CLUSTERING IN BIOINFORMATICS
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1 CLUSTERING IN BIOINFORMATICS CSE/BIMM/BENG 8 MAY 4, 0
2 OVERVIEW Define the clustering problem Motivation: gene expression and microarrays Types of clustering Clustering algorithms Other applications of clustering CSE/BIMM/BENG 8 MAY 4, 0
3 THE CLUSTERING PROBLEM Motivation: Find patterns in a sea of data Input: A (large) number of datapoints: N A measure of distance between any two data points d ij Output: Groupings (clustering) of the elements into K (the number can be userspecified or automatically determined) similarity classes Sometimes there is also an objective measure that the obtained clustering seeks to minimize. CSE/BIMM/BENG 8 MAY 4, 0
4 A MARQUEE APPLICATION: MICROARRAY ANALYSIS What do newly sequenced genes do? Simply comparing the new gene sequences to known DNA sequences often does not necessarily reveal the function of a gene: for 40% of sequenced genes, functionality cannot be ascertained by only comparing to sequences of other known genes Genes that perform similar or complementary function to known genes (reference) will be expressed (transcribed) at high levels together with known genes Genes that perform antagonistic functions (e.g. down-regulation) may be expressed at high levels at an earlier or later time point when compared to known genes E.g. what happens to gene expression in cancer cells? Expression level is estimated by measuring the amount of mrna for that particular gene A gene is active if it is being transcribed More mrna usually indicates more gene activity CSE/BIMM/BENG 8 MAY 4, 0
5 A MICROARRAY EXPERIMENT Produce cdna from mrna (cdna is more stable) Label cdna with a fluorescent dye or biotin for detection Different color labels are available to compare many samples at once Wash cdna over the microarray containing thousands of high density probes that hybridize to complementary strands in the sample and immobilize them on the surface. For biotin-labeled samples, stain with the biotin-specific fluorescently labeled antibody Read the microarray, using a laser or a high-resolution CCD Illumination reveals transcribed/co-expressed genes CSE/BIMM/BENG 8 MAY 4, 0
6 Green: expressed only in control Red: expressed only in an experimental cell Yellow: equally expressed in both samples Black: NOT expressed in either control or sample CSE/BIMM/BENG 8 MAY 4, 0
7 Track the sample over a period of time to observe changes in gene expression over time Track two samples under the same conditions to look for differential expression Each box represents one gene s expression over time CSE/BIMM/BENG 8 MAY 4, 0
8 MICROARRAY DATA Microarray data are usually transformed into a (relative, normalized) intensity matrix Can also be represented as a bit matrix (log of relative intensity) The intensity matrix allows biologists to infer correlations between different genes (even if they are dissimilar) and to understand how genes functions might be related Care must be taken to normalize the data appropriately, e.g. different time points can come from different arrays. CSE/BIMM/BENG 8 MAY 4, 0
9 Gene Time Time Time Which genes are similar? What defines co-expression? How to measure the distance/ similarity? INTENSITY TABLE EUCLIDEAN DISTANCE IN D-DIMENSIONS D(x, y) = d (x i y i ) i= CSE/BIMM/BENG 8 MAY 4, 0
10 FINDING SIMILAR GENES ,4,9,0 0,6, ,5,8 PAIRWISE DISTANCES REARRANGED DISTANCES CSE/BIMM/BENG 8 MAY 4, 0
11 CLUSTERING PRINCIPLES Homogeneity: elements of the same cluster are maximally close to each other Separation: elements in separate clusters are maximally far apart from each other One is actually implied by the other (in many cases) Generally speaking, this is a hard problem. min clustering α x,y the same cluster d(x, y) β x,y different clusters d(x, y) RELATIVE IMPORTANCE CSE/BIMM/BENG 8 MAY 4, 0
