A Biochemical Calculus Based on Strategic Graph Rewriting Oana Andrei 1 ene Kirchner 2 H´ el` 1 INRIA Nancy - Grand-Est & LORIA 2 INRIA Bordeaux - Sud-Ouest France Algebraic Biology’08 Short communication O. Andrei & H. Kirchner (INRIA) A Biochemical Calculus... Algebraic Biology’08 1 / 13
Biological Intuition 1 Extending the chemical model 2 A high-level biochemical calculus 3 Conclusions and Perspectives 4 O. Andrei & H. Kirchner (INRIA) A Biochemical Calculus... Algebraic Biology’08 2 / 13
Biological Intuition Graphs are suitable for describing the structure complex systems and graph transformations for modeling their dynamic evolution. We are interested in a particular representation of molecular complexes as graphs and of reaction patterns as graph transformations [DL04]: the behavior of a protein is given by its functional domains / sites on the surface two proteins can interact by binding or changing the states of sites bound proteins form complexes that have a graph-like structure membranes can also form molecular complexes, called tissues O. Andrei & H. Kirchner (INRIA) A Biochemical Calculus... Algebraic Biology’08 3 / 13
Biological Intuition Graphs are suitable for describing the structure complex systems and graph transformations for modeling their dynamic evolution. We are interested in a particular representation of molecular complexes as graphs and of reaction patterns as graph transformations [DL04]: the behavior of a protein is given by its functional domains / sites on the surface two proteins can interact by binding or changing the states of sites bound proteins form complexes that have a graph-like structure membranes can also form molecular complexes, called tissues Port graphs are graphs with multiple edges and loops [AK08], where nodes have explicit connection points, called ports the edges attach to ports of nodes. O. Andrei & H. Kirchner (INRIA) A Biochemical Calculus... Algebraic Biology’08 3 / 13
Biological Intuition Molecular graph Port graph protein node site port with maximum degree 1 bond edge O. Andrei & H. Kirchner (INRIA) A Biochemical Calculus... Algebraic Biology’08 4 / 13
Biological Intuition Molecular graph Port graph protein node site port with maximum degree 1 bond edge transformation of molecular complexes � molecular graph rewrite rule � port graph rewrite rule � port graph O. Andrei & H. Kirchner (INRIA) A Biochemical Calculus... Algebraic Biology’08 4 / 13
Biological Intuition Molecular graph Port graph protein node site port with maximum degree 1 bond edge transformation of molecular complexes � molecular graph rewrite rule � port graph rewrite rule � port graph Port graphs represent a unifying structure for representing both molecular complexes and the reaction patterns between them. O. Andrei & H. Kirchner (INRIA) A Biochemical Calculus... Algebraic Biology’08 4 / 13
Example A molecular graph G representing the initial state of the system modeling a fragment of the EGFR signaling cascade, and a subsequent state modeled by G ′ : 1.2:EGF.EGF 3.4:EGF.EGF 1:EGF 2:EGF 3:EGF 4:EGF 1 1 2 2 2 2 2 2 1 2 2 1 1 1 1 1 2 2 2 2 1 1 5:EGFR 1 1 6:EGFR 5:EGFR 6:EGFR 3 3 3 3 4 4 4 4 1 2 1 2 7:SHC 7:SHC G G' Some reaction patterns: i:EGF k:EGF.EGF k:EGF.EGF i:EGF.EGF i:EGF.EGF 2 2 2 1 2 2 2 2 1 r1 r2 2 r3 2 1 1 1 1 4 1 4 1 1 2 1 4 2 2 4 4 2 2 4 i.j:EGF.EGF j:EGFR j:EGFR i:EGFR j:EGFR i:EGFR j:EGFR j:EGF O. Andrei & H. Kirchner (INRIA) A Biochemical Calculus... Algebraic Biology’08 5 / 13
Extending the chemical model The chemical model of computation – the Γ language [BM86]: a chemical solution where molecules interact freely according to (conditional) reaction rules multisets for chemical solutions multiset rewrite rules for reaction rules extensions: ◮ the CHemical Abstract Machine (CHAM) [BB92], ◮ the γ -calculus and HOCL [BFR06] O. Andrei & H. Kirchner (INRIA) A Biochemical Calculus... Algebraic Biology’08 6 / 13
A high-level biochemical calculus A rewriting calculus [CK01] for molecular graphs with higher-order capabilities: first citizens: molecular graphs, abstractions (molecular graph rewrite rules), and rule application. abstractions may introduce other abstractions (the right-hand side of an abstraction may contain other abstractions) control mechanisms encoded as entities of the calculus (as strategies) extends the chemical model (Γ, CHAM, the γ -calculus) with high-level features by considering the structure of port graphs for data and for the computation rules. O. Andrei & H. Kirchner (INRIA) A Biochemical Calculus... Algebraic Biology’08 7 / 13
