Anim. Blood Grps biochem. Genet. 9 (1978) 55-58 SHORT COMMUNICATION

A note on deviation from Hardy-Weinberg proportions due to differences in gene frequencies between parental males and females

E. Andresen

Department of Animal Genetics, Royal Veterinary and Agricultural University, Copenhagen, Denmark

Received: 6 February 1978

Key-words: Hardy-Weinberg equilibrium, heterozygous excess

Deviations from Hardy-Weinberg proportions are always due to lack of fulfilment of the necessary requirements for the law. Some prerequisites may be taken for granted and, thus, may be less emphasized in an actual study. This can lead to unexplained results or erroneous conclusions. For example, the frequencies of genotypes among the offspring clearly depend on the actual frequencies of the corresponding genes among the parental gametes. Thus, if there is a difference in gene frequency between males and females of the parental generation the Hardy- Weinberg proportions of genotypes are not expected among the offspring (Wright, 1969). This may be one of several reasons for 'the often peculiar structure' of populations of domestic animals and game species and, consequently, for their exclusion from various comparative population studies such as exemplified by Frydenberg & Simonsen (1973).

The present note is concerned with an apparent heterozygous excess within population material which, however, is caused by differences in gene frequencies between males and females of the parental generation of the population studied. A more sophisticated treatment of this topic has been presented by Robertson (1965).

Consider a genetic system comprizing two alleles without dominance, A1 and AZ, and the corresponding genotypes A1A1, AIAZ, and AZAZ. Assume further the following gene frequencies within parental males and females: pr(Al) = 0.3 and pF(A1) = 0.7. Thus, among the offspring the following genotype frequencies are expected:

A'A' : py X pF = 0.21

= 0.58 A'A* : pJa X (l-pF) + (l-pM) X PF

55

E. ANDRESEN

A'AZ : (l-px) x (1-pp) = 0.21

rf, for example, 300 offspring were produced the following distribution is the most likely to be observed: 63 A1A1, 174 A l A ? , and 63 AZAZ. The corresponding gene frequencies, p(A 1 ) and q(Az), are obtained by simple gene counting.

Supposing that this population sample had been collected without knowledge of its actual derivation the calculated gene frequencies would lead to the following expected genotype frequencies using the conventional procedure:

A'A' : p' Y N = 75 (obs.: 63)

A*.4? : 2pq X N = 150 (obs.: 174)

A'A' : q? X N = 75 (obs.: 63)

Since the difference between observed and expected numbers is highly significant (jC,df = 7.68, P < O . O l ) , the erroneous assumption might be that the 'excess' of heterozygotes is due to 'heterozygous advantage'. Moreover, if the presumed excess is believed to be due to a balanced polymorphism, the relevant selection coeffi- cients (s) can be calculated since p = q = 0.5 and s(A1AZ) = 0. Hence, the expected number of both A'AI and AZA? individuals would be half the number of observed heterozygotes, i.e. % X 174 = 87. The reduction of the Iatter number to the observed numbers of 63 A'A1 and 63 AZA2 individuals would give the presumed selection coefficients of s (A1A1) = s (A2A.7) = approx. 0.28.

In the example chosen there is a considerable difference between the gene fre- quencies in parental males and females. In most situations the difference is likely to be much smaller. Therefore, the corresponding discrepancy between observed numbers of individuals and those expected according to Hardy-Weinberg propor- tions will usually not in itself be significant, but the mean from several such samples may be. In this context the same locus is considered in each of the samples. How- ever, the statement is also true for one sample of a population tested with respect to Hardy-Weinberg proportions at several, neutral loci.

General equations for expected genotype frequencies can be derived. Let dp = plr - pF denote the difference in the frequency of gene A1 between parental males and females (i.e., generation 0). (Notice that d denotes a difference in this context, not a change; cf. Falconer 1960.) Random matiiig would therefore lead to the following genotype frequencies among the offspring (i.e., generation 1):

A'A' (p+l/,dp) X (p-xdp) = p'-(l/,dp)' = P* --'/$(Ap)*

A'A? (P+%dP) x (q+%dp) + (P--l/z.4P> x (S-WP) = 2Pq+x(dP> '

A"? : (q+'/zdp) x (q-$$Ap) = q'-(Kdp)' = q' --'/$(LIP>'

56 Anim. Blood Grps biochem. Genet. 9 (1978)

DEVIATION FROM HARDY-WEINBERG PROPORTIONS

These equations are analogous to those derived by Bundgaard & Christiansen (1972). Corresponding equations can be derived for multiple allelic systems. Con- sider for example a three-allelic system compnzhg the genes A1, A2, A3 and the corresponding frequencies, p, q, and r computed from one generation. The terms dp, Aq, and Ar denote the differences between the frequency of the corresponding alleles in parental males and females. Hence, random mating of such parents (i.e., generation 0) is expected to lead to the following frequencies among the offspring (i.e., generation 1):

A'A' = p'-%(dp)' ; A'A2 = 2pq--l/zAp x dq

AZA' = q?-l/( 4 dq)' ; A'A3 = 2pr --l/zdp X dr

A3A3 = r2 -g(dr)z ; A2A3 = 2qr --1/2dq X dr

The three expressions for deviation of the heterozygotes from the Hardy-Wein- berg expectations in generation 1 have negative signs, but the total sum is positive and, therefore, represents the heterozygous 'excess'. Robertson (1965) has shown that on average the apparent proportional excess at diallelic and multiple allelic loci is 1/8M + 1/8F, where M and F are the number of male and female parents.

It is noteworthy that the gene and genotype frequencies are expected to be the same in males and females within the offspring generation (i.e., generation 1) although there may be considerable differences between the parents. Thus, an additional generation of random mating is expected to ensure Hardy-Weinberg proportions in the two sexes, i.e. in generation 2.

Failure to fit observed frequencies to Hardy-Weinberg expectations has many causes of which only one has been considered in this note. However, this cause is probably more frequent than usually envisaged. In fact it is likely that even for strictly neutral polymorphisms within comparatively small samples of farm animals with few breeding males relative to females, heterozygous excess over the Hardy- Weinberg proportion is the rule rather than the exception, most exceptions being due to sampling error. Thus, awareness of this phenomenon might contribute to elucidate part of the presumed heterozygous excess reported in the literature on genetic polymorphisms in farm animals. However, specific references to pertinent reports are avoided at this place since only the authors can distinguish between the possible causes in each particular study.

References

Bundgaard, J. & F. B. Christiansen, 1972. Dynamics of polymorphisms. I. Selection com- ponents in an experimental population of Drosophila melanogaster. Genetics 71: 439-460.

Falconer, D. S., 1960. Introduction to quantitative genetics. Oliver & Boyd, Edinburgh & London.

Frydenberg, 0. & V. Simonsen, 1973. Genetics of Zoarces populations. V. Amount of protein polymorphism and degree of genic heterozygosity. Heredim 75: 221-232.

57 Anirn. Blood Grps biochem. Genet. 9 (1978)

E. ANDRESEN

Robertson, A., 1965. The interpretation of genotypic ratios in domestic animal populations.

Wright, S., 1969. Evolution and the genetics of populations. Vol. 11. The theory of gene Anim. Prod. 7: 319-324.

frequencies. The University of Chicago Press, Chicago and London.

58 Anim. Blood Grps biochem. Genet. 9 (1978)