Inferring Quantitative Models of Regulatory Networks From Expression Data
Inferring gene regulatory networks with non-stationary dynamic Bayesian networks
description
Transcript of Inferring gene regulatory networks with non-stationary dynamic Bayesian networks
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Inferring gene regulatory networks with non-stationary dynamic Bayesian networks
Dirk Husmeier Frank Dondelinger
Sophie Lebre
Biomathematics & Statistics Scotland
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Overview
• Introduction
• Non-homogeneous dynamic Bayesian network for non-stationary processes
• Flexible network structure
• Open problems
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Can we learn signalling pathways from postgenomic data?
From Sachs et al Science 2005
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Network reconstruction from postgenomic data
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Friedman et al. (2000), J. Comp. Biol. 7, 601-620
Marriage between
graph theory
and
probability theory
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Bayes net
ODE model
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A
CB
D
E F
NODES
EDGES
Graph theory
•Directed acyclic graph (DAG) representing conditional independence relations.
Probability theory
•It is possible to score a network in light of the data: P(D|M), D:data, M: network structure.
•We can infer how well a particular network explains the observed data.
),|()|(),|()|()|()(
),,,,,(
DCFPDEPCBDPACPABPAP
FEDCBAP
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[A]= w1[P1] + w2[P2] + w3[P3] +
w4[P4] + noise
BGe (Linear model)
A
P1
P2
P4
P3
w1
w4
w2
w3
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BDe (Nonlinear discretized model)
P1
P2
P1
P2
Activator
Repressor
Activator
Repressor
Activation
Inhibition
Allow for noise: probabilities
Conditional multinomial distribution
P
P
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Model Parameters q
Integral analytically tractable!
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BDe: UAI 1994
BGe: UAI 1995
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Dynamic Bayesian network
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Example: 2 genes 16 different network structures
Best network: maximum score
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Identify the best network structure
Ideal scenario: Large data sets, low noise
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Uncertainty about the best network structure
Limited number of experimental replications, high noise
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Sample of high-scoring networks
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Sample of high-scoring networks
Feature extraction, e.g. marginal posterior probabilities of the edges
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Sample of high-scoring networks
Feature extraction, e.g. marginal posterior probabilities of the edges
High-confident edge
High-confident non-edge
Uncertainty about edges
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Can we generalize this scheme to more than 2 genes?
In principle yes.
However …
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Number of structures
Number of nodes
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Configuration space of network structures
Find the high-scoring structures
Sampling from the posterior distribution
Taken from the MSc thesis by Ben Calderhead
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Madigan & York (1995), Guidici & Castello (2003)
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Configuration space of network structures
MCMC Local change
If accept
If accept with probability
Taken from the MSc thesis by Ben Calderhead
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Overview
• Introduction
• Non-homogeneous dynamic Bayesian networks for non-stationary processes
• Flexible network structure
• Open problems
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Dynamic Bayesian network
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Example: 4 genes, 10 time points
t1 t2 t3 t4 t5 t6 t7 t8 t9 t10
X(1) X1,1 X1,2 X1,3 X1,4 X1,5 X1,6 X1,7 X1,8 X1,9 X1,10
X(2) X2,1 X2,2 X2,3 X2,4 X2,5 X2,6 X2,7 X2,8 X2,9 X2,10
X(3) X3,1 X3,2 X3,3 X3,4 X3,5 X3,6 X3,7 X3,8 X3,9 X3,10
X(4) X4,1 X4,2 X4,3 X4,4 X4,5 X4,6 X4,7 X4,8 X4,9 X4,10
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t1 t2 t3 t4 t5 t6 t7 t8 t9 t10
X(1) X1,1 X1,2 X1,3 X1,4 X1,5 X1,6 X1,7 X1,8 X1,9 X1,10
X(2) X2,1 X2,2 X2,3 X2,4 X2,5 X2,6 X2,7 X2,8 X2,9 X2,10
X(3) X3,1 X3,2 X3,3 X3,4 X3,5 X3,6 X3,7 X3,8 X3,9 X3,10
X(4) X4,1 X4,2 X4,3 X4,4 X4,5 X4,6 X4,7 X4,8 X4,9 X4,10
Standard dynamic Bayesian network: homogeneous model
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Limitations of the homogeneity assumption
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Our new model: heterogeneous dynamic Bayesian network. Here: 2 components
t1 t2 t3 t4 t5 t6 t7 t8 t9 t10
X(1) X1,1 X1,2 X1,3 X1,4 X1,5 X1,6 X1,7 X1,8 X1,9 X1,10
X(2) X2,1 X2,2 X2,3 X2,4 X2,5 X2,6 X2,7 X2,8 X2,9 X2,10
X(3) X3,1 X3,2 X3,3 X3,4 X3,5 X3,6 X3,7 X3,8 X3,9 X3,10
X(4) X4,1 X4,2 X4,3 X4,4 X4,5 X4,6 X4,7 X4,8 X4,9 X4,10
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t1 t2 t3 t4 t5 t6 t7 t8 t9 t10
X(1) X1,1 X1,2 X1,3 X1,4 X1,5 X1,6 X1,7 X1,8 X1,9 X1,10
X(2) X2,1 X2,2 X2,3 X2,4 X2,5 X2,6 X2,7 X2,8 X2,9 X2,10
X(3) X3,1 X3,2 X3,3 X3,4 X3,5 X3,6 X3,7 X3,8 X3,9 X3,10
X(4) X4,1 X4,2 X4,3 X4,4 X4,5 X4,6 X4,7 X4,8 X4,9 X4,10
Our new model: heterogeneous dynamic Bayesian network. Here: 3 components
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Learning with MCMC
q
k
h
Number of components (here: 3)
Allocation vector
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Non-homogeneous model
Non-linear model
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[A]= w1[P1] + w2[P2] + w3[P3] +
w4[P4] + noise
BGe: Linear model
A
P1
P2
P4
P3
w1
w4
w2
w3
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BDe: Nonlinear discretized model
P1
P2
P1
P2
Activator
Repressor
Activator
Repressor
Activation
Inhibition
Allow for noise: probabilities
Conditional multinomial distribution
P
P
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Pros and cons of the two models
Linear Gaussian model
• Restriction to linear processes
• Original data no information loss
Multinomial model
• Nonlinear model
• Discretization information loss
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Can we get an approximate nonlinear model without data discretization?
