Gibbs biclustering of microarray data Yves Moreau & Qizheng Sheng Katholieke Universiteit Leuven...

Gibbs biclustering of microarray data

Yves Moreau& Qizheng Sheng

Katholieke Universiteit LeuvenESAT-SCD (SISTA)

on leave at Center for Biological Sequence analysis,

Danish Technical University

April 18, 2023 CBS Microarray Course 2

Clustering

Form coherent groups of Genes Patient samples (e.g., tumors) Drug or toxin response

Study these groups to get insight into biological processes Diagnostic and prognostic classes Genes in same clusters can have

same function or same regulation Clustering algorithms

Hierarchical clustering K-means Self-Organizing Maps ...


What’s wrong with clustering?

Clustering is a long-solved problem ?!?

Many problems with current clustering algorithms PCA does not do any form of grouping Hierarchical clustering does not produce distinct groups

Only a tree; it is then up to the user to pick nodes from the tree

K-means does not tell you how many clusters really are present in the data

...


A wish list for clustering We expect a lot from a clustering algorithm

Fast and not memory hungry Can run easily on a large microarray data set

10-100.000 genes, >100 experiments Partitioning of genes into distinct groups and automatically

determine the “right” number of groups Robust

If you remove some genes and some experiments, you want to obtain roughly the same groups

Rejection of outliers (genes that do not clearly belong to any group)

Probabilistic cluster membership One gene can belong to several clusters

Incorporation of biological knowledge into account Maybe you want some known genes to cluster together Meaning of the clusters?

Heterogeneous microarray data sources


Biclustering microarray data


Microarray cost per expression measurement Budgets and expertise

Publicly available microarray data Need for exchange standards & repositories

Big consortia set up big microarray projects Genome projects “transcriptome” projects (=

compendia)

Change in microarray projects ( sequence analysis) Analyze public data first to generate an hypothesis Design and perform your own microarray experiment

From genome projects to transcriptome projects


Data becomes more heterogeneous Gene clustering

Group genes that behave similarly over all conditions

Gene biclustering Group genes that behave similarly

over a subset of conditions “Feature selection” More suitable

for heterogeneous compendium

Why biclustering?


Probabilistic graphical models

Biostatistics

Bayesian statsClusteringDecision support

Genetics

Linkage analysisPhylogeny

Sequence analysis

Modeling protein familiesGene predictionRegulatory sequence analysis

Expression analysis

ClusteringGenetic network inference

Graphicalmodels


Distribution of expression values for a given gene

HighMediumLow

Bicluster Discretized microarray

data set

Discretizing microarray data Microarray data is

continuous Discretize by equal

frequency

gen

es

conditions


Bicluster


Likelihood

0

1

Background Pattern


Likelihood0

1

.9.9.9.9.9

.9.05.9.9.9

.9.9.9.9.9.05.9.9.9.9

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( | , , )P D g c


Likelihood0

1

.9.05.05.05.9

.05.9.9.05.05

.05.05.05.05.05

.05.05.9.9.05

( | ', , )

( | , , )

P D g c

P D g c

Get the right genes


Likelihood0

1

.9.9.05.05.9

.9.05.05.9.9

.9.9 .05 .05.9.05.9.05 .05.9

.9.9 .05 .05.05

( | , ', )

( | , , )

P D g c

P D g c

Get the right conditions


Likelihood0

1

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.2.6.2.2.2

( | , , ')

( | , , )

P D g c

P D g c

Get the right frequency pattern


Optimizing the bicluster Find the right bicluster

Genes Conditions Pattern

For a given choice of genes and conditions, the “best” pattern is given by the frequencies found in the extracted pattern No more need to optimize over the pattern

Maximum likelihood: find genes and conditions that maximize

Gibbs sampling: find genes and conditions that optimize

( | , )P D g c

( , | )P g c D


Gibbs sampling


Markov Chain Monte-Carlo Markov chain with transition matrix T

)|( 1 iXjXPT ttij

A C G TA 0.0643 0.8268 0.0659 0.0430

C 0.0598 0.0484 0.8515 0.0403

G 0.1602 0.3407 0.1736 0.3255

T 0.1507 0.1608 0.3654 0.3231

X=A

X=C X=G

X=T


Markov Chain Monte-Carlo Markov chains can sample from complex distributions

ACGCGGTGTGCGTTTGACGAACGGTTACGCGACGTTTGGTACGTGCGGTGTACGTGTACGACGGAGTTTGCGGGACGCGTACGCGCGTGACGTACGCGTGAGACGCGTGCGCGCGGACGCACGGGCGTGCGCGCGTCGCGAACGCGTTTGTGTTCGGTGCACCGCGTTTGACGTCGGTTCACGTGACGCGTAGTTCGACGACGTGACACGGACGTACGCGACCGTACTCGCGTTGACACGATACGGCGCGGCGGGCGCGGACGTACGCGTACACGCGGGAACGCGCGTGTTTACGACGTGACGTCGCACGCGTCGGTGTGACGGCGGTCGGTACACGTCGACGTTGCGACGTGCGTGCTGACGGAACGACGACGCGACGCACGGCGTGTTCGCGGTGCGG

