Mathematical and computational methods for understanding the dynamics of biochemical networks

Transcription

1 Mathematical and computational methods for understanding the dynamics of biochemical networks Gheorghe Craciun Department of Mathematics and Department of Biomolecular Chemistry University of Wisconsin - Madison

2 Outline Examples Bistable biochemical networks The global attractor conjecture

3 Hanahan and Weinberg, The Hallmarks of Cancer, Cell, 2000.

4 Metabolic Pathways Roche Applied Science

5 Outline Examples Bistable biochemical networks The global attractor conjecture

6 Bistability and Biochemical Switching J. Bailey, Complex biology with no parameters, Nature, 2001: does a cell generally operate at one steady state, or can a cell switch under certain circumstances from one steady state to another?

7 Bistability and hysteresis in the embrionic cell cycle of Xenopus oocites J. Pomerening, E. Sontag, J. Ferrell, Building a cell cycle oscillator: hysteresis and

8 Bistabili ty and Biochemic al Oscillati ons J. Pomerening, E. Sontag, J. Ferrell, Building a cell cycle oscillator: hysteresis and bistability in the activation of Cdc2,

9 Given a reaction network, how can we decide if it has the capacity for bistability?

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16 Bistabili ty cp cs cs2 0

17 Vocabulary:

18 Vocabulary:

19 The Species-Reaction Graph

20 Another SpeciesReaction Graph

21 Theorem: Consider a reaction network for which the species-reaction graph has no cycles. Then, regardless of parameter values, the corresponding system of massaction differential equations does not have the capacity for bistability.

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23 Important remark: Enzyme catalysis generates cycles in the species-reaction graph.

24 More vocabulary: classifying cycle o-cycles, e-cycles, and s-cycles

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26 o-cycles and e-cycles An o-cycle is a cycle containing an odd An e-cycle is a cycle containing an even number of c-pairs. number of c-pairs. e-cycle

27 Theorem: Consider a reaction network for which each cycle in the species-reaction graph is an o-cycle. Then, regardless of parameter values, the corresponding system of mass-action differential equations does not have the capacity for bistability.

28 scycles

29 scycles

30 scycles

31 Two cycles split a c-pair if their combined arcs contain the c-pair and if at least one of the cycles contains only one arc of the c-pair.

32 Theorem: Consider a reaction network for which the species-reaction graph has the following properties: (i) Each cycle is an o-cycle or an scycle (or both). (ii) No c-pair is split by two e-cycles. Then, regardless of parameter values, the corresponding system of mass-action differential equations does not have the capacity for bistability.

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35 Bistability impossible, regardless of parameter value

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37 s-cycle Bistability impossible, regardless of parameter value

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39 s-cycle Bistability impossible, regardless of parameter values

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41 Theorem: Consider a reaction network for which the species-reaction graph has the following properties: (i) Each cycle is an o-cycle or an scycle (or both). (ii) No c-pair is split by two e-cycles. Then, regardless of parameter values, the corresponding system of mass-action differential equations does not have the capacity for bistability.

42 e-cycle

43 ecycle

44 Does have capacity for

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47 No bistability, regardless of parameter values!

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49 Theorem: Consider a reaction network for which the species-reaction graph has the following properties: (i) Each cycle is an o-cycle or an scycle (or both). (ii) No c-pair is split by two e-cycles. Then, regardless of parameter values, the corresponding system of mass-action differential equations does not have the capacity for bistability.

50 e-cycle

51 e-cycle e-cycles 1 and 2 split the red c-pair. Does have capacity for

52 Outline Examples Bistable biochemical networks The global attractor conjecture

53 Mass action kinetics

54 Mass action kinetics

55 Mass action kinetics

56 Mass action kinetics

57 Invariant subspaces c3 c1 c2

58 Invariant subspaces c3 c1 c2

59 Invariant subspaces c3 c1 c2

60 Invariant subspaces c3 c1 c2

61 Deficiency theory (Feinberg, Horn, and Jackson) Definition. The deficiency of a reaction network is n l s, where n = the number of complexes l = the number of linkage classes s = the dimension of the stoichiometric subspace

62 Deficiency zero theory (Feinberg, Horn, and Jackson) Theorem. If a reaction network has deficiency zero, then there exists a unique positive equilibrium in each stoichiometric subspace, and this equilibrium is complex balanced.

