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Background Constraint Programing in Community Networks Experiments and Results Conclusions Constraint Programming in Community-based Gene Regulatory Network Inference Ferdinando Fioretto Enrico Pontelli Dept. Computer Science, New Mexico

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constraint Programming in Community-based Gene Regulatory Network Inference

Ferdinando Fioretto Enrico Pontelli

  • Dept. Computer Science, New Mexico State University
  • Sept. 24, 2013
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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Talk Outline

1

Background

2

Constraint Programing in Community Networks

3

Experiments and Results

4

Conclusions

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Gene Regulatory Networks

A cell contains different entities (including pro- teins, RNA) which interact and perform specific functions.

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Gene Regulatory Networks

A cell contains different entities (including pro- teins, RNA) which interact and perform specific functions. DNA transcription

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Gene Regulatory Networks

A cell contains different entities (including pro- teins, RNA) which interact and perform specific functions. DNA transcription mRNA translation

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Gene Regulatory Networks

Some proteins (Transcriptor Factors (TF)) can regulate the production of other proteins. Done by enhancing or inhibiting DNA transcription or mRNA translation. The unit of encapsulation of these interactions are the coding regions of the DNA: the genes. A Gene Regulatory Network is the set of the interactions among genes.

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Gene Regulatory Networks

Modeling

A GRN is described by a weighted directed graph G = (V, E). V is the set of genes of the network. E ⊆ V × V × [0, 1] is the set of the regulatory interactions. Each regulatory interaction s → t is associated with a confidence value ωs→t ∈ [0, 1]. Example

G1 regulates G2. G2 regulates G5. G3 is regulated by G4. G4 regulates G2 and is regulated by G5.

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Gene Regulatory Network Inference

GRN inference from high-throughput data

Motivation: Key to understand important genetic diseases, such as cancer. Crucial to devise effective medical interventions.

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Gene Regulatory Network Inference

Current Methods and Challenges

Methods proposed:

Correlation-based. Information-theoretic based. Boolean Networks. Bayesian Networks. Regression-based. Stochastics.

Based on different assumptions. Exhibits peculiar limitations.

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Gene Regulatory Network Inference

Current Methods and Challenges

Methods proposed:

Correlation-based. Information-theoretic based. Boolean Networks. Bayesian Networks. Regression-based. Stochastics.

Based on different assumptions. Exhibits peculiar limitations. Solutions proposed:

Integrating heterogeneous data into the inference model. Meta-approaches using multiple inference models (Community Networks (CN)).

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Gene Regulatory Network Inference

Community Networks

community network

G1 G2 GJ

edge ranking

Borda voting score: ω

s

#

→t = 1

|G|

|G|

  • j=1

ω j

s

#

→t

ω j

s # →t : the ranked interaction s → t

by the j-th method in G.

  • D. Marbach et al. “Wisdom of crowds for robust gene network inference”.

Nature Methods, 9(8):796–804, Aug. 2012.

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Gene Regulatory Network Inference

Our Approach

CN approach for an “initial analysis” of the GRN.

Community prediction collective agreements.

Integrate additional biological knowledge (when available).

Leverage specific GRN properties.

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Gene Regulatory Network Inference

Our Approach

CN approach for an “initial analysis” of the GRN.

Community prediction collective agreements.

Integrate additional biological knowledge (when available).

Leverage specific GRN properties.

Why CP ?

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constraint Programming

Constraint Satisfaction Problem (CSP)

Variables X: xi = position of the queen in the ith column. Domains D: Dxi = {1, . . . , n}. Constraints C: ∀i, ∀j with i < j: xi = xj xi + i = xj + j xi − j = xj − j Search = Labeling + Constraint Propagation

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constraint Programming

Constraint Satisfaction Problem (CSP)

Variables X: xi = position of the queen in the ith column. Domains D: Dxi = {1, . . . , n}. Constraints C: ∀i, ∀j with i < j: xi = xj xi + i = xj + j xi − j = xj − j Search = Labeling + Constraint Propagation

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constraint Programming

Constraint Satisfaction Problem (CSP)

Variables X: xi = position of the queen in the ith column. Domains D: Dxi = {1, . . . , n}. Constraints C: ∀i, ∀j with i < j: xi = xj xi + i = xj + j xi − j = xj − j Search = Labeling + Constraint Propagation

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constraint Programming

Constraint Satisfaction Problem (CSP)

