Rules with Constraints

1. Rules withConstraints Mathias John, Cédric Lhoussaine, Joachim Niehren, Cristian Versari ESOP 2011

2. Stochastic Modeling • chemical reactions: • initial solution: k1 2* Mon 1* Dim Mon Mon Dim k2 2* 4* 0* Mon Mon Mon 0* 1* 2* Dim Dim Dim • ctmc: r1 r2 r3 r4

3. Combinatorial Explosion of Reaction Sets • [Goldstein04], [Hlavacek06] • example: monomer with single modification side for phosphorylation and methylation • sides act mostly independently • All these reactions do the very same thing k k Dim_P_P Mon_P Dim_P Mon k k Dim_P Dim Dim_M_P Dim_M [Hlavacek06] Rules for Modeling Signal-Transduction Systems, Science Signaling. [Goldstein04] MathematicandComputational Models of Immune-Receptor Signalling, Nature Reviews, Immunolgy.

4. View on Chemical Species • example: monomer with single modification side for phosphorylation and methylation M N P F F F binding site to other monomer Mon Mon Mon Mon modification site Mon_P Mon_M Mon = = = F – free, N - not modified, P – phosphorylated, M - methylated

5. View on Chemical Solutions bond denoted by shared identifier b bond b b N N N N = = Dim Mon Mon Mon Mon = Dim_M_P … M P b b Mon Mon

6. Reaction with Structured Species • accessing structures with variables --> decrease reaction number x – variable, i.e. for all values k N P x x Mon_P Mon Mon Mon k Dim_P Dim k k Dim_P_P Dim_P k Dim_M_P Dim_M

7. Rules with constraints

8. Examples for Using Constraints • occasional dependence of binding sites • increased compactness • in combination with rate expressions x2 x1 b b x2 k F x1 F Mon Mon Mon Mon x N N x F F F F x1 != M & x2 != M Mon Mon Mon Mon k x in {M,P} if x=M then kM else kP x in {M,P}

9. Space kd neighbors(p1, p2) x1 x2 x1 x2 x x F F F F F F Mon Mon Mon Mon Mon Mon p2 p1 p1 p1 p2 p2 p1 p2 p1 p2 f(kd,p1, p2)

10. Expressing the Pi-Calculus • all pi-calculus models can now be formulated in rules • idea: • given a set of process definitions (pi-calculus) • turn interaction capabilities into rules z x y x A C A B A(x) = x!().0 + x!(y).0 B(y) = y?().B’() C(z) = z?().C’() k k B’ C’ x = y x = z k = rate constant of x

11. Semantics: Enumerating Reactions rule set current state (solution) k 2* 2* P M N N P N N N F x F F x F F x Mon Mon Mon Mon Mon Mon Mon Mon one match: replace variables by values (F->x) k replace variables also in constraints & rate expressions then solve / evaluate bond creation: introduce names not present in solution

12. The Price: Analysis Complexity • size of CTMC -> simulation only • complexity of single simulation step mn, where n maixmal number of reactants • our idea: use constraint solvers • good news: element constraints • first prototype implementation: promising results