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Stochastic Chemical Reaction Networks Matthew Douglas Johnston - PowerPoint PPT Presentation

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Background Stochastic Models Interesting Systems Stochastic Chemical Reaction Networks Matthew Douglas Johnston University of Waterloo Fall 2010 Matthew Douglas Johnston Stochastic Chemical Reaction Networks Background Stochastic Models

  1. Background Chemical Reactions Stochastic Models Standard Model Interesting Systems Stoichiometric Compatibility Classes Consider the general system k i → C ′ C i − i , i = 1 , . . . , r . This system is governed by the system of autonomous, polynomial, ordinary differential equations r i − z i ) x z i . � k i ( z ′ x = ˙ (1) i =1 We have the following important components: we sum over r reactions, k i is the reaction rate, ( z ′ i − z i ) is the reaction vector, Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  2. Background Chemical Reactions Stochastic Models Standard Model Interesting Systems Stoichiometric Compatibility Classes Consider the general system k i → C ′ C i − i , i = 1 , . . . , r . This system is governed by the system of autonomous, polynomial, ordinary differential equations r i − z i ) x z i . � k i ( z ′ x = ˙ (1) i =1 We have the following important components: we sum over r reactions, k i is the reaction rate, ( z ′ i − z i ) is the reaction vector, and x z i = � m j =1 ( x j ) z ij is the mass-action term. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  3. Background Chemical Reactions Stochastic Models Standard Model Interesting Systems Stoichiometric Compatibility Classes Consider the (reversible) system k 1 A 1 2 A 2 . ⇄ k 2 Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  4. Background Chemical Reactions Stochastic Models Standard Model Interesting Systems Stoichiometric Compatibility Classes 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 , x 2 ˙ 2 − 2 where x 1 and x 2 are the concentrations of A 1 and A 2 respectively. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  5. Background Chemical Reactions Stochastic Models Standard Model Interesting Systems Stoichiometric Compatibility Classes 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 , x 2 ˙ 2 − 2 where x 1 and x 2 are the concentrations of A 1 and A 2 respectively. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  6. Background Chemical Reactions Stochastic Models Standard Model Interesting Systems Stoichiometric Compatibility Classes 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 , x 2 ˙ 2 − 2 where x 1 and x 2 are the concentrations of A 1 and A 2 respectively. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  7. Background Chemical Reactions Stochastic Models Standard Model Interesting Systems Stoichiometric Compatibility Classes 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 , x 2 ˙ 2 − 2 where x 1 and x 2 are the concentrations of A 1 and A 2 respectively. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  8. Background Chemical Reactions Stochastic Models Standard Model Interesting Systems Stoichiometric Compatibility Classes 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 , x 2 ˙ 2 − 2 where x 1 and x 2 are the concentrations of A 1 and A 2 respectively. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  9. Background Chemical Reactions Stochastic Models Standard Model Interesting Systems Stoichiometric Compatibility Classes What kind of properties does this system have? � ˙ � − 1 x 1 � � � 1 � x 2 = k 1 x 1 + k 2 2 x 2 ˙ 2 − 2 Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  10. Background Chemical Reactions Stochastic Models Standard Model Interesting Systems Stoichiometric Compatibility Classes What kind of properties does this system have? � ˙ � − 1 � 0 x 1 � � � 1 � � �� x 2 = k 1 x 1 + k 2 = 2 x 2 ˙ 2 − 2 0 The (positive) equilibrium set is given by � � � k 1 x ∈ R 2 E = > 0 | x 2 = x 1 . k 2 Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  11. Background Chemical Reactions Stochastic Models Standard Model Interesting Systems Stoichiometric Compatibility Classes What kind of properties does this system have? � ˙ � − 1 x 1 � � � 1 � x 2 = k 1 x 1 + k 2 2 x 2 ˙ 2 − 2 The (positive) equilibrium set is given by � � � k 1 x ∈ R 2 E = > 0 | x 2 = x 1 . k 2 For any k 1 , k 2 , x 1 , x 2 we have f ( x ) ∈ S where �� �� 1 S = span . − 2 Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  12. Background Chemical Reactions Stochastic Models Standard Model Interesting Systems Stoichiometric Compatibility Classes (x 0 +S) E x 2 x 1 Figure: Previous system with k 1 = k 2 = 1. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  13. Background Chemical Reactions Stochastic Models Standard Model Interesting Systems Stoichiometric Compatibility Classes (x 0 +S) E x 2 x 1 Figure: Previous system with k 1 = k 2 = 1. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  14. Background Chemical Reactions Stochastic Models Standard Model Interesting Systems Stoichiometric Compatibility Classes (x 0 +S) E x 2 x 1 Figure: Previous system with k 1 = k 2 = 1. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  15. Background Chemical Reactions Stochastic Models Standard Model Interesting Systems Stoichiometric Compatibility Classes (x 0 +S) E x 2 x 1 Figure: Previous system with k 1 = k 2 = 1. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  16. Background Chemical Reactions Stochastic Models Standard Model Interesting Systems Stoichiometric Compatibility Classes The restriction of solutions is a general property. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  17. Background Chemical Reactions Stochastic Models Standard Model Interesting Systems Stoichiometric Compatibility Classes The restriction of solutions is a general property. Definition The stoichiometric subspace S ⊂ R m is given by z ′ � � S = span i − z i | i = 1 , . . . , r . Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  18. Background Chemical Reactions Stochastic Models Standard Model Interesting Systems Stoichiometric Compatibility Classes The restriction of solutions is a general property. Definition The stoichiometric subspace S ⊂ R m is given by z ′ � � S = span i − z i | i = 1 , . . . , r . Theorem Solutions x ( t ) of (1) are restricted to stoichiometric compatibility classes such that x ( t ) ∈ ( S + x 0 ) ∩ R m ∀ t ≥ 0 . + Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  19. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation 1 Background Chemical Reactions Standard Model Stoichiometric Compatibility Classes 2 Stochastic Models Small-Scale Considerations Gillespie Algorithm Chemical Master Equation 3 Interesting Systems Lotka-Volterra System The Block Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  20. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation What about cases where the number of reactant molecules n i is small (e.g. biological cells)? Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  21. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation What about cases where the number of reactant molecules n i is small (e.g. biological cells)? A few considerations: Differences between states is large - i.e. continuity of concentrations breaks down. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  22. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation What about cases where the number of reactant molecules n i is small (e.g. biological cells)? A few considerations: Differences between states is large - i.e. continuity of concentrations breaks down. Each occurrence of a reaction matters - i.e. we cannot average into a lump parameter k i . Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  23. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation What about cases where the number of reactant molecules n i is small (e.g. biological cells)? A few considerations: Differences between states is large - i.e. continuity of concentrations breaks down. Each occurrence of a reaction matters - i.e. we cannot average into a lump parameter k i . We cannot tell when reactions will occur - i.e. the model is stochastic/probabilistic instead of deterministic. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  24. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation There are two general approaches to analyzing purely stochastic chemical kinetics systems: Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  25. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation There are two general approaches to analyzing purely stochastic chemical kinetics systems: 1 Evaluating sample trajectories/realizations. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  26. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation There are two general approaches to analyzing purely stochastic chemical kinetics systems: 1 Evaluating sample trajectories/realizations. 2 Analyzing the chemical master equation (models the probability distribution over the admissible states as a function of time). Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  27. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation There are two general approaches to analyzing purely stochastic chemical kinetics systems: 1 Evaluating sample trajectories/realizations. 2 Analyzing the chemical master equation (models the probability distribution over the admissible states as a function of time). Evaluating sample trajectories is simple to do numerically but not particularly insightful. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  28. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation There are two general approaches to analyzing purely stochastic chemical kinetics systems: 1 Evaluating sample trajectories/realizations. 2 Analyzing the chemical master equation (models the probability distribution over the admissible states as a function of time). Evaluating sample trajectories is simple to do numerically but not particularly insightful. Solving the chemical master equation is typically several orders of magnitude beyond impossible. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  29. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation An algorithm for evaluating sample trajectories was developed by Doob (1945) and Gillespie (1977) [1]: Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  30. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation An algorithm for evaluating sample trajectories was developed by Doob (1945) and Gillespie (1977) [1]: 1 Initialize reactant numbers n i (0). Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  31. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation An algorithm for evaluating sample trajectories was developed by Doob (1945) and Gillespie (1977) [1]: 1 Initialize reactant numbers n i (0). 