SELECTION IN EXPERIMENTAL POPULATIONS. I. LETHAL GENES WYATT W. ANDERSON Department of Biology, Yale Uniuersity, New Hauen, Connecticut 06520 Received April 16, 1969 THEmain concerns of population genetics are the frequencies of genes in popu- lations and the forces that alter them. Central in importance among these forces is natural selection. The analysis of selection in experimental populations began with the work of L’H~RITIERand TEISSIERin 1934 and is today a standard laboratory procedure. Despite the attention of geneticists for thirty years, how- ever, certain important aspects of selection remain undemonstrated. The purpose of this article is to investigate in detail two particular aspects of selection: how it varies, and how it is partitioned among its more important components. Lethal genes have been chosen for this inquiry into the mechanisms of selection, since for them the algebra of selection is greatly simplified. We shall consider selection which acts on two alleles, one of them lethal, at a single locus on one of the autosomes. A lethal allele is one which, when homozy- gous, causes death before reproduction. The selection against the homozygotes for the lethal allele is thus complete; our task is to estimate the selection on heterozy- gotes for the lethal allele (LETH) and a non-lethal allele (NL) . COMPONENTS OF SELECTION We shall first develop an algebraic model for the study of lethal genes. The model will involve discrete generations; that is, it will be constructed with specific, non-overlapping periods for reproduction. Let us call the total selective value of organisms carrying two non-lethal alleles, W.“W” is assigned to the heterozy- gotes in accordance with the usual convention in discussions o€ lethal genes. It measures the average number of off spring produced by heterozygotes relative to the number produced by organisms carrying two non-lethal alleles. The homo- zygotes for a lethal produce no offspring, so their selective value is 0. The number of offspring, of course, determines the number of alleles present in the next generation. Thus the selective values determine the frequencies of the alleles. In order to reproduce, a fly must live to reproductive age; thus, the total selective value W is composed of all the various types of selection which occur from the formation of zygotes in one generation to the formation of zygotes in the next. The terms “fitness” and “adaptive value” are frequently used as synonyms for selective value. Consider a population in which we determine the frequencies of the genotypes among the adults of each generation before they are transferred to fresh food and Genetics 62: 65%672 July 1969. 654 W. W. ANDERSON allowed to lay eggs or bear young. This is, in fact, the common laboratory pro- cedure for experimental populations where generations are discrete. Now, the overall fitness W extends over the entire interval from the lormation of zygotes in one generation to their formation in the next. We are, however, considering the general model where the populations are sampled and the frequencies of the alleles determined some time during the zygote-to-zygote cycle. The results of selection may not then be described by W alone, because mating occurs during the interval between samples and separates the action of the principal components of selection. The selection which occurs between the scoring of the adults and the occurrence of mating alters the genotype frequencies. Mating then redistributes the alleles among the genotypes. The selection which next acts on the zygotes will have a different effect than if the two phases of selection were not separated by mating, since the genotype frequencies on which the selection acts are different in the two situations. PROUT(1965) first pointed out this important point. Follow- ing his notation, we separate W into two components: E (for Early), that part of the selective value which operates from the formation of zygotes to our determi- nation of allele frequencies in the adults; and L (for Late), that part of the selec- tive value which operates from the determination of allele frequencies in the adults to the formation of the next generation's zygotes. The overall selective value is equal to the product of its components: W = E.L. Now E and L are not independent. E is the expected number of NL/LETH zygotes which survive to reproductive age, relative to the number of NL/NL zygotes which do so. L is tlie expected number of zygotes generated by the heterozygotes, relative to the num- ber generated by the NL/NL genotype, given that the heterozygous parents have survived to reproduce. Hence W is the product of its components by the definition of conditional expectations, not the law for combining independent ones. The E component of the selective value consists almost entirely of selection by viability from egg to adult. The L component consists largely of differences in mating activity and fecundity, but also includes any viability differences during the usually short reproductive phase. THE ASSUMPTIONS The following assumptions will be incorporated in the model. 