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Background Original Results Conclusions and Future Work Doctoral Dissertation: Topics in Chemical Reaction Modeling Matthew D. Johnston Department of Applied Mathematics University of Waterloo December 9, 2011 MD Johnston Topics in

  1. Background Chemical Reaction Networks Original Results Weakly Reversible Networks Conclusions and Future Work Complex Balanced Networks Consider the reversible system k 1 A 1 2 A 2 . ⇄ k 2 This has the governing dynamics � ˙ � − 1 � � � � x 1 1 x 2 = k 1 x 1 + k 2 2 , − 2 x 2 ˙ 2 where x 1 and x 2 are the concentrations of A 1 and A 2 respectively. MD Johnston Topics in Chemical Reaction Modeling

  2. Background Chemical Reaction Networks Original Results Weakly Reversible Networks Conclusions and Future Work Complex Balanced Networks Consider the reversible system k 1 A 1 2 A 2 . ⇄ k 2 This has the governing dynamics � ˙ � − 1 � � � � x 1 1 x 2 = k 1 x 1 + k 2 2 , − 2 x 2 ˙ 2 where x 1 and x 2 are the concentrations of A 1 and A 2 respectively. MD Johnston Topics in Chemical Reaction Modeling

  3. Background Chemical Reaction Networks Original Results Weakly Reversible Networks Conclusions and Future Work Complex Balanced Networks Consider the reversible system k 1 A 1 2 A 2 . ⇄ k 2 This has the governing dynamics � ˙ � − 1 � � � � x 1 1 x 2 = k 1 x 1 + k 2 2 , − 2 x 2 ˙ 2 where x 1 and x 2 are the concentrations of A 1 and A 2 respectively. MD Johnston Topics in Chemical Reaction Modeling

  4. Background Chemical Reaction Networks Original Results Weakly Reversible Networks Conclusions and Future Work Complex Balanced Networks Consider the reversible system k 1 A 1 2 A 2 . ⇄ k 2 This has the governing dynamics � ˙ � − 1 � � � � x 1 1 x 2 = k 1 x 1 + k 2 2 , − 2 x 2 ˙ 2 where x 1 and x 2 are the concentrations of A 1 and A 2 respectively. MD Johnston Topics in Chemical Reaction Modeling

  5. Background Chemical Reaction Networks Original Results Weakly Reversible Networks Conclusions and Future Work Complex Balanced Networks Consider the reversible system k 1 A 1 2 A 2 . ⇄ k 2 This has the governing dynamics � ˙ � − 1 � � � � x 1 1 x 2 = k 1 x 1 + k 2 2 , − 2 x 2 ˙ 2 where x 1 and x 2 are the concentrations of A 1 and A 2 respectively. MD Johnston Topics in Chemical Reaction Modeling

  6. Background Chemical Reaction Networks Original Results Weakly Reversible Networks Conclusions and Future Work Complex Balanced Networks Figure: Previous system with k 1 = k 2 = 1. MD Johnston Topics in Chemical Reaction Modeling

  7. Background Chemical Reaction Networks Original Results Weakly Reversible Networks Conclusions and Future Work Complex Balanced Networks Figure: Previous system with k 1 = k 2 = 1. MD Johnston Topics in Chemical Reaction Modeling

  8. Background Chemical Reaction Networks Original Results Weakly Reversible Networks Conclusions and Future Work Complex Balanced Networks Figure: Previous system with k 1 = k 2 = 1. MD Johnston Topics in Chemical Reaction Modeling

  9. Background Chemical Reaction Networks Original Results Weakly Reversible Networks Conclusions and Future Work Complex Balanced Networks The dynamical behaviour of chemical reaction networks is heavily dependent on the structure of its reaction graph . MD Johnston Topics in Chemical Reaction Modeling

  10. Background Chemical Reaction Networks Original Results Weakly Reversible Networks Conclusions and Future Work Complex Balanced Networks The dynamical behaviour of chemical reaction networks is heavily dependent on the structure of its reaction graph . The particular class of networks I have been interested in are weakly reversible networks . MD Johnston Topics in Chemical Reaction Modeling

