SocalBSI 2008: Clustering Microarray Datasets Sagar Damle, Ph.D. Candidate, Caltech

1. SocalBSI 2008:Clustering Microarray DatasetsSagar Damle, Ph.D. Candidate, Caltech Distance Metrics: Measuring similarity using the Euclidean and Correlation distance metrics Principle Components Analysis: Reducing the dimensionality of microarray data Clustering Agorithms: Kmeans Self-Organizing Maps (SOM) Hierarchical Clustering

2. MATRIXgenes,conditions = Expression datasetthe first genevector = (x11, x12, x13, x14… x1n)the leftmost condition vector = (x11, x21, x31 … xm1) Columns (conditions [timepoints, or tissues]) x11 , x12 , x13 , … x1n x21 x31 , … Xm1 … xmn Rows (genes)

3. Similarity measures • Clustering identifies group of genes with “similar” expression profiles • How is similarity measured? • Euclidian distance • Correlation coefficient • Others: Manhattan, Chebychev, Euclidean Squared

4. In an experiment with 10 conditions, the gene expression profiles for two genes X, and Y would have this form X = (x1, x2, x3, …, x10) Y = (y1, y2, y3, …, y10)

5. d(Ga, Gb) = sqrt( (x1-y1)2 + (x2 -y2)2 ) Similarity measure - Euclidian distance Gb: (x1, x2) Ga: (y1, y2) In general: if there are M experiments: X = (x1, x2, x3, …, xm) Y = (y1, y2, y3, …, ym)

6. Similarity measure – Pearson Correlation Coefficient X = (x1, x2, x3, …, xm), Y = (y1, y2, y3, …, ym) • D = 1 - r • r = [Z(X)*Z(Y)] (dot product of the z-scores of vectors X and Y) • r = |Z(X)| |Z(Y)| cos(T) • When two unit vectors are completely correlated, r=1 and D=0 • When two unit vectors are non correlated, r=0 and D = 1 • Dot product review: http://mathworld.wolfram.com/DotProduct.html

7. Euclidian vs Pearson Correlation • Euclidian distance – takes into account the magnitude of the expression • Correlation distance - insensitive to the amplitude of expression, takes into account the trends of the change. • Common trends are considered biologically relevant, the magnitude is considered less important Gene Y Gene X

8. What euclidean distance sees What correlation distance sees

9. Principle Components Analysis (PCA) • A method for projecting microarray data onto a reduced (2 or 3 dimensional) easily visualized space Definition: Principle Components - A set of variables that define a projection that encapsulates the maximum amount of variation in a dataset and is orthogonal (and therefore uncorrelated) to the previous principle component of the same dataset. • Example Dataset: Thousands of genes probed in 10 conditions. • The expression profile of each gene is presented by the vector of its expression levels: X = (X1, X2, X3, X4, X5) • Imagine each gene X as a point in a 5-dimentional space. • Each direction/axis corresponds to a specific condition • Genes with similar profiles are close to each other in this space • PCA- Project this dataset to 2 dimensions, preserving as much information as possible

11. PCA transformation of a microarray dataset Visual estimation of the number of clusters in the data

12. 1-page tutorial on singular value decomposition (PCA) http://web.mit.edu/be.400/www/SVD/Singular_Value_Decomposition.htm

13. Cluster analysis  Function • Places genes with similar expression patterns in groups. • Sometimes genes of unknown function will be grouped with genes of known function. • The functions that are known allow the investigator to hypothesize regarding the functions of genes not yet characterized. • Examples: • Identify genes important in cell cycle regulation • Identify genes that participate in a biosynthetic pathway • Identify genes involved in a drug response • Identify genes involved in a disease response

14. Clustering yeast cell cycle dataset VS gene tree ordering

15. How to choose the number of clusters needed to informatively partition the data Trial and error: Try clustering with a different number of clusters, and compare your results • Criteria for comparison: Homogeneity vs Separation • Use PCA (Principle Component Analysis) to visually determine how well the algorithm grouped genes • Calculate the mean distance between all genes within a cluster (it should be small) and compare that to the distance between clusters (which should be large)