Variables X: xi = position of the queen in the ith column. Domains D: Dxi = {1, . . . , n}. Constraints C: ∀i, ∀j with i < j: xi = xj xi + i = xj + j xi − j = xj − j Search = Labeling + Constraint Propagation

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constraint Programming

Constraint Satisfaction Problem (CSP)

Variables X: xi = position of the queen in the ith column. Domains D: Dxi = {1, . . . , n}. Constraints C: ∀i, ∀j with i < j: xi = xj xi + i = xj + j xi − j = xj − j Search = Labeling + Constraint Propagation

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constraint Programming

Constraint Satisfaction Problem (CSP)

Variables X: xi = position of the queen in the ith column. Domains D: Dxi = {1, . . . , n}. Constraints C: ∀i, ∀j with i < j: xi = xj xi + i = xj + j xi − j = xj − j Search = Labeling + Constraint Propagation

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constraint Programming

Constraint Satisfaction Problem (CSP)

Variables X: xi = position of the queen in the ith column. Domains D: Dxi = {1, . . . , n}. Constraints C: ∀i, ∀j with i < j: xi = xj xi + i = xj + j xi − j = xj − j Search = Labeling + Constraint Propagation

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constraint Programming

Constraint Satisfaction Problem (CSP)

Variables X: xi = position of the queen in the ith column. Domains D: Dxi = {1, . . . , n}. Constraints C: ∀i, ∀j with i < j: xi = xj xi + i = xj + j xi − j = xj − j Search = Labeling + Constraint Propagation

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constraint Programming

Constraint Satisfaction Problem (CSP)

Variables X: xi = position of the queen in the ith column. Domains D: Dxi = {1, . . . , n}. Constraints C: ∀i, ∀j with i < j: xi = xj xi + i = xj + j xi − j = xj − j Search = Labeling + Constraint Propagation Solution = assignment for X satisfying all c ∈ C

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Gene Regulatory Network Inference

Our Approach

CN approach for an “initial analysis” of the GRN.

Community prediction collective agreements.

Integrate additional biological knowledge (when available).

Leverage specific GRN properties.

Why CP ?

Separation between prediction methods and model. Declaratively. Constraint expressions allow incremental model refinement.

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constrained Community Networks

CSP Modeling

GRN inference (GRNi) problem: Given a set of n genes, a GRNi is a CSP X, D, C X = x1, . . . , xn2−n (regulatory relations, exuding self regulations). D = D1, . . . , Dn2−n, with each Dk = {0, . . . , 100} (possible confidence values). C is a list of constraints expressing properties of the GRNs. Notation: xs→t: “s regulates t” and Ds→t its domain. d(xs→t): the value assigned to xs→t.

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constrained Community Networks

CSP Modeling

A solution to the GRNi defines a GRN prediction G = (V, E) V = {1, . . . , n}, E = {s, t, w | d(xs→t) > 0}, where w = d(xs→t)/100.

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constrained Community Networks

E.coli2 size 10 (from DREAM3)

G2 G10 G9 G1 G5 G8 G7 G6 G4 G3

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constrained Community Networks

E.coli2 size 10 CN prediction

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Analysis and Domains Reduction

The pre resolution phase

Leverage the collection of GRN predictions G by:

(i.) Reducing the size of the solution search space. (ii.) Integrate the Gj ∈ G taking into account their discrepancies.

Set up domains of each variable xs→t ∈ X, such that: Ds→t = Ds→t ∩ Bs→t

where: Bs→t =

  • ω

s

#

→t

  • if σs→t < θd
  • σs→t =

1 |G|

2

  • |G|
  • j=1

|G|

  • i=j+1
  • ω j

s

#

→t − ω i s

#

→t

  • θd ∈ [0, 1] is a “disagreement threshold”.
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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Analysis and Domains Reduction

The pre resolution phase

Leverage the collection of GRN predictions G by:

(i.) Reducing the size of the solution search space. (ii.) Integrate the Gj ∈ G taking into account their discrepancies.

