333
Ann. Hum. Genet., Lo&. (1975), 38, 333 Prinled in &eat Brtiain
Total number of individuals affected by deleterious mutant genes in a finite population BY WEN-HSIUNG LI Center for Demographic and Population Genetics, University of Texas Health Science Center, Houston, Texas 77025
A mutant gene which causes some genetic disease in a heterozygous or homozygous condition creates a burden to a society. It is therefore important to know the total number of individuals affected before the gene is eliminated from the population and the number of generations it may persist in the population. These two problems have recently been studied by Kimura & Ohta (1969), Nei (1971a, b ) , Li & Nei (1972), and Li (1973). However, only the case where initially there is only one mutant gene in the population has received detailed treatments. It is not clear how these quantities increase with the initial frequency of the mutant gene. Since in many human populations there are deleterious genes persisting with appreciably high frequencies, more detailed studies should be done. Also, the effect of inbreeding has not been considered in previous studies. The purpose of this paper is to study these problems. A general approach to these problems has been provided by Maruyama & Kimura (1971). The approach by Nagylaki (1974) provides an excellent alternative. I n this paper we shall follow the method of Maruyama & Kimura (1971). Since the basic principles have already been given by Li & Nei (1972), we shall give only a brief summary of the results obtained in this study.
TOTAL NUMBER O F MUTANT INDIVIDUALS
Consider a population in which a mutant gene a and its wild-type allele A are segregating a t a locus. Let x be the frequency of a and a be the average inbreeding coefficient of the population. Let f ( x ) be either the number of mutant heterozygotes or that of mutant homozygotes in a particular generation. Explicitly, f( 2 ) = 2N( 1 - a ) x( 1 - x ) for the number of heterozygotes, and f ( x ) = N [ x a ax(1 - x ) ]for the number of homozygotes, where N is the actual population size. In the following we shall denote the effective size of the population by N, (see Wright, 1969, Crow & Kimura, 1970, for definition and explanation). Now let P(")(p)be the nth moment of the total number of mutant heterozygotes or homozygotes in the whole process starting from gene frequency p at t = 0 (and continuing over all subsequent generations as long as mutants survive). Following Maruyama & Kimura (1971), i t can be shown that F ( n ) ( p ) is given by
+
where
22-2
WEN-HSIUNQ LI
334
and and K ( z ) = nf(x)P - l ) ( x ) with F(O)(x)= 1 . I n the above formula, M,, and V,, denote the mean and variance of gene frequency change per generation. Explicitly,
M,,
= -sx(l-x) [(l-a) h+a+(1-2h) (1-a)x]
if the relative fitnesses of genotypes A A , Aa and aa are 1, 1 -h and 1 - s, respectively, and V,, = x( 1 - x)/(2Ne).Formula ( 1 ) is identical with that of Maruyama (unpublished). For n = 1 , formula (1) can be shown to be equivalent to formula (5)of Li & Nei (1972).Li & Nei (1972) have derived simpler formulae for the case of a = 0 andp = 1/(2N).However, the error function in their paper was defined erroneously. It should read
instead of
Total number of mutant heterozygotes As mentioned earlier, f(x) = 2N( 1 - a ) z(1 - x) in this case. Putting this into formula ( l ) ,we and second ( H 2 ( p ) )moments of the total number of mutant heteroobtain the first (Ill@)) zygotes.
The variance is given by H2(p)- Hq(p). The relationship between H,(p) and the heterozygous effect of mutant genes is given in Fig. 1 for two initial gene frequencies, two mean inbreeding coefficients and two population sizes. I n Fig. 1 , s = 1 (lethal mutation), N, = 0.75N, solid lines stand for a = 0, and dashed lines for a = 0.002. We consider, instead of the initial frequency p , the initial number of mutant genes r = 2Np, which makes comparison of H,(r) corresponding to different population sizes easier. (Note that H , ( l ) now denotes the value of H, for r = 1 instead of p = 1 . ) It is seen that the ratio of Hl for r = 25 to that of H, for r = 1 is virtually the same for all types of mutation when the population size is around 1000, but when the population size is around 10,000 the ratio H1(25)/H1(1) for an overdominant mutation (i.e. hs < 0) is much smaller than that for a partially recessive mutation (i.e. hs > 0). For a partially recessive mutation, the value of H , for a given r increases slowly with increasing N and becomes linearly proportional to r as N approaches infinity (see Nei, 1 9 7 1 ~ )On . the other hand, for an over-dominant mutation, the value of Hl for a given r increases rapidly with increasing N and becomes independent of r as N approaches infinity. It is interesting to note that for an overdominant mutation, the effect of an increase in population size on H, is much more profound than that of an increase in initial number of mutant genes. For example, If, for r = 25 is 35,000 if N = 1000 but H, for r = 1 is 123,000 if
335
Number of mutant individuals
-0.04
-0.02
0
0.02
Selection coefficients for heterozygotes
Fig. 1. Relationship between the heterozygous effect (h) of a mutant gene and the expected total number of mutant heterozygotes (HI)for two population sizes ( N , = 0.76N),two initial numbers of mutant genes (r) and two average inbreeding coefficients (01). It is assumed that the mutant homozygote is lethal, i.e. s = 1. Solid lines stand for u = 0 while dashed lines for u = 0.002.
