1. The Paradox of Chemical Reaction Networks : Robustness in the face of total uncertainty By David Angeli: Imperial College, London University of Florence, Italy

2. Definition of CRN List of Chemical Reactions: The S i for i = 1,2,...,n are the chemical species. The non-negative integers ,  are the stoichiometry coefficients.

3. Example of CRN E + S0  ES0  E + S1  ES1  E + S2 F + S2  FS2  F + S1  FS1  F + S0 S0 F E FS2FS1 ES1ES0 S1S2

4. Discrete Modeling Framework Stochastic: Discrete event systems: PETRI NETS Reaction rates: mass-action kinetics Problem Problem : Markov Chain with huge number of states S0 F E FS2FS1 ES1ES0 S1S2

5. Continuous Modeling Framework Deterministic: Continuous concentrations, ODE models variance is neglegible Large molecule numbers: variance is neglegible

6. Isolated vs. Open systems Thermodynamically isolated systems: Reaction rates derived from a potential. Every reaction is reversible. Steady-states are thermodynamic equilibria: detailed balance Passive circuits analog of CRNs. Entropy acts as a Lyapunov function. Open systems: Some species are ignored: clamped concentrations. Partial stoichiometry. Arbitrary kinetic coefficients. No obvious Lyapunov function. Possibility of “complex” behaviour.

7. Relating Dynamics and Topology structuredynamics How does structure affect dynamics ? robust How robust is the net to parameter variations ? convergeoscillate Does the reaction converge or oscillate ? Qualitative tools: can work regardless of specific parameters values. robustness How to define robustness ? Consistent qualitative behavior regardless of Parameters or kinetics.

8. MAPK random simulation

9. More random simulations

10. What is Persistence Notion introduced in mathematical ecology: Notion introduced in mathematical ecology: non extinction of species non extinction of species For positive systems For positive systems it amounts to: it amounts to: For systems with bounded solutions equivalently:For systems with bounded solutions equivalently:

11. Why persistence ? Possibility of studying reduced order modelsPossibility of studying reduced order models if persistence fails if persistence fails Interest in itself: availability of all species at all timesInterest in itself: availability of all species at all times Technical results which require persistence:Technical results which require persistence: 1. Irreducibility of associated incidence graph (useful to study strong monotonicity properties) (useful to study strong monotonicity properties) 2. Feinberg deficiency zero theorem, gives GAS under persistence under persistence

12. Some Preliminary Considerations Highly uncertain system: Kinetic constants may be unknown Need for purely topological sufficient or necessary conditions for persistence ECOLOGY vs. CHEMISTRY Once a speciesSpecies can trasform disappears it is goneinto one another forever No mass conservation !Conservation laws !

13. S0 F E FS2FS1 ES1ES0 S1S2 Petri Nets Background Bipartite graph: PLACES (round nodes) TRANSITIONS (boxes) Incidence matrix = Stoichiometry matrix = S P-semiflow: non-negative integer row vector v such that v S = 0 v S = 0 T-semiflow: non-negative integer column vector v with S v = 0 S v = 0 Support of v: set of places i (transitions) such that v_i>0

14. Necessary conditions for persistence Let r(x) denote the vector of reaction rateLet r(x) denote the vector of reaction rate We assume that for x>>0, r(x)>>0We assume that for x>>0, r(x)>>0 Under persistence, the average of r(x(t)) isUnder persistence, the average of r(x(t)) is strictly positive and belongs to the kernel of S strictly positive and belongs to the kernel of S Hence, Persistence implies existence of aHence, Persistence implies existence of a T-semiflow whose support coincides with the T-semiflow whose support coincides with the set of all transitions. set of all transitions. This kind of net is called: CONSISTENT This kind of net is called: CONSISTENT

15. Petri Net approach to persistence S0 F E FS2FS1 ES1ES0 S1S2 Assume that x(t n ) approaches The boundary. Let  be the set of i such that x i (t n )  0 Then  is a SIPHON SIPHON: Input transitions Included in Output transitions

16. Structurally non-emptiable siphons structurally non-emptiable A siphon is structurally non-emptiable if it contains the support of a positive conservation law S0 F E FS2FS1 ES1ES0 S1S2 P-semiflows: E+ES0+ES1 F+FS2+FS1 S0+S1+S2+ES0+ES1+FS2+FS1 Minimal Siphons: { E, ES0, ES1 } { F, FS2, FS1 } { S0, S1, S2, ES0, ES1, FS2, FS1 } All siphons are SNEPERSISTENCE All siphons are SNE  PERSISTENCE

17. Network compositions Full MAPK cascade 22 chemical species 7 minimal siphons 7 P-semiflows whose supports coincide whose supports coincide with the minimal siphons with the minimal siphons

18. Hopf’s bifurcations Symbolic linearization: Characteristic polynomial Hurwitz determinant H n-1 = 0 is a necessary condition for Hopf’s bifurcation (n=6).

19. Hurwitz determinant a i are polynomials of degree i in the kinetic parameters det(H 5 ) is a polynomial of degree 15 in the kinetic parameters (12 parameters + 5 concentrations) Number of monomials is unknown Letting all kinetic constants = 1 except for k 1 k 3 k 5 k 7 yields 68.425 monomials all with a + coefficient

20. Remarks This is much stronger than: det(H n-1 ) is positive definite. Purely algebraic and graphic criterion for ruling out Hopf’s bifurcations expected. Notion of negative loop in the presence of conservation laws.

21. Conclusions CRN theory: open problems and challenges At the cross-road of many fields: - dynamical systems - biochemistry - graph theory - linear algebra HAPPY 60 EDUARDO