� � � � � � Syntax M the class of molecular graphs Abstractions: A ::= | ⇒ ⇒ � � � � ... ... ... ... M M M A + M Objects of the calculus: G ::= X | M | A | G G | ε State or simple world: V ::= Y | [ G ] O. Andrei & H. Kirchner (INRIA) A Biochemical Calculus... Algebraic Biology’08 8 / 13
Syntax Box-based representation of a simple world consisting of the abstractions A 1 , A 2 , and A 3 , and the molecular graphs M 1 and M 2 : A 3 M 2 A 1 M 1 A 2 for [ A 2 M 1 A 1 A 3 M 2 ]. O. Andrei & H. Kirchner (INRIA) A Biochemical Calculus... Algebraic Biology’08 9 / 13
Reduction Semantics ( Heating ) [ X A M ] �− → [ X A @ M ] (1) ( Application / Success ) A @ M �− → G if M → A G (2) ( Application / Fail ) A @ M �− → A M otherwise (3) O. Andrei & H. Kirchner (INRIA) A Biochemical Calculus... Algebraic Biology’08 10 / 13
Reduction Semantics ( Heating ) [ X A M ] �− → [ X A @ M ] (1) ( Application / Success ) A @ M �− → G if M → A G (2) ( Application / Fail ) A @ M �− → A M otherwise (3) By introducing an explicit object (node) for failure, stk, we gain in expressivity: ( Application / Fail ′ ) A @ M �− → stk if M is A − irreducible (4) O. Andrei & H. Kirchner (INRIA) A Biochemical Calculus... Algebraic Biology’08 10 / 13
Reduction Semantics ( Heating ) [ X A M ] �− → [ X A @ M ] (1) ( Application / Success ) A @ M �− → G if M → A G (2) ( Application / Fail ) A @ M �− → A M otherwise (3) By introducing an explicit object (node) for failure, stk, we gain in expressivity: ( Application / Fail ′ ) A @ M �− → stk if M is A − irreducible (4) Possible extension: consider a structure of all possible results for application O. Andrei & H. Kirchner (INRIA) A Biochemical Calculus... Algebraic Biology’08 10 / 13
Strategies Instead of this highly non-deterministic (and possibly non-terminating) behaviour of abstraction application, one may want to introduce some control to compose or choose the abstractions to apply, possibly exploiting failure information. Strategies as abstractions: � id X ⇒ X � X ⇒ stk fail � seq ( S 1 , S 2 ) X ⇒ S 2 @( S 1 @ X ) � first ( S 1 , S 2 ) X ⇒ ( S 1 @ X ) (stk ⇒ ( S 2 @ X ))@( S 1 @ X ) � try ( S ) first ( S , id) � repeat ( S ) try ( seq ( S , repeat ( S ))) O. Andrei & H. Kirchner (INRIA) A Biochemical Calculus... Algebraic Biology’08 11 / 13
Improving the calculus using strategies 1 Failure catching: if S @ M reduces to the failure construct stk, then the strategy try (stk ⇒ S M ) restores the initial entities subjects to reduction. (Heating’) [ X S M ] �− → [ X seq ( S , try (stk ⇒ S M ))@ M ] 2 Persistent strategies: S ! applies S to an object and, if successful, replicates itself. S ! � seq ( S , first (stk ⇒ stk , Y ⇒ Y S !)) O. Andrei & H. Kirchner (INRIA) A Biochemical Calculus... Algebraic Biology’08 12 / 13
Conclusions and Perspectives Conclusions: we defined a higher-order calculus with high-level capabilities for modeling interactions in molecular complexes. from the verification point of view we have: ◮ classical rewriting techniques for checking properties of the modeled systems: verification of confluence, termination for port graph rewriting (under strategies) ◮ ideas for runtime verification of properties in such systems O. Andrei & H. Kirchner (INRIA) A Biochemical Calculus... Algebraic Biology’08 13 / 13
Conclusions and Perspectives Conclusions: we defined a higher-order calculus with high-level capabilities for modeling interactions in molecular complexes. from the verification point of view we have: ◮ classical rewriting techniques for checking properties of the modeled systems: verification of confluence, termination for port graph rewriting (under strategies) ◮ ideas for runtime verification of properties in such systems Perspectives: verification interactions between abstractions control mechanisms O. Andrei & H. Kirchner (INRIA) A Biochemical Calculus... Algebraic Biology’08 13 / 13
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