16. Mathematical evaluation of clustering solution Merits of a ‘good’ clustering solution: • Homogeneity: • Genes inside a cluster are highly similar to each other. • Average similarity between a gene and the center (average profile) of its cluster. • Separation: • Genes from different clusters have low similarity to each other. • Weighted average similarity between centers of clusters. • These are conflicting features: increasing the number of clusters tends to improve with-in cluster Homogeneity on the expense of between-cluster Separation

17. Performance on Yeast Cell Cycle Data CAST* “True” CLICK GeneCluster Separation K-means Homogeneity 698 genes, 72 conditions (Spellman et al. 1998). Each algorithm was run by its authors in a “blind” test. *Ben-Dor, Shamir, Yakhini 1999

18. Clustering Algorithms • K–means • SOMs • Hierarchical clustering

19. K-MEANS • The user sets the number of clusters- k • Initialization: each gene is randomly assigned to one of the k clusters • Average expression vector is calculated for each cluster (cluster’s profile) • Iterate over the genes: • For each gene- compute its similarity to the cluster profiles. • Move the gene to the cluster it is most similar to. • Recalculated cluster profiles. • Score current partition: sum of distances between genes and the profile of the cluster they are assigned to (homogeneity of the solution). • Stop criteria: further shuffling of genes results in minor improvement in the clustering score

22. Mean profile Standard deviation in each condition K-MEANS example: 4 clusters (too many?)

23. Evaluating Kmeans Cluster 1 Cluster 3 Mis-classified Cluster 4 Cluster 2

24. K-means example: 3 clusters (looks right)

25. Kmeans clustering: K=2 (too few)

26. SOMs (Self-Organizing Maps)less clustering and more data organizing • User sets the number of clusters in a form of a rectangular grid (e.g., 3x2) – ‘map nodes’ • Imagine genes as points in (M-dimensional) space • Initialization: map nodes are randomly placed in the data space

27. Genes – data points Clusters – map nodes

28. SOM - Scheme • Randomly choose a data point (gene). • Find its closest map node • Move this map node towards the data point • Move the neighbor map nodes towards this point, but to lesser extent (thinner arrows show weaker shift) • Iterate over data points

29. Each successive gene profile (black dot) has less of an influence on the displacement of the nodes. • Iterate through all profiles several times (10-100) • When positions of the cluster nodes have stabilized, assign each gene to its closest map node (cluster)

32. {1,2,3,4,5} {1,2,3} {4,5} {1,2} g1 g2 g3 g4 g5 Hierarchical Clustering • Goal#1: Organize the genes in a structure of a hierarchical tree • 1) Initial step: each gene is regarded as a cluster with one item • 2) Find the 2 most similar clusters and merge them into a common node (red dot) • 3) Merge successive nodes until all genes are contained in a single cluster • Goal#2: Collapse branches to group genes into distinct clusters

33. Which genes to cluster? • Apply filtering prior to clustering – focus the analysis on the ‘responding genes’ • The application of controlled statistical tests to identify ‘responding genes’ usually ends up with too few genes that do not allow for a global characterization of the response. • Variance: filter out genes that do not vary greatly among the conditions of the experiment. • Non-varying genes skew clustering results, especially when using a correlation coefficient • Fold change: choose genes that change by at least M-fold in at least L conditions.

34. Clustering – Tools • Cluster (Eisen) – hierarchical clustering • http://rana.lbl.gov/EisenSoftware.htm • GeneCluster (Tamayo) – SOM • http://bioinfo.cnio.es/wwwsomtree/ • TIGR MeV – K-Means, SOM, hierarchical, QTC, CAST • http://www.tm4.org/mev.html • Expander – CLICK, SOM, K-means, hierarchical • http://www.cs.tau.ac.il/~rshamir/expander/expander.html • Many others (e.g. GeneSpring) • http://www.agilent.com/chem/genespring

35. Analysis Strategy • Transform Dataset Using PCA • Cluster • Parameters to test: • Distance Metric • Number of clusters • Separation & Homogeneity • Assign biological meaning to clusters

36. Original presentation created by Rani Elkon and posted at: http://www.tau.ac.il/lifesci/bioinfo/teaching/2002-2003/DNA_microarray_winter_2003.html