### 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%.

Original language | English (US) |
---|---|

Journal | Bioinformatics |

Volume | 24 |

Issue number | 13 |

DOIs | |

State | Published - Jan 1 2008 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Statistics and Probability
- Biochemistry
- Molecular Biology
- Computer Science Applications
- Computational Theory and Mathematics
- Computational Mathematics

### Cite this

*Bioinformatics*,

*24*(13). https://doi.org/10.1093/bioinformatics/btn163

**Biomolecular network motif counting and discovery by color coding.** / Alon, Noga; Dao, Phuong; Hajirasouliha, Iman; Hormozdiari, Fereydoun; Sahinalp, S. Cenk.

Research output: Contribution to journal › Article

*Bioinformatics*, vol. 24, no. 13. https://doi.org/10.1093/bioinformatics/btn163

}

TY - JOUR

T1 - Biomolecular network motif counting and discovery by color coding

AU - Alon, Noga

AU - Dao, Phuong

AU - Hajirasouliha, Iman

AU - Hormozdiari, Fereydoun

AU - Sahinalp, S. Cenk

PY - 2008/1/1

Y1 - 2008/1/1

N2 - 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%.

AB - 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%.

UR - http://www.scopus.com/inward/record.url?scp=84975860562&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84975860562&partnerID=8YFLogxK

U2 - 10.1093/bioinformatics/btn163

DO - 10.1093/bioinformatics/btn163

M3 - Article

C2 - 18586721

AN - SCOPUS:84975860562

VL - 24

JO - Bioinformatics

JF - Bioinformatics

SN - 1367-4803

IS - 13

ER -