Jul 29, 1982 - model with modified fitness functions (e.g., see Yokoyama [13]). All of these ... and two affected parents are given by 1, 1 - w, and (1 - w)2, respectively. Thus ... where a = P - 2cD, 13 = Q+ 1/2cD - w Y5, y = R + ½12cD, 8 = S - 2cD ... + 1/2(X3 + X6 + X9), S = X10 + 1/2(X4 + X7 + Xg), D = .... X - v(1 - s)Is. (8).
Am J Hum Genet 35:675 -686, 1983
Social Selection in Human Populations: Fitness Modification of Offspring by an Affected Parent at Two Loci SHOZO YOKOYAMA
SUMMARY
The effect of social reaction to a disease at one locus (the B locus) on the frequencies of deleterious genes at that locus and another locus (the A locus) has been studied. Social reaction is formulated so that affected parents at the B locus reduce or increase their offsprings' fitnesses. It has been shown that the genotype frequencies at the two loci depend strongly on the social reaction, but, because the magnitude of linkage disequilibrium is very small, the gene frequency at each locus can be treated as if each locus were segregating independently. Consequently, the model can be extended so that many rare diseases can be considered
simultaneously.
INTRODUCTION
Disease incidences in a population depend on the etiology of the disease, the fitness of an individual, and the demographic structure of the population. In the studies of Huntington disease, Wallace [1] pointed out that the fitness of an individual is determined not only by the disease, but also by other factors such as cultural and social reactions to the disorder. At present, such social effects have been studied at least for three traits: Huntington disease [2-4], schizophrenia [5-7], and mental retardation [8, 9]. Most of these studies demonstrate that two individuals with the same genotype (or phenotype) have different fitnesses due to cultural and social reactions to the presence of affected family members. Social selection consists of two fitness components: (1) an intrinsic deleterious effect of a disease that is applicable only to an affected individual and (2) a fitness modification of both affected and normal individuals due to sociocultural reaction to an affected family member or more distant relatives. Mathematical analyses Received July 29, 1982. This research was supported by U.S. Public Health Service grants GM-28672 and MH-31302 with Washington University. 1 Department of Psychiatry, Washington University School of Medicine and The Jewish Hospital of St. Louis, 216 S. Kingshighway, St. Louis, MO 63110. © 1983 by the American Society of Human Genetics. All rights reserved. 0002-9297/83/3504-0013$02.00
675
676 YOKOYAMA show that the change in gene frequencies and the equilibrium frequency are affected strongly not only by the intrinsic deleterious effect of a disease but also by the behavioral response to the presence of a family member with the disease [10-15]. Furthermore, when the frequency of deleterious genes is low, the social selection can be treated approximately as the traditional individual selection model with modified fitness functions (e.g., see Yokoyama [13]). All of these studies consider various single-locus models, and virtually nothing is known about the effect of the sociocultural reaction to genetic diseases when multiple loci are considered. Here I shall study a two-loci model. The results obtained are of considerable importance to understand interactions of two diseases with respect to their incidences. A TWO-LOCI MODEL
Assume that alleles A1 and A2 are segregating at the A locus and alleles B1 and B2 are segregating at the B locus. Let p and q be the frequencies of A2 and B2 in adults, respectively. Let c be the recombination fraction between the two loci. Let us denote that the frequencies of the four gametes, A1B1, A1B2, A2B1, and A2B2, by P, Q, R, and S, respectively. Furthermore, we denote the respective 10 genotypes, AIB11A1B1, A1B1/A1B2, AIB1A2B1, A1B1/A2B2, A1B2/A1B2, A1B2/A2B1, A1B2/A2B2, A2B1/A2BI, A2B21A2B2, and A2B2/A2B2, in adults, by X1, X2, . . X10, respectively. in the following, we study the fitness modification due to both social reactions to the disease at the B locus and intrinsic deleterious effects of the disease at the two loci. The analyses can be extended to the case where the social reactions stem from both loci (see DISCUSSION). For simplicity, we assume that the trait at the B locus is recessive (i.e., only B2B2 individuals develop the disease). For the A locus, we assume that A2A2 always develops a disease and that AIA, is normal. Furthermore, h is the degree of dominance of the trait, so that, with probability h, A1A2 will be affected by the disease. Let s and t be the respective magnitudes of fitness loss of affected individuals due to the traits at the A and B loci. Let us also assume that the fitness of an individual is determined multiplicatively by the two loci. Here we are concerned only with the effect of social reaction to affected parents. The relative fitness of an individual with two normal parents, one affected parent, and two affected parents are given by 1, 1 - w, and (1 - w)2, respectively. Thus, roughly speaking, w is the magnitude of fitness loss when an individual has one affected parent. After recombination, mating, and selection, the frequencies of 10 genotypes in the next generation are given by: .
