J. Math. Biology 5, 43-53 (1977)
Journal of
9 by Springer-Verlag 1977
An Explicit Formula for Frequency Changes in Genetic Algebras Ivar Heuch* Lehrstuhl f~r Biomathematik, Universitat T~bingen, D-7400 Tt~bingen, Federal Republic of Germany
Sgmmary. A general situation in population genetics is considered with any hereditary system described by a genetic algebra. It is assumed that there is random mating, no selection, and infinite population size. A formula is derived for the distribution of genetic types in the general jth generation given the distribution in the initial generation. Special attention is paid to the case of one locus with polyploidy and chromosome segregation. The calculations are carried out as far as possible in the situation with triploid gametes. 1. Introduction
Suppose that the genotype composition of a certain population in a fixed initial generation is known. It is then a basic problem in mathematical population genetics to construct an expression for the genotype distribution of this population in the general jth generation. The possibility of solving this problem depends very much on the kind of inheritance considered, and also on population structure and mating system. Even in the simple situations with non-overlapping generations, infinite populations, random mating, and no selection, the task of finding the desired expression can be formidable. For example, Geiringer's classical extensive papers on linked loci [2] and polyploidy [3] provide explicit expressions only in a few particular cases. None the less, Lyubich [12] succeeded in constructing such an expression for the situation with any number of linked autosomal loci. He applied the method of linear algebras in conjunction with differential operators as introduced by Reiersol [14]. While this method is well suited to the study of this particular type of inheritance, it seems that the problem must be attacked in a different way if we want a general formula which applies to most distinct situations of interest. Our derivation of a formula of this kind is motivated by the presentation in [12; Section l 1]. We retain the general representation of hereditary systems by commutative, non-assodative linear algebras as introduced by Ethedngton [1 ]. But * Work supported by the Alexander yon Humboldt Foundation.
44
I. Heuch
rather than specifying any particular system of this type, we postulate that the cases studied correspond to the special class of genetic algebras. This will be true [6] if we can construct a canonical basis Co, (71. . . . . Cn with a multiplication table of this form: =
CoC, =
Co + ~=I
CtC~ =
~
1
i
,,;
k=i
~'tjkCk,
1
Journal of
9 by Springer-Verlag 1977
An Explicit Formula for Frequency Changes in Genetic Algebras Ivar Heuch* Lehrstuhl f~r Biomathematik, Universitat T~bingen, D-7400 Tt~bingen, Federal Republic of Germany
Sgmmary. A general situation in population genetics is considered with any hereditary system described by a genetic algebra. It is assumed that there is random mating, no selection, and infinite population size. A formula is derived for the distribution of genetic types in the general jth generation given the distribution in the initial generation. Special attention is paid to the case of one locus with polyploidy and chromosome segregation. The calculations are carried out as far as possible in the situation with triploid gametes. 1. Introduction
Suppose that the genotype composition of a certain population in a fixed initial generation is known. It is then a basic problem in mathematical population genetics to construct an expression for the genotype distribution of this population in the general jth generation. The possibility of solving this problem depends very much on the kind of inheritance considered, and also on population structure and mating system. Even in the simple situations with non-overlapping generations, infinite populations, random mating, and no selection, the task of finding the desired expression can be formidable. For example, Geiringer's classical extensive papers on linked loci [2] and polyploidy [3] provide explicit expressions only in a few particular cases. None the less, Lyubich [12] succeeded in constructing such an expression for the situation with any number of linked autosomal loci. He applied the method of linear algebras in conjunction with differential operators as introduced by Reiersol [14]. While this method is well suited to the study of this particular type of inheritance, it seems that the problem must be attacked in a different way if we want a general formula which applies to most distinct situations of interest. Our derivation of a formula of this kind is motivated by the presentation in [12; Section l 1]. We retain the general representation of hereditary systems by commutative, non-assodative linear algebras as introduced by Ethedngton [1 ]. But * Work supported by the Alexander yon Humboldt Foundation.
44
I. Heuch
rather than specifying any particular system of this type, we postulate that the cases studied correspond to the special class of genetic algebras. This will be true [6] if we can construct a canonical basis Co, (71. . . . . Cn with a multiplication table of this form: =
CoC, =
Co + ~=I
CtC~ =
~
1
i
,,;
k=i
~'tjkCk,
1