12 BECAUSE d(x, y)+ d(x, y) x,y the same cluster x,y different clusters = x,y d(x, y) =D = const WE CAN SIMPLIFY min clustering α x,y the same cluster d(x, y) β x,y different clusters d(x, y) TO AN EQUIVALENT EXPRESSION THAT ONLY DEPENDS ON INTRA-CLUSTER DISTANCES CSE/BIMM/BENG 8 MAY 4, 0
13 POOR CLUSTERING EXAMPLE This clustering violates both principles: Points in the same cluster are far apart Points in different cluster are close together CSE/BIMM/BENG 8 MAY 4, 0
14 BETTER CLUSTERING EXAMPLE This clustering appears sensible. But we need to use an objective metric to optimize cluster assignment. CSE/BIMM/BENG 8 MAY 4, 0
15 CLUSTERING TECHNIQUES Agglomerative: Start with every element in its own cluster, and iteratively join clusters together Divisive: Start with one cluster and iteratively divide it into smaller clusters Hierarchical: Organize elements into a tree, leaves represent genes and the length of the paths between leaves represents the distances between genes. Similar genes lie within the same subtrees Generally, finding the exact solution to a clustering problem is NP hard. CSE/BIMM/BENG 8 MAY 4, 0
16 K-MEANS CLUSTERING A technique to partition a set of N points into K clusters Each cluster is represented with a mean (a centroid) hence Kmeans Input: A set V with N points (v, v... v n ), the desired number of clusters K and a distance measure between any two points d(v,w) Output: A set X of K cluster centers that minimize the squared error distortion D(V,X) over all possible choices of X. D(V,X)= N N i= min k d (v i,x k ) CSE/BIMM/BENG 8 MAY 4, 0
17 K-MEANS CLUSTERING For K=, the problem becomes trivial: the centroid of all points is the solution for Euclidean distances. For K the problem becomes NP-complete An efficient heuristic exists Lloyd s algorithm. x = N i v i CSE/BIMM/BENG 8 MAY 4, 0
18 LLOYD S ALGORITHM. Arbitrarily assign the K cluster centers (this can significantly influence the outcome). while cluster centers keep changing A. Compute the distance from each data point to the current cluster center C i ( i K) and assign the point to the nearest cluster B. After the assignment of all data points, compute new centers for each cluster by taking the centroid of all the points in that cluster 3. Output cluster centers and assignments CSE/BIMM/BENG 8 MAY 4, 0
19 K-MEANS EXECUTION EXAMPLE STEP.00 y.00 Center Center x CSE/BIMM/BENG 8 MAY 4, 0
20 K-MEANS EXECUTION EXAMPLE STEP.00 y.00 Center Center x CSE/BIMM/BENG 8 MAY 4, 0
21 K-MEANS EXECUTION EXAMPLE STEP 3.00 y.00 Center Center x CSE/BIMM/BENG 8 MAY 4, 0
22 K-MEANS EXECUTION EXAMPLE STEP 4.00 y.00 Center Center x CSE/BIMM/BENG 8 MAY 4, 0
23 K-MEANS EXECUTION EXAMPLE STEP 5.00 y.00 Center Center x CSE/BIMM/BENG 8 MAY 4, 0
24 K-MEANS EXECUTION EXAMPLE STEP 6.00 y.00 Center Center x CSE/BIMM/BENG 8 MAY 4, 0
25 K-MEANS EXECUTION EXAMPLE K= K=3 K=3 (different starting points) y.00 Center Center y.00 Center 3 3 Center Center 3 y.00 Center 3 3 Center Center x x x CSE/BIMM/BENG 8 MAY 4, 0
26 HOW TO CHOOSE K? The simplest approach is to start with K= and increase K until the squared error distortion (SED) stops decreasing The problems is that K=N always achieves the value of 0 (each point is a cluster), so we always keep increasing K. Generally, need to add further constraints (e.g. model complexity) to obtain non-trivial results 5.5 SED K CSE/BIMM/BENG 8 MAY 4, 0
27 CONSERVATIVE K-MEANS ALGORITHM Lloyd algorithm is fast but in each iteration it moves many data points, not necessarily causing better convergence. A more conservative method would be to move one point at a time only if it improves the overall clustering cost The smaller the clustering cost of a partition of data points is the better that clustering is Different methods (e.g. the squared error distortion) can be used to measure this clustering cost CSE/BIMM/BENG 8 MAY 4, 0
28 K-MEANS GREEDY ALGORITHM ProgressiveGreedyK-Means(k) Select an arbitrary partition P into k clusters while forever bestchange 0 for every cluster C for every element i not in C if cost(p) cost(pi C) > bestchange bestchange cost(p) cost(pi C) i * I C * C if bestchange > 0 Change partition P by moving i * to C * else return P CSE/BIMM/BENG 8 MAY 4, 0
