MASATOSHI Division of Genetics, National Institute of Radiological Sciences, Chiba

Transcription

1 JAP. J. GENET. Vol. 39, No. 1: 7-25 (1964) EFFECTS OF LINKAGE AND EPISTASIS ON THE EQUILIBRIUM FRE- QUENCIES OF LETHAL GENES. II. NUMERICAL SOLUTIONS Received May 27, 1964 MASATOSHI NET Division of Genetics, National Institute of Radiological Sciences, Chiba As indicated elsewhere (Nei 1964) it is not easy to obtain general algebraic solutions for the equilibrium frequencies of lethal genes with linkage and epistasis unless there is linkage equilibrium for different loci. Accordingly, numerical solutions have been undertaken with the aid of an electronic computer. In addition with these solutions the changes in population fitness and linkage disequilibrium under selection have been examined. Several unforeseen results are obtained. The results of this investigation have been orally presented at a meeting of the Genetics Society of Japan (Nei 1963). METHODS Consider two loci each with two alleles, A, - A2 and B, - B2. Then, there occur four types of gametes, i. e., AB,, A,B2, A2B and A2B2. If the frequencies of these four types of gametes are represented by P, Q, R, and S, respectively, the frequencies of the nine possible genotypes in a randomly mating population are given by the appropriate terms in the expanision of (P+ Q+ R+ S)2. We assume that A2 and B2 are lethal and cause the individuals homozygous for either or both of them to die. Thus, only four genotypes, i, e., A,A,B,B A,A2B,B A,A,B,B2, and A,A2B,B2, survive. We denote the selective values of these four genotypes by a, b, c, and d, respectively. Under the condition specified above, the gamete frequencies in the (t+l)th generation are expressed in terms of the gamete frequencies in the preceding generation as follows (cf. Nei 1964) : =[P(aPt+cQ,+bR,+dS)-rdD,]/`V Q~+, = I Q, (cp, +dr,) +rdd]/w, Rt+,=[Rt (bpt+dq)+rdd~]/w, S~+,= [SdP~-rdD,i/W where r represents the recombination value between the two loci, D, the linkage disequilibrium in the tth generation or PS-QR,, and \V the mean selective value or P, (ap, +2 dr, +2 cq) +2 d (PS, +QR3. The equilibrium gamete frequencies for a given set of genotype fitnesses are then obtained numerically by solving the following simultaneous equations:

2 8 M. NEI P+-P=0 R+-R=0 Q~+~-Q~=O S+-S=0 This method was employed by Lewontin in a recent paper (1964) for non-lethal genes. But it is of interest not only to have equilibrium frequencies but also to see how the gamete and gene frequencies approach the equilibria. We, therefore, examined the complete process of gamete and gene frequency changes by using the formulas given above. Note that the gene frequencies are obtained from the gamete frequencies by the following relations: p~=p+q, q =R+S p2=p+r q2=q+s where p q~, p2, and q2 stand for the frequencies of genes A A2, B and B2 respectively. Starting from a given set of initial values of gamete frequencies, computation was made until the population reached an apparent equilibrium, i. e., until the rates of change per generation of P, Q, R, S, D, and W all became less than lor4, except in a few cases where the critical amount of change was set to be 10 or Thus, the equilibrium values obtained under this criterion may not be the true values in the strict sense, but they should not be far therefrom, because different sets of initial values of P, Q, R, and S gave the same or similar values for a given set of fitness and recombination values. For these computations a digital computer known as HIPAC 103 was used. As the values of a, b, c, and d, it would be desirable to use those values which have been obtained experimentally. For example, Oshima (1963) showed in an experiment with Drosophila melanogaster that the viability of double lethal heterozygotes in the coupling phase is higher than that of single lethal heterozygotes by about 7 percent and than that of normal heterozygotes by about 5 percent. However, it was found from the preliminary computation that if we use viability difference of this order of magnitude, a long time is required for a population to reach an equilibrium, and thus computation becomes costly. are given in Table 1. It may be noted that our primary purpose is to examine the effects of linkage and epistasis on the equilibrium frequencies of lethal genes. Rather extreme fitness differences may reveal the effects clearer. The sets of selective values given in Table 1 are divided into three groups according to the value of ad-bc. In the first group ad-bc is 0, where linkage equilibrium is maintained permanently if the original population is in linkage So, in most cases we used those values which Table 1. Selective values employed

