Mining Correlated Motifs with Wildcards in PPI Networks: Exact and Optimal
A protein-protein interaction (PPI) network is a graph whose nodes are protein sequences and whose edges represent physical interactions between proteins. Correlated Motif Mining (CMM) seeks pairs of sequence motifs whose co-occurrence in interacting proteins reveals binding sites and interaction mechanisms. Formally, in the $(\ell,d)$-CMM problem, we are given a graph $G=(V,E)$, with $V\subseteqΣ^*$ for an alphabet $Σ$, and integers $0\leq d<\ell$, and we seek a pair $\{X,Y\}$ of motifs, each of length $\ell$ and containing $d$ wildcards, that maximizes a support measure $f(G,\{X,Y\})$; e.g., the count $f_E$ of edges $\{u,v\}\in E$ such that $X$ matches a substring of $u$ and $Y$ matches a substring of $v$, or vice versa. Although existing approaches have produced biologically meaningful results, they are either inefficient or heuristic-based, as the underlying optimization problem is NP-hard. Here, we revisit $(\ell,d)$-CMM. For $d=0$, we give an exact $\mathcal{O}(N^2)$-time and $\mathcal{O}(N)$-space algorithm, where $N$ is the total length of the node strings, and prove that this time is conditionally optimal under the Strong Exponential-Time Hypothesis. For any $d>0$, we give an exact $\mathcal{O}\!\left(\binom{\ell}{d}^2N^2\right)$-time and $\mathcal{O}(N)$-space algorithm, and observe that the problem is W[1]-hard parameterized by $d$. The latter algorithm improves the state of the art by exponentially large factors in both time and space. We further demonstrate experimentally that our algorithms are practical on real-world PPI networks and find significantly higher-scoring motif pairs than the state of the art. Moreover, we show that our algorithms find many top-scoring motif pairs that the state of the art misses entirely. Taken together, our results suggest that exact CMM with wildcards is more tractable in practice than previously assumed.