1. Mutation will be ignored, as is justified in constructing a model for short- term changes (of the order of a few years at best) at a single gene locus. 2. The population size is assumed to be sufficiently large that random sampl- ing effects will not be important, an assumption which is valid for the type of data we shall analyze. 3. The selective values of the genotypes are assumed to be constant, or very nearly so. This assumption is certainly not always justified, but we shall incor- porate it in order to simplify the analysis. Later I shall outline a technique to reveal some types of variation in selection from generation to generation. 4. Mating is assumed to be random, and it is assumed that the effects of differ- ential mating activity and differential fecundity are simply to alter the effective SELECTION ON LETHAL GENES 655 frequencies of alleles in the populations. Differential mating activity effectively alters the frequencies of the alleles as if there were more individuals with the favored genotype in the population. If the females lay approximately the same number of eggs from matings with either of the male genotypes, then differential fecundity between the female genotypes will likewise simply increase the effec- tive frequencies of the favored alleles in the “mating pool” of gametes. If different types of mdings result in differentnumbers of zygotes, however, a more complex analysis must be undertaken (see BODMER1965). 5. Selection will initially be assumed to be the same, or at least very similar, in the two sexes. Later this assumption will be specifically evaluated for lethal genes and shown to be reasonable. 6. Segregation of the alleles at meiosis is assumed to be normal in both sexes; there is assumed to be no “meiotic drive” ( SANDLERand NOVITSKI1957). THE ALGEBRA Let; Q be the frequency of the lethal allele and P the frequency of the non-lethal alleles. The algebra will begin, among the adults, at the point where the frequen- cies of the alleles are determined. There are only two genotypes among the adults: individuals with two non-lethal alleles and heterozygotes for the lethal. Only the heterozygotes carry the lethal allele, and in them only one of their two alleles is lethal. Hence the frequency of the heterozygotes is 24. NL/NL NL/LETH Genotypes 1-2Qt 2Qt Genotype frequencies among the adults at generation t. 1 L “L” selection l+Qt 2QtL Genotype frequencies 1+qt (L-1) 1+2Qt(L--l) after “L” selection. 1+Qt (L-2) QtL Allele frequencies in the and Qmat = (I) = 1+2Qt(L-1) 1+2Qt (L-I) “mating” pool of gametes. Random mating occurs, and zygotes are formed. Then selection acts through the “E” component. NL/NL NL/LETH LET€X/LETH Genotypes P‘mat 2PmatQmat Pmat Genotype frequencies after random mating 1 E 0 “E” selection P‘mat 2PmatQmatE 0 Genotype frequencies P’matf2PmatQmatE P2mat+2PmatQmatE ‘after “E)’ selection It is at this point that allele frequencies are again scored; tl3.e full cycle of one generation has passed. - QmatE (2) Qt Pmat+ 2QmatE Substituting for Pmatand Qmat,and letting E.L = W where the product appears, QtW (3) Qt+l = I+Qt (L-2),+2QtW * 656 W. W. ANDERSON Inverting both sides of this equation, we transform this non-linear difference equation into one which is linear in a simple function of the Q’s. 1 1 L-2+2W 1- . -+ W (4) ot+1- ot~ W Solving (4), we obtain 1 (L-2+2W) (L-2-2W) - - W-1 = Qt(&-W-1 ) (5) Qt provided W = E.L# 1. The values of 1/Q converge monotonically to a final value by a factor of 1/W per generation; (l/W) gives the convergence over t genera- tions. If W>1, then 1/W<1 and the right hand side of (5) approaches zero as t w-1 the equilibrium increases. Hence - + L-2+2W as t+ 00 and Qt + Qt W-1 L-2f2W ’ gene frequency. If W<1, then 1/W>1 and the right hand side of (5) increases without bound with increasing t; this requires that Qt+O, so that the lethal allele will be eliminated. These equilibrium properties are just one case of the general formulation for two alleles at a single autosomal locus (FISHER1922): an equi- librium at intermediate frequencies of the alleles is possible if and only if the selective value of heterozygote is larger than that of either homozygote. Simplifying (5), Wt(W-1) Qo (6) Qt= W-1 + (L-2f2W) (Wt-l)Qo ‘ The time required to change the frequency of the lethal allele from Qo to Qt is (Qt(l+QoK)) loge Qo (1+QtK) t= where K = L-2+2W and W#l.