Biomolecular network motif counting and discovery by color coding

Abstract

Protein–protein interaction (PPI) networks of many organisms share global topological features such as degree distribution, k-hop reachability, betweenness and closeness. Yet, some of these networks can differ significantly from the others in terms of local structures: e.g. the number of specific network motifs can vary significantly among PPI networks.Counting the number of network motifs provides a major challenge to compare biomolecular networks. Recently developed algorithms have been able to count the number of induced occurrences of subgraphs with k≤ 7 vertices. Yet no practical algorithm exists for counting non-induced occurrences, or counting subgraphs with k≥ 8 vertices. Counting non-induced occurrences of network motifs is not only challenging but also quite desirable as available PPI networks include several false interactions and miss many others.In this article, we show how to apply the ‘color coding’ technique for counting non-induced occurrences of subgraph topologies in the form of trees and bounded treewidth subgraphs. Our algorithm can count all occurrences of motif G′ with k vertices in a network G with n vertices in time polynomial with n, provided k=O(log n). We use our algorithm to obtain ‘treelet’ distributions for k≤ 10 of available PPI networks of unicellular organisms (Saccharomyces cerevisiae Escherichia coli and Helicobacter Pyloris), which are all quite similar, and a multicellular organism (Caenorhabditis elegans) which is significantly different. Furthermore, the treelet distribution of the unicellular organisms are similar to that obtained by the ‘duplication model’ but are quite different from that of the ‘preferential attachment model’. The treelet distribution is robust w.r.t. sparsification with bait/edge coverage of 70% but differences can be observed when bait/edge coverage drops to 50%.Contact:cenk@cs.sfu.ca

Randomised Enumeration of Small Witnesses Using a Decision Oracle

“…Positive results in this setting typically exploit structural properties of the graphs involved (e.g. small treewidth) to design (approximate) counting algorithms for inputs with these properties, avoiding any dependence on N [2,3,18].…”

Subgraph Counting: Color Coding Beyond Trees

“…In a different work, Alon et al [1] explored its applications to approximate subgraph counting (most commonly known as motif counting) in computational biology. They were motivated by the fact that subgraph counting is an important primitive for characterizing biological networks [?].…”

“…The search space of non-induced subgraphs is larger and furthermore, these counts are more robust with respect to perturbations of the data graph [1].…”

“…In a different work, Alon et al [1] applied the color coding technique for counting the occurrences of treelets (tree queries) in biological networks. Color coding allowed them to handle tree queries up to size 10 in protein interaction networks, extending beyond the reach of previously known approaches [25,18,17].…”

“…To the best of our knowledge, the previously-mentioned algorithm [1] is the best known algorithm for treewidth 2 queries, and we use it as our baseline. We rephrase this algorithm within our framework and devise a distributed implementation.…”