INTERACTION BETWEEN NATURAL SELECTION FOR HETEROZYGOTES AND DIRECTIONAL SELECTION

INTERACTION BETWEEN NATURAL SELECTION FOR HETEROZYGOTES AND DIRECTIONAL SELECTION MARGRITH WEHRLI VERGHESE 1228 Kingston Ridge Driue, Cary, N.C. 27511 Manuscript received May 5, 1973 Revised copy received August 16, 1973 ABSTRACT When directional selection for an additively inherited trait is opposed by natural selection favoring heterozygous genotypes a selection plateau may be reached where genetic variance is present. The amount of response when this plateau is reached is a simple function of the selection response in the first generation and the intensity of natural selection. When selection is practiced in small populations, the sizes of the initial equilibrium gene frequencies are at least as important as the intensity of natural selection in determining the probability of fixing desirable alleles. HE aim of artificial selection is to improve some particular trait and thus it Tis generally directed toward extreme phenotypic values. Natural selection, however, often favors intermediate expression of metric traits unless these traits are very closely associated with fitness. Therefore, plateaus observed in artificial selection experiments could result from a conflict between directional and opposing natural selection rather than from a loss of additive genetic variance. A considerable amount of theory exists, treating artificial selection and natural selection separately, but only rarely is their action considered jointly. Natural selection for intermediate phenotypes can be discussed in terms of the following two models: (a) the overdominance model where extreme phenotypes are presumably less fit because they are more homozygous than intermediate phenotypes (LERNER 1954), and (b) the optimum model where intermediate phenotypes have higher fitness than extremes because of some causal relationship between a particular metric trait and fitness (FISHER 1930; WRIGHT 1935; HALDANE 1954). When strictly additive gene action is assumed for the metric trait, the optimum model inevitably leads to gene fixation (FISHER 1930; ROBERT- SON 1956) while the overdominance model can maintain stable gene frequency equilibria (ROBERTSON 1956). JAMES (1962) considered optimum and directional selection simultaneously and found that an equilibrium between the two opposing systems of selection could be established where genetic variance would still be present. It is to be expected that such an equilibrium between directional and natural selection for intermediates also exists where overdominance is the determining factor in fitness. Genetics 76: 163-168 January, 1974

164 M. W. VERGHESE MODEL We have a random mating base population at equilibrium which is to undergo artificial selection to increase some trait 2. The usual assumptions are made that the phenotypic values of this trait are distributed normally around a mean fi and variance of upz. Parents are selected by truncation on individual measurements. The inheritance of this trait is based on a simple additive model with two alleles per locus, all loci having equal effects. Furthermore, the loci are pleiotropic and also affect fitness. Gene action on fitness, however, is overdominant. With random mating, we have the following situation: Genotype Frequency Genotypic value for z i Selective values under truncation 1 + &q 1 + &(q-'/e) 1 - -W Fitness scores 1 -SI 1 1 -s2 where i is the selection differential in standard deviations. Considering truncation selection on z alone, the change in gene frequency after one generation of selection is AP = % UPq (1) -a where U = i -. Thus the change in the population mean p after the first t generations of selection is (GRIFFING 1960) and AG, =the selection response in the first generation of truncation selection. Artificial selection experiments are generally conducted under ideal laboratory conditions and animals which are not selected as parents are usually discarded. Thus mortality data can be obtained only over a short lifespan and often show very little correlated responses to directional selection for other quantitative traits. Reproductive fitness, i.e., the ability to produce offspring, however, can change considerably during such selection experiments (e.g., VERGHESE and NORDSKGG 1968). We restrict our model to the case where natural selection acts only through differential reproductive capacity of the parents. Consequently, artificial selection acts prior to natural selection. It seem reasonable to asume that the number of progeny produced by a gven mating should be proportional to the product of the fitness scores of the two parents. We can then obtain the overall selective value for any genotype by multiplying its fitness score with its probability of being selected as a breeder, so that w,, = (1Suq) (I+,), w,, = I+u(q - '/e). w,, = (l--up) (1-s,). This is basically an application of the model that KEMPTHORNE and POLLACK (1970) used in discussing the validity of FISHER'S fundamental theorem of natural selection. Now we can calculate the gene frequency changes resulting from the combination of natural and artificial selection as Ap = Pqfi - P9(S,+S,) (P-80) - P24zu(s,+s,) 2 (3) W $2 where Po = - s, ts., and W = p2wi, $2pqW,, + q2w2,.

