MasterMPRIC2-19ComputationalMethodsinSystemsandSyntheticBiology ModelReductionsbySubgraphEpimorphisms

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

Model Reductions by Subgraph Epimorphisms

Master MPRI C2-19

Computational Methods in Systems and Synthetic Biology

Fran¸cois Fages

Inria Saclay, Lifeware team, France

after Steven Gay’s PhD Thesis (former MPRI C2-19 student 2008-2009)

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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 modelsrelated by reduction/refinement relations.

To understand a system is not to know everything about it, but to know abstraction levels that aresufficient

for answering given questions about it

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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:

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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: Model Repositories

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: Model Repositories

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: Model Repositories

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: Model Repositories

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: Model Repositories

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: Model Repositories

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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Detect Reductions from the Structure of the Reactions ?

Mathematical methods based on slow/fast reactions currently do not apply on a large scale(Thesis subject using tropical algebra)

⇒Purely graphical method based on the reactant/product structureabstracting from names, kinetics and stoichiometry ?

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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Detect Reductions from the Structure of the Reactions ?

Mathematical methods based on slow/fast reactions currently do not apply on a large scale(Thesis subject using tropical algebra)

⇒Purely graphical method based on the reactant/product structureabstracting from names, kinetics and stoichiometry ? 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

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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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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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 detect graph reductions efficiently ? Without loosing time in symmetrical solutions ?

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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.

Subgraph isomorphisms(SISO) allow delete 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 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.

Practical significance of NP-completeness ?

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.

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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.

Practical significance of NP-completeness ?

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.

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Constraint Satisfaction Problems

Aconstraint satisfaction problem(CSP for short) is a triple P = (X,D,C), where:

X is a set ofvariables

D is a family ofdomains (e.g.N) indexed by variables inX :

∀x ∈X,Dx is a finite set.

C is a set of constraints, eachc ∈C is defined by its arity ar(c)∈N, a tuple of variables~x(c)∈X^ar(c) and a relation R(c)⊆Qar(c)

i=1 D_~_x_i_(c).

Anassignmentν :X −→D_x is a solutionof P when

∀c ∈C,(ν(x1), . . . , ν(xn))∈R(c), with~x(c) = (x1, . . . ,xn).

Example

For graph morphisms problems, associate one variable per vertex in the source graph, with the target graph vertices as domain.

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Constraint Satisfaction Problems

Aconstraint satisfaction problem(CSP for short) is a triple P = (X,D,C), where:

X is a set ofvariables

D is a family ofdomains (e.g.N) indexed by variables inX :

∀x ∈X,Dx is a finite set.

C is a set of constraints, eachc ∈C is defined by its arity ar(c)∈N, a tuple of variables~x(c)∈X^ar(c) and a relation R(c)⊆Qar(c)

i=1 D_~_x_i_(c).

Anassignmentν :X −→D_x is a solutionof P when

∀c ∈C,(ν(x1), . . . , ν(xn))∈R(c), with~x(c) = (x1, . . . ,xn).

Example

For graph morphisms problems, associate one variable per vertex in the source graph, with the target graph vertices as domain.

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Domain Filtering Algorithms for Basic Constraints

A constraint is a reactive agent that reduces the domains of its variables upon domain reduction events.

Example (c =relation(~x,R))

where~x is a tuple of variables andR is a relation given in extension, is defined by:

ar(c) = arity of~x = arity ofR= n

~x(c) =~x R(c) =R ∩Qn

i=1D_x_i,

Example (c =element(i,(f1, . . . ,f_n),x) meaningf_i =x) ar(c) =n+ 2

~x(c) = (i,f₁, . . . ,f_n,x)

R(c) ={(ι, ν1, . . . , νn, ξ)∈Di×(Qn

i=1D_f_i)×Dx |νι =ξ}

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Domain Filtering Algorithms for Basic Constraints

A constraint is a reactive agent that reduces the domains of its variables upon domain reduction events.

