Cluster analysis Cluster analysis Function 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

Cluster analysis software is developed to group genes with similar patterns of expression.

In this example, the columns represent different timepoints, and the rows represent the results for a single gene.

The products of the genes expressed in a single cluster may have related or similar functions.

FUNCTIONAL GENOMICSCLUSTER ANALYSIS OF MICROARRAY DATA

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CLUSTER ANALYSISCLUSTER ANALYSIS

OUTLINE OF TALK

Clustering - GoalClustering - Goal

Partition of the genes in the dataset into distinct sets (clusters), according to similarity in their expression profiles across the probed conditions

MATRIXMATRIXgenes,conditionsgenes,conditions = Expression dataset = Expression datasetthe first genevector = (xthe first genevector = (x1111, x, x1212, x, x1313, x, x1414… x… x1n1n))

the leftmost condition vector = (xthe leftmost condition vector = (x1111, x, x2121, x, x3131 … x … xm1m1))R

ows

(gen

es)

Columns (conditions [timpepoints, or tissues])

x11 , x12 , x13 , … x1n

x21

x31 ,…Xm1 … xmn

Clustering yeast cell cycle dataset VS gene tree ordering

Clustering – why?Clustering – why? Reduce the dimensionality of the

problem – identify the major patterns in the dataset

Similar expression profiles suggest functional relationship Functional annotation of ESTs Links among pathways

Related functions suggest coordinated regulatory control Dissection of regulatory networks

Clustering identifies group of genes with “similar” expression profiles

How is similarity measured? Euclidian distance Correlation coefficient Others

Similarity measuresSimilarity measures

In an experiment with 10 conditions, the gene expression profiles for two genes X, and Y would have this form

X = (x1, x2, x3, …, xm)

Y = (y1, y2, y3, …, ym)

Similarity measure - Euclidian distanceSimilarity measure - Euclidian distance

In general: if there are M experiments:

X = (x1, x2, x3, …, xm)

Y = (y1, y2, y3, …, ym)

Similarity measure – Correlation Similarity measure – Correlation CoefficientCoefficient

X = (x1, x2, x3, …, xm)

Y = (y1, y2, y3, …, ym)

-1 ≤ S(X,Y) ≤ 1

Euclidian vs CorrelationEuclidian vs 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 → correlation

Gene X

Gene Y

What correlation distance sees

What euclidean distance sees

PCAPCA

A technique for projecting the expression data set onto a reduced (2 or 3 dimensional) easily visualized space

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

PCA transformation of a microarray PCA transformation of a microarray datasetdataset

Visual estimation of the number of clusters in the data

Clustering AlgorithmsClustering Algorithms

K–meansSOMsHierarchical clustering

K-MEANSK-MEANS1. The user sets the number of clusters- k2. Initialization: each gene is randomly assigned

to one of the k clusters3. Average expression vector is calculated for

each cluster (cluster’s profile) 4. 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.

5. Score current partition: sum of distances between genes and the profile of the cluster they are assigned to (homogeneity of the solution).

6. Stop criteria: further shuffling of genes results in minor improvement in the clustering score

How to choose the number of clusters How to choose the number of clusters needed to informatively partition the data needed to informatively partition the data

Try several parameters (number of desired clusters,distance metric) and compare the clustering

solutions Criteria for comparison: Homogeneity vs

SeparationUse PCA (Principle Component Analysis)

K-MEANS example: 4 clustersK-MEANS example: 4 clusters

Mean profile

Standard deviation in each condition

Evaluating KmeansEvaluating Kmeans

Cluster 3

Cluster 1

Cluster 4

Cluster 2

Mis-classified

K-means example: 3 K-means example: 3 clustersclusters

Too few clusters: K=2Too few clusters: K=2

SOMs (Self-Organizing SOMs (Self-Organizing Maps)Maps)

User sets the number of clusters in a form of a rectangular grid (e.g., 3x2) – ‘map nodesmap nodes’

