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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 Si for i = 1,2,...,n are the chemical species. The non-negative integers , are the stoichiometry coefficients.
3. F FS1 FS2 S0 S1 S2 ES0 ES1 E Example of CRN E + S0  ES0  E + S1  ES1  E + S2 F + S2  FS2  F + S1  FS1  F + S0
4. F FS1 FS2 S0 S1 S2 ES0 ES1 E Discrete Modeling Framework Stochastic: Discrete event systems: PETRI NETS Reaction rates: mass-action kinetics Problem : Markov Chain with huge number of states
5. Continuous Modeling Framework Deterministic: Continuous concentrations, ODE models 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 How does structure affect dynamics? How robust is the net to parameter variations ? Does the reaction converge or oscillate? Qualitative tools: can work regardless of specific parameters values. 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: non extinction of species For positive systems it amounts to: For systems with bounded solutions equivalently:
11. Petri Nets Background F FS1 FS2 S0 S1 S2 ES0 ES1 E Bipartite graph: PLACES (round nodes) TRANSITIONS (boxes) P-semiflow: non-negative integer row vector v such that v S = 0 T-semiflow: non-negative integer column vector v with S v = 0 Support of v: set of places i (transitions) such that v_i>0 Incidence matrix = Stoichiometry matrix = S
12. Necessary conditions for persistence Let r(x) denote the vector of reaction rate We assume that for x>>0, r(x)>>0 Under persistence, the average of r(x(t)) is strictly positive and belongs to the kernel of S Hence, Persistence implies existence of a T-semiflow whose support coincides with the set of all transitions. This kind of net is called: CONSISTENT
13. F FS1 FS2 S0 S1 S2 ES0 ES1 E Petri Net approach to persistence SIPHON: Input transitions Included in Output transitions Assume that x(tn) approaches The boundary. Let S be the set of i such that xi(tn) 0 Then S is a SIPHON
14. F FS1 FS2 S0 S1 S2 ES0 ES1 E Structurally non-emptiable siphons A siphon is structurally non-emptiable if it contains the support of a positive conservation law 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 SNE PERSISTENCE
15. Network compositions Full MAPK cascade 22 chemical species 7 minimal siphons 7 P-semiflows whose supports coincide with the minimal siphons
16. Hopf’s bifurcations Symbolic linearization: Characteristic polynomial Hurwitz determinant Hn-1 = 0 is a necessary condition for Hopf’s bifurcation (n=6).
17. Hurwitz determinant ai are polynomials of degree i in the kinetic parameters det(H5) 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 k1 k3 k5 k7 yields 68.425 monomials all with a + coefficient
18. Remarks This is much stronger than: det(Hn-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.
19. Conclusions CRN theory: open problems and challenges At the cross-road of many fields: - dynamical systems - biochemistry - graph theory - linear algebra HAPPY 60 EDUARDO