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Background Stochastic Extinction Future Work SCU Colloquium Series: Extinction and Persistence in Discrete Chemical Reaction Networks Matthew Douglas Johnston Assistant Professor San Jose State University One Washington Square San Jose, CA

  1. Background Overview Stochastic Extinction Deterministic Model Future Work Stochastic Model Protein activation model: ( A inactive, B active) x A = − α x A x B + β x B ˙ x B = ˙ α x A x B − β x B . Conservation law: (Total proteins constant) x A + ˙ ˙ x B = 0 = ⇒ x A + x B = x A (0) + x B (0) = C . Reduces 2 -D system to 1 -D! � x A = − α x A ( C − x A ) + β ( C − x A ) ˙ = ⇒ x B = C − x A Matthew Douglas Johnston Extinction in Discrete CRNs

  2. Background Overview Stochastic Extinction Deterministic Model Future Work Stochastic Model X 3 X 1 X 2 Figure: State space is partitioned (invariant spaces) Matthew Douglas Johnston Extinction in Discrete CRNs

  3. Background Overview Stochastic Extinction Deterministic Model Future Work Stochastic Model X 3 X 1 X 0 X 2 Figure: State space is partitioned (invariant spaces) Matthew Douglas Johnston Extinction in Discrete CRNs

  4. Background Overview Stochastic Extinction Deterministic Model Future Work Stochastic Model X 3 X* X 1 X 0 X 2 Figure: State space is partitioned (invariant spaces) Matthew Douglas Johnston Extinction in Discrete CRNs

  5. Background Overview Stochastic Extinction Deterministic Model Future Work Stochastic Model Stochastic Model: If molecular counts are low (e.g. biochemical/genetic models), more realistic to model systems stochastically . Keep track of reactant numbers : X i ∈ { 0 , 1 , 2 , . . . } Reactions occur discretely and at separate times Modeled as a continuous time Markov chain (CTMC): �� t r � � · ( y ′ X ( t ) = X (0) + λ i ( X ( s )) ds i − y i ) Y i 0 k =1 Matthew Douglas Johnston Extinction in Discrete CRNs

  6. Background Overview Stochastic Extinction Deterministic Model Future Work Stochastic Model Stochastic Model: If molecular counts are low (e.g. biochemical/genetic models), more realistic to model systems stochastically . Keep track of reactant numbers : X i ∈ { 0 , 1 , 2 , . . . } Reactions occur discretely and at separate times Modeled as a continuous time Markov chain (CTMC): �� t r � � · ( y ′ X ( t ) = X (0) + λ i ( X ( s )) ds i − y i ) Y i 0 k =1 Y i ( · ) a unit-rate Poisson process Matthew Douglas Johnston Extinction in Discrete CRNs

  7. Background Overview Stochastic Extinction Deterministic Model Future Work Stochastic Model Protein activation model: � A + B − → 2 B B − → A Deterministic Stochastic B B * 4 * 4 * * * * * 3 * 3 * * * * 2 2 * * * * * * 1 S+x 0 * * 1 * * * * A * * * * * * A 0 4 0 1 2 3 2 3 4 1 Moral of the story: State space is the same but discretized ! Matthew Douglas Johnston Extinction in Discrete CRNs

  8. Background Overview Stochastic Extinction Deterministic Model Future Work Stochastic Model Two approaches for analyzing Continuous Time Markov Chains. 1 Generate sample paths : Think numerical integration with noise Commonly simulated with Gillespie’s Algorithm (can be computationally-intensive!) (Gillespie, 1976 [7]) 2 Evolve probability distribution : Tracks probability of being in a given state at a given time Evolution given by Chemical Master Equation (linear ODE but very high-dimensional!) Matthew Douglas Johnston Extinction in Discrete CRNs

  9. Background Overview Stochastic Extinction Deterministic Model Future Work Stochastic Model 3.5 3 2.5 2 A B 1.5 1 0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Sample paths are generated using Gillespie’s Algorithm [7] (think deterministic plus noise). Matthew Douglas Johnston Extinction in Discrete CRNs

