Stochastic Modelling of CRNs Combinatorial and algebraic approaches - PowerPoint PPT Presentation

Background Chemical Reaction Networks Connections Between Approaches Stochastic vs. Deterministic Modelling Future Work? Steady States vs. Stationary Distributions A + B − → 2 B Consider following model: − → A B Matthew Douglas Johnston Stochastic CRNs

Background Chemical Reaction Networks Connections Between Approaches Stochastic vs. Deterministic Modelling Future Work? Steady States vs. Stationary Distributions A + B − → 2 B Consider following model: − → A B Stochastic Deterministic B 4 * * * * * * * 3 * * * 2 * * * * * 1 * * * * * * * * * A 0 4 1 2 3 Matthew Douglas Johnston Stochastic CRNs

Background Chemical Reaction Networks Connections Between Approaches Stochastic vs. Deterministic Modelling Future Work? Steady States vs. Stationary Distributions A + B − → 2 B Consider following model: − → A B Stochastic Deterministic B 4 * * * * * * * 3 * * * 2 * * * * * 1 * * * * * * * * * A 0 4 1 2 3 Moral of the story: State space is the same but discretized ! Matthew Douglas Johnston Stochastic CRNs

Background Chemical Reaction Networks Connections Between Approaches Stochastic vs. Deterministic Modelling Future Work? Steady States vs. Stationary Distributions k Consider the simple bacterial growth model : B − → 2 B . Matthew Douglas Johnston Stochastic CRNs

Background Chemical Reaction Networks Connections Between Approaches Stochastic vs. Deterministic Modelling Future Work? Steady States vs. Stationary Distributions k Consider the simple bacterial growth model : B − → 2 B . Deterministic ODE: db ( t ) = kb ( t ) , b (0) = b 0 dt System has the solution b ( t ) = b 0 e kt Matthew Douglas Johnston Stochastic CRNs

Background Chemical Reaction Networks Connections Between Approaches Stochastic vs. Deterministic Modelling Future Work? Steady States vs. Stationary Distributions k Consider the simple bacterial growth model : B − → 2 B . Deterministic ODE: db ( t ) = kb ( t ) , b (0) = b 0 dt System has the solution b ( t ) = b 0 e kt Stochastic CTMC: � t � � B ( t ) = B (0) + Y k B ( s ) ds . 0 What does it mean to have a “ solution ”? Matthew Douglas Johnston Stochastic CRNs

Background Chemical Reaction Networks Connections Between Approaches Stochastic vs. Deterministic Modelling Future Work? Steady States vs. Stationary Distributions There are two primary approaches to analyzing CTMCs. Matthew Douglas Johnston Stochastic CRNs

Background Chemical Reaction Networks Connections Between Approaches Stochastic vs. Deterministic Modelling Future Work? Steady States vs. Stationary Distributions There are two primary approaches to analyzing CTMCs. 1) Generate sample paths: Think numerical integration with noise! Commonly simulated with Gillespie’s Algorithm (can be computationally-intensive!) (Gillespie, 1977, [4]) Matthew Douglas Johnston Stochastic CRNs

Background Chemical Reaction Networks Connections Between Approaches Stochastic vs. Deterministic Modelling Future Work? Steady States vs. Stationary Distributions There are two primary approaches to analyzing CTMCs. 1) Generate sample paths: Think numerical integration with noise! Commonly simulated with Gillespie’s Algorithm (can be computationally-intensive!) (Gillespie, 1977, [4]) 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 Stochastic CRNs

Background Chemical Reaction Networks Connections Between Approaches Stochastic vs. Deterministic Modelling Future Work? Steady States vs. Stationary Distributions B → 2 B 1) Generate sample paths : 70 60 50 Colony size 40 30 20 10 0 1 2 3 4 5 Time Three stochastic (coloured) realizations compared to the deterministic solution (black) Matthew Douglas Johnston Stochastic CRNs