  11. Background Chemical Reaction Networks Original Results Weakly Reversible Networks Conclusions and Future Work Complex Balanced Networks The dynamical behaviour of chemical reaction networks is heavily dependent on the structure of its reaction graph . The particular class of networks I have been interested in are weakly reversible networks . k (1 , 2) C 1 − → C 2 k (4 , 5) C 4 C 5 ⇄ k (3 , 1) տ ւ k (2 , 3) k (5 , 4) C 3 MD Johnston Topics in Chemical Reaction Modeling

  12. Background Chemical Reaction Networks Original Results Weakly Reversible Networks Conclusions and Future Work Complex Balanced Networks The dynamical behaviour of chemical reaction networks is heavily dependent on the structure of its reaction graph . The particular class of networks I have been interested in are weakly reversible networks . k (1 , 2) C 1 − → C 2 k (4 , 5) C 4 C 5 ⇄ k (3 , 1) տ ւ k (2 , 3) k (5 , 4) C 3 / A network is weakly reversible if a path from one another complex to another implies a path back. MD Johnston Topics in Chemical Reaction Modeling

  13. Background Chemical Reaction Networks Original Results Weakly Reversible Networks Conclusions and Future Work Complex Balanced Networks The dynamical behaviour of chemical reaction networks is heavily dependent on the structure of its reaction graph . The particular class of networks I have been interested in are weakly reversible networks . k (1 , 2) C 1 − → C 2 k (4 , 5) C 4 C 5 ⇄ k (3 , 1) տ ւ k (2 , 3) k (5 , 4) C 3 / A network is weakly reversible if a path from one another complex to another implies a path back. MD Johnston Topics in Chemical Reaction Modeling

  14. Background Chemical Reaction Networks Original Results Weakly Reversible Networks Conclusions and Future Work Complex Balanced Networks The dynamical behaviour of chemical reaction networks is heavily dependent on the structure of its reaction graph . The particular class of networks I have been interested in are weakly reversible networks . k (1 , 2) C 1 − → C 2 k (4 , 5) C 4 C 5 ⇄ k (3 , 1) տ ւ k (2 , 3) k (5 , 4) C 3 / A network is weakly reversible if a path from one another complex to another implies a path back. MD Johnston Topics in Chemical Reaction Modeling

  15. Background Chemical Reaction Networks Original Results Weakly Reversible Networks Conclusions and Future Work Complex Balanced Networks The dynamical behaviour of chemical reaction networks is heavily dependent on the structure of its reaction graph . The particular class of networks I have been interested in are weakly reversible networks . k (1 , 2) C 1 − → C 2 k (4 , 5) C 4 C 5 ⇄ k (3 , 1) տ ւ k (2 , 3) k (5 , 4) C 3 / A network is weakly reversible if a path from one another complex to another implies a path back. MD Johnston Topics in Chemical Reaction Modeling

  16. Background Chemical Reaction Networks Original Results Weakly Reversible Networks Conclusions and Future Work Complex Balanced Networks An important subset of weakly reversible networks are complex balanced networks . MD Johnston Topics in Chemical Reaction Modeling

  17. Background Chemical Reaction Networks Original Results Weakly Reversible Networks Conclusions and Future Work Complex Balanced Networks An important subset of weakly reversible networks are complex balanced networks . A network is complex balanced at x ∗ if n n k ( j , i )( x ∗ ) z j = ( x ∗ ) z i � � k ( i , j ) , for i = 1 , . . . , n . j =1 j =1 MD Johnston Topics in Chemical Reaction Modeling

  18. Background Chemical Reaction Networks Original Results Weakly Reversible Networks Conclusions and Future Work Complex Balanced Networks An important subset of weakly reversible networks are complex balanced networks . A network is complex balanced at x ∗ if n n k ( j , i )( x ∗ ) z j = ( x ∗ ) z i � � k ( i , j ) , for i = 1 , . . . , n . j =1 j =1 MD Johnston Topics in Chemical Reaction Modeling

  19. Background Chemical Reaction Networks Original Results Weakly Reversible Networks Conclusions and Future Work Complex Balanced Networks An important subset of weakly reversible networks are complex balanced networks . A network is complex balanced at x ∗ if n n k ( j , i )( x ∗ ) z j = ( x ∗ ) z i � � k ( i , j ) , for i = 1 , . . . , n . j =1 j =1 MD Johnston Topics in Chemical Reaction Modeling

