Coalgebraic Structure of Genetic Inheritance - Jianjun Tian; Bai-Lian Li (Mathematical Biosciences and Engineering)
Although in the broadly defined genetic algebra, multiplicationsuggests a forward direction of from parents to progeny, when looking from the reversedirection, it also suggests to us a new algebraic structure ---coalgebraic structure, which we call genetic coalgebras. It isnot the dual coalgebraic structure and can be used in theconstruction of phylogenetic trees. Mathematically, to constructphylogenetic trees means we need to solve equations x[n]=a, or x(n)=b. It is generally impossible tosolve these equations in algebras. However, we can solve themin coalgebras in the sense of tracing back for their ancestors. Athorough exploration of coalgebraic structure in genetics isapparently necessary. Here, we develop a theoretical frameworkof the coalgebraic structure of genetics. From biological viewpoint, we defined various fundamental concepts andexamined their elementary properties that contain geneticsignificance. Mathematically, by genetic coalgebra, we mean anycoalgebra that occurs in genetics. They are generallynoncoassociative and without counit; and in the case ofnon-sex-linked inheritance, they are cocommutative. Eachcoalgebra with genetic realization has a baric property. We havealso discussed the methods to construct new genetic coalgebras,including cocommutative duplication, the tensor product, linearcombinations and the skew linear map, which allow us to describecomplex genetic traits. We also put forward certain theoremsthat state the relationship between gametic coalgebra and gameticalgebra. By Brower's theorem in topology, we prove the existenceof equilibrium state for the in-evolution operator.