KELLER CE: Caribe Cuna moon-child and its heredity. JHered44:163-171, 1953. 29. WOOLF CM, DUKEPOO FC: Hopi Indians, inbreeding, and albinism.
Am J Hum Genet 33:407-417, 1981
Social Selection in Human Populations. I. Modification of the Fitness of Offspring by an Affected Parent SHOZO YOKOYAMA'
SUMMARY
The concept of social selection for deleterious genes has been introduced by considering two alleles at one locus. A social selection model is constructed by assuming that the fitness of an individual is determined by his or her own as well as the parental phenotypes. It is shown that the equilibrium gene frequency depends on the loss of fitness of an individual due to the trait (-y), due to affected parents (,/), and the probability that the heterozygote develops the trait (h). With mutational changes from the wild-type allele to the deleterious gene at a rate of a per generation, the equilibrium frequency of deleterious genes is approximately a/hs for o < h < land a/sforh = 0,wheres = -y + /8(1 - y)/2.Implications of the social selection model have been discussed for several diseases in man.
INTRODUCTION
Two individuals with identical genotypes or phenotypes may have different fitnesses because of abnormal sibs, parents, or other relatives. In a study of Huntington disease, Reed and Neel [1] estimated the relative fitness of affected to nonaffected sibs to be 1.01 to 1. This and other studies [2, 3] show that this disease is either nearly neutral or slightly favored in heterozygous condition. Relative to the general population, however, the fitness of the heterozygotes was .81, and, therefore, the fitness of the normal sib is about .80. This clearly shows that even if the individual has a normal genotype, mate-finding and fertility are considerably affected by the Received June 19, 1980; revised September 15, 1980. This work was supported by grants GM-28672 from the National Institutes of Health and MH-31302 from the U.S. Public Health Service in connection with Washington University, St. Louis, Missouri. Department of Psychiatry, Washington University, School of Medicine, and Jewish Hospital of St. Louis, 216 South Kingshighway, St. Louis, MO 63110. © 1981 by the American Society of Human Genetics. 0002-9297/81/3303-0009$02.00
407
YOKOYAMA 408 existence of the affected individuals in the family. Similar phenomena have been observed for schizophrenia [4, 5], although the genetics involved are not clear. We have shown that the frequency of Huntington disease in the population is determined mainly by such social effects [6]. In addition, we have shown that marriage frequency of a mentally normal individual is affected strongly by his or her retarded parent and sibs [7]. So far, other diseases have not been studied systematically, but the social impact of a disease on the fitness of an individual may be very important in determining the equilibrium gene frequency of many disorders in human populations. This kind of fitness interaction among relatives has been called social selection, and mathematical analyses considering rare lethal genes have been initiated [6,
8, 9]. The
evolutionary consequences of fitness interaction among relatives
has also
been considered in the study of altruism, in which the concept of kin selection has an important role [10-16]. There is a fundamental difference between the concept of social selection and that of kin selection. In kin selection, the focus of interest is a behavioral characteristic such as altruism. But, in social selection, emphasis is placed on the population dynamics of genetic or common diseases and the effect of social factors on the fitness of an individual. The important difference between these two concepts becomes apparent when one realizes that kin selection requires the coexistence of an altruist and a recipient. For example, the loss of fitness of an altruistic parent cannot be determined until he or she has offspring, so that viability selection in the usual sense is not included. Thus, existing theories of kin selection are not helpful in studies of the population dynamics of genetic or common diseases and social factors in human populations. Furthermore, in population genetics theory, the fitness of an individual has traditionally been quantified by using only his or her genotype or phenotype. Because of the social impact of a disease, family structure also must be considered.