  10. Background Overview Stochastic Extinction Deterministic Model Future Work Stochastic Model 3.5 3 2.5 2 A B 1.5 1 0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Sample paths are generated using Gillespie’s Algorithm [7] (think deterministic plus noise). Matthew Douglas Johnston Extinction in Discrete CRNs

  11. Background Overview Stochastic Extinction Deterministic Model Future Work Stochastic Model 3.5 3 2.5 2 A B 1.5 1 0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Sample paths are generated using Gillespie’s Algorithm [7] (think deterministic plus noise). Matthew Douglas Johnston Extinction in Discrete CRNs

  12. Background Overview Stochastic Extinction Deterministic Model Future Work Stochastic Model Time = 0 Time = 0.5 1 0.1 0.8 0.08 Probability Probability 0.6 0.06 0.4 0.04 0.2 0.02 0 0 0 10 20 30 40 0 10 20 30 40 molecules of A molecules of A Time = 1 Time = 5 0.1 0.2 0.08 0.15 Probability Probability 0.06 0.1 0.04 0.05 0.02 0 0 0 10 20 30 40 0 10 20 30 40 molecules of A molecules of A Probability evolution can be determined through the Chemical Master Equation (Kolmogorov’s forward equations) Matthew Douglas Johnston Extinction in Discrete CRNs

  13. Background Overview Stochastic Extinction Deterministic Model Future Work Stochastic Model Deterministic and stochastic models converge probabilistically in scaling as V → ∞ (Kurtz, 1971 [11]): a ( t ) = A ( t ) b ( t ) = B ( t ) and . V V V = 1 4 3 2 A B 1 0 0 1 2 3 4 5 Matthew Douglas Johnston Extinction in Discrete CRNs

  14. Background Overview Stochastic Extinction Deterministic Model Future Work Stochastic Model Deterministic and stochastic models converge probabilistically in scaling as V → ∞ (Kurtz, 1971 [11]): a ( t ) = A ( t ) b ( t ) = B ( t ) and . V V V = 10 4 3 2 A B 1 0 0 1 2 3 4 5 Matthew Douglas Johnston Extinction in Discrete CRNs

  15. Background Overview Stochastic Extinction Deterministic Model Future Work Stochastic Model Deterministic and stochastic models converge probabilistically in scaling as V → ∞ (Kurtz, 1971 [11]): a ( t ) = A ( t ) b ( t ) = B ( t ) and . V V V = 100 4 3 2 A B 1 0 0 1 2 3 4 5 Matthew Douglas Johnston Extinction in Discrete CRNs

  16. Background Overview Stochastic Extinction Deterministic Model Future Work Stochastic Model Deterministic and stochastic models converge probabilistically in scaling as V → ∞ (Kurtz, 1971 [11]): a ( t ) = A ( t ) b ( t ) = B ( t ) and . V V V = 1000 4 3 2 A B 1 0 0 1 2 3 4 5 Matthew Douglas Johnston Extinction in Discrete CRNs

  17. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) 1 Background Overview Deterministic Model Stochastic Model 2 Stochastic Extinction Background Network Properties Conditions (technical details!) 3 Future Work Matthew Douglas Johnston Extinction in Discrete CRNs

  18. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) CAUTION: Discretization can create extinction events not permissible in the deterministic model! Stochastic Protein activation model: B * 4 * * * * � A + B − → 2 B * 3 * * * * B − → A 2 * * * * * It is possible to irreversibly lose the 1 * * * * * last activated protein through * * * * A reaction B − → A . 0 3 4 1 2 Matthew Douglas Johnston Extinction in Discrete CRNs

  19. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) CAUTION: Discretization can create extinction events not permissible in the deterministic model! Stochastic Protein activation model: B * 4 * * * * � A + B − → 2 B * 3 * * * * B − → A 2 * * * * * It is possible to irreversibly lose the 1 * * * * * − → ← − last activated protein through . * * * * A reaction B − → A . 0 3 4 1 2 Matthew Douglas Johnston Extinction in Discrete CRNs

