Clustering analysis of gene expression data

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1 Clustering analysis of gene expression data Chapter 11 in Jonathan Pevsner, Bioinformatics and Functional Genomics, 3 rd edition (Chapter 9 in 2 nd edition)

Human T cell expression data The matrix contains 47 expression samples from Lukk et al, Nature Biotechnology 2010 All samples are from T cells in different individuals Only

2 Human T cell expression data The matrix contains 47 expression samples from Lukk et al, Nature Biotechnology 2010 All samples are from T cells in different individuals Only the top 3000 genes with the largest variability were used The value is log2 of gene s expression level in a given sample as measured by the microarray technology T cells a T cell

3 How to interpret the expression data if you still have many genes and many samples?

4 Clustering to the rescue!

5 Clustering is a part of Machine Learning Supervised Learning: A machine learning technique whereby a system uses a set of training examples to learn how to correctly perform a task Example: a bunch of cancer expression samples each annotated with cancer type/tissue. Goal: predict cancer type based on expression pattern Unsupervised Learning (including clustering): In machine learning, unsupervised learning is a class of problems in which one seeks to determine how the data are organized. One only has unlabeled examples. Example: a bunch of breast cancer expression samples. Identify several different (yet unknown) subtypes with different likely outcomes

6 What is clustering? The goal of clustering is to group data points that are close (or similar) to each other Usually we need to identify such groups (or clusters) in an unsupervised manner Sometimes we take into account prior information (Bayesian methods) Need to define distance d ij between objects i and j In our case objects could be either genes or samples Easy in 2 dimensions but hard in 3000 dimensions Need to somehow reduce dimensionality x x x x x x x x x

7 How to define distance? Correlation coefficient distance d( X, Y) 1 ( X, Y) 1 Cov( X, Y ) ( Var( X ) Var( Y )

8 Reminder: Principal Component Analysis

9 Multivariable statistics and Principal Component Analysis (PCA) A table of n observations in which p variables were measured p x p symmetric matrix R of corr. coefficients PCA: Diagonalize matrix R

10 Trick: Rotate Coordinate Axes Suppose we have a population measured on p random variables X 1,,X p. Note that these random variables represent the p-axes of the Cartesian coordinate system in which the population resides. Our goal is to develop a new set of p axes (linear combinations of the original p axes) in the directions of greatest variability: X 2 X 1 This is accomplished by rotating the axes. Adapted from slides by Prof. S. Narasimhan, Computer Vision course at CMU

11 Principle Component Analysis (PCA) p x p symmetric matrix R of corr. coefficients R=n 1 Z *Z is a square of the matrix Z of standardized r.v.: all eigenvalues of R are non negative Diagonal elements=1 tr(r)=p Can be diagonalized: R=V*D*V where D is the diagonal matrix d(1,1) largest eig. value, d(p,p) the smallest one The meaning of V(i,k) contribution of the data type i to the k th eigenvector tr(d)=p, the largest eigenvalue d(1,1) absorbs a fraction =d(1,1)/p of all correlations can be ~100% Scores: Y=Z*V: n x p matrix. Meaning of Y(,k) participation of the sample # in the k th eigenvector

12 Back to clustering

aspirates = biopsy for breast cancer Black dots cell nuclei.

13 Let s work with real cancer data! Data from Wolberg, Street, and Mangasarian (1994) Fine needle aspirates = biopsy for breast cancer Black dots cell nuclei. Irregular shapes/sizes may mean cancer 212 cancer patients and 357 healthy (column 1) 30 other properties (see table)

14 Common types of clustering algorithms Hierarchical if don t know in advance # of clusters Agglomerative: start: N clusters, merge into 1 cluster Divisive: start with 1 cluster and breaks it up into N Non hierarchical algorithms Principal Component Analysis (PCA) plot pairs of top eigenvectors of the covariance matrix Cov(X i, X j ) and uses visual information to group K means clustering: Iteratively apply the following two steps: Calculate the centroid (center of mass) of each cluster Assign each to the cluster to the nearest centroi

15 Credit: XKCD comics

16 UPGMA algorithm Hierarchical agglomerative clustering algorithm UPGMA = Unweighted Pair Group Method with Arithmetic mean Iterative algorithm: Start with a pair with the smallest d(x,y) Cluster these two together and replace it with their arithmetic mean (X+Y)/2 Recalculate all distances to this new cluster node Repeat until all nodes are merged

17 Output of UPGMA algorithm

19 Matlab demo

20 Human T cell expression data The matrix contains 47 expression samples from Lukk et al, Nature Biotechnology 2010 All samples are from T cells in different individuals Only the top 3000 genes with the largest variability were used The value is log2 of gene s expression level in a given sample as measured by the microarray technology T cells a T cell

21 Choices of distance metrics in clustergram( RowPDistValue, ColumnPDistValue,)

22 Choices of hierarchical clustering algorithm in clustergram( linkage, )

23 Clustering group exercise Each group will analyze a cluster of genes identified in the T cell expression table Analyze the table of top 100 genes by variance in 47 samples Cluster them using: Group 1: UPGMA = linkage, average, RowPDistValue, euclidian, Group 2: linkage, single, RowPDistValue, cityblock, Group 3: linkage, average, RowPDistValue, correlation, Group 4: UPGMA = linkage, single, RowPDistValue, euclidian, Use clustergram(, 'Standardize','Row', linkage, as specified for your group, RowPDistValue as specified for your group, 'RowLabels',gene_names1,'ColumnLabels, array_names)