3 EQUILIBRIUM FREQUENCIES OF LETHAL GENES. II. 9 equilibrium (Nei 1964). In the second group ad-bc > 0 and in the third group ad-bc <0. As will be seen later, the sign of ad-bc conforms to the sign of the linkage disequilibrium maintained in the equilibrium population. The last column of Table 1 shows the dominance x dominance epistatic comparison when the population is in linkage equilibrium. Other orthogonal comparisons such as additive and additive x additive comparisons are all dependent on the gene frequencies, so that they are not listed. Five different recombination values were used, namely, 0.5, 0.2, 0.1, 0.05, and 0.01; in some cases only two levels, i. e., 0.5 and 0.01 were employed. The sets of initial values of P, Q, R, and S are given in Table 2, together with Table 2. Initial values of gamete and gene frequencies those of q q2, and D. These sets of initial values are divided into two groups. In the first group q, = q2 = 0.5, while in the second q, = q2 = The second group was set up to simulate the probable population dynamics of lethal genes in nature. In nature any lethal gene must have arisen by mutation, and thus the initial gene frequency is necessarily small. If the new mutant gene shows overdominance, then the gene is able to increase in frequency. The first group of initial values was employed to investigate the effects of linkage and epistasis more clearly, just as Drosophila experiments with cage populations. It should be noted that computations were not made for all combinations of the fitness values and initial values of gamete frequencies given in Tables 1 and 2. In the following the set of fitness values (a b c d) will be called the fitness vector and denoted by a. RESULTS (1) Changes in gamete and gene frequencies towards equilibria (i) a=( ); a=( ) If the fitness vector is ( ), gene action is complementary in the classical sense. In this case ad-bc=0, so that if the original linkage disequilibrium, Do, is 0, then there is no stable equilibrium, since the equilibrium frequencies q, and q2 are given by

4 10 M. NET q ' _d-c 2d--c q2 2 d_-b d -b respectively, and d = c = b in this case (Nei 1964). Thus, the will be eventually eliminated (Figure 1). In this situation and gene frequencies are independent of the recombination loci. lethal genes A2 the changes in value between the and gamete B2 two Fig. 1. Changes in gamete and gene frequencies for a = ( ). p=1-q. If, however, there is a negative linkage disequilibrium, the average fitness of A,A2 or B,B2 can be larger than that of AA, or BB, since, if Q=R or p,=p2=p and q~=q2=q, WA,A,-WA,A2-WB,B,--WB,B2=D (pq-d)/p2q. So, computation was made with the initial values of Po=0, Q0=0.5, Ro=0.5, So=0, and Do= with r=0.01. The results of this computation are given in Figure 1. Inspection of this Figure shows that the shapes of the curves of gamete and gene frequency changes towards equilibria (fixation) are considerably changed ; the rates of change per generation in gamete and gene frequencies in early generations are smaller than those of the case of Do=0. This reduction in the rates of gamete and gene frequency changes seems to be caused by the marginal overdominance mentioned above. Nevertheless, this overdominance does not appear to be able to form a stable equilibrium except the case of r=0. Probably the breakdown of linkage disequilibrium is too large to maintain the overdominance permanently. It is worthwhile to note that in the case of Do= the rates of change in gamete and gene frequencies in later generations are so large, that at about the 100th generation the gamete and gene frequencies are almost the same for both Do=0 and Do= If the fitness vector is ( ), the equilibrium frequencies of A2 and B2 under linkage equilibrium are q, =1 /3 and 42=0 respectively. Numerical computation with several sets of recombination values and initial values of P, Q, R, and S gave nearly the same result as in the case of a=( ), although q, converged to 1/3 as expected and generally reached the equilibrium value more rapidly than q2. In one

