A non-linear genetic model

Summary

A model, developed by Seyffert and Forkmann (1976), simulates quantitative characters by genes with biochemically definable action. This model, however, possesses a number of shortcomings which have been overcome by a modified model of the form:

$${\text{y}}_{\left[ {{\text{X}}_{\text{1}} ,{\text{X}}_{\text{2}} , \ldots {\text{X}}_{\text{k}} } \right]} = {\text{Y}}^{\text{k}} {\text{ }}\mathop \Pi \limits_{{\text{i = l}}}^{\text{k}} {\text{ }}\left[ {{\text{1}} - \left( {{\text{1}} - {\text{r}}_{\text{i}} } \right){\text{exp}}\left( { - {\text{c}}_{\text{i}} {\text{x}}_{\text{i}} } \right)} \right]$$

where ${\text{y}}_{\left[ {{\text{X}}_{\text{1}} ,{\text{ X}}_{\text{2}} , \ldots {\text{X}}_{\text{k}} } \right]}$ is the score of the genotype [x₁,... x_k], x_i is the number of positive alleles (0,1,2) at locus i, and Y, r_i, c_i are fitted constants. As well as having a better fit to the data published by Seyffert and Forkmann for the anthocyanin content of flowers of Matthiola incana, this modified model has implications concerning heterosis, multiple allelism and optimum genotypes.