1. Invariant grids: method of complexity reduction in reaction networks Andrei Zinovyev Institut Curie, Paris Institut des Hautes Études Scientifiques

2. 2 - European conference on complex systems, Paris, November 2005 - Stoichiometric equations  s1 A 1 + … +  sn A n   s1 A 1 + … +  sn A n n – number of species, s – number of reactions c1 c2 cncn

3. 3 - European conference on complex systems, Paris, November 2005 - What is Model Reduction ? 1 Shorten list of species – eliminate some – create “integrated” components 2 Shorten list of reactions – eliminate some – “freeze” fast reactions 3 Decompose motion into fast and slow

4. 4 - European conference on complex systems, Paris, November 2005 - Approaching steady state

5. 5 - European conference on complex systems, Paris, November 2005 - Positively Invariant Manifold Steady state fast motion slow motion Why Invariant? once the point on the manifold, the trajectory will stay on it until the equilibrium 

6. 6 - European conference on complex systems, Paris, November 2005 - Why for do we need invariant manifold? Model reduction: “Macroscopic” system description x  R N – detailed description y  R m – “macroscopic” description. m<<N

7. 7 - European conference on complex systems, Paris, November 2005 - Why for do we need invariant manifold? Dynamics visualization

8. 8 - European conference on complex systems, Paris, November 2005 - Other useful non-invariant manifolds Quasy steady-state Fast variables are steady Quasi-equilibrium Manifolds maximizing entropy Intrinsic low-dimensional manifold Decomposition of Jacobian fields

9. 9 - European conference on complex systems, Paris, November 2005 - Projector P c on (some) manifold induces new (reduced) dynamics J induced dynamics P c J tangent space  = (1-P c )J - invariance defect   xx

10. 10 - European conference on complex systems, Paris, November 2005 - Quasi-equilibrium manifold is not necessarily invariant entropy S  max macroscopic (reduced) variables

11. 11 - European conference on complex systems, Paris, November 2005 - Class of dissipative systems c G c eq Lyapunov function

12. 12 - European conference on complex systems, Paris, November 2005 - Thermodynamic projector J P c J The induced dynamics is dissipative only if

13. 13 - European conference on complex systems, Paris, November 2005 - Correction of invariance defect 0.2 0.4 0.6 0.8 1.0 0.05 0.100.15 0.20 C1C1 C3C3 equilibrium invariant manifold initial approximation corrections (1-P c )J = 0 invariance equation Newton iterations

14. 14 - European conference on complex systems, Paris, November 2005 - Invariant grid EQUILIBRIUM tangent space J invariance defect is corrected for every node independently

15. 15 - European conference on complex systems, Paris, November 2005 - Invariant grid J TxTx x 0 +ker P C

16. 16 - European conference on complex systems, Paris, November 2005 - Growing Invariant Flag Phase space

17. 17 - European conference on complex systems, Paris, November 2005 - Entropic scalar product 0 1 2 -2 equilibrium natural parameter entropy

18. 18 - European conference on complex systems, Paris, November 2005 - Hydrogen burning model reaction 1: H 2  2H 2: O 2  2O 3: H 2 O  H + OH 4: H 2 + O  H + OH 5: O 2 + H  O + OH 6: H 2 + O  H 2 O Conservation laws: 2c H2 +2c H2O +c H +c OH = b H 2c O2 +c H2O +c O +c OH = b O H 2 H O 2 H2OH2O O OH

19. 19 - European conference on complex systems, Paris, November 2005 - One-dimensional dynamics equilibrium

20. 20 - European conference on complex systems, Paris, November 2005 - Separation of times is the eigen value of symmetrised matrix

21. 21 - European conference on complex systems, Paris, November 2005 - Two-dimensional dynamics

22. 22 - European conference on complex systems, Paris, November 2005 - Visualizing functions: concentration of H “Fast” coordinate

23. 23 - European conference on complex systems, Paris, November 2005 - Visualizing functions: concentration of H 2 “Slow” coordinate

24. 24 - European conference on complex systems, Paris, November 2005 - Visualizing functions: concentration of OH

25. 25 - European conference on complex systems, Paris, November 2005 - Visualizing functions: Entropy and entropy production Entropy Entropy production

26. 26 - European conference on complex systems, Paris, November 2005 - Visualizing functions: Separation of relaxation times 2 / 1 3 / 2

27. 27 - European conference on complex systems, Paris, November 2005 - Open system as “closed system in a flow” closed system flow Entropy does not increase everywhere Non-uniqueness of stationary states, auto-oscillations, etc. inertial manifold often exists

28. 28 - European conference on complex systems, Paris, November 2005 - Zero-order approximation Construct the invariant manifold  for 

29. 29 - European conference on complex systems, Paris, November 2005 - First-order approximation New invariance equation: “Fast” and “slow” flow:  ’’

30. 30 - European conference on complex systems, Paris, November 2005 - Conclusions Invariant grids: constructive method for chemical kinetics class of dissipative systems extension to open systems Use of thermodynamics: metrics in the phase space unique thermodynamic projector Possibility to visualize and explore system dynamics globally

31. 31 - European conference on complex systems, Paris, November 2005 - Papers Gorban A, Karlin I, Zinovyev A. Constructive Methods of Invariant Manifolds for Kinetic Problems 2004. Physics Reports 396, pp.197-403. Gorban A, Karlin I, Zinovyev A. Invariant Grids for Reaction Kinetics 2004. Physica A, V.333, pp.106-154

32. 32 - European conference on complex systems, Paris, November 2005 - People Doctor Iliya Karlin ETH, Zurich Professor Alexander Gorban University of Leicester, UK