5 EQUILIBRIUM FREQUENCIES OF LETHAL GENES. II. 11 case, however, a very interesting result was obtained. Namely, when r=0.01, Po=0.99, Qp=0, Ro=0, and So=0.01, the value of S gradually increased in early generations up to 0.28 at the 10th generation and then began to decrease very slowly, becoming almost 0 at the 400th generation (Figure 2). In conformity with this, the frequency of lethal gene B2, q27 increased in early generations from 0.01 to 0.3 and then began to decrease. The peculiar behavior of S and q2 stems from the fact that in early generations ab gemetes have a selective advantage over AB gametes, since they unite with AB gametes more frequently than with ab themselves at fertilization, but in later generations ab gametes increase by recombination and these have a selective advantage over ab gametes. Fig. 2. Changes in gamete and gene frequencies for a=( ), P~=1--q~. p2=1-q2. (ii) a=( ) In this case q, and q2 are both 1 /3 under linkage equilibrium. If Do * 0, the resultant equilibrium values of gamete and gene frequencies are dependent on the sign of Do and the recombination value. When r=0.5, the equilibrium values are the same as those for Do=0 irrespective of the initial value, but as the recombination value decreases the frequencies of lethal genes increase, as will be examined in detail in the next section. The form of approach of gamete and gene frequencies towards equilibria is also influenced by the recombination value. Figure 3 shows the change in gamete and gene frequencies for r=0.5 and 0.01 in the case of D0=0.25. It is seen that the rates of change in both gamete and gene frequencies are much greater for r=0.5 than for r=0.01 in the early generations and there are great differences between the equilibrium values for the two cases. Figure 4 again shows the effect of recombination value but in this case for Do= If r is of the order of 0.1, the equilibrium value of P is larger than that of Q, but if r is as small as 0.01, it becomes much smaller than this. The number of generations required for the population to reach an (apparent) equilibrium is also influenced by the values of Do and r. If D =0.25

6 12 M. NET Fig. 3. Changes in a=( ) Fig. 4. Changes (1224) Fig. 5. Changes a=(1 2 gamete and gene Do=.25, p=1-q. in gamete and gene D0=-.25. p =1-q. in 2 4) gamete r=.01 and. p= gene 1-q. frequencies for q=q.=q2 frequencies for a= q=q~=q2 frequencies for q=q1 =q2. and r=0.01, the number of generations is 12, while if Do= and r=0.1 it is about 140. In the above examples the initial gene frequencies were 0.5 for both loci, though linkage phase was not the same. The effect of initial gene frequencies on the equilibrium gamete and gene frequencies does not appear to be so important as the linkage phase or the sign of Do, but they affect the form of the changes in gamete and gene frequencies and the number of generations required for approaching equilibria (cf. Figure 5). (iii) a=( ) The value of ad-bc is not 0 in this case, so that linkage equilibrium can not be maintained permanently under selection (Nei 1964). Therefore, even if Do=0, the equilibrium gamete and gene frequencies are not independent of the recombination value. This situation can be observed from Figure 6, where the changes in gamete and gene frequencies for r=0.5 and 0.01 are given. Further, the form of the changes and gene frequencies in gamete is not the same for the two cases; with r=0.5 these frequencies directly converge to their equilibrium, but with r=0.01 they pass the minimum or maximum point before they reach an equilibrium. Thus, for example, q, = q2 = q

7 EQUILIBRIUM FREQUENCIES OF LETHAL GENES. II. 13 Fig. 6. Changes a=(1 1 in gamete 1 2); Da=O. and p=1-q. (=1-p) first decreases to about 0.25 and then begins to increase, approaching the equilibrium value 0.33 at the 27th generation. In the case of a=( ) the equilibrium gamete and gene frequencies were affected by the sign of Do. In the present case, however, they are not affected and seem to depend only on the recombination value. Initial gene frequencies do not affect the equilibrium value either. The initial linkage phase and gene frequencies, however, influence the form of the changes in gamete and gene frequencies (Figure 7). If Do is negative and r is small, gamete and gene frequencies change little in the early generations. (iv) a=( ); a=(21 1 4) It may be anticipated that the fitness vector a=( ) forms no stable equilibrium unless r=0, since even under a=(1 111) no such equilibrium was obtained. This anticipation proved to be true with the computation with several sets of initial values of P, Q, R, and S. However, it was found that if the initial linkage phase is of repulsion and the recombination value is as small as 0.01, the rates of change in gamete and gene frequencies are very small in the early generations and become large just before reaching the equilibrium (Figure 8). This indicates that if linkage is tight in the repulsion phase the lethal genes are retained in the population time before they are eliminated. gene frequencies q =q1=q2 Fig. 7. Changes in gamete and gene frequencies for a= (1112);r =.01. p =1-q. q= q~=q2. for a long period of The properties of gamete and gene frequency changes for a=( ) are roughly a mixture of those of a=( ) and a=( ). The only new feature is that an extremely long time is required for the population to reach equilibrium if qo (initial value)=0.01, Do=0, r=0.5 or qo=0.5, D0=-0.25, r=0.01. In these two cases the rates of change in gamete frequencies were very small from the beginning for

