Chapter Abstract:
In this thesis, an algebra is constructed for studying the dynamics of Mendelian populations. The symbols of the algebra represent groups of individuals or populations. T...Show MoreMetadata
Chapter Abstract:
In this thesis, an algebra is constructed for studying the dynamics of Mendelian populations. The symbols of the algebra represent groups of individuals or populations. The indexed symbol ?h J i k, for example, represents a population in which two gene loci are under consideration (the number of loci corresponds to the number of pairs of indices). The number of allelomorphs in each locus is completely arbitrary, as is the recombination value for the loci. The different components of a population symbol, represented by fixing the indices at specific values, are numbers whose values correspond to the fractions of the population with certain genetic formulae. It is convenient in some cases to consider as populations symbols whose components are negative or even complex. Such symbols cannot, of course, represent an actual group of individuals and are called unrealizable populations, but their use sometimes facilitates the solution of problems. Addition of two population symbols, R?h j i k + S?h j i k, results in a third population symbol which is defined in such a way as to represent the population obtained by merely combining the original populations in fractional proportions corresponding to the scalar coefficients R and S. Cross multiplication of population symbols ?h j i k ? ?h j i k gives a population symbol which is defined in such a way as to represent the expected offspring population when the two original populations are crossmated at random. When two gene loci are considered, this is realized by the mathematical definition ?h j i k ? ?h j i k = 1/2[p0?h ? i ? + p1?h ? ? i][p0?j ? k ? + p1?j ? ? k] + 1/2[p0?j ? k ? + p1?j ? ? k][p0?h ? i ? + p1?h ? ? i] in which p 1 = 1 - p0 is the recombination value for the two loci, and replacing an index by a dot indicates summation of the population symbol on that index. Cross multiplication is defined analogously for n loci. It is shown that this algebra is commutative on addition and multiplication, distributive, and ass...
Page(s): 891 - 920
Copyright Year: 1993
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