N = 10,000, assuming s = 1 , hs = - 0.02, a = 0 and N, = 0.75N. This is because the expected extinction time of an overdominant mutant gene increases with increasing population size much more rapidly than with increasing initial number of mutant genes, as will be seen later. Fig. 1 shows that the effect of inbreeding on Hl is small when the mutant gene is partially recessive. For an overdominant mutation, the effect of inbreeding on Hl increases with increasing population size and increasing degree of overdominance. However, except when the population size or the degree of overdominance is large, the effect of inbreeding is usually small. These results suggest that the effect of inbreeding is very similar to the effect of reduction in effective population size. A comparison of Fig. 1 in this paper and Fig. 1 in Li & Nei (1972)(inthe former N, = 0.75N while N, = N in the latter) also reveals this property. We shall pursue this point further when we consider the total number of mutant homozygotes below. The standard error of the total number of heterozygotes will be discussed later. Total number of mutant homozygotes
In this case, f(x)= N [ x ~ + c c x ( ~ - xIt ) ] follows . from formula ( 1 ) that the first (Ml(p))and second (M2(p))moments of the total number of mutant homozygotes are given by
336
WEN-HSIUNQ LI
For a completely recessive deleterious mutation, the expected total number of mutant homozygotes is given by
2Np
r 2s'
Ml@) = -= 2s
(6)
if the absolute fitnesses of A A and Aa are equal to 1, The derivation is briefly as follows. Under our assumption, the expected number of mutant genes in a population changes only when mutant homozygotes appear. Let nt be the number of mutant homozygotes at generation t . When a mutant homozygote appears, it dies with probability s and survives with probability 1- s ; in the former case, two mutant genes are eliminated while in the latter no genes are eliminated. Therefore, Crz02snt = r, that is, Ml(r) = Ent = r/(2s). Note that this formula is independent of the population size and the degree of inbreeding. It is also clear that it applies under any population structure since our arguments do not depend on any assumption on population structure. When r = 1, formula (6) is identical with that of Li & Nei (1972).Values obtained from formula (6) agree rather well with numerical integrations of formula (4). The expected total number of mutant homozygotesM,(r)is given in Table 1 for various values of hs, r , N and a, assuming s = 1 and N, = 0.75N. If he > 0 , Ml(r) increases very slowly with increasing r and is usually small, particularly in large populations. This is because in this case mutant genes are mostly eliminated in heterozygous condition. In the case of completely recessive mutations, Ml(r) increases linearly with increasing r and is independent of N and a. The values for Ml(r) obtained by numerical integration of formula (4) agree rather well with those obtained from formula (6) (values in parentheses). On the other hand, for an overdominant mutation (h< 0 ) ,Ml(r) increases rapidly both with increasing population size and increasing degree of overdominance. In this case, as in the case of total number of mutant heterozygotes, the effect of an increase in population size on Ml(r)is much more profound than the effect of an increase in initial number of mutant genes. For example, if N is 10,000, MI for r = 1 is 1200, but if N is 1000, Ml(r) is only 47 even if r = 25, assuming s = 1, hs = -0.02, a = 0, and N, = 0.75N. Note that inbreeding increases the total number of mutant homozygotes if the mutant gene is partially recessive while it reduces this number if there is overdominance, particularly in large populations. This is very similar to the effect of reduction in effective population size. The reduction in Ml(r) due to inbreeding is substantial if the population size and degree of overdominance are large. EXTINCTION TIME
Following Maruyama & Kimura (1971),it can be shown that the first and second moments of extinction time, excluding the cases of eventual fixation, are given by
When a = 0 formula ( 7 ) is identical with that of Kimura & Ohta (1969).