Set up domains of each variable xs→t ∈ X, such that: Ds→t = Ds→t ∩ Bs→t

where: Bs→t =

  • ω

s

#

→t − σs→t

2 , ω

s

#

→t, ω s

#

→t + σs→t

2

  • if σs→t ≥ θd

∧ 0.1 < ω

s# →t

< 0.9

  • σs→t =

1 |G|

2

  • |G|
  • j=1

|G|

  • i=j+1
  • ω j

s

#

→t − ω i s

#

→t

  • θd ∈ [0, 1] is a “disagreement threshold”.
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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constraints

Sparseness

Elements of a GRN are considered to be controlled by a small number of genes: GRN are sparse. Combining predictions in a CN does not guarantee sparseness. Enforce a sparseness constraint by: atleast k ge(kl, X, θl) :

  • {xi ∈ X | d(xi) > θl}
  • ≥ kl

and atmost k ge(km, X, θm) :

  • {xi ∈ X | d(xi) > θm}
  • ≤ km

with kl,m > 0 and 0 ≤ θl,m ≤ 100, and where d(xi) indicates the value of an assignment for xi

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constraints

Sparseness

atleast k ge(10, X, 65) ∩ atmost k ge(25, X, 65)

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constraints

Sparseness

atleast k ge(10, X, 65) ∩ atmost k ge(25, X, 65)

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constraints

Redundant edge

Several state-of-the art inference methods rely on techniques which cannot discriminate causality (e.g., M.I., Correlation). Given a collection of predictions G ={G1, . . . , GJ} for a GRN G=(V, E) and a non-empty set of non causal based methods H ⊆ G, an edge t → s is redundant if: ∀ Gi ∈ G \ H . ω i

s→t > ω i t→s + β

If an edge t → s is redundant we call the edge s → t required. Let XR be the set of all the required and redundant variables, red edge(xs→t, xt→s, θR, θr) : xs→t > θR ∧ xt→s < θr

with θR, θr ∈ N, and 0 ≤ θR ≤ 100.

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constraints

Redundant edge

∀xs→t, xt→s ∈ XR red edge(xs→t, xt→s, 75, 50)

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constraints

Redundant edge

∀xs→t, xt→s ∈ XR red edge(xs→t, xt→s, 75, 50)

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constraints

Sparseness + Redundant edge

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constraints

Transcriptor Factor

Information about DNA-binding motifs often available from public sources (e.g., BDB, Gene Ontology). Existing methods do not often allow integration of such information (treated in postprocess). A gene s ∈ V is a transcriptor factor (TF) if it regulates the production of other genes. Express this property on the out-degree of s: tf(s) : atleast k ge(ks, Xs, θs)

where: Xs = {xs→t ∈ X | t ∈ V} k is the co-expressing degree (the number of genes targeted by the TF).

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constraints

Transcriptor Factor

atleast k ge(2, Ni, 85) with Ni = {xi→s | (∀Gj ∈ G) ω j

i→s > 0.10}, (i = 1, 5, 9)

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constraints

Transcriptor Factor

atleast k ge(2, Ni, 85) with Ni = {xi→s | (∀Gj ∈ G) ω j

i→s > 0.10}, (i = 1, 5, 9)

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constraints

Co-transcriptor Factors

Multiple TFs cooperate to regulate a specific gene (Co-regulators). Let s′, s′′ ∈ V be two TFs, which are co-regulators.

coregulator(k, X, θ) : ∀xs′→t′, xs′′→t′′ ∈ X | {(s′, s′′, t′) | s′ =s′′ ∧ t′ =t′′ ∧ d(xs′→t′)>θ ∧ d(xs′′→t′′)>θ} | ≥ k with k ∈ N and 0 < θ < 1

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constraints

Co-transcriptor Factors

coregulator(1, V, 75), with s′ =1, s′′ =5

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Constraints

Co-transcriptor Factors

coregulator(1, V, 75), with s′ =1, s′′ =5

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

GRN Consensus

We implement two solution strategy prop-labeling (DFS) and a Monte Carlo (MC) based prop-labeling tree exploration. No consensus on objective function to drive the solution search. We propose 3 metric to generate a GRN consensus Constrained Community Network (CCN). Given a set S of m solutions, the consensus value a∗

k associated

with the variable xk is computed by:

Max Frequency: a∗

k = arg max a∈S|xk

(freq(a, k)) Average: a∗

k = 1

m

m

  • i=1

ai

k.

Weighted average: a∗

k =

1

  • a∈S|xk

freq(a, k)2

  • a∈S|xk

freq(a, k)2a.

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Experiments

Community Networks

The CN was built from 4 top ranking methods of last DREAM competitions:

1

TIGRESS (Regression model)

2

Genie3 (Random Forest approach)

3

Infleator (MCZ + tlCLR + linear ODE)

4

CLR (Mutual Information model)

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Experiments

Datasets and validation

Benchmarks: DREAM{3,4} (110 GRNs of various sizes). Subnetworks from GRNs of E. coli and S. cerevisiae. Datasets:

steady state expressions for wild types steady state expressions measured after gene knockouts. time-series data.