  20. Background Chemical Reaction Networks Original Results Weakly Reversible Networks Conclusions and Future Work Complex Balanced Networks An important subset of weakly reversible networks are complex balanced networks . A network is complex balanced at x ∗ if n n k ( j , i )( x ∗ ) z j = ( x ∗ ) z i � � k ( i , j ) , for i = 1 , . . . , n . j =1 j =1 Theorem (Lemma 4C and Theorem 6A, [1]) If a chemical reaction network is complex balanced , then there exists within each invariant space of the network a unique positive equilibrium concentration , and this equilibrium concentration is locally asymptotically stable relative to that invariant space. MD Johnston Topics in Chemical Reaction Modeling

  21. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks 1 Background Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks 2 Original Results Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks 3 Conclusions and Future Work MD Johnston Topics in Chemical Reaction Modeling

  22. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Linearization of Complex Balanced Network The proof of Theorem 1 was dependent on the Lyapunov function m � x i [ln( x i ) − ln( x ∗ i ) − 1] + x ∗ L ( x ) = i i =1 where x ∗ ∈ R m > 0 is a positive equilibrium concentration. MD Johnston Topics in Chemical Reaction Modeling

  23. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Linearization of Complex Balanced Network The proof of Theorem 1 was dependent on the Lyapunov function m � x i [ln( x i ) − ln( x ∗ i ) − 1] + x ∗ L ( x ) = i i =1 where x ∗ ∈ R m > 0 is a positive equilibrium concentration. For complex balanced networks, it can be shown that d dt L ( x ) = ∇ L ( x ) · f ( x ) < 0 for all x ∈ R m > 0 such that f ( x ) � = 0 . MD Johnston Topics in Chemical Reaction Modeling

  24. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Linearization of Complex Balanced Network The proof of Theorem 1 was dependent on the Lyapunov function m � x i [ln( x i ) − ln( x ∗ i ) − 1] + x ∗ L ( x ) = i i =1 where x ∗ ∈ R m > 0 is a positive equilibrium concentration. For complex balanced networks, it can be shown that d dt L ( x ) = ∇ L ( x ) · f ( x ) < 0 for all x ∈ R m > 0 such that f ( x ) � = 0 . This means that all trajectories converge locally to their respective equilibrium concentrations! MD Johnston Topics in Chemical Reaction Modeling

  25. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks This problem has not been attempted by the method of linearization about equilibrium concentrations before. MD Johnston Topics in Chemical Reaction Modeling

  26. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks This problem has not been attempted by the method of linearization about equilibrium concentrations before. For the method of linearization, we consider the decomposition f ( x ) = D f ( x ∗ )( x − x ∗ ) + O (( x − x ∗ ) 2 ) and the corresponding linear problem d y dt = A y where y = x − x ∗ and A = D f ( x ∗ ). MD Johnston Topics in Chemical Reaction Modeling

  27. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks This problem has not been attempted by the method of linearization about equilibrium concentrations before. For the method of linearization, we consider the decomposition f ( x ) = D f ( x ∗ )( x − x ∗ ) + O (( x − x ∗ ) 2 ) and the corresponding linear problem d y dt = A y where y = x − x ∗ and A = D f ( x ∗ ). This method has the advantage of guaranteeing local exponential convergence to x ∗ if D f ( x ∗ ) has non-degenerate eigenvalues with negative real part. MD Johnston Topics in Chemical Reaction Modeling

  28. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Theorem A complex balanced mass-action system satisfies the following properties about any positive equilibrium concentration x ∗ ∈ R m > 0 : MD Johnston Topics in Chemical Reaction Modeling

  29. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Theorem A complex balanced mass-action system satisfies the following properties about any positive equilibrium concentration x ∗ ∈ R m > 0 : 1 the local stable manifold W s loc coincides locally with the compatible linear invariant space MD Johnston Topics in Chemical Reaction Modeling

  30. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Theorem A complex balanced mass-action system satisfies the following properties about any positive equilibrium concentration x ∗ ∈ R m > 0 : 1 the local stable manifold W s loc coincides locally with the compatible linear invariant space; and 2 the local centre manifold W c loc coincides locally with the tangent plane to the equilibrium set at x ∗ MD Johnston Topics in Chemical Reaction Modeling

