A Graphical Method For Reducing and Relating Models in Systems - - PowerPoint PPT Presentation

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A Graphical Method For Reducing and Relating Models in Systems Biology

Fran¸ cois Fages Joint work with Steven Gay, Sylvain Soliman ECCB’10 special issue, Bioinformatics 26(18):575-581 (2010)

The French National Institute for Research in Computer Science and Control INRIA, Paris-Rocquencourt, France

TSB, 30 May 2011, Grenoble

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From Models to Metamodels

In Systems Biology, models are built with two contradictory perspectives:

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From Models to Metamodels

In Systems Biology, models are built with two contradictory perspectives:

◮ models for representing knowledge:

the more detailed the better

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From Models to Metamodels

In Systems Biology, models are built with two contradictory perspectives:

◮ models for representing knowledge:

the more detailed the better

◮ models for making predictions:

the more abstract the better ! get rid of useless details

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From Models to Metamodels

In Systems Biology, models are built with two contradictory perspectives:

◮ models for representing knowledge:

the more detailed the better

◮ models for making predictions:

the more abstract the better ! get rid of useless details These two perspectives can be reconciled by organizing models in a hierarchy of models related by reduction/refinement relations. To understand a system is not to know everything about it, but to know abstraction levels that are sufficient for answering given questions about it

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State of the Art: Model Repositories

biomodels.net: plain list of 241 curated models in SBML format

◮ MAPK signaling cascade

009 Huan: three-level cascade of double phosphorylations 010 Khol: reduced model without dephosphorylation catalysts 011 Levc: same model as 009 Huan with different parameter values and different molecule names 027 Mark, 028 Mark, 029 Mark, 030 Mark, 031 Mark: reduced one-level models with different levels of details

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State of the Art: Model Repositories

biomodels.net: plain list of 241 curated models in SBML format

◮ MAPK signaling cascade

009 Huan: three-level cascade of double phosphorylations 010 Khol: reduced model without dephosphorylation catalysts 011 Levc: same model as 009 Huan with different parameter values and different molecule names 027 Mark, 028 Mark, 029 Mark, 030 Mark, 031 Mark: reduced one-level models with different levels of details

◮ Circadian clock:

074 Lelo, 021 Lelo, 170 Weim, 171 Lelo, ...

◮ Calcium oscillation: 122 Fish, 044 Borg, 117 Dupo,... ◮ Cell cycle: 056 Chen, 144 Calz, 007 Nova, 169 Agud,...

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State of the Art: Model Repositories

biomodels.net: plain list of 241 curated models in SBML format

◮ MAPK signaling cascade

009 Huan: three-level cascade of double phosphorylations 010 Khol: reduced model without dephosphorylation catalysts 011 Levc: same model as 009 Huan with different parameter values and different molecule names 027 Mark, 028 Mark, 029 Mark, 030 Mark, 031 Mark: reduced one-level models with different levels of details

◮ Circadian clock:

074 Lelo, 021 Lelo, 170 Weim, 171 Lelo, ...

◮ Calcium oscillation: 122 Fish, 044 Borg, 117 Dupo,... ◮ Cell cycle: 056 Chen, 144 Calz, 007 Nova, 169 Agud,...

• Relations between molecule names may be given in annotations
• No relations between models (given in the articles at best)

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Our Contribution

A graphical method for infering model reduction relationships between SBML models, automatically from the structure of the reactions, abstracting from names, kinetics and stoichiometry.

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Our Contribution

A graphical method for infering model reduction relationships between SBML models, automatically from the structure of the reactions, abstracting from names, kinetics and stoichiometry. State-of-the-art mathematical methods for model reductions based

• n kinetics (time/phase decompositions with slow/fast reactions)

are far too restrictive to be applicable on a large scale

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Our Contribution

A graphical method for infering model reduction relationships between SBML models, automatically from the structure of the reactions, abstracting from names, kinetics and stoichiometry. State-of-the-art mathematical methods for model reductions based

• n kinetics (time/phase decompositions with slow/fast reactions)

are far too restrictive to be applicable on a large scale

Example (Hierarchy of MAPK models in biomodels.net computed from the structure of their reactions)

009_Huan 010_Khol 011_Levc 027_Mark 029_Mark 031_Mark 026_Mark 028_Mark 030_Mark 049_Sasa 146_Hata

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Reaction Graphs (Petri net structure)

Definition

A reaction graph is a bipartite graph (S, R, A) where S is a set of species, R is a set of reactions and A ⊆ S × R ∪ R × S.