Number of mutant individuals
337
Table 1. Expected total number of mutant homozygotes aflected by deleterious mutant genes for two population sizes ( N ) , two initial numbers of mutant genes ( r ) , and two average inbreeding coeficients (a) r = 25
r=x A
I
\
A
r
>
N = 1000 N = IO,OOO N = 1000 N = 10,000 &&&& Partial Recessive
a=o
a=o.ooz
0.04
0.07 0.15 0.26
0.08
0.009
0.03
3.0
0.17 0.27
0.032
0.067
5.4
0.089
0.135
7.95
0'02
0'01
Recessive Overdominance
a = o a=o.ooz
h8
u=o
a=o~oo2
3.3 5.7 8.2
u=o
0.43 1.17 2.82
0.45
0.47
0.46
0.46
12.8
12.8
12.5
(0.5)
(0.5)
(0.5)
(0.5)
(12.5)
(12.5)
(12.5)
- 0.01
0.97
0.92
8.5
22
- 0'02 - 0.04
2.3
2.1
1200
21
17
43 267
0'0
23 5 400 47 3 2 x 1 0 ~ 35x10" 323
196 24,000 47x10'~
u=o'oo2
0.98 2.07 4'0 12.5
(12'5)
119 8000 53x10~
Note. It is assumed that N , = o.75N. The values in parentheses are obtained from formula (6).
105
104
VI
.-6 Y
e 103
M
.-YE .-6 Y
.5
102
w X
10
-0.04
-0.02
0
0.02
Selection coefficients for heterozygotes
Fig. 2. Relationship between the heterozygous effect (hs) of a mutant gene and the expected extinction time (to)for two population sizes ( N a = 0.75N) two average inbreeding coefficients ( a ) and two initial numbers of mutant genes ( T ) . It is assumed that the mutant homozygote is lethal i.e. s = 1. Solid lines stand for u = 0 while dashed lines for 01 = 0.002.
WEN-HSIUNQ LI
338
Table 2. The means and standard errors of the total numbers of heterozygotes and homozygotes affected by a single deleterious mutant gene and the extinction time No. of heterozygotes
ha Partial Recessive
0.04 0'02 0'01
Recessive Overdominance
0'0
- 0'01 - 0'02
-0.04
16 (63) (87) 24 (103) 29 (124) 21
34 (151) 42 (185) 65 (289)
No. of homozygotes
0.41 (2.9) 0.65 (4-5) 0.83 (5.6) 1.0 (7.1) 1'4 (9.2) 1'9 (11.9) 3'5 (20.7)
Note. The values in parentheses denote the standard errors. It is assumed that N = and 8 = 1.
Extinction time
4'5 (7'7) 5 ' 0 (9'5) 5'2 (10.7) 5.6 (11.4) 6.0 (12.9) 6-5(14.7) 8.0 (20) 100, N ,
= 75,a = o
Fig. 2 shows the relationship between the expected extinction time and the heterozygous effect of the mutant genes for two initial gene frequencies, two average inbreeding coefficients and two population sizes. It is clear that the relationship betweent, and hs for given r , a,N and s is very similar to that between HI and hs in Fig. 1. The effects of population size, initial number of genes, and inbreeding on to@) are very similar to those on HI@). STANDARD ERRORS
Table 2 shows the means and standard errors of the total number of mutant heterozygotes and homozygotes, and extinction time for various values of hs, assuming a = 0, N = 100 and N , = 75. Note that the standard errors of these quantities are quite large and increase with increasing heterozygote fitness. Therefore, the mean values of these quantities are subject to large errors. Numerical results indicate that inbreeding reduces the standard errors of these quantities but only very slightly if a is around 0.002. However, if a is very large, the standard errors of these quantities are expected to be much smaller. For example, if a = 1 and s = 1, then these standard errors are zero. DISCUSSION
I n the present paper we have assumed that the selection intensity and the population size remain constant and there is no further mutation of the same type. For a discussion of these assumptions readers may refer to the paper by Li t Nei (1972). I n the previous sections we have seen that the effect of inbreeding on the total number of individuals affected and the extinction time is small, except when the population size or the degree of overdominance is large. Note that the amounts of inbreeding in most human populations are usually smaller than the amount 0.002 considered in this study (cf. Cavalli-Sforza & Bodmer, 1971) and also the effect of inbreeding on these quantities decreases as the selection coefficient against the mutant homozygotes decreases. We may therefore conclude that the formulae developed for randomly mating populations hold approximately except for mutations with large degrees of overdominance in large populations. Formula (6) provides a simple principle for estimating the burden to a society created by completely recessive deleterious genes. As an application, let us consider the sickle-cell anaemia
Number of mutant
individuals
339
