%0 Journal Article %1 karlin1983deleterious %A Karlin, Samuel %A Tavaré, Simon %D 1983 %J Stochastic Processes and their Applications %K deleterious_alleles diffusion_approximation killing selection %N 1 %P 107 %R 10.1016/0304-4149(83)90050-9 %T A diffusion process with killing: the time to formation of recurrent deleterious mutant genes %U http://dx.doi.org/10.1016/0304-4149(83)90050-9 %V 14 %X In a diploid population with two alleles $(a,A)$ at a single locus it is assumed that there are initially only AA or aA individuals; the focus of the paper is the time to first appearance of aa individuals. All states with any aa are treated as an absorbing state and a limiting process constructed which is a diffusion with killing or loss to the absorbing state. The infinitesimal generator of the diffusion is $Lu=\textstyle12(xu''+u'-x^2u)$, where $x$ is nonnegative and $\nu$ is a parameter which determines the mutation rate from A to a. The cases $0<\nu<1$ and $\nu1$ give a regular or entrance boundary at $x=0$, respectively. For large mutation rates $(\nu1)$, the mean killing time is found exactly when the initial number of aA is small. The cumulative number of aA before the appearance of any aa and the probability that the diffusion is killed in a given set are calculated. Numerical values of the expected population at which detection of aa occurs are found for various $\nu$. For small mutation rates $(0<\nu<1)$, a reflecting boundary condition is imposed at the origin. For $0<\nu<1$, quasifixation probabilities are determined and employed to find expressions for the probability of detection as a function of population size and the initial number of heterozygotes. In the corrigenda a misprint in a formula for the mean time to appearance of a recessive homozygote in a diploid population (two alleles, single locus) is corrected.

@article{karlin1983deleterious, abstract = {In a diploid population with two alleles $(\text a,\text A)$ at a single locus it is assumed that there are initially only AA or aA individuals; the focus of the paper is the time to first appearance of aa individuals. All states with any aa are treated as an absorbing state and a limiting process constructed which is a diffusion with killing or loss to the absorbing state. The infinitesimal generator of the diffusion is $Lu={\textstyle\frac 1{2}}(xu''+\nu u'-x^2u)$, where $x$ is nonnegative and $\nu$ is a parameter which determines the mutation rate from A to a. The cases $0<\nu<1$ and $\nu\geq 1$ give a regular or entrance boundary at $x=0$, respectively. For large mutation rates $(\nu\geq 1)$, the mean killing time is found exactly when the initial number of aA is small. The cumulative number of aA before the appearance of any aa and the probability that the diffusion is killed in a given set are calculated. Numerical values of the expected population at which detection of aa occurs are found for various $\nu$. For small mutation rates $(0<\nu<1)$, a reflecting boundary condition is imposed at the origin. For $0<\nu<1$, quasifixation probabilities are determined and employed to find expressions for the probability of detection as a function of population size and the initial number of heterozygotes. In the corrigenda a misprint in a formula for the mean time to appearance of a recessive homozygote in a diploid population (two alleles, single locus) is corrected. }, added-at = {2010-09-04T19:27:19.000+0200}, author = {Karlin, Samuel and Tavar{\'e}, Simon}, biburl = {https://www.bibsonomy.org/bibtex/2b8117daf78c38333c6f26b2b96f0f426/peter.ralph}, coden = {STOPB}, description = {MR: Publications results for "Anywhere=(deleterious AND recessive)"}, doi = {10.1016/0304-4149(83)90050-9}, interhash = {922da89d5130b9e3fe990664d7234d4c}, intrahash = {b8117daf78c38333c6f26b2b96f0f426}, issn = {0304-4149}, journal = {Stochastic Processes and their Applications}, keywords = {deleterious_alleles diffusion_approximation killing selection}, mrclass = {92A10 (60J70)}, mrnumber = {676277 (84g:92021b)}, mrreviewer = {H. C. Tuckwell}, number = 1, pages = 107, sjournal = {Stochastic Process. Appl.}, timestamp = {2010-09-04T19:27:19.000+0200}, title = {A diffusion process with killing: the time to formation of recurrent deleterious mutant genes}, url = {http://dx.doi.org/10.1016/0304-4149(83)90050-9}, volume = 14, year = 1983 }