y
x
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Can we get an approximate nonlinear model without data discretization?
Idea: piecewise linear model
y
x
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t1 t2 t3 t4 t5 t6 t7 t8 t9 t10
X(1) X1,1 X1,2 X1,3 X1,4 X1,5 X1,6 X1,7 X1,8 X1,9 X1,10
X(2) X2,1 X2,2 X2,3 X2,4 X2,5 X2,6 X2,7 X2,8 X2,9 X2,10
X(3) X3,1 X3,2 X3,3 X3,4 X3,5 X3,6 X3,7 X3,8 X3,9 X3,10
X(4) X4,1 X4,2 X4,3 X4,4 X4,5 X4,6 X4,7 X4,8 X4,9 X4,10
Inhomogeneous dynamic Bayesian network with common changepoints
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Inhomogenous dynamic Bayesian network with node-specific changepoints
t1 t2 t3 t4 t5 t6 t7 t8 t9 t10
X(1) X1,1 X1,2 X1,3 X1,4 X1,5 X1,6 X1,7 X1,8 X1,9 X1,10
X(2) X2,1 X2,2 X2,3 X2,4 X2,5 X2,6 X2,7 X2,8 X2,9 X2,10
X(3) X3,1 X3,2 X3,3 X3,4 X3,5 X3,6 X3,7 X3,8 X3,9 X3,10
X(4) X4,1 X4,2 X4,3 X4,4 X4,5 X4,6 X4,7 X4,8 X4,9 X4,10
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NIPS 2009
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Overview
• Introduction
• Non-homogeneous dynamic Bayesian network for non-stationary processes
• Flexible network structure
• Open problems
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Non-stationarity in the regulatory process
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Non-stationarity in the network structure
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ICML 2010
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Flexible network structure with regularization
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Flexible network structure with regularization
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Flexible network structure with regularization
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Morphogenesis in Drosophila melanogaster
• Gene expression measurements over 66 time steps of 4028 genes (Arbeitman et al., Science, 2002).
• Selection of 11 genes involved in muscle development.
Zhao et al. (2006),
Bioinformatics 22
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Transition probabilities: flexible structure with regularization
Morphogenetic transitions: Embryo larva larva pupa pupa adult
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Comparison with:
Dondelinger, Lèbre & Husmeier Ahmed & Xing
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Collaboration with Frank Dondelinger and Sophie Lèbre
NIPS 2010
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Method based on homogeneous DBNs
Method based on differential equations
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Sample of high-scoring networks
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Sample of high-scoring networks
Feature extraction, e.g. marginal posterior probabilities of the edges
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Method based on homogeneous DBNs
Method based on differential equations
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Overview
• Introduction
• Non-homogeneous dynamic Bayesian network for non-stationary processes
• Flexible network structure
• Open problems
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Exponential versus binomial prior distribution
Exploration of various information sharing options
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How to deal with static data?
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Change-point process
Free allocation
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Allocation sampler versus change-point process
• More flexibility, unrestricted mixture model.
• Not restricted to time series
• Higher computational costs
• Incorporates plausible prior knowledge for time series.
• Reduced complexity• Less universal, not
applicable to static data
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Marco GrzegorczykUniversity of Dortmund
Germany
Frank Dondelinger Biomathematics & Statistics Scotland
United Kingdom
Sophie LèbreUniversité de Strasbourg
France
Acknowledgements
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Further details for discussion during
question time
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Details on exponential prior
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Hierarchical Bayesian model
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Hierarchical Bayesian model
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MCMC scheme (for symmetric proposal distributions)
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Details on other priors
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where
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Partition function
Ignoring the fan-in restriction:
Number of genes
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Simulation study• We randomly generated 10 networks with 10
nodes each.• Number of regulators for each node drawn from
a Poisson distribution with mean=3.• 5 time series segments• Network changes: number of changes drawn
from a Poisson distribution.• For each segment: time series of length 50
generated from a linear regression model, interaction parameters drawn from N(0,1), iid Gaussian noise from N(0,1).
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Synthetic simulation study
No information sharing between
adjacent segments
Information sharing between adjacent
segments
Frank Dondelinger, Sophie Lèbre, Dirk Husmeier: ICML 2010