2 Determine time τ until next reaction. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  32. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation An algorithm for evaluating sample trajectories was developed by Doob (1945) and Gillespie (1977) [1]: 1 Initialize reactant numbers n i (0). 2 Determine time τ until next reaction. 3 Determine next reaction. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  33. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation An algorithm for evaluating sample trajectories was developed by Doob (1945) and Gillespie (1977) [1]: 1 Initialize reactant numbers n i (0). 2 Determine time τ until next reaction. 3 Determine next reaction. 4 Step forward τ , update system and return to step 2. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  34. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation An algorithm for evaluating sample trajectories was developed by Doob (1945) and Gillespie (1977) [1]: 1 Initialize reactant numbers n i (0). 2 Determine time τ until next reaction. 3 Determine next reaction. 4 Step forward τ , update system and return to step 2. Typically carried out for a finite number of iterations or for a fixed amount of time. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  35. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation Consider the Michaelis-Menton enzyme reaction k + 1 k 2 S + E SE → P + E ⇄ k − 1 Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  36. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation Consider the Michaelis-Menton enzyme reaction k + 1 k 2 S + E SE → P + E ⇄ k − 1 Models the conversion of some substrate S into some product P via the enzyme E . Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  37. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation Consider the Michaelis-Menton enzyme reaction k + 1 k 2 S + E SE → P + E ⇄ k − 1 Models the conversion of some substrate S into some product P via the enzyme E . The deterministic model is a limiting case for n i → ∞ keeping n i / V constant. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  38. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation Figure: Comparison of deterministic and stochastic Michaelis-Menton enzyme mechanism. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  39. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation Figure: Comparison of deterministic and stochastic Michaelis-Menton enzyme mechanism. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  40. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation Figure: Comparison of deterministic and stochastic Michaelis-Menton enzyme mechanism. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  41. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation Figure: Comparison of deterministic and stochastic Michaelis-Menton enzyme mechanism. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  42. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation Evaluating sample trajectories is illustrative but not particularly enlightening. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  43. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation Evaluating sample trajectories is illustrative but not particularly enlightening. We are typically interested in questions such as: Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  44. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation Evaluating sample trajectories is illustrative but not particularly enlightening. We are typically interested in questions such as: 1 What is the probability that X i = n i , X i ≥ n i , etc., at time t ? Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  45. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation Evaluating sample trajectories is illustrative but not particularly enlightening. We are typically interested in questions such as: 1 What is the probability that X i = n i , X i ≥ n i , etc., at time t ? 2 Does the system have steady states? Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  46. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation Evaluating sample trajectories is illustrative but not particularly enlightening. We are typically interested in questions such as: 1 What is the probability that X i = n i , X i ≥ n i , etc., at time t ? 2 Does the system have steady states? If we sample enough trajectories we can build curves of best fit and confidence intervals but we will still miss many details. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  47. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation Evaluating sample trajectories is illustrative but not particularly enlightening. We are typically interested in questions such as: 1 What is the probability that X i = n i , X i ≥ n i , etc., at time t ? 2 Does the system have steady states? If we sample enough trajectories we can build curves of best fit and confidence intervals but we will still miss many details. We can do better - in fact, we can model the evolution of these probabilities explicitly ! Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  48. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation We will let P ( n ; t ) = P ( X 1 = n 1 , . . . , X m = n m ; t ) and m k i n j ! � λ i ( n ) = V | z i |− 1 ( n j − z ij )! j =1 denote the transition probability from one state to another. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  49. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation We will let P ( n ; t ) = P ( X 1 = n 1 , . . . , X m = n m ; t ) and m k i n j ! � λ i ( n ) = V | z i |− 1 ( n j − z ij )! j =1 denote the transition probability from one state to another. The chemical master equation is given by dP ( n ; t ) � λ i ( n + z i − z ′ i ) P ( n + z i − z ′ � = i ; t ) − P ( n ; t ) λ i ( n ) dt i ∈I i ∈O Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  50. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation We will let P ( n ; t ) = P ( X 1 = n 1 , . . . , X m = n m ; t ) and m k i n j ! � λ i ( n ) = V | z i |− 1 ( n j − z ij )! j =1 denote the transition probability from one state to another. The chemical master equation is given by dP ( n ; t ) � λ i ( n + z i − z ′ i ) P ( n + z i − z ′ � = i ; t ) − P ( n ; t ) λ i ( n ) dt i ∈I i ∈O where I are the reactions which lead into a given state... Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  51. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation We will let P ( n ; t ) = P ( X 1 = n 1 , . . . , X m = n m ; t ) and m k i n j ! � λ i ( n ) = V | z i |− 1 ( n j − z ij )! j =1 denote the transition probability from one state to another. The chemical master equation is given by dP ( n ; t ) � λ i ( n + z i − z ′ i ) P ( n + z i − z ′ � = i ; t ) − P ( n ; t ) λ i ( n ) dt i ∈I i ∈O where I are the reactions which lead into a given state and O are the reactions which lead from a given state. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  52. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation Reconsider the earlier system k 1 A 1 ⇄ 2 A 2 . k 2 Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  53. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation Reconsider the earlier system k 1 A 1 ⇄ 2 A 2 . k 2 Consider the states (2 , 0), (1 , 2), and (0 , 4)... Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  54. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation Reconsider the earlier system k 1 A 1 ⇄ 2 A 2 . k 2 Consider the states (2 , 0), (1 , 2), and (0 , 4), for which we have dP (2 , 0; t ) = 2 k 2 V P (1 , 2; t ) − 2 k 1 P (2 , 0; t ) dt dP (1 , 2; t ) = 12 k 2 � k 1 + 2 k 2 � P (0 , 4; t ) + 2 k 1 P (2 , 0; t ) − P (1 , 2; t ) dt V V dP (0 , 4; t ) = k 1 P (1 , 2; t ) − 12 k 2 P (0 , 4; t ) . dt V Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  55. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation Reconsider the earlier system k 1 A 1 ⇄ 2 A 2 . k 2 Consider the states (2 , 0), (1 , 2), and (0 , 4), for which we have dP (2 , 0; t ) = 2 k 2 V P (1 , 2; t ) − 2 k 1 P (2 , 0; t ) dt dP (1 , 2; t ) = 12 k 2 � k 1 + 2 k 2 � P (0 , 4; t ) + 2 k 1 P (2 , 0; t ) − P (1 , 2; t ) dt V V dP (0 , 4; t ) = k 1 P (1 , 2; t ) − 12 k 2 P (0 , 4; t ) . dt V This can be solved explicitly! Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  56. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation Nice features about the CME: Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  57. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation Nice features about the CME: 1 It is linear ! Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  58. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation Nice features about the CME: 1 It is linear ! 2 If it can be solved, it completely describes every aspect of the mechanism. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  59. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation Nice features about the CME: 1 It is linear ! 2 If it can be solved, it completely describes every aspect of the mechanism. Less-than-nice features about the CME: Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  60. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation Nice features about the CME: 1 It is linear ! 2 If it can be solved, it completely describes every aspect of the mechanism. Less-than-nice features about the CME: 1 It is typically massive (for unbounded systems, it is infinite-dimensional). Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  61. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation Nice features about the CME: 1 It is linear ! 2 If it can be solved, it completely describes every aspect of the mechanism. Less-than-nice features about the CME: 1 It is typically massive (for unbounded systems, it is infinite-dimensional). 2 Mass-action term must be computed for each state. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  62. Background Small-Scale Considerations Stochastic Models Gillespie Algorithm Interesting Systems Chemical Master Equation Nice features about the CME: 1 It is linear ! 2 If it can be solved, it completely describes every aspect of the mechanism. Less-than-nice features about the CME: 1 It is typically massive (for unbounded systems, it is infinite-dimensional). 2 Mass-action term must be computed for each state. 3 Connections between states can be complicated near the boundary. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  63. Background Lotka-Volterra System Stochastic Models The Block Interesting Systems 1 Background Chemical Reactions Standard Model Stoichiometric Compatibility Classes 2 Stochastic Models Small-Scale Considerations Gillespie Algorithm Chemical Master Equation 3 Interesting Systems Lotka-Volterra System The Block Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  64. Background Lotka-Volterra System Stochastic Models The Block Interesting Systems Modelling chemical kinetics systems stochastically can qualitatively change the dynamics of a system. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  65. Background Lotka-Volterra System Stochastic Models The Block Interesting Systems Modelling chemical kinetics systems stochastically can qualitatively change the dynamics of a system. Consider the Lotka-Volterra predator-prey system ( A 1 is the prey, A 2 is the predator) k 1 A 1 − → 2 A 1 k 2 A 1 + A 2 − → 2 A 2 k 3 A 2 − → O . Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  66. Background Lotka-Volterra System Stochastic Models The Block Interesting Systems Modelling chemical kinetics systems stochastically can qualitatively change the dynamics of a system. Consider the Lotka-Volterra predator-prey system ( A 1 is the prey, A 2 is the predator) k 1 A 1 − → 2 A 1 k 2 A 1 + A 2 − → 2 A 2 k 3 A 2 − → O . For the rate constant values k 1 = k 2 = k 3 = 1 the large-scale deterministic system has a unique positive equilibrium x ∗ 1 = x ∗ 2 = 1 which is a centre, however... Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  67. Background Lotka-Volterra System Stochastic Models The Block Interesting Systems Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  68. Background Lotka-Volterra System Stochastic Models The Block Interesting Systems The stable equilibrium concentration is no longer stable! (In fact, none of the stable periodic orbits are stable.) Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  69. Background Lotka-Volterra System Stochastic Models The Block Interesting Systems The stable equilibrium concentration is no longer stable! (In fact, none of the stable periodic orbits are stable.) Oscillatory behaviour remains but appears almost chaotic . Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  70. Background Lotka-Volterra System Stochastic Models The Block Interesting Systems The stable equilibrium concentration is no longer stable! (In fact, none of the stable periodic orbits are stable.) Oscillatory behaviour remains but appears almost chaotic . Furthermore, extinction events which were not possible in the continuous, deterministic system are now possible. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  71. Background Lotka-Volterra System Stochastic Models The Block Interesting Systems The stable equilibrium concentration is no longer stable! (In fact, none of the stable periodic orbits are stable.) Oscillatory behaviour remains but appears almost chaotic . Furthermore, extinction events which were not possible in the continuous, deterministic system are now possible. Although it is unlikely for either the predator or the prey to go extinct, it is irreversible — carried over a long enough time scale, extinction is the inevitable outcome of the system! Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  72. Background Lotka-Volterra System Stochastic Models The Block Interesting Systems Horn and Jackson consider the following system in their seminal paper “General Mass Action Kinetics” [2]: 1 2 A 1 + A 2 − → 3 A 1 ǫ ↑ ↓ ǫ 1 3 A 2 ← − A 1 + 2 A 2 . Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  73. Background Lotka-Volterra System Stochastic Models The Block Interesting Systems Horn and Jackson consider the following system in their seminal paper “General Mass Action Kinetics” [2]: 1 2 A 1 + A 2 − → 3 A 1 ǫ ↑ ↓ ǫ 1 3 A 2 ← − A 1 + 2 A 2 . The system exhibits varying behaviour depending on the value of ǫ : Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  74. Background Lotka-Volterra System Stochastic Models The Block Interesting Systems Horn and Jackson consider the following system in their seminal paper “General Mass Action Kinetics” [2]: 1 2 A 1 + A 2 − → 3 A 1 ǫ ↑ ↓ ǫ 1 3 A 2 ← − A 1 + 2 A 2 . The system exhibits varying behaviour depending on the value of ǫ : 1 ǫ ≥ 1 / 6: one stable equilibrium. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  75. Background Lotka-Volterra System Stochastic Models The Block Interesting Systems Horn and Jackson consider the following system in their seminal paper “General Mass Action Kinetics” [2]: 1 2 A 1 + A 2 − → 3 A 1 ǫ ↑ ↓ ǫ 1 3 A 2 ← − A 1 + 2 A 2 . The system exhibits varying behaviour depending on the value of ǫ : 1 ǫ ≥ 1 / 6: one stable equilibrium. 2 0 < ǫ < 1 / 6: two stable and one unstable equilibria. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  76. Background Lotka-Volterra System Stochastic Models The Block Interesting Systems Horn and Jackson consider the following system in their seminal paper “General Mass Action Kinetics” [2]: 1 2 A 1 + A 2 − → 3 A 1 ǫ ↑ ↓ ǫ 1 3 A 2 ← − A 1 + 2 A 2 . The system exhibits varying behaviour depending on the value of ǫ : 1 ǫ ≥ 1 / 6: one stable equilibrium. 2 0 < ǫ < 1 / 6: two stable and one unstable equilibria. 3 ǫ = 0: two stable boundary equilibria. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

  77. Background Lotka-Volterra System Stochastic Models The Block Interesting Systems (a) (b) (c) Figure: Block system with (a) ǫ ≥ 1 / 6, (b) 0 < ǫ < 1 / 6, and (c) ǫ = 0. Matthew Douglas Johnston Stochastic Chemical Reaction Networks

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