ESTIMATION OF AVERAGE DOMINANCE OF GENES 507 



r 



where ^^ indicates summation over all pairs of loci for which the initial 



r 



linkage phase was coupling and > , summation over pairs for which the 



'77k 



initial phase was repulsion. 



When the associations between alleles at two loci are at equilibrium with 

 respect to coupling and repulsion phases, either because the loci are not 

 linked or because there has been sufficient opportunity for recombination 



P = qjqh r = qj{\ - qf,) 



s= (l-q,) qk 1= {\-q,)i\- q,) 

 and [pt — rs) — 0. Thus assuming no linkages (3) and (4) reduce to 



as indicated in the preceding section. If, in addition, gene frequencies at all 

 segregating loci are assumed to be one-half, o^^ and a\a reduce to the values 

 assigned them in Table 30.4. 



If there are linkages and equilibrium has not been reached, {pi — rs) will 

 be negative if the initial phase was repulsion, positive if the initial phase was 

 coupling. Thus covariances from repulsion and coupling linkages will tend to 

 cancel in al,. In fact if one assumes that enough loci are involved so that the 

 number of linked pairs must be high and that there is no reason why the 

 closer linkages should be predominantly in one phase, one is tempted to con- 

 clude that the sum of co variance will not be very important in a%. 



On the other hand the covariance term is always positive in al,i, being a 

 function of (pt — rs) for coupling and of (rs — pt) for repulsion.^ Thus pres- 

 ence of any linkage, regardless of whether the two phases are equally frequent , 

 will cause o-^, to be greater than I,q(l — q)ahi", except in the improbable 

 event that a for either or both members of pairs of linked loci is zero. And un- 

 less all linkages were in the coupling phase (in which case the ratio of a;, to 

 ^2^(1 — q)u- would be the same as of o-^, to Zq(l — q)a-u^ and hence the 

 ratio of o-^ to ala unaffected by the linkages) crl,i/2al would overestimate 

 Zq(l — q)a?u-/'2,q(\ — g)^- so long as equilibrium in linkage associations had 

 not been attained through recombination. However, as stated in the preced- 

 ing section, the linkage bias becomes progressively smaller as equilibrium is 

 approached. 



For purposes of illustration, consider application of the formulae in a 

 simple hypothetical situation. Assume that Experiment III is applied as 

 first described, with variable parents taken from an Fo, and that the quanti- 



4. This assumes generality of dominance of the more favorable allele — that a will al- 

 most always be positive. 



