dorsal/arxiv
View SchemaVector space of DNA genomic sequences on a Genetic Code Galois Field
| Authors | Robersy Sanchez, Eberto R Morgado, Ricardo Grau |
|---|---|
| Categories | |
| ArXiv ID | q-bio/0510029 |
| URL | https://arxiv.org/abs/q-bio/0510029 |
Abstract
A new N-dimensional vector space of DNA sequences over the Galois field of the 64 codons (GF(64)) was recently presented. Now, in order to include deletions and insertions (indel mutations), we have defined a new Galois field over the set of elements X1X2X3 (C125), where Xi belong to {O, A, C, G, C}. We have called this set, the extended triplet set and the elements X1X2X3, the extended triplets. The order of the bases is derived from the Z64-algebra of the genetic code -recently published-. Starting from the natural bijection phi: GF(5^3)-> C125 between the polynomial representation of elements from GF(5^3) and the elements X1X2X3, a novel Galois field over the set of elements X1X2X3 is defined. Taking the polynomial coefficients a0, a1, a2 belong to GF(5) and the bijective function f: GF(5) ->{O, A, C, G, C}, where f(0) = O, f(1) = A, f(2) = C, f(3) = G, f(4) = U, bijection phi is induced such that phi(a0 + a1x + a2x^2) = (f(a1), f(a2), f(a0)) = (X1X2X3). Next, by means of the bijection phi we define sum "+" and product "*" operations in the set of codons C125, in such a way that the resultant field (C125, +, *) turns isomorphic to the Galois Field GF(5^3). This field allows the definition of a novel N-dimensional vector space (S) over the field GF (5^3) on the set of all 125^N sequences of extended triplets in which all possible DNA sequence alignments of length N are included. Here the "classical gap" produced by alignment algorithms corresponds to the neutral element "O". It is verified that the homologous (generalized) recombination between two homologous DNA duplexes involving a reciprocal exchange of DNA sequences -e.g. between two chromosomes that carry the same genetic loci- algebraically corresponds to the action of two automorphism pairs (or two translation pairs) over two paired DNA duplexes.
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"abstract": "A new N-dimensional vector space of DNA sequences over the Galois field of\nthe 64 codons (GF(64)) was recently presented. Now, in order to include\ndeletions and insertions (indel mutations), we have defined a new Galois field\nover the set of elements X1X2X3 (C125), where Xi belong to {O, A, C, G, C}. We\nhave called this set, the extended triplet set and the elements X1X2X3, the\nextended triplets. The order of the bases is derived from the Z64-algebra of\nthe genetic code -recently published-. Starting from the natural bijection phi:\nGF(5^3)-\u003e C125 between the polynomial representation of elements from GF(5^3)\nand the elements X1X2X3, a novel Galois field over the set of elements X1X2X3\nis defined. Taking the polynomial coefficients a0, a1, a2 belong to GF(5) and\nthe bijective function f: GF(5) -\u003e{O, A, C, G, C}, where f(0) = O, f(1) = A,\nf(2) = C, f(3) = G, f(4) = U, bijection phi is induced such that phi(a0 + a1x +\na2x^2) = (f(a1), f(a2), f(a0)) = (X1X2X3). Next, by means of the bijection phi\nwe define sum \"+\" and product \"*\" operations in the set of codons C125, in such\na way that the resultant field (C125, +, *) turns isomorphic to the Galois\nField GF(5^3). This field allows the definition of a novel N-dimensional vector\nspace (S) over the field GF (5^3) on the set of all 125^N sequences of extended\ntriplets in which all possible DNA sequence alignments of length N are\nincluded. Here the \"classical gap\" produced by alignment algorithms corresponds\nto the neutral element \"O\". It is verified that the homologous (generalized)\nrecombination between two homologous DNA duplexes involving a reciprocal\nexchange of DNA sequences -e.g. between two chromosomes that carry the same\ngenetic loci- algebraically corresponds to the action of two automorphism pairs\n(or two translation pairs) over two paired DNA duplexes.",
"arxiv_id": "q-bio/0510029",
"authors": [
"Robersy Sanchez",
"Eberto R Morgado",
"Ricardo Grau"
],
"categories": [
"q-bio.GN",
"q-bio.QM"
],
"title": "Vector space of DNA genomic sequences on a Genetic Code Galois Field",
"url": "https://arxiv.org/abs/q-bio/0510029"
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