Homozygote Incidence

Transcription

1 Am. J. Hum. Genet. 41: , 1987 The Effects of Genetic Screening and Assortative Mating on Lethal Recessive-Allele Frequencies and Homozygote Incidence R. B. CAMPBELL Department of Mathematics and Computer Science, University of Northern Iowa, Cedar Falls SUMMARY The widespread use of genetic screening, along with mating and reproductive patterns reflecting that information, can significantly alter the genetic structure of populations. Both allele frequencies and mortality could be significantly reduced if carriers of lethal recessive alleles were withdrawn from the mating pool. But schemes to mask deleterious alleles in heterozygous condition could significantly increase the deleterious-allele frequencies while resulting in only a slight reduction in mortality. The immediate and equilibrium consequences of such mating strategies may be quite disparate. INTRODUCTION It is a standard result that the number of deaths resulting from a deleterious mutation is equal to one if the mutation manifests partial dominance and is equal to two if the mutation is recessive (Li 1955; Li 1975). However, this result assumes a large random mating population, so that allele-frequency changes are attributable to deaths of genetically inferior individuals. By weighting the presence of different genetic types in the mating pool, mechanisms such as assortative mating and sexual selection (Karlin and O'Donald 1978) can, without concomitant deaths, change allele frequencies. Reproductive overcompensation has been observed; in theory it could increase deleterious-allele frequencies, although an excess sufficient to achieve this has not been observed (Koeslag and Schach 1984). The effect of amniocentesis and selective abortion Received July 15, Address for correspondence and reprints: Dr. R. B. Campbell, Department of Mathematics and Computer Science, University of Northern Iowa, Cedar Falls, IA X 1987 by the American Journal of Human Genetics. All rights reserved /87/ $

2 672 CAMPBELL on allele frequencies and mortality has been studied by Hagy and Kidwell (1972). Treating genetic diseases may cause mortality to increase (Turner 1968). Modified reproductive behavior based on disease incidence in relatives can alter gene frequencies (Yokoyama 1980). The present study considers the implications of assortative mating (coupled with differential reproduction) that is based on genetic screening for allele frequencies and mortality. (Because screening occurs before mutation, mortality can never be zero.) Only a one-locus model is considered, because the diversity of known genetic disorders (McKusick 1983), coupled with theoretical studies that indicate that there should be many rare deleterious alleles at loci where genetic disorders occur (Hartl and Campbell 1982), precludes the possibility of screening for all genetic disorders. The deleterious mutations are assumed to be lethal but recessive. The immediate consequences on mortality of introducing genetic screening-based mating, as well as the associated equilibrium allele frequencies and mortality are contrasted with those of a naive (panmictic) population. The models are also studied when only part of the population is screened (or there are systematic errors in the screening), as well as when there are transient errors in the screening procedure. Three classes of mating systems based on the knowledge of the genotypes of individuals are studied. The first is a panmictic population, which serves as the control. It is characterized as naive because genotypes are not known, and hence individuals mate and reproduce at random. The second seeks to reduce genetic deaths by withdrawing deleterious alleles from the mating pool. It entails for any allele frequency a reduction in both deaths and allele frequencies below what random mating would provide. Two examples from this class are presented. The third class seeks to reduce genetic deaths by arranging that deleterious alleles occur in heterozygous individuals. It may be characterized as maximal mating, since it excludes as few individuals as possible from the mating pool. In it genetic deaths are reduced vis-a-vis random mating, but deleterious-allele frequencies increase. THE CONTROL NAIVE POPULATION The standard for comparison, and point of departure for introduction of other mating strategies, is a large panmictic population with mutation rate u for the lethal recessive allele. The equilibrium allele frequency is /u-, and the incidence of mortality is Nu individuals per generation, where N is the population size (Crow and Kimura 1970). This assumes that the occurrence of inviable progeny does not affect future reproduction of the parents. If parents of inviable progeny withdraw from the mating pool, these results do not change significantly; but if every couple has two viable progeny, allele frequencies will increase to \4j732u and the mortality rate will double (Campbell, submitted). ASSORTATIVE MATING THAT DECREASES ALLELE FREQUENCIES An extreme form of assortative mating with differential reproduction would bar all carriers of the deleterious allele from mating. In this case the allele frequency would be equal to the mutation rate u and the mortality would be