Background Chemical Reaction Networks Connections Between Approaches Stochastic vs. Deterministic Modelling Future Work? Steady States vs. Stationary Distributions 2) Evolve probability distribution: 1 0.18 0.16 0.8 0.14 0.12 0.6 0.1 p n (0) p n (1) 0.08 0.4 0.06 0.04 0.2 0.02 0 0 10 12 14 16 18 20 10 20 30 40 50 60 n n 0.1 0.07 0.06 0.08 0.05 0.06 p n (2) 0.04 p n (3) 0.04 0.03 0.02 0.02 0.01 0 10 20 30 40 50 60 0 10 20 30 40 50 60 n Matthew Douglas Johnston Stochastic CRNs

Background Chemical Reaction Networks Connections Between Approaches Stochastic vs. Deterministic Modelling Future Work? Steady States vs. Stationary Distributions Of course, we should be careful... consider adding bacterial 2 k death to the model: B − → 2 B . Matthew Douglas Johnston Stochastic CRNs

Background Chemical Reaction Networks Connections Between Approaches Stochastic vs. Deterministic Modelling Future Work? Steady States vs. Stationary Distributions Of course, we should be careful... consider adding bacterial 2 k k death to the model: ∅ ← − B − → 2 B . Matthew Douglas Johnston Stochastic CRNs

Background Chemical Reaction Networks Connections Between Approaches Stochastic vs. Deterministic Modelling Future Work? Steady States vs. Stationary Distributions Of course, we should be careful... consider adding bacterial 2 k k death to the model: ∅ ← − B − → 2 B . Deterministic model is (as before): db ( t ) = kb ( t ) , b (0) = b 0 dt Matthew Douglas Johnston Stochastic CRNs

Background Chemical Reaction Networks Connections Between Approaches Stochastic vs. Deterministic Modelling Future Work? Steady States vs. Stationary Distributions Of course, we should be careful... consider adding bacterial 2 k k death to the model: ∅ ← − B − → 2 B . Deterministic model is (as before): db ( t ) = kb ( t ) , b (0) = b 0 dt 25 20 Colony Size 15 10 5 0 0 2 4 6 8 10 Time Matthew Douglas Johnston Stochastic CRNs

Background Chemical Reaction Networks Connections Between Approaches Stochastic vs. Deterministic Modelling Future Work? Steady States vs. Stationary Distributions Of course, we should be careful... consider adding bacterial 2 k k death to the model: ∅ ← − B − → 2 B . Deterministic model is (as before): db ( t ) = kb ( t ) , b (0) = b 0 dt 25 20 Colony Size 15 10 More on 5 . ← this later! 0 0 2 4 6 8 10 Time Matthew Douglas Johnston Stochastic CRNs

Background Chemical Reaction Networks Connections Between Approaches Stochastic vs. Deterministic Modelling Future Work? Steady States vs. Stationary Distributions We want to compare the long-term behaviour of the ODE and CTMC models! Matthew Douglas Johnston Stochastic CRNs

Background Chemical Reaction Networks Connections Between Approaches Stochastic vs. Deterministic Modelling Future Work? Steady States vs. Stationary Distributions We want to compare the long-term behaviour of the ODE and CTMC models! Deterministic model: Solutions tend asymptotically and irreversibly toward stable fixed points , stable limit cycles , infinity, etc. � dx ( t ) � Solutions stay at fixed points = 0 . dt Matthew Douglas Johnston Stochastic CRNs

Background Chemical Reaction Networks Connections Between Approaches Stochastic vs. Deterministic Modelling Future Work? Steady States vs. Stationary Distributions We want to compare the long-term behaviour of the ODE and CTMC models! Deterministic model: Solutions tend asymptotically and irreversibly toward stable fixed points , stable limit cycles , infinity, etc. � dx ( t ) � Solutions stay at fixed points = 0 . dt Stochastic model: Solutions do not stay at predicted fixed points! (Dynamic equilibrium...) Randomness destroys this notion of stability. :-( Matthew Douglas Johnston Stochastic CRNs

