Section 9 — Population Genetics 



9.12. An Instance of Interaction of Genotype and 

 Environment at the Population-level. F. E. 



Binet (Werribee, Australia). 



The interaction of genotype and enviionment 

 is usually considered on the level of the individu- 

 al; it denotes the phenomenon of different en- 

 vironmental effects on the phenotypic expression 

 of different genotypes. 



In the terminology of Analysis of Variance this 

 may be expressed as the "non-additivity" of the 

 "average phenotypic value", and the "environ- 

 mental contribution". 



Considering a population as a whole, another 

 influence of its environment on its phenotypic 

 composition needs consideration. Even without 

 any difference in overall allele-composition the 

 zygotic genotype-composition of two populations 

 may be different. In presence of any dominance 

 and/or epistasis different "average phenotypic 

 values" follow from that difference. 



Environmentally conditioned mating-systems 

 assemble identical gene-material into different 

 zygotic genotype-collections. The conditioning 

 factor is ecological (in wild populations) or 

 human action (in domesticated populations). 



This lecture attempts to account for findings 

 (Clifford and Binet, 1954), contrary to antici- 

 pations, based on Anderson's (1949) models. 



We measured certain characters of trees in 

 three stands; their estimated dispersion-matrices 

 define quadratic forms, whose cartesian images 

 are hyperellipsoids, with centroids, representing 

 (as position-vectors) the estimated mean-vectors. 

 In this representation (cf. Pearson, 1901) Ander- 

 son's model leads to anticipating the major axis 

 of the hybrid-stand's hyperellipsoid to lie on the 

 line connecting the parent-stands' centroids. 

 Contrariwise, we found the major axes of the 

 three hyperellipsoids nearly parallel. 



It is submitted that these findings do not 

 contradict Anderson's theory on the formation 

 of the "recombination-spindle" and on subse- 

 quent "introgression"; rather, it appears that in 

 the "hybridized habitat" natural selection may 

 favour the hybrid individuals so strongly over 

 the pure offspring of both parents that the hybrid 

 stand consists almost exclusively of hybrids. From 

 those hybrid stands whose representative hyper- 

 ellipsoids are such recombination-spindles, such 

 pure offspring have not been eliminated by natu- 

 ral selection. We assume that in our case the 

 effect of natural selection resulted in a hybrid 

 stand of a very similar constitution to one at 

 which a breeder, crossing individuals from 

 different parent strains only, would aim. 



Anderson, E. Introgressive Hybridization, 

 New York, 1949: Wiley. 



Clifford, H. T., and F. E. Binet. A quantita- 

 tive study of a presumed hybrid swarm between 

 Eucalyptus elaeophora and E. goniocalyx. Aust. 

 J. Bot. 2, 325-36 (1954). 



Pearson, K. The lines of closest fit to a system 

 of points. Phil. Mag. 2, 559 (1901). 



9.13. Selection ton aid an Optimum and Linkage 

 Disequilibrium. S. Wright (Madison, U.S.A.). 



The author (1935, 1937) gave an approximate 

 formula for change of gene frequency from 

 selection under random mating, assuming ran- 

 dom combination among multiple interacting 



factors (/\q cj(\-q) 2"' where the Ws 



are selective values of total genotypes, assumed 

 constant). This is only approximate because 

 selection with interaction usually maintains 

 deviations for randomness. Exact treatment 

 (1944, 1952) of an extreme two locus case 

 (W 7 falling off symmetrically from an optimum: 

 W = 1 for two plus factors, ( \-s) for one or three, 

 ( M4s) for none or four, recombination c), showed 

 considerable linkage disequilibrium at (1/2, 1/2) 

 \{ s = c (0.1465 AB or ab 0.3535 Ab or aB) but 

 little if s/2c is small (gamete frequencies about 



1/4(1 



2c 



) ). This paper considers three linked 



loci, ABC, both recombination fractions c, com- 

 plete interference, W = 1 for three plus factors 

 and (\-s), (l-4s), (l-9.s) for increasing deviations. 

 If s = c, and sc is small, the equilibrium fre- 

 quencies are 0.0495 ABC (and abc), 0.1387 A Be, 

 etc., 0.1731 AbC (and aBc) but 0.1097, 0.1278 

 and 0.1347 respectively if s = 0.1c. If s = c, 

 the disequilibrium (lAbiaB-^ABiab), for loci A,B 

 in presence of C,c is 59.7 per cent of that with 

 no C,c segregation while disequilibrium of loci 

 A,C in presence of B,b is only 26.5 per cent of 

 that with BB or bb. The approximate formula 

 for /\ q may be considered reasonably satisfac- 

 tory, in spite of the extreme interactions of the 

 optimum model, with respect to selection co- 

 efficients of lower order than the recombination 

 fractions. 



9.14. Monte Carlo Investigation of Interaction 

 between Linkage and Selection under Dominance 

 Model. N. R. Bohidar, D. G. Patel and 

 R. L. Hurst (Logan, U.S.A.). 



The mathematics of simulation of genetic 



147 



