Angelika Wörz-Busekros

Polyploidy with an arbitrary mixture of chromosome- and chromatid segregation

SummaryThe gametic algebraG is constructed for a random mating population of 2r-ploid individuals which differ in a single locus with the allelesA anda. It is assumed that every kind of segregation between chromosome- and chromatid segregation occurs with a given probability. This algebraG is a convex combination of 2r+1 genetic algebras which have a common canonical basis. The train roots of these algebras are calculated and shown to be monotonically descending. The algebraG possesses a one-dimensional manifold of idempotents. With a generalization of Gonshors theorem on the convergence of the sequence of plenary powers of an element of unit weight it is shown that for every initial gametic distribution the distribution in the following generations converges towards an equilibrium state whose coordinates are polynomials in the frequency of the alleleA in the initial generation.

Solutions to a degenerate system of parabolic equations from marine biology

SummaryA system of parabolic and ordinary differential equations ut = a2uxx + F(u, v, w), vt = a2vxx + G(u, v, w),wx = − k(u)w is studied which has been proposed by Radach and Maier-Reimer for the dynamics of phytoplankton and nutrient in dependence of light intensity. It is shown that there is a unique solution to this system satisfying given initial and boundary conditions. The solution depends continuously on the data. For specific nonlinearities F, G, and k bounds for the solutions are given.

The zygotic algebra for sex linkage II

SummaryThe zygotic algebra 3 for a finite number of sex-linked characters with arbitrary segregation rates is defined in two equivalent ways. A sufficient condition for the existence and uniqueness of non-trivial idempotents in a baric ideal ℬ of 3 is given. A convergence theorem for the sequence of plenary powers of elements of ℬ of unit weight is proved. In the case of Mendelian segregation in the male sex the conditions for uniqueness of idempotents and for convergence are the same for symmetric inheritance and sex-linked inheritance. The special cases of Mendelian and additive segregation rates in the females are discussed in greater detail.

Dynamics of a continuous culture with catalysis of the growth-limiting substrate by an enzyme of the cells

In the present paper we discuss the behaviour of solutions of a dynamical system describing the growth of cells in a well-mixed continuous culture where the supply of the growth-limiting nutrient depends on the activity of an enzyme outside the cell membrane. It turns out that for positive dilution rates there exists an exponentially attractive two-dimensional simplex. Furthermore, the reversed system restricted to this simplex is quasimonotone. In every case all trajectories tend to an equilibrium state.

Construction of New Algebras

Suppose we have a “complex” genetic situation in which each particular feature can be described by an algebra.

The Non Commutative Case

Non commutative algebras occur in genetic applications if not all individuals of a population mate randomly, e.g. in populations with a sex linked character or in partially selfing populations, cf. section 8. But also from a theoretical point of view it may be of interest to establish how far the theory can be carried over to non commutative algebras.

Concluding Remarks and Bernstein Algebras

In the preceding two sections we have, with the exception of migration, considered all essential genetic situations which, have so far been described by algebras.

Train Algebras, Genetic Algebras and Special Train Algebras

In the following we assume that a is an n-dimensional commutative, not necessarily associative algebra over a field k. Then the aforementioned right and left properties coincide and we can neglect such attributes.

Idempotents and Sequences of Powers in Train Algebras, Genetic Algebras and Algebras with Genetic Realization

For the structure of an algebra the existence of idempotents, i.e. of elements e ≠ 0 with e2 = e, is of particular interest. But also from the biological aspect the existence of such elements is of interest because the equilibria of a population which can be described by an algebra correspond to idempotents of this algebra.