Model Reductions by Subgraph Epimorphisms

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Model reductions Graph Reductions Graph Morphisms Algorithms Evaluation in BioModels Well-formed Reactions

Model Reductions by Subgraph Epimorphisms

Formal Methods in Systems and Synthetic Biology Porquerolles 2016

Fran¸cois Fages Inria Saclay, EPI Lifeware, France

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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 model structuresof increasing details

related byreduction/refinement relations.

This hierarchy of models is orthogonal to the hierarchy of semantics (formalisms) for interpreting each model.

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From Models to Metamodels

models for representing knowledge:

the more detailed the better

models for making predictions:

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From Models to Metamodels

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 model structuresof increasing details

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From Models to Metamodels

the more detailed the better models for making predictions:

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State of the Art: Flat Model Repositories with Annotations

biomodels.net: plain list of583 manually curated and 796 non 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 Huanwith 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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State of the Art: Flat Model Repositories with Annotations

009 Huan: three-level cascade of double phosphorylations 010 Khol: reduced model without dephosphorylation catalysts

011 Levc: same model as 009 Huanwith different parameter values and different molecule names

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State of the Art: Flat Model Repositories with Annotations

009 Huan: three-level cascade of double phosphorylations 010 Khol: reduced model without dephosphorylation catalysts 011 Levc: same model as 009 Huanwith different parameter values and different molecule names

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State of the Art: Flat Model Repositories with Annotations

027 Mark, 028 Mark, 029 Mark, 030 Mark, 031 Mark:

reduced one-level models with different levels of details

Circadian clock:

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State of the Art: Flat Model Repositories with Annotations

reduced one-level models with different levels of details Circadian clock:

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: Flat Model Repositories with Annotations

reduced one-level models with different levels of details Circadian clock:

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 inannotations

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Can we Detect Reductions from the Network Structure ?

Mathematical methods based on slow/fast reactions currently do not apply on a large scale.

Can we design a purely graphical method based on the

reactant/product structureabstracting from names, kinetics and stoichiometry ? [Steven Gay Thesis]

Example (Hierarchy of MAPK models in biomodels.net automatically computed from the structure of the 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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Can we Detect Reductions from the Network Structure ?

Mathematical methods based on slow/fast reactions currently do not apply on a large scale.

Can we design a purely graphical method based on the

reactant/product structureabstracting from names, kinetics and stoichiometry ? [Steven Gay Thesis]

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

009_Huan

010_Khol 011_Levc

027_Mark

029_Mark 031_Mark

026_Mark

146_Hata

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Reaction Hypergraphs (Petri net structure)

Definition

Areaction hypergraph is a bipartite graph (S,R,A) whereS is a set ofspecies,R is a set ofreactionsand A⊆S×R∪R×S.

Example (E +S ES →E+P and E+S →E+P)

E c ES

d S

p P

S c P

E

not contained in the previous graph no subgraph isomorphism

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Model Reductions by Graph Operations

In our setting, a model reduction is afinite sequenceof 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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Model Reductions by Graph Operations

1 Species deletion

3 Species merging

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Model Reductions by Graph Operations

1 Species deletion

3 Species merging

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Model Reductions by Graph Operations

1 Species deletion

3 Species merging

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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:

How to eliminate those symmetries for detecting graph reductions ?

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Subgraph Epimorphisms

Definition

Asubgraph morphism(SMOR) fromG = (S,A) to G⁰= (S⁰,A⁰) is a functionµ:S₀ −→S⁰, with S₀ ⊆S such that

∀(x,y)∈A∩(S0×S₀) (µ(x), µ(y))∈A⁰.

Asubgraph epimorphism (SEPI) is a surjective SMOR

∀x⁰ ∈S⁰ ∃x ∈S0 µ(x) =x⁰,

∀(x⁰,y⁰)∈A⁰ ∃(x,y)∈A µ(x) =x⁰ µ(y) =y⁰.

Theorem (Gay Soliman FagesBioinformatics 26(18):i575i581, 2010) There exists a subgraph epimorphism from G to G⁰

if and only if there exists a graphical reduction from G to G⁰ by species/reactions deletions and mergings.

Subgraph isomorphisms(SISO) allow delete operations only Graph epimorphisms(GEPI) allow merge operations only

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Subgraph Epimorphisms

Definition

∀(x,y)∈A∩(S0×S₀) (µ(x), µ(y))∈A⁰.

∀x⁰ ∈S⁰ ∃x ∈S0 µ(x) =x⁰,

∀(x⁰,y⁰)∈A⁰ ∃(x,y)∈A µ(x) =x⁰ µ(y) =y⁰.

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Subgraph Epimorphisms

Definition

∀(x,y)∈A∩(S0×S₀) (µ(x), µ(y))∈A⁰.

∀x⁰ ∈S⁰ ∃x ∈S0 µ(x) =x⁰,

∀(x⁰,y⁰)∈A⁰ ∃(x,y)∈A µ(x) =x⁰ µ(y) =y⁰.

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Subgraph Epimorphisms

Definition

∀(x,y)∈A∩(S0×S₀) (µ(x), µ(y))∈A⁰.

∀x⁰ ∈S⁰ ∃x ∈S0 µ(x) =x⁰,

∀(x⁰,y⁰)∈A⁰ ∃(x,y)∈A µ(x) =x⁰ µ(y) =y⁰.

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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 SEPI Existence Problem

Input: two reaction graphs

Output: whether there exists a SEPI (i.e. a graphical model reduction) from the first graph to the second.