8 14 M. NET Fig. 8. Changes in gamete and gene frequencies for a=( ). p=1-q. q=q1=q2. we the and became less than 10 at about the 15th generation. Further computations, however, indicated that the gamete frequencies had not reached equilibrium but were changing with the rate of change of the order of 3 x 10-'. This rate of change gradually increased in later generations, and the population was considered to be changing towards the common equilibrium (see Table 4), though several thousand generations appeared to be necessary for reaching the equilibrium point. (v) a=( ); a=( ) In the foregoing two subsections we considered the case of ad-bc > 0, and now are concerned with the case of ad-bc <0. As was the case with ad-bc > 0, equilibrium gamete and gene frequencies are not affected by the initial linkage phase but by the recombination value. One of the new features in this case is that the number of generations required for the population to reach the equilibrium is longer in the coupling than in the repulsion phase in contrast with the case of ad-bc > 0 (Figure 9). This is owing to the fact that if ad-bc < 0 the gametes in the repulsion phase are favored, so that much time is required to reorganize the linkage phase from coupling to repulsion when loaded in coupling. Figure 9 also indicates that there can be both maximum and minimum points for gene frequencies in this case. (2) Equilibrium gamete and gene frequencies As was shown in the foregoing section, there are no stable equilibria for lethal genes for a=(1 111) and a=( ), even if the assumption of linkage equilibrium is removed. In the case of a=( ) a stable equilibrium is possible only for locus A,-A2, and this equilibrium is determined independently value and initial linkage phase. Fig. 9. Changes in gamete for a=(1 222);r= and gene.01, p =1-q frequencies. q=q.=q2. of the recombination In the case of a= ( ), however, the equilibrium gamete and gene frequencies

9 EQUILIBRIUM FREQUENCIES OF LETHAL GENES. II. 15 are dependent both on the recombination value and initial linkage phase. This dependence is shown numerically in Table 3. As was already indicated, the equilibrium frequencies for A2 and B2 when D0=0 are both 1/3 and independent of the recombination value. When D0*0, the equilibrium frequencies are dependent on r. If r= 0.5, they are the same as those of Do=0, but as r gets small the equilibrium gene frequencies increase, though there is little difference between the values for r=0.5 and 0.2. It is also noted that the rate of increase of equilibrium values due to linkage is not the same for positive and negative values of Do. It appears that the equilibrium values depend only on the sign of Do and not on its absolute value, because computation with several different sets of initial values of qo and Do gave the same equilibrium values as given in Table 3. Table 3. Equilibrium values of gamete and gene frequencies, and population fitness for a = (l 2 2 4) linkage disequilibrium, The equilibrium values of gamete frequencies (P, O, R, and S) also depend on the recombination value and linkage phase. When Do is positive, P and S increase as r becomes small. Consequently, the equilibrium value of linkage disequilibrium, D, increases in a positive direction with decreasing r. On the other hand, if Do is negative, P and S decrease and b increases in a negative direction with decreasing r. If a=( ), the equilibrium values of gene frequencies are independent of