Example (c =relation(~x,R))

where~x is a tuple of variables andR is a relation given in extension, is defined by:

ar(c) = arity of~x = arity ofR= n

~x(c) =~x R(c) =R ∩Qn

i=1D_x_i,

Example (c =element(i,(f1, . . . ,f_n),x) meaningf_i =x) ar(c) =n+ 2

~x(c) = (i,f₁, . . . ,f_n,x)

R(c) ={(ι, ν1, . . . , νn, ξ)∈D_i ×(Qn

i=1D_f_i)×Dx |νι =ξ}

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SEP Existence Problem as a CSP

Variablesfor the vertices ofG and G⁰ and for the edges of G⁰ X ={x_v |v ∈V(G)} ] {y_v⁰ |v⁰ ∈V(G⁰)} ] {y_e⁰ |e⁰ ∈E(G⁰)}, D(xv) =V(G_⊥⁰ ), D(yv⁰) ={1, . . . ,|V(G)|}, D(ye⁰) ={1, . . . ,|E(G)|}

Theconstraintsare :

{relation((xu,xv),E(G_⊥⁰ ))|(u,v)∈E(G)}

∪{element(yv⁰,x(V(G)),v⁰)|v⁰ ∈V(G⁰)}

∪{element(y_(u⁰_,v⁰₎, π₁(x(E(G))),u⁰)|(u⁰,v⁰)∈E(G⁰)}

∪{element(y(u⁰,v⁰), π2(x(E(G))),v⁰)|(u⁰,v⁰)∈E(G⁰)} wherex(V(G)) ={x_v |v∈V(G)},

x(E(G)) ={(xu,x_v) |(u,v)∈E(G)},

andπ1, π2 map the first and second projection functions on lists.

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SEP Existence Problem as a CSP

Variablesfor the vertices ofG and G⁰ and for the edges of G⁰ X ={x_v |v ∈V(G)} ] {y_v⁰ |v⁰ ∈V(G⁰)} ] {y_e⁰ |e⁰ ∈E(G⁰)}, D(xv) =V(G_⊥⁰ ), D(yv⁰) ={1, . . . ,|V(G)|}, D(ye⁰) ={1, . . . ,|E(G)|}

Theconstraintsare :

{relation((xu,xv),E(G_⊥⁰))|(u,v)∈E(G)}

∪{element(yv⁰,x(V(G)),v⁰)|v⁰ ∈V(G⁰)}

∪{element(y_(u⁰_,v⁰₎, π₁(x(E(G))),u⁰)|(u⁰,v⁰)∈E(G⁰)}

∪{element(y(u⁰,v⁰), π2(x(E(G))),v⁰)|(u⁰,v⁰)∈E(G⁰)} wherex(V(G)) ={x_v |v∈V(G)},

x(E(G)) ={(xu,x_v) |(u,v)∈E(G)},

andπ1, π2 map the first and second projection functions on lists.

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Search Strategy

Enumerating source vertex variables is sufficient to enforce surjectivity and decide SEPI satisfiability.

In practice it is more efficient toenumerate the antecedent variables of the reduced graph

and finally instanciate the remaining source vertex variables to⊥. functionVariableValueSelectionantecedents(D)

if ∃v⁰ ∈V(G⁰)|y_v⁰ ={d₁,d₂, . . .}then return(yv⁰,d1)

else if ∃(u⁰,v⁰)∈E(G⁰)|y_(u⁰_,v⁰₎ ={d₁,d₂, . . .} then return(y_(u⁰_,v⁰₎,d₁)

else if ∃v ∈V(G)| |D_x_v|>1∧Dxv 6={⊥} then return(xv,⊥)

end if end function

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Search Strategy

Enumerating source vertex variables is sufficient to enforce surjectivity and decide SEPI satisfiability.

