Bull. Org. mond. Bull. Wld Hit/h Org.SanteJ 1971, 45, 201-208 Equilibrium under Inbreeding in Balanced Polymorphisms I. BARRAI 1 AND S. P. H. MANDEL 2 The results of a study on frequencies of genes causing lethal or sublethal disease are reported in relation to the level ofconsanguinity and heterozygote advantage in populations. The theoretical background of the population dynamics of genes under heterozygote advantage and in the presence of inbreeding is examined and the theoretical models are applied to the estimation of the level of heterosis for lethal and sublethal genes that are responsible for important public health problems in certain populations; these problems include fl-thalassaemia and sickle-cell anaemia. Estimates obtained from the proposed models show a high degree of internal consistency under widely different conditions; data on sickle-cell anaemia from several African populations and on f-thalassaemia from Italy and Malta were analysed. The hypothesis that rare lethal genes, such as those for Tay- Sachs disease, are maintained in the human population by heterozygote advantage is examined and discussed. In the past few decades there has been a general The closer the consanguinity between parents, the decrease in morbidity and mortality due to infectious higher is the value of F. A second type of genetic and parasitic diseases. As a result, diseases of morbidity is determined by genes that confer an genetic origin, or those with a genetic component, advantage on the individuals carrying them in single have shown an increase in relative frequency with dose in the heterozygote, but which are deleterious respect to total morbidity and have thus become or lethal to the individual in the homozygous form; comparatively of greater importance from the public sickle-cell anaemia and fl-thalassaemia are diseases health point of view. These diseases of genetic origin of this kind. These genes, which may reach pre- are maintained in the human population essentially valence levels as high as 20 %, are called "poly- by two basic mechanisms: recurrent mutation and morphic" genes, and the corresponding situation at heterozygote advantage. Genetic morbidity due to the population level is known as a balanced poly- recurrent mutation is associated with a class of morphism. genes having a relatively low prevalence. These A more detailed account of the relationship be- so-called " oligomorphic " genes give rise to con- tween human genetics and public health, some of the ditions such as phenylketonuria, galactosaemia, essential features of which have been mentioned and Tay-Sachs disease, the incidence of which briefly here, is given in a recent review of this subject is determined by the homozygous state of their by Barrai (1970). specific alleles. That is to say, the genes in question Recently it has been suggested that some oligo- are recessive, and the affected individuals are often morphic genes that are deleterious or lethal in found to be the offspring of consanguineous mar- homozygotes, such as those responsible for Tay- riages. Consanguineous marriage generates inbreed- Sachs disease and for cystic fibrosis of the pancreas, ing of the progeny, the intensity of which is measured may be maintained in human populations by hetero- by a coefficient of inbreeding (F), which is an expres- zygote advantage. For Tay-Sachs disease, this sion of the level of homozygosity in the progeny. hypothesis was prompted by the possibility that these genes might have increased in frequency in 1 Chief Medical Officer, Human Genetics, World Health some groups that separated in historical times from Organization, Geneva, Switzerland. an & Aron- 2Chief Statistician, Mathematics-Statistics, World Health ancestral population (Myrianthopoulos Organization, Geneva, Switzerland. son, 1966; Smith & Shaw, 1969). 2717 - 201 - 202 I. BARRAI & S. P. H. MANDEL Since the genes under consideration are recessive Table 1. Array of genotype frequencies under and have low frequencies that would respond to inbreeding inbreeding, it is convenient, in examining this Equivalent I Equivalent hypothesis, to consider the properties of a diallelic Genotype random mating selfing Fitness system at a single locus under inbreeding and fraction - fraction selection. The elements of such a system are two allelic genes, symbolically represented by A and a, AA (1-q)2 (1-F) (1-q)F Wii and the possible genotypes that may result from Aa 2q(1 -q) (1-F) 0 W12 them, namely the two homozygotes AA and aa and the heterozygote Aa. These alleles have a character- aa q2 (1 - F) qF W22 istic prevalence in any given population of interest, which may be represented by, say, q for allele a so