24 Matlab code load expression_table.mat gene_variation=std(exp_t')'; [a,b]=sort(gene_variation,'descend'); ngenes=100; exp_t1=exp_t(b(1:ngenes),:); gene_names1=gene_names(b(1:ngenes)); %%% for group 1 CGobj1 = clustergram(exp_t1, 'Standardize','Row', 'RowLabels', gene_names1,'columnlabels',array_names) set(cgobj1,'rowlabels',gene_names1,'columnlabels', array_names,'linkage', 'average','rowpdist','euclidean');

25 Cluster analysis group exercise Which biological functions are overrepresented in different clusters? Pick a cluster: Select a node on the tree of rows, Right click Choose export group info into the workspace Name it gene_list Run the following two Matlab commands to display genes g1=gene_list.rownodenames; for m=1:length(g1); disp(g1{m}); end;

26 Search for shared biological functions copy the list of displayed genes go to "Start Analysis" on paste gene list into the box in the left panel select ENSEMBL_GENE_ID select gene list radio button pick "Functional Annotation Clustering" for gene list broken into multiple groups (clusters) each with related biological functions Select "Functional Annotation Clustering" rectangular button below to display annotation results Write down the # of genes in the cluster and the top functions in two most interesting clusters

27 Before clustering

28 UPGMA hierarchical clustering, Euclidian distance

29 UPGMA hierarchical clustering, correlation distance

30 Credit: XKCD comics

31 Basic concepts of network analysis

32 Degree of a node its # of neighbors 2 Degree K 2 =2 1 Degree K 1 =4

33 Directed networks have in and out degrees In degree K in =2 Out degree K out =5

34 How to find important nodes? By their degree Hubs = important Degree centrality Google tweaked it in 1990s Hubs pointed by other hubs = more important 2?? 1?

35 How Google PageRank algorithm works? Too many hits to a typical query: need to rank One could rank the importance of webpages by K in, but: Too democratic: It doesn t take into account importance of webpages sending links it s easy to trick and artificially boost the rank Google s solution: simulate the behavior of many random surfers and then count the number of hits for every webpage Popular pages send more surfers your way the Google weight is proportional to K in weighted by popularity

36 Mathematics of the Google Google solves a self consistent Eq.: G i ~ j T ij G j find the principal eigenvector T ij = A ij /K out (j) Problem: surfers trapped infinite loops with one entry and no exit Model with random jumps: G i ~ (1 ) j T ij G j + j G j =0.15 meaning that an average web surfer (circa 1995) on average jumped around 1/ 6 webpages before going somewhere else

37 How to find important nodes? By their degree Hubs = important Degree centrality By their connectivity Connectors = important Betweenness centrality

38 Betweenness centrality: definition Take a node i There are (N 1)*(N 2)/2 pairs of other nodes For each pair find the shortest path on the network If more than one shortest path, sample them equally Betweenness centrality C(I = the probability that a randomly selected pair has shortest path going through i

39 Clustering coefficient: Are my friends also friends with each other

40 Clustering coefficient: what fraction of your neighbors are connected to each other? For the entire network: C=mean(C i )=Σ i C i /N Network Science: Graph Theory 2012

41 Why it matters for expression data analysis?

T cell expression data The matrix contains 47 expression samples from Lukk et al, Nature Biotechnology 2011 All samples are normal T cells from different

42 T cell expression data The matrix contains 47 expression samples from Lukk et al, Nature Biotechnology 2011 All samples are normal T cells from different individuals Only the top 3000 genes with the largest variability were used The value is log2 of gene s expression level in a given sample as measured by microarray technology

43 Before clustering

44 UPGMA hierarchical clustering, Euclidian distance

45 Another way to analyze expression data: Co expression networks

of N(N 1)/2 pairs of genes i and j calculate the correlation coefficient ρ ij =σ ij /σ i σ j of gene levels across K samples Put a threshold, e.g. ρ ij >0.

46 How to construct a co-expression network? A co-expression network Samples Start with a matrix of log2 of expression levels of N genes in K samples (conditions): for our T cell data N=3000, K=47 For each of N(N 1)/2 pairs of genes i and j calculate the correlation coefficient ρ ij =σ ij /σ i σ j of gene levels across K samples Put a threshold, e.g. ρ ij >0.85, or otherwise select the most correlated pairs of genes (~4500 in our case). Now you have a weighted network. Identify densely interconnected functional modules in this network. Modules can be used to infer unknown functions of genes via Guilt by Association principle. Functional modules

47 Co expression network analysis exercise Start Gephi and open coexpression_network_random_start.gephi Run Layout Fruchterman Reingold Speed 10.0 Run Average degree, Network diameter, Modularity in the Statistics tab in the right panel. Color nodes by modularity class : Appearance Nodes Attribute Palette Icon Size nodes first by degree. Appearance Nodes Attribute Circles Icon Nodes in large tightly connected clusters have large degree Then size nodes by betweenness centrality Appearance Nodes Attribute Circles Icon These are coordinator nodes connecting different clusters to each other. Potentially biologically interesting

48 Credit: XKCD comics

Dimension reduction : PCA and Clustering

Dimension reduction : PCA and Clustering By Hanne Jarmer Slides by Christopher Workman Center for Biological Sequence Analysis DTU The DNA Array Analysis Pipeline Array design Probe design Question Experimental