Example (E + S ⇋ ES → E + P)

E c ES d S p P

Example (E + S → E + P )

not a motif in the previous graph there is no subgraph isomorphism

S c P E

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Model Reductions as Graph Operations

In our setting, a model reduction is a finite sequence of graph reduction operations of four types:

• 1. Species deletion

deletion of one species vertex with all its incoming/outgoing arcs

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Model Reductions as Graph Operations

In our setting, a model reduction is a finite sequence of graph reduction operations of four types:

• 1. Species deletion

deletion of one species vertex with all its incoming/outgoing arcs

• 2. Reaction deletion

idem

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Model Reductions as Graph Operations

In our setting, a model reduction is a finite sequence of graph reduction operations of four types:

• 1. Species deletion

deletion of one species vertex with all its incoming/outgoing arcs

• 2. Reaction deletion

idem

• 3. Species merging

replacement of two species vertices by one species vertex with all their incoming/outgoing arcs

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Model Reductions as Graph Operations

In our setting, a model reduction is a finite sequence of graph reduction operations of four types:

• 1. Species deletion

deletion of one species vertex with all its incoming/outgoing arcs

• 2. Reaction deletion

idem

• 3. Species merging

replacement of two species vertices by one species vertex with all their incoming/outgoing arcs

• 4. Reactions merging

idem

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Example of the Michaelis-Menten Reduction

E c ES d S p P c+p ES E d S P c+p ES E d S P c+p ES E S P

merge(c,p)

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Example of the Michaelis-Menten Reduction

E c ES d S p P c+p ES E d S P c+p ES E d S P c+p ES E S P

merge(c,p) delete(d)

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Example of the Michaelis-Menten Reduction

E c ES d S p P c+p ES E d S P c+p ES E d S P c+p ES E S P

merge(c,p) delete(d) delete(ES)

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Commutation Properties of Delete/Merge Operations

The merge and delete operations enjoy the following commutation and association properties:

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Subgraph Epimorphisms

Definition

A subgraph morphism µ from G = (S, A) to G ′ = (S′, A′) is a function µ : S0 − → S′, with S0 ⊆ S such that

◮ ∀(x, y) ∈ A ∩ (S0 × S0) (µ(x), µ(y)) ∈ A′.

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Subgraph Epimorphisms

Definition

A subgraph morphism µ from G = (S, A) to G ′ = (S′, A′) is a function µ : S0 − → S′, with S0 ⊆ S such that

◮ ∀(x, y) ∈ A ∩ (S0 × S0) (µ(x), µ(y)) ∈ A′.

A subgraph epimorphism is a surjective subgraph morphism

◮ ∀x′ ∈ S′ ∃x ∈ S0 µ(x) = x′, ◮ ∀(x′, y′) ∈ A′ ∃(x, y) ∈ A µ(x) = x′ µ(y) = y′.

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Subgraph Epimorphisms

Definition

A subgraph morphism µ from G = (S, A) to G ′ = (S′, A′) is a function µ : S0 − → S′, with S0 ⊆ S such that

◮ ∀(x, y) ∈ A ∩ (S0 × S0) (µ(x), µ(y)) ∈ A′.

A subgraph epimorphism is a surjective subgraph morphism

◮ ∀x′ ∈ S′ ∃x ∈ S0 µ(x) = x′, ◮ ∀(x′, y′) ∈ A′ ∃(x, y) ∈ A µ(x) = x′ µ(y) = y′.

Theorem

There exists a subgraph epimorphism from G to G ′ if and only if there exists a graphical reduction from G to G ′ (by species/reactions deletions and mergings)

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Subgraph Epimorphisms

Definition

A subgraph morphism µ from G = (S, A) to G ′ = (S′, A′) is a function µ : S0 − → S′, with S0 ⊆ S such that

◮ ∀(x, y) ∈ A ∩ (S0 × S0) (µ(x), µ(y)) ∈ A′.

A subgraph epimorphism is a surjective subgraph morphism

◮ ∀x′ ∈ S′ ∃x ∈ S0 µ(x) = x′, ◮ ∀(x′, y′) ∈ A′ ∃(x, y) ∈ A µ(x) = x′ µ(y) = y′.

Theorem

There exists a subgraph epimorphism from G to G ′ if and only if there exists a graphical reduction from G to G ′ (by species/reactions deletions and mergings) Subgraph isomorphisms correspond to delete operations only Graph epimorphisms correspond to merge operations only

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Model Reductions as Subgraph Epimorphisms

Example (Michaelis-Menten reduction)

Subgraph epimorphism: E → C S → A P → B c → r p → r d → ⊥ ES → ⊥ E c ES d S p P A r C B Equivalent to the graphical reduction: merge(c,p), delete(d), delete(ES)

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The Subgraph Epimorphism Problem

Input: two reaction graphs Output: whether there exists a subgraph epimorphism (i.e. a graphical model reduction) from the first graph to the second.

