## Coalgebraic Structure of Genetic Inheritance

### Jianjun (Paul) Tian (Mathematical Biosciences Institute, The Ohio State University)

#### Abstract

Although in the broadly defined genetic algebra, multiplication suggests a forward direction from parents to progeny, when looking from the reverse direction, it also suggests to us a new algebraic structure - coalgebraic structure, which we call genetic coalgebras. It is not the dual coalgebraic structure and can be used in the construction of phylogenetic trees. Mathematically, to construct phylogenetic trees means we need to solve equations x^[n]=a, or x^(n)=a. It is generally impossible to solve these equations in algebras. However, we can solve them in coalgebras in the sense of tracing back for their ancestors. A thorough exploration of coalgebraic structure in genetics is apparently necessary. Here, we develop a theoretical framework of the coalgebraic structure of genetics. From biological viewpoint, we defined various fundamental concepts and examined their elementary properties that contain genetic significance. Mathematically, by genetic coalgebra, we mean any coalgebra that occurs in genetics. They are generally noncoassociative and without counit; and in the case of non-sex-linked inheritance, they are cocommutative. Each coalgebra with genetic realization has a baric property. We have also discussed the methods to construct new genetic coalgebras, including cocommutative duplication, the tensor product, linear combinations and the skew linear map, which allow us to describe complex genetic traits. We also put forward certain theorems that state the relationship between gametic coalgebra and gametic algebra. By Brower's theorem in topology, we prove the existence of equilibrium state for the in-evolution operator. (The paper is available http://math.asu.edu/~mbe/, Vol.1, 2. pp.243-266)

Note: Joint work with Bai-Lian Li, Department of Botany and Plant Sciences, University of California, Riverside.