  20. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) V = 1 5 4 3 A 2 B 1 0 0 2 4 6 8 10 On the unbounded interval [0 , ∞ ) extinction is inevitable . The deterministic and stochastic models do not agree . Matthew Douglas Johnston Extinction in Discrete CRNs

  21. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Extinction phenomena in well-known in many stochastic models: Population biology models (e.g. Lotka-Volterra, etc.) Disease spread models (e.g. SIS, SIR, SIRS models etc.) Chemical reaction models (e.g. Keizer’s paradox (1987 [10]), Anderson et al. (2014 [1]), Brijder (2015 [2]), etc.) Alternative statistics required (e.g. expected time to extinction, quasi-stationary distribution, etc.) Matthew Douglas Johnston Extinction in Discrete CRNs

  22. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) For realistic biochemical networks, it may not be obvious that a stochastic extinction event occurs! Signaling Pathway: (Shinar and Feinberg, 2010 [12]) 1 3 5 X p XD X XT 2 4 6 8 X p + Y X p Y X + Y p 7 9 11 XD + Y p XDY p XD + Y 10 Matthew Douglas Johnston Extinction in Discrete CRNs

  23. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) For realistic biochemical networks, it may not be obvious that a stochastic extinction event occurs! Signaling Pathway: (Shinar and Feinberg, 2010 [12]) 1 3 5 X p XD X XT 2 4 6 8 X p + Y X p Y X + Y p 7 9 11 XD + Y p XDY p XD + Y 10 Stochastic system inevitably converges to an extinction state ( X p = X tot , Y p = Y tot , rest = 0) (Anderson et al. , 2014 [1]) Matthew Douglas Johnston Extinction in Discrete CRNs

  24. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Objective: We want to derive network-based conditions for extinction and/or persistence of chemical species. Tools: Use elements from Chemical Reaction Network Theory (CRNT) and Petri Net Theory . CRNT has been used extensively for deterministic models: Deficiency Zero Theorem [3, 8, 9] Deficiency One Theorem/Algorithm [4, 6, 5] Matthew Douglas Johnston Extinction in Discrete CRNs

  25. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Technical details ahead! Matthew Douglas Johnston Extinction in Discrete CRNs

  26. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Technique: Construct conditions for system to exhibit non-extinction (and then violate them!) We are going to need two things : 1 Sequence of reactions corresponding to modes of stoichiometric balance (a priori known) 2 Sequence of reactions corresponding to admissible recurrent behavior under the assumption of non-exinction. Matthew Douglas Johnston Extinction in Discrete CRNs

  27. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Technique: Construct conditions for system to exhibit non-extinction (and then violate them!) We are going to need two things : 1 Sequence of reactions corresponding to modes of stoichiometric balance (a priori known) 2 Sequence of reactions corresponding to admissible recurrent behavior under the assumption of non-exinction. Matthew Douglas Johnston Extinction in Discrete CRNs

  28. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Observation #1: State of system at time t ≥ 0 is given by X ( t ) = X (0) + Γ N ( t ) where Γ is stoichiometric matrix and N ( t ) contains reaction counts. If a state recurs (i.e. X ( t ) = X (0)) then we have N ( t ) ∈ ker(Γ) ∩ R r ≥ 0 . Generators of cone ker(Γ) ∩ R r ≥ 0 are known ( stoichiometric flux modes ) Matthew Douglas Johnston Extinction in Discrete CRNs

  29. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Signaling Pathway: 1 3 5 X p XD X XT 2 4 6 8 X p + Y X p Y X + Y p 7 9 11 XD + Y p XDY p XD + Y 10   − 1 1 0 0 0 0 0 0 − 1 1 1 1 − 1 − 1 1 0 0 0 0 0 0 0     0 0 1 − 1 − 1 0 0 0 0 0 0     0 0 0 0 1 − 1 1 0 0 0 0   Γ =   0 0 0 0 0 − 1 1 0 0 0 0     0 0 0 0 0 1 − 1 − 1 0 0 0     0 0 0 0 0 0 0 1 − 1 1 0   − 1 − 1 0 0 0 0 0 0 0 0 1 Matthew Douglas Johnston Extinction in Discrete CRNs

