SCU Colloquium Series: Extinction and Persistence in Discrete - PowerPoint PPT Presentation

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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