Xl' = a2/w, X2' = 2a4/W, X3' = 2(1 - hs)aoy/W, X4'I= 2(1 - hs)otb/W,
SOCIAL SELECTION IN HUMAN POPULATIONS
X5'
677
= (1 - t)12/W,
X6' = 2(1 -hs)p3ylW, X7' = 2(1 -hs)(1 - t)P/W, X8' = (1 - s)2IW, X9 f= 2(1 - s)-y8IW, X10' = (1 - s)(l -t)82/W
()
where a = P - 2cD, 13 = Q + 1/2cD - w Y5, y = R + ½12cD, 8 = S - 2cD - W YIO, P = X1 + 1/2(X2 + X3 + X4), Q = X5 + 1/2(X2 + X6 + X7), R = X8 - X9, Y5 = X5 + 1/2(X3 + X6 + X9), S = X10 + 1/2(X4 + X7 + Xg), D = + X7/2, Y10 = X10 + X7/2, and W = (a + 13 + y + 5)2- s{2h [(a + 13)(y + 5) - tpa] + (y + 5)2 - t 82} _ t (a + 8)2. In equation (1), D is the product of the coupling gametic frequencies minus that of the repulsion gametic frequencies and is known as a measure of linkage disequilibrium (e.g., see Lewontin and Kojima [16]). In general, the frequency of deleterious alleles in a population is low; therefore, I consider irreversible mutations from AI to A2 and BI to B2. I assume that these mutation rates at the two loci are equal and denote them by v. Then, the change in the genotype frequencies due to selection and mutation are given by:
AX1 = [a2 - (1 + 4v)X W]/W AX2 = [2at- (X2 - 2vXI + 3vX2)W]IW AX3 = [2(1 -hs)ay - (X3 - 2vX1 + 3vX3)W]/W AX4 = [2(1 -hs)otb - (X4 -VX2 -VX3 + 2vX4)W]/W (2) AX5 = [(1 - t) 12 - (X5 - vX2 + 2vX5)W]/W AX6 = [2(1 -hs) Ady- (X6 -V X2 -V X3 + 2v X6)W]IW AX7 = [2(1 -hs)(1 - t) 8 -(X7 -v X4 - 2vX5 - VX6 + VX7)W]/W AX8 = [(1 -)sY2 - (X8 -V X3 + 2v X8)W]IW AX9 = [2(1 - s) y - (X9 - vX4 - vX6 - 2vX8 + VX9)W]/W AX10 = [(1 -s)(l - t) 82 - (X10 - VX7- VXg)W]/W . EQUILIBRIUM UNDER SELECTION AND MUTATIONS
Equation (2) is too complex to derive the exact equilibrium frequencies of different genotypes analytically, but approximate equilibrium frequencies can be obtained if we consider the cases h > 0, h = 0, and h < 0, separately. Thus, we approximate the system (2) considering (i) dominant, (ii) recessive, and (iii) overdominant diseases at the A locus.
YOKOYAMA
678
Dominant Diseases (h > 0) Without any interaction between the loci, it is expected that p vlhs and q v7t (e.g., see Crow and Kimura [17]). Thus, ignoring higher-order terms than v2, system (2) is reduced approximately to -
{X2 [/4X2
AX1
V/2X3
+
+
X5 + t(WX2 + X5
l/X22)
-
t)X5]
- hsX3 + 4v - W(2 +
AX2
4v + 2w X5}IW, [X2(-2X5 - '/2X2 - '/2X3 + hsX3 + ¼14tX22- 5v + 3wX5) + 2X5 + X6 + cD + 2v - 2WX5]/W,
AX3
{-X2
-
X5
+ hs
X6
-
[/2 (1
-
(X3
hs) X3
X4
+
+
2v]
+
X6) - cD
+
X6
-
+ cD
-hs (X3 + X6 + cD) + 2v}/W,
[-hsX4 (1 hs)cD + vX21/W, {X2 [(1 -t)(¼14X2 + X5 WX5) + V] {X2 [(1/2 (1 hs) X3 + VI -X6/W,
AX4
-
-
AX5
-
AX6
-
X5}IW,
-
where W 1 - hs (X3 + X4 + X6) - tX2 (AX2 + Xs) - 2w Xs(1 - '2tX2). Setting equation (3) equal to zero, it is easy to obtain the following equilibrium genotype frequencies -
X2 A/4v/[t - w(l - t)] X3- ~2vlhs, X4 ~ (vlhs) V4v/[t + w(1 -t)] Xs vOl -t)/[t + w(1 - t)], X6 ~ (v/hs) V'4v/[t + w(1 -t)] X1 1 X2 X3-X4 X5-X6, -
X8
X7
X9
Xl0
=
(4)
°
It should be noted that equation (3) implies D = 0. But, this simply says that the magnitude of linkage disequilibrium is smaller than vX§. Furthermore, using formula (4), the approximate equilibrium gene frequencies p and q can also be obtained.
pA= R q = Q + S
-
+
S~ (X3
+
X4 + X6)/2 vlhs.