Here we study the effect of social selection on the equilibrium frequency of rare deleterious genes. For simplicity, we consider the case in which the modification of the fitness of offspring by one affected parent is exactly halfway between that of two normal parents and two affected parents. Deterministic analysis of the equilibrium populations will be considered in later papers. Studies of sib-to-sib fitness interaction and other aspects of social selection are also left to future studies. GENE FREQUENCY CHANGE UNDER FITNESS MODIFICATION BY AFFECTED PARENTS
Assume that a mutant gene Al and its wild-type allele A2 are segregating with respective
ofA IA1 A IA2, and frequencies p and q in the adult population. Let us denoteA the frequencies a disease and develops Assume that always and adults 1 w, u, v, IA respectively. by A2A2in of the trait so
that A2A2 is normal. Furthermore, assume that h is the degree of dominance that, with probability h, A IA2 will also be affected by the disease. If sex differences are ignored, there are three parental combinations: both parents are normal, one parent is affected, and both are affected. An individual can therefore be classified into nine groups, that is, three genotypes and three parental combinations, before an individual develops the trait. Denote a individuals of genotypesAAJ by (A A )0, (AA.),, and (A A )2 when they have zero, one, group ofaffected parents, respectively. Let y lge the magnitude ofitness loss of an individual and two
SOCIAL SELECTION
409
due to the trait. Furthermore, let f/2 and 1 be the magnitude of fitness loss when an individual has one and two affected parents, respectively. We assume that the fitness of an individual is determined multiplicatively by the individual's and parental phenotypes. In table 1, the disease status of A A individuals and associated fitnesses are shown. Thus, the relative fitnesses of a normal individual with two normal parents, one affected parent, and two affected parents are given by 1, 1 - 13/2, and 1 - 1, respectively. Similarly, the relative fitnesses of an affected individual with two normal parents, one affected parent, and two affected parents are given by 1 - y, (1 - -y)(l - 13/2), and (1 - y)(1 - 13), respectively. It is assumed that the gene frequency is changed by the processes of random mating, reproduction, and selection, in that order. The process of mutational change is initially ignored, but will be considered in EQUILIBRIUM UNDER SELECTION AND MUTATION. Let us now consider the process of mating. Since the offspring's fitness is determined by his or her own and parental phenotypes, the parental phenotype must be specified. Let us denote an individual whose genotype is A1A1, with normal and affected phenotypes by (AAJ)N and (A Aj)A' respectively. Considering the gene frequency change from the newly formed zygote to adulthood, it is easy to see that the frequency of (A IA )A is given by u. Similarly, the frequencies of (A IA2)N, (A lA2)A, and (A 2A2)N are given by (1 - h)v/(1 - hy), (1 - y)hv/ (1 - hy), and w, respectively. At this point, it is appropriate to point out the difference between social selection and kin selection models. In the kin selection model, the loss of fitness, -y, of an altruistic parent cannot be determined until he or she has offspring. Therefore, the frequencies of altruistic and nonaltruistic parents do not depend on the selection coefficients -y or 13. For example, the frequencies of altruistic and nonaltruistic parents of the genotype A A2 are given by hv and (1 - h)v, respectively (see also Cavalli-Sforza and Feldman [16]). Mating type frequencies and the probabilities of their offspring are shown in table 2. After the development of the disease and selection, the frequencies of the three genotypes in the next generation are given by
p2(1 V = 2pq(1 U =
w =
- y)(1 - H1)/W, -
(1.1)
hy)( -H2)/W,
q2(1 - H3)/W ,
where Hi = 1 {u/p + [h(l - y)v]/[2(1 H2 = (Hi + H3)12, W = p2(l - -/)(I
-hy)p]}, H3
-HI)
=
{[h(1 -y)v]/[2(1
1
+ 2pq(1 - hy)(l
TABLE 1 DEVELOPMENT OF A DISEASE AND FITNESS DETERMINATION OF A Iaj -A. INDIVIDUALS Genotype
(A )A) 0............ (
A.
(1.2) (1.3)
Phenotype
Fitness
Normal
1
Affected
1 - y
Normal
Affected (1 { Normal
I - y)(I
/2
-,8//2)
1- /
Affected (I - -y)(1 - /3)
-H2)
-
hy)q]},
+ q2(1 - H).
410
YOKOYAMA "T
I
'11
3
v
z
Q z
lz
C',
w
0 0
L
-J
LF
J
0
m
X
m
co
3
Pw
w H.
