SBFM’12 Formal model reduction Jérôme Feret Laboratoire d’Informatique de l’École Normale Supérieure INRIA, ÉNS, CNRS 29 March 2012
Overview 1. Context and motivations 2. Handmade ODEs 3. Abstract interpretation framework 4. Kappa 5. Concrete semantics 6. Abstract semantics 7. Conclusion Jérôme Feret 2 29 March 2012
Signalling Pathways Eikuch, 2007 Jérôme Feret 3 29 March 2012
Bridge the gap between. . . dx 1 dt = − k 1 · x 1 · x 2 + k − 1 · x 3 dx 2 dt = − k 1 · x 1 · x 2 + k − 1 · x 3 dx 3 dt = k 1 · x 1 · x 2 − k − 1 · x 3 + 2 · k 2 · x 3 · x 3 − k − 2 · x 4 dx 4 3 − k 2 · x 4 + v 4 · x 5 dt = k 2 · x 2 p 4 + x 5 − k 3 · x 4 − k − 3 · x 5 dx 5 dt = · · · . . . dx n dt = − k 1 · x 1 · c 2 + k − 1 · x 3 Oda, Matsuoka, Funahashi, Kitano, Molecular Systems Biology, 2005 Jérôme Feret 4 29 March 2012
Rule-based models E G dx 1 dt = − k 1 · x 1 · x 2 + k − 1 · x 3 1 / 2 1 b y 1 dx 2 r z 1 1 / 2 dt = − k 1 · x 1 · x 2 + k − 1 · x 3 1 / 3 a dx 3 dt = k 1 · x 1 · x 2 − k − 1 · x 3 + 2 · k 2 · x 3 · x 3 − k − 2 · x 4 1 / 2 l R 1 / 3 1 d x z 2 y 2 1 / 2 dx 4 3 − k 2 · x 4 + v 4 · x 5 dt = k 2 · x 2 p 4 + x 5 − k 3 · x 4 − k − 3 · x 5 1 / 3 Y 68 r dx 5 dt = · · · So Y 7 1 y 3 z 3 1 / 2 Y 48 . . pi . 1 / 2 Sh dx n dt = − k 1 · x 1 · c 2 + k − 1 · x 3 Interaction ODEs CTMC map Jérôme Feret 5 29 March 2012
Complexity walls Jérôme Feret 6 29 March 2012
A breach in the wall(s) ? Jérôme Feret 7 29 March 2012
Overview 1. Context and motivations 2. Handmade ODEs (a) a system with a switch 3. Abstract interpretation framework 4. Kappa 5. Concrete semantics 6. Abstract semantics 7. Conclusion Jérôme Feret 8 29 March 2012
A system with a switch Jérôme Feret 9 29 March 2012
A system with a switch ( u , u , u ) − → ( u , p , u ) k c ( u , p , u ) − → ( p , p , u ) k l ( u , p , p ) − → ( p , p , p ) k l ( u , p , u ) − → ( u , p , p ) k r ( p , p , u ) − → ( p , p , p ) k r Jérôme Feret 9 29 March 2012
A system with a switch ( u , u , u ) − → ( u , p , u ) k c ( u , p , u ) − → ( p , p , u ) k l ( u , p , p ) − → ( p , p , p ) k l ( u , p , u ) − → ( u , p , p ) k r ( p , p , u ) − → ( p , p , p ) k r d [( u , u , u )] = − k c · [( u , u , u )] dt d [( u , p , u )] = − k l · [( u , p , u )] + k c · [( u , u , u )] − k r · [( u , p , u )] dt d [( u , p , p )] = − k l · [( u , p , p )] + k r · [( u , p , u )] dt d [( p , p , u )] = k l · [( u , p , u )] − k r · [( p , p , u )] dt d [( p , p , p )] = k l · [( u , p , p )] + k r · [( p , p , u )] dt Jérôme Feret 9 29 March 2012
