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 1 Fran¸ cois Fages - TSB’11 Grenoble
From Models to Metamodels In Systems Biology, models are built with two contradictory perspectives: 2 Fran¸ cois Fages - TSB’11 Grenoble
From Models to Metamodels In Systems Biology, models are built with two contradictory perspectives: ◮ models for representing knowledge: the more detailed the better 3 Fran¸ cois Fages - TSB’11 Grenoble
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 4 Fran¸ cois Fages - TSB’11 Grenoble
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 5 Fran¸ cois Fages - TSB’11 Grenoble
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 6 Fran¸ cois Fages - TSB’11 Grenoble
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 ,... 7 Fran¸ cois Fages - TSB’11 Grenoble
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) 8 Fran¸ cois Fages - TSB’11 Grenoble
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. 9 Fran¸ cois Fages - TSB’11 Grenoble
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 on kinetics (time/phase decompositions with slow/fast reactions) are far too restrictive to be applicable on a large scale 10 Fran¸ cois Fages - TSB’11 Grenoble
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 on 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 028_Mark 030_Mark 049_Sasa 011_Levc 146_Hata 026_Mark 010_Khol 029_Mark 031_Mark 027_Mark 11 Fran¸ cois Fages - TSB’11 Grenoble
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 ) p c E ES P d S Example ( E + S → E + P ) not a motif in the previous graph E there is no subgraph isomorphism c S P 12 Fran¸ cois Fages - TSB’11 Grenoble
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 13 Fran¸ cois Fages - TSB’11 Grenoble
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 14 Fran¸ cois Fages - TSB’11 Grenoble
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 15 Fran¸ cois Fages - TSB’11 Grenoble
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 16 Fran¸ cois Fages - TSB’11 Grenoble
Example of the Michaelis-Menten Reduction p c+p c E ES P merge(c,p) d E ES P S S d c+p E E ES P c+p S P S d ES 17 Fran¸ cois Fages - TSB’11 Grenoble
Example of the Michaelis-Menten Reduction p c+p c E ES P merge(c,p) d E ES P S S d c+p E E ES P c+p S P S d ES delete(d) 18 Fran¸ cois Fages - TSB’11 Grenoble
Example of the Michaelis-Menten Reduction p c+p c E ES P merge(c,p) d E ES P S S d c+p E E ES P c+p S P S d ES delete(d) delete(ES) 19 Fran¸ cois Fages - TSB’11 Grenoble
Commutation Properties of Delete/Merge Operations The merge and delete operations enjoy the following commutation and association properties: 20 Fran¸ cois Fages - TSB’11 Grenoble
Subgraph Epimorphisms Definition A subgraph morphism µ from G = ( S , A ) to G ′ = ( S ′ , A ′ ) is a function µ : S 0 − → S ′ , with S 0 ⊆ S such that ◮ ∀ ( x , y ) ∈ A ∩ ( S 0 × S 0 ) ( µ ( x ) , µ ( y )) ∈ A ′ . 21 Fran¸ cois Fages - TSB’11 Grenoble
Subgraph Epimorphisms Definition A subgraph morphism µ from G = ( S , A ) to G ′ = ( S ′ , A ′ ) is a function µ : S 0 − → S ′ , with S 0 ⊆ S such that ◮ ∀ ( x , y ) ∈ A ∩ ( S 0 × S 0 ) ( µ ( x ) , µ ( y )) ∈ A ′ . A subgraph epimorphism is a surjective subgraph morphism ◮ ∀ x ′ ∈ S ′ ∃ x ∈ S 0 µ ( x ) = x ′ , ◮ ∀ ( x ′ , y ′ ) ∈ A ′ ∃ ( x , y ) ∈ A µ ( x ) = x ′ µ ( y ) = y ′ . 22 Fran¸ cois Fages - TSB’11 Grenoble
Subgraph Epimorphisms Definition A subgraph morphism µ from G = ( S , A ) to G ′ = ( S ′ , A ′ ) is a function µ : S 0 − → S ′ , with S 0 ⊆ S such that ◮ ∀ ( x , y ) ∈ A ∩ ( S 0 × S 0 ) ( µ ( x ) , µ ( y )) ∈ A ′ . A subgraph epimorphism is a surjective subgraph morphism ◮ ∀ x ′ ∈ S ′ ∃ x ∈ S 0 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) 23 Fran¸ cois Fages - TSB’11 Grenoble
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