Sylvia Richardson, with Alex Lewin Department of Epidemiology and Public Health,

Imperial College

Bayesian modelling of differential gene expression data

In collaboration with Anne Mette Hein and Clare Marshall (Imperial)

Philippe Broët (INSERM) and Peter Green (Bristol)Helen Causton and Tim Aitman (Hammersmith)

Outline

• Introduction: what is gene expression data ?

• Array effects (normalisation)

• Exchangeable model for the gene specific

variances and Bayesian model checks

• Differential expression

• Rank statistics

• Mixture models for differential expression

Protein-encoding genes are transcribed into mRNA (messenger), and the mRNA is translated to make proteins, the building blocks of living cells

DNA Microarrays can be used to measure the relative abundance of mRNA, providing information on gene expression in a particular cell type, under particular conditions

The fundamental principle used to measure the expression is that of hybridisation

What is gene expression data ?

Introduction 1

The Principle of Hybridisation

20µm

Millions of copies of a specificoligonucleotide sequence element

Image of Hybridised Array

Approx. ½ million differentcomplementary oligonucleotides

Single stranded, labeled RNA sample

Oligonucleotide element

**

**

*

1.28cm

Hybridised Spot

Slide courtesy of Affymetrix

Expressed genes

Non-expressed genes

Zoom Image of Hybridised Array

Introduction 2

Hybridisation indicates that the corresponding gene is present because :

The binding is highly specific The sequences represented on the array are

designed to be unique in the genome

The expression level of thousands of genes are measured on a single microarray

gene expression profile

There are different types of arrays :we consider the case of oligonucleotide arrays where one extract per slide is hybridised

Introduction 3

Variation and uncertainty

Gene expression data (e.g. Affymetrix) is the result of multiple sources of variability

• Treatment• Response to genetic and environmental

conditions

• Biological heterogeneity• Sample preparation• Array manufacture• Hybridisation process• Imaging

signal

Different components of noise

Introduction 4

Variation and uncertainty

Gene expression data (e.g. Affymetrix) is the result of multiple sources of variability

• Treatment• Response to genetic and environmental

conditions

• Biological heterogeneity inherent• Sample preparation• Array manufacture• Hybridisation process• Imaging

signal

Good practice minimizes these sources of variation

Low-level Model(how is the measured expression related to the

signal)

Normalisation(to make samples comparable)

Differential Expression

ClusteringPartition Model

Gene expression analysis is a multi-step process

We aim to integrate all the steps in a common statistical framework

Introduction 5

Bayesian hierarchical model framework

Ability to model various sources of variability:e.g. detailed modelling of experimental variability: within array, between array, estimation of gene specific variability …

Building of all these features into a common model

uncertainty is propagated Ability to borrow / share information in appropriate ways to get better estimates

Introduction 6

Data Set and Biological question

Previous Work (Tim Aitman, Anne Marie Glazier)Deficiency in gene Cd36 found to be associated

with insulin resistance in SHR (spontaneously hypertensive rat)

Good animal model to tease out genes implicated in this syndrome

Microarray Study • 3 SHR compared with 3 transgenic rats• 3 wildtype mice compared with 3 knockout mice• Two tissues: fat and heart• Affymetrix chips U34A-C and U74A-C

( 12000 genes)

I Additive (log scale) model for expression

Notation• ygr = gene expression measurements (log scale) for

gene g, g=1, …, N, replicate r, r = 1,…, R Additive model:

ygr = g + r(g) + gr

Here:

-- g is the expression level of the gth gene,

-- r(g) is the array effect (normalisation term) possibly dependent on g through the expression level g ,

constraints needed to ensure identifiability:

Σr r(g) = 0-- gr an error term, mean 0, Var(gr ) = σg

2 Model 1

6 equal size groupsof genes defined by mean expression level

Exploratory analysis of array effect

Wildtype mouse fat data on 3 arrays

Estimate a constant array effect ri

for each group i, i =1:6

ygr = g + ri + gr

Each array corresponds to a different colour

ri, r =1:3, i=1:6

Model 2

Flexible model of the array effect

The exploratory analysis suggests to model the array effect as a (smooth) function of g

Piecewise polynomial with unknown break points:

r(g) = quadratic in g for ark-1 ≤ g ≤ ark

with coeff (brk(1), brk

(2) ), k =1, … # knots

Location of break points are not fixed

All coefficients brk(j) are given centred normal priors,

Σr r(g) = 0 constraintModel 3

Non linear fit of array effect as a function of level g

Model 4

Before (ygr)

After (ygr- r(g) )

Wildtype Knockout

Effect of normalisation on density

Model 5

• 2nd level of the model : Exchangeable hierarchical prior:

g2 lognormal (μ, τ), g = 1, … N

The hyper parameters μ and τ can be influential• 3rd level of the model

μ N( c, d)

τ lognormal (e, f)

where the constants c, d, e and f are chosen so that the third level priors are vague

Hierarchical structure for gene variability in each condition

Model 6

μ, τygr

br ar

g

r(g)

r = 1:R

g = 1:G

g2

br ar

Graphicalrepresentationof the 3-levelmodel

Model 7

• Variances are estimated using information from all G x R measurements (~12000 x 3) rather than just 3

• Variances are stabilised and shrunk towards average variance

Smoothing of the gene specific variances

The implementation of the model uses WinBugsModel 8

• Check different possible assumptions on gene variances, e.g.

equal : gr N(0, 2) or exchangeable variances ?• Predict sample variance Sg

2 new (our checking function)

from the model specification• Compare predicted Sg

2 new with observed Sg2 obs

Bayesian p-value Prob( Sg2 new > Sg

2 obs )