A system with a switch ( u , u , u ) − → ( u , p , u ) k c ( u , p , u ) − → ( p , p , u ) k l ( u , p , p ) − → ( p , p , p ) k l ( u , p , u ) − → ( u , p , p ) k r ( p , p , u ) − → ( p , p , p ) k r d [( u , u , u )] = − k c · [( u , u , u )] dt d [( u , p , u )] = − k l · [( u , p , u )] + k c · [( u , u , u )] − k r · [( u , p , u )] dt d [( u , p , p )] = − k l · [( u , p , p )] + k r · [( u , p , u )] dt d [( p , p , u )] = k l · [( u , p , u )] − k r · [( p , p , u )] dt d [( p , p , p )] = k l · [( u , p , p )] + k r · [( p , p , u )] dt Jérôme Feret 9 29 March 2012
Two subsystems Jérôme Feret 10 29 March 2012
Two subsystems Jérôme Feret 10 29 March 2012
Two subsystems [( u , u , u )] = [( u , u , u )] [( u , u , u )] = [( u , u , u )] ∆ ∆ [( u , p , ? )] = [( u , p , u )] + [( u , p , p )] [( ? , p , u )] = [( u , p , u )] + [( p , p , u )] ∆ ∆ [( p , p , ? )] = [( p , p , u )] + [( p , p , p )] [( ? , p , p )] = [( u , p , p )] + [( p , p , p )] d [( u , u , u )] d [( u , u , u )] = − k c · [( u , u , u )] = − k c · [( u , u , u )] dt dt d [( u , p , ? )] d [( ? , p , u )] = − k l · [( u , p , ? )] + k c · [( u , u , u )] = − k r · [( ? , p , u )] + k c · [( u , u , u )] dt dt d [( p , p , ? )] d [( ? , p , p )] = k l · [( u , p , ? )] = k r · [( ? , p , u )] dt dt Jérôme Feret 10 29 March 2012
Dependence index The states of left site and right site would be independent if, and only if: [( ? , p , u )] + [( ? , p , p )] = [( p , p , p )] [( ? , p , p )] [( p , p , ? )] . Thus we define the dependence index as follows: ∆ = [( p , p , p )] · ([( ? , p , u )] + [( ? , p , p )]) − [( ? , p , p )] · [( p , p , ? )] . X We have: dX k l + k r � dt = − X · � + k c · [( p , p , p )] · [( u , u , u )] . So the property ( X = 0 ) is not an invariant. Jérôme Feret 11 29 March 2012
Conclusion We can use the absence of flow of information to cut chemical species into self-consistent fragments of chemical species: − some information is abstracted away: we cannot recover the concentration of any species; + flow of information is easy to abstract; We are going to track the correlations that are read by the system. Jérôme Feret 12 29 March 2012
Overview 1. Context and motivations 2. Handmade ODEs 3. Abstract interpretation framework (a) Concrete semantics (b) Abstraction 4. Kappa 5. Concrete semantics 6. Abstract semantics 7. Conclusion Jérôme Feret 13 29 March 2012
Differential semantics Let V , be a finite set of variables ; and F , be a C ∞ mapping from V → R + into V → R , as for instance, ∆ = { [( u , u , u )] , [( u , p , u )] , [( p , p , u )] , [( u , p , p )] , [( p , p , p )] } , • V [( u , u , u )] � → − k c · ρ ([( u , u , u )]) [( u , p , u )] � → − k l · ρ ([( u , p , u )]) + k c · ρ ([( u , u , u )]) − k r · ρ ([( u , p , u )]) ∆ [( u , p , p )] � → − k l · ρ ([( u , p , p )]) + k r · ρ ([( u , p , u )]) • F ( ρ ) = [( p , p , u )] � → k l · ρ ([( u , p , u )]) − k r · ρ ([( p , p , u )]) [( p , p , p )] � → k l · ρ ([( u , p , p )]) + k r · ρ ([( p , p , u )]) . The differential semantics maps each initial state X 0 ∈ V → R + to the maximal solution X X 0 ∈ [ 0, T max X 0 [ → ( V → R + ) which satisfies: � T X X 0 ( T ) = X 0 + F ( X X 0 ( t )) · dt. t = 0 Jérôme Feret 14 29 March 2012
