Comp. Genomics

Comp. Genomics

Recitation 10

Clustering and analysis of microarrays

Exercise 1

•A microarray that contains probes for allthe N metabolic enzymes of the bacteriumD.Angerous was used for the followingtime-series microarray experiment: Thebacteria population were exposed to adrug, and gene expression was measuredevery hour for M hours.

•The expression values are discretized to{-1,0,1}

Exercise 1

•Find the longest expression pattern that iscommon to at least k enzymes. Eachenzyme may start the pattern at adifferent time.

-1

K=3

Solution

•Treat each expression vector as a string

•Create a generalized suffix tree O(MN)

•Find longest k-common substring

Exercise 2

•Expression of N genes was measuredunder a certain condition using amicroarray.

•No discretization was performed.

•Give a polynomial time algorithm forclustering these genes into exactly kclusters.

•The objective function is

Pictorially

Expression level

If {G3,G4,G5}is a cluster, its contribution to the

objective function is d(G3,G5)

Solution

•Create a weighted directed graph, everygene is a node and the edge from i to j

has weight d(i,j-1) if i’s expression islower than j’s (otherwise ∞)

The path in the graph that corresponds to this clustering

is G1G3G6. The value of the objective function is

d(G1,G2)+d(G3,G5)+0

Solution

•Next:

•Find the shortest path that visits exactly knodes

•Dynamic programming:

Start from k because if l<k Pl(k-1)=∞

Exercise 3

•A microarray experiment with N genesand M conditions was conducted

•Describe a polynomial algorithm thatdetermines whether the genes can beclustered into 2 clusters such that themaximum distance d(Gi,Gj) in each cluster< W

Illustration

0 1 1

0 0 1

111

1 1 0

W=2

Solution

•Create a graph with a node for every gene

•Add an edge (i,j) if d(i,j)> W

•Check if the resulting graph is bipartite:Run BFS, if you discover an edge (u,v) toa gray node and the depths of u and v areboth even or both odd, answer: “no”.

Solution

Not Bipartite

Exercise 4

•We are given a microarray with N genesand M experiments

•We want to cluster the genes into kclusters such that the distance betweengenes that belong to the same cluster willbe < W

•Can you give a polynomial algorithm thatsolves this problem?

Solution

•Probably not

•More specifically, if we could solve thisproblem in polynomial time, we couldsolve a large class of problem that arewidely believed to be unsolvable inpolynomial time

Solution

•How can we show that we can probablynot find a solution in polynomial time?

•We will take a problem for which this hasalready been shown

•We will construct a polynomial timereduction to our problem

•So, if our problem could be solvedefficiently the “hard” problem could alsobe solved efficiently

Graph description

The following graph can describe ourproblem:

There’s an edge (Gi,Gj) ifthe distance between Gi andGj is less than W

Graph description

Clustering with k=3:

3COL

3-Colorability: Given a graph G, can we dyeits vertices with 3 different colors such thatno two adjacent nodes have the samecolor?

Comparing the problems

•What is common to both these problems?

•In both we “cluster” the nodes

•What are the differences?

•First, in 3COL there are only 3 clustersinstead of k

•Second, the elements that belong to thesame group in 3COL must not have edgesbetween them

Reduction

•Now that we understand the differences,we can take a graph G that is an input to3COL, and transform it to a graph G’ anda constant k that are the input to the k-clustering problem

•We assume that we have a polynomial k-clustering algorithm, and we apply it to(G’,k) and translate the solution to 3COL

Reduction

•Given the first difference that we noted,what should be the value of k?

•We set k to 3, i.e. the algorithm shouldfind exactly 3 clusters

•How do we change G to get G’?

•G’ has the complement edges of G

Example

Proof

 Suppose that G is 3 colorable. LetV1,V2,V3 be the groups of nodes that canbe colored by distinct colors. There areno edges between any pair of nodes in V1,and therefore it forms a legal cluster in G’.Similarly, the nodes of V2 and V3 formclusters. Since V1UV2UV3 contains all thenodes all the genes are clustered in the 3corresponding clusters.

Proof, second direction

Suppose that G’ contains a clustering to 3legal clusters. These clusters correspond to3 nodes sets in G such that within each setthere are no edges between pairs of nodes.Therefore, assigning a different color toevery set is a 3-coloring.