Imagine genes as points in (M-dimensional) space

Initialization: map nodes are randomly placed in the data space

Genes – data points

Clusters – map nodes

SOM - SchemeSOM - 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

• 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)

Hierarchical ClusteringHierarchical 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 g1 g2 g3 g4 g5

{1,2}

{4,5}

{1,2,3}

{1,2,3,4,5}

Mathematical evaluation of Mathematical evaluation of clustering solutionclustering 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

“True”CAST*

GeneCluster

K-means

CLICK

Homogeneity

Separa

tion

Performance on Yeast Cell Cycle Data

*Ben-Dor, Shamir, Yakhini

1999

698 genes, 72 conditions (Spellman et al. 1998). Each algorithm was run by its authors in a “blind” test.

Overall strategy:

PCA-transformationClustering and evaluation of clusteringCheck for bio-significance

Which genes to cluster? Which genes to cluster? Apply filtering prior to clustering – focus the

analysis on the ‘responding genes’ Applying 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.

Fold change: choose genes that changed by at least M-folds in at least L conditions

Variance: filter out genes that do not vary greatly among the conditions of the experiment.

Try various filtering schemes to find the setting that gives the best biological results

Clustering – ToolsClustering – Tools

Cluster (Eisen) – hierarchical GeneCluster (Tamayo) – SOM TIGR MeV – K-Means, SOM,

hierarchical, QTC, CAST Expander – CLICK, SOM, K-means,

hierarchical Many others (e.g. GeneSpring)

CSLA Workshop, Day2CSLA Workshop, Day2

Presentation created by Rani Elkon and posted at:

http://www.tau.ac.il/lifesci/bioinfo/teaching/2002-2003/DNA_microarray_winter_2003.html

Ascribing Biological Meaning to Ascribing Biological Meaning to ClustersClusters

Identify over-represented functional categories in the clusters (i.e., cluster contains much more genes of specific biological process than expected by chance)

Requirements for systematic analysis: Controlled vocabulary for describing biological

processes (protein biosynthesis\translation, apoptosis\programmed cell death)

Standard assignment of genes into functional categories

Gene Ontology (GO) projectGene Ontology (GO) project Purpose:

1) Define controlled terms (ontologies) for description of gene products from 3 aspects:

Biological process (DNA repair, mitosis) Molecular function (protein serine/threonine kinase activity,

transcription factor activity) Cellular component (nucleus, ribosome)

2) Establish a unified framework for organism-independent gene annotation

Characteristics:1) A gene can have multiple associations in each ontology2) GO terms are organized in hierarchical structures called

directed acyclic graphs (DAGs)- The most general classifications are at top levels

of the graph- More specialized classifications at lower levels

Hierarchical classification scheme for proteins that function in M-phase of mitosis

Each gene can be a member of more than one GO classifications

Online Databases that Online Databases that annotate genes by GOannotate genes by GO

Human Entrez http://www.ncbi.nih.gov/entrez/query.fcgi?db=gene GOA http://www.ebi.ac.uk/GOA/

Mouse – Mouse Genome Informatics (MGI) http://www.informatics.jax.org/

Rat – Rat Genome Database http://rgd.mcw.edu/

Fly – FlyBase http://flybase.bio.indiana.edu/

Arabidopsis – TAIR http://www.arabidopsis.org/

Yeast – Sacchromaces Genome Database http://www.yeastgenome.org/

Affymetrix chips – Netaffx http://www.affymetrix.com

Example: Cluster 3, 95 genesExample: Cluster 3, 95 genes

Identifying enriched GO categories Identifying enriched GO categories in clustersin clusters

In the previous example: Total number of chip’s genes with annotation = 5000 Total number of chip’s genes associated with metabolism

GO category = 3,600 (72%) Number of annotated genes in cluster 3 = 73 Number of metabolic genes in cluster 3 = 50 (68%)

Statistical tests are essential to determine whether enrichment of a certain class of proteins is significant

AcknowledgementsAcknowledgements

SOM Figures in this presentations were taken from presentation of Benedikt Brors