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Reconstructing gene regulatory networks
with probabilistic models
Marco GrzegorczykDirk Husmeier
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Regulatory network
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Network unknown
High-throughput experiments
Postgenomic
data
Machine learning
Statistics
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Overview
• Introduction
• Bayesian networks
• Comparative evaluation
• Integration of biological prior knowledge
• A non-homogeneous Bayesian network for non-stationary processes
• Current work
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Overview
• Introduction
• Bayesian networks
• Comparative evaluation
• Integration of biological prior knowledge
• A non-homogeneous Bayesian network for non-stationary processes
• Current work
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Elementary molecular biological processes
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Description with differential equations
Rates
Concentrations
Kinetic parameters q
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Given: Gene expression time series
Can we infer the correct gene regulatory network?
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Parameters q known: Numerically integrate the differential equations for different hypothetical networks
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Model selection for known parameters q
Gene expression time series predicted with different modelsMeasured gene
expression time series
Highest likelihood: best model
Compare
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Model selection for unknown parameters q
Gene expression time series predicted with different modelsMeasured gene
expression time series
Highest likelihood: over-fitting
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Bayesian model selection
Select the model with the highest posterior probability:
This requires an integration of the whole parameter space:
This integral is usually intractable
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Marginal likelihoods for the alternative pathways
Computational expensive, network reconstruction ab initio unfeasible
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Overview
• Introduction
• Bayesian networks
• Comparative evaluation
• Integration of biological prior knowledge
• A non-homogeneous Bayesian network for non-stationary processes
• Current work
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Objective: Reconstruction of regulatory networks ab initio
Higher level of abstraction: Bayesian networks
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Bayesian networks
A
CB
D
E F
NODES
EDGES
•Marriage between graph theory and probability theory.
•Directed acyclic graph (DAG) representing conditional independence relations.
•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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Bayes net
ODE model
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[A]= w1[P1] + w2[P2] + w3[P3] +
w4[P4] + noise
Linear model
A
P1
P2
P4
P3
w1
w4
w2
w3
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Nonlinear discretized model
P1
P2
P1
P2
Activator
Repressor
Activator
Repressor
Activation
Inhibition
Allow for noise: probabilities
Conditional multinomial distribution
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Model Parameters q
Integral analytically tractable!
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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
Limted 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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Complete enumeration unfeasible Hill climbing
increasesAccept move when
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Configuration space of network structures
Local optimum
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Configuration space of network structures
MCMC Local change
If accept
If accept with probability
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Algorithm converges to
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Madigan & York (1995), Guidici & Castello (2003)
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Configuration space of network structures
Problem: Local changes small steps slow convergence, difficult to cross valleys.
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Configuration space of network structures
Problem: Global changes large steps low acceptance slow convergence.
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Configuration space of network structures
Can we make global changes that jump onto other peaks and are likely to be accepted?
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Conventional scheme New scheme
MCMC trace plots
Plot of against iteration number
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Overview
• Introduction
• Bayesian networks
• Comparative evaluation
• Integration of biological prior knowledge
• A non-homogeneous Bayesian network for non-stationary processes
• Current work
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Cell membran
nucleus
Example: Protein signalling pathway
TF
TF
phosphorylation
-> cell response
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Evaluation on the Raf signalling pathway
From Sachs et al Science 2005
Cell membrane
Receptor molecules
Inhibition
Activation
Interaction in signalling pathway
Phosphorylated protein
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Flow cytometry data
• Intracellular multicolour flow cytometry experiments: concentrations of 11 proteins
• 5400 cells have been measured under 9 different cellular conditions (cues)
• Downsampling to 100 instances (5 separate subsets): indicative of microarray experiments
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Simulated data or “gold standard” from the literature
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Simulated data or “gold standard” from the literature
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Simulated data or “gold standard” from the literature
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From Perry Sprawls
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ROC curve
5 FP counts
BN
GGM
RN
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ROC curveFP
TP
Four different evaluation criteria
DGE UGE
TP for fixed FP
Area under the curve (AUC)
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Synthetic data, observations
Relevance networksBayesian
networksGraphical Gaussian models
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Synthetic data, interventions
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Cytometry data, interventions
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Overview
• Introduction
• Bayesian networks
• Comparative evaluation
• Integration of biological prior knowledge
• A non-homogeneous Bayesian network for non-stationary processes
• Current work
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Can we complement microarray data with prior knowledge from public data bases like KEGG?
KEGG pathwayMicroarray data
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How do we extract prior knowledge from a collection of KEGG pathways?
