Left-Symmetric Algebras Arising from Modified DNA Insertion Operations

DNA recombination is a fundamental biological process that encodes genetic information for organism development and function. In this study, we construct left-symmetric algebras arising from DNA insertion operations. That is, we define a modified insertion operation by weighting the simplified insertion. It generalizes the left-symmetric algebra constructed from the simplified DNA insertion operation. We prove that the algebra F (R) (over a field 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 a certain multiplicative condition for all positive integers m, n, and p. A key example of such a function is f (m, n) =expg (m, n), where g (m, n) =k·mn, and k is a fixed positive number, which effectively models length-dependent DNA insertion dynamics. This work contributes an algebraic framework that may be useful for quantitative modeling of DNA recombination processes.