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Using Bayesian Networks to
Analyze Expression Data
N. Friedman M. Linial I. Nachman D. Pe’er
Hebrew University, Jerusalem
.
Central Dogma
Transcription
Translation
mRNA
Gene
Cells express different subset of the genes
In different tissues and under different conditions
Protein
Microarrays (aka “DNA chips”)
 New
technological breakthrough:
 Measure RNA expression levels of thousands
of genes in one experiment
 Measure expression on
a genomic scale
 Opens up new
experimental designs
 Many major labs are using,
or will use this technology
in the near future
The Problem
Experiments
j
Genes
i
Goal:
Aij - the mRNA level of gene j in experiment i
 Learn regulatory/metabolic networks
 Identify causal sources of the biological
phenomena of interest
Our Approach
 Characterize
statistical relationships between
expression patterns of different genes
 Beyond pair-wise interactions


Many interactions are explained by intermediate factors
Regulation involves combined effects of several geneproducts
We build on the language of Bayesian networks
Network: Example
Noisy stochastic process:
Example: Pedigree Homer
 A node represents
an individual’s
genotype
Bart
 Modeling

Marge
Lisa
Maggie
assumptions:
Ancestors can effect descendants' genotype only by
passing genetic materials through intermediate
generations
Ancestor
Network Structure
Parent
Generalizing to DAGs:

A child is conditionally
independent from its
non-descendents, given the
value of its parents
Y1
Y2
X
Often a natural assumption
for causal processes

if we believe that we capture
the relevant state of each
intermediate stage.
Non-descendent
Descendent
Local Probabilities
 Associated
with each variable Xi is a conditional
probability distribution P(Xi|Pai:)
X P(Y |X)
 Discrete variables:
X
x 0.9 0.1
Multinomial distribution
x
Y
variables:
Choice: for example linear gaussian
P(Y | X)
 Continuous
X
Y
0.3 0.7
Bayesian Network Semantics
B
E
R
A
C
Qualitative part
DAG specifies
conditional
independence
statements
Quantitative part
+
local
probability
models
=
Unique joint
distribution
over domain
 Compact


& efficient representation:
 k parents  O(2kn) vs. O(2n) params
parameters pertain to local interactions
P(C,A,R,E,B) = P(B)*P(E|B)*P(R|E,B)*P(A|R,B,E)*P(C|A,R,B,E)
versus
P(C,A,R,E,B) = P(B)*P(E) * P(R|E) * P(A|B,E) * P(C|A)
Why Bayesian Networks?
Bayesian Networks:
 Flexible representation of dependency structure
of multivariate distributions
 Natural for modeling processes with local
interactions
Learning of Bayesian Networks
 Can learn dependencies from observations
 Handles stochastic processes:


“true” stochastic behavior
noise in measurements
Modeling Regulatory Interactions
Variables of interest:
 Expression levels of genes
 Concentration levels of proteins (proteomics!)
 Exogenous variables: Nutrient levels, Metabolite
Levels, Temperature,
 Phenotype information
 …
Bayesian Network Structure:
 Capture dependencies among these variables
Examples
Measured expression
level of each gene
Gene interaction
Random variables
Probabilistic
dependencies
Interactions are represented by a graph:
 Each gene is represented by a node in the graph
 Edges between the nodes represent direct
dependency
X
A
B
A
B
More Complex Examples
 Dependencies
can be mediated through other
nodes
B
A
A
C
C
Common cause
 Common
B
Intermediate gene
effects can imply conditional dependence
A
B
C
Outline of Our Approach
Bayesian Network
Learning Algorithm
Expression data
B
E
R
A
C
Use learned network to make predictions about
structure of the interactions between genes
Experiment
Data from Spellman et al.
(Mol.Bio. of the Cell 1998)
 Contains 76 samples of all the
yeast genome:
 Different methods for
synchronizing cell-cycle in
yeast
 Time series at few minutes
(5-20min) intervals
 Spellman et al. identified 800
cell-cycle regulated genes.
Methods
 Treat
samples as IID (ignoring temporal order)
Experiment 1:
 Discretized into three levels of expression
0
-
-0.5
+
0.5
Log(ratio to control)
 Learn
multinomial probabilities
Experiment 2:
 Learn linear interactions (w/ Gaussian noise)
No prior biological knowledge was used
Network Learned
Challenge: Statistical Significance
Sparse Data
 Small number of samples
 “Flat posterior” -- many networks fit the data
Solution
 estimate confidence in network features
 Two types of features
 Markov neighbors: X directly interacts with Y
 Order relations: X is an ancestor of Y
Confidence Estimates
B
E
Bootstrap approach
[FGW, UAI99]
D1
Learn
R
A
C
E
D
resample
D2
Learn
R
A
C
...
Dm
E
R
B
A
Learn
C
1
C (f )   1f  Gi 
m i 1
m
Estimate:
B
Testing for Significance
 We
run our procedure on randomized data where
we reshuffled the order of values for each gene
 Histograms of number of Markov features at each
confidence level
Original Data
Randomized Data
Testing for Significance
 We
run our procedure on randomized data where
we reshuffled the order of values for each gene
Markov w/ Gaussian Models
Features with Confidence above t
4000
500
450
3500
Random
Real
Random
Real
400
350
3000
300
2500
250
200
2000
150
1500
100
50
1000
500
0
0.1
0.2
0.3
0.4
0.5
0.6
t
0
0.1
0.2
0.7
0.8
0.3
0.9
0.4
0.5
1
0.6
0.7
0.8
0.9
1
Testing for Significance
Markov w/ Multinomial Models
Features with Confidence above t
250
1400
Random
Real
Random
Real
200
1200
1000
150
800
100
600
50
400
0
0.1
200
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
t
0.7
0.8
0.9
1
0.5
0.6
0.7
0.8
0.9
1
Local Map
Finding Key Genes
Key gene: a gene that preceeds many other genes
YLR183C
 MCD1 Mitotic Chromosome Determinant;
 RAD27 DNA repair protein
 CLN2 role in cell cycle START
 SRO4 involved in cellular polarization during budding
 YOX1 Homeodomain protein that binds leu-tRNA gene
 POL30 required for DNA replication and repair
 YLR467W
 CDC5
 MSH6 Homolog of the human GTBP protein
 YML119W
 CLN1 role in cell cycle START

Future Work
 Finding
suitable local distribution models
 Correct handling of hidden variables

Can we recognize hidden causes of coordinated
regulation events?
 Incorporating

prior knowledge
Incorporate large mass of biological knowledge, and
insight from sequence/structure databases
 Abstraction

Combine with cluster analysis of higher confidence
conclusions