Biological Networks Analysis

Transcription

1 iological Networks nalysis Introduction and ijkstra s algorithm Genome 559: Introduction to Statistical and omputational Genomics Elhanan orenstein

2 The clustering problem: partition genes into distinct sets with high homogeneity and high separation Hierarchical clustering algorithm: 1. ssign each object to a separate cluster.. Regroup the pair of clusters with shortest distance. 3. Repeat until there is a single cluster. Many possible distance metrics K-mean clustering algorithm: 1. rbitrarily select k initial centers. ssign each element to the closest center Voronoi diagram quick review 3. Re-calculate centers (i.e., means) 4. Repeat and 3 until termination condition reached

3 iological networks What is a network? What networks are used in biology? Why do we need networks (and network theory)? How do we find the shortest path between two nodes?

4 Networks vs. Graphs Network theory Social sciences iological sciences Mostly 0 th century Modeling real-life systems Measuring structure & topology Graph theory omputer science Since 18 th century!!! Modeling abstract systems Solving graphrelated questions

5 What is a network? map of interactions or relationships collection of nodes and links(edges)

6 What is a network? map of interactions or relationships collection of nodes and links(edges)

7 Edges: Types of networks irected/undirected Weighted/non-weighted Simple-edges/Hyperedges Special topologies: irected cyclic Graphs (G) Trees ipartite networks

8 Transcriptional regulatory networks Reflect the cell s genetic regulatory circuitry Nodes: transcription factors and genes; Edges:from TF to the genes it regulates irected; weighted?; almost bipartite erived through: hromatin IP Microarrays omputationally

9 Metabolic networks Reflect the set of biochemical reactions in a cell Nodes: metabolites Edges: biochemical reactions irected; weighted?; hyperedges? erived through: Knowledge of biochemistry Metabolic flux measurements Homology? S. erevisiae 106 metabolites 1149 reactions

10 Protein-protein interaction (PPI) networks Reflect the cell s molecular interactions and signaling pathways (interactome) Nodes:proteins Edges: interactions(?) Undirected High-throughput experiments: Protein omplex-ip (o-ip) Yeast two-hybrid omputationally S. erevisiae 4389 proteins interactions

11 Other networks in biology/medicine

12 Non-biological networks omputer related networks: WWW; Internet backbone ommunications and IP Social networks: Friendship (facebook; clubs) itations / information flow o-authorships (papers) o-occurrence (movies; Jazz) Transportation: Highway systems; irline routes Electronic/Logic circuits Many many more

13 Why networks? Networks as models Networks as tools Simple, visual representation of complex systems Focus on organization (rather than on components) Problem representation (more common than you think) lgorithm development iscovery (topology affects function) Predictive models iffusion models (dynamics)

14 TheSeven ridges of Königsberg Published by Leonhard Euler, 1736 onsidered the first paper in graph theory Leonhard Euler

15 The shortest path problem Find the minimal number of links connecting node to node in an undirected network How many friends between you and someone on F (6 degrees of separation) Erdös number, Kevin acon number How far apart are genes in an interaction network What is the shortest (and likely) infection path Find the shortest (cheapest) path between two nodes in a weighted directed graph GPS; Google map

16 ijkstra slgorithm Edsger Wybe ijkstra "omputer Science is no more about computers than astronomy is about telescopes."

17 Solves the single-source shortest path problem: Find the shortest path from a single source to LLnodes in the network Works on both directed and undirected networks Works on both weighted and non-weighted networks pproach: Iterative Maintain shortest path to each intermediate node Greedy algorithm ijkstra salgorithm but still guaranteed to provide optimal solution!!!

18 1. Initialize: ijkstra s algorithm i. ssign a distance value,, to each node. Set to zero for startnode and to infinity for all others. ii. Mark all nodes as unvisited. iii. Set startnode as current node.. For each of the current node s unvisited neighbors: i. alculate tentative distance, t, through current node. ii. If t smaller than (previously recorded distance): t iii. Mark current node as visited (note: shortest dist. found). 3.Set the unvisited node with the smallest distance as the next "current node" and continue from step. 4.Once all nodes are marked as visited, finish.

19 ijkstra s algorithm simple synthetic network F 3 E 1 1. Initialize: i. ssign a distance value,, to each node. Set to zero for startnode and to infinity for all others. ii. Mark all nodes as unvisited. iii. Set startnode as current node.. For each of the current node s unvisited neighbors: i. alculate tentative distance, t, through current node. ii. If t smaller than (previously recorded distance): t iii. Mark current node as visited (note: shortest dist. found). 3.Set the unvisited node with the smallest distance as the next "current node" and continue from step. 4.Once all nodes are marked as visited, finish.

20 ijkstra s algorithm Initialization Mark (start) as current node E F : E 1 F

21 ijkstra s algorithm heck unvisited neighbors of E F vs. 9 5 : vs E 1 F

22 ijkstra s algorithm Update Record path E F 0 9, : 0 3 1, E 1 F

23 Mark as visited ijkstra s algorithm E F 0 9, : 0 3 1, E 1 F

24 ijkstra s algorithm Mark as current (unvisited node with smallest ) E F 0 9, : 0 3 1, E 1 F

25 ijkstra s algorithm heck unvisited neighbors of E F vs. 9 9, vs : 0 3 1, E 1 F 3+ vs.

26 ijkstra s algorithm Update distance Record path E F 0 9,9,7, : 0 3 1, E,5 1 F

27 ijkstra s algorithm Mark as visited Note: istance to is final!! E F : 0 9 3,9,7 1, ,6 E,5 5 1 F

28 ijkstra s algorithm Mark E as current node heck unvisited neighbors of E E F : 0 9 3,9,7 1, ,6 E,5 5 1 F

29 ijkstra s algorithm Update Record path E F 0 9,9,7, : 0 3 1, E,5 1 F,17

30 Mark E as visited ijkstra s algorithm E F 0 9,9,7, : 0 3 1, E,5 1 F,17

31 ijkstra s algorithm Mark as current node heck unvisited neighbors of E F : 0 9 3,9,7 1, ,6 E,5 5 1 F,17

32 ijkstra s algorithm Update Record path (note: path has changed) E F : ,9,7, ,6 E,5 5 1 F,17,11

33 Mark as visited ijkstra s algorithm E F 0 9,9,7, : 0 3 1, E,5 1 F,17,11

34 ijkstra s algorithm Mark as current node heck neighbors E F 0 9,9,7, : 0 3 1, E,5 1 F,17,11

35 ijkstra s algorithm No updates.. Mark as visited E F 0 9,9,7, : 0 3 1, E,5 1 F,17,11

36 Mark F as current ijkstra s algorithm E F 0 9,9,7, : 0 3 1, E,5 1 F,17,11

37 Mark F as visited ijkstra s algorithm E F 0 9,9,7, : 0 3 1, E,5 1 F,17,11 11

38 We now have: Shortest path from to each node (both length and path) Minimum spanning tree We are done! 9,9,7,6 5 E F : 0 3 1, E,5 1 Will we always get a tree? an you prove it? F,17,11

39 omputational Representation of Networks List of edges: (ordered) pairs of nodes [ (,), (,), (,), (,) ] onnectivity Matrix Name: ngr: p1 p Object Oriented Name: ngr: p1 Name: ngr: Name: ngr: p1 Which is the most useful representation?

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