gene Hb S in the United States black population. The sickle-cell mutant heterozygotes in this population seem to be slightly disadvantageous compared with normal homozygotes (Rucknagel & Neel, 1961), but no reliable estimate of heterozygote fitness is available. For simplicity, we shall assume that the absolute fitness of heterozygotes is 1. This may not represent an overestimate because the United States black population is still growing. However, this mutant gene is expected to become extinct if the mutant homozygous condition remains disadvantageous and no immigrants with Hb S genes come to the United States. The present frequency of this gene in the United States black population is about 0.05 (Motulsky, Fraser & Felsenstein, 1971; Cavalli-Sforza & Bodmer, 1971) and the size of this population is about 22,500,000, therefore, the total number of Hb 8 genes is about 2,250,000. The fitness of sickle-cell homozygotes is not zero, but no reliable estimate is available. We therefore consider two extreme values s = 0.5 and s = 1, and one intermediate value s = 0.8. By formula (6), the total number of mutant homozygotes, who will suffer from this disease before the mutant genes disappear, is estimated to be about 2,250,000 for s = 0.5, 1,406,250 for s = 0.8 and 1,125,000 for s = 1. At any rate, a large number of individuals will be affected in the future. Furthermore, Li & Nei’s (1972) computation suggests that it will take at least several thousand years before this gene is eliminated from the United States population. SUMMARY
The means and standard errors of the total numbers of heterozygotes and homozygotes affected by deleterious mutant genes and the extinction time are studied by using diffusion methods. For an overdominant mutation, the effects of an increase in population size on these quantities are much more profound than that of an increase in initial number of mutant genes whereas for a partially recessive mutation the situation is reversed. For a completely recessive mutation, the expected total number of mutant homozygotes is independent of the population size and degree of inbreeding, though the expected total number of heterozygotes and the average extinction time are dependent on these factors, particularly the population size. The effect of inbreeding on these quantities is very similar to that of reduction in effective population size and is usually small a t the prevailing level of inbreeding, except for mutations with large degrees of overdominance in large populations. The standard errors of these quantities are large. The expected total number of sickle-cell mutant homozygotes in the U.S. population has been computed. The author is very grateful to Dr M. Nei for his encouragement and help through the course of this study. He also wishes to thank Drs W. J. Schull, T. Maruyama, T. Nagylaki and K. Weiss for many discussions and valuable comments. This research was supported by U.S.National Institutes of Health Research Grants GM 20293 and GM 19513. REFERENCES
CAVALLI-SFORZA, L. L. & BODMER, W. F. (1971). The Genetic8 of Human POptdatiOn8. San Francisco: W. H. Freeman. CROW,J. F. & KIMUF~A, M. (1970). A n Introduction to Population Genetic8 Thewy. New York: Harper and Row. KIMURA, M. & OHTA,T. (1969).The average number of generations until extinction of an individual mutant gene in a h i t e population. Genetic8 63, 701-9. LI, W-H. (1973). Total number of individuals affected by a single sex-linked deleterious mutation in a finite population. American Journal of Human Genetics 25, 698-603.
340
WEN-HSIUNG LI
LI, W-H. & NEI,M. (1972). Total number of individuals affected by a single deleterious mutation in a k i t e population. American Journal of Human Genetics 24, 667-79. MARWAMA,T. BE KIMURA, M. (1971). Some methods for treating continuous stochastic processes in population genetics. Japneae Journcll of Genetics 46, 407-10.
MOTULSKY,A. G., FRASER, G. R. L FELSENSTEIN, F. (1971). Public health and long-term genetic implications of intrauterine diagnosis and selective abortion. Birth Defects: Original Article Series 7 , 22-32. NAGYLAKI, T. (1974). The moments cf stocha.stic integrals and the distribution of sojourn time. Proceedings of the National Academy of Sciences, 71, 746-9. NEI,M. (1971a). Total number of individuals affected by a single deleterious mutation in large populations. Thewetical Population Biology, 2, 426-30. NEI, M. (1971 b). Extinction time of deleterious mutant genes in large populations. Theoreticd Population Biology 2, 419-25. RUCKNAGEL, D. L. & NEEL,J. V. (1961). The hemoglobinopathies. In A. C.Steinberg (ed). Progress in Medical Genetics 1, 158-260. WRIGHT,S. (1969). Evolution and the Genetics of Populations. Vol. 2. The Theory of Qene Frequencies.Chicago : The University of Chicago Press.