10 16 M. NET the initial linkage phase and determined solely by the recombination value, as was already mentioned. In this case the effects of linkage on the equilibrium gene frequencies are greater than in the case of a=( ), since even with r=0.2 the frequencies are affected appreciably and with r=0.01 the equilibrium values are 54 percent larger than those with r=0.5 compared with the 35 percent increase for a= ( ) (Tables 3 and 4). The equilibrium values of gamete frequencies P and S are increased with decreasing r, while Q and ft are decreased, so that linkage disequilibrium increases in a positive direction. Note that D~0 even with r=0.5. In the case of a=(2 11 4) the effects of linkage on the equilibrium gamete and gene frequencies are almost the same as those for a=( ) and nothing is to be added except that the effects are greater in magnitude in this case (Table 4). Table 4. Equi and librium values of gamete and gene frequencies, 1 inkage population fitness for u=( ) and a=( ) (i) a=( ) disequilibrium, The equilibrium values of gene and gamete frequencies for a=( ) and a = ( ) are again affected by the recombination value and not by the initial linkage phase. In these cases, however, the effects are not so large as in the previous case (Table 5). The gamete frequencies are distorted in the direction of a negative linkage disequilibrium. This distortion, too, is not so great as in the previous case. (3) Linkage disequilibrium The equilibrium values of linkage disequilibrium for various fitness vectors are given in Tables 3, 4, and 5. Inspection of these Tables shows that the equilibrium value is positive if ad-bc> 0, and negative if ad-bc <0, the absolute value increasing with decrease of r. The value of D is not 0 even if Do=0. This was the case also with such fitness vectors as ( ) and ( ), which were used in the preliminary computation. This dependence of f on the value of ad-bc is due

11 EQUILIBRIUM FREQUENCIES OF LETHAL GENES. II. 17 Table 5. Equi and librium values of gamete and gene frequencies, 1 in kage disequilibrium, population fitness for a = (l 2 2 2) and a = (1 2 (i 1 n=(l ) to the fact that if ad-bc > 0 the gametes in the coupling phase are favored under selection and if ad-bc < 0 the gametes in the repulsion phase are favored. The value of i3 for the case of ad-bc=0 is dependent on the values of r and Do. If r=0.5, i3 is 0 irrespective of the value of Do, while if r <0.5, the sign of f conforms to that of Do, i. e., D > 0 for D0> 0, 13<0 for Do <0, and D=0 for Do =0. The dependence of f on the r and Do is again explained in terms of the selective difference between the gametes in the coupling and repulsion phases. The approach of linkage disequilibrium to the equilibrium value is rather complicated. In the case of a = (l 1 1 1) the linkage disequilibrium is broken down very rapidly if r=0.5, but maintained for a considerable time if r is small. For example, if r=0.01, the value of D at the 100th generation was still of the order of for Do=0.25 and of the order of for Do= The breakdown of linkage disequilibrium is slower for the coupling than for the repulsion phase. The change in linkage disequilibrium for a=( ) is shown in Figure 10. It is seen from this Figure that the pattern of approach of D to the equilibrium value is nearly the same both for Do=0.25 and Do= -0.25, though the time necessary for the approach is generally longer for the latter than for the former. If, however, Do is as small as , then D does not necessarily converge directly to the equilibrium value but may pass the maximum point before it converges, as is seen from the case of r=0.5. In the case of a=( ), D reaches the equilibrium value as given in Table 4 without much complication, depending only on the recombination value. The fitness vector a=( ) gives a little different type of change in D. If linkage is loaded in the repulion phase the rate of change in D in early generations is very

12 18 M. NET small and gradually increases in later generations. This tendency is strengthened much more for a=( ), and in this case a high amount of linkage disequilibrium can be maintained for a long time (see the previous section). The linkage disequilibrium for a=( ) approaches the equilibrium value almost directly, if Do is large. However, if this is small and positive, D first increases and then begins to decrease (Figure 11). The fitness vector ( ) gives almost the same pattern of change of D. Fig. 10. Changes positive the case in linkage disequilibrium for a=( ). part refers to the case of D0=.0099 and that of Do= See Table 2. The in chain line in the the negative part Fig. 11. Changes in linkage disequilibrium for a=(1 2 2 the positive part refers to the case of D0= ). and The that in broken the line negative in part the case of Do=.00; q0=.01. See Table 2. (4) Population fitness The equilibrium population fitnesses (mean selective values) for a=( ) and a=(21 1 2) are obviously 1 and 2 respectively and independent of the recombination value, since the population is eventually composed only of genotype A,A,B,B,. The