H --4
:
:
:
:
:
CZ
-,Z
:
:
z -
V)
ll-s
Vz 6L
It I
?-
u
z uw
1-1
Cy Cd
?I -a .
.
4
uz
.
.
.
-C
11
z
P
z
SOCIAL SELECTION
411
Then, it is easy to show that
pq P = - -=(y(l +
Hj(1
-
h)p
+
hyq
hy)(½2
-
p)
-
(I -y) [H[(l q]}) .
+
-
y)p
-
+
(1
-
hy)(½2
-
p)]
(2)
Using equations (1) and (2), we can derive the equilibrium gene frequencies. It can also be shown that a sufficient condition for both alleles to be maintained in the population is
_2 y I:< - -y*(3
2-y
1
-
2-y
It turns out that this is the stability condition for any value of h. Detailed analyses on these points will be discussed elsewhere. EQUILIBRIUM UNDER SELECTION AND MUTATION
When the forces of selection and mutation are balanced, the equilibrium gene frequency is generally low. Therefore, I assume that the frequency of allele A is low so that the relationship /3 > - 2y/(l - -y) holds [see equation (3)]. When 0 < h < 1, equation (2) is approximated by
Ap
-hpq[,y + 3(1 - -y)v/(4p)]
-l-hpq[y + 3(1 - -y)/2],
(4)
since v/p = v/(u + '/2v) 2 with good accuracy. Now suppose that irreversible mutation from A2 to A1 occurs with rate a per generation. Then the change in the frequency of A1 is given approximately by -
-hspq + aq ,
Ap
(5)
where s = y + 13(1 - -y)/2. Note that the range of our interest is /3 > -2y/(I - y) and, therefore, s > 0. At equilibrium
P
(6)
s *
When h = 0, equation (2) reduces to P
-
pq[yp + f3u(-y + 1/2P)] -pq[yp 1 - YP I - /3u(1I - y/p)-pq[,yp +
f3u/2p]
+
flu(-y + 1/2p)] (7)
.
In equation (7), the quantity u/2p cannot be ignored. From equation (1.1), '
p2(1
-
/)(I
-
flU/p)
.
(8)
When the frequency of allele A1 is low, the change in the frequencies of A1A, is also small, so u with good accuracy. Hence, from equation (8),
u'
-
U
p
p(l A
-
y)
(9)
412
YOKOYAMA
Therefore, from equations (7) and (9),
.\p
-
-p2q[-y + (3(1
-
-Sp2q sy)/2]
(10)
Thus, under selection and mutation,
'AP
-
-sp2q + aq
(11)
= 0.5 (see Nei [21]). Thus, when h = 0.0 and s < 0.5, the mean and variance of gene frequencies can be obtained numerically using equation (14).
413
SOCIAL SELECTION From equations (15) and (16), it is easy to compute the mean (j5) variance (VP) of p: A
P= A
P=
a
and hs
Vp
=
a
~~~4Ne(/is)2
F(2N ea+½) \12-Nis(2Nea)
an d P
(f1)2
= S
for/h#0, 0
(17)
= ° for/0
(18)
(see Nei[21]). The equilibrium frequency of recessive genes (18) can further be approximated by \la-Is for large Ne and a l,2rNe/s for small Ne [21]. Thus when A, allele is dominant or intermediate (0 < h < 1), the equilibrium gene frequency in a finite population is identical with that in an infinitely large population. When Al allele is recessive (h = 0), however, the equilibrium gene frequency is much smaller in a small population, as is well known for lethal genes. Equations (17) and (18) show that the variance of the gene frequency at equilibrium is inversely related to the effective population size. These analyses also show that the effect of social selection on the equilibrium gene frequency cannot be ignored, because both the mean and variance of depend on s = y + /3(1 - y)/2. BothjA and VP increase when /8 decreases. The number of mutants that are introduced into the population in each generation is 2Na, where N is the actual population size. This is much smaller than unity for a small population. Even if the number of new mutants is larger than unity, they may be lost from the population because of random genetic drift. This may happen often for isolated populations. It is, therefore, of interest to consider the probability that the mutant gene is temporarily lost from the population. The stationary distribution of gene frequencies is useful for this purpose, because the probabilityf0, that A l allele is temporarily lost from the population may be given by 0Jl/(2N)/(p)dp [17]. As an example, consider the case of h # 0 and also assume that Ne = N. Then, using equation (15), A
- (4Nhs)4Na
fo F(4Na)
M
e-4Nhspp4Na-l dp 0
=
1
eF(4Na)
fhs et t4Na 0
I
(19) e(9