  30. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Signaling Pathway: 1 3 5 X p XD X XT 2 4 6 8 X p + Y X p Y X + Y p 7 9 11 XD + Y p XDY p XD + Y 10   − 1 1 0 0 0 0 0 0 − 1 1 1 1 − 1 − 1 1 0 0 0 0 0 0 0     0 0 1 − 1 − 1 0 0 0 0 0 0     0 0 0 0 1 − 1 1 0 0 0 0   Γ =   0 0 0 0 0 − 1 1 0 0 0 0     0 0 0 0 0 1 − 1 − 1 0 0 0     0 0 0 0 0 0 0 1 − 1 1 0   − 1 − 1 0 0 0 0 0 0 0 0 1 Matthew Douglas Johnston Extinction in Discrete CRNs

  31. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Signaling Pathway: 1 3 5 X p XD X XT 2 4 6 8 X p + Y X p Y X + Y p 7 9 11 XD + Y p XDY p XD + Y 10   − 1 1 0 0 0 0 0 0 − 1 1 1 1 − 1 − 1 1 0 0 0 0 0 0 0     0 0 1 − 1 − 1 0 0 0 0 0 0     0 0 0 0 1 − 1 1 0 0 0 0   Γ =   0 0 0 0 0 − 1 1 0 0 0 0     0 0 0 0 0 1 − 1 − 1 0 0 0     0 0 0 0 0 0 0 1 − 1 1 0   − 1 − 1 0 0 0 0 0 0 0 0 1 Matthew Douglas Johnston Extinction in Discrete CRNs

  32. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Signaling Pathway: 1 3 5 X p XD X XT 2 4 6 8 X p + Y X p Y X + Y p 7 9 11 XD + Y p XDY p XD + Y 10 Stoichiometric Flux Modes: Trivial, e.g. (1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0) ∈ ker(Γ) Non-trivial, e.g. (0 , 0 , 1 , 0 , 1 , 1 , 0 , 1 , 1 , 0 , 1) ∈ ker(Γ) Matthew Douglas Johnston Extinction in Discrete CRNs

  33. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Signaling Pathway: 1 3 5 X p XD X XT 2 4 6 8 X p + Y X p Y X + Y p 7 9 11 XD + Y p XDY p XD + Y 10 Stoichiometric Flux Modes: Trivial, e.g. (1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0) ∈ ker(Γ) Non-trivial, e.g. (0 , 0 , 1 , 0 , 1 , 1 , 0 , 1 , 1 , 0 , 1) ∈ ker(Γ) Matthew Douglas Johnston Extinction in Discrete CRNs

  34. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Signaling Pathway: 1 3 5 X p XD X XT 2 4 6 8 X p + Y X p Y X + Y p 7 9 11 XD + Y p XDY p XD + Y 10 Stoichiometric Flux Modes: Trivial, e.g. (1 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0) ∈ ker(Γ) Non-trivial, e.g. (0 , 0 , 1 , 0 , 1 , 1 , 0 , 1 , 1 , 0 , 1) ∈ ker(Γ) Matthew Douglas Johnston Extinction in Discrete CRNs

  35. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Objective: S1 Construct another sequence of balancing reactions by assuming recurrence! S2 Compare resulting vector of counts to known stoichiometric flux modes (i.e. generators of ker(Γ) ∩ R r ≥ 0 ). Technical assumptions: A1 Assume there is a reaction from a non-terminal complex which is recurrent (i.e. may always fire). A2 Assume system has finite state space . Matthew Douglas Johnston Extinction in Discrete CRNs

  36. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Observation #2: If a reaction can occur from one complex, it can occur from any complex further down a directed path ! 1 3 5 X p XD X XT 2 4 6 8 X p + Y X p Y X + Y p 7 9 11 XD + Y p XDY p XD + Y 10 Can only become “trapped” on terminal components . Matthew Douglas Johnston Extinction in Discrete CRNs

  37. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Observation #2: If a reaction can occur from one complex, it can occur from any complex further down a directed path ! 1 3 5 X p XD X XT 2 4 6 8 X p + Y X p Y X + Y p 7 9 11 XD + Y p XDY p XD + Y 10 Can only become “trapped” on terminal components . Matthew Douglas Johnston Extinction in Discrete CRNs

