Algebraic structure of genetic inheritance

Abstract

In this paper we will explore the nonassociative algebraic structure that naturally occurs as genetic information gets passed down through the generations. While modern understanding of genetic inheritance initiated with the theories of Charles Darwin, it was the Augustinian monk Gregor Mendel who began to uncover the mathematical nature of the subject. In fact, the symbolism Mendel used to describe his first results (e.g., see his 1866 paper Experiments in Plant-Hybridization) is quite algebraically suggestive. Seventy four years later, I.M.H. Etherington introduced the formal language of abstract algebra to the study of genetics in his series of seminal papers [Genetic algebras. Proc. Roy. Soc. Edinburgh, 59:242–258, 1939], [Duplication of linear algebras. Proc. Edinburgh Math. Soc. (2), 6:222–230, 1941.], [Non-associative algebra and the symbolism of genetics. Proc. Roy. Soc. Edinburgh, 61:24–42, 1941.]. In this paper we will discuss the concepts of genetics that suggest the underlying algebraic structure of inheritance, and we will give a brief overview of the algebras which arise in genetics and some of their basic properties and relationships. With the popularity of biologically motivated mathematics continuing to rise, we offer this survey article as another example of the breadth of mathematics that has biological significance. The most comprehensive reference for the mathematical research done in this area (through 1980) is Wörz-Busekros [Algebras in Genetics. Lecture Notes in Biomathematics, vol. 36, Springer-Verlag, New York, 1980].