In practice it is more efficient toenumerate the antecedent variables of the reduced graph

and finally instanciate the remaining source vertex variables to⊥.

functionVariableValueSelectionantecedents(D) if ∃v⁰ ∈V(G⁰)|y_v⁰ ={d₁,d₂, . . .}then

return(yv⁰,d1)

else if ∃(u⁰,v⁰)∈E(G⁰)|y_(u⁰_,v⁰₎ ={d₁,d₂, . . .} then return(y_(u⁰_,v⁰₎,d₁)

else if ∃v ∈V(G)| |D_x_v|>1∧Dxv 6={⊥} then return(xv,⊥)

end if end function

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Evaluation on biomodels.net

Implemented in Biocham v3 (not yet in v4) in GNU-Prolog http://gprolog.inria.frand MiniSAT http://minisat.se

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

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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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Problem of ODE Models badly Transcribed in SBML

E.g. Biomodels 8 originiates from an ODE model of a simple mitotic oscillator [Goldbeter 91]

It wasquickly transcribed in SBML with synthesis and degradation reactions and keeping terms for molecules eliminated by invariants M⁰ = 1−M X⁰= 1−X.

Can we infer the right structure of the SBML reactions from the ODE? infer hidden molecules andwell-formed reactions...

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Problem of ODE Models badly Transcribed in SBML

E.g. Biomodels 8 originiates from an ODE model of a simple mitotic oscillator [Goldbeter 91]

It wasquickly transcribed in SBML with synthesis and degradation reactions and keeping terms for molecules eliminated by invariants M⁰ = 1−M X⁰= 1−X.

Can we infer the right structure of the SBML reactions from the ODE? infer hidden molecules andwell-formedreactions...

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Infering Reaction Systems from ODEs

dAi/dt =fi(A) is the ODE semantics offi for => Ai

Definition

A reaction with inhibitors m,f for r / m => p, over molecular species{x₁, . . . ,x_s} is well-formed if the following conditions hold:

1 f(x1, . . . ,x_s) is partially differentiable, non-negative on R^s+;

2 x_i ∈r if and only if∂f/∂x_i(~x)>0 for some value~x ∈R^s+;

3 xi ∈m if and only if∂f/∂xi(~x)<0 for some value~x ∈R^s+. A molecular species can be both areactant and aninhibitor. Example

To ˙x =−k one can associate the well-formed reaction system l*x for x => 2*x,k+l*x for x =>

This is the result computed in BIOCHAM byload ode. Note thatk+l∗x6= 0 when x = 0

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Infering Reaction Systems from ODEs

dAi/dt =fi(A) is the ODE semantics offi for => Ai Definition

3 xi ∈m if and only if∂f/∂xi(~x)<0 for some value~x ∈R^s+.

A molecular species can be both areactant and aninhibitor. Example

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Infering Reaction Systems from ODEs

3 xi ∈m if and only if∂f/∂xi(~x)<0 for some value~x ∈R^s+. A molecular species can be both areactantand aninhibitor.

Example

To ˙x =−k one can associate the well-formed reaction system

l*x for x => 2*x,k+l*x for x =>

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Infering Reaction Systems from ODEs

3 xi ∈m if and only if∂f/∂xi(~x)<0 for some value~x ∈R^s+. A molecular species can be both areactantand aninhibitor.

Example

This is the result computed in BIOCHAM byload ode.

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Strict Kinetics and Positive Systems

A reactionf for r / m => p isstrict if its kinetics

f(x1, . . . ,x_s) = 0 wheneverx_j = 0 for anyx_j such that r(xj)>0.

A dynamical system overR^k is called positive ifR^k+ is an invariant set for the system, i.e.,∀x₀≥0,t ≥0x(t,x0)≥0

Proposition (Positivity)

Well-formed and strict reactions define positive ODEs Proof: We have ˙x_j=Pn

i=1(pi(xj)−r_i(xj))×f_i and since the system is well-formed, thefi are all non-negative. The only negative terms thus haveri(xj)>0 and from the strictness condition this entails thatfi= 0 whenx_j = 0. Hence ˙x_j ≥0 wheneverx_j = 0 since it is a sum of

non-negative terms. Thereforexj cannot become negative when its initial value is non-negative, and since this holds for allj, the system is positive.