13 EQUILIBRIUM FREQUENCIES OF LETHAL GENES. II. 19 equilibrium population fitness for a=( ) is also independent of the recombination value because of the fixation of the B, gene. For other fitness vectors, however, the equilibrium values of the population fitness, W, is dependent on the recombination value, as will be seen from Tables 3, 4, and 5. The population fitness increases as r decreases for all cases examined, except the case of a= ( ) with D0=0. In the case of a=( ) W depends on r only when Do * 0, and this dependence is not the same for D0> 0 and Do <0; if r is the same, W is higher when D0> 0 than when Do <0. Further, it is noted that W increases in parallel with q. The increase in population fitnese with decreasing r was also noted by Lewontin (1964) for non-lethal genes. The change in population fitness under selection is of particular interest, since according to Fisher's (1930) fundamental theorem o f natural selection there is no decrease of population fitness. Our computation has, however, indicated that this is not necessarily true for the case where linkage and epistasis are present. Namely, with some fitness vectors the population fitness gradually shown in Figure 12. This Figure reveals that the change in population fitness is highly dependent on the recombination value. In most cases such a decrease of W as this is associated with the breakdown of linkage disequilibrium, as was theoretically shown by Ko j ima and Kelleher (1961). Very rarely, however, a decrease in population fitness accompanied the build-up of linkage disequilibrium, though the amount of this type of decrease Table 6). was very small (see Another new feature of the change of W when linkage and epistasis are present is that the maximum point of W does not necessarily represent the population equilibrium. For instance, with a= ( ) and Do= -0.25, W reaches the equilibrium point after passing the maximum point unless r=0.5 (Figure 13). This indicates that the equilibrium values of gamete or gene frequencies can not be decreased under selection, as is Fig. 12. Changes in population fitness for a = (1 11 2) ; Po=.5, Qo= 0, Rp=O, S=.5. Table 6. Changes in population fitness and linkage disequilibrium for a = ( ); Do=O, r=0.2

14 20 M. NEI obtained by maximizing W as in the case of linkage equilibrium. In general the rate of change of W per generation is larger in early generations than in later generations when r is 0.5, but if r becomes small the rate of change of W is often large in later generations rather than in early generations. This situation can be observed from Figure 14, where the changes of W are shown for a = ( ). In the case of r=0.01 the population fitness hardly changes for the first 90 generations and then shows a rapid change to reach the equilibrium at about the 130th generation. Fig. 13. Changes in R0=.5, S o= population 0. fitness for a = (1224); Pa=O, Q.o=.5, Fig, 14. Changes in population Ra=O, S0=.5. fitness for a=(1 112); Po=.5, Qo=O, DISCUSSION In the computations given above two different sets of initial gene frequencies, i. e., 0.5 and 0.01, were used both for A2 and B2. The results obtained with q, = q2 0.5 may be applied directly to those experiments in which the initial gene frequencies

15 EQUILIBRIUM FREQUENCIES OF LETHAL GENES. II. 21 are artificially set up to be 0.5 or to the progeny of a hybrid between two different strains in which the gene frequencies have become 0.5 (by chance) because of the heterozygousness of its parents. This latter situation may occur in nature or in plant and animal breeding. In order to simulate the population dynamics of new lethal mutant genes such gene frequencies as 0.01 may not represent the real initial values, since these must be usually much smaller. When gene frequencies are very small, however, their changes are determined largely stochastically (cf. Fisher 1930). If the gene frequencies are increased to a sufficient amount by means of stochastic processes, then our deterministic models may apply. In the genetic models we considered the recombination value was assumed to be the same for males and females. In some genera such as Drosophila this is not the case. If, however, the selective value of each genotype and the inital gamete frequencies are the same for males amd females, the same genetic models as considered here will apply, employing the average recombination value of males and females (Nei unpublished). Nevertheless, the selective values assigned to the four viable genotypes in our computation are highly artificial. The real fitness differences among genotypes must be much smaller than those in our models. Thus, the conclusions to be drawn from this investigation must be guarded; and the results have only qualitative significance. Our computation has indicated that there are non-trivial equilibria for two lethal gene frequencies, if the two single or the double heterozygotes or both show a higher selective value than the normal homozygotes. Marginal overdominance can be incorporated temporarily by linkage disequilibrium even for the case where the fitnesses of heterozygotes and lethal-free homozygotes are the same, but this type of overdominance will not lead to stable equilibria. If there is no epistasis between loci or there is no linkage disequilibrium, only one stable equilibrium is possible for each of the two lethal genes, disregarding such complicating factors as genotype x environment interaction, etc. If, however, both epistasis and linkage disequilibrium are present, then the equilibrium gene frequencies are functions of the recombination value between the two loci. The equilibrium gene frequencies increase as the recombination value decreases irrespective of the linkage phase involved. Further, if ad-be = 0 and r <0.5, there seems to be three equilibria for a given value of r. The stability of these equilibria has not been tested in the strict sense but suggested by the convergence of different sets of initial values of gamete frequencies to the same values. The dominance x dominance epistatic comparisons are given in Table 1, but it appears that these values are scarcely related with any of the population parameters examined here. It was previously noted that with some fitness vectors the population fitness increases in parallel with the lethal gene frequencies as linkage becomes tight. This suggests that, if there arises any mechanism, possibly by mutation, in some individuals of the population which reduces the recombination value, then those individuals