OVERDOMINANCE AND SELECTION LIMITS If the selective values are small we can use the relative fitness values of Thus U w'll = (1 + 2) Apz (I-s1), WIZ = 1, wzz V = (1--) (l-sz). 2 p; - pq(s,+s,) (P-fiJ - Pq-(S,+s,) b+po - 2Pfio) 2 2 w where W' = p2w',, + Zpqw',, + q2wz,. The fiist two terms in equations (3) and (3a) represent the action of artificial and natural selection separately, while the third term stands for an interaction between the two forces of selection. The progress in the fiist generation of selection is obtained by adding the changes of all individual loci as - (4) U 165 (34 = &$z for small S. SlSZ where S = -. SI+% It is obvious that the term for natural selection in equations (3) and (3a) becomes increasingly important as artificial selection forces the gene frequency away from its original equilibrium. By using equation (3a) we can obtain a rather simple expression for a new gene frequency e, where natural selection exactly cancels the effect of artificial selection. Setting Ap=O leads to This equilibrium is stable whenever Wll is less than one, i.e., s,>v/2. We can therefore say that natural selection can indeed oppose artificial selection to such an extent that no further progress is possible in spite of the presence of additive genetic variance. Since the interaction term in equations (3) and (3a) is always negative we can obtain an upper limit to the response to directional selection by neglecting this term. The total possible response is when we neglect the interaction term, we get The quantity S is the difference between the fitness of the heterozygote and the mean fitness of the equilibrium base population. It is directly proportional to the intensity of natural selection I on trait z. HALDANE (1954) defined I as the logarithm of the ratio of the fitness of the optimal

166 M. W. VERGHESE phenotype to the mean fitness of the whole population. Following Ro rson S tions we obtain I = Sh2/2. We can express the selection limit in terms of I as (1956) deriva- A; = AGlh2/21. (10) JAMES (1962) determined Ai=AGl@lhz using the optimum model for fitness. The two models therefore yield the same results only when hz=l, i.e. when there is no environmental influence on trait z. In small populations there is a defiite possibility that undesirable alleles may become fixed in spite of strong selection pressure against them because of the importance of random fluctuations in gene frequencies. KIMURA (1957) developed a formula for the probability of fixation, u(p), for an allele with initial frequency p which takes into account the effects of restricted population size as well as those of systematic selection pressures. u(p) is then the expected proportion of equivalent loci which will become fixed for the favored allele in any one line or the expected number of replicate lines in which the same allele will be fixed (ROBERTSON 1960). The general formula for u(p) is M(Ap) is the expected change in gene frequency per generation and V(Ap) the variance of Ap. Taking M(Ap) from equation (3a) and V(Ap) = pq/2n where N is the effective population size, G(p) is For very small values of N(s,+s,) u(fjij, giving and Nu, a first order approximation can be obtained for Thus, for intermediate gene frequencies, ROBERTSON S (1960) formula for the expected selection response (equivalent to equation (13) above with S=0) would give only a slight overestimate since the term NSu is very small. However, if initial gene frequencies are considerably greater than one-half, natural selection may actually increase the probability of fixation of favorable genes and thus increase the selection limit. Under these circumstances it could be advantageous to reduce the population size in artificial selection programs. But when initial gene frequencies are less than one-half, natural selection could depress the response to directional selection drastically, since it would increase the chance of losing favorable alleles. In order to investigate how many generations of selection are necessary for a certain amount of progress, we need to express the changes in gene frequencies as functions of time. We can then calculate an approximate value for the total response as the sum of the responses in every generation t by assuming that pq decreases by 1/2N and pq(1-2p) by 3/2N per generation (ROBERTSON 1960). The total change in gene frequency after t generations of selection is (neglecting the interaction term of equation (3))

