1. Chemical Reaction Networks :
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. Example of CRN E + S0  ES0  E + S1  ES1  E + S2
F + S2  FS2  F + S1  FS1  F + S0 S0 F E
FS2
FS1
ES1
ES0 S1 S2

4. Discrete Modeling Framework
Stochastic:
Discrete event systems: PETRI NETS
Reaction rates: mass-action kinetics S0 F E
FS2
FS1
ES1
ES0 S1 S2
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 Bipartite graph: PLACES (round nodes)F E
FS2
FS1
ES1
ES0 S1 S2
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. Petri Net approach to persistence
SIPHON:
Input transitions
Included in
Output transitions S0 F E
FS2
FS1
ES1
ES0 S1 S2
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. Structurally non-emptiable siphons
A siphon is structurally non-emptiable if it contains
the support of a positive conservation law S0 F E
FS2
FS1
ES1
ES0 S1 S2
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 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. HAPPY 60 EDUARDO Conclusions CRN theory: open problems and challenges
At the cross-road of many fields:
- dynamical systems
- biochemistry
- graph theory
- linear algebra
HAPPY 60 EDUARDO