To see the effect of social selection on this probability, let us simply assume that 4Na = 1. Then,fo = 1 - exp (-2hs). When -y = 0.01 and h = 1,fo = 0.0295, 0.1122, and 0.4025 for ,/ = .01, . 1, and .5, respectively. Thus the probability that the population is monomorphic increases as the value of /3 increases, as expected. The same trend holds for recessive genes. Similarly, it is easy to see that the value offo decreases as N increases. DISCUSSION
In the social selection model, the fitness of an individual is described by both individual selection in the usual sense (y) and the social effect of a trait (/8). One special model of the fitness modification of an individual by an affected parent under random mating is presented here. As already noted, we previously have considered the population dynamics of rare lethal genes under social selection without assuming random mating [4, 8, 9]. In these papers, social selection is modeled by considering that whenever normal individuals have at least one affected nuclear family member the chance of successful mating or producing progeny is reduced. The results obtained in these papers agree with those presented here: that a low mutation rate, a small population size, and strong social selection along with positive value of /8 lowers the equilibrium frequency of lethal genes.
414
YOKOYAMA
In the development of the social selection model, the clear distinction between the concept of social selection and that of kin selection is needed. In the equivalent version of the kin selection model, the fitness loss of an altruistic parent cannot be determined until he or she has offspring, and, therefore, the usual viability selection cannot be considered, but the social selection model includes viability selection. Because of this difference, it is harder to maintain a polymorphism under social selection than under kin selection. Stability analyses also show the same result, which will be presented elsewhere. As already noted, when the frequency of mutant genes is small or /3 > -2y/ (1 - My), the relative fitnesses of A1A , AIA2, and A2A2 individuals are given by 1 - s, 1 - hs, and 1, respectively. Thus, the present social selection model can easily be extended in many directions by using available theories in population genetics. It is possible to consider the gene frequency not only in equilibrium populations but also in transient populations. The present analyses, however, add other difficulties in the interpretation of the data. Even for a simple situation like the present one, the estimation of s can be done correctly only by considering family structure. Furthermore, even if the value of s is estimated correctly, it cannot be used for other populations without checking their social structure. In the present model, random mating is assumed. If this assumption is relaxed and complexity in the mating system is introduced, a new formulation of N, is required. Theoretical analysis of social selection is still in its initial stage so that no one has attempted to systematically measure the consequences of social factors on the equilibrium gene frequencies of different diseases in man. Studies of Huntington disease show that individual selection is weak and nearly neutral or slightly favors the heterozygotes. With this fitness pattern, Huntington disease should steadily increase in frequency without social selection. For this disease, the magnitude of 0.2 [6]. For dominant disease, setting h = 1 in equation social impact may be,8/ the (6), equilibrium gene frequency is given by f = a/s. Furthermore, assuming that the heterozygote is selectively neutral, that is, -y = 0, then s = 0.1. Therefore, = 10-4 for a mutation rate of a = 10-5. A more appropriate approach to Huntington disease may be to consider nonrandom mating, but the equilibrium gene frequency obtained is about the same [6]. It should be noted that the mean age of onset of Huntington disease is close to age 30 (e.g., see [22]). Thus, most of the social effects are due to the influence of parents on their offspring. Detection of social effects of schizophrenia may also be of interest, even though most studies are restricted to sib-to-sib fitness interaction. Using the 1961 census of Canada, Buck et al. [5] estimated the loss of the relative fitness of a normal female 0.09. Similarly, Lindelius [4] showed with a schizophrenic sib to be about 0.07 that the reproductive rate of married nonschizophrenic sibs is slightly less than 85% of that of the general population. However, Erlenmeyer-Kimling and Paradowski [23] did not observe such a difference using U.S. data. Since the genetics of schizophrenia are not known, it is not clear what proportion of the differences in reproductive rates are due to social or genetic factors. If schizophrenia is a polygenic trait, the normal sibs of schizophrenics may have phenotypic scores close to that of A