that the prevalence of the complementary allele A is p=l-q. other is equivalent to a selfing fraction. The resulting The objectives and rationale of mathematical array is shown in Table 1. models describing the population dynamics of The whole mathematical treatment that follows is genetical systems are discussed in considerable detail based on the assumption that we are dealing with an by Li (1967). The fundamental mathematical pro- infinite population in which the level of inbreeding is position, which forms the basis of all such models, constant through generations. However, even in a is the celebrated theorem of Hardy & Weinberg, finite population with variable F, the variation in which asserts that, in a population of diploid indivi- inbreeding will be small, provided that the level of duals with random mating and in the absence of inbreeding is low. selection, zygotes AA, Aa, and aa will be formed in The change in gene frequency for allele a in any the ratio (1- q) 2: 2q (1- q): q2, respectively. one generation is given by the following equation In real population situations, however, matings (Li, 1955): do not occur at random. From the point of view of ~dWR dWil population genetics, these departures from the ran- Aq = q(l1-q +2(2) dom mating situation are adequately described by 2W dq dq J the concept of inbreeding. Moreover, with respect to most genetic loci, it cannot realistically be assumed where WR is the fitness of the random mating that all genotypes are equally well adapted to their fraction of the system, WI is the fitness of the inbred environment. The term selection describes such fraction, and W is the average fitness of the entire differences between genotypes in fitness or adapta- population so that: tion to the environment. Fitness can be defined as the relative ability of a genotype to survive from zlq= q(l-q){(W22-WI,) F+(1 -F) birth to maturity and to produce offspring; sym- bolically, the values of fitness of the three genotypes [W12-Wll-q (2W12-W11-W22)]), (2) can be represented by non-negative numbers W11, W,2, and W22, whose magnitudes are proportional which can be written in the form: to the ratios of the fitnesses of the genotypes AA, Aa, and aa, respectively. aq= q(l(-q) (2W123W1)_W22) (I-F) (3) If we incorporate into the basic mathematical model of Hardy & Weinberg the complicating where phenomena of selection and inbreeding, and if we W12_W11_ (W12_W22)F now let F represent the average inbreeding level in qF:-- (4) the population, where F is a number between 0 (I _F) (2W12_Wn_W22) and 1, it is possible to generate an array describing (Malecot, 1948, p. 44). the structure of the population with respect to genotype, gene prevalence, and fitness. The frequen- The equilibrium condition is given by Aq=0, cies of the various genotypes can be partitioned into which is satisfied by the trivial solutions q=0 and two main fractions, one of which is equivalent to the q=1, and by the solution q =q, which corresponds random mating part of the population, while the to a non-trivial equilibrium if, and only if, 0<qF < 1. EQUILIBRIUM UNDER INBREEDING IN BALANCED POLYMORPHISMS 203 (Note that under F = 0, the equation stable non-trivial equilibrium exists if, and only if, W12 W'11 W12-Wul qF F < ---12 (9) 2W12 -Wi11-W22 2 is obtained, which is the usual random mating It may be noted that the equilibrium gene fre- solution.) It follows that a polymorphic equilibrium quency of the gene corresponding to the less-fit exists if, and only if, one of the following pairs of homozygote under inbreeding is smaller than the inequalities holds: corresponding random mating value since W, 2-WjLj> FMW 2-W22) (5) 15 ~~~(W11-W22)F W12-W22> F(W12-W11)( qF_-O-qF - (1-F) (2W12-W11-W22)>0 (10) or and that this difference increases with increasing W12-WI, <F(W12-W22) . (6 values of Snc is an function. W12 -W22 <F(W12-W1Jj F, -F increasing Moreover, equation (3) can be written in the form 2 In the limiting case where F - W Wn it follows XV12-W22 Aq= W- (7) at once from equation (4) that qF tends to 0. It is interesting to note that in the limiting case where where F= 1 we have A={2W12-W11-W22} (1-F). (8) q(1-q) (W22-Wn) Then a=W]l+(W22-Wnl)q (11) d(Aq)=d A-{W[q2-q+(4F-q) x W2 so that a non-trivial equilibrium exists only when dq WU=W22; such an equilibrium exists, in this case, (I -2q)]-q(l-q) (l-q) dW at all values of q, and hence the equilibrium is dq neutral. Otherwise, the frequency of allele a changes A2 with respect to the frequency of A, so that after (q n generations of selfing one would observe W FF_qF) q0W 22 at the equilibrium point q = qF. (I(-qo) Wln +q0W 22 Now, clearly, qt2_qF=qF(qF-1)<O at any non- and clearly q.