Ann. Hum. Genet., Lo&. (1975), 38, 333 Prinled in &eat Brtiain
Total number of individuals affected by deleterious mutant genes in a finite population BY WEN-HSIUNG LI Center for Demographic and Population Genetics, University of Texas Health Science Center, Houston, Texas 77025
A mutant gene which causes some genetic disease in a heterozygous or homozygous condition creates a burden to a society. It is therefore important to know the total number of individuals affected before the gene is eliminated from the population and the number of generations it may persist in the population. These two problems have recently been studied by Kimura & Ohta (1969), Nei (1971a, b ) , Li & Nei (1972), and Li (1973). However, only the case where initially there is only one mutant gene in the population has received detailed treatments. It is not clear how these quantities increase with the initial frequency of the mutant gene. Since in many human populations there are deleterious genes persisting with appreciably high frequencies, more detailed studies should be done. Also, the effect of inbreeding has not been considered in previous studies. The purpose of this paper is to study these problems. A general approach to these problems has been provided by Maruyama & Kimura (1971). The approach by Nagylaki (1974) provides an excellent alternative. I n this paper we shall follow the method of Maruyama & Kimura (1971). Since the basic principles have already been given by Li & Nei (1972), we shall give only a brief summary of the results obtained in this study.
TOTAL NUMBER O F MUTANT INDIVIDUALS
Consider a population in which a mutant gene a and its wild-type allele A are segregating a t a locus. Let x be the frequency of a and a be the average inbreeding coefficient of the population. Let f ( x ) be either the number of mutant heterozygotes or that of mutant homozygotes in a particular generation. Explicitly, f( 2 ) = 2N( 1 - a ) x( 1 - x ) for the number of heterozygotes, and f ( x ) = N [ x a ax(1 - x ) ]for the number of homozygotes, where N is the actual population size. In the following we shall denote the effective size of the population by N, (see Wright, 1969, Crow & Kimura, 1970, for definition and explanation). Now let P(")(p)be the nth moment of the total number of mutant heterozygotes or homozygotes in the whole process starting from gene frequency p at t = 0 (and continuing over all subsequent generations as long as mutants survive). Following Maruyama & Kimura (1971), i t can be shown that F ( n ) ( p ) is given by
+
where
22-2
WEN-HSIUNQ LI
334
and and K ( z ) = nf(x)P - l ) ( x ) with F(O)(x)= 1 . I n the above formula, M,, and V,, denote the mean and variance of gene frequency change per generation. Explicitly,
M,,
= -sx(l-x) [(l-a) h+a+(1-2h) (1-a)x]
if the relative fitnesses of genotypes A A , Aa and aa are 1, 1 -h and 1 - s, respectively, and V,, = x( 1 - x)/(2Ne).Formula ( 1 ) is identical with that of Maruyama (unpublished). For n = 1 , formula (1) can be shown to be equivalent to formula (5)of Li & Nei (1972).Li & Nei (1972) have derived simpler formulae for the case of a = 0 andp = 1/(2N).However, the error function in their paper was defined erroneously. It should read
instead of
Total number of mutant heterozygotes As mentioned earlier, f(x) = 2N( 1 - a ) z(1 - x) in this case. Putting this into formula ( l ) ,we and second ( H 2 ( p ) )moments of the total number of mutant heteroobtain the first (Ill@)) zygotes.
The variance is given by H2(p)- Hq(p). The relationship between H,(p) and the heterozygous effect of mutant genes is given in Fig. 1 for two initial gene frequencies, two mean inbreeding coefficients and two population sizes. I n Fig. 1 , s = 1 (lethal mutation), N, = 0.75N, solid lines stand for a = 0, and dashed lines for a = 0.002. We consider, instead of the initial frequency p , the initial number of mutant genes r = 2Np, which makes comparison of H,(r) corresponding to different population sizes easier. (Note that H , ( l ) now denotes the value of H, for r = 1 instead of p = 1 . ) It is seen that the ratio of Hl for r = 25 to that of H, for r = 1 is virtually the same for all types of mutation when the population size is around 1000, but when the population size is around 10,000 the ratio H1(25)/H1(1) for an overdominant mutation (i.e. hs < 0) is much smaller than that for a partially recessive mutation (i.e. hs > 0). For a partially recessive mutation, the value of H , for a given r increases slowly with increasing N and becomes linearly proportional to r as N approaches infinity (see Nei, 1 9 7 1 ~ )On . the other hand, for an over-dominant mutation, the value of Hl for a given r increases rapidly with increasing N and becomes independent of r as N approaches infinity. It is interesting to note that for an overdominant mutation, the effect of an increase in population size on H, is much more profound than that of an increase in initial number of mutant genes. For example, If, for r = 25 is 35,000 if N = 1000 but H, for r = 1 is 123,000 if
335
Number of mutant individuals
-0.04
-0.02
0
0.02
Selection coefficients for heterozygotes
Fig. 1. Relationship between the heterozygous effect (h) of a mutant gene and the expected total number of mutant heterozygotes (HI)for two population sizes ( N , = 0.76N),two initial numbers of mutant genes (r) and two average inbreeding coefficients (01). It is assumed that the mutant homozygote is lethal, i.e. s = 1. Solid lines stand for u = 0 while dashed lines for u = 0.002.