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Total number of times the gene pair [i,j ] is included in the extracted pathways
Total number of edges i j that appear in the extracted pathways
=
Example: Extract 20 pathways, 10 contain [i,j ], 8 contain i j
B = 8/10 = 0.8i,j
Relative frequency of edge occurrence
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Prior knowledge from KEGG
Raf network
0.25
00.5
0
0.5
0.87
0
1
0.5
0 0
0.5
0
10.71
0
0
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Prior distribution over networks
Deviation between the network M and the prior knowledge B:
Prior knowledge ε [0,1]
Graph ε {0,1}
Hyperparameter
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Hyperparameter β trades off data versus prior knowledge
KEGG pathwayMicroarray data
β
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Hyperparameter β trades off data versus prior knowledge
KEGG pathwayMicroarray data
β small
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Hyperparameter β trades off data versus prior knowledge
KEGG pathwayMicroarray data
β large
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Sample networks and hyperparameters from the posterior distribution
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Revision
Prior distribution
Marginal likelihood
Integral analytically tractable for Bayesian networks
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Application to the Raf pathway:
Flow cytometry data and KEGG
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ROC curveFP
TP
Four different evaluation criteria
DGE UGE
TP for fixed FP
Area under the curve (AUC)
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β
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Overview
• Introduction
• Bayesian networks
• Comparative evaluation
• Integration of biological prior knowledge
• A non-homogeneous Bayesian network for non-stationary processes
• Current work
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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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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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We have to learn from the data:
• Number of different components
• Allocation of time points
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Two MCMC strategies
q
k
h
Number of components (here: 3)
Allocation vector
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Synthetic study: posterior probability of the number of components
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Circadian clock in Arabidopsis thaliana Collaboration with the Institute of Molecular Plant
Sciences (Andrew Millar)
• Focus on 9 circadian genes.•2 time series T20 and T28 of microarray gene expression data from Arabidopsis thaliana.• Plants entrained with different light:dark cycles10h:10h (T20) and 14h:14h (T28)
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macrophage
cytomegalovirus
Interferon gamma
Macrophage
Cytomegalovirus (CMV)
Interferon gamma IFNγ
InfectionTreatment
Collaboration with DPM
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macrophage
IFNγ12 hour time course measuring total RNA
0 1 2 3 4 5 6 7 8 9 10 11 12
72 Agilent Arrays
Time series statistical analysis (using EDGE)
Clustering Analysis
30 min sampling
24 samples per group:
• Infection with CMV
• Pre-treatment with IFNγ
• IFNγ + CMV
CMV
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Posterior probability of the number of components
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IRF1
IRF2
IRF3
Literature “Known” interactions between three cytokines: IRF1, IRF2 and IRF3
Evaluation: Average marginal posterior probabilities of
the edges versus non-edges
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Sample of high-scoring networks
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IRF1
IRF2
IRF3
Gold standard known Posterior probabilities of true interactions
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AUROC scores
New modelBGeBDe
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Collaboration with the Institute of Molecular Plant
Sciences at Edinburgh University
2 time series T20 and T28 of microarray gene expression data from Arabidopsis thaliana.
- Focus on: 9 circadian genes: LHY, CCA1, TOC1, ELF4,
ELF3, GI, PRR9, PRR5, and PRR3
- Both time series measured under constant light condition
at 13 time points: 0h, 2h,…, 24h, 26h
- Plants entrained with different light:dark cycles
10h:10h (T20) and 14h:14h (T28)
Circadian rhythms in Arabidopsis thaliana
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Gene expression time series plots (Arabidopsis data T20 and T28)
T28 T20
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Posterior probability of the number of components
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Predicted network
Blue – activation
Red – inhibition
Black – mixture
three different line widths - thin = PP>0.5- medium = PP>0.75- fat = PP>0.9
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Overview
• Introduction
• Bayesian networks
• Comparative evaluation
• Integration of biological prior knowledge
• A non-homogeneous Bayesian network for non-stationary processes
• Current work
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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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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
Heterogeneous dynamic Bayesian network
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Heterogenous dynamic Bayesian network with node-specific breakpoints
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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Evaluation on synthetic data
X
Y(1) Y(2) Y(3)
f: three phase-shifted sinusoids
BGe
Heterogeneous BNet without/with nodespecific
breakpoints
AUROC
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Four time series for A. thaliana under different experimental conditions (KAY,KDE,T20,T28)
Blue – activation
Red – inhibition
Black – mixture
three different line widths - thin = PP>0.5- medium = PP>0.75- fat = PP>0.9
Network obtained for merged data
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KAY_LL KDE_LL T20 T28
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datadata data datadata data
Monolithic Separate
Propose a compromise between the two
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M1 M221
D1 D2
M*
MII
DI
. . .
Compromise between the two previous ways of combining the data
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Original work with Adriano:
Poor convergence and mixing due too strong coupling effects.
Marco’s current work:
Improve convergence and mixing by weakening the coupling.
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Mean absolute deviation of edge posterior probabilities (independent BN inference)
KAY KDE T20 T28
KAY --- 0.14 0.15 0.14
KDE 0.14 --- 0.19 0.15
T20 0.15 0.19 --- 0.10
T28 0.14 0.15 0.10 ---
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Mean absolute deviation of edge posterior probabilities (coupled BN inference)
KAY KDE T20 T28
KAY --- 0.11 0.12 0.11
KDE 0.11 --- 0.13 0.11
T20 0.12 0.13 --- 0.06
T28 0.11 0.11 0.06 ---
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Mean absolute deviation of edge posterior (independent BN - coupled BN)
KAY KDE T20 T28
KAY --- 0.03 0.03 0.03
KDE 0.03 --- 0.05 0.03
T20 0.03 0.05 --- 0.04
T28 0.03 0.03 0.04 ---
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Summary
• Differential equation models
• Bayesian networks
• Comparative evaluation
• Integration of biological prior knowledge
• A non-homogeneous Bayesian network for non-stationary processes
• Current work
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Adriano Werhli
Marco Grzegorzcyk
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Thank you!
Any questions?