16 22 M. NET having reduced recombination values are selectively favored. Thus, natural selection tends to reduce the recombination value between two lethal loci. Of course, close linkage is not favorable for changing environments, so that there must be a balance between recombination reducing and enhancing mechanisms in nature. Furthermore, if there are many loci having lethal alleles on a chromosome, those genes which are closely linked are expected to be of higher frequency than those loosely linked. These two agencies lead to the expectation that the distribution of lethal genes experimentally located on a chromosome is not at random but clustered, even if the mutation rate is the same for all loci. Spiess et al. (1963) and Watanabe (1963) have shown that this is actually the case in Drosophila melanogaster, though the mutation rate was not examined. Our numerical computation has shown that the equilibrium gene frequencies are determined independently of the initial gene frequencies, though affected by the linkage phase if ad-bc=0. However, the initial gene frequencies affect the rate of change in gene frequencies per generation appreciably. The most extreme example is the case of a=( ) and qo = 0.01 with D0=0. In this case the rate of change in gene frequencies after 15 generations was so small that the population remained practically constant for a long time thereafter. In practice, it would be experimentally impossible to differentiate this quasi-equilibrium from a true equilibrium. The equilibrium linkage phase is of interest, because it affects the frequency of lethal chromosomes and the allelic rate of lethal genes, which are important genetic parameters in experimental population genetics. In the case of two lethal loci, the frequency of lethal chromosomes, which are tested by isogenization techniques in Drosophila, is given by and the allelic rate by 1-P=1- (1-q~)(1-q2) -D [l-(l-1)(l-l) qq-q,g2d-d2]/(1-p)2 Thus, if gene frequencies remain constant, the frequency of lethal chromosomes is higher when D is negative than when D is positive, while the allelic rate is affected only slightly. Further, for a given frequency of lethal chromosomes the allelic rate is higher when D is positive than when D is negative. The frequency of lethal chromosomes and the allelic rate can also be related to the value of ad-bc, since the equilibrium linkage phase depends on this value. Cannon (1963) has noted in Drosophila melanogaster that the (apparent) equilibrium frequency of the cesro genes is dependent on whether they are introduced in the coupling phase or repulsion phase. This may correspond to the case of ad-bc=0 in our models, although the selective mechanism of the cesro genes must be more complicated than in our models because they are not lethal genes. At any rate, these results indicate that the initial state of gamete frequencies affects the final state of

17 EQUILIBRIUM FREQUENCIES OF LETHAL GENES. II. 23 gamete and gene frequencies. According to Crow (1958), the genetic load is defined as the proportion by which the population fitness is decreased in comparison with an optimum genotype. Lewontin (1964) has shown that this genetic load is decreased as the recombination value decreases for non-lethal genes. The results of our computations on lethal genes also indicate the decrease of genetic load due to the linkage effect, since the population fitness increases with decreasing r. This linkage effect may have a role in the retention of numerous recessive lethal genes in the population. The linkage of epistatic recessive lethal genes may also affect the genetic load manifested under sudden inbreeding (inbreeding load). In the case of two lethal genes the genetic load manifested under complete inbreeding is proportional to 1-P. This quantity is greater when the linkage between the two loci is tight than when it is loose if D is negative, while the situation is reversed if f is positive (see Tables 3, 4, and 5). Consequently, the lower the random mating load (genetic load for random mating population) the higher the inbreeding load for a given set of fitness values if f is negative, while the two kinds of genetic load both decrease with decreasing r if b is positive, since the random mating load decreases with decreasing r irrespective of the linkage phase. In this investigation no models involving more than two loci have been considered, but it seems that the qualitative aspect of the conclusions reached above will apply also to the case of more than two loci, since no new factors are introduced. Further, the results obtained here may be applied also to the artificial selection for a quantitative character in which the effects of genes on the phenotypic value are of the same type as the models considered in this paper. mean value of the character. SUMMARY In this case W represents the (1) The effects of linkage and epistasis on the equilibrium frequencies of lethal genes were studied by numerical means, making use of an electronic computer. The changes in gamete and gene frequencies, linkage disequilibrium, and population fitness under selection were also examined. The selective values of A,A,B,B A,A2B,B A,A,B,B2, and A,A2B,B2 are denoted by a, b, c, and d respectively, those of the remaining genotypes being all 0, where A2 and B2 represent lethal genes. (2) If a=b=c=d, there are no stable equilibria for two lethal genes even if there is linkage disequilibrium. If a is smaller than b and c or both, there occur non-trivial equilibria for the two genes jointly. The equilibium frequencies of lethal genes increases the recombination value between the two loci decreases, unless adbc=0 and the original population is in linkage equilibrium. If ad-bc=0 and the original population is not in linkage equilibrium the equilibrium gene frequencies depend on the sign of the original linkage disequilibrium as well as the recombination

