Representation and calculation of selfed population by group ring

Abstract

A method to represent and calculate frequencies of zygotes in selfed population is presented. This is an alternative to the conventional matrix method and reduces the amount of calculation to a considerable extent. Explanations are given that the phenomenon of genetic recombination is represented by a set of symbols called group, among which certain relations hold, and that a selfed population is conveniently described by a quantity called group ring, certain properties of which are used to reduce the difficulty of calculation of population frequencies. A case of a selfed population with two linked genes under seed selection is taken as an example, and the steps to obtain the final algebraic expressions of frequencies are illustrated. The expressions are obtained as functions of probabilities of crossing overs, number of generations and viability constants of zygotes. A numerical calculation is carried out for the case, in which some of the viability constants are equal to zero and the recombination probability is 0.3. The results obtained may be used for calculations of two linked genes for various combinations of values of recombination fractions, viability constants and number of generations. Derivation of zygotic frequencies of selfed population with more than two linked genes is possible by the use of the mathematical tools explained in the paper. After such derivations, one will also be able to make numerical calculations for any number of linked loci.