  38. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Observation #2: If a reaction can occur from one complex, it can occur from any complex further down a directed path ! 1 3 5 X p XD X XT 2 4 6 8 X p + Y X p Y X + Y p 7 9 11 XD + Y p XDY p XD + Y 10 Can only become “trapped” on terminal components . Matthew Douglas Johnston Extinction in Discrete CRNs

  39. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Observation #2: If a reaction can occur from one complex, it can occur from any complex further down a directed path ! 1 3 5 X p XD X XT 2 4 6 8 X p + Y X p Y X + Y p 7 9 11 XD + Y p XDY p XD + Y 10 Can only become “trapped” on terminal components . Matthew Douglas Johnston Extinction in Discrete CRNs

  40. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Observation #2: If a reaction can occur from one complex, it can occur from any complex further down a directed path ! 1 3 5 X p XD X XT 2 4 6 8 X p + Y X p Y X + Y p 7 9 11 XD + Y p XDY p XD + Y 10 Can only become “trapped” on terminal components . Matthew Douglas Johnston Extinction in Discrete CRNs

  41. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Observation #2: If a reaction can occur from one complex, it can occur from any complex further down a directed path ! 1 3 5 X p XD X XT 2 4 6 8 X p + Y X p Y X + Y p 7 9 11 XD + Y p XDY p XD + Y 10 Can only become “trapped” on terminal components . Matthew Douglas Johnston Extinction in Discrete CRNs

  42. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Observation #2: If a reaction can occur from one complex, it can occur from any complex further down a directed path ! 1 3 5 X p XD X XT 2 4 6 8 X p + Y X p Y X + Y p 7 9 11 XD + Y p XDY p XD + Y 10 Can only become “trapped” on terminal components . Matthew Douglas Johnston Extinction in Discrete CRNs

  43. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Observation #3: If a reaction can occur from one complex, it can occur from any complex with equal or lower multiplicity in each species! 1 3 5 X p XD X XT 2 4 6 8 X p + Y X p Y X + Y p 7 9 XD + Y p XDY p 11 XD + Y 10 Notice X + Y p “dominates” X and XD + Y “dominates” XD . Matthew Douglas Johnston Extinction in Discrete CRNs

  44. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Observation #3: If a reaction can occur from one complex, it can occur from any complex with equal or lower multiplicity in each species! 1 3 5 X p XD X XT 2 4 6 8 X p + Y X p Y X + Y p 7 9 XD + Y p XDY p 11 XD + Y 10 Notice X + Y p “dominates” X and XD + Y “dominates” XD . Matthew Douglas Johnston Extinction in Discrete CRNs

  45. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Observation #3: If a reaction can occur from one complex, it can occur from any complex with equal or lower multiplicity in each species! 1 3 5 X p XD X XT 2 4 6 8 X p + Y X p Y X + Y p 7 9 XD + Y p XDY p 11 XD + Y 10 Notice X + Y p “dominates” X and XD + Y “dominates” XD . Matthew Douglas Johnston Extinction in Discrete CRNs

  46. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Observation #3: If a reaction can occur from one complex, it can occur from any complex with equal or lower multiplicity in each species! 1 3 5 X p XD X XT 2 4 6 8 X p + Y X p Y X + Y p 7 9 XD + Y p XDY p 11 XD + Y 10 Notice X + Y p “dominates” X and XD + Y “dominates” XD . Matthew Douglas Johnston Extinction in Discrete CRNs

  47. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Observation #3: If a reaction can occur from one complex, it can occur from any complex with equal or lower multiplicity in each species! 1 3 5 X p XD X XT 2 4 6 8 X p + Y X p Y X + Y p 7 9 XD + Y p XDY p 11 XD + Y 10 Notice X + Y p “dominates” X and XD + Y “dominates” XD . Matthew Douglas Johnston Extinction in Discrete CRNs

