SYSTEMS OF MATING 45 



d = s = q 



<y 2 o = o* = ?(i - q) 



r os = [q- (hl2) - ? 2 ]/[<7(l - q)] 

 = 1 - (h/ho) 



This correlation coefBcient is usually designated by F, the inbreeding coefficient 

 (or fixation index) . As the proportion of heterozygosis in a random-breeding popula- 

 tion is h Q = 2q{\ — q), the amount of heterozygosis, relative to that in a random- 

 breeding population with the same gene frequencies (the panmictic index P) is equal 

 to 1 - F. 



The frequencies of genotypes can be written in any of the following forms, of which 



the first is that used above. 



Deviations from Deviations from Weighted 



array of fixed lines panmixia average 



AA q - Pq{\ - q) q 2 + Fq(\ - q) Pq 2 + Fq 



Aa 2Pq(l - q) 2q{\ - q) - 2Fq{\ - q) 2Pq{\ - q) 



aa (1 - q) - Pq{\ - q) (1 - q) 2 + Fq{\ - q) P (1 - q) 2 + F(l - q) 



It becomes obvious on applying path analysis that the correlation, F, between 

 uniting gametes is a function merely of the breeding system without any assumptions 

 with respect to number or relative frequencies of alleles or of any values which might be 

 assigned them. It is thus a somewhat unusual sort of correlation coefficient, a point 

 that has made some difficulty. 701 It remains to be shown how it is related to hetero- 

 zygosis in the case of multiple alleles. 



Under Mendelian heredity, any set of multiple alleles a 1 , a 2 , . . . a k may be treated 



formally as a pair of alleles by any sort of dichotomy: a 1 vs. a 2 ' •••*, a 1 - 2 vs. a 3 - - k , etc. 



The expressions for the genotypic frequencies in the case of two alleles still hold. 



Genotype Frequency Genotype Frequency 



aV Pqf + (1 - P)?i o» ; V'" P(qi + q 2 ) 2 + (1 - P){qx + q 2 ) 



a l a 2 - k 2Pq 1 (\ - ?1 ) a 1 - 2 * 1 -" 2P( 9l + q 2 )(\ - q x - q 2 ) 



fl 2... fcfl2 ...k P(1 _ qi) 2 + (1 _ P)(l _ qx ) a 3...ft a 3...fc P (i _ ?1 _ q2 )2 + (1 _ P)(l - qx - q 2 ) 



Since the value of P for any given pair of neutral alleles is independent of genie 

 frequencies and depends merely on the breeding history, it may be assumed that the 

 value of P is the same for any dichotomy of a neutral, multiple allelic series. Thus, in 

 such a series, the frequency of any homozygote, a l a\ is of the form Pq? + (1 — P)q t . 

 By subtracting the frequencies of aW and a 1 ' a 1 ' from that for a Ui a iJ , it may be seen that 

 the frequency of any heterozygote, a { a j , is of the form 2Pq l q j , and total heterozygosis 

 is given by h = IP 2 Mr Tnus p = ^l h o irrespective of the number of alleles. 



If now any arbitrary values V 1} V 2 , V k are assigned to the alleles, the mean and vari- 

 ance for each and the correlation between uniting gametes may be calculated. The 

 symbol, /, is used here for proportional frequency. 



o = / = 2 Vifi = I ViK 



a 2 = a 2 = a 2 = 2 V?f K - (2 VJ,) 2 

 r os =[2 2 Vfth - (2 VJ.nia 2 , 