-
SOCIAL SELECTION
415
schizophrenics and, therefore, lower reproductive fitness. Furthermore, schizophrenia seems to be heterogeneous and people's response to the trait may also be variable. Reed and Reed [24] have compared the reproduction of 208 normal persons from sibships that include a retardate to that of 208 normal first cousins from sibships without a retardate. They have shown that the reproductive fitness is the same for the two groups. In this study, mental retardation is defined as an IQ score less than 70. However, the etiology of mental retardation is heterogeneous, and severe mental retardation is more likely to be caused by a chromosomal aberration or by a single Mendelian gene [25]. The number of children per person will reflect genetic and environmental components that contribute to fertility. Furthermore, an individual's mating choice may depend strongly on the severity of the disease in the family. For these reasons, using the extensive data compiled by Reed and Reed [24], we compared the marriage frequencies of normal individuals with or without mental retardates in their nuclear families [7]. The retardates were divided into three classes, that is, IQ scores of less than 20, 20-49, and 50-69. We have shown that marriage frequency of a female decreases significantly when both parents are retarded (IQ scores are 50-69), and that of a male is also reduced when at least one parent is retarded or when his sib is severely retarded (IQ score is less than 20). This analysis suggests a significant difference in the marriage frequencies of the two sexes in the families of retardates. If different levels of severity among retardates is neglected, the effect of social impact due to the trait becomes small. The maintenance of elevated incidences of different diseases in different ethnic groups has often been studied, and social selection may be important in understanding this problem. As an example, consider the albinism that is a recessive genetic disease. The incidence of albinism is generally low in most human populations and has been estimated at about one in 20,000 individuals in various European countries [26]. But, the incidences of albinism among Amerindians in the southwestern United States or Panama and among the Brandywine triracial isolate in Maryland are surprisingly high. For example, observed frequencies range from one in 143 to one in 3,750 [27, 28]. For Hopi Indians, Woolf and Dukepoo [29] proposed a ''cultural selection"' hypothesis to explain the extremely high frequency of the disease. According to them, albino men stay at home because of their sensitivity to sunlight, while other men leave for work, and, thus, the former have a greater chance of passing their genes onto the next generation. However, Witkop et al. [30] showed that "cultural selection" is not a workable hypothesis. Our preliminary theoretical studies of this trait show that when the selective disadvantage of an albino is small a small amount of fitness gain due to parental care can enhance the frequency of albino genes considerably [31]. All of these examples indicate that the fitness of an individual is determined not only by his or her own genotype or phenotype but also by the presence of other affected family members. If this is generally true, it is important to determine the directions and magnitudes of the social effect of different diseases. So far, this kind of study has been neglected, but it seems to be essential for assessing mutational damage in human populations.