Validation: AUROC score. CCNs generated via MC search with 1, 000 samplings.

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Experiments

Settings

Domains Setup. θd = 1 |ECN| X

(s,t,w)∈ECN

σs→t Sparseness constraint. atleast k ge(kl, X, θl) \ atmost k ge(km, X, θm)

Ordered ECN

1 g1 → g3 0.998 2 g1 → g8 0.981 . . . n g4 → g6 0.856 . . . n log(n) g7 → g3 0.633 . . .

kl ≤ |{xi|xi 2 X ^ max(Dxi) > θl}| km ≥ |{xi|xi 2 X ^ min(Dxi) > θm}| Redundant edge constraint. 8 Gi 2 G \ H . ω i

s→t > ω i t→s + β

1 |G||ERR| X

Gi2G\H

(ω i

s→t − ω i t→s))

red edge(xs→t, xt→s, θR, θr) 1 |G \ H||EREQ| X

Gi2G\H

ω i

s→t

1 |G \ H||ERED| X

Gi2G\H

ω i

t→s

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Results

CCN with sparsity and redundant edge constraints

AUROC % improvement 5 10 15

b f a w b f a w b f a w b f a w b f a w DREAM3 10 DREAM4 10 DREAM3 50 DREAM3 100 DREAM4 100

s,r s,r,t

Average AUC score improvements (in percentage) w.r.t. CN rank

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Experiments

Integrating GRN knowledge: TFs

Domains Setup. θd = 1 |ECN| X

(s,t,w)∈ECN

σs→t Sparseness constraint. atleast k ge(kl, X, θl) \ atmost k ge(km, X, θm)

Ordered ECN

1 g1 → g3 0.998 2 g1 → g8 0.981 . . . n g4 → g6 0.856 . . . n log(n) g7 → g3 0.633 . . .

kl ≤ |{xi|xi 2 X ^ max(Dxi) > θl}| km ≥ |{xi|xi 2 X ^ min(Dxi) > θm}| Redundant edge constraint. 8 Gi 2 G \ H . ω i

s→t > ω i t→s + β

1 |G||ERR| X

Gi2G\H

(ω i

s→t − ω i t→s))

red edge(xs→t, xt→s, θR, θr) 1 |G \ H||EREQ| X

Gi2G\H

ω i

s→t

1 |G \ H||ERED| X

Gi2G\H

ω i

t→s

Transcription Factor constraint. atleast k ge(blog(n)c, X, θ)

Ordered ECN

1 g1 → g3 0.998 2 g1 → g8 0.981 . . . n g4 → g6 0.856 . . .

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Results

CCN with additional GRN knowledge integration

AUROC % improvement 5 10 15

b f a w b f a w b f a w b f a w b f a w DREAM3 10 DREAM4 10 DREAM3 50 DREAM3 100 DREAM4 100

s,r s,r,t

Average AUC score improvements (in percentage) w.r.t. CN rank

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Results

CCN with additional GRN knowledge integration

AUROC % improvement 5 10 15

b f a w b f a w b f a w b f a w b f a w DREAM3 10 DREAM4 10 DREAM3 50 DREAM3 100 DREAM4 100

s,r s,r,t

Average AUC score improvements (in percentage) w.r.t. CN rank

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Conclusions

CP-based approach to infer GRNs by integrating several methods in a CN. Introduces a set of constraints able to:

1

enforce the satisfaction of GRNs specific properties;

2

take account of the community predictions agreements and methods limitations.

No assumptions on datasets nor on the type of inference methods. Take Home Message:

GRN knowledge integration offer improvements in prediction accuracy. Constraints are a powerful tool to model and integrate GRN properties.

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Background Constraint Programing in Community Networks Experiments and Results Conclusions

Conclusions

CP-based approach to infer GRNs by integrating several methods in a CN. Introduces a set of constraints able to:

1

enforce the satisfaction of GRNs specific properties;

2

take account of the community predictions agreements and methods limitations.

No assumptions on datasets nor on the type of inference methods. Take Home Message:

GRN knowledge integration offer improvements in prediction accuracy. Constraints are a powerful tool to model and integrate GRN properties.

Thank you!