A quick review. The clustering problem: Hierarchical clustering algorithm: Many possible distance metrics K-mean clustering algorithm:
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1 The clustering problem: partition genes into distinct sets with high homogeneity and high separation Hierarchical clustering algorithm: 1. Assign 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. Arbitrarily select k initial centers. Assign each element to the closest center Voronoi diagram A quick review 3. Re-calculate centers (i.e., means) 4. Repeat and 3 until termination condition reached
2 Biological Networks Analysis Introduction Genome 373 Genomic Informatics Elhanan Borenstein
3 Why we need networks (and systems biology)? VS.
4 Biological 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?
5 What is a network? A collection of nodes and links (edges) A map of interactions or relationships
6 A long history of network/graph theory!! Network theory Social sciences (and Biological sciences) Mostly 0 th century Modeling real-life systems Measuring structure & topology Graph theory Computer science Since 18 th century!!! Modeling abstract systems Solving graphrelated questions
7 Leonhard Euler The Seven Bridges of Königsberg Published by Leonhard Euler, 1736 Considered the first paper in graph theory
8 Leonhard Euler The Seven Bridges of Königsberg Published by Leonhard Euler, 1736 Considered the first paper in graph theory
9 Edge properties and special topologies Edges: Directed/undirected Weighted/non-weighted Simple-edges/Hyperedges Special topologies: Trees Directed Acyclic Graphs (DAG) Bipartite networks
10 Transcriptional regulatory networks Reflect the cell s genetic regulatory circuitry Nodes: transcription factors and genes; Edges: from TF to the genes it regulates Directed; weighted?; almost bipartite Derived through: Chromatin IP Microarrays Computationally
11 Metabolic networks Reflect the set of biochemical reactions in a cell Nodes: metabolites Edges: biochemical reactions Directed; weighted?; hyperedges? Derived through: Knowledge of biochemistry Metabolic flux measurements Homology S. Cerevisiae 106 metabolites 1149 reactions
12 Protein-protein interaction (PPI) networks Reflect the cell s molecular interactions and signaling pathways (interactome) Nodes: proteins Edges: interactions(?) Undirected High-throughput experiments: Protein Complex-IP (Co-IP) Yeast two-hybrid Computationally S. Cerevisiae 4389 proteins interactions
13 Other networks in biology/medicine
14 Non-biological networks Computer related networks: WWW; Internet backbone Communications and IP Social networks: Friendship (facebook; clubs) Citations / information flow Co-authorships (papers) Co-occurrence (movies; Jazz) Transportation: Highway systems; Airline routes Electronic/Logic circuits Many many more
15 The Bacon Number Game Tropic Thunder (008) Tropic Thunder Iron Man Proof Flatliners Tom Cruise Robert Downey Jr. Gwyneth Paltrow Hope Davis Kevin Bacon Tropic Thunder Iron Man Frost/Nixon Tom Cruise Robert Downey Jr. Frank Langella Kevin Bacon
16 The Paul Erdos Number Game
17
18 The shortest path problem Find the minimal number of links connecting node A to node B in an undirected network How many friends between you and someone on FB (6 degrees of separation, Erdös number, Kevin Bacon number) How far apart are two 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
19 Dijkstra s Algorithm Edsger Wybe Dijkstra "Computer Science is no more about computers than astronomy is about telescopes."
20 Solves the single-source shortest path problem: Find the shortest path from a single source to ALL nodes in the network Works on both directed and undirected networks Works on both weighted and non-weighted networks Approach: Iterative : maintain shortest path to each intermediate node Greedy algorithm Dijkstra s algorithm but still guaranteed to provide optimal solution!!
21 Dijkstra s algorithm 1. Initialize: i. Assign a distance value, D, to each node. Set D to zero for start node and to infinity for all others. ii. iii. Mark all nodes as unvisited. Set start node as current node.. For each of the current node s unvisited neighbors: i. Calculate tentative distance, D t, through current node. ii. iii. If D t smaller than D (previously recorded distance): D D t 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.
22 Dijkstra s algorithm A simple synthetic network 9 B D 5 A F 3 C E 1 1. Initialize: i. Assign a distance value, D, to each node. Set D to zero for start node and to infinity for all others. ii. Mark all nodes as unvisited. iii. Set start node as current node.. For each of the current node s unvisited neighbors: i. Calculate tentative distance, D t, through current node. ii. If D t smaller than D (previously recorded distance): D D 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.
23 Dijkstra s algorithm Initialization Mark A (start) as current node A B C D E F 0 9 B D 5 D: 0 A 3 1 C E 1 F
24 Dijkstra s algorithm Check unvisited neighbors of A A B C D E F vs. 9 B D 5 D: 0 A vs. C E 1 F
25 Dijkstra s algorithm Update D Record path A B C D E F 0 9,9 B D D: 0 A 3 1 C, E 1 F
26 Mark A as visited Dijkstra s algorithm A B C D E F 0 9,9 B D D: 0 A 3 1 C, E 1 F
27 Dijkstra s algorithm Mark C as current (unvisited node with smallest D) A B C D E F 0 9,9 B D D: 0 A 3 1 C, E 1 F
28 Dijkstra s algorithm Check unvisited neighbors of C A B C D E F vs. 9 9,9 B D vs D: 0 A 3 1 C, E 1 F 3+ vs.
29 Dijkstra s algorithm Update distance Record path A B C D E F 0 9,9,7 B,6 D D: 0 A 3 1 C, E,5 1 F
30 Dijkstra s algorithm Mark C as visited Note: Distance to C is final!! A B C D E F D: 0 A 9 3,9,7 1 B C,3 D ,6 E,5 5 1 F
31 Dijkstra s algorithm Mark E as current node Check unvisited neighbors of E A B C D E F D: 0 A 9 3,9,7 1 B C,3 D ,6 E,5 5 1 F
32 Dijkstra s algorithm Update D Record path A B C D E F 0 9,9,7 B,6 D D: 0 A 3 1 C, E,5 1 F,17
33 Mark E as visited Dijkstra s algorithm A B C D E F 0 9,9,7 B,6 D D: 0 A 3 1 C, E,5 1 F,17
34 Dijkstra s algorithm Mark D as current node Check unvisited neighbors of D A B C D E F D: 0 A 9 3,9,7 1 B C,3 D ,6 E,5 5 1 F,17
35 Dijkstra s algorithm Update D Record path (note: path has changed) A B C D E F D: 0 A 9 1 3,9,7 B C,3 D ,6 E,5 5 1 F,17,11
36 Mark D as visited Dijkstra s algorithm A B C D E F 0 9,9,7 B,6 D D: 0 A 3 1 C, E,5 1 F,17,11
37 Dijkstra s algorithm Mark B as current node Check neighbors A B C D E F 0 9,9,7 B,6 D D: 0 A 3 1 C, E,5 1 F,17,11
38 Dijkstra s algorithm No updates.. Mark B as visited A B C D E F 0 9,9,7 B,6 D D: 0 A 3 1 C, E,5 1 F,17,11
39 Mark F as current Dijkstra s algorithm A B C D E F 0 9,9,7 B,6 D D: 0 A 3 1 C, E,5 1 F,17,11
40 Mark F as visited Dijkstra s algorithm A B C D E F 0 9,9,7 B,6 D D: 0 A 3 1 C, E,5 1 F,17,11 11
41 We now have: Shortest path from A to each node (both length and path) Minimum spanning tree We are done! 9,9,7 B,6 D 5 A B C D E F D: 0 A 3 1 C, E,5 1 Will we always get a tree? Can you prove it? F,17,11
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