N = 10,000, assuming s = 1 , hs = - 0.02, a = 0 and N, = 0.75N. This is because the expected extinction time of an overdominant mutant gene increases with increasing population size much more rapidly than with increasing initial number of mutant genes, as will be seen later. Fig. 1 shows that the effect of inbreeding on Hl is small when the mutant gene is partially recessive. For an overdominant mutation, the effect of inbreeding on Hl increases with increasing population size and increasing degree of overdominance. However, except when the population size or the degree of overdominance is large, the effect of inbreeding is usually small. These results suggest that the effect of inbreeding is very similar to the effect of reduction in effective population size. A comparison of Fig. 1 in this paper and Fig. 1 in Li & Nei (1972)(inthe former N, = 0.75N while N, = N in the latter) also reveals this property. We shall pursue this point further when we consider the total number of mutant homozygotes below. The standard error of the total number of heterozygotes will be discussed later. Total number of mutant homozygotes
In this case, f(x)= N [ x ~ + c c x ( ~ - xIt ) ] follows . from formula ( 1 ) that the first (Ml(p))and second (M2(p))moments of the total number of mutant homozygotes are given by
336
WEN-HSIUNQ LI
For a completely recessive deleterious mutation, the expected total number of mutant homozygotes is given by
2Np
r 2s'
Ml@) = -= 2s
(6)
if the absolute fitnesses of A A and Aa are equal to 1, The derivation is briefly as follows. Under our assumption, the expected number of mutant genes in a population changes only when mutant homozygotes appear. Let nt be the number of mutant homozygotes at generation t . When a mutant homozygote appears, it dies with probability s and survives with probability 1- s ; in the former case, two mutant genes are eliminated while in the latter no genes are eliminated. Therefore, Crz02snt = r, that is, Ml(r) = Ent = r/(2s). Note that this formula is independent of the population size and the degree of inbreeding. It is also clear that it applies under any population structure since our arguments do not depend on any assumption on population structure. When r = 1, formula (6) is identical with that of Li & Nei (1972).Values obtained from formula (6) agree rather well with numerical integrations of formula (4). The expected total number of mutant homozygotesM,(r)is given in Table 1 for various values of hs, r , N and a, assuming s = 1 and N, = 0.75N. If he > 0 , Ml(r) increases very slowly with increasing r and is usually small, particularly in large populations. This is because in this case mutant genes are mostly eliminated in heterozygous condition. In the case of completely recessive mutations, Ml(r) increases linearly with increasing r and is independent of N and a. The values for Ml(r) obtained by numerical integration of formula (4) agree rather well with those obtained from formula (6) (values in parentheses). On the other hand, for an overdominant mutation (h< 0 ) ,Ml(r) increases rapidly both with increasing population size and increasing degree of overdominance. In this case, as in the case of total number of mutant heterozygotes, the effect of an increase in population size on Ml(r)is much more profound than the effect of an increase in initial number of mutant genes. For example, if N is 10,000, MI for r = 1 is 1200, but if N is 1000, Ml(r) is only 47 even if r = 25, assuming s = 1, hs = -0.02, a = 0, and N, = 0.75N. Note that inbreeding increases the total number of mutant homozygotes if the mutant gene is partially recessive while it reduces this number if there is overdominance, particularly in large populations. This is very similar to the effect of reduction in effective population size. The reduction in Ml(r) due to inbreeding is substantial if the population size and degree of overdominance are large. EXTINCTION TIME
Following Maruyama & Kimura (1971),it can be shown that the first and second moments of extinction time, excluding the cases of eventual fixation, are given by
When a = 0 formula ( 7 ) is identical with that of Kimura & Ohta (1969).