AC

GT

%

Position


Gibbs sampling Markov chain for Gibbs sampling

1

1 1

0 0 0

( | , )1

( | , )1 1

( | , )1 1 1

( , , ) ( | , ) ( | , ) ( | , )

( , , )

( , , )

( , , )

( , , )

lim ( , , ) ( , , )

lim ( )

i i

i i

i i

P A B b C ci i i i

P B A a C ci i i i

P C A a B bi i i i

k k kk

kk

P A B C P A B C P B A C P C A B

a b c

a a b c

b a b c

c a b c

P A B C P A B C

P A

( ); lim ( ) ( ); lim ( ) ( )k kk k

P A P B P B P C P C


Gibbs sampling True target distribution (2D normal N(,))

0 1 0.5true

0 0.5 1


Gibbs sampling First 20 Gibbs sampling iterates (conditionals are 1D

normals)


Gibbs sampling Burn-in samples (1000 samples)

0 1 0.5true

0 0.5 1

0.3634 1.1243 0.7443burn-in

0.4190 0.7443 1.3724


Gibbs sampling Samples after Markov chain convergence (samples 1000-

2000)

0 1 0.5true

0 0.5 1

0.3634 1.1243 0.7443burn-in

0.4190 0.7443 1.3724

0.0187 1.0282 0.5052converged

0.0443 0.5052 1.0621


Data augmentation Gibbs sampling

Introducing unobserved variables often simplifies the expression of the likelihood

A Gibbs sampler can then be set up

Samples from the Gibbs sampler can be used to estimate parameters

( , | ) ( | , ) ( | , )

( | , , ) ( | , , )

model parameters, missing data, data

i ji ji j

P M D P M D P M D

P M D P M M D

M D

PME

1

1( | ) ( , | )

Nk

kM

E D P M D dMdN


Pros and cons Gibbs sampling

Explore the space of configuration of a probabilistic model of the data according to the probability of each configuration

Based on incrementaly perturbing the configuration one variable at a time, preferably choosing more likely configurations

Pros Clear probabilistic interpretation Bayesian framework “Global optimization”

Cons Mathematical details not easy to work out Relatively slow


Gibbs biclustering


Gibbs samplingCurrent configuration

1 1( 1| , , )?P g g c D2 2( 1| , , )?P g g c D

Next gene configuration

3 3( 1| , , )?P g g c D


Updated gene configuration

Next complete configuration iterate many times


Gibbs biclustering( , | ) ( | , , ) ( | , , )i ji ji j

P g c D P g g c D P c c g D


Simulated data


Remarks Gibbs biclustering allows noisy patterns Optimized configuration is obtained by averaging successive

iterated configurations

Biclustering is oriented Find subset of samples for which a subset of genes is

consistenly expressed across genes Find subset of genes that are consistently expressed across a

subset of samples

Searching for multiple patterns For gene biclustering, remove the data of

the genes from the current bicluster Search for a new pattern Stop if only empty pattern repeatedly found


Multiple biclusters


Leukemia fingerprints


Mixed-Lineage Leukemia Armstrong et al., Nature Genetics, 2002

Mixed-Lineage Leukemia (MLL) is a subtype of ALL Caused by chromosomal rearrangement in MLL gene Poorer prognosis than ALL

Microarray analysis shows that MLL is distinct from ALL

FLT3 tyrosine kinase distinguishes most strongly between MLL, ALL, and AML Candidate drug target


PCA Features


Biclustering leukemia data Bicluster patients

Find patients for which a subset of genes has a consistent expression profile across this group of patients

Discovery set 21 ALL, 17 MLL, 25 AML

Validation set 3 ALL, 3 MLL, 3 AML


Discovering ALL Bicluster 1: 18 out of 21 ALL patients


Discovering MLL Bicluster 2: 14 out of 17 MLL patients


Discovering AML Bicluster 3: 19 out of 25 AML patients


Rescoring ALL


Rescoring MLL


Rescoring AML

K.U.Leuven ESAT-SCD-Bioi

Qizheng Sheng

Gibbs biclustering of microarray data Yves Moreau & Qizheng Sheng Katholieke Universiteit Leuven...

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