Genetic algebra

A genetic algebra has the mathematical structure of an algebra and can be used for the mathematical modeling of inheritance in genetics .

motivation

Some facts in genetics can be described with certain mathematical structures , so-called algebras . The following simple example is intended to explain why these structures appear suitable for modeling genetic facts.

In a (very simple) population there are only two different gametes and . crossed with should again result in gametes of the type , the analogue applies to . If, on the other hand , one crosses with , half gametes of the type and half of the type should result. This can also be expressed formally as 'multiplication' and 'addition', the intersection of with for example through ${\displaystyle a_{1}}$ ${\displaystyle a_{2}}$ ${\displaystyle a_{1}}$ ${\displaystyle a_{1}}$ ${\displaystyle a_{1}}$ ${\displaystyle a_{2}}$ ${\displaystyle a_{1}}$ ${\displaystyle a_{2}}$ ${\displaystyle a_{1}}$ ${\displaystyle a_{2}}$ ${\displaystyle a_{1}}$ ${\displaystyle a_{2}}$

${\displaystyle a_{1}.a_{2}={1 \over 2}a_{1}+{1 \over 2}a_{2}}$

A mathematical structure in which this 'multiplication' and this 'addition' can be precisely defined is the non- associative algebra with ${\displaystyle G=\{\alpha _{1}a_{1}+\alpha _{2}a_{2}|\alpha _{1},\alpha _{2}\in \mathbb {R} \}}$

• ${\displaystyle a_{1}.a_{1}=a_{1}}$
• ${\displaystyle a_{2}.a_{2}=a_{2}}$
• ${\displaystyle a_{1}.a_{2}=a_{2}.a_{1}={1 \over 2}a_{1}+{1 \over 2}a_{2}}$

the gametic algebra of simple Mendelian inheritance is called.

This type of algebraic description makes it easier to consider various questions in genetics, such as: B .:

• What population results from repeated cross-breeding of a population with itself?
• Do states of equilibrium exist in a population, and if so, which ones?

In connection with genetics, there are special non-associative algebras such as Baric algebras , algebras with genetic realization , train algebras and genetic algebras . These algebras are not among the better known non-associative algebras of the Lie or Jordan algebras .

definition

A commutative, non-associative algebra A over a field K is called genetic algebra if a base exists such that the multiplication constants are defined by ${\displaystyle u_{1},u_{2},\dots ,u_{n}}$ ${\displaystyle \gamma _{ijk}}$

${\displaystyle u_{i}u_{j}=\sum _{k=1}^{n}\gamma _{ijk}u_{k}\qquad i,j=1,\ldots ,n,}$

have the following properties:

a) ${\displaystyle \gamma _{111}=1}$

b) for ${\displaystyle \gamma _{1jk}=0}$ ${\displaystyle k<j}$

c) for and ${\displaystyle \gamma _{ijk}=0}$ ${\displaystyle i,j>1}$ ${\displaystyle k\leq \max\{i,j\}}$

The base is called the canonical base . ${\displaystyle u_{1},u_{2},\dots ,u_{n}}$

properties

• Every genetic algebra is a Baric algebra .

Further definitions

In a non-associative algebra, the product of more than two elements of the algebra is not uniquely determined by their order. The special products defined below have interesting genetic interpretations.

Let A be an algebra , ${\displaystyle x\in A}$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle x^{n}}$is called the nth major right power of x, where:

${\displaystyle x^{1}=x}$ and ${\displaystyle x^{n}=x^{n-1}\cdot x}$

In the same way, one defines main powers on the left , in the commutative case one speaks only of main powers .

${\displaystyle x^{[n]}}$is called the nth plenary power of x, where:

${\displaystyle x^{[1]}=x}$ and ${\displaystyle x^{[n]}=x^{[n-1]}\cdot x^{[n-1]}}$

The genetic interpretation of the main potencies is as follows: If you cross a population, which is represented by, with yourself, you get a population, which is represented by. If one crosses the resulting population with the original one, then it arises . The sequence of populations that is created by repeating this process is thus represented by the sequence of the main powers of x. ${\displaystyle x\in A}$ ${\displaystyle x\cdot x=x^{2}}$ ${\displaystyle x^{2}\cdot x=x^{3}}$

If, on the other hand, one crosses a population repeatedly with oneself, then the sequence of populations created in this way can be described by the associated sequence of plenary powers.

literature

• Harald Geppert and Siegfried Koller : Erbmathematik . Quelle and Meyer, Leipzig 1938
• Otfried Mittmann : Hereditary biological questions in mathematical treatment . De Gruyter, Berlin 1940
• Erna Weber : Mathematical Foundations of Genetics . Gustav Fischer , Jena 1967
• Rudolf Lidl and Günter Pilz: Applied Abstract Algebra II . Bibliographisches Institut, Mannheim Vienna Zurich 1982 ISBN 3-411-01621-3 .
• Angelika Wörz-Busekros: Algebras in Genetics . Springer-Verlag, Berlin Heidelberg New York 1980 ISBN 3-540-09978-6 .
• H. Gonshor: Contributions to genetic algebras . Proc. Edinb. Math. Soc. (2), 17 (1971), 289-298.