%0 Journal Article %1 karlin1981detection %A Karlin, S %A Tavaré, S %D 1981 %J Theor Popul Biol %K deleterious_alleles diffusion_approximation hitting_times penetrance recessive_alleles %N 2 %P 215-229 %T The detection of particular genotypes in finite populations. II. The effects of partial penetrance and family structure %U http://www.ncbi.nlm.nih.gov/pubmed/7256681 %V 19 %X The detection problem introduced by Robertson (1978) concerns the time taken to form the first recessive homozygote in finite populations. The diallelic model previously studied assumes that heterozygote carriers, Au, are indistinguishable from the normal homozygote, AA. Various distributional properties of the time to detection or loss of the u-allele were determined using the traditional method of diffusion approximation to the relevant Wright-Fisher discrete process. The novelty in the diffusion structure here rests on the appearance of a killing rate corresponding to the a&genotype detection events; see Karlin and TavarC (1980). The analysis was extended in Karlin and Tavar6 (I98 la), referred to henceforth as Part I, to take account of various forms and levels of natural selection effects. In this part we will study the effects of partial pefietrance in heterozygotes or, equivalently, the effect of partial detection, resulting from a screening program that can in some cases detect the presence of the u-allele in heterozygous form. These considerations are appropriate in view of recent progress in biochemical techniques which permit greater ability to detect differences in phenotypically similar genotypes. Utilizing the advancing technology, increasingly more genetic disease screening programs (e.g., sickle cell anemia, a number of thalassemia disorders, Tay Sachs syndrome, phenylketonuria) are available for purposes of identifying heterozygous carriers. In this perspective we envision the detection problem under the conditions where recessive homozygotes are instantly detected as before, but in addition, a heterozygous carrier can be ascertained or will express itself with probability a. If a is “significant” then the recessive allele will usually be first detected among heterozygotes. For a “very small,” the heterozygotes are indistinguishable from normal homozygotes and allele a will usually be revealed with the appearance of a recessive homozygote. The distribution of the detection time differs sharply in these two cases. The detailed comparisons and contrasts of these models are set forth in the following section. Another class of models takes account of family structure, motivated prin- cipally by artificial selection schemes (Section 2). We have also tried to assess the effect on detection of examining more individuals than are used as parents in the subsequent generation. If we use N parents and examine M offspring (usually M > N) then the detection probability is increased, as expected. A quantitative assessment of this procedure is developed. We also consider a model of N independent breeding individuals per generation which produce families of r offspring. All offspring are examined and the process is terminated if a recessive homozygote appears. This model can easily be extended to allow random family sizes. The presence of family structure shows qualitatively the same distributional properties of the detection times as the case of families with one offspring. We also treat a mixed reproduction scheme in which any undetected recessive homozygotes are replaced by a random sample from a large population with fixed proportions of AA and Au individuals. One aim of this study is to assess the stability of the order of magnitude NV3 generations until detection that was first noted by Robertson (1978). As will be seen, in several of the problems studied here the time scale N1’3 is no longer the only scaling that leads to an appropriate diffusion process with killing; further, the different orders of magnitude are often reflected by processes whose infinitesimal parameters are functionally different. In this analysis it is our intention to discuss qualitative aspects of these variants; the methods described in Karlin and Tavare (1980) can readily be evaluated in the present context. Derivations of a variety of differential equations satisfied by relevant probabilistic functions are given in Karlin and TavarC (198 lb). Further discussion of the background of this problem appears in Part I.

@article{karlin1981detection, abstract = {The detection problem introduced by Robertson (1978) concerns the time taken to form the first recessive homozygote in finite populations. The diallelic model previously studied assumes that heterozygote carriers, Au, are indistinguishable from the normal homozygote, AA. Various distributional properties of the time to detection or loss of the u-allele were determined using the traditional method of diffusion approximation to the relevant Wright-Fisher discrete process. The novelty in the diffusion structure here rests on the appearance of a killing rate corresponding to the a&genotype detection events; see Karlin and TavarC (1980). The analysis was extended in Karlin and Tavar6 (I98 la), referred to henceforth as Part I, to take account of various forms and levels of natural selection effects. In this part we will study the effects of partial pefietrance in heterozygotes or, equivalently, the effect of partial detection, resulting from a screening program that can in some cases detect the presence of the u-allele in heterozygous form. These considerations are appropriate in view of recent progress in biochemical techniques which permit greater ability to detect differences in phenotypically similar genotypes. Utilizing the advancing technology, increasingly more genetic disease screening programs (e.g., sickle cell anemia, a number of thalassemia disorders, Tay Sachs syndrome, phenylketonuria) are available for purposes of identifying heterozygous carriers. In this perspective we envision the detection problem under the conditions where recessive homozygotes are instantly detected as before, but in addition, a heterozygous carrier can be ascertained or will express itself with probability a. If a is “significant” then the recessive allele will usually be first detected among heterozygotes. For a “very small,” the heterozygotes are indistinguishable from normal homozygotes and allele a will usually be revealed with the appearance of a recessive homozygote. The distribution of the detection time differs sharply in these two cases. The detailed comparisons and contrasts of these models are set forth in the following section. Another class of models takes account of family structure, motivated prin- cipally by artificial selection schemes (Section 2). We have also tried to assess the effect on detection of examining more individuals than are used as parents in the subsequent generation. If we use N parents and examine M offspring (usually M > N) then the detection probability is increased, as expected. A quantitative assessment of this procedure is developed. We also consider a model of N independent breeding individuals per generation which produce families of r offspring. All offspring are examined and the process is terminated if a recessive homozygote appears. This model can easily be extended to allow random family sizes. The presence of family structure shows qualitatively the same distributional properties of the detection times as the case of families with one offspring. We also treat a mixed reproduction scheme in which any undetected recessive homozygotes are replaced by a random sample from a large population with fixed proportions of AA and Au individuals. One aim of this study is to assess the stability of the order of magnitude NV3 generations until detection that was first noted by Robertson (1978). As will be seen, in several of the problems studied here the time scale N1’3 is no longer the only scaling that leads to an appropriate diffusion process with killing; further, the different orders of magnitude are often reflected by processes whose infinitesimal parameters are functionally different. In this analysis it is our intention to discuss qualitative aspects of these variants; the methods described in Karlin and Tavare (1980) can readily be evaluated in the present context. Derivations of a variety of differential equations satisfied by relevant probabilistic functions are given in Karlin and TavarC (198 lb). Further discussion of the background of this problem appears in Part I.}, added-at = {2015-02-18T18:13:07.000+0100}, author = {Karlin, S and Tavar{\'e}, S}, biburl = {https://www.bibsonomy.org/bibtex/2b43fd1ad22dd898c5c4396a4764c9b2e/peter.ralph}, interhash = {78848e45fe1b155a25f3146d7ed4ff48}, intrahash = {b43fd1ad22dd898c5c4396a4764c9b2e}, journal = {Theor Popul Biol}, keywords = {deleterious_alleles diffusion_approximation hitting_times penetrance recessive_alleles}, month = apr, number = 2, pages = {215-229}, pmid = {7256681}, timestamp = {2015-02-18T18:13:07.000+0100}, title = {The detection of particular genotypes in finite populations. {II}. {The} effects of partial penetrance and family structure}, url = {http://www.ncbi.nlm.nih.gov/pubmed/7256681}, volume = 19, year = 1981 }