  31. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Theorem A complex balanced mass-action system satisfies the following properties about any positive equilibrium concentration x ∗ ∈ R m > 0 : 1 the local stable manifold W s loc coincides locally with the compatible linear invariant space; and 2 the local centre manifold W c loc coincides locally with the tangent plane to the equilibrium set at x ∗ ; and 3 for any M > 0 satisfying max { Re ( λ i ) | Re ( λ i ) < 0 } < − M < 0 there exists a k ≥ 1 such that � x ( t ) − x ∗ � ≤ ke − Mt � x 0 − x ∗ � , ∀ t ≥ 0 for all x 0 sufficiently close to x ∗ in the compatible linear invariant space. MD Johnston Topics in Chemical Reaction Modeling

  32. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Consider the network 1 / 2 2 A 1 + A 2 − → 3 A 1 1 / 8 ↑ ↓ 1 3 A 2 ← 1 / 4 A 1 + 2 A 2 . − MD Johnston Topics in Chemical Reaction Modeling

  33. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Consider the network 1 / 2 2 A 1 + A 2 − → 3 A 1 1 / 8 ↑ ↓ 1 3 A 2 ← 1 / 4 A 1 + 2 A 2 . − This system is governed by the dynamics � 1 2 + 1 � dx 1 4 x 2 4 x 1 x 2 + x 2 dt = ( x 2 − 2 x 1 ) 1 � 1 2 + 1 � dx 2 4 x 2 4 x 1 x 2 + x 2 dt = (2 x 1 − x 2 ) . 1 MD Johnston Topics in Chemical Reaction Modeling

  34. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Figure: Vector field plot of previous system. Invariant spaces satisfy x 1 + x 2 = constant , and equilibria are along the line x 2 = 2 x 1 . MD Johnston Topics in Chemical Reaction Modeling

  35. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Consider the equilibrium concentration x ∗ 1 = 1 / 2, x ∗ 2 = 1. MD Johnston Topics in Chemical Reaction Modeling

  36. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Consider the equilibrium concentration x ∗ 1 = 1 / 2, x ∗ 2 = 1. This network has the Jacobian � − 5 5 � D f ( x ∗ ) = 4 8 . 5 − 5 4 8 MD Johnston Topics in Chemical Reaction Modeling

  37. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Consider the equilibrium concentration x ∗ 1 = 1 / 2, x ∗ 2 = 1. This network has the Jacobian � − 5 5 � D f ( x ∗ ) = 4 8 . 5 − 5 4 8 This matrix has the eigenvalue/eigenvector pairs λ 1 = − 15 / 8, v 1 = [ − 1 1] T , and λ 2 = 0, v 2 = [1 2] T . MD Johnston Topics in Chemical Reaction Modeling

  38. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Consider the equilibrium concentration x ∗ 1 = 1 / 2, x ∗ 2 = 1. This network has the Jacobian � − 5 5 � D f ( x ∗ ) = 4 8 . 5 − 5 4 8 This matrix has the eigenvalue/eigenvector pairs λ 1 = − 15 / 8, v 1 = [ − 1 1] T , and λ 2 = 0, v 2 = [1 2] T . As expected, the local stable manifold is tangent to the invariant space x 1 + x 2 = constant , and the local centre manifold is tangent to the equilibrium set x 2 = 2 x 1 ! MD Johnston Topics in Chemical Reaction Modeling

  39. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks ke -Mt ||x 0 - x*|| ||x(t) - x*|| t Figure: The exponential bound is shown with values M = 1 . 85 and k = 1 . 25. For an initial condition chosen sufficiently close to x ∗ , we can see that the correspondence between the convergence of x ( t ) to x ∗ and the upper bound is nearly exact. MD Johnston Topics in Chemical Reaction Modeling

  40. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Global Attractor Conjecture A significant amount of research has been conducted on the question of persistence of chemical reaction networks. MD Johnston Topics in Chemical Reaction Modeling

  41. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Global Attractor Conjecture A significant amount of research has been conducted on the question of persistence of chemical reaction networks. Persistence is the property that if all species are initially present then none will tend toward zero . MD Johnston Topics in Chemical Reaction Modeling

