Polynomial dynamics of human blood genotypes frequencies

Abstract

The frequencies of human blood genotypes in the ABO and Rh systems differ between populations. Moreover, in a given population, these frequencies typically evolve over time. The possible reasons for the existing and expected differences in these frequencies (such as disease, random genetic drift, founder effects, differences in fitness between the various blood groups etc.) are the focus of intensive research. To understand the effects of historical and evolutionary influences on the blood genotypes frequencies, it is important to know how these frequencies behave if no influences at all are present. Under this assumption the dynamics of the blood genotypes frequencies is described by a polynomial dynamical system defined by a family of quadratic forms on the 17-dimensional projective space. To describe the dynamics of such a polynomial map is a task of substantial computational complexity.We give a complete analytic description of the evolutionary trajectory of an arbitrary distribution of human blood variations frequencies with respect to the clinically most important ABO and RhD antigens. We also show that the attracting algebraic manifold of the polynomial dynamical system in question is defined by a binomial ideal.