TY - JOUR
T1 - Complete Numerical Solution of the Diffusion Equation of Random Genetic Drift
JF - Genetics
JO - Genetics
SP - 973
LP - 985
DO - 10.1534/genetics.113.152017
VL - 194
IS - 4
AU - Zhao, Lei
AU - Yue, Xingye
AU - Waxman, David
Y1 - 2013/08/01
UR - http://www.genetics.org/content/194/4/973.abstract
N2 - A numerical method is presented to solve the diffusion equation for the random genetic drift that occurs at a single unlinked locus with two alleles. The method was designed to conserve probability, and the resulting numerical solution represents a probability distribution whose total probability is unity. We describe solutions of the diffusion equation whose total probability is unity as complete. Thus the numerical method introduced in this work produces complete solutions, and such solutions have the property that whenever fixation and loss can occur, they are automatically included within the solution. This feature demonstrates that the diffusion approximation can describe not only internal allele frequencies, but also the boundary frequencies zero and one. The numerical approach presented here constitutes a single inclusive framework from which to perform calculations for random genetic drift. It has a straightforward implementation, allowing it to be applied to a wide variety of problems, including those with time-dependent parameters, such as changing population sizes. As tests and illustrations of the numerical method, it is used to determine: (i) the probability density and time-dependent probability of fixation for a neutral locus in a population of constant size; (ii) the probability of fixation in the presence of selection; and (iii) the probability of fixation in the presence of selection and demographic change, the latter in the form of a changing population size.
ER -