  42. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Global Attractor Conjecture A significant amount of research has been conducted on the question of persistence of chemical reaction networks. Persistence is the property that if all species are initially present then none will tend toward zero . This is a very important property of networks! (e.g. Chemical reactors using up reactants, biological circuits losing intermediates, etc.) MD Johnston Topics in Chemical Reaction Modeling

  43. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Reconsider complex balanced networks : it is known that d dt L ( x ) = ∇ L ( x ) · f ( x ) < 0 for all x ∈ R m > 0 such that f ( x ) � = 0 . MD Johnston Topics in Chemical Reaction Modeling

  44. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Reconsider complex balanced networks : it is known that d dt L ( x ) = ∇ L ( x ) · f ( x ) < 0 for all x ∈ R m > 0 such that f ( x ) � = 0 . QUESTION: Is this sufficient to show x ∗ is globally asymptotically stable relative to its corresponding linear invariant space? MD Johnston Topics in Chemical Reaction Modeling

  45. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Reconsider complex balanced networks : it is known that d dt L ( x ) = ∇ L ( x ) · f ( x ) < 0 for all x ∈ R m > 0 such that f ( x ) � = 0 . QUESTION: Is this sufficient to show x ∗ is globally asymptotically stable relative to its corresponding linear invariant space? ANSWER: Only if the network is also persistent (since L ( x ) is not radially unbounded with respect to R m > 0 ). MD Johnston Topics in Chemical Reaction Modeling

  46. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks One approach to persistence is identifying certain special regions of the boundary and showing they must repel trajectories. MD Johnston Topics in Chemical Reaction Modeling

  47. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks One approach to persistence is identifying certain special regions of the boundary and showing they must repel trajectories. The main result of [3] is the following: Theorem (Theorem 3.7, [3]) Consider a mass-action system with bounded solutions. Suppose that every semilocking set I is weakly dynamically nonemptiable . Then the system is persistent. MD Johnston Topics in Chemical Reaction Modeling

  48. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks One approach to persistence is identifying certain special regions of the boundary and showing they must repel trajectories. The main result of [3] is the following: Theorem (Theorem 3.7, [3]) Consider a mass-action system with bounded solutions. Suppose that every semilocking set I is weakly dynamically nonemptiable . Then the system is persistent. Weak dynamical non-emptiability is a technical condition and is a generalization of dynamical non-emptiability introduced in [4]. MD Johnston Topics in Chemical Reaction Modeling

  49. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Another approach to persistence is the idea of stratifying the state space R m > 0 , which was introduced in [5] for detailed balanced networks (a subset of complex balanced networks). MD Johnston Topics in Chemical Reaction Modeling

  50. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Another approach to persistence is the idea of stratifying the state space R m > 0 , which was introduced in [5] for detailed balanced networks (a subset of complex balanced networks). We generalized this to complex balanced networks in [6] and proved the following: Theorem (Theorem 3.12, [6]) Consider a complex balanced system and an arbitrary permutation operator µ . If S µ ∩ L I � = ∅ then there exists an α ∈ R m ≤ 0 satisfying α i < 0 for i ∈ I α i = 0 for i �∈ I and such that � α, f ( x ) � ≤ 0 for every x ∈ S µ . MD Johnston Topics in Chemical Reaction Modeling

  51. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Another approach to persistence is the idea of stratifying the state space R m > 0 , which was introduced in [5] for detailed balanced networks (a subset of complex balanced networks). We generalized this to complex balanced networks in [6] and proved the following: Theorem (Theorem 3.12, [6]) Consider a complex balanced system and an arbitrary permutation operator µ . If S µ ∩ L I � = ∅ then there exists an α ∈ R m ≤ 0 satisfying α i < 0 for i ∈ I α i = 0 for i �∈ I and such that � α, f ( x ) � ≤ 0 for every x ∈ S µ . MD Johnston Topics in Chemical Reaction Modeling

  52. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Another approach to persistence is the idea of stratifying the state space R m > 0 , which was introduced in [5] for detailed balanced networks (a subset of complex balanced networks). We generalized this to complex balanced networks in [6] and proved the following: Theorem (Theorem 3.12, [6]) Consider a complex balanced system and an arbitrary permutation operator µ . If S µ ∩ L I � = ∅ then there exists an α ∈ R m ≤ 0 satisfying α i < 0 for i ∈ I α i = 0 for i �∈ I and such that � α, f ( x ) � ≤ 0 for every x ∈ S µ . MD Johnston Topics in Chemical Reaction Modeling

