Algebraic Theory of DNA Recombination

In this paper we investigate the structure and representation of n-ary algebras arising from DNA recombination, where n is a number of DNA segments participating in recombination. Our methods involve a generalization of the Jordan formalization of observables in quantum mechanics in n-ary splicing algebras. We show that the splicing algebras are an n-ary envelope for algebras of DNA recombination. We have constructed the basis for free algebra of the variety of the n-ary splicing algebras and found the defining identities for n-ary splicing operations. Using the relationship between algebras and its enveloping algebras, we have constructed the basis of the free algebra of the variety of nary algebras of DNA recombination. It is proved that every polynomial identity satisfied by n-ary DNA recombination, with no restriction on the degree, is consequence of n-ary commutativity and the special n-ary identity of the degree 3n-2. We obtain a criterion, analogous to the Specht-Wever Lie criterion, for determining whether an element of n-ary free splicing algebra is an element of the algebra of DNA recombination. Using this criterion, it is proved that all n-ary algebras of DNA recombination are special by module of n-ary splicing algebra (analog of the Poincare-Birkhoff-Witt theorem). The skew-symmetrization of n-ary splicing operation converts the splicing algebras into n-ary skew-symmetric algebras. In case n = 2, they are Lie algebras. It is proved that every polynomial identity of these algebras, with no restriction on the degree, is consequence of centrally metabelian identity. In case n >2, they are nilpotent of index 3. The non-homologous recombination is formalized by the algebras of simplified insertions. It is shown that all identities of the algebra of simplified insertion follow from the right-symmetric identity. We construct an infinite series of relations in the algebra of simplified insertion which hold for the words of length n and are not consequences of the right-symmetric identity