Number of mutant individuals
337
Table 1. Expected total number of mutant homozygotes aflected by deleterious mutant genes for two population sizes ( N ) , two initial numbers of mutant genes ( r ) , and two average inbreeding coeficients (a) r = 25
r=x A
I
\
A
r
>
N = 1000 N = IO,OOO N = 1000 N = 10,000 &&&& Partial Recessive
a=o
a=o.ooz
0.04
0.07 0.15 0.26
0.08
0.009
0.03
3.0
0.17 0.27
0.032
0.067
5.4
0.089
0.135
7.95
0'02
0'01
Recessive Overdominance
a = o a=o.ooz
h8
u=o
a=o~oo2
3.3 5.7 8.2
u=o
0.43 1.17 2.82
0.45
0.47
0.46
0.46
12.8
12.8
12.5
(0.5)
(0.5)
(0.5)
(0.5)
(12.5)
(12.5)
(12.5)
- 0.01
0.97
0.92
8.5
22
- 0'02 - 0.04
2.3
2.1
1200
21
17
43 267
0'0
23 5 400 47 3 2 x 1 0 ~ 35x10" 323
196 24,000 47x10'~
u=o'oo2
0.98 2.07 4'0 12.5
(12'5)
119 8000 53x10~
Note. It is assumed that N , = o.75N. The values in parentheses are obtained from formula (6).
105
104
VI
.-6 Y
e 103
M
.-YE .-6 Y
.5
102
w X
10
-0.04
-0.02
0
0.02
Selection coefficients for heterozygotes
Fig. 2. Relationship between the heterozygous effect (hs) of a mutant gene and the expected extinction time (to)for two population sizes ( N a = 0.75N) two average inbreeding coefficients ( a ) and two initial numbers of mutant genes ( T ) . It is assumed that the mutant homozygote is lethal i.e. s = 1. Solid lines stand for u = 0 while dashed lines for 01 = 0.002.
WEN-HSIUNQ LI
338
Table 2. The means and standard errors of the total numbers of heterozygotes and homozygotes affected by a single deleterious mutant gene and the extinction time No. of heterozygotes
ha Partial Recessive
0.04 0'02 0'01
Recessive Overdominance
0'0
- 0'01 - 0'02
-0.04
16 (63) (87) 24 (103) 29 (124) 21
34 (151) 42 (185) 65 (289)
No. of homozygotes
0.41 (2.9) 0.65 (4-5) 0.83 (5.6) 1.0 (7.1) 1'4 (9.2) 1'9 (11.9) 3'5 (20.7)
Note. The values in parentheses denote the standard errors. It is assumed that N = and 8 = 1.
Extinction time
4'5 (7'7) 5 ' 0 (9'5) 5'2 (10.7) 5.6 (11.4) 6.0 (12.9) 6-5(14.7) 8.0 (20) 100, N ,
= 75,a = o
Fig. 2 shows the relationship between the expected extinction time and the heterozygous effect of the mutant genes for two initial gene frequencies, two average inbreeding coefficients and two population sizes. It is clear that the relationship betweent, and hs for given r , a,N and s is very similar to that between HI and hs in Fig. 1. The effects of population size, initial number of genes, and inbreeding on to@) are very similar to those on HI@). STANDARD ERRORS
Table 2 shows the means and standard errors of the total number of mutant heterozygotes and homozygotes, and extinction time for various values of hs, assuming a = 0, N = 100 and N , = 75. Note that the standard errors of these quantities are quite large and increase with increasing heterozygote fitness. Therefore, the mean values of these quantities are subject to large errors. Numerical results indicate that inbreeding reduces the standard errors of these quantities but only very slightly if a is around 0.002. However, if a is very large, the standard errors of these quantities are expected to be much smaller. For example, if a = 1 and s = 1, then these standard errors are zero. DISCUSSION
I n the present paper we have assumed that the selection intensity and the population size remain constant and there is no further mutation of the same type. For a discussion of these assumptions readers may refer to the paper by Li t Nei (1972). I n the previous sections we have seen that the effect of inbreeding on the total number of individuals affected and the extinction time is small, except when the population size or the degree of overdominance is large. Note that the amounts of inbreeding in most human populations are usually smaller than the amount 0.002 considered in this study (cf. Cavalli-Sforza & Bodmer, 1971) and also the effect of inbreeding on these quantities decreases as the selection coefficient against the mutant homozygotes decreases. We may therefore conclude that the formulae developed for randomly mating populations hold approximately except for mutations with large degrees of overdominance in large populations. Formula (6) provides a simple principle for estimating the burden to a society created by completely recessive deleterious genes. As an application, let us consider the sickle-cell anaemia
Number of mutant
individuals
339