X5 + (X2 + X4 + X6)12 '-V/[t + w(1 - t)]
(5-1) 2 (5-2)
679
SOCIAL SELECTION IN HUMAN POPULATIONS
These formulas are very close to the results from the traditional single-locus model. The only difference is the value of qc, which contains the effect of social reaction to the disease. Thus, two loci A and B can be treated almost independently. To see this, it may be of interest to derive the change in the gene frequencies at each locus. From equation (3),
1/2(AX3
Ap
+ AX4 + AX6)
-hsp + v,
AX5
Aq
+
(6-1)
1/2(AX2 + AX4 + AX6)
t)]q2
- [t + w(l -
+ v,
(6-2)
since q X2/2. Equation (6-2) implies that when w 0, the approximate formulas (5-1) and (9-1) work very well. From table 1, when h = 0. 1, f = 0.0093 and 0.000498 for s = 0.01 and 0.2, respectively, whereas the respective approximate frequencies are 0.01 and 0.0005 from formula (5-1). When, h = 0.0, p = 0.0316 and 0.00707 in table 1, which are identical to vl7s in formula (9-1). An approximate formula for j in formula (12) is slightly different from the values in table 1. When h = -0.05, j = 0.0603 (for s = 0.01) and 0.0464 (for s = 0.2), whereas j5 2h'I[2 + (2 + s)h'] = 0.0476 (for s = 0.01) and 0.0474 (for s = 0.2). However, the differences are small. Thus, the present approximate approach works extremely well. DISCUSSION
It has been shown that the change in the genotype frequencies depends on the magnitude of social reaction to the disease, whereas the gene frequencies at the two loci are not affected by each other. Thus, the gene frequency at the A locus is not affected by social reaction to the disease at the B locus. This invariant property of the equilibrium gene frequency can be seen in formulas (6-1), (101), and (13-1). These approximations have been obtained by assuming that the values of p and q are low. It is not clear whether this property holds when the deleterious alleles are not so rare.
SOCIAL SELECTION IN HUMAN POPULATIONS
683
To check this point, let us study the genotype and gene frequencies in a population during the process of achieving a new equilibrium from the old one due to the change in trend of sociocultural reaction to the disease. Considering recessive diseases, two sets of changes in the sociocultural reaction are considered: (1) (s = t = 0.01 and w = 0.20) (s = t = 0.01 and w = -0.05) and (2) (s = t = 0.20 and w = 0.20) -* (s = t = 0.20 and w = -0.10). In these numerical analyses, it is assumed that c = 0.01 and v = 10-5. Thus, a rather tight linkage is assumed. For the first case, w < -t/(l - t) in the new environment, and, therefore, the value of q will become unity eventually [see equation (10-2)]. In this example, drastic changes occur in X5 and X7. The former is changed from .0004 to .9378 and the latter from .004 to .0612, as we can see in table 2. Note that X7 is the frequency of A1B2/A2B2, which contains A2 allele. Nevertheless, the frequency of A2 alleles is kept as .0316 during the entire process. For the second case, the increase in gene frequency q is not much, but the values of p again remains constant throughout the process. Therefore, the changes in the gene frequencies at the two loci are almost independent from each other. The numbers of generations required for these changes are 2,316 and 3,391 for the first and second cases, respectively. We assumed that only one of the two loci causes social reactions that lead to the fitness modification of offspring. The analyses can easily be extended to the case where diseases at both loci create fitness modification of the offspring due to social reactions. As an example, consider recessive diseases. From formula (8), the four gamete frequencies at equilibrium are given by
QVv/I[t+w(1 -t)], R
S
-
VS
v/\/s[t + w(l - t)]
and PA 1 -Q - R - S. Let us now consider that disease at the A locus also induces social reaction so that the fitness of an offspring is reduced by the amount u when one parent is affected at that locus. Then, the gamete frequencies are modified to:
\/v/[t + w(l - t)], R Vv/[s + u(1 vlV'[s + u(l - s)][t + w(1 -t)], Q
S
1 - Q - R - S. Suppose that the individual selection and social and P reaction are the same for the two loci, that is, u = w and s = t, then X2 2Vvl7F, X3 2\/vlF, X4 2vIF, X5 v/F, X6 2vlF, X7 2(vlF)VvF, X8 v/F, X9 2(vIF)V1771, and X10 (vF)2, where F = t + w(l - t). Under this condition, the frequency that an individual is affected at at least one of the loci is given by X5 + X7 + X8 + Xg + X10 2vIF. -
=
-
-
=
684
YOKOYAMA
o
*z
z O
m
OOo
X
o
al
C:0
00
oc
z
0000
-
N000 0 0 00
00000
z zaf
o
o
0 000
-
C 0 0C
F~~~~~~~71
CN00