3 GENETIC SCREENING AS SELECTION 673 Nu2. Equilibrium for the allele frequency and mortality are achieved in the first generation, but 2NX& individuals would be barred from reproduction in the first generation, as opposed to only 2Nu being barred in subsequent generations. If screening were only performed (or accurate; unscreened individuals mate as if wild type) looa% of the time, only looa% of the deleterious alleles would be removed by screening each generation, and the allele frequency would decay more slowly from (1 - a) V- in the first generation to u/a, with -(1 - a)2nu deaths decreasing to Nu21(a2) deaths. The number of individuals who are precluded from mating would decrease from 2Na V in the first generation to 2Nu at equilibrium. (These formulas do not hold for small values of a.) Allele frequencies are below the control level, so an occasional generation in which screening does not work will not raise mortality above the control level. A less extreme variation would allow random mating but only bar reproduction by those couples in which both individuals carry the bad allele. The changes in allele frequencies satisfy (ignoring higher-order terms in u) the equation p,=p(l - 2p) + u(i - p)2 - p2u ) I 4p2-2pu(l - 2p) where p is the frequency of the deleterious allele and the prime (') designates the next generation. From equation (1) we can read off immediately that the initial allele frequency of \/u- will decrease to the equilibrium level i/_u_12). Since mating is random, the number of individuals barred from reproducing (a number that is twice the number of double heterozygous matings) will decrease from 4Nu to 2Nu. The number of deaths will decrease from 2NuVu to \/;2NuV-. If only looa% of the deleterious alleles are detected, the equation governing the change in allele frequencies is {p + u(1 - p) - 4p2a2[(l + u)/2] - (1 - a2)4p2[(i + u)/2]2 p - 4p(l - 2p)u[(l + u)/2] - (1-2p)2u2} (2) {1-4p2a2 - (1 - a2)4p2 [(1 + u)/2]2-4p(l - 2p)u[(1 + u)/2] - (1-2p)2u2} The initial allele frequency of i- will decrease to \/ +7i2h. The number of individuals barred from reproducing will decrease from a24nu to a24nu/(1 + a2). The number of deaths will decrease from N(l - a2)u to N(l - a2)u/(1 + a2). (These formulas have ignored higher-order terms in u, which assumes that a is not too small.) ASSORTATIVE MATING THAT MASKS DELETERIOUS ALLELES If a lethal recessive allele is rare, it is possible to mate all carriers with individuals homozygous for the wild type and then let the remaining wild-type

4 674 CAMPBELL 674 CMBL individuals mate with themselves. The result of such a scheme would be no change in allele frequencies, since no individuals homozygous for the lethal recessive allele would be formed, if there were no mutation. Mutation will increase the frequency of the deleterious allele and, to a lesser degree, allow the formation of individuals homozygous for the deleterious allele. The net effect will be an increase in the frequency of the deleterious allele. If the frequency of the deleterious allele is >.25, some carriers of the deleterious allele will not be able to find homozygous wild-type mates and a modification of the model will be necessary. Consider first the case when the frequency of the deleterious allele is <.25. The frequency of heterozygous individuals will then be <.5; hence, all can find a wild-type mate, and all changes in allele frequencies will be due to new mutations. The equation governing allele frequencies (ignoring higher-order terms in u) is p + u(l -p) - 2pu (3) l-2pu.(3 From equation (3) it is manifest that the initial allele frequency will increase slowly from V-, that the initial number of deaths is 2uV-, and that no individuals are barred from reproduction. It is also manifest that there is no equilibrium solution with p <.25, and it is easily demonstrated that p will increase to >.25. In that circumstance another equation governs, one that reflects the case that only as many carriers of the deleterious allele as there are wild-type individuals mate; i.e., the proportion of lethal recessive alleles in the mating pool (excluding new mutations) will be.25. If p >.25, the relevant equation (ignoring higher-order terms in u) is r u -.5u (4) 1 -.5u The resultant allele frequency is always >.25, so once the frequency of the deleterious allele is >.25, this equation describes the situation. The equilibrium allele frequency is given by equation (4); it is The number of deaths is -.5Nu, since every mating pair contains one premutation deleterious allele. The number of individuals who are excluded from mating (i.e., the excess of heterozygotes over wild-type homozygotes) is 3Nu. The circumstance in which only looa% of the deleterious alleles are detected is not as accessible as those of the previous models. If ap <.25, all individuals will be mated and the transition equation for p is [p + u(l - p) - 2ap(l + u)(u{[l - (1 + a)p]l + 2ap)} p= + (1 - a)p/(l - 2ap)) - [u + (1 - a)pl(l - 2ap)]2] (5) [1-2ap(I + u)(u{[l - (1 + a)p]l(l - 2ap)} + (1 - a)p(lo - 2ap)) - [u + (1 - a)pl(l - 2ap)]2]

