Singular solutions of the diffusion equation of population genetics.

The forward diffusion equation for gene frequency dynamics is solved subject to the condition that the total probability is conserved at all times. This can lead to solutions developing singular spikes (Dirac delta functions) at the gene frequencies 0 and 1. When such spikes appear in solutions they signal gene loss or gene fixation, with the "weight" associated with the spikes corresponding to the probability of loss or fixation. The forward diffusion equation is thus solved for all gene frequencies, namely the absorbing frequencies of 0 and 1 along with the continuous range of gene frequencies on the interval (0,1) that excludes the frequencies of 0 and 1. Previously, the probabilities of the absorbing frequencies of 0 and 1 were found by appeal to the backward diffusion equation, while those in the continuous range (0,1) were found from the forward diffusion equation. Our unified approach does not require two separate equations for a complete dynamical treatment of all gene frequencies within a diffusion approximation framework. For cases involving mutation, migration and selection, it is shown that a property of the deterministic part of gene frequency dynamics determines when fixation and loss can occur. It is also shown how solution of the forward equation, at long times, leads to the standard result for the fixation probability.

[1]  W. Feller.  THE PARABOLIC DIFFERENTIAL EQUATIONS AND THE ASSOCIATED SEMI-GROUPS OF TRANSFORMATIONS , 1952 .

[2]  S. Lang,et al.  An Introduction to Fourier Analysis and Generalised Functions , 1959 .

[3]  M. Kimura.  Stochastic processes and distribution of gene frequencies under natural selection. , 1955, Cold Spring Harbor symposia on quantitative biology.

[4]  W. Godwin.  Article in Press , 2000 .

[5]  Rory A. Fisher,et al.  XXI.—On the Dominance Ratio , 1923 .

[6]  E. Hill.  Journal of Theoretical Biology , 1961, Nature.

[7]  R. Punnett,et al.  The Genetical Theory of Natural Selection , 1930, Nature.

[8]  C. Gardiner.  Handbook of Stochastic Methods , 1983 .

[9]  W. Feller.  Diffusion processes in one dimension , 1954 .

[10]  S Wright,et al.  The Differential Equation of the Distribution of Gene Frequencies. , 1945, Proceedings of the National Academy of Sciences of the United States of America.

[11]  J. Keller,et al.  Asymptotic analysis of stochastic models in population genetics , 1975 .

[12]  J. S. Gale,et al.  Theoretical Population Genetics , 1990, Springer Netherlands.

[13]  A. Messiah.  Quantum Mechanics , 1961 .

[14]  R. A. Fisher,et al.  On the dominance ratio , 1990 .

[15]  T. Maruyama,et al.  Stochastic Problems in Population Genetics , 1977 .

[16]  Motoo Kimura,et al.  Some Genetic Problems in Natural Populations , 1956 .

[17]  M Kimura,et al.  SOLUTION OF A PROCESS OF RANDOM GENETIC DRIFT WITH A CONTINUOUS MODEL. , 1955, Proceedings of the National Academy of Sciences of the United States of America.

[18]  G. Barton.  Elements of Green's Functions and Propagation: Potentials, Diffusion, and Waves , 1989 .

[19]  M. Kimura,et al.  An introduction to population genetics theory , 1971 .

[20]  H. L. Le Roy,et al.  Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability; Vol. IV , 1969 .

[21]  S. Wright,et al.  Evolution in Mendelian Populations. , 1931, Genetics.