic population;