# Finding Disjoint Motifs in a Gene solved by 251

April 20, 2013, 2:30 p.m. by Rosalind Team

## Disjoint Motifs

In this problem, we will consider an algorithmic (but not particularly practical) variant of motif finding for multiple motifs. Say we have two motifs corresponding to the coding regions of genes, and we want to know whether these motifs can be found in genes occupying the same region of the genome. To prevent exons from coinciding, we further stipulate that the two motifs are nonoverlapping.

In this problem, we will ask whether two disjoint motifs can be located in a given string. We considered a similar problem in “Interleaving Two Motifs”, which asked us to find a shortest possible string containing two motifs; however, in that problem, the motifs were allowed to coincide.

## Problem

Given three strings $s$, $t$, and $u$, we say that $t$ and $u$ can be interwoven into $s$ if there is some substring of $s$ made up of $t$ and $u$ as disjoint subsequences.

For example, the strings "$\color{blue}\text{ACAG}$" and "$\color{red}\text{CCG}$" can be interwoven into "$\color{black}\text{G}\color{blue}\text{A}\color{red}\text{C}\color{blue}\text{CA}\color{red}\text{C}\color{blue}\text{G}\color{red}\text{G}\color{black}\text{TT}$". However, they cannot be interwoven into "$\color{black}\text{G}\color{blue}\text{A}\color{red}\text{C}\color{blue}\text{CA}\color{red}\text{C}\color{black}\text{AAAA}\color{blue}\text{G}\color{red}\text{G}\color{black}\text{TT}$" because of the appearance of the four 'A's in the middle of the subsequences. Similarly, even though both "$\textrm{ACACG}$" is a shortest common supersequence of $\textrm{ACAG}$ and $\textrm{CCG}$, it is not possible to interweave these two strings into "$\textrm{ACACG}$" because the two desired subsequences must be disjoint; see “Interleaving Two Motifs” for details on finding a shortest common supersequence of two strings.

Given: A text DNA string $s$ of length at most 10 kbp, followed by a collection of $n$ ($n \leq 10$) DNA strings of length at most 10 bp acting as patterns.

Return: An $n \times n$ matrix $M$ for which $M_{j, k} = 1$ if the $j$th and $k$th pattern strings can be interwoven into $s$ and $M_{j, k} = 0$ otherwise.

## Sample Dataset

GACCACGGTT
ACAG
GT
CCG


## Sample Output

0 0 1
0 1 0
1 0 0


## Citation

This problem follows Jones & Pevzner, An Introduction to Bioinformatics Algorithms, Problem 6.31