• Distribution of p-values Uniform if model is ‘true’• Easily implemented in MCMC algorithm

Bayesian Model Checking

Model 9

μ, τ

ygr

br ar

g

r(g)

r = 1:R

g = 1:G

g2

br ar

Bayesian modelchecking

ygrg

2new new

Model 10

Bayesian predictive p-valuesExchangeable variance model is supported by the data

Equal variance model has too little variability for the data

• Gene expression data can be used in several types of analysis:

-- Comparison of gene expression

under different experimental

conditions,

-- Classification of gene

expression profiles

-- Exploration of patterns in gene

expression matrices

-- Association of gene expression

with factors, e.g. prognosis

II-- Analysing gene expression dataGene expression data matrix

Samples

Gen

esCond 1

yg1r

Cond 2

yg2r

Differential expression model

The quantity of interest is the difference between conditions for each gene: dg , g = 1, …,N

Joint model for the 2 conditions :

yg1r = g - ½ dg + 1r(g) + g1r , r = 1, … R1

yg2r = g + ½ dg + 2r(g) + g2r , r = 1, … R2

• g is now the overall gene effect over the conditions

• Same assumptions for the distribution of σ2gs and the

modelling of sr(g) as before, s = 1, 2

• All hyper parameters are indexed by the condition

(1)

Differential 2

Possible statistics for differential expression

• The differences dg (≈ log fold change) or

the standardised differences

dg* = dg / (σ2 g1 / R1 + σ2 g2 / R2 )½

• We obtain the joint distribution of all {dg } or {dg* }

Process the output to have the distributions of the ranks

{ r (dg ), g = 1, …., N }

Model the distribution of dg flexibly to allow a mixture of

(small) subgroups of genes with ‘extreme’ dg (H1)

and a (large) group of genes with dg around 0 (H0)

Differential 3

Ranks of modelled log fold change dg

150 genes with lowest rank

Even genes with median rank less than 100 can have large uncertainty

3 wildtype micecompared to3 knockout mice

2.5% - 97.5% rank intervals for each gene

Differential 4

Posterior probabilities for each gene to be ranked

In the bottom 100 In the top 100

log fold change dg Differential 5

Mixture modelling the distribution of dg

• Mixture model framework but with an ‘unknown’ number of components

• Fully Bayesian hierarchical framework (the number of components is a random variable)

• Based on Green (1995) and Richardson and Green (1997) papers

• MCMC algorithms• Applied in an experimental context concerning

bladder cancer

(Broët, Richardson and Radvanyi; Journal of Computational Biology, 9, 671-683, 2002) Mixture 1

Bayesian mixture model

Normal mixtures with an unknown number of states

A gene can be in different states:

down regulated, …, unaffected, …, up-regulated

Bayesian estimation of posterior distribution:• for number of states (besides the unaffected one)• for the allocation of genes to the states

classification of states in the ‘extreme components’

or in the central one, based on their posterior probability

the central state corresponding to dg ≈ 0

Mixture 2

Mixture model specification

dg ~ w0 N(0, λ2η0

2) + j=1:k wj N(μj, ηj2)

Prior setting

λ2 > 1 expresses that the central component is expected to have a larger variance

μj+ > 0 , ordered, uniform on upper range

μj- < 0 , ordered, uniform on lower range

{η02 , ηj

2 } exchangeable, Gamma distributed

Weights {w0 , wj, j = 1:k} ~ Dirichlet, uniform or other alternativesk, unknown number of components

μj

Mixture 3

Biological Context

• Data from the Curie Institute/Research Section

(F. Radvanyi team, CNRS/Curie)

• The cell line: T24 bladder cell lines • The transfection: FGFR2 cDNA (fibroblast growth factor

receptor 2)• Comparison of 2 cell lines: Unmodified and Modified

(transfected by FGFR2 cDNA)• Material: Nylon microarray, 4608 gene expression from the

same batch, 33P labelling, 4 replicates, same experimenter, same day

Aim: to study transcriptional changes induced by a

defined DNA transfection in a cell line

Mixture 4

Comparison of gene expression in two bladder cancer cell lines (unmodified versus transfected)

Distribution of dg and QQ plot

dg : Differential expression for gene g after taking intoaccount array and cell line main effects. Negative values of dg correspond to up-regulated genes.

Posterior distribution of the number of mixture components

Components left of central one

P(0) = 0.00 P(1) = 0.11 P(2) = 0.73

P(3) = 0.14 P(4) = 0.01 P(5) = 0.00

Components right of central one

P(0) = 0.00 P(1) = 0.46 P(2) = 0.49

P(3) = 0.04 P(4) = 0.052 P(5) = 0.00

Support for a model with 5 components,

2 on the left and 2 on the right of the central one Mixture 5

Estimate of the posterior probability for each gene to be in each of the 5 components

Classification

• Classification based on maximum a posteriori rule:

Group 1 2 3 4 5

Nb 9 26 4540 29 4 Mean -0.82 -0.51 0 0.51 0.86

SD 0.16 0.08 0.14 0.10 0.17

Weight 0.2% 0.8% 98% 0.8% 0.2%

This analysis highlights 9 up-regulated genes (expressed after transfection of the receptor) and 4 down-regulated ones, with an indication of 2 further subgroups showing some evidence of differential expression

• Model different sources of variability into a single model

• Borrow information from all genes to estimate gene specific variances, non linear array effect

• Exploit the joint distribution of the differential expression measure through ranks or mixture models

• Further work is under way – on modelling of the low level Affy probe data – on more general clustering algorithms

Summary