Background Chemical Reaction Networks Connections Between Approaches Stochastic vs. Deterministic Modelling Future Work? Steady States vs. Stationary Distributions To determine where trajectories of CTMC congregate, look for convergence in probability distribution ! Matthew Douglas Johnston Stochastic CRNs

Background Chemical Reaction Networks Connections Between Approaches Stochastic vs. Deterministic Modelling Future Work? Steady States vs. Stationary Distributions Distribution at infinity is known as a stationary distribution (denoted π ( X ) where X is the state). Matthew Douglas Johnston Stochastic CRNs

Background Chemical Reaction Networks Connections Between Approaches Stochastic vs. Deterministic Modelling Future Work? Steady States vs. Stationary Distributions Distribution at infinity is known as a stationary distribution (denoted π ( X ) where X is the state). Analogous to fixed points : dP ( X ( t ) = X ) = 0 = ⇒ Prob( X ( t ) = X ) = π ( X ) dt Stationary distributions are asymptotically stable = ⇒ initial distributions tend to them. Stationary distributions have support on the absorbing components of the chain. Matthew Douglas Johnston Stochastic CRNs

Background Chemical Reaction Networks Connections Between Approaches Stochastic vs. Deterministic Modelling Future Work? Steady States vs. Stationary Distributions Distribution at infinity is known as a stationary distribution (denoted π ( X ) where X is the state). Analogous to fixed points : dP ( X ( t ) = X ) = 0 = ⇒ Prob( X ( t ) = X ) = π ( X ) dt Stationary distributions are asymptotically stable = ⇒ initial distributions tend to them. Stationary distributions have support on the absorbing components of the chain. Interpret carefully! Stable fixed points are (usually!) peaks of distribution. Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour 1 Background Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions 2 Connections Between Approaches Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour 3 Future Work? Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour How are the deterministic and stochastic models related ? Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour How are the deterministic and stochastic models related ? Scaling limit: Do stochastic solutions “converge” in some sense to deterministic solutions? In what sense? Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour How are the deterministic and stochastic models related ? Scaling limit: Do stochastic solutions “converge” in some sense to deterministic solutions? In what sense? Similar Behaviour: Which networks exhibit parallel long-term behaviour? Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour How are the deterministic and stochastic models related ? Scaling limit: Do stochastic solutions “converge” in some sense to deterministic solutions? In what sense? Similar Behaviour: Which networks exhibit parallel long-term behaviour? Different Behaviour: Which networks exhibit disparate long-term behaviour? Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour Intuitively, we expect the stochastic model to scale to the deterministic model as number of molecules is increased . Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour Intuitively, we expect the stochastic model to scale to the deterministic model as number of molecules is increased . Define X V ( t ) = X ( t ) / V ( V scaling constant) and consider � t r X V ( t ) ≈ 1 1 � � κ k � X V ( s ) y k ds � V | y k | ( y ′ k − y k ) . V X (0) + V Y k ˆ 0 k =1 Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour Intuitively, we expect the stochastic model to scale to the deterministic model as number of molecules is increased . Define X V ( t ) = X ( t ) / V ( V scaling constant) and consider � t r X V ( t ) ≈ 1 1 � � κ k � X V ( s ) y k ds � V | y k | ( y ′ k − y k ) . V X (0) + V Y k ˆ 0 k =1 Applying law of large numbers as V → ∞ gives integral form: �� t r � � κ k x ( s ) y k ds · ( y ′ x ( t ) = x (0) + k − y k ) . 