OVERDOMINANCE AND SELECTION LIMITS 167 = N(u+s,--s,)pq[ l--fl- --It] 1 2N To approximate the probability of fixation of p, we set t=m in equation (14). Hence This is equivalent to equation (13) without the interaction term. The half-life, i.e., the number of generations needed to achieve one-half of the total possible change in gene frequency can be worked out for each specific case from equation (14). When,Bo = i/z, the second term in that equation is zero and the half-life is the usual l.4n generations. As N gets large, i.e., (l-i/2n) (1-3/2N), the half-life also reduces to 1.4N. DISCUSSION It is often argued that even small amounts of overdominance at many loci would result in an unbearable segregational load and thus such a population would soon die out. If, however, we consider a fixed amount of resources available which can support a fixed number of individuals, we are really dealing with relative, and not absolute, fitness values. This implies, of course, that the reproductive potential of the population is great enough to maintain a steady population size. Under these circumstances, it is possible to have overdominance at many loci (e.g., SVED, REED and BODMER 1967). We have seen before, however, that stable gene frequency equilibria between natural and artificial selection can be maintained only if the reproductive disadvantage of the most favored genotype is greater than the selective advantage that this genotype gains through directional selection. Therefore, it should theoretically be possible to overcome natural selection with sufficiently intense artificial selection. An increase in the intensity of artificial selection would, however, have to be associated either with increasing the reproductive rate or decreasing the number of selected parents. Since the decline in fitness with directional selection is Zg2/h2 (g = selection response in terms of additive genetic standard deviations; ROBERTSON 1956), it may not be possible to enlarge the population, and a reduction of the number of selected parents in the early stages of selection could increase the loss of favorable alleles. Thus selection limits are often reached in practice because it is not possible to increase the intensity of artificial selection. The decline in fitness at the new gene frequency equilibrium is >h4/4z and if a satisfactory population size can be maintained with this fitness, the total possible response of AGlh2/2Z should be realized. If cessation of selection response is due to opposing natural selection as described here, the return in one generation of relaxed selection is a fraction S of the total progress made, assuming that the initial changes in gene frequency were small.

168 M. W. VERGHESE The extreme influence of initial equilibrium gene frequencies on the selection limit in small populations is remarkable. More than the intensity of natural selection, it is the difference in fitness values between the two homozygous genotypes that determines the probability of fixation of the favored allele. This is the same phenomenon that ROBERTSON (1962) observed when dealing with natural selection only. If population sizes are not kept rather large in the initial generations of selection, desirable alleles maintained at low frequencies by natural selection may be lost. However, when the selection limit is reached at a point where $ is considerably larger than one-half, we could force fixation of favorable alleles by reducing the number of parents while maintaining sufficiently intense directional selection pressure. The model described here is, of course, a gross simplification of actual situations in that it assumes that all loci have equal effects on trait z as well as on fitness. In practice, there could be some loci with large overdominance for fitness and small effects on trait 5. Fixation of favorable alleles at such loci could cost too much in terms of reduced fitness and might not be desirable. The ultimate goal in an artificial selection program is to obtain the maximum possible response without reducing fitness to a level where the desired population size can no longer be maintained. To attain this, we must try to find the best possible combinations of population size and selection intensity. In view of the above analysis, the maintenance of a sufficiently large breeding population at the beginning of a selection proogram is imperative when the original equilibrium gene frequencies are unknown. LITERATURE CITED FISHER, R. A., 1930 The genetical theory of natural selection. Oxford, England, Clarendon Press. GRIFFING, B., 1960 Theoretical consequences of truncation selection based on the individual phenotype. Australian J. Biol. Sci. 13: 307-343. HALDANE, J. B. S., 1954 The measurement of natural selection. Proc. 9th Intern. Congr. Genetics Part 1: 480-487. JAMES, J. W., 1962 Conflict between directional and centripetal selection. Heredity 17: 487-4+99. KEMPTHORNE, 0. and E. POLLACK, 1970 Concepts of fitness in Mendelian populations. Genetics 64: 125-16. KIMURA, M., 1957 Some problems of stochastic processes in genetics. Ann. Math. Statist. 28: 882-901. LERNER, I. M., 1954 Genetic homeostasis. London, England, Oliver and Boyd. ROBERTSON, A., 1956 The effect of selection against extreme deviants based on deviation or on homozygods. J. Genetics 54: 236-24.8. -, 1960 A theory of limits in artificial selection programmes. Roy. Soc. Proc. 153 B: 234449. -, 1962 Selection for heterozygotes in small populations. Genetics 47: 1291-1300. SVED, J. A., T. E. REED and W. F. BODMER, 1967 The number of balanced polymorphisms that can be maintained in a natural population. Genetics 55: 469-481. VERGHESE WEHRLI, M. and A. W. NORDSKOG, 1968 Correlated responses in reproductive fitness to selection in chickens. Genet. Res. 11 : 221-238. WRIGHT, S., 1935 Evolution in populations in approximate equilibrium. J. Genetics 30: 257-266. Corresponding editor: T. PROUT