  48. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Observation #3: If a reaction can occur from one complex, it can occur from any complex with equal or lower multiplicity in each species! 1 3 5 X p XD X XT 2 4 6 8 X p + Y X p Y X + Y p 7 9 XD + Y p XDY p 11 XD + Y 10 Notice X + Y p “dominates” X and XD + Y “dominates” XD . Matthew Douglas Johnston Extinction in Discrete CRNs

  49. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Observation #3: If a reaction can occur from one complex, it can occur from any complex with equal or lower multiplicity in each species! 1 3 5 X p XD X XT 2 4 6 8 X p + Y X p Y X + Y p 7 9 XD + Y p XDY p 11 XD + Y 10 Notice X + Y p “dominates” X and XD + Y “dominates” XD . Matthew Douglas Johnston Extinction in Discrete CRNs

  50. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Reconstruct the network to capture these new connections ! 9 11 XD + Y p XDY p XD + Y 10 D D 1 3 5 X p XD X XT 2 4 D 7 8 X + Y p X p Y X p + Y 6 Only one terminal component in new graph. Matthew Douglas Johnston Extinction in Discrete CRNs

  51. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Reconstruct the network to capture these new connections ! 9 11 XD + Y p XDY p XD + Y 10 D D 1 3 5 X p XD X XT 2 4 D 7 8 X + Y p X p Y X p + Y 6 Only one terminal component in new graph. Matthew Douglas Johnston Extinction in Discrete CRNs

  52. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Reconstruct the network to capture these new connections ! 9 11 XD + Y p XDY p XD + Y 10 D D 1 3 5 X p XD X XT 2 4 D 7 8 X + Y p X p Y X p + Y 6 Only one terminal component in new graph. Matthew Douglas Johnston Extinction in Discrete CRNs

  53. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Reconstruct the network to capture these new connections ! 9 11 XD + Y p XDY p XD + Y 10 D D 1 3 5 X p XD X XT 2 4 D 7 8 X + Y p X p Y X p + Y 6 Only one terminal component in new graph. Matthew Douglas Johnston Extinction in Discrete CRNs

  54. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Reconstruct the network to capture these new connections ! 9 11 XD + Y p XDY p XD + Y 10 D D 1 3 5 X p XD X XT 2 4 D 7 8 X + Y p X p Y X p + Y 6 Only one terminal component in new graph. Matthew Douglas Johnston Extinction in Discrete CRNs

  55. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Reconstruct the network to capture these new connections ! 9 11 XD + Y p XDY p XD + Y 10 D D 1 3 5 X p XD X XT 2 4 D 7 8 X + Y p X p Y X p + Y 6 Only one terminal component in new graph. Matthew Douglas Johnston Extinction in Discrete CRNs

  56. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Reconstruct the network to capture these new connections ! 9 11 XD + Y p XDY p XD + Y 10 D D 1 3 5 X p XD X XT 2 4 D 7 8 X + Y p X p Y X p + Y 6 Only one terminal component in new graph. Matthew Douglas Johnston Extinction in Discrete CRNs

  57. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Reconstruct the network to capture these new connections ! 9 11 XD + Y p XDY p XD + Y 10 D D 1 3 5 X p XD X XT 2 4 D 7 8 X + Y p X p Y X p + Y 6 Only one terminal component in new graph. Matthew Douglas Johnston Extinction in Discrete CRNs

  58. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Question: How do we construct a meaningful sequence of reactions? Basic Idea: (Brijder, 2015 [2]) 1 Fix a subset of reactions which guarantee full access to reaction graph. 2 Use A1 to move to non-terminal region of reaction graph. 3 Follow regimented sequence of reactions from non-terminal complex to terminal component, and then repeat. 4 Use A2 to construct a vector of counts v . Matthew Douglas Johnston Extinction in Discrete CRNs

  59. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Question: How do we construct a meaningful sequence of reactions? Basic Idea: (Brijder, 2015 [2]) 1 Fix a subset of reactions which guarantee full access to reaction graph. 2 Use A1 to move to non-terminal region of reaction graph. 3 Follow regimented sequence of reactions from non-terminal complex to terminal component, and then repeat. 4 Use A2 to construct a vector of counts v . Matthew Douglas Johnston Extinction in Discrete CRNs

