Microarray analysis

Algorithms in Computational BiologySpring 2006

Written by Itai Sharon

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Microarrays

Measure the expression of genes in the cell “Count” the number of mRNA molecules that attach

to biological probes

Expression data is gathered for many (thousands) of genes at once

Data is gathered for several experiments Either in several time stamps or different conditions

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NM1N

ij

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2M21

1M131211

ee

e

e

ee

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Genes N

sExperiment M

Relative expression of gene i in experiment j

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Detecting Patterns in Expression Data

Genes may have similar expression patterns because They are part of the same complex (protein-protein

interactions) They are part of the same pathway They have similar regulatory elements They have similar functions (part of a fail-safe

mechanism)

A popular solution: clustering (we saw already) Hierarchical clustering, K-means, agglomerative,...

Today: dimensionality reduction PCA SVD

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Why Dimensionality Reduction

Using irrelevant data may harm accuracy

Clustering algorithms do not perform well in high dimensional data

Visualizing high dimensional data

),( ii yx '

ix

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Principle Components Analysis (PCA)

PCA seeks for a linear projection that best describes the data in a least mean squares sense

Finds a set of principle components (PCs) A PC defines a projection that encapsulates the

maximum amount of variation in a dataset Each PC is orthogonal to all other PCs

Reduce dimensionality by picking the most informative PCs Namely, for reducing from dimension d to dimension

d’, pick the d’ most informative PCs

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PCA - Steps

Input: a dataset

Subtract the mean from each dimension

Compute the covariance matrix for the d dimensions The covariance of two variables X and Y:

The covariance matrix:

n

i

ii

n

YYXXYX

1 )1(

)()(),cov(

),cov(),(),( YXXYYX

id

iin sssssS ,..., },,...,{ 11

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PCA – Steps (cont.)

Compute the eigenvectors and eigenvalues of the covariance matrix

Choose the most informative PCs, construct a feature vector Eigenvectors with highest eigenvalues carry the

most information Feature vector is simply the combination of all

eigenvectors chosenFeatureVector = (eig1, eig2, …, eigd’)

Transform dataset to the new axis system For sS:

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T

s

s

s

eig

eig

eig

storFeatureVecs2

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'

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1

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When Things Get Messy…

PCA is fine when initial dimension is not too big Space and time complexity are of O(d2) - size of

covariance matrix

Otherwise – we have a problem… E.g. when d=104 time/space complexity is O(108)…

Luckily an alternative exists: SVD

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Eigengenes, Eigenarrays and SVD

The idea: Use the singular value decomposition (SVD)

theorem for transforming the dataset from the gene/array space to the eigengene/eigenarray space

Eigengenes, eigenarrays and eigenvalues: Each dimension is represented by an

eigengene/eigenarray/eigenvalue triplet Eigenvalues are used for ranking dimensions

Paper: Alter et. Al., 2000

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Singular Value Decomposition (SVD)

Theorem: if E is a real M by N matrix, then there exist orthogonal matrices

s.t.

Where

and

and ],...,[ MxM1 MuuU NxN1 ],...,[ NvvV

TVWUE

),...,( 1 pdiagW

),min( ,0...21 nmpp

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SVD

i is the ith singular value of E. ui and vi are the ith left singular vector and right singular vector of E, respectively.

It holds that

Efficient algorithms for calculating the SVD exist

),min(:1 NMivuE

uvEi

iiT

ii

i

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Orthogonality of Decomposition

MMM

M

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u

uuu

1

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121

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MNM

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1

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vv

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1 ,..., ],,...,[ MiM uuuuuU

iN

iiN vvvvvV ,..., ],,...,[ 11

TVWUE

),...,( 1 pdiagW

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0

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MMMMM

MMM

uu

uu

uuu

WU

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11

Orthogonality of Decomposition

MNM

ij

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VWU

1

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p

k

kj

kikij vue

1

p

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vuE1

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SVD and Microarray analysis

Reduction from the N genes x M arrays to p eigengenes x p eigenarrays space W is the eigenexpression matrix U represents the expression of genes over

eigenarrays V represents the expression of eigengenes over

arrays

The “fraction of eigenexpression”:

“Shannon entropy” of the dataset:

p

kkiip

1

2

p

kkk pp

pd

1

1)log()log(

10

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Example: Cell cycle of Saccharomyces Cerevisiae

Data is available for 5981 genes over 14 time steps (with ½ hour intervals)

784 genes were classified as cell-cycle regulated (with no missing values)

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Data Sorting

For eigengenes 1 and 2, plot the correlation of each gene g1 with both on a 2-D plot X-axis represents the correlation with 2, Y-axis

relates to 1. Sort by angular distance

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Further Reading

PCA: L. Smith: A Tutorial on Principal Components

Decomposition

Eigengenes, eigenvectors and SVD: O. Alter, P. Brown & D. Botstein: Singular Value

Decomposition for Genome-wide Expression Data Processing and Modeling, PNAS 97:18, 2000