Evolution Algebras and Non-Mendelian Genetics

In this chapter, we shall apply evolution algebra theory to non-Mendelian genetics. In the first section, we give a brief reflection of how non-Mendelian genetics motivated the development evolution algebras. In section 2, we review the basic biological components of non-Mendelian genetics and the inheritance of organelle genes; we also give a general algebraic formulation of non-Mendelian genetics. In section 3, we use evolution algebras to study the heteroplasmy and homoplasmy of organelle populations, and show that concepts of algebraic transiency and algebraic persistency relate to biological transitory and stability, respectively. Coexistence of triplasmy in tissues of patients with sporadic mitochondrial disorders is studied as well. In section 4, we apply evolution algebra theory to the study of asexual progenies of Phytophthora infestans, an important agricultural pathogen.