Genotype×environment interactions

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Abstract
An algebraic formulation, alternative to that of Mather and Jones (1958) and hierarchial rather than factorial in nature, is presented for describing the differences among the phenotypes produced by a number of genotypes each grown in each of a number of environments. This formulation does not include terms representing statistical interactions between genotypes and environments : it depends instead on comparisons between the different genotypes in their variation over the relevant ranges of environments. The two-line case is considered and the condition established for linearity of the regression of genotypeenvironment interactions (g in Mather and Jones' formulation) on overall effect of the environment (e in Mather and Jones' formulation). This condition is the same as that derived in a different way by Mather and Caligari. It is shown that genes which respond to the environmental differences but which do not differ between the genotypes under observation will nevertheless affect the relation of genotypeenvironment interaction to the overall effect of the environment. In particular, different values may be obtained for the regression of g on e solely because of the effects of genes in which the genotypes under comparison do not differ. The multi-line case is developed as a simple extension of the two-line situation. Again the condition for linearity of the regressions is obtained and shown to be the same as that derived earlier and differently by Mather and Caligari. The value of genetically non-identical, but closely related, material for the biological measurement of the environment is discussed, and is shown to depend on the variation in the responses of the various genes to the environmental changes. Where these responses are similar any reasonably related material will give an adequate representation; but where the various genes may respond differently, the choice of material for the biological measurement of the environment is much more critical. Multiplicative action of genotype and environment is considered. Not only must the plot of g for each genotype against the general e give a set of straight lines passing through the origin, but the plot of the phenotype from each genotype against the average of all phenotypes, environment by environment, must also give a set of straight lines passing through the origin. Thus multiplicative action should not be difficult to detect by routine biometric analysis. The environment is made up of factors which may vary independently of one another and to whose changes the different genes affecting a character will not respond equally. The case is considered of two environmental factors and two gene-pairs, one reacting to each of the factors. The condition is derived for a straight regression line of g on e. It is shown that if one pair of genotypes, say AB and ab, give a straight regression line, the other genotypes, Ab and aB, will give a box like relation of g to e. Thus recombination converts a straight regression line into a regression box (fig. 2). This basic result still obtains where sets of genes rather than single genes react to change in an environmental factor and also where genes can be pleiotropic in their responses in that they respond to changes in both environmental factors. Finally, the consequences are considered of incomplete sets of genotypes being observed in incomplete sets of environments. Use of an incomplete set of environments does not destroy any straight regression line that may exist for g on e. With an incomplete set of genotypes, the straight regression may be preserved or it may be lost according to which genotype is missing; but where it is lost a near straight regression line is obtained for another genotype (fig. 4). In short a linear relation of g to e will seldom be lost or even obscured by failure to raise all possible genotypes in all possible environments.