18 24 M. NEI value. It does not appear that the initial gene frequencies affect the equilibrium values. (3) Gene frequencies do not necessarily approach the equilibrium values straightly but in some situations the lethal gene frequencies first increase considerably and then begin to decrease, depending on the initial gamete frequencies. If the recombination value is large, the rate of change in gene frequencies is usually larger in early generations than in later generations, but if the recombination value is small, there occur situations where little changes of gene frequencies are observed in early generations. (4) The equilibrium value of linkage disequilibrium has the same sign as that of ad-bc, if ad-bc~0. If ad-bc=0, the sign of the equilibrium value conforms to the sign of the original linkage disequilibrium. The equilibrium values of linkage disequilibrium increases as linkage becomes tight for a given set of selective values. The forms of changes in gamete frequencies and linkage disequilibrium are again affected considerably by the recombination value and initial gamete frequencies. (5) In some situations the population fitness decreases appreciably. This decrease is usually associated with the breakdown of linkage disequilibrium, but not always so. (6) It has been indicated that the frequency of lethal chromosomes and the allelic rate of lethal genes are differentially affected by linkage disequilibrium. The effect of linkage on the genetic load manifested under inbreeding is also discussed. Further, the applicability of the results obtained to the artificial selection for quantitative characters is pointed out. ACKNOWLEDGMENT The author is indebted to Dr. M. Kimura for his criticism during preparation of the manuscript. He also wishes to thank Drs. W. J. Schull and T. Mukai for reading the manuscript. LITERATURE CITED Cannon, G. B., 1963 The effects of natural selection on linkage disequilibrium and relative fitness in experimental populations of Drosophila melanogaster. Genetics 48: Crow, J. F., 1958 Some possibilities for measuring selection intensities in man. Human Biol. 30: Fisher, R. A., 1930 The Genetical Theory o f Natural Selection. Clarendon Press, Oxford. Kojima, K., and T. M. Kelleher, 1961 Changes of mean fitness in random mating populations when epistasis and linkage are present. Genetics 46: Lewontin, R. C., 1964 The interaction of selection and linkage. I. General considerations; heterotic models. Genetics 49: Nei, M., 1963 Effects of linkage and epistasis on the equilibrium frequencies of lethal genes (Abstract). Jap. J. Genet. 38: 197. Nei, M., 1964 Effects of linkage and epistasis on the equilibrium frequencies of lethal genes. I. Linkage equilibrium. Jap. J. Genet. 39: 1-6. Oshima, C., 1963 The persistence of some recessive lethal genes in natural populations of

19 EQUILIBRIUM FREQUENCIES OF LETHAL GENES. II. 25 Drosophila melanogaster. III. Difference between lethal and semi-lethal genes. Proc. Jap. Acad. 39: Spiess, E. B., R. B. Helling, and M. R. Capenos, 1963 Linkage of autosomal lethals from a laboratory population of Drosophila melanogaster. Genetics 48: Watanabe, T. K., 1963 Study on deleterious genes in a natural population of Drosophila melanogaster (Abstract). Jap. J. Genet. 38: 213.