RECURRENT SELECTION AND OVERDOMINANCE 463 



of the two gametic arrays provides expected frequencies of aa, aA, A A in 

 Fi(P, X Pj) with respect to n loci. 



Fi'^n (l — r) {1 — iv) aa-{-n[v {\ —w) + w ( 1 — v) ] aA -\-nvwA A 

 Define' phenotypes: 



aifll C2O2 • • • Ondn = T 



a A = T -\- d -^ kd 



A A = T -\-2d 



Fi = niv-\-w){d-^kd) -n2iw{kd) +T (1) 



This is the regression of Fi phenotype on gene frequencies of parents and is 

 independent of degree of inbreeding of parents. 



If P, is homozygous it has ;/(! — v)aa and nvAA loci. 



P, = nvi2d) +T, v= {P,-T)/n2d 



(2) 

 Pj=nw{2d) ^-T , w= {Pj-T) /ti2d 



Substituting for v and iv in (1) 

 Fi= {\ + k + kT/nd) (P,-\-Pj)/2-ik/ n2d) iP,Pj)-kr~/ n2d-kT (3) 



This is the regression of Fi phenotype on phenotypes of homozygous parents, 

 the equation of a surface in three dimensions, Fi, P,, Pj. The surface is a 

 plane if 62 = k/n2d is zero, if ^ = 0, if there is no dominance, no inter- 

 action of P, with Pj. Then, Fj = (P, + P,)/2, by setting ^ = in (3). 

 Taking P, constant as the common tester of one column of the regression 



table, 



Fi= [i(l + yfe) -k(P,-T)/n2d]Pj + C' (4) 



bp is the coefficient of Pj, within brackets, 



bj,= i-k/n2d)P,-j-^{\-^k) -{-kT/)i2d . lik = 0, bp=\ 



Regression of bp on P, is 62 = —k/u2d. Since P, == iiv{2d) -\- T, bp = 

 i(l + k) - kv. If bp - 0, 



v^ {\Jrk)/2k (5) 



With no dominance, 



^ = 0, I' =1/0 at equilibrium 



1. T, d, and kd are defined here in units of bushels/acre or pounds/plot, for example. 

 Then, k = kd/d is in units of (bu./A)/(bu./A), likewise bp, but 62 is in units of l/(bu./A), 

 making the whole term boPiFj in bu./A. 



In terms of selective values it is convenient to define ds in terms of number of progeny 

 surviving to breed. Then, ks may be greater or less than k, depending on artificial breeding 

 plan. If roan in Shorthorn cattle is intermediate, k is essentially zero, but if roan is favored 

 in artificial selection over red and over white, kg > I and there is overdominance with re- 

 spect to artificial selective values. 



With corn yield no single locus is identified, no heterozygote may be favored to pro- 

 vide ke > I, except that k > I. Then, ks may depend on gene frequency and on rate of 

 culUng. 