29 CONCLUSION: LLOYD S IN MORE EFFICIENT, BOTH IN RUN-TIME AND IN BEST FOUND SED CSE/BIMM/BENG 8 MAY 4, 0
30 Euclidean distance is not necessarily the best measure for coexpression. CSE/BIMM/BENG 8 MAY 4, 0
31 HIERARCHICAL CLUSTERING Instead of grouping into discrete clusters, produces a classification tree, also called a dendrogram A more intuitive example is probably obtained from molecular sequence data (an early example of clustering applications) We have a collection of aligned nucleotide sequences from different species, and wish to construct their evolutionary hierarchy/history a phylogeny. CSE/BIMM/BENG 8 MAY 4, 0
32 HIERARCHICAL CLUSTERING Consider the following distance matrix on 5 nucleotide (partial mitochondrial genome) sequences. The values are p-distances defined as the number of nucleotide differences normalized by the length of the sequence. Human Chimpanzee Gorilla Orangutan Gibbon Human Chimpanzee Gorilla Orangutan Gibbon CSE/BIMM/BENG 8 MAY 4, 0
33 CLUSTERING PROCEDURE At each step, we select the two closest sequences and join them to form a clade. We then replace the two just joined sequences with their ancestor This reduces the size of the data matrix by one We need to compute the distances from the new ancestor to the remaining sequences Human Chimpanzee Gorilla Orangutan Gibbon Human Chimpanzee Gorilla Orangutan Gibbon CSE/BIMM/BENG 8 MAY 4, 0
34 UPDATING DISTANCES There are multiple strategies for computing the distances to the new ancestral sequence a that joins sequences m and n Single Linkage Complete Linkage UPGMA Unweighted Pair Group Method with Arithmetic Mean WPGMA Weighted Pair Group Method with Arithmetic Mean d(x, a) =min[d(x, m),d(x, n)] d(x, a) = max [d(x, m),d(x, n)] d(x, a) = d(x, a) = d(x, m)+d(x, n) s(m)d(x, m)+s(n)d(x, n) s(m)+s(n) s(n) counts the number of actual sequences represented by node n. CSE/BIMM/BENG 8 MAY 4, 0
35 EXAMPLE CONTINUED Use complete linkage. Joining human and chimp... Human Chimpanzee Gorilla Orangutan Gibbon Human Chimpanzee Gorilla Orangutan Gibbon Human-Chimpanzee Gorilla Orangutan Gibbon Human-Chimpanzee Gorilla Orangutan Gibbon CSE/BIMM/BENG 8 MAY 4, 0
36 Human-Chimpanzee Gorilla Orangutan Gibbon Human-Chimpanzee Gorilla Orangutan Gibbon Human-Chimpanzee-Gorilla Orangutan Gibbon Human-Chimpanzee-Gorilla Orangutan Gibbon CSE/BIMM/BENG 8 MAY 4, 0
37 Orangutan Gibbon Human-Chimpanzee-Gorilla Orangutan Gibbon - - Gibbon Hum-Chimp-Gor-Orang Gibbon - CSE/BIMM/BENG 8 MAY 4, 0
38 A NOTE ON WPGMA Gorilla Orangutan Gibbon Human-Chimpanzee Gorilla Orangutan Gibbon Orangutan Gibbon Human-Chimpanzee-Gorilla Orangutan Gibbon - - d(hcg-orang) = /3 [ d(hc-orang) + d (Gor-Orang)] CSE/BIMM/BENG 8 MAY 4, 0
39 BACK TO MICROARRAYS... Clustering plots can be interpreted as gene/condition hierarchy CSE/BIMM/BENG 8 MAY 4, 0
40 A FEW OTHER APPLICATIONS CSE/BIMM/BENG 8 MAY 4, 0
41 Use clustering of similar sequences in protein databases to reduce complexity and speed up comparisons. Each cluster of similar sequences is represented by a single sequence. Complexity reduction is an important application of clustering CSE/BIMM/BENG 8 MAY 4, 0
42 The structure of proteins interactions can be represented by a graph Node = proteins, Edges = interactions Look for clusters (densely connected components) in graphs CSE/BIMM/BENG 8 MAY 4, 0
43 Hierarchical clustering to improve protein structure prediction by merging the predictions made by a large number of alternative conformation models CSE/BIMM/BENG 8 MAY 4, 0
44 FURTHER READING... CSE/BIMM/BENG 8 MAY 4, 0
45 DEFINES THE CONCEPT OF ELEMENT BELONGS TO A PARTITION WITH A PROBABILITY BUILD A MINIMUM SPANNING TREE AND DELETE LONGEST EDGES TO CREATE PARTITIONS CSE/BIMM/BENG 8 MAY 4, 0
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