Theorem

The subgraph epimorphism problem is NP-complete.

Proof (article submitted with Christine Solnon):

by reduction of the Set Covering Problem.

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Implementation in Constraint Logic Programming

Constraint model:

◮ variable Xu for each vertex u ∈ V with domain V ′ ∪ {⊥}

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Implementation in Constraint Logic Programming

Constraint model:

◮ variable Xu for each vertex u ∈ V with domain V ′ ∪ {⊥} ◮ morphism requirement (arc preservation) implemented with

relation constraint (Xu, Xv) ∈ A′ for all (u, v) ∈ A

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Implementation in Constraint Logic Programming

Constraint model:

◮ variable Xu for each vertex u ∈ V with domain V ′ ∪ {⊥} ◮ morphism requirement (arc preservation) implemented with

relation constraint (Xu, Xv) ∈ A′ for all (u, v) ∈ A

◮ the surjectivity constraint is implemented with antecedent

variables Av = u ⇒ Xu = v

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Implementation in Constraint Logic Programming

Constraint model:

◮ variable Xu for each vertex u ∈ V with domain V ′ ∪ {⊥} ◮ morphism requirement (arc preservation) implemented with

relation constraint (Xu, Xv) ∈ A′ for all (u, v) ∈ A

◮ the surjectivity constraint is implemented with antecedent

variables Av = u ⇒ Xu = v

◮ Redundant constraint all different({Ai})

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Implementation in Constraint Logic Programming

Constraint model:

◮ variable Xu for each vertex u ∈ V with domain V ′ ∪ {⊥} ◮ morphism requirement (arc preservation) implemented with

relation constraint (Xu, Xv) ∈ A′ for all (u, v) ∈ A

◮ the surjectivity constraint is implemented with antecedent

variables Av = u ⇒ Xu = v

◮ Redundant constraint all different({Ai})

Enumeration strategy:

◮ on antecedent variables {Ai} ◮ before vertex variables {Xj} ◮ variables with least domain size first

Implemented in Biocham http://contraintes.inria.fr/biocham using Gnu-Prolog http://gprolog.inria.fr

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Evaluation on biomodels.net

◮ Search of a subgraph epimorphism between all pairs of models

with a time out of 20mn 90% of comparisons took less than 5s

◮ Computes model hierarchies where each node represents a

model: M − → M′ means a reduction from M to M′ was found. M ← → M′ means M and M′ are isomorphic.

◮ 9% of false positive found between different model classes

typically involving very small models recognized as patterns in larger models (e.g. double phosphorylations) 1.2% of false positive after removal of small models

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MAPK Hierarchy

009_Huan 010_Khol 011_Levc 027_Mark 029_Mark 031_Mark 026_Mark 028_Mark 030_Mark 049_Sasa 146_Hata

Models 009 (Huang 1996), 010 (Kholodenko 2000) and 011 (Levchenko 2000) are three-level cascade models. Models 026 to 031 (Markevitch 2004) are one-level. Models 049 (Sasagawa 2005) is a larger model (216 reactions), some computations timed out.

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Circadian Clock Models Hierarchy

021_Lelo 170_Weim 022_Ueda 034_Smol 055_Lock 073_Lelo 078_Lelo 074_Lelo 083_Lelo 089_Lock 171_Lelo

Models 073, 078 isomorphic [Leloup et al. 03] different parameter values. Models 074, 083 isomorphic, refinement with ErvErbα False negative: models 021, 171 have the same structure but with different encodings in SBML (functions vs species)

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Calcium Oscillation Models Hierarchy

039_Marh 098_Gold 115_Somo 117_Dupo 166_Zhu 043_Borg 044_Borg 045_Borg 058_Bind 122_Fish 145_Wang

Models 098, 115, 117 are very small two-species oscillators. Model 122 (Fisher et al. 2006) NFAT, NFκB and side calcium

• scillation.

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Cell Cycle Models Hierarchy

007_Nova 008_Gard 168_Obey 056_Chen 169_Agud 196_Sriv 109_Habe 111_Nova 144_Calz

Not satisfactory: these ODE models have been transcribed in SBML without writing all reactants in the reaction rules Species eliminated by conservation laws are encoded in the kinetics and not visible in the rules Events (cell division) are not reflected in the reaction graph.