  60. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Question: How do we construct a meaningful sequence of reactions? Basic Idea: (Brijder, 2015 [2]) 1 Fix a subset of reactions which guarantee full access to reaction graph. 2 Use A1 to move to non-terminal region of reaction graph. 3 Follow regimented sequence of reactions from non-terminal complex to terminal component, and then repeat. 4 Use A2 to construct a vector of counts v . Matthew Douglas Johnston Extinction in Discrete CRNs

  61. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Question: How do we construct a meaningful sequence of reactions? Basic Idea: (Brijder, 2015 [2]) 1 Fix a subset of reactions which guarantee full access to reaction graph. 2 Use A1 to move to non-terminal region of reaction graph. 3 Follow regimented sequence of reactions from non-terminal complex to terminal component, and then repeat. 4 Use A2 to construct a vector of counts v . Matthew Douglas Johnston Extinction in Discrete CRNs

  62. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Question: How do we construct a meaningful sequence of reactions? Basic Idea: (Brijder, 2015 [2]) 1 Fix a subset of reactions which guarantee full access to reaction graph. 2 Use A1 to move to non-terminal region of reaction graph. 3 Follow regimented sequence of reactions from non-terminal complex to terminal component, and then repeat. 4 Use A2 to construct a vector of counts v . Matthew Douglas Johnston Extinction in Discrete CRNs

  63. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Observation #4: We can fix the reactions to form a rooted tree ! 9 11 XD + Y p XDY p XD + Y 10 D D 1 3 5 X p XD X XT 2 4 D 7 8 X + Y p X p Y X p + Y 6 NOTE: Unique path from each non-terminal complex to terminal component ! Matthew Douglas Johnston Extinction in Discrete CRNs

  64. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Basic Idea (con’t) Following rooted tree until a state recurs ! (i.e. X ( t ) = X (0)) (This is guaranteed by the finite state space assumption.) Resulting vector of counts must satisfy... 1 Has support contained in a given rooted tree . 2 Is weighted toward terminal component . 3 No restrictions within terminal components (if applicable). Matthew Douglas Johnston Extinction in Discrete CRNs

  65. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Conditions for extinction: This can give conditions for extinction ! Example 1: 1 3 2 X 1 X 1 + X 2 2 X 2 2 Stoic. Modes: ker(Γ) = span { (1 , 1 , 0) , (0 , 1 , 1) } Rooted Tree: v = ( v 1 , 0 , v 3 ) , 0 ≤ v 1 ≤ v 3 Matthew Douglas Johnston Extinction in Discrete CRNs

  66. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Conditions for extinction: This can give conditions for extinction ! Example 1: 1 3 2 X 1 X 1 + X 2 2 X 2 2 Stoic. Modes: ker(Γ) = span { (1 , 1 , 0) , (0 , 1 , 1) } Rooted Tree: v = ( v 1 , 0 , v 3 ) , 0 ≤ v 1 ≤ v 3 Matthew Douglas Johnston Extinction in Discrete CRNs

  67. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Conditions for extinction: This can give conditions for extinction ! Example 1: 1 3 2 X 1 X 1 + X 2 2 X 2 2 Stoic. Modes: ker(Γ) = span { (1 , 1 , 0) , (0 , 1 , 1) } Rooted Tree: v = ( v 1 , 0 , v 3 ) , 0 ≤ v 1 ≤ v 3 Matthew Douglas Johnston Extinction in Discrete CRNs

  68. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Conditions for extinction: This can give conditions for extinction ! Example 1: 1 3 2 X 1 X 1 + X 2 2 X 2 2 Stoic. Modes: ker(Γ) = span { (1 , 1 , 0) , (0 , 1 , 1) } Rooted Tree: v = ( v 1 , 0 , v 3 ) , 0 ≤ v 1 ≤ v 3 v �∈ ker(Γ) = ⇒ Extinction! Matthew Douglas Johnston Extinction in Discrete CRNs

