348 GORDON E. DICKERSON 



Drosophila suggests the need for further investigation of the importance of 

 diversity of origin for attainment of maximum heterosis in crosses. 



Use of an Fi Cross as the Tester 



Hull (1945) also has suggested selection to complement the Fi cross of two 

 homozygous lines as a means for developing new lines to replace the poorer 

 ones presently used in successful corn hybrids. Here, expected rate of im- 

 provement in performance of the 3-line cross (AP/) would be a composite 

 of that expected from selection for cross performance with a homozygous 

 tester, and with a non-inbred strain in which gene frequency is ^ at each 

 locus: 



AP/= [q\'^qd\-k)-\-{\-qT)''^qM + k) + 2qA\-qT)'^qf\d (9) 



where qr is average proportion of loci homozygous for the more favorable 

 allele in the lines represented in the Fi tester; qi, qj, and q/ are average fre- 

 quencies of the more favorable alleles at loci that are A A, aa, and Aa, 

 respectively, in the Fi tester. The Fi tester is A A at qr, aa at (1 — 5^)^ and 

 Aa at 2^1.(1 — qr) of the loci. Hence, 



Aqi = ^ qrqii^ - 9,)(1 -k) , A9, = :r— (1 - 9r)9;(l - qj)(l-\-k) 



and 



sd 



A?/ = 7i V2gr(l - 9r) -9/(1 - (//) . 



Substituting in equation (9), we obtain as estimated progress per cycle: 



Ai'/= \q%qS\-q){\-ky-^{\-q^yq^{\-q){\-^ky~ 



sd^- (10) 

 + [2q^{\-q^)]^/'^-qf{\-q;)]j— 



^ (J p 



Apparently one might expect that selection to complement an Fi tester (of 2 

 homozygous lines) would be about one-half as effective as selection to fit a 

 single homozygous tester. 



In selection for complementary strains in livestock, the Fi tester may be a 

 cross of two partially inbred lines, M and X. Selection of a population, L, to 

 complement M-N would tend to improve the L{M'N) cross at a rate inter- 

 mediate between one-half that for reciprocal selection (APr)/2 and that for 

 use of an Fi cross of homozygous lines as the tester (AP/), depending on the 

 degree of inbreeding in lines M and N . 



If M and .Y were being selected to complement each other, gene frequency 

 in the {M'N) linecross tester would tend to be lowered from equilibrium 

 (1 -f k)/2k toward ^ as the limit. Consequently, rate of improvement in the 

 L{M-N) cross from selecting L to fit Af •A'^ should approach that expected 

 from selecting in population L to complement a non-inbred tester in which 