63 Deficiency theory (Feinberg, Horn, and Jackson) Definition. The deficiency of a reaction network is n l s, where n = the number of complexes l = the number of linkage classes s = the dimension of the stoichiometric subspace

64 Deficiency zero theory (Feinberg, Horn, and Jackson) Theorem. If a reaction network has deficiency zero, then there exists a unique positive equilibrium in each stoichiometric subspace, and this equilibrium is complex balanced. Theorem. If a reaction network has complex balanced equilibrium, then, for each stoichiometric subspace, there exists a strict Lyapunov function with a minimum at that equilibrium.

65 Complex balanced reaction networks Deficiency zero reaction networks

66 Lyapunov function and boundary equilibria

67 Lyapunov function and boundary equilibria

68 Global Attractor Conjecture Conjecture. If a reaction network has a complex balanced equilibrium, then any trajectory with positive initial condition converges to that equilibrium.

69 Global Attractor Conjecture Conjecture. If a reaction network has a complex balanced equilibrium, then any trajectory with positive initial condition converges to that equilibrium. How do we prove this? No boundary equilibria. (Angeli, De Leenheer, Sontag) Even if there exist boundary equilibria, they are repelling.

70 Global Attractor Conjecture Conjecture. If a reaction network has a complex balanced equilibrium, then any trajectory with positive initial condition converges to that equilibrium. Theorem 1. If a reaction network has a complex balanced equilibrium, then no trajectory with positive initial condition can converge to a vertex of its stoichiometric compatibility class.

71 Global Attractor Conjecture Conjecture. If a reaction network has a complex balanced equilibrium, then any trajectory with positive initial condition converges to that equilibrium. Theorem 2. If a reaction network has a detailed balanced equilibrium, then no trajectory with positive initial condition can converge to a bounded facet of its stoichiometric compatibility class.

72 Complex balanced reaction networks Detail balanced reaction networks Deficiency zero reaction networks

73 Global Attractor Conjecture Conjecture. If a reaction network has a complex balanced equilibrium, then any trajectory with positive initial condition converges to that equilibrium. Theorem 3. If a reaction network has stoichiometric subspace of dimension two, then any trajectory with positive initial condition converges to that equilibrium.

74 Global Attractor Conjecture Conjecture. If a reaction network has a complex balanced equilibrium, then any trajectory with positive initial condition converges to that equilibrium. Theorem 4. If a reaction network has zero repelling subnetworks, then no trajectory with positive initial condition can converge to a boundary point.

75 Complex balanced reaction networks Detail balanced reaction networks Deficiency zero reaction networks Reaction networks that have zero repelling subnetworks

76 Summary Conjecture: the complex balancing equilibrium is global attractor. The complex balancing equilibrium always exists if the deficiency is zero. Proof, under additional assumptions: either the rate constants satisfy additional conditions, or the reaction network has special structure.

77 Collaborato Martin Feinberg, Department of Chemical Engineering and Department of rs Mathematics, Ohio State University. David Anderson, Department of Mathematics, University of Wisconsin Madison. Fedor Nazarov, Department of Mathematics, University of Wisconsin Madison. Bernd Sturmfels, Department of Mathematics, UC Berkeley. Anne Shiu, Department of Mathematics, Department of Mathematics, UC Berkeley. Alicia Dickenstein, Department of Mathematics, University of Buenos Aires. Support: NSF and DOE BACTER Institute.

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80 Global Attractor Conjecture Conjecture. If a reaction network has a complex balanced equilibrium, then any trajectory with positive initial condition converges to that equilibrium.

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82 Example

83 Given a reaction network, does it have multiple equilibria?

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85 p(c,k)

86 The Jacobian Criterion Theorem. For a fully diffusive reaction network, the associated polynomial function p(c,k) is injective for all k if and only if the determinant of the Jacobian does not vanish for any k.

87 Exampl e p(c,k ) Coefficients of monomials in the determinant of the Jacobian of p(c,k): The determinant of the Jacobian does not vanish!

