# Computation of the Likelihood in Biallelic Diffusion Models Using Orthogonal Polynomials

## Abstract

**:**

## 1. Introduction

## 2. Mutation and Drift Diffusion

#### 2.1. Moran and Diffusion Models

**mutation**) at a rate of $\mu ={\mu}_{0}+{\mu}_{1}$, a random individual i is picked to mutate to type one with probability $\alpha ={\mu}_{1}/\mu $ or to type zero with probability $\beta ={\mu}_{0}/\mu $; or (ii) (

**genetic drift**) at a rate of one, a random individual i is replaced by another random individual j. Setting $\theta =\mu N$, the rate of change of the allelic proportion x of the mean per unit time is caused by mutation:

#### 2.2. Solution of the Mutation-Drift Diffusion Using Modified Jacobi Polynomials

#### 2.2.1. Relationship of the Forward and Backward Diffusion Equation; Sturm–Liouville Form

#### 2.2.2. Modified Jacobi Polynomials

#### 2.2.3. Series Expansion; Approximation of Functions by Orthogonal Polynomials

#### 2.2.4. Example: A Change in the Scaled Mutation Rate with Modified Jacobi Polynomials

#### 2.3. Statistics of Site Frequency Spectra

#### 2.3.1. Equilibrium

#### 2.3.2. Outside Equilibrium

## 3. Selection and Drift Diffusion with Mutations from the Boundaries

#### 3.1. Pure Drift within the Polymorphic Region

#### 3.1.1. Equilibrium of Mutations from the Boundaries and Drift; Outgroup Information

#### 3.1.2. Equilibrium of Mutations from the Boundaries and Drift; No Outgroup Information

#### 3.1.3. Example for the Use of Gegenbauer Polynomials: Evolve and Resequence

**Figure 1.**Distribution of the allelic proportion x starting from a $dbeta(x\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}3,2)$ distribution (thick line). The thin lines represent the loss of variation through genetic drift at generations $t/N=(0.05,0.15,0.25,0.35,0.45)$.

#### 3.2. Selection and Drift

## 4. Conclusions

## Acknowledgments

## Appendix. The Oblate Spheroidal Wave Function

## Conflicts of Interest

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Vogl, C. Computation of the Likelihood in Biallelic Diffusion Models Using Orthogonal Polynomials. *Computation* **2014**, *2*, 199-220.
https://doi.org/10.3390/computation2040199

**AMA Style**

Vogl C. Computation of the Likelihood in Biallelic Diffusion Models Using Orthogonal Polynomials. *Computation*. 2014; 2(4):199-220.
https://doi.org/10.3390/computation2040199

**Chicago/Turabian Style**

Vogl, Claus. 2014. "Computation of the Likelihood in Biallelic Diffusion Models Using Orthogonal Polynomials" *Computation* 2, no. 4: 199-220.
https://doi.org/10.3390/computation2040199