CSCI1950 Z Computa3onal Methods for Biology Lecture 24. Ben Raphael April 29, hgp://cs.brown.edu/courses/csci1950 z/ Network Mo3fs

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1 CSCI1950 Z Computa3onal Methods for Biology Lecture 24 Ben Raphael April 29, 2009 hgp://cs.brown.edu/courses/csci1950 z/ Network Mo3fs Subnetworks with more occurrences than expected by chance. How to find? Exhaus3ve: Count all k node subgraphs. Heuris3c methods: sampling, greedy, etc. Approximate coun3ng via randomized algorithms. 1

2 Network Mo3fs Subnetworks with more occurrences than expected by chance. How to assess sta3s3cal significance? Compare number of occurrences to random network. Random Networks Occurrence of mo3fs depend strongly on network topology. What is an appropriate ensemble of random networks? (null model) 2

3 Random Networks One parameter governing occurrence of mo3fs is degree distribu3on. hgps://nwb.slis.indiana.edu/community/?n=customfillings.analysisofbiologicalnetworks Preserving Degree Distribu3on How to sample a graph with the same degree sequence? Method of Newman, Strogatz and Watts (2001) 1. Assign indegree i(v) and outdegree o(v) to vertex v according to degree sequence. 2. Randomly pair o(v) and i(w). 3

4 Network Mo3fs Transcrip3onal regulatory network of E. coli: 116 transcrip3on factors ~700 genes (operons) 577 interac3ons. Shen Orr et al E. coli Network Mo3fs Enumerated all 3 and 4 node mo3fs. Looked for iden3cal rows in adjacency matrix (SIM) Used clustering algorithm to iden3fy DOR. Shen Orr et al

5 Coun3ng Subnetworks G = (V,E). V = n. E = m. Network centric approach Count/enumerate all subgraphs with k ver3ces. Imprac3cal for large n, m, k Query based approach Enumerate query graphs Q. For each Q, count occurrences. (Subgraph isomorphism) Q could be a non induced subgraph. Coun3ng non induced subgraphs Suppose want to count paths in G = (V,E). Idea: use color coding to count colorful paths Dynamic programming solu3on (Whiteboard) Can extend dynamic program to count trees and bounded treewidth graphs. 5

6 Rela3on between Forward and Viterbi VITERBI Ini0aliza0on: V 0 (0) = 1 V k (0) = 0, for all k > 0 Itera0on: FORWARD Ini0aliza0on: f 0 (0) = 1 f k (0) = 0, for all k > 0 Itera0on: V j (i) = e j (x i ) max k V k (i 1) a kj f l (i) = e l (x i ) Σ k f k (i 1) a kl Termina0on: Termina0on: P(x, π*) = max k V k (N) P(x) = Σ k f k (N) a k0 Importance of Network Mo3fs Building block of networks. Indicate modular structure of biological networks. Appearance of some mo3fs might be explained by par3cular dynamics (e.g. feedforward and feedback loops) Healthy skep3cism about all these claims, par3cularly because data is incomplete. 6

7 Network Integra3on Given: G = (V,E) interac3on network. V = genes E = protein DNA or proteinprotein interac3ons Normalized expression z score z ij for gene i in condi3on/sample j. Goal: Find ac3ve subnetworks. Subgraphs whose genes are are differen3ally expressed in many condi3ons. Ideker, et al. (2002); Chuang et al. (2007) (Whiteboard) Network Integra3on Given: G = (V,E) interac3on network. V = genes E = protein DNA or proteinprotein interac3ons M = [ z ij ] z scores of gene i in condi3on/sample j. Ideker, et al. (2002); Chuang et al. (2007) Goal: Find A* = argmax r A A: connected subgraph 7

8 Finding High scoring subnetwork Simulated Annealing: Global op3miza3on method. Iden3fy set of ac3ve nodes. G w = working subgraph induced by ac3ve nodes. Based on idea of random, local search similar to MCMC. Temperature func3on controls when moves to subop3mal neighbors are permitng. Temperature decreased during search, so that eventually segle in local op3mum. Results 8

9 Future: Knockout Experiments & Reverse Engineering Input: Signal Output: Gene/protein expression. Given input output rela3onship for normal ( wild type ) and mutant ( knockout ) cells, what can one infer about the network? Topology: hard or impossible de novo: too many combina3ons. New interac3ons or signs of exis3ng interac3ons. Future: Engineering Networks Engineer biological networks to perform new tasks. Change metabolic networks to create cells that produce new products. 9

10 Sources Shen Orr, S.S., Milo, R., Mangan, S., et al Network mo3fs in the transcrip3onal regula3on network of Escherichia coli. Nature Gene;cs 31, Newman, M.E.J., Strogatz, S.H., and WaGs, D.J Random graphs with arbitrary degree distribu3ons and their applica3ons. Phys. Rev. E 64, Ideker T, Ozier O, Schwikowski B, Siegel AF. Discovering regulatory and signalling circuits in molecular interac3on networks. Bioinforma;cs. 2002;18 Suppl 1:S Chuang HY, Lee E, Liu YT, Lee D, Ideker T Network based classifica3on of breast cancer metastasis. Mol Syst Biol. 2007;3: