Left symmetric algebras from DNA insertion

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 $$x\Rightarrow y:=f(\mid x\mid,\ \mid y \mid)\sum\limits_{i=0}^{q} y_{1}y_{2}\cdots y_{i} x y_{i+1}\cdots y_{q},$$ 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.