0 k =1 Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour Intuitively, we expect the stochastic model to scale to the deterministic model as number of molecules is increased . Define X V ( t ) = X ( t ) / V ( V scaling constant) and consider � t r X V ( t ) ≈ 1 1 � � κ k � X V ( s ) y k ds � V | y k | ( y ′ k − y k ) . V X (0) + V Y k ˆ 0 k =1 Applying law of large numbers as V → ∞ gives integral form: �� t r � � κ k x ( s ) y k ds · ( y ′ x ( t ) = x (0) + k − y k ) . 0 k =1 Convergence holds on compact time intervals [0 , T ]. [7] Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour Consider the stochastic model 2 / V A + B − → 2 B 1 B − → A and initial conditions A (0) = 3 V and B (0) = V so that A V (0) = 3 and B V (0) = 1. Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour Consider the stochastic model 2 / V A + B − → 2 B 1 B − → A and initial conditions A (0) = 3 V and B (0) = V so that A V (0) = 3 and B V (0) = 1. Corresponding ODE model is 2 − → 2 B A + B 1 B − → A with initial conditions a (0) = 3 and b (0) = 1. Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour V=10 V=1 4.5 4.5 4 4 3.5 3.5 3 3 A 2.5 2.5 A B 2 2 B 1.5 1.5 1 1 0.5 0.5 0 0 0 1 2 3 4 5 0 1 2 3 4 5 V=100 V=1000 4.5 4.5 4 4 3.5 3.5 3 3 A 2.5 A 2.5 B B 2 2 1.5 1.5 1 1 0.5 0.5 0 0 0 1 2 3 4 5 0 1 2 3 4 5 Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour Also consider the Lotka-Volterra model : Stochastic Deterministic 100 100 A − → 2 A A − → 2 A 10 / V 10 − → 2 B − → 2 B A + B A + B 100 100 B − → ∅ B − → ∅ . where A = prey and B = predator. Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour Also consider the Lotka-Volterra model : Stochastic Deterministic 100 100 A − → 2 A A − → 2 A 10 / V 10 − → 2 B − → 2 B A + B A + B 100 100 B − → ∅ B − → ∅ . where A = prey and B = predator. Deterministic model has isolated limit cycles which obey 100(ln( a ( t )) + ln( b ( t ))) − 10( a ( t ) + b ( t )) = constant . Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour V=1 V=10 25 300 250 20 200 15 150 10 100 5 50 0 0 0 5 10 15 20 25 30 0 50 100 150 200 250 300 V=200 V=100 4000 2200 2000 3500 1800 3000 1600 1400 2500 1200 2000 1000 800 1500 600 1000 400 200 500 400 600 800 1000 1200 1400 1600 1800 2000 500 1000 1500 2000 2500 3000 3500 4000 Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour So stochastic models are basically deterministic plus noise . Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour So stochastic models are basically deterministic plus noise . Not so fast! Consider toy model more closely: A + B − → 2 B B − → A Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour So stochastic models are basically deterministic plus noise . Not so fast! Consider toy model more closely: A + B − → 2 B B − → A Stochastic State space is: B 4 * * * * * * 3 * * * * 2 * * * * * * 1 * * * * * * * * A 0 2 3 4 1 Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour So stochastic models are basically deterministic plus noise . Not so fast! Consider toy model more closely: A + B − → 2 B B − → A Stochastic State space is: B 4 * * * * * * 3 * * * * 2 * * * * * * 1 * * * * ← − Irreversible transition! * * * * A 0 2 3 4 1 Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour Similar phenomenon occurs for Lotka-Volterra model: A − → 2 A A + B − → 2 B B − → ∅ . Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour Similar phenomenon occurs for Lotka-Volterra model: A − → 2 A A + B − → 2 B B − → ∅ . There are two extinction possibilities: If the prey dies first , the predator soon goes extinct as well. If the predator dies first , the prey grows unbounded. Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour Similar phenomenon occurs for Lotka-Volterra model: A − → 2 A A + B − → 2 B B − → ∅ . There are two extinction possibilities: If the prey dies first , the predator soon goes extinct as well. If the predator dies first , the prey grows unbounded. Note: Extinction cannot occur in deterministic model! (Boundary is not accessible .) Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour How do we reconcile this with the previous convergence results? Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour How do we reconcile this with the previous convergence results? Convergence only guaranteed on compact time intervals [0 , T ]—even though on the interval [0 , ∞ ) extinction is inevitable! Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour How do we reconcile this with the previous convergence results? Convergence only guaranteed on compact time intervals [0 , T ]—even though on the interval [0 , ∞ ) extinction is inevitable! 40 35 V=1 4.5 30 4 25 3.5 3 20 2.5 B 15 2 A 10 1.5 5 1 0.5 0 0 5 10 15 20 25 30 35 0 0 1 2 3 4 5 Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour We seek network conditions which guarantee models behave analogously/disparately on the interval [0 , ∞ ). Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour We seek network conditions which guarantee models behave analogously/disparately on the interval [0 , ∞ ). Two examples: 1 Same behaviour: Complex balanced systems (weakly reversible, deficiency zero networks) 2 Different behaviour: Absolute concentration robust systems (single terminal component, deficiency one networks) Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour First, we introduce some elements from Chemical Reaction Network Theory (CRNT). Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour First, we introduce some elements from Chemical Reaction Network Theory (CRNT). Connectivity of ( S , C , R ): Reversible Weakly Reversible C 1 C 1 C 3 C 3 C 2 C 2 Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour We have the following definitions: Linkage class : Connected component C 2 C 5 C 7 C 3 C 6 C 1 C 4 C 8 Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour We have the following definitions: Linkage class : Connected component C 2 C 5 C 7 C 3 C 6 C 1 C 4 C 8 Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour We have the following definitions: Linkage class : Connected component Strong linkage class : Maximal weakly reversible component (subset of nodes) C 2 C 5 C 7 C 3 C 6 C 1 C 4 C 8 Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour We have the following definitions: Linkage class : Connected component Strong linkage class : Maximal weakly reversible component (subset of nodes) C 2 C 5 C 7 C 3 C 6 C 1 C 4 C 8 Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour We have the following definitions: Linkage class : Connected component Strong linkage class : Maximal weakly reversible component (subset of nodes) Terminal strong linkage class : SLC with no outward edges C 2 C 5 C 7 C 3 C 6 C 1 C 4 C 8 Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour We have the following definitions: Linkage class : Connected component Strong linkage class : Maximal weakly reversible component (subset of nodes) Terminal strong linkage class : SLC with no outward edges C 2 C 5 C 7 C 3 C 6 C 1 C 4 C 8 Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour The deficiency of a chemical reaction network is δ = n − ℓ − s where n is number of complexes (nodes), ℓ is number of linkage classes, and s = dim( S ). Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour The deficiency of a chemical reaction network is δ = n − ℓ − s where n is number of complexes (nodes), ℓ is number of linkage classes, and s = dim( S ). A + B 2 B ⇄ Example 1: B A ⇄ n = 4, ℓ = 2, s = 1 = ⇒ δ = 1 Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour The deficiency of a chemical reaction network is δ = n − ℓ − s where n is number of complexes (nodes), ℓ is number of linkage classes, and s = dim( S ). A + B 2 B ⇄ Example 1: B A ⇄ n = 4, ℓ = 2, s = 1 = ⇒ δ = 1 A + B C ⇄ Example 2: B A ⇄ n = 4, ℓ = 2, s = 2 = ⇒ δ = 0 Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour Theorem (Deterministic δ = 0 (Horn, Jackson, Feinberg, 1972 [3, 5, 6])) Suppose that ( S , C , R ) satisfies the following: the deficiency is zero (i.e. δ = 0 ) the network is weakly reversible. Then there exists within each invariant linear space of the system a unique steady state which is locally asymptotically stable . Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour Theorem (Deterministic δ = 0 (Horn, Jackson, Feinberg, 1972 [3, 5, 6])) Suppose that ( S , C , R ) satisfies the following: the deficiency is zero (i.e. δ = 0 ) the network is weakly reversible. Then there exists within each invariant linear space of the system a unique steady state which is locally asymptotically stable . Positive steady state c ∈ R m > 0 is guaranteed to be complex balanced , i.e. for every y ∗ ∈ C r r κ k ( c ) y k = � � κ k ( c ) y k . k =1 k =1 y k = y ∗ y ′ k = y ∗ Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour Theorem (Stochastic δ = 0 (Anderson, Craciun, Kurtz, 2011 [1])) Suppose that ( S , C , R ) satisfies the following: the deficiency is zero (i.e. δ = 0 ) the network is weakly reversible. Then there is a stationary distribution which is a product of Poissons with marginal means c i , i.e. m c X i � i X ∈ Γ π ( X ) = M X i ! , i =1 where Γ a closed irreducible component and M is a normalizing constant. Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour The stochastic network really is just deterministic plus noise ! Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour The stochastic network really is just deterministic plus noise ! S + E ⇄ SE ⇄ P + E Example: S ⇄ ∅ ⇄ E . Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour The stochastic network really is just deterministic plus noise ! S + E ⇄ SE ⇄ P + E Example: S ⇄ ∅ ⇄ E . Network is weakly reversible, deficiency zero = ⇒ Stochastic process converges to distribution with marginal means equal to deterministic system In general, precise distribution/independence depends on state space . Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour Now re-consider the example α A + B − → 2 B β B − → A . Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour Now re-consider the example α A + B − → 2 B β B − → A . Deterministic model: da ( t ) = − db ( t ) = β b ( t ) − α a ( t ) b ( t ) = b ( t ) ( β − α a ( t )) . dt dt Model has absolute concentration robustness since a ∗ = β/α (stable steady state) regardless of initial conditions. Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour Now re-consider the example α A + B − → 2 B β B − → A . Deterministic model: da ( t ) = − db ( t ) = β b ( t ) − α a ( t ) b ( t ) = b ( t ) ( β − α a ( t )) . dt dt Model has absolute concentration robustness since a ∗ = β/α (stable steady state) regardless of initial conditions. Stochastic model: Extinction of B is inevitable on the interval [0 , ∞ ) = ⇒ different long-term behaviour. Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour Can we characterize a general class of networks with this distinction in long-term behaviour? Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour Can we characterize a general class of networks with this distinction in long-term behaviour? Theorem (Deterministic ACR (Shinar, Feinberg, 2010 [8])) Suppose that ( S , C , R ) satisfies the following: The deficiency is one (i.e. δ = 1 ); System admits a positive steady state ; and There are non-terminal complexes which differ only in S Then the network exhibits ACR in S. Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour Example 1: Re-consider α A + B − → 2 B β B − → A . Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour Example 1: Re-consider α A + B − → 2 B β B − → A . Can check deficiency is one and there is a positive steady state . Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour Example 1: Re-consider α A + B − → 2 B β B − → A . Can check deficiency is one and there is a positive steady state . Non-terminal complexes are: { A + B , B } ⇒ ACR in A . which differ only in A = Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour Example 2: Consider the following [8]: Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour k 1 k 3 [ ] T k 5 XD X XT X p k 2 [ ] k 4 D k 6 k 8 X p Y X p +Y X+Y p k 7 k 9 k 11 XD+Y XDY XD+Y p p k 10 Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour k 1 k 3 [ ] T k 5 XD X XT X p k 2 [ ] k 4 D k 6 k 8 X p Y X p +Y X+Y p k 7 k 9 k 11 XD+Y XDY XD+Y p p k 10 We have that: network is deficiency of one ; permits positive steady states ; and non-terminal complexes (blue) XD and XD + Y p differ in Y p . It follows that the network is ACR in Y p (also stable). Matthew Douglas Johnston Stochastic CRNs

Background Scaling Limit Connections Between Approaches Classes with Similar Behaviour Future Work? Classes with Different Behaviour What can we say about the corresponding stochastically modelled systems ? Matthew Douglas Johnston Stochastic CRNs