Extracting Best Consensus Motifs from Positive and Negative Examples

摘要

We define the best consensus motif (BCM) problem motivated by the problem of extracting motifs from nucleic acid and amino acid sequences. A type over an alphabet sum is a family #OMEGA# of subsets of sum. A motif #pi# of type #OMEGA# is a string #pi# = #pi#_1 ...#pi#_n of motif components, each of which stands for an element in #OMEGA#. The BCM problem for #OMEGA# is, given a yes-no sample S={(#alpha#~((1)),#beta#~((1)),...,(#alpha#~((m)), #beta#~((m)))} of pairs of strings in sum with #alpha#~((i)) not = #beta#~((i)) for 1 <=i <=m, to find a motif #pi# of type #OMEGA# that maximizes the number of good pairs in S, where (#alpha#~((i)), #beta#~((i)) is good for #pi# if #pi# accepts #alpha#~((i)) and rejects #beta#~((i)). We prove that the BCM problem is NP-complete even for a very simple type #OMEGA#_1=2~sum - {#PHI#}, which is used, in practice, for describing protein motifs in the PROSITE database. We also show that the NP-completeness of the problem does not change for the type #OMEGA#_infinity = #OMEGA#_1 U {sum~+} U {sum~([i,j]) | 1 <=i<=j}, where sum~([i,j]) is the set of strings over sum of length between i and j. Furthermore, for the BCM problem for #OMEGA#_1, we provide a polynomal-time greedy algorithm based on the probabilistic method. Its performance analysis shows an explicit approximation ratio of the algorithm.