  53. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Another approach to persistence is the idea of stratifying the state space R m > 0 , which was introduced in [5] for detailed balanced networks (a subset of complex balanced networks). We generalized this to complex balanced networks in [6] and proved the following: Theorem (Theorem 3.12, [6]) Consider a complex balanced system and an arbitrary permutation operator µ . If S µ ∩ L I � = ∅ then there exists an α ∈ R m ≤ 0 satisfying α i < 0 for i ∈ I α i = 0 for i �∈ I and such that � α, f ( x ) � ≤ 0 for every x ∈ S µ . MD Johnston Topics in Chemical Reaction Modeling

  54. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks The strata S µ represent a partition of the state space R m > 0 . MD Johnston Topics in Chemical Reaction Modeling

  55. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks The strata S µ represent a partition of the state space R m > 0 . S 4 S 2 x* S 3 S 1 MD Johnston Topics in Chemical Reaction Modeling

  56. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks The strata S µ represent a partition of the state space R m > 0 . S 4 S 2 x* S 3 S 1 This result guarantees trajectories within strata are kept away from portions of the boundary intersecting the strata! MD Johnston Topics in Chemical Reaction Modeling

  57. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks The strata S µ represent a partition of the state space R m > 0 . S 4 S 2 x* S 3 S 1 This result guarantees trajectories within strata are kept away from portions of the boundary intersecting the strata! MD Johnston Topics in Chemical Reaction Modeling

  58. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks The strata S µ represent a partition of the state space R m > 0 . S 4 S 2 x* S 3 S 1 This result guarantees trajectories within strata are kept away from portions of the boundary intersecting the strata! Supplemental conditions guarantee persistence. MD Johnston Topics in Chemical Reaction Modeling

  59. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Conjugacy of Chemical Reaction Networks A lot of recent research has been conducted on determining conditions under which two networks with disparate network structure share the same qualitative dynamics . MD Johnston Topics in Chemical Reaction Modeling

  60. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Conjugacy of Chemical Reaction Networks A lot of recent research has been conducted on determining conditions under which two networks with disparate network structure share the same qualitative dynamics . We can often relate the dynamics of networks based on properties of their reaction graphs. MD Johnston Topics in Chemical Reaction Modeling

  61. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Conjugacy of Chemical Reaction Networks A lot of recent research has been conducted on determining conditions under which two networks with disparate network structure share the same qualitative dynamics . We can often relate the dynamics of networks based on properties of their reaction graphs. In the case where one network has “good” structure and the other does not, we can extend the dynamics to the poorly structured network! MD Johnston Topics in Chemical Reaction Modeling

  62. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks It is known that two chemical reaction networks can give rise to the same set of differential equations . MD Johnston Topics in Chemical Reaction Modeling

  63. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks It is known that two chemical reaction networks can give rise to the same set of differential equations . Consider the chemical reaction networks 1 1 N : 2 A 1 − → 2 A 2 − → A 1 + A 2 1 N ′ : 2 A 1 2 A 2 . ⇄ 0 . 5 MD Johnston Topics in Chemical Reaction Modeling

  64. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks It is known that two chemical reaction networks can give rise to the same set of differential equations . Consider the chemical reaction networks 1 1 N : 2 A 1 − → 2 A 2 − → A 1 + A 2 1 N ′ : 2 A 1 2 A 2 . ⇄ 0 . 5 Both of these networks are governed by x 1 = − 2 x 2 1 + x 2 ˙ 2 x 2 = 2 x 2 1 − x 2 ˙ 2 . MD Johnston Topics in Chemical Reaction Modeling

  65. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks It is known that two chemical reaction networks can give rise to the same set of differential equations . Consider the chemical reaction networks 1 1 N : 2 A 1 − → 2 A 2 − → A 1 + A 2 1 N ′ : 2 A 1 2 A 2 . (Well structured!) ⇄ 0 . 5 Both of these networks are governed by x 1 = − 2 x 2 1 + x 2 ˙ 2 x 2 = 2 x 2 1 − x 2 ˙ 2 . MD Johnston Topics in Chemical Reaction Modeling