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Conclusion

Purely structural model reduction method that correctly identifies model reduction relationships in biomodels.net (as long as the SBML rules do not omit species hidden in the kinetic expressions)

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Conclusion

Purely structural model reduction method that correctly identifies model reduction relationships in biomodels.net (as long as the SBML rules do not omit species hidden in the kinetic expressions)

◮ Independent of annotations, kinetics and stoichiometry.

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Conclusion

Purely structural model reduction method that correctly identifies model reduction relationships in biomodels.net (as long as the SBML rules do not omit species hidden in the kinetic expressions)

◮ Independent of annotations, kinetics and stoichiometry. ◮ Graph-theoretic definition of model reduction: graph

reduction operations equivalent to subgraph epimorphisms

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Conclusion

Purely structural model reduction method that correctly identifies model reduction relationships in biomodels.net (as long as the SBML rules do not omit species hidden in the kinetic expressions)

◮ Independent of annotations, kinetics and stoichiometry. ◮ Graph-theoretic definition of model reduction: graph

reduction operations equivalent to subgraph epimorphisms

◮ NP-complete problem but efficient constraint logic program to

solve it on real-size models

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Conclusion

Purely structural model reduction method that correctly identifies model reduction relationships in biomodels.net (as long as the SBML rules do not omit species hidden in the kinetic expressions)

◮ Independent of annotations, kinetics and stoichiometry. ◮ Graph-theoretic definition of model reduction: graph

reduction operations equivalent to subgraph epimorphisms

◮ NP-complete problem but efficient constraint logic program to

solve it on real-size models

◮ Implemented in Biocham 3.0 modeling environment

http://contraintes.inria.fr/biocham

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Conclusion

Purely structural model reduction method that correctly identifies model reduction relationships in biomodels.net (as long as the SBML rules do not omit species hidden in the kinetic expressions)

◮ Independent of annotations, kinetics and stoichiometry. ◮ Graph-theoretic definition of model reduction: graph

reduction operations equivalent to subgraph epimorphisms

◮ NP-complete problem but efficient constraint logic program to

solve it on real-size models

◮ Implemented in Biocham 3.0 modeling environment

http://contraintes.inria.fr/biocham

◮ New method for querying model repositories by the

structure of the models

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On-going work

◮ Rewriting of cell cycle models in SBML to better reflect their

dynamics in the structure of the rules.

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On-going work

◮ Rewriting of cell cycle models in SBML to better reflect their

dynamics in the structure of the rules.

◮ Theory of subgraph epimorphisms

Maximum common epimorphic subgraph (model intersection) Minimum common epimorphic supergraph (model union)

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On-going work

◮ Rewriting of cell cycle models in SBML to better reflect their

dynamics in the structure of the rules.

◮ Theory of subgraph epimorphisms

Maximum common epimorphic subgraph (model intersection) Minimum common epimorphic supergraph (model union)

◮ Search of protein circuit motifs as common epimorphic

subgraphs

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On-going work

◮ Rewriting of cell cycle models in SBML to better reflect their

dynamics in the structure of the rules.

◮ Theory of subgraph epimorphisms

Maximum common epimorphic subgraph (model intersection) Minimum common epimorphic supergraph (model union)

◮ Search of protein circuit motifs as common epimorphic

subgraphs

◮ Finding mathematical conditions on the kinetics for the graph

reduction operations:

◮ species deletions for species in excess ◮ reaction deletions for slow reverse reactions ◮ species mergings for fast equilibria (QSSA) ◮ reaction mergings for limiting reactions

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Acknowledgements

◮ Contraintes group at INRIA Paris-Rocquencourt on this topic:

Gr´ egory Batt, Sylvain Soliman, Elisabetta De Maria, Aur´ elien Rizk, Steven Gay, Faten Nabli, Xavier Duportet, Janis Ulhendorf, Dragana Jovanovska

◮ ANR EraSysBio C5Sys (follow up of FP6 Tempo) on cancer

chronotherapies, coord. Francis L´ evi, INSERM; Jean Clairambault INRIA; Coupled models of cell and circadian cycles, p53/mdm2, cytotoxic drugs.

◮ INRIA/INRA project Regate coord. F. Cl´

ement INRIA; E. Reiter, D. Heitzler Modeling of GPCR Angiotensine and FSH signaling networks

◮ ANR project Calamar, coord. C. Chaouiya, D. Thieffry Univ.

Marseille, L. Calzone Curie Institute Modularity and Compositionality in regulatory networks.

◮ OSEO Biointelligence, coord. Dassault-Syst`

emes, Technology transfer of Biocham concepts and tools