modelling gene expression dynamics with gaussian processes · di erential equations: pol-ii to mrna...
TRANSCRIPT
![Page 1: Modelling gene expression dynamics with Gaussian processes · Di erential equations: Pol-II to mRNA dynamics. Gaussian processes: ... M.Rattray, N.D. Lawrence \Fast non-parametric](https://reader035.vdocuments.net/reader035/viewer/2022063010/5fc36c8a76d85279b24608ac/html5/thumbnails/1.jpg)
Modelling gene expression dynamicswith Gaussian processes
Regulatory Genomics and EpigenomicsMarch 10th 2016
Magnus RattrayFaculty of Life Sciences
University of Manchester
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Talk Outline
Introduction to Gaussian process regression
Example 1. Hierarchical models: batches and clusters
Example 2. Branching models: perturbations and bifurcations
Example 3. Differential equations: Pol-II to mRNA dynamics
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Gaussian processes: flexible non-parametric models
Probability distributions over functions
f (t) ∼ GP(mean(t), cov(t, t ′)
)Covariance function k = cov(t, t ′) defines typical properties,
I Static . . . Dynamic
I Smooth . . . Rough
I Stationary. . . non-Stationary
I Periodic. . . Chaotic
The covariance function has parameters tuning these properties
Bayesian Machine Learning perspective: Rasmussen & Williams“Gaussian Processes for Machine Learning” (MIT Press, 2006)
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Gaussian processes
Samples from a 25-dimensional multivariate Gaussian distribution:
n
fn
5 10 15 20 25
−1
−0.5
0.5
1
1.5
2
n
m
5 10 15 20 25
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[f1, f2, . . . , f25] ∼ N (0,C )
Learning and Inference in Computational Systems Biology, MIT Press
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Gaussian processes
Take dimension →∞
−3 −2 −1 1 2 3
−2.5
−2
−1.5
−1
−0.5
0.5
1
1.5
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2.5
−3 −2 −1 1 2 3
−3
−2
−1
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f ∼ GP(0, k) k(t, t ′)
= exp
(−(t − t ′)2
l2
)
Learning and Inference in Computational Systems Biology, MIT Press
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Gaussian processes for inference: Bayesian Regression
2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0
2
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1
2
Figures from James Hensman
![Page 7: Modelling gene expression dynamics with Gaussian processes · Di erential equations: Pol-II to mRNA dynamics. Gaussian processes: ... M.Rattray, N.D. Lawrence \Fast non-parametric](https://reader035.vdocuments.net/reader035/viewer/2022063010/5fc36c8a76d85279b24608ac/html5/thumbnails/7.jpg)
Regression example
2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0
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Figures from James Hensman
![Page 8: Modelling gene expression dynamics with Gaussian processes · Di erential equations: Pol-II to mRNA dynamics. Gaussian processes: ... M.Rattray, N.D. Lawrence \Fast non-parametric](https://reader035.vdocuments.net/reader035/viewer/2022063010/5fc36c8a76d85279b24608ac/html5/thumbnails/8.jpg)
Regression example
2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0
2
1
0
1
2
Figures from James Hensman
![Page 9: Modelling gene expression dynamics with Gaussian processes · Di erential equations: Pol-II to mRNA dynamics. Gaussian processes: ... M.Rattray, N.D. Lawrence \Fast non-parametric](https://reader035.vdocuments.net/reader035/viewer/2022063010/5fc36c8a76d85279b24608ac/html5/thumbnails/9.jpg)
Regression example
2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0
2
1
0
1
2
Figures from James Hensman
![Page 10: Modelling gene expression dynamics with Gaussian processes · Di erential equations: Pol-II to mRNA dynamics. Gaussian processes: ... M.Rattray, N.D. Lawrence \Fast non-parametric](https://reader035.vdocuments.net/reader035/viewer/2022063010/5fc36c8a76d85279b24608ac/html5/thumbnails/10.jpg)
Ex1. Hierarchical models: batches and clusters
0 5 10 15 2021012
0 5 10 15 20 0 5 10 15 20 0 5 10 15 20
0 5 10 15 2021012
0 5 10 15 20 0 5 10 15 20 0 5 10 15 20
Data from Kalinka et al. ”Gene expression divergence recapitulatesthe developmental hourglass model” Nature 2010
Joint work with James Hensman and Neil Lawrence
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Usual processing options for time course batches
Lumped
0 5 10 15 20
2
1
0
1
2
Averaged
0 5 10 15 20
2
1
0
1
2
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Hierarchical Gaussian process
gene:
g(t) ∼ GP(0, kg (t, t ′)
)replicate:
fi (t) ∼ GP(g(t), kf (t, t ′)
)
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Hierarchical Gaussian process
3
2
1
0
1
2
J. Hensman, N.D. Lawrence, M.Rattray ”Hierarchical Bayesianmodelling of gene expression time series across irregularly sampledreplicates and clusters” BMC Bioinformatics 2013
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Hierarchical Gaussian process for clustering
An extended hierarchy
h(t) ∼ GP(
0, kh(t, t ′))
cluster
gi (t) ∼ GP(h(t), kg (t, t ′)
)gene
fir (t) ∼ GP(gi (t), kf (t, t ′)
)replicate
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Hierarchical Gaussian process for clustering
2
1
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CG11786
2
1
0
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CG12723
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1
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1
2
CG13196
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1
0
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2
CG13627
2
1
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CG4386
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Osi15
0 5 10 15 202
1
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2
0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20
Time (hours)
Norm
alis
ed
gen
e e
xp
ress
ion
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Hierarchical Gaussian process for clustering
Modifying an existing algorithm to include this model of replicateand cluster structure leads to more meaningful clustering
MF BP CC L N. clust.
agglomerative HGP 0.46 0.16 0.50 7360.8 50agglomerative GP 0.39 0.13 0.36 6203.7 128Mclust (concat.) 0.39 0.07 0.25 1324.0 26
Mclust (averaged) 0.40 0.08 0.24 -736.2 20
Variational Bayes algorithm is more efficient, allowing Bayesianclustering of >10K profiles with a Dirichlet Process prior
J. Hensman, M.Rattray, N.D. Lawrence “Fast non-parametricclustering of time-series data” IEEE TPAMI 2015
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Clustering with a periodic covariance
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Ex2. Branching models: perturbations and bifurcations
0 5 10 15Time (hrs)
−4
−3
−2
−1
0
1
2
3
4
5
Sim
ulated gene expression
Simulated control data
Simulated perturbed data
Joint work with Jing Yang, Chris Penfold and Murray Grant
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Samples from a branching model
Sample 1
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Samples from a branching model
Sample 2
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Samples from a branching model
Sample 3
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Samples from a branching model
Sample 4
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Samples from a branching model
Sample 5
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Joint distribution to two functions crossing at tp
f ∼ GP(0,K ) , g ∼ GP(0,K ) , g(tp) = f (tp)
Σ =
(Kff Kfg
Kgf Kgg
)=
(K (T,T)
K(T,tp)K(tp ,T)k(tp ,tp)
K(T,tp)K(tp ,T)k(tp ,tp)
K (T,T)
)(1)
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Joint distribution of two datasets diverging at tp
y c(tn) ∼ N (f (tn), σ2)
yp(tn) ∼ N (f (tn), σ2) for tn ≤ tp
yp(tn) ∼ N (g(tn), σ2) for tn > tp
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Posterior probability of the perturbation time tp
p(tp|y c(T), yp(T)) ' p(y c(T), yp(T)|tp)∑t=tmaxt=tmin
p(y c (T), yp(T)|t)
0 5 10 15Time (hrs)
−4
−3
−2
−1
0
1
2
3
4
5
Sim
ulated gene expression
Simulated control data
Simulated perturbed data
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Comparison to DE thresholding approach
Alternative: use first point where some DE threshold is passedWe tested several DE metrics in the nsgp package
0 5 10 15 20
0
5
10
15
20
Esti
mate
d p
ert
urb
ati
on
tim
es
DEtime
Mean
Median
MAP
0 5 10 15 20
nsgp
EMLL0.5
EMLL1
Testing perturbation times
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Investigating a plant’s response to bacterial challenge
Infection with virulent Pseudomonas syringage pv. tomato DC3000vs. disarmed strain DC3000hrpA
0 10 17
Times (hrs)
8
9
10
11
12
13
Gen
e e
xp
ressio
n
CATMA1c72124
Condition 1
Condition 2
0 10 17
Times (hrs)
CATMA1a09335
Condition 1
Condition 2
Yang et al. “Inferring the perturbation time from biological timecourse data” Bioinformatics (accepted) preprint: arxiv 1602.01743
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Current work: modelling branching in single-cell data
12
48
1632
64start
5
10
15
20
25
30
35
40
1
2
3
4
5
6
7
with A. Boukouvalas, M. Zweissele, J. Hensman and N. Lawrence
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Ex 3. Linking Pol-II activity to mRNA profiles
040
80160
320640
1280
t (min)
mRNA
a
b
c
d
e
f
040
80160
320640
1280
t (min)
Pol−II
0
2
4
6
Pol II
ENSG00000163659 (TIPARP)
0 40 80 160 320 64012800
50
100
mR
NA
(F
PK
M)
t (min)0 50 100
0
0.2
0.4
0.6
0.8
1
Delay (min)
Joint work with Antti Honkela,Jaakko Peltonen, Neil Lawrence
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Linking Pol-II activity to mRNA profiles
dm(t)
dt= βp(t −∆)− αm(t)
I m(t) is mRNA concentration (RNA-Seq data)
I p(t) is mRNA production rate (3’ pol-II ChIP-Seq data)
I α is degradation rate (mRNA half-life t1/2 = 2/α)
I ∆ is processing delay
We model p(t) ∼ GP(0, kp) as a Gaussian process (GP)
Likelihood can be worked out exactly
Bayesian MCMC used to estimate parameters α, β, ∆ and GPcovariance (2) and noise variance (1) parameters
![Page 32: Modelling gene expression dynamics with Gaussian processes · Di erential equations: Pol-II to mRNA dynamics. Gaussian processes: ... M.Rattray, N.D. Lawrence \Fast non-parametric](https://reader035.vdocuments.net/reader035/viewer/2022063010/5fc36c8a76d85279b24608ac/html5/thumbnails/32.jpg)
Linking Pol-II activity to mRNA profiles
dm(t)
dt= βp(t −∆)− αm(t)
I m(t) is mRNA concentration (RNA-Seq data)
I p(t) is mRNA production rate (3’ pol-II ChIP-Seq data)
I α is degradation rate (mRNA half-life t1/2 = 2/α)
I ∆ is processing delay
We model p(t) ∼ GP(0, kp) as a Gaussian process (GP)
Likelihood can be worked out exactly
Bayesian MCMC used to estimate parameters α, β, ∆ and GPcovariance (2) and noise variance (1) parameters
![Page 33: Modelling gene expression dynamics with Gaussian processes · Di erential equations: Pol-II to mRNA dynamics. Gaussian processes: ... M.Rattray, N.D. Lawrence \Fast non-parametric](https://reader035.vdocuments.net/reader035/viewer/2022063010/5fc36c8a76d85279b24608ac/html5/thumbnails/33.jpg)
Example fits
040
80160
320640
1280
t (min)
mRNA
a
b
c
d
e
f
040
80160
320640
1280
t (min)
Pol−II a: Early pol-II, no production delay
0
2
4
Pol II
ENSG00000166483 (WEE1)
0 40 80 160 320 64012800
20
40
mR
NA
(F
PK
M)
t (min)0 50 100
0
0.2
0.4
0.6
0.8
1
Delay (min)
![Page 34: Modelling gene expression dynamics with Gaussian processes · Di erential equations: Pol-II to mRNA dynamics. Gaussian processes: ... M.Rattray, N.D. Lawrence \Fast non-parametric](https://reader035.vdocuments.net/reader035/viewer/2022063010/5fc36c8a76d85279b24608ac/html5/thumbnails/34.jpg)
Example fits
040
80160
320640
1280
t (min)
mRNA
a
b
c
d
e
f
040
80160
320640
1280
t (min)
Pol−II b: Early pol-II, delayed production
0
1
2
3
Pol II
ENSG00000019549 (SNAI2)
0 40 80 160 320 64012800
2
4
6
8
mR
NA
(F
PK
M)
t (min)0 50 100
0
0.2
0.4
0.6
0.8
1
Delay (min)
![Page 35: Modelling gene expression dynamics with Gaussian processes · Di erential equations: Pol-II to mRNA dynamics. Gaussian processes: ... M.Rattray, N.D. Lawrence \Fast non-parametric](https://reader035.vdocuments.net/reader035/viewer/2022063010/5fc36c8a76d85279b24608ac/html5/thumbnails/35.jpg)
Example fits
040
80160
320640
1280
t (min)
mRNA
a
b
c
d
e
f
040
80160
320640
1280
t (min)
Pol−II c: Later pol-II, delayed production
0
2
4
6
Pol II
ENSG00000163659 (TIPARP)
0 40 80 160 320 64012800
50
100
mR
NA
(F
PK
M)
t (min)0 50 100
0
0.2
0.4
0.6
0.8
1
Delay (min)
![Page 36: Modelling gene expression dynamics with Gaussian processes · Di erential equations: Pol-II to mRNA dynamics. Gaussian processes: ... M.Rattray, N.D. Lawrence \Fast non-parametric](https://reader035.vdocuments.net/reader035/viewer/2022063010/5fc36c8a76d85279b24608ac/html5/thumbnails/36.jpg)
Example fits
040
80160
320640
1280
t (min)
mRNA
a
b
c
d
e
f
040
80160
320640
1280
t (min)
Pol−II d: Late pol-II, no delay
0
0.5
1
Pol II
ENSG00000162949 (CAPN13)
0 40 80 160 320 64012800
5
10
15
mR
NA
(F
PK
M)
t (min)0 50 100
0
0.2
0.4
0.6
0.8
1
Delay (min)
![Page 37: Modelling gene expression dynamics with Gaussian processes · Di erential equations: Pol-II to mRNA dynamics. Gaussian processes: ... M.Rattray, N.D. Lawrence \Fast non-parametric](https://reader035.vdocuments.net/reader035/viewer/2022063010/5fc36c8a76d85279b24608ac/html5/thumbnails/37.jpg)
Example fits
040
80160
320640
1280
t (min)
mRNA
a
b
c
d
e
f
040
80160
320640
1280
t (min)
Pol−II e: Late pol-II, no delay
0
0.5
1
Pol II
ENSG00000117298 (ECE1)
0 40 80 160 320 64012800
20
40
mR
NA
(F
PK
M)
t (min)0 50 100
0
0.2
0.4
0.6
0.8
1
Delay (min)
![Page 38: Modelling gene expression dynamics with Gaussian processes · Di erential equations: Pol-II to mRNA dynamics. Gaussian processes: ... M.Rattray, N.D. Lawrence \Fast non-parametric](https://reader035.vdocuments.net/reader035/viewer/2022063010/5fc36c8a76d85279b24608ac/html5/thumbnails/38.jpg)
Example fits
040
80160
320640
1280
t (min)
mRNA
a
b
c
d
e
f
040
80160
320640
1280
t (min)
Pol−II f: Late pol-II, delayed production
0
1
2
3
Pol II
ENSG00000181026 (AEN)
0 40 80 160 320 64012800
10
20
30
mR
NA
(F
PK
M)
t (min)0 50 100
0
0.2
0.4
0.6
0.8
1
Delay (min)
![Page 39: Modelling gene expression dynamics with Gaussian processes · Di erential equations: Pol-II to mRNA dynamics. Gaussian processes: ... M.Rattray, N.D. Lawrence \Fast non-parametric](https://reader035.vdocuments.net/reader035/viewer/2022063010/5fc36c8a76d85279b24608ac/html5/thumbnails/39.jpg)
Large processing delays observed in 11% of genes
Delay (min)
# of
gen
es
0 40 80 >120
050
100
1500
1550
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Delay linked with gene length and intron structure
0 20 40 60 80t (min)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Fra
ctio
n of
gen
es w
ith ∆
>t
02
46
810
12−
log 1
0(p−
valu
e)
m<10000(N=282)m>10000(N=1504)
p−value
0 20 40 60 80t (min)
0.00
0.05
0.10
0.15
0.20
0.25
Fra
ctio
n of
gen
es w
ith ∆
>t
02
46
810
−lo
g 10(p
−va
lue)
f<0.20(N=1443)f>0.20(N=343)
p−value
∆: delay m: gene length f: final intron length / gene length
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Delayed genes show evidence of late-intron retention
DLX
3
0246
10 m
in
0
5
10
20 m
in
0102030
40 m
in
05
101520
80 m
in
02468
10
160
min
02468
320
min
10 20 30 40 50 60t (min)
−0.
010.
000.
010.
020.
030.
04M
ean
pre−
mR
NA
end
acc
umul
atio
n in
dex
01
23
45
67
−lo
g 10(p
−va
lue)
∆ > t∆ < tp−value
Left: density of RNA-Seq reads uniquely mapping to the introns inthe DLX3 gene
Right: Differences in the mean pre-mRNA accumulation index inlong delay genes (blue) and short delay genes (red)
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Comparison of processing and transcription times
RNA processing delay ∆ (min)
Tran
scrip
tiona
l del
ay (
min
)
1 3 10 30 100
13
1030
100
Transcription time = length/velocityestimate from Danko Mol Cell 2013
Transcription time > processing delayin 87% of genes
Honkela et al. “Genome-wide modeling of transcription kineticsreveals patterns of RNA production delays” PNAS 2015
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Extension - inferring time-varying degradation rates
dm(t)
dt= βp(t)− α(t)m(t)
0 2 4 6 8 10Times
0.5
0.0
0.5
1.0
1.5
2.0
2.5
Degra
dati
on r
ate
D
0 2 4 6 8 10Times
5
0
5
10
15
20
25
30
35
Gene e
xpre
ssio
n
mRNA measurementspre-mRNA measurements
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Conclusion
I Gaussian process models provide a good mix of flexibility andtractability in temporal and spatial modelling:
I Hierarchical modelling, e.g. replicates, clusters, speciesI Periodic models without strong sinuisoidal assumptionsI Tractable under branching and bifurcationsI Tractable under linear operations on functions
I Easy addition and multiplication of kernels
I Good approximate inference algorithms for non-Gaussian data
I Codebase is growing making methods increasingly flexible andcomputationally efficient (e.g. GPy, GPflow and some R)
Funding: BBSRC (EraSysBio+ SYNERGY), EU FP7 (RADIANT)
Collaborators: Neil Lawrence, James Hensman, Antti Honkela,Jaakko Peltonen, Jing Yang, Alexis Boukouvalas, Max Zweissele
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Our existential crisis