5 GENETIC SCREENING AS SELECTION 675 This does not entail approximations ignoring higher-order terms, but approximations yielding approximate solutions have eluded us. Hence, a qualitative remark-that when only some deleterious alleles are detected the results should be intermediate between those produced by random mating and those that occur when all deleterious alleles are detected-will have to suffice. DISCUSSION As tests to detect carriers of recessive deleterious genes are developed and become readily available to the public, it is appropriate to ask what effect this may have on the incidence of diseases in the population. Even eugenic laws would not be able to achieve strict adherence to any of the mating systems described above, but a philosophical consensus among the population could effectively implement one of the regimes. It must be an individual decision whether to undergo genetic screening-and, if one is found to be a carrier, whether to abstain from reproduction or seek a partner who does not carry the allele. But a concurrence of individual actions could effectively provide a policy for a population. (Other breeding strategies, some involving amniocentesis and abortion, are of course available.) Reproductive compensation has not been discussed except under random mating because under assortative mating based on genetic screening as discussed above deaths only occur from mutations or errors in screening, and hence couples with defective progeny should not further reproduce once they know that they carry a bad gene. Whether all couples have the same (average) number of progeny or whether parents of defective progeny withdraw from the mating pool will only negligibly affect the results. The results that occur when only a fraction of the deleterious alleles are detected could reflect either that individuals are not being screened or that there are errors in the screening procedure. The results are included for completeness and indeed describe a more realistic scenario; but because they lie between the random-mating and accurate-screening values, they shall not be given further discussion. Amid the diversity shown in table 1, the greatest uniformity is in the equilibrium number of individuals barred from reproducing (i.e., the chastity rate), which is of the order Nu, for both mating systems that withdraw and mask deleterious genes. This is the primary means by which deleterious genes are removed from the populations, and this number corresponds to the mortality of the control group. Indeed, more individuals must be barred from reproducing than would be required to maintain the equilibrium by mortality, but the goal of reducing mortality is achieved with only a small reduction in the breeding pool. The exclusion of all carriers from reproducing entails that many more individuals are barred from reproducing during the first generation after panmixia than are ever barred under the other systems. But this expense is rewarded with immediate attainment of equilibrium that has the lowest allele frequency and mortality thereafter. Hence, the high initial cost belies the subsequent advantages of this system. Barring at-risk couples from reproducing is an intermediate strategy. For the

