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<p><b>Abstract</b>—Contact maps are a model to capture the core information in the structure of biological molecules, e.g., proteins. A contact map consists of an ordered set <tmath>S</tmath> of elements (representing a protein's sequence of amino acids), and a set <tmath>A</tmath> of element pairs of <tmath>S</tmath>, called <it>arcs</it> (representing amino acids which are closely neighbored in the structure). Given two contact maps <tmath>(S,A)</tmath> and <tmath>(S_p,A_p)</tmath> with <tmath>|A|\geq |A_p|</tmath>, the C<scp>ontact</scp> M<scp>ap</scp> P<scp>attern</scp> M<scp>atching</scp> (CMPM) problem asks whether the "pattern”<tmath>(S_p,A_p)</tmath> "occurs” in <tmath>(S,A)</tmath>, i.e., informally stated, whether there is a subset of <tmath>|A_p|</tmath> arcs in <tmath>A</tmath> whose arc structure coincides with <tmath>A_p</tmath>. CMPM captures the biological question of finding structural motifs in protein structures. In general, CMPM is NP-hard. In this paper, we show that CMPM is solvable in <tmath>O(|A|^6|A_p|^2)</tmath> time when the pattern is <tmath>\{<,{\hbox{{\rlap{)}\kern -1pt{\hbox{(}}}}}\}{\hbox{-}}{\rm{structured}}</tmath>, i.e., when each two arcs in the pattern are disjoint or crossing. Our algorithm extends to other closely related models. In particular, it answers an open question raised by Vialette that, rephrased in terms of contact maps, asked whether CMPM for <tmath>\{<,{\hbox{{\rlap{)}\kern -1pt{\hbox{(}}}}}\}{\hbox{-}}{\rm{structured}}</tmath> patterns is NP-hard or solvable in polynomial time. Our result stands in sharp contrast to the NP-hardness of closely related problems. We provide experimental results which show that contact maps derived from real protein structures can be processed efficiently.</p>
Pattern matching, algorithm design and analysis, biology and genetics.

J. Gramm, "A Polynomial-Time Algorithm for the Matching of Crossing Contact-Map Patterns," in IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol. 1, no. , pp. 171-180, 2004.
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