On DNA Codes

Abstract

We develop and study the concept of similarity functions for q-ary sequences. For the case q = 4, these functions can be used for a mathematical model of the DNA duplex energy [1,2], which has a number of applications in molecular biology. Based on these similarity functions, we define a concept of DNA codes [1]. We give brief proofs for some of our unpublished results [3] connected with the well-known deletion similarity function [4][5][6]. This function is the length of the longest common subsequence; it is used in the theory of codes that correct insertions and deletions [5]. Principal results of the present paper concern another function, called the similarity of blocks. The difference between this function and the deletion similarity is that the common subsequences under consideration should satisfy an additional biologically motivated [2] block condition, so that not all common subsequences are admissible. We prove some lower bounds on the size of an optimal DNA code for the block similarity function. We also consider a construction of close-to-optimal DNA codes which are subcodes of the parity-check one-error-detecting code in the Hamming metric [7]. 0032-9460/05/4104-0349 c 2005 Pleiades Publishing, Inc.

On critical relative distance of DNA codes for additive stem similarity

“…Identity (5) implies the symmetry property of hybridization energy between DNA sequences x and y [7]- [13]. Example 1: In [18] we considered constant weights w = w(a, b) ≡ 1, a, b ∈ A, for which the additive stem 1-similarity S 1 (x, y), 0 ≤ S 1 (x, y) ≤ S 1 (x, x) = n − 1, is the abovementioned number of stems in the longest common Hamming subsequence between x and y.…”

“…. x n ) are identically distributed in accordance with the Markov chain (13) and, in virtue of (11), the corresponding reverse complement codewords x = (x nxn−1 . .…”

“…When these collections consist entirely of pairs of mutually reverse complementary DNA strands they are called DNA tag-antitag systems [4] and DNA codes [7]- [13].…”

“…Later attempts included deletion similarity [8], which was earlier introduced by Levenshtein [17] and block similarity [12]- [13]. Both functions are non-additive which allowed for consideration of such cases as shifts of DNA sequences along each other.…”

New results on DNA codes for additive stem similarity

“…When these collections consist entirely of pairs of mutually reverse complementary DNA strands they are called DNA tag-antitag systems [3] and DNA codes [5]- [9].…”