## 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

## References

- Parsch, J.; Novozhilov, S.; Saminadin-Peter, S.; Wong, K.; Andolfatto, P. On the utility of short intron sequences as a reference for the detection of positive and negative selection in Drosophila. Mol. Biol. Evol.
**2010**, 27, 1226–1234. [Google Scholar] [CrossRef] [PubMed] - Fisher, R. The Genetical Theory of Natural Selection; Clarendon Press: Oxford, UK, 1930. [Google Scholar]
- Wright, S. Evolution in Mendelian populations. Genetics
**1931**, 16, 97–159. [Google Scholar] [PubMed] - Moran, P. Random processes in genetics. Proc. Camb. Philos. Soc.
**1958**, 54, 60–71. [Google Scholar] [CrossRef] - Kimura, M. The Neutral Theory of Molecular Evolution; Cambridge University Press: Cambridge, UK, 1983. [Google Scholar]
- Kimura, M. Population Genetics, Molecular Evolution, and the Neutral Theory: Selected Papers; University of Chicago Press: Chicago, IL, USA, 1994. [Google Scholar]
- Evans, S.; Shvets, Y.; Slatkin, M. Non-equilibrium theory of the allele frequency spectrum. Theor. Popul. Biol.
**2007**, 71, 109–119. [Google Scholar] [CrossRef] [PubMed] - Zivkovic, D.; Stephan, W. Analytical results on the neutral non-equilibrium allele frequency spectrum based on diffusion theory. Theor. Popul. Biol.
**2011**, 79, 184–191. [Google Scholar] [CrossRef] [PubMed] - Ewens, W. Mathematical Population Genetics, 2nd ed.; Springer: New York, NY, USA, 2004. [Google Scholar]
- Wolfram Research, Inc. Mathematica, Version 10.0. Champaign, IL, USA, 2014. Available online: http://wolfram.com/ (accessed on 6 November 2014).
- Matlab 8.4. The MathWorks Inc.: Natick, MA, USA, 2014. Available online: http://www.mathworks.de/ (accessed on 6 November 2014).
- Song, Y.; Steinrücken, M. A simple method for finding explicit analytic transition densities of diffusion processes with general diploid selection. Genetics
**2012**, 190, 1117–1129. [Google Scholar] [CrossRef] [PubMed] - Baake, E.; Bialowons, R. Ancestral Processes with Selection: Branching and Moran Models; Banach Center Publications: Bielefeld, Germany, 2008; Volume 80, pp. 33–52. [Google Scholar]
- Etheridge, A.; Griffiths, R. A coalescent dual process in a Moran model with genic selectio. Theor. Popul. Biol.
**2009**, 75, 320–330. [Google Scholar] [CrossRef] [PubMed] - Vogl, C.; Clemente, F. The allele-frequency spectrum in a decoupled Moran model with mutation, drift, and directional selection, assuming small mutation rates. Theor. Popul. Genet.
**2012**, 81, 197–209. [Google Scholar] [CrossRef] [PubMed] - Hein, J.; Schierup, M.; Wiuf, C. Gene Genealogies, Variation, and Evolution: A Primer in Coalescent Theory; Oxford University Press: Oxford, UK, 2005. [Google Scholar]
- Hazewinkel, M. Sturm-Liouville Theory. In Encyclopedia of Mathematics; Springer: New York, NY, USA, 2001. [Google Scholar]
- Griffiths, R.; Spanò, D. Diffusion processes and coalescent trees. In Probability and Mathematical Genetics: Papers in Honour of Sir John Kingman; Cambridge University Press: Cambridge, UK, 2010; pp. 358–375. [Google Scholar]
- Handbook of Mathematical Functions, 9th ed.; Abramowitz, M.; Stegun, I. (Eds.) Dover: New York, NY, USA, 1970.
- Kimura, M. Solution of a process of random genetic drift with a continuous model. Proc. Natl. Acad. Sci. USA