˙

x=−k is not the ODE semantics of strict well-formed reactions

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Strict Kinetics and Positive Systems

A dynamical system overR^k is called positive ifR^k+ is an invariant set for the system, i.e.,∀x₀ ≥0,t≥0x(t,x0)≥0

Well-formed and strict reactions define positive ODEs

Proof: We have ˙x_j=Pn

i=1(pi(xj)−r_i(xj))×f_i and since the system is well-formed, thefi are all non-negative. The only negative terms thus haveri(xj)>0 and from the strictness condition this entails thatfi= 0 whenx_j = 0. Hence ˙x_j ≥0 wheneverx_j = 0 since it is a sum of

˙

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Strict Kinetics and Positive Systems

i=1(pi(xj)−r_i(xj))×f_i and since the system is well-formed, thefi are all non-negative. The only negative terms thus haveri(xj)>0 and from the strictness condition this entails thatfi= 0 whenx_j = 0. Hence ˙x_j≥0 wheneverx_j = 0 since it is a sum of

˙

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Strict Kinetics and Positive Systems

i=1(pi(xj)−r_i(xj))×f_i and since the system is well-formed, thefi are all non-negative. The only negative terms thus haveri(xj)>0 and from the strictness condition this entails thatfi= 0 whenx_j = 0. Hence ˙x_j≥0 wheneverx_j = 0 since it is a sum of

˙

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Reaction Inference Algorithm

input: ODE systemOover variables for molecular concentrations, partial has pos valtest

1 rewriteOinto additive normal form 2 compute the setT of all terms appearing inO 3 letR:=∅

4 for each non-decomposable termtinT, 1 letr:=∅ ,p:=∅ ,m:=∅

2 for each variablex wheret occurs with integer coefficient c in

˙ x inO,

1 ifc<0 thenr(x) :=−c,

2 ifc>0 thenp(x) :=c,

3 for each variablex such thatr(x) = 0 and partial has pos val(t,x),

1 r(x) := 1,

2 p(x) :=p(x) + 1,

4 for each variablex such thatpartial has pos val(−t,x),

1 m(x) := 1,

5 R:=R∪ {r /m−→^t p},

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Inference of Molecules Eliminated by Invariants

input: ODE systemOover variables{x₁, . . . ,x_s},

1 iteratively replace inOany expression of the form−x+ybyy−x,

2 for each expression of the formk−x−yinOwherekis a numerical constant or a parameter, andxand yare variables,

1 introduce a new variablez with time derivative ˙z =−x˙−y˙, and functional dependency equationz =k−x−y,

2 substitute any occurrence ofk−x−y in O byz,

3 substitute any occurrence ofk+v−x−y inO for any expressionv, by v+z,

4 substitute any occurrence ofk−x+w−y inO for any w, by v+z,

3 for each expressionk−xappearing inOwherekis a constant or a parameter andxa variable, 1 introduce a new variablez with time derivative ˙z =−x˙ and

functional dependency equationz =k−x,

2 substitute any occurrence ofk−x inO byz,

3 substitute any occurrence ofk+v−x in O for any expression v, by z+v,

output: ODE systemOover variables{x₁, . . . ,x_s}and hidden molecule variables{z₁, . . . ,z_k}, together with

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Conclusion

To a large extend, model reductions can be detected in model repositories by checking the existence of SEPI between the reaction hypergraphs

The SEPI existence problem between two graphs is

NP-complete but constraint programming and SAT solvers apply with few timeouts

Automatic creation of model hierarchies (metamodels) pauseitem Works only for well-written SBML models (well-formed reaction systems)

Automatic inference of well-formed reaction systems from ODEs (Biocham load odecommand, used to import MatLab models in SBML)

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Conclusion

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Conclusion

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Conclusion

Automatic inference of well-formed reaction systems from ODEs (Biochamload ode command, used to import MatLab models in SBML)