Abstract
Background The deviation of the observed frequency of a word w from its expected frequency in a given sequence x is used to determine whether or not the word is avoided. This concept is particularly useful in DNA linguistic analysis. The value of the deviation of w, denoted by textit{dev}(w), effectively characterises the extent of a word by its edge contrast in the context in which it occurs. A word w of length k>2 is a rho -avoided word in x if textit{dev}(w) le rho , for a given threshold rho < 0. Notice that such a word may be completely absent from x. Hence, computing all such words naïvely can be a very time-consuming procedure, in particular for large k.Results In this article, we propose an mathcal {O}(n)-time and mathcal {O}(n)-space algorithm to compute all rho -avoided words of length k in a given sequence of length n over a fixed-sized alphabet. We also present a time-optimal mathcal {O}(sigma n)-time algorithm to compute all rho -avoided words (of any length) in a sequence of length n over an integer alphabet of size sigma . In addition, we provide a tight asymptotic upper bound for the number of rho -avoided words over an integer alphabet and the expected length of the longest one. We make available an implementation of our algorithm. Experimental results, using both real and synthetic data, show the efficiency and applicability of our implementation in biological sequence analysis.ConclusionsThe systematic search for avoided words is particularly useful for biological sequence analysis. We present a linear-time and linear-space algorithm for the computation of avoided words of length k in a given sequence x. We suggest a modification to this algorithm so that it computes all avoided words of x, irrespective of their length, within the same time complexity. We also present combinatorial results with regards to avoided words and absent words.
Highlights
The deviation of the observed frequency of a word w from its expected frequency in a given sequence x is used to determine whether or not the word is avoided
We present a time-optimal O(σ n)-time algorithm to compute all ρ-avoided words in a sequence of length n over an integer alphabet of size σ
We presented a time-optimal O(σ n)-time algorithm to compute all ρ -avoided words in a sequence of length n over an integer alphabet
Summary
The deviation of the observed frequency of a word w from its expected frequency in a given sequence x is used to determine whether or not the word is avoided. Words which may be present in a genome or in genomic sequences of a specific role (e.g., protein coding segments, regulatory elements, conserved non-coding elements etc.) but they are strongly underrepresented—as we can estimate on the basis of the frequency of occurrence of their longest proper factors— may be of particular importance They can be words of nucleotides which are hardly tolerated because they negatively influence the stability of the chromatin or, more generally, the functional genomic conformation; they can represent targets of restriction endonucleases which may be found in bacterial and viral genomes; or, more generally, they may be short genomic regions whose presence in wide parts of the genome are not tolerated for less known reasons. The understanding of such avoidances is becoming an interesting line of research (for recent studies, see [7, 8])
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