416
YOKOYAMA
ACKNOWLEDGMENTS Comments by Drs. K. Morgan, T. Reich, and the reviewers were greatly appreciated. REFERENCES 1. REED TE, NEEL JV: Huntington's chorea in Michigan. 2. Selection and mutation. AmJ Hum Genet 11:107-136, 1959 2. WALLACE DC: Huntington's chorea in Queensland. A not uncommon disease. Med J Aust 2:299-307, 1972 3. SHOKEIR MHK: Investigation of Huntington's disease in the Canadian prairies. II. Fecundity and fitness. Clin Genet 7:349-353, 1975 4. LINDELius R: A study of schizophrenia. Acta Psychiatr Scand[Suppl] 216:1970 5. BUCK C, HOBBS GE, SIMPSON H, WANKLIN JM: Fertility of the sibs of schizophrenic patients. Br J Psychiatry 127:235-239, 1975 6. YOKOYAMA S, TEMPLETON A: The effect of social selection on the population dynamics of Huntington's disease. Ann Hum Genet 43:413-417, 1980 7. YOKOYAMA S, RICE JP, YOKOYAMA R: The effect of social selection due to mental retardation on the marriage frequency of normal individuals. Submitted for publication 8. YOKOYAMA S: The effect of social selection on population dynamics of rare deleterious genes. Heredity (Lond). In press, 1980 9. YOKOYAMA S: The effect of variable progeny size and social selection on population dynamics of rare lethal genes. Soc Biol. In press, 1980 10. HAMILTON WD: The genetical evolution of altruistic behavior. I, II. J Theor Biol 7:1-52, 1964 11. LEVITT PR: General kin selection models for genetic evolution of sib altruism in diploid and haplodiploid species. Proc Natl Acad Sci USA 72:4531-4535, 1975 12. MATESSI C, JAYAKAR SD: Conditions for the evolution of altruism under Darwinian selection. Theor Popul Biol 9:360-387, 1976 13. CHARLESWORTH B: Some models of the evolution of altruistic behavior between siblings. J Theor Biol 72:297-319, 1978 14. WADE MJ: Kin selection: a classical approach and a general solution. ProcNatlAcadSci USA 75:6154-6158, 1978 15. YOKOYAMA S, FELSENSTEIN J: A model of kin selection for an altruistic trait considered as a quantitative character. Proc Natl Acad Sci USA 75:420-422, 1978 16. CAVALLI-SFORZA LL, FELDMAN MW: Darwinian selection and "altruism." Theor Popul Biol 14:268-280, 1978 17. CROW JF, KIMURA M: An Introduction to Population Genetics Theory. New York, Harper & Row, 1970 18. WRIGHT S: The distribution of gene frequencies in population. Proc NatlAcad Sci USA 23:307-320, 1937 19. ROBERTSON A: Selection for heterozygotes in small populations. Genetics 47:1291-1300,
1962 20. NEI M: The frequency distribution of lethal chromosomes in finite populations. Proc Natl Acad Sci USA 60:517-524, 1968 21. NEI M: Molecular Population Genetics and Evolution. New York, North Holland/Elsevier, 1975 22. REED TE, CHANDLER JH: Huntington's chorea in Michigan. I. Demography and genetics. Am J Hum Genet 10:201-224, 1958 23. ERLENMEYER-KIMLING L, PARADOWSKI W: Selection and schizophrenia. Am Nat 100: 651-665, 1966 24. REED EW, REED SC: MentalRetardation: A Family Study. Philadelphia and London, WB Saunders, 1965 25. MORTON NE, RAO DC, LANG-BROWN H, MACLEAN CJ, BART RD, LEW R: Colchester revisited: a genetic study of mental defect. J Med Genet 14:1-9, 1977
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26. PEARSON K, NETTLESHIP E, USHER CH: A Monograph on Albinism in Man. Drapers' Company research memoirs. Biometric Ser. 6, 8, and 9. London, Dolan, 1911-1913 27. WOOLF CM: Albinism among Indians in Arizona and New Mexico. Am J Hum Genet 17:23-35, 1965 28. KELLER CE: Caribe Cuna moon-child and its heredity. JHered44:163-171, 1953 29. WOOLF CM, DUKEPOO FC: Hopi Indians, inbreeding, and albinism. Science 164:30-37, 1969 30. WITKOP CJ, NISWANDER JD, BERGSMA DR, WORKMAN PL, WHITE JG: Tyrosinase positive oculocutaneous albinism among the Zuni and the Brandywine triracial isolate. Biochemical and clinical characteristics and fertility. AmJPhysAnthropol36:397-406, 1972 31. YOKOYAMA S, MORGAN K: Social selection as a probable mechanism for the high frequency of albinisms among Amerindian populations. In preparation
SIXTH INTERNATIONAL CONGRESS OF HUMAN GENETICS will be held in Jerusalem, Israel, September 13-18, 1981. Application forms available from the Congress Secretariat: P.O.B. 16271, Tel Aviv, Israel.
