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Structural Monotonicity and Convergence in Chemical Reaction Networks

Structural Monotonicity and Convergence in Chemical Reaction Networks. By David Angeli University of Firenze. MAPK: signaling pathways. F. FS1. FS2. S0. S1. S2. ES0. ES1. E. Chemical Reaction Networks. E + S0  ES0  E + S1  ES1  E + S2.

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Structural Monotonicity and Convergence in Chemical Reaction Networks

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  1. Structural Monotonicity and Convergence in Chemical Reaction Networks By David Angeli University of Firenze

  2. MAPK: signaling pathways

  3. F FS1 FS2 S0 S1 S2 ES0 ES1 E Chemical Reaction Networks E + S0  ES0  E + S1  ES1  E + S2 F + S2  FS2  F + S1  FS1  F + S0

  4. Chemical Reaction Networks a11S1+a12S2+…+a1nSn  b11S1+b12S2+…+b1nSn a21S1+a22S2+…+a2nSn  b21S1+b22S2+…+b2nSn am1S1+am2S2+…+amnSn bm1S1+bm2S2+…+bmnSn With aij and bij non-negative integers. Reversible reactions: Stoichiometry matrix obtained by the above List by letting: []ij=-aij+bij

  5. F FS1 FS2 S0 S1 S2 ES0 ES1 E Choosing a Modeling Framework Stochastic: Discrete event systems: PETRI NETS Reaction rates: mass-action kinetics Problem : Markov Chain with huge number of states

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

  7. Chemical Master Equation S’ =  R(S) Reaction Rates : • Ri(S) is an analytic function • Ri(S) only depends upon reagents concentration • of the ith reaction • Ri(S) is strictly increasing with respect to its • arguments and zero at zero • Mass-action kinetics: polynomial dependence • (of degree equal to aij ) for ith reaction and jth species

  8. Relating Dynamics and Topology • How does structure affect dynamics? • How robust is the net to parameter variations ? • Does the network converge or oscillate ? Not much is known !!! Feinberg: convergence for weakly reversible Deficiency zero CRN IDEA: Investigate monotonicity of CRN

  9. Monotone dynamical systems • The more of x(0), the more of x(t) for t > 0 • Trivial for scalar systems • Need a precise notion of order POSITIVE CONE Closed set K aK  K for all a>0 K+K  K K  –K = {0} PARTIAL ORDER x  x x  y & y  z  x z x  y & y  x  x=y

  10. Strict notions of order x  y iff x-y K x>y iff x-y  K and xy x>>y iff x-y  int(K) x>>y z>y Not true: x>z x y z

  11. Monotonicity and Strong monotonicity MONOTONE For all x1  x2 and all t  0 : x(t,x1)  x(t,x2) strongly monotone monotone non monotone STRONGLY MONOTONE For all x1 > x2 and all t > 0 : x(t,x1) >> x(t,x2)

  12. Some classical results Hirsch’s Generic Convergence Theorem: For strongly monotone dynamical systems with bounded trajecories almost all solutions converge to equilibria. Smale’s example: Any n-dimensional dynamical systems can be embedded in an n+1 dimensional cooperative system (Chaos or oscillations are possible but non generic ) JiFa’s Theorem: For strongly cooperative systems with a positive first integral, evolving in the positive orthant, all solutions converge to the unique equilibrium Smillie’s Theorem: Global convergence to equilibria for tridiagonal strongly cooperative systems with bounded solutions

  13. Checking monotonicity If K is the positive orthantmonotonicity is equivalent to:  fi /  xj 0 for all i  j (Jacobian is Metzler ) Sufficient condition for strong monotonicity: Jacobian is Metzler and irreducible General orthant: Sign-definite Jacobian and parity condition on negative edges for any loop in the graph associated to the system General cone: x1  x2  f(x1)-f(x2)  TCx1-x2(K)

  14. Structural monotonicity GOAL: Find conditions which guarantee monotonicity of a CRN irrespective of kinetic constants (and reaction rates) PROBLEM: choosing the cone. Once a cone is given check Monotonicity is an easy task. 1st APPROACH: look for cones which are independent of reaction rates. 2nd APPROACH: the cone depends upon kinetic constants 3rd APPROACH: look for change of coordinate which make a CRN cooperative

  15. A factorization approach x=[ S, P, E, F, ES, FP ]’ E + S  ES  E + P P + F  FP  F + S R (x) = [ k1 E.S – k-1 ES, k2 ES, k3 F.P-k-3 FP, k4 FP ] Stoichiometry matrix 1 2 x’=  R(x)

  16. Changing variables x = x0 +1 z z’=2 R(x0+1 z ) Easy factorizations: Jacobian Matrix =  . I = I .  2DR/dx 1 Species and Reaction coordinates

  17. SR-GRAPH F F 4 FP 3 FP S P S P 1 ES 2 ES E E S-GRAPH 4 3 1 2 A Graphical criterion R-GRAPH

  18. E-loops and O-loops [Craciun & Feinberg] O-LOOP E-LOOP Even (or Odd ) number of Pair of edges of the same color Along a loop.

  19. Orthant Monotonicity F 4 FP 3 S P 1 ES 2 E • A CRN is Orthant Monotone in • Species coordinates if and only if : • Each reaction is linked to at most two • species • 2. Each loop in the SR-Graph is an E-loop • A CRN is Orthant Monotone in • Reaction coordinates if and only if : • Each species is linked to at most two • reactions • 2. Each loop in the SR-Graph is an E-loop

  20. Open problems 1. No general method or algorithm for choosing a good factorization of  2. No general method or algorithm for selecting a cone and/or a change of variables which makes a system monotone 3. Narrow the search to physically meaningful cones

  21. Convergence in Reaction Coordinates z’=R(x0+z) No Conservation Laws ! Dually to first integrals: Translation invariance of the flow Solutions are not necessarily bounded

  22. Positive translation invariance K K Assume that v>>0 exists so that v=0 Pick arbitrary initial conditions z1 and z2 v [Angeli &Sontag] Boundedness modulo translation implies convergence to a unique equilibrium

  23. Ker[ ]  K = {0} Pass to the quotient system: under the equivalence class z1z2 iff z1-z2Ker[] The quotient system is strongly monotone with respect to A proper positivity cone Moreover, solutions of the quotient systems are always Bounded (quotient space is isomorphic to stoichiometry Class of the CRN in species coordinates) Generic convergence theorem applies !

  24. Strong monotonicity and persistence Strong monotonicity is related to irreducibility of the jacobian matrix Usually a mild condition, in the interior of the positive orthant However, reaction rates are not strictly monotone on the boundary of the positive orthant Persistence: -limit sets do not touch the boundary

  25. 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

  26. 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 Positive First Integrals: 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

  27. Example: single phosphorilation F 4 FP 3 S P 1 ES 2 E 4 3 1 2 E + S  ES  E + P P + F  FP  F + S Global Convergence Unique equilibrium Ker[  ]=[1,1,1,1]’ Strong Monotonicity Minimal Siphons: {E ES} {F,FP} {S, ES, F, FP} PERSISTENCE + IRREDUCIBILITY Conservation Laws: E+ES, F+FP S+ES+F+FP

  28. Conclusions • Need for systematic analysis tools for CRN • Monotonicity in species and reaction coordinates • More general factorizations of stoichiometry matrix • Analysis of persistence through Petri-Nets invariants THANKS to my Coworkers: Eduardo Sontag Patrick De Leenheer

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