gene Hb S in the United States black population. The sickle-cell mutant heterozygotes in this population seem to be slightly disadvantageous compared with normal homozygotes (Rucknagel & Neel, 1961), but no reliable estimate of heterozygote fitness is available. For simplicity, we shall assume that the absolute fitness of heterozygotes is 1. This may not represent an overestimate because the United States black population is still growing. However, this mutant gene is expected to become extinct if the mutant homozygous condition remains disadvantageous and no immigrants with Hb S genes come to the United States. The present frequency of this gene in the United States black population is about 0.05 (Motulsky, Fraser & Felsenstein, 1971; Cavalli-Sforza & Bodmer, 1971) and the size of this population is about 22,500,000, therefore, the total number of Hb 8 genes is about 2,250,000. The fitness of sickle-cell homozygotes is not zero, but no reliable estimate is available. We therefore consider two extreme values s = 0.5 and s = 1, and one intermediate value s = 0.8. By formula (6), the total number of mutant homozygotes, who will suffer from this disease before the mutant genes disappear, is estimated to be about 2,250,000 for s = 0.5, 1,406,250 for s = 0.8 and 1,125,000 for s = 1. At any rate, a large number of individuals will be affected in the future. Furthermore, Li & Nei’s (1972) computation suggests that it will take at least several thousand years before this gene is eliminated from the United States population. SUMMARY
The means and standard errors of the total numbers of heterozygotes and homozygotes affected by deleterious mutant genes and the extinction time are studied by using diffusion methods. For an overdominant mutation, the effects of an increase in population size on these quantities are much more profound than that of an increase in initial number of mutant genes whereas for a partially recessive mutation the situation is reversed. For a completely recessive mutation, the expected total number of mutant homozygotes is independent of the population size and degree of inbreeding, though the expected total number of heterozygotes and the average extinction time are dependent on these factors, particularly the population size. The effect of inbreeding on these quantities is very similar to that of reduction in effective population size and is usually small a t the prevailing level of inbreeding, except for mutations with large degrees of overdominance in large populations. The standard errors of these quantities are large. The expected total number of sickle-cell mutant homozygotes in the U.S. population has been computed. The author is very grateful to Dr M. Nei for his encouragement and help through the course of this study. He also wishes to thank Drs W. J. Schull, T. Maruyama, T. Nagylaki and K. Weiss for many discussions and valuable comments. This research was supported by U.S.National Institutes of Health Research Grants GM 20293 and GM 19513. REFERENCES
CAVALLI-SFORZA, L. L. & BODMER, W. F. (1971). The Genetic8 of Human POptdatiOn8. San Francisco: W. H. Freeman. CROW,J. F. & KIMUF~A, M. (1970). A n Introduction to Population Genetic8 Thewy. New York: Harper and Row. KIMURA, M. & OHTA,T. (1969).The average number of generations until extinction of an individual mutant gene in a h i t e population. Genetic8 63, 701-9. LI, W-H. (1973). Total number of individuals affected by a single sex-linked deleterious mutation in a finite population. American Journal of Human Genetics 25, 698-603.
340
WEN-HSIUNG LI
LI, W-H. & NEI,M. (1972). Total number of individuals affected by a single deleterious mutation in a k i t e population. American Journal of Human Genetics 24, 667-79. MARWAMA,T. BE KIMURA, M. (1971). Some methods for treating continuous stochastic processes in population genetics. Japneae Journcll of Genetics 46, 407-10.
MOTULSKY,A. G., FRASER, G. R. L FELSENSTEIN, F. (1971). Public health and long-term genetic implications of intrauterine diagnosis and selective abortion. Birth Defects: Original Article Series 7 , 22-32. NAGYLAKI, T. (1974). The moments cf stocha.stic integrals and the distribution of sojourn time. Proceedings of the National Academy of Sciences, 71, 746-9. NEI,M. (1971a). Total number of individuals affected by a single deleterious mutation in large populations. Thewetical Population Biology, 2, 426-30. NEI, M. (1971 b). Extinction time of deleterious mutant genes in large populations. Theoreticd Population Biology 2, 419-25. RUCKNAGEL, D. L. & NEEL,J. V. (1961). The hemoglobinopathies. In A. C.Steinberg (ed). Progress in Medical Genetics 1, 158-260. WRIGHT,S. (1969). Evolution and the Genetics of Populations. Vol. 2. The Theory of Qene Frequencies.Chicago : The University of Chicago Press.
