Algebraic model of non-Mendelian inheritance (Q542555)

Evolution algebra theory is used to study non-Mendelian inheritance, particularly organelle heredity and population genetics of Phytophthora infectans. The algebras can explain a puzzling feature of establishment of homoplasmy from a heteroplasmic cell population and the coexistence of mitochondrial triplasmy; they also can predict all mechanisms to form the homoplasmy of cell populations, which are hypothetical mechanisms in current mitochondrial disease research. The paper is focused on two particular genetic phenomena to show how evolution algebras work for them. One is the organelle population genetics, and the other is Phytophthora infestans population genetics. The author shows that concepts of algebraic transiency and algebraic persistency catch essences of biological transitory and stability, respectively. Coexistence of triplasmy in tissues of patients with sporadic mitochondrial disorders is also studied. Four different evolution algebras derived from the study of homoplasmy. They are not the same in the skeleton classification of algebras. Therefore, their dynamics, which are actually genetic evolution processes, are different. However, we need to look for what are the biological evidences for defining these different algebras. By \textit{F. Ling} and \textit{T. Shibata} [Mol. Biol. Cell, 15, 310--322 (2004)] several hypothetical mechanisms were put forward for establishment of homoplasmy, and their hypothetical mechanisms exactly correspond to four different algebraic structures presented in the article. Therefore, the author obtains an algebraic structure of mitochondrial genetic dynamics. Besides the experimental results, it is predicted that there are several transient phases. These transient phases are algebraic transient elements, that are important for medical treatments. The presented evolution algebra theory is applied to the study of algebraic structures of asexual progenies of Phytophthora infestans, and the following conclusion are made: (1) Evolution algebra theory can predict the existence of intermediate transient races. Intermediate transient races correspond to algebraic transient elements. They are biologically unstable, and will extinct or disappear by producing other races after a certain period of time. (2) Evolution algebra theory states that biologically stable races correspond to algebraic persistent elements. This predicts the periodicity of reproduction of stable races. (3) Evolution algebra can re-recover progeny subpopulations. Furthermore, because these progeny subpopulations correspond to simple subalgebras, each race in the same subpopulation shares the same dynamics of reproduction and spreading. (4) Evolution algebra theory provides a way to compute the frequency of each race in progeny populations given the reproduction rates, which are algebra structural constants. Practically, these frequencies can be measured, and so reproduction rates can be computed by formulas in evolution algebras. Therefore, evolution algebras will be a helpful tool to study many aspects of asexual reproduction processes, like that of Oomycetes, Phytophthora.