6 676 CAMPBELL TABLE 1 INITIAL AND EQUILIBRIUM ALLELE FREQUENCY, MORTALITY, AND RATE OF EXCLUSION FROM MATING FOR MATING SYSTEMS BASED ON GENETIC SCREENING Allele frequency: MATING SYSTEM Panmictic Individuals Pairs Maximal PARAMETER Control Excluded Excluded Mating - Initial......u <u Equilibrium u... V u Vu2.25 Mortality: Initial... Nu Nu2 2NuVl 2Nu\& Equilibrium Nu Nu2 2NuVu.5Nu Chastity rate: Initial NIu 4Nu 0 Equilibrium Nu 2Nu 3Nu initial generation after panmixia, allele frequency, mortality, and chastity rate are intermediate between those resulting when carriers are excluded from reproducing and those resulting from maximal mating. (The apparent equality in table 1 is the result of approximation.) At equilibrium the three indices are also intermediate. Contrast with the control shows a small decrease in allele frequency, a large reduction in mortality, and introduction of a chaste fraction. Maximal mating looks good initially, with little change in allele frequency, a significant reduction in mortality, and no chastity. (Indeed, although the deleterious allele is rare, it should not be difficult to find wild-type mates without disrupting other matings.) But with the passage of time the allele frequency increases to substantially above the control level, mortality increases to the order of magnitude of the control, and the highest chastity rate of all is obtained. (Furthermore, all wild types must mate with carriers; hence, mating becomes a serious constraint.) The initial appeal of this mating scheme belies its denouement. One aspect of these mating systems that merits consideration is their vulnerability. If genetic screening procedures were to become unavailable, what would the consequences be? The population would have to return to random mating. The systems that withdraw deleterious alleles from the mating pool achieve allele frequencies lower than the control level; hence, the mortality on resumption of panmixia would be lower than if the mating system had never been implemented. But the strategy of masking deleterious alleles allows a significant increase in allele frequency, so that a return to panmixia would be accompanied by >.0625N lethal homozygotes. Although all the systems are rather nice while everything is working, allowing the possibility that there might be a failure in screening makes the option of masking deleterious alleles less desirable. The difference between immediate and equilibrium consequences of mating strategies is illustrated by these examples. An individual may only care what

7 GENETIC SCREENING AS SELECTION 677 the consequences of his mating are (e.g., one avoids consanguineous marriages to avoid having genetically defective progeny without considering what longterm effect leaving one's deleterious genes in the breeding pool will have), but public health officials, at least, should be aware of what current breeding strategies bode for the future. The above examples illustrate that immediate benefit may indeed belie ill in the future and vice versa. It is tempting to seek an explanation for the high incidence of Tay-Sachs disease in terms of the masking of deleterious alleles. Genetic screening has certainly not been available, but individuals who have lost a child to the disease might actively investigate the family history of prospective mates for their viable children. Some degree of masking would have been possible, and the observed frequency of.02 in Ashkenazi Jews is much less than the maximum level of >.25 that could be obtained with accurate screening. But equation (5) requires that a =.95 in order to have p =.01 with u =.00001; hence, the masking of alleles cannot be a sufficient explanation-but may be included as a contributory factor. REFERENCES Campbell, R. B Mating structure and the cost of deleterious mutations. I. The effect of postponing inbreeding (submitted). Crow, J. F., and M. Kimura An introduction to population genetics theory. Harper & Row, New York. Hagy, G. W., and J. F. Kidwell Effect of amniocentesis, selective abortion, and reproductive compensation on the incidence of autosomal recessive diseases. J. Hered. 63: Hartl, D. L., and R. B. Campbell Allele multiplicity in simple Mendelian disorders. Am. J. Hum. Genet. 34: Karlin, S., and P. O'Donald Some population genetic models combining sexual selection with assortative mating. Heredity 41: Koeslag, J. H., and S. R. Schach Tay-Sachs disease and the role of reproductive compensation in the maintenance of ethnic variations in the incidence of autosomal recessive disease. Ann. Hum. Genet. 48: Li, C. C Population genetics. University of Chicago Press, Chicago. Li, W.-H Total number of individuals affected by deleterious mutant genes in a finite population. Ann. Hum. Genet. 38: McKusick, V. A Mendelian inheritance in man: catalogs of autosomal dominant, autosomal recessive, and X-linked phenotypes. Johns Hopkins University Press, Baltimore. Turner, J. R. G How does treating congenital diseases affect the genetic load? Eugenics Q. 15: Yokoyama, S The effect of social selection on population dynamics of rare deleterious genes. Heredity 45: