beyond static networks: a bayesian non-parametric approach

32
Beyond Static Networks: A Bayesian Non-Parametric Approach Michael P.H. Stumpf Theoretical Systems Biology Group, Department of Life Sciences, Imperial College London www.theosysbio.bio.ic.ac.uk 22nd July 2013

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Page 1: Beyond Static Networks: A Bayesian Non-Parametric Approach

Beyond Static Networks: A Bayesian Non-ParametricApproach

Michael P.H. Stumpf

Theoretical Systems Biology Group, Department ofLife Sciences, Imperial College Londonwww.theosysbio.bio.ic.ac.uk

22nd July 2013

Page 2: Beyond Static Networks: A Bayesian Non-Parametric Approach

Networks: Mapping Processes and Understanding

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Beyond Static Networks M Stumpf 1 of 13

Page 3: Beyond Static Networks: A Bayesian Non-Parametric Approach

Networks: Mapping Processes and UnderstandingGraphical models of gene regulatory networks

• Provides aconvenientrepresentation

• Numerousapproachesfor learning

• Large p, smalln

GaussianGraphical Models

Bayesian Networks

Dynamic Bayesian Networks

Biological networks Tom Thorne 3 / 35

Beyond Static Networks M Stumpf 1 of 13

Page 4: Beyond Static Networks: A Bayesian Non-Parametric Approach

Biology is Dynamic — Networks Change with Time

A

AP

AP

B

• Inferred regulatory network structures represent correlations rather thandirect interactions.

• Gene products may require activation and need to be transported into thenucleus to influence regulation; or complexes formed by signallingcascades may be required to activate transcription.

• Many factors that are not a part of a traditional regulatory network modelcan also influence regulatory interactions.

• These relationships may change depending on external signals or otherfactors.

Beyond Static Networks M Stumpf 2 of 13

Page 5: Beyond Static Networks: A Bayesian Non-Parametric Approach

Biology is Dynamic — Networks Change with Time

A AP

AP

B

• Inferred regulatory network structures represent correlations rather thandirect interactions.

• Gene products may require activation and need to be transported into thenucleus to influence regulation; or complexes formed by signallingcascades may be required to activate transcription.

• Many factors that are not a part of a traditional regulatory network modelcan also influence regulatory interactions.

• These relationships may change depending on external signals or otherfactors.

Beyond Static Networks M Stumpf 2 of 13

Page 6: Beyond Static Networks: A Bayesian Non-Parametric Approach

Biology is Dynamic — Networks Change with Time

A

AP

AP

B

• Inferred regulatory network structures represent correlations rather thandirect interactions.

• Gene products may require activation and need to be transported into thenucleus to influence regulation; or complexes formed by signallingcascades may be required to activate transcription.

• Many factors that are not a part of a traditional regulatory network modelcan also influence regulatory interactions.

• These relationships may change depending on external signals or otherfactors.

Beyond Static Networks M Stumpf 2 of 13

Page 7: Beyond Static Networks: A Bayesian Non-Parametric Approach

Biology is Dynamic — Networks Change with Time

A

AP

AP

B

• Inferred regulatory network structures represent correlations rather thandirect interactions.

• Gene products may require activation and need to be transported into thenucleus to influence regulation; or complexes formed by signallingcascades may be required to activate transcription.

• Many factors that are not a part of a traditional regulatory network modelcan also influence regulatory interactions.

• These relationships may change depending on external signals or otherfactors.

Beyond Static Networks M Stumpf 2 of 13

Page 8: Beyond Static Networks: A Bayesian Non-Parametric Approach

Capturing Biological Dynamics — ChangepointModels for Networks

• We can include hidden factors that my change the regulatory interactions takingplace in our model by allowing the regulatory network structure to vary betweentimepoints and/or conditions.

• In changepoint models the time series is divided into a number of segments,allowing a different network structure in each.

• Using Bayesian inference it is possible to infer the posterior distribution ofchangepoint positions.

Time point 1 2 3 4 5 6 7 8 9 10

S. Lebre, J. Becq, F. Devaux, M. P. H. Stumpf, G. Lelandais, Statistical inference of the time-varying structure ofgene-regulation networks. BMC Systems Biology, 4:130, 2010.

Beyond Static Networks M Stumpf 3 of 13

Page 9: Beyond Static Networks: A Bayesian Non-Parametric Approach

Capturing Biological Dynamics — ChangepointModels for Networks

• We can include hidden factors that my change the regulatory interactions takingplace in our model by allowing the regulatory network structure to vary betweentimepoints and/or conditions.

• In changepoint models the time series is divided into a number of segments,allowing a different network structure in each.

• Using Bayesian inference it is possible to infer the posterior distribution ofchangepoint positions.

Time point 1 2 3 4 5 6 7 8 9 10

S. Lebre, J. Becq, F. Devaux, M. P. H. Stumpf, G. Lelandais, Statistical inference of the time-varying structure ofgene-regulation networks. BMC Systems Biology, 4:130, 2010.

Beyond Static Networks M Stumpf 3 of 13

Page 10: Beyond Static Networks: A Bayesian Non-Parametric Approach

Modelling Gene Expression Networks

Given gene expression time series data over m genes at n timepoints, we denote the observations as the n ×m matrix

X = (x1, . . . , xn)T ,

where xj = (xj1, . . . , xjm)T , the column vector of expression levels for

each of the m genes at time point j .

We formulate our model as a Hierarchical Dirichlet Process HiddenMarkov Model, a stochastic process whereby a set of hidden statess1, . . . , sn govern the parameters of some emission distribution Fover a sequence of time points 1 . . . n.Each observation xj is then generated from a corresponding emissiondistribution F (θk), where sj = k . For our emission distributions, F , weuse a Bayesian Network model over the m variables to represent theregulatory network structures corresponding to each hidden state.

Beyond Static Networks M Stumpf 4 of 13

Page 11: Beyond Static Networks: A Bayesian Non-Parametric Approach

Bayesian Networks

Conditional probability distribution represented by a Directed AcyclicGraph (DAG)

X1 X2

X3

X4 X5

X6

P(X1, . . . ,X6) =P(X1)P(X2)P(X3|X1,X2)P(X4|X3)P(X5|X3)P(X6|X4,X5)

Order sampling:

p(@) ∼∏

u∈NG

∑pa@G (u)

p(u, paG)

Beyond Static Networks M Stumpf 5 of 13

Page 12: Beyond Static Networks: A Bayesian Non-Parametric Approach

What We Want to Know is Often Not Measured:Hidden Markov Models

• Here we measure transcriptomic data, whereas the action is alldue to proteins and their interactions among themselves and withDNA/RNA.

• We measure mRNA expression (yi ), which is influenced by anetwork (si ) that is not or cannot be observed directly.

• The transition probability between different states (networks) isgiven by πkl = p(sj = l |sj−1 = k).

s1

y1

θs1

πs1

s2

y2

θs2

s3

y3

θs3

. . . sT

. . . yT

θsT

Beyond Static Networks M Stumpf 6 of 13

Page 13: Beyond Static Networks: A Bayesian Non-Parametric Approach

What We Want to Know is Often Not Measured:Hidden Markov Models

• Here we measure transcriptomic data, whereas the action is alldue to proteins and their interactions among themselves and withDNA/RNA.

• We measure mRNA expression (yi ), which is influenced by anetwork (si ) that is not or cannot be observed directly.

• The transition probability between different states (networks) isgiven by πkl = p(sj = l |sj−1 = k).

s1

y1

θs1

πs1

s2

y2

θs2

s3

y3

θs3

. . . sT

. . . yT

θsT

Beyond Static Networks M Stumpf 6 of 13

Page 14: Beyond Static Networks: A Bayesian Non-Parametric Approach

What We Want to Know is Often Not Measured:Hidden Markov Models

• Here we measure transcriptomic data, whereas the action is alldue to proteins and their interactions among themselves and withDNA/RNA.

• We measure mRNA expression (yi ), which is influenced by anetwork (si ) that is not or cannot be observed directly.

• The transition probability between different states (networks) isgiven by πkl = p(sj = l |sj−1 = k).

s1

y1

θs1

πs1

s2

y2

θs2

s3

y3

θs3

. . . sT

. . . yT

θsT

Beyond Static Networks M Stumpf 6 of 13

Page 15: Beyond Static Networks: A Bayesian Non-Parametric Approach

The Chinese Restaurant Process

. . .

θ1 θ2 θ3 θ4

H

θ5

Analogy for the Dirichlet process due to Pitman and Dubins

D. Aldous, Exchangeability and Related Topics. In l’Ecole d’ete de probabilites de Saint-Flour, XIII, pages 1-198. 1983

Beyond Static Networks M Stumpf 7 of 13

Page 16: Beyond Static Networks: A Bayesian Non-Parametric Approach

The Chinese Restaurant Process

. . .

θ1 θ2 θ3 θ4

H

θ5

Analogy for the Dirichlet process due to Pitman and Dubins

D. Aldous, Exchangeability and Related Topics. In l’Ecole d’ete de probabilites de Saint-Flour, XIII, pages 1-198. 1983

Beyond Static Networks M Stumpf 7 of 13

Page 17: Beyond Static Networks: A Bayesian Non-Parametric Approach

The Chinese Restaurant Process

. . .

θ1 θ2 θ3 θ4

H

θ5

Analogy for the Dirichlet process due to Pitman and Dubins

D. Aldous, Exchangeability and Related Topics. In l’Ecole d’ete de probabilites de Saint-Flour, XIII, pages 1-198. 1983

Beyond Static Networks M Stumpf 7 of 13

Page 18: Beyond Static Networks: A Bayesian Non-Parametric Approach

The Chinese Restaurant Process

. . .

θ1 θ2 θ3 θ4

H

θ5

Analogy for the Dirichlet process due to Pitman and Dubins

D. Aldous, Exchangeability and Related Topics. In l’Ecole d’ete de probabilites de Saint-Flour, XIII, pages 1-198. 1983

Beyond Static Networks M Stumpf 7 of 13

Page 19: Beyond Static Networks: A Bayesian Non-Parametric Approach

The Chinese Restaurant Process

. . .

θ1 θ2 θ3 θ4

H

θ5

Analogy for the Dirichlet process due to Pitman and Dubins

D. Aldous, Exchangeability and Related Topics. In l’Ecole d’ete de probabilites de Saint-Flour, XIII, pages 1-198. 1983

Beyond Static Networks M Stumpf 7 of 13

Page 20: Beyond Static Networks: A Bayesian Non-Parametric Approach

Systems at Different Times are Related: TheChinese Restaurant Franchise

α

θ2 θ1 θ1 θ3 θ2 θ2

θ1 θ2 θ3 θ ′ ∼ H

γ

Beyond Static Networks M Stumpf 8 of 13

Page 21: Beyond Static Networks: A Bayesian Non-Parametric Approach

Systems at Different Times are Related: TheChinese Restaurant Franchise

α

θ2 θ1 θ1 θ3 θ2 θ2

θ1 θ2 θ3 θ ′ ∼ H

γ

Beyond Static Networks M Stumpf 8 of 13

Page 22: Beyond Static Networks: A Bayesian Non-Parametric Approach

Systems at Different Times are Related: TheChinese Restaurant Franchise

α

θ2 θ1 θ1 θ3 θ2 θ2

θ1 θ2 θ3 θ ′ ∼ H

γ

Beyond Static Networks M Stumpf 8 of 13

Page 23: Beyond Static Networks: A Bayesian Non-Parametric Approach

Each hidden state k possesses a Dirichlet Process Gk from which πk ·is drawn, and a common (discrete) base measure G0 is sharedbetween the Gk , so that

Gk ∼ DP(α,G0).

Thus transitions are made into a discrete set of states shared acrossall of the Gk , and drawn from G0. The base measure G0 is in turndrawn from a Dirichlet Process,

G0 ∼ DP(γ,H) H is the prior for the emission distributions Fk .

Using the stick breaking construction for G0 and drawing θl ∼ H, wehave

G0 =

∞∑l

βlδθl with β ∼ GEM(γ),

and so

Gk =

∞∑l

πklδθl with πk ∼ DP(α,β).

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Biological Systems do Not Change Wildly(Assumption!): Hidden States are Correlated

s1 s2 s3 s4 s5 s6 s7 s8 s9

Observations y1 y2 y3 y4 y5 y6 y7 y8 y9

Time point 1 2 3 4 5 6 7 8 9

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Systems at Different Times are Related: HDP-HMM

H θk

γ β

α πk·

s0 s1 s2 sn

y1 y2 yn

∞• Base measure H• Shared state

distribution β• Transition

distributions πi,·

• State sequences0, . . . , sn

• Observationsy1, . . . , yn

Each si is a Bayesian network and H specifies the prior over the parameters for theemission distribution.

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Systems at Different Times are Related: HDP-HMM

To sample from the hidden state sequence we have used a Gibbssampling procedure based on the conditional probabilities for thehidden state si given the remaining hidden states s−i , updating eachhidden state individually in a sweep over the n states,

p(sj = k |s−j ,α,β, κ) ∝(N−j

sj−1k + αβk + κδsj−1(k))N−jksj+1

+ αβsj+1 + κδsj+1(k) + δsj−1(k)δsj+1(k)

α+ N−jk· + κ+ δsj−1(k)

p(X j·|X i· : si = k , i 6= j,Fk), (1)

where N−jkl indicates the number of observed transitions from state k

to state l within the hidden state sequence s−j , and N−jk· the total

number of transitions from state k within s−j .Beyond Static Networks M Stumpf 11 of 13

Page 27: Beyond Static Networks: A Bayesian Non-Parametric Approach

D. melanogaster development

Expression data for D. melanogaster midgut development. Taken at11 time points, during which larval midgut becomes adult midgut.

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A. thaliana diurnal cycle

Expression data for A. thaliana over 24 hours, with a light and darkphase.

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The S. cerevisae Cell Cycle

Expression data for S. cerevisae over two cell cycles, at 25 timepoints.

1 2

3 4

Fre

quen

cy

0 10 20 30 40 50 60 70 80 90 105 1200.

00.

20.

40.

60.

81.

0

T. Pramila, W. Wu, S. Miles, W.S. Noble et al., The Forkhead transcription factor Hcm1 regulates chromosome segregationgenes and fills the S-phase gap in the transcriptional circuitry of the cell cycle. Genes Dev Aug 15;20(16):2266-78, 2006.

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Candida glabrata osmotic stress response

SPT16

FPS1

EMC6

SMX3

ISD11

MKS1

CAGL0K04235g

FPS1

VMA22 SRB8

SMX3

CAGL0H00704g

ISD11

BUD31 CAGL0K06127g

YJR085C

CUE2

2

1Time point (mins)

Fre

quen

cy

0.0

0.2

0.4

0.6

0.8

1.0

15 30 60 90 120 150 180 240

Two distinct regulatoryarchitectures appearto control theexpression of thegenes involved inosmotic stressresponse in C.glabrata.

TemporalDependencies

T<30min:

ISD11→ SMX3

T>30min:

ISD11→ BUD31

SMX3→ BUD31

Interactions change withtime and may becontingent on pastinteractions.

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Candida glabrata osmotic stress response

SPT16

FPS1

EMC6

SMX3

ISD11

MKS1

CAGL0K04235g

FPS1

VMA22 SRB8

SMX3

CAGL0H00704g

ISD11

BUD31 CAGL0K06127g

YJR085C

CUE2

2

1Time point (mins)

Fre

quen

cy

0.0

0.2

0.4

0.6

0.8

1.0

15 30 60 90 120 150 180 240

Two distinct regulatoryarchitectures appearto control theexpression of thegenes involved inosmotic stressresponse in C.glabrata.

TemporalDependencies

T<30min:

ISD11→ SMX3

T>30min:

ISD11→ BUD31

SMX3→ BUD31

Interactions change withtime and may becontingent on pastinteractions.

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Acknowledgements

Imperial College London

• ThomasThorne

• JustinaZurauskine

• Paul Kirk• Daniel Silk

Thorne & Stumpf, Bioinformatics, 2012, 28:3298Thorneet al., MolBiosystems, 2013, 9:1736-1742

Exter University• Andrew McDonagh• Melanie Puttnam• Lauren Ames• Ken Haynes

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