Overview 1. Context and motivations 2. Handmade ODEs 3. Abstract interpretation framework (a) Concrete semantics (b) Abstraction 4. Kappa 5. Concrete semantics 6. Abstract semantics 7. Conclusion Jérôme Feret 15 29 March 2012
Abstraction An abstraction ( V ♯ , ψ, F ♯ ) is given by: • V ♯ : a finite set of observables, • ψ : a mapping from V → R into V ♯ → R , • F ♯ : a C ∞ mapping from V ♯ → R + into V ♯ → R ; such that: • ψ is linear with positive coefficients, • the following diagram commutes: F ( V → R + ) − → ( V → R ) ψ � ψ ℓ ∗ ℓ ∗ � F ♯ ( V ♯ → R + ) → ( V ♯ → R ) − i.e. ψ ◦ F = F ♯ ◦ ψ . Jérôme Feret 16 29 March 2012
Abstraction example ∆ = { [( u , u , u )] , [( u , p , u )] , [( p , p , u )] , [( u , p , p )] , [( p , p , p )] } • V [( u , u , u )] � → − k c · ρ ([( u , u , u )]) [( u , p , u )] � → − k l · ρ ([( u , p , u )]) + k c · ρ ([( u , u , u )]) − k r · ρ ([( u , p , u )]) ∆ • F ( ρ ) = [( u , p , p )] � → − k l · ρ ([( u , p , p )]) + k r · ρ ([( u , p , u )]) · · · • V ♯ ∆ = { [( u , u , u )] , [( ? , p , u )] , [( ? , p , p )] , [( u , p , ? )] , [( p , p , ? )] } [( u , u , u )] � → ρ ([( u , u , u )]) [( ? , p , u )] � → ρ ([( u , p , u )]) + ρ ([( p , p , u )]) ∆ • ψ ( ρ ) = [( ? , p , p )] � → ρ ([( u , p , p )]) + ρ ([( p , p , p )]) . . . [( u , u , u )] � → − k c · ρ ♯ ([( u , u , u )]) [( ? , p , u )] � → − k r · ρ ♯ ([( ? , p , u )]) + k c · ρ ♯ ([( u , u , u )]) ∆ • F ♯ ( ρ ♯ ) = [( ? , p , p )] � → k r · ρ ♯ ([( ? , p , u )]) . . . (Completeness can be checked analytically.) Jérôme Feret 17 29 March 2012
Abstract differential semantics Let ( V , F ) be a concrete system. Let ( V ♯ , ψ, F ♯ ) be an abstraction of the concrete system ( V , F ) . Let X 0 ∈ V → R + be an initial (concrete) state. We know that the following system: � T F ♯ � Y ψ ( X 0 ) ( t ) � Y ψ ( X 0 ) ( T ) = ψ ( X 0 ) + · dt t = 0 has a unique maximal solution Y ψ ( X 0 ) such that Y ψ ( X 0 ) = ψ ( X 0 ) . Theorem 1 Moreover, this solution is the projection of the maximal solution X X 0 of the system � T F � X X 0 ( t ) � X X 0 ( T ) = X 0 + · dt. t = 0 (i.e. Y ψ ( X 0 ) = ψ ( X X 0 ) ) Jérôme Feret 18 29 March 2012
Overview 1. Context and motivations 2. Handmade ODEs 3. Abstract interpretation framework 4. Kappa 5. Concrete semantics 6. Abstract semantics 7. Conclusion Jérôme Feret 19 29 March 2012
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