POPULATION GENETICS AND EVOLUTIONARY CHANGE 431 



Hardy-Weinberg Formula 



We have just noted that offspring derived from a gene pool in which 

 half the genes are dominant, half recessive, are expected to consist of Y^ 

 homozygous dominants, % heterozygotes, and '4 homozygous recessives. 

 In terms of the genes we have used for illustration we may write this as: 

 IMM + 2Mm + \mm. 



Suppose we now write MM as Af", and mm as m'-. Our statement then 

 becomes: M'- + 2Mm + m'-. Such a statement should begin to stir dor- 

 mant memories of something we encountered in high school algebra, or 

 more recently as college freshmen. Probably memories will be still further 

 stimulated if we express it with rt's and 6's: or + lab + b'-. Maybe we can 

 now recall that this is the result of multiplying (c; + 6) by itself: {a + bY. 

 In other words, a- + lab + b- is the expansion to the second power of the 

 binomial {a + /?)• Evidently the 1:2:1 ratio we have been discussing is a 

 special case of such an expansion. 



Instead of employing «'s and b's, we may follow custom and use p's and 

 ^'s, those letters we are proverbially admonished to "mind." 



Let p = the frequency of gene M 

 q = the frequency of gene m 



Then if random mating occurs, the offspring resulting can be calculated 

 by use of the formula 



(p + qf = p2 + 2pq + q\ 



We may note that this formula is an algebraic equivalent of the small 

 checkerboard diagram (p. 430). Along the left side of the latter we listed 

 the genes carried in the sperm cells together with fractions expressing their 

 frequency: '^/^M + ^■^m. This is equivalent io p + q. Along the top we 

 listed the genes carried in the ova together with fractions expressing their 

 frequency: HM + %m. This, also, is equivalent to p + q. Filling in the 

 squares of the checkerboard involved multiplying the frequencies of the 

 two kinds of genes carried in the sperm cells by the frequencies of the two 

 kinds of genes carried in the ova: (^{.M + y.,m){V>M + V.m). This is 

 equivalent io [p + q) (p + q) or (p + q)'-. Obviously, then, the binomial 

 is squared because two parents are involved in the production of offspring. 



This formula is referred to as the Hardy-Weinberg formula, from the 

 names of the two men who first realized its application to the problems of 

 population genetics. 



Let us apply the formula to the situation we have just been discussing, 