Theorem (Gay Martinez Fages Soliman SolnonDiscrete Applied Mathematics 162:214228, 2014)

The SEPI existence problem between two graphs is NP-complete.

Proof by reduction of the Set Covering Problem.

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Practical significance of NP-completeness ?

Wrong answer:

Cannot solve NP-complete problems on large size instances Correct answer:

Cannot solve NP-complete problems on some (small size) instances:

For any algorithm there exists an infinite set of instances for which the computation time is exponential in their size. Says nothing about the existence of efficient algorithms for solving an infinite class of (practical) instances possibly of very large size.

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Practical significance of NP-completeness ?

Wrong answer:

Cannot solve NP-complete problems on large size instances

Correct answer:

For any algorithm there exists an infinite set of instances for which the computation time is exponential in their size. Says nothing about the existence of efficient algorithms for solving an infinite class of (practical) instances possibly of very large size.

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Practical significance of NP-completeness ?

Wrong answer:

Cannot solve NP-complete problems on large size instances Correct answer:

For any algorithm there exists an infinite set of instances for which the computation time is exponential in their size.

Says nothing about the existence of efficient algorithms for solving an infinite class of (practical) instances possibly of very large size.

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Implementation with Constraint Logic Program

Constraint model:

target variableXu for each vertexu ∈V with domain V ∪ {⊥}

antecedent variable Av for each vertexv ∈V⁰ with domainV morphism requirement (arc preservation) implemented with tabular constraint (Xu,Xv)∈Afor all (u,v)∈A

the surjectivity constraint is implemented with antecedent variables and reified constraint A_v =u ⇒X_u=v Redundant constraint alldifferent({A_i})

Enumeration strategy:

on antecedent variables {A_i} before vertex variables {X_j}

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Exercise in BIOCHAM-web about SEPI Reductions

draw reactions for the three different Trypsin reaction models

search all reductionsbetween the Michaelis-Menten reductions of Trypsin models

search reduction between some MAPK models of BioModels

check the differences between the original and ODE-curated versions of models 021 and 171

check SEPI model reductions between them.

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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 take less than 5s

Computes model hierarchies where each node represents a model:

M −→M⁰ means a reduction fromM toM⁰ was found. M ←→M⁰ means M andM⁰ are isomorphic.

9% of false positive found between different model classes typically involving very small modelsrecognized as motifs in larger models (e.g. double phosphorylations)

1.2% of false positive after removal of small models

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Evaluation on biomodels.net

M −→M⁰ means a reduction fromM toM⁰ was found.

M ←→M⁰ means M andM⁰ are isomorphic.

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Evaluation on biomodels.net

M −→M⁰ means a reduction fromM toM⁰ was found.

M ←→M⁰ means M andM⁰ are isomorphic.

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MAPK Hierarchy

009_Huan

010_Khol 011_Levc

027_Mark

029_Mark 031_Mark

026_Mark

146_Hata

Models 009 (Huang 1996), 010 (Kholodenko 2000) and 011 (Levchenko 2000) arethree-level cascade models.

Models 026 to 031 (Markevitch 2004) areone-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 (different parameter values).

Models 074, 083 isomorphic, refinement of 073 078 with RevErbα False negative: models 021, 171 have the same ODEs but with different reaction encodings(variable eliminated by invariants)

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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 oscillation.

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Cell Cycle Models Hierarchy

007_Nova

008_Gard 168_Obey

056_Chen

169_Agud 109_Habe 111_Nova 196_Sriv

144_Calz

Not satisfactory: these ODE models have been transcribed in SBML withill-formed reactions (missing reactants, same ODEs) Species eliminated byconservation lawsare encoded by terms in the kinetics and invisible in the reaction

Events(for cell division) are invisible in the reaction hypergraph.

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Automatic Rewriting of SBML Models in Biocham

1 load sbml(file)

2 export ode(file)

3 load ode(file)

4 export sbml(file)

E.g. Biomodels 8 mitotic oscillator [Goldbeter 91]

Inference of molecules eliminated by invariants in the original SBML file: Mi = 1−M Xi = 1−X.

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Automatic Rewriting of SBML Models in Biocham

1 load sbml(file)

2 export ode(file)

3 load ode(file)

4 export sbml(file)

E.g. Biomodels 8 mitotic oscillator [Goldbeter 91]

Inference of molecules eliminated by invariants in the original SBML file: Mi = 1−M Xi = 1−X.

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Conclusion

Model reductions can be detected in model repositories by checking the existence of a SEPIs between reaction hypergraphs

Automatic creation of model hierarchies (metamodels) Automatic check for the existence of more/less detailed models

Automatic curation of SBML writing of reaction models using Biocham’s load odealgorithm: decreases from 65% to 28% the number of non well-formed models in BioModels

The SEPI existence problem between two graphs is

NP-complete but CLP or SAT solvers apply with few timeouts (e.g. on Schoeberl model of MAPK)

The enumeration using CLP or SAT of all minimal siphons (NP complete) in all BioModels reaction models takes a few seconds [Nabli Martinez FF Soliman 15 Constraints]

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Conclusion

Automatic creation of model hierarchies (metamodels)

Automatic check for the existence of more/less detailed models

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Conclusion

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Conclusion

Automatic curation of SBML writing of reaction models using Biocham’s load odealgorithm: decreases from 65% to 28%

the number of non well-formed models in BioModels

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Conclusion

the number of non well-formed models in BioModels The SEPI existence problem between two graphs is

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Conclusion

the number of non well-formed models in BioModels The SEPI existence problem between two graphs is

The enumeration using CLP or SAT of all minimal siphons (NP complete) in all BioModels reaction models takes a few