  69. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Example 2: 1 3 X 1 + X 2 2 X 1 2 X 2 2 Stoic. modes: ker(Γ) = span { (1 , 1 , 0) , (2 , 0 , 1) } Rooted Tree: v = ( v 1 , 0 , v 3 ) , 0 ≤ v 1 ≤ v 3 Matthew Douglas Johnston Extinction in Discrete CRNs

  70. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Example 2: 1 3 X 1 + X 2 2 X 1 2 X 2 2 Stoic. modes: ker(Γ) = span { (1 , 1 , 0) , (2 , 0 , 1) } Rooted Tree: v = ( v 1 , 0 , v 3 ) , 0 ≤ v 1 ≤ v 3 Matthew Douglas Johnston Extinction in Discrete CRNs

  71. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Example 2: 1 3 X 1 + X 2 2 X 1 2 X 2 2 Stoic. modes: ker(Γ) = span { (1 , 1 , 0) , (2 , 0 , 1) } Rooted Tree: v = ( v 1 , 0 , v 3 ) , 0 ≤ v 1 ≤ v 3 Matthew Douglas Johnston Extinction in Discrete CRNs

  72. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Example 2: 1 3 X 1 + X 2 2 X 1 2 X 2 2 Stoic. modes: ker(Γ) = span { (1 , 1 , 0) , (2 , 0 , 1) } Rooted Tree: v = ( v 1 , 0 , v 3 ) , 0 ≤ v 1 ≤ v 3 v �∈ ker(Γ) = ⇒ Extinction! Matthew Douglas Johnston Extinction in Discrete CRNs

  73. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Signaling Pathway: 9 11 XD + Y p XDY p XD + Y 10 D D 1 3 5 X p XD X XT 2 4 D 7 8 X + Y p X p Y X p + Y 6 Stoic. modes: ker(Γ) = (0 , 0 , 1 , 0 , 1 , 1 , 0 , 1 , 1 , 0 , 1) + cycles Rooted tree: v = ( v 1 , 0 , v 3 , 0 , v 5 , v 6 , 0 , v 8 , 0 , v 10 , 0), 0 ≤ v 10 ≤ v 1 ≤ v 3 ≤ v 5 . . 0 ≤ v 6 ≤ v 8 ≤ v 3 ≤ v 5 Matthew Douglas Johnston Extinction in Discrete CRNs

  74. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Signaling Pathway: 9 11 XD + Y p XDY p XD + Y 10 D D 1 3 5 X p XD X XT 2 4 D 7 8 X + Y p X p Y X p + Y 6 Stoic. modes: ker(Γ) = (0 , 0 , 1 , 0 , 1 , 1 , 0 , 1 , 1 , 0 , 1) + cycles Rooted tree: v = ( v 1 , 0 , v 3 , 0 , v 5 , v 6 , 0 , v 8 , 0 , v 10 , 0), 0 ≤ v 10 ≤ v 1 ≤ v 3 ≤ v 5 . . 0 ≤ v 6 ≤ v 8 ≤ v 3 ≤ v 5 Matthew Douglas Johnston Extinction in Discrete CRNs

  75. Background Background Stochastic Extinction Network Properties Future Work Conditions (technical details!) Signaling Pathway: 9 11 XD + Y p XDY p XD + Y 10 D D 1 3 5 X p XD X XT 2 4 D 7 8 X + Y p X p Y X p + Y 6 Stoic. modes: ker(Γ) = (0 , 0 , 1 , 0 , 1 , 1 , 0 , 1 , 1 , 0 , 1) + cycles Rooted tree: v = ( v 1 , 0 , v 3 , 0 , v 5 , v 6 , 0 , v 8 , 0 , v 10 , 0), 0 ≤ v 10 ≤ v 1 ≤ v 3 ≤ v 5 . . 0 ≤ v 6 ≤ v 8 ≤ v 3 ≤ v 5 Matthew Douglas Johnston Extinction in Discrete CRNs

  76. Background Stochastic Extinction Future Work 1 Background Overview Deterministic Model Stochastic Model 2 Stochastic Extinction Background Network Properties Conditions (technical details!) 3 Future Work Matthew Douglas Johnston Extinction in Discrete CRNs

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