  66. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks It is known that two chemical reaction networks can give rise to the same set of differential equations . Consider the chemical reaction networks 1 1 N : 2 A 1 − → 2 A 2 − → A 1 + A 2 1 N ′ : 2 A 1 2 A 2 . (Well structured!) ⇄ 0 . 5 Both of these networks are governed by x 1 = − 2 x 2 1 + x 2 ˙ 2 (Known dynamics!) x 2 = 2 x 2 1 − x 2 ˙ 2 . MD Johnston Topics in Chemical Reaction Modeling

  67. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks It is known that two chemical reaction networks can give rise to the same set of differential equations . Consider the chemical reaction networks 1 1 N : 2 A 1 − → 2 A 2 − → A 1 + A 2 1 N ′ : 2 A 1 2 A 2 . (Well structured!) ⇄ 0 . 5 Both of these networks are governed by x 1 = − 2 x 2 1 + x 2 ˙ 2 (Known dynamics!) x 2 = 2 x 2 1 − x 2 ˙ 2 . MD Johnston Topics in Chemical Reaction Modeling

  68. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks A complete theoretical analysis of dynamical equivalence was conducted in [6]. MD Johnston Topics in Chemical Reaction Modeling

  69. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks A complete theoretical analysis of dynamical equivalence was conducted in [6]. The problem of finding dynamically equivalent networks using computer software has been investigated in [7] and subsequent papers. MD Johnston Topics in Chemical Reaction Modeling

  70. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks A complete theoretical analysis of dynamical equivalence was conducted in [6]. The problem of finding dynamically equivalent networks using computer software has been investigated in [7] and subsequent papers. The procedure uses mixed-integer linear programming (MILP) methods to find dynamically equivalent networks with a maximal and minimal number of reactions. MD Johnston Topics in Chemical Reaction Modeling

  71. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks QUESTION #1: Can we find more general conditions under which two networks have the same qualitative dynamics? MD Johnston Topics in Chemical Reaction Modeling

  72. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks QUESTION #1: Can we find more general conditions under which two networks have the same qualitative dynamics? QUESTION #2: Given a network with “bad” structure, can we find a network with “good” structure with related dynamics? MD Johnston Topics in Chemical Reaction Modeling

  73. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks We extended the notion of dynamical equivalence to linear conjugacy with the following result: Theorem (Theorem 3.2 of [2] and Theorem 2 of [4]) Consider the kinetics matrix A k corresponding to N and suppose that there is a kinetics matrix A b with the same structure as N ′ and a vector c ∈ R n > 0 such that Y · A k = T · Y · A b where T = diag { c } . Then N is linearly conjugate to N ′ with kinetics matrix A ′ k = A b · diag { Ψ( c ) } . MD Johnston Topics in Chemical Reaction Modeling

  74. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Two networks N and N ′ are said to be linearly conjugate to one another if there is a linear mapping which takes the flow from one in the flow of the other. MD Johnston Topics in Chemical Reaction Modeling

  75. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Two networks N and N ′ are said to be linearly conjugate to one another if there is a linear mapping which takes the flow from one in the flow of the other. The qualitative dynamics (e.g. number and stability of equilibria, persistence, boundedness, etc.) of linearly conjugate networks are identical! (i.e. The answer to Question #1 is YES.) MD Johnston Topics in Chemical Reaction Modeling

  76. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks Two networks N and N ′ are said to be linearly conjugate to one another if there is a linear mapping which takes the flow from one in the flow of the other. The qualitative dynamics (e.g. number and stability of equilibria, persistence, boundedness, etc.) of linearly conjugate networks are identical! (i.e. The answer to Question #1 is YES.) Linear conjugacy includes dynamical equivalence as a special case (identity transformation). MD Johnston Topics in Chemical Reaction Modeling

  77. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks PROBLEM: If we are just given a single network N , can we find a linearly conjugate network N ′ with known dynamics? MD Johnston Topics in Chemical Reaction Modeling

  78. Background Linearization of Complex Balanced Networks Original Results Global Attractor Conjecture Conclusions and Future Work Conjugacy of Chemical Reaction Networks PROBLEM: If we are just given a single network N , can we find a linearly conjugate network N ′ with known dynamics? The space of possible networks is typically too large to consider by hand. MD Johnston Topics in Chemical Reaction Modeling

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