88 Which polynomial dynamical systems can arise from massaction kinetics?

89 Which polynomial dynamical systems can arise from masstheorem. An n dimensional polynomial dynamical system can arise action kinetics? from mass action kinetics if and only if it can be written as dc1/dt = P1(c1,,cn) c1q1(c1,,cn) dc2/dt = P2(c1,,cn) c2q2(c1,,cn) dcn/dt = Pn(c1,,cn) cnqn(c1,,cn) for some polynomials Pi and Qi with non negative coefficients.

90 Not mass-action kinetics: the Lorenz equations dc1/dt = P1(c1,,cn) c1q1(c1,,cn) dc2/dt = P2(c1,,cn) c2q2(c1,,cn) dcn/dt = Pn(c1,,cn) cnqn(c1,,cn)

91 The positive quadrant is an invariant dc /dt =subspace P (c,,c ) c Q (c,,c ) 1 c3 1 1 n 1 1 n dc2/dt = P2(c1,,cn) c2q2(c1,,cn) dcn/dt = Pn(c1,,cn) cnqn(c1,,cn) c1 c2 1

92 Acknowledgments Fedor Nazarov - Department of Mathematics, University of Wisconsin-Madison Anne Shiu - Department of Mathematics, University of California at Berkeley Alicia Dickenstein - Department of Mathematics, University of Buenos Aires, Argentina Bernd Sturmfels - Department of Mathematics, University of California at Berkeley Casian Pantea - Department of Mathematics, University of Wisconsin-Madison Maya Mincheva - BACTER Institute, University of Wisconsin-Madison Martin Feinberg - Dept. of Chemical and Biomolecular Engineering, Ohio State University Support: NSF, DOE.

93 Entrapped species models Theorem. Consider some reaction network which contains some entrapped species. If the corresponding fully diffusive network does not admit multiple equilibria, then the reaction network does not admit multiple equilibria.

94 Methotrexate: How a Cancer Drug Works After Voet & Voet,

95 Methotrexate: How a Cancer Drug Works Methotrexate After Voet & Voet,

96 Inside the DHFR Red Dot (No Methotrexate!) E + H 2F EH 2F + NH Mechanism and rate constants from Appleman et al., J. Biol. Chem., 265, , (1990). E = Human DHFR NH = NADPH EH 2F 24 ENHH 2F ENHH 2F ENH 4F ENH 4F EN + H 4F E + NH ENH ENH + H 2F ENHH 2F E + N EN EH 4F E + H 4F ENH + H 4F ENHH 4F ENHH 4F EH 4F + NH EH 4F + N ENH 4F EH 2F + N ENH 2F EN + H 2F ENH 2F 1.3 H2F = H2folate N =

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98 y t i l i b a t s i B t o n. d e d u l c e pr

99 Bistability Inside the DHFR Red Dot (No Methotrexate) E + H 2F EH 2F + NH EH 2F 24 ENHH 2F ENHH 2F ENH 4F ENH 4F EN + H 4F E + NH ENH ENH + H 2F ENHH 2F E + N EN EH 4F E + H 4F ENH + H 4F ENHH 4F ENHH 4F EH 4F + NH EH 4F + N ENH 4F EH 2F + N ENH 2F EN + H 2F ENH 2F 1.3 H2F + NH + H N H4F +

100 Bistability Inside the DHFR Red Dot (No Methotrexate) E + H 2F EH 2F + NH 24 EH 2F ENHH 2F ENHH 2F ENH 4F ENH 4F EN + H 4F E + NH ENH ENH + H 2F ENHH 2F E + N EN EH 4F E + H 4F ENH + H 4F ENHH 4F ENHH 4F EH 4F + NH EH 4F + N ENH 4F EH 2F + N ENH 2F EN + H 2F ENH 2F 1.3

101 Bistability Inside the DHFR Red Dot (No Methotrexate) E + H 2F EH 2F + NH 24 EH 2F ENHH 2F ENHH 2F ENH 4F ENH 4F EN + H 4F E + NH ENH ENH + H 2F ENHH 2F E + N EN EH 4F E + H 4F ENH + H 4F ENHH 4F ENHH 4F EH 4F + NH EH 4F + N ENH 4F EH 2F + N ENH 2F EN + H 2F ENH 2F 1.3