**1955**, 41, 144–150. [Google Scholar] [CrossRef] [PubMed] - Vogl, C. Estimating the Scaled Mutation Rate and Mutation Bias with Site Frequency Data. Theor. Popul. Biol.
**2014**, in press. [Google Scholar] [CrossRef] [PubMed] - McKane, A.; Waxman, D. Singular solutions of the diffusion equation of population genetics. J. Theor. Biol.
**2007**, 247, 849–858. [Google Scholar] [CrossRef] [PubMed] - Tran, T.; Hofrichter, J.; Jost, J. An introduction to the mathematical structure of the Wright-Fisher model of population genetics. Theory Biosci.
**2013**, 132, 73–82. [Google Scholar] [CrossRef] [PubMed] - Clemente, F.; Vogl, C. Unconstrained evolution in short introns?—An analysis of genome-wide polymorphism and divergence data from Drosophila. J. Evol. Biol.
**2012**, 25, 1975–1990. [Google Scholar] [PubMed] - Clemente, F.; Vogl, C. Evidence for complex selection on four-fold degenerate sites in Drosophila melanogaster. J. Evol. Biol.
**2012**, 25, 2582–2595. [Google Scholar] [CrossRef] [PubMed] - Ewens, W. A note on the sampling theory for infinite alleles and infinite sites models. Theor. Popul. Biol.
**1974**, 6, 143–148. [Google Scholar] [CrossRef] - Watterson, G. On the number of segregating sites in genetical models without recombination. Theor. Popul. Biol.
**1975**, 7, 256–276. [Google Scholar] [CrossRef] - Sawyer, S.; Hartl, D. Population genetics of polymorphism and divergence. Genetics
**1992**, 132, 1161–1176. [Google Scholar] [PubMed] - RoyChoudhury, A.; Wakeley, J. Sufficiency of the number of segregating sites in the limit under finite-sites mutation. Theor. Popul. Biol.
**2010**, 78, 118–122. [Google Scholar] [CrossRef] [PubMed] - Vogl, C. Biallelic Mutation-Drift Diffusion in the Limit of Small Scaled Mutation Rates. ArXiv E-Prints
**2014**. [Google Scholar] - Gutenkunst, R.; Hernandez, R.; Williamson, S.; Bustamante, C. Inferring the Joint Demographic History of Multiple Populations from Multidimensional SNP Frequency Data. PLoS Genet.
**2009**, 5, e1000695. [Google Scholar] [CrossRef] [PubMed] - Tobler, R.; Franssen, S.; Kofler, R.; Orozco-Terwengel, P.; Nolte, V.; Hermisson, J.; Schlötterer, C. Massive habitat-specific genomic response in D. melanogaster populations during experimental evolution in hot and cold environments. Mol. Biol. Evol.
**2014**, 31, 364–375. [Google Scholar] [PubMed] - Williamson, E.; Slatkin, M. Using maximum likelihood to estimate population size from temporal changes in allele frequencies. Genetics
**1999**, 152, 755–761. [Google Scholar] [PubMed] - Anderson, E.; Williamson, E.; Thompson, E. Monte Carlo evaluation of the likelihood for N
_{e}from temporally spaced samples. Genetics**2000**, 156, 2109–2118. [Google Scholar] [PubMed] - R Core Team. R: A Language and Environment for Statistical Computing; ISBN 3-900051-07-0. R Foundation for Statistical Computing: Vienna, Austria, 2013. [Google Scholar]
- Falloon, P.; Abbott, P.; Wang, J. Theory and computation of spheroidal wave functions. J. Phys. A Math. Gen.
**2003**, 36, 5477–5495. [Google Scholar] [CrossRef] - Beaumont, M.; Zhang, W.; Balding, D. Approximate Bayesian Computation in Population Genetic. Genetics
**2002**, 162, 2025–2035. [Google Scholar] [PubMed] - Stratton, J. Spheroidal Wave Functions; The Technology Press of the Massachusetts Institute of Technology: Cambridge, MA, USA, 1954. [Google Scholar]
- Meixner, J.; Schäfke, F. Mathieusche Funktionen und Sphäroidfunktionen; Springer: Berlin, Germany, 1954. (In German) [Google Scholar]
- Flammer, C. Spheroidal Wave Functions; Stanford University Press: Palo Alto, CA, USA, 1957. [Google Scholar]
- Li, L.W.; Leong, M.S.; Yeo, T.S.; Kooi, P.S.; Tan, K.Y. Computations of spheroidal harmonics with complex arguments: A review with an algorithm. Phys. Rev. E
**1998**, 58, 6792–6806. [Google Scholar] [CrossRef] - Falloon, P.E. Theory and Computation of Spheroidal Harmonics with General Arguments. Master’s Thesis, Department of Physics, The University of Western Australia, Crawley, Australia, 2001. [Google Scholar]

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