Left symmetric algebras from DNA insertion

Abstract

DNA recombination is a fundamental biological process that encodes genetic information for organism development and function. In this study, we construct the left symmetric algebras arising from the operation of DNA insertion. We define a new operation of insertion by modifying the simplified insertion $$x\Rightarrow y:=f(\mid x\mid ,\mid y\mid )\sum _{i=0}^q{y}_1{y}_2\cdots y_{i}x{y}_{i+1}\cdots y_q,$$ where $x=x_1{x}_2\cdots x_p$, $y=y_1{y}_2\cdots y_q$, and $\mid x\mid ,\mid y\mid$ denote the lengths of $x$ and $y$, respectively. We prove that the algebra $\mathbb{F}(R)$ (over a field $\mathbb{F}$ of characteristic $0$, with $R$ being an infinite free semigroup generated by DNA nucleotides $\{A,G,C,T\}$) forms a left symmetric algebra if and only if the function $f$ satisfies the condition $$f(m,n)f(m+n,p)=f(n,p)f(m,n+p)=f(m,p)f(n,m+p),$$ where $m,n,p\in \mathbb{N}$. A key example of such a function is $f(m,n)=\exp \{g(m,n)\}$, where $g(m,n)=k\cdot mn,$ and $k$ is a fixed positive number, which effectively models length-dependent DNA insertion dynamics. This work enriches the theory of non-associative algebras and provides a mathematical framework for quantitative analysis of DNA recombination processes.