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

Background Connections Between Approaches Future Work?

Stochastic Modelling of CRNs

“Combinatorial and algebraic approaches to chemical reaction networks workshop” at University of Portsmouth Matthew Douglas Johnston Van Vleck Visiting Assistant Professor University of Wisconsin-Madison (Modified from a Systems Biology of Madison talk given by David F. Anderson on April 29, 2014) June 23, 2014

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work?

1 Background

Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work?

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

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work?

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 Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

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 Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

A chemical reaction network is given by a triple of (finite) sets (S, C, R):

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

A chemical reaction network is given by a triple of (finite) sets (S, C, R): Species set: S = {X1, . . . , Xm}

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

A chemical reaction network is given by a triple of (finite) sets (S, C, R): Species set: S = {X1, . . . , Xm} Reaction set: R = {R1, . . . , Rr} where Rk = Ck → C ′

k

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

A chemical reaction network is given by a triple of (finite) sets (S, C, R): Species set: S = {X1, . . . , Xm} Reaction set: R = {R1, . . . , Rr} where Rk = Ck → C ′

k

Complex set: C = r

k=1{Ck, C ′ k} where Ck = m j=1 yijXj

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

A chemical reaction network is given by a triple of (finite) sets (S, C, R): Species set: S = {X1, . . . , Xm} Reaction set: R = {R1, . . . , Rr} where Rk = Ck → C ′

k

Complex set: C = r

k=1{Ck, C ′ k} where Ck = m j=1 yijXj

Chemical reaction network theory (CRNT) attempts to extract dynamical behaviour from network structure (minimal dependence

• n parameters).

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

Dynamics may be modelled deterministically or stochastically.

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

Dynamics may be modelled deterministically or stochastically. Deterministic: Keep track of reactant concentrations: xi ∈ R≥0 Reactions occur continuously and simultaneously Modelled with autonomous system of ODEs

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

Dynamics may be modelled deterministically or stochastically. Deterministic: Keep track of reactant concentrations: xi ∈ R≥0 Reactions occur continuously and simultaneously Modelled with autonomous system of ODEs Stochastic: Keep track of reactant numbers: Xi ∈ {0, 1, 2, . . .} Reactions occur discretely and at separate times Modelled as a continuous time Markov chain (CTMC)

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

Deterministic mass action kinetics: Rate of reaction is proportional to product of concentrations, e.g. X + Y = ⇒ [rate] ∝ [X][Y ]

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

Deterministic mass action kinetics: Rate of reaction is proportional to product of concentrations, e.g. X + Y = ⇒ [rate] ∝ [X][Y ] Stochastic mass action kinetics: Probability a reaction occurs if proportional to combinatorial ways constituent species could meet, e.g. ∅ = ⇒ [Prob] ∝ 1 X = ⇒ [Prob] ∝ X X + Y = ⇒ [Prob] ∝ X · Y 2X = ⇒ [Prob] ∝ X(X − 1)

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

Deterministic mass action kinetics: Rate of reaction is proportional to product of concentrations, e.g. X + Y = ⇒ [rate] ∝ [X][Y ] Stochastic mass action kinetics: Probability a reaction occurs if proportional to combinatorial ways constituent species could meet, e.g. ∅ = ⇒ [Prob] ∝ 1 X = ⇒ [Prob] ∝ X X + Y = ⇒ [Prob] ∝ X · Y 2X = ⇒ [Prob] ∝ X(X − 1)

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

Consider stoichiometric subspace defined as S = span{y′

k − yk | (Ck → C ′ k) ∈ R}.

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

Consider stoichiometric subspace defined as S = span{y′

k − yk | (Ck → C ′ k) ∈ R}.

Deterministic model: (integral form) x(t) = x(0) +

r

• k=1

t κkx(s)yk ds

• (y′

k − yk)

reactions push trajections along S

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

Consider stoichiometric subspace defined as S = span{y′

k − yk | (Ck → C ′ k) ∈ R}.

Deterministic model: (integral form) x(t) = x(0) +

r

• k=1

t κkx(s)yk ds

• (y′

k − yk)

reactions push trajections along S Stochastic model: (RTC form, Kurtz [7]) X(t) = X(0) +

r

• k=1

Yk t λk(X(s)) ds

• · (y′

k − yk)

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

Consider stoichiometric subspace defined as S = span{y′

k − yk | (Ck → C ′ k) ∈ R}.

Deterministic model: (integral form) x(t) = x(0) +

r

• k=1

t κkx(s)yk ds

• (y′

k − yk)

reactions push trajections along S Stochastic model: (RTC form, Kurtz [7]) X(t) = X(0) +

r

• k=1

Yk t λk(X(s)) ds

• · (y′

k − yk)

Yk(·) unit-rate Poisson processes, λk(·) propensities

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

Consider stoichiometric subspace defined as S = span{y′

k − yk | (Ck → C ′ k) ∈ R}.

Deterministic model: (integral form) x(t) = x(0) +

r

• k=1

t κkx(s)yk ds

• (y′

k − yk)

reactions push trajections along S Stochastic model: (RTC form, Kurtz [7]) X(t) = X(0) +

r

• k=1

Yk t λk(X(s)) ds

• · (y′

k − yk)

Yk(·) unit-rate Poisson processes, λk(·) propensities reactions increment trajectories along S

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

Consider following model: A + B − → 2B B − → A

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

Consider following model: A + B − → 2B B − → A

Deterministic

A B 1 2 3 4 1 2 3 4

* * * * * * * * * * * * * * * * * * * * * * * *

Stochastic

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

Consider following model: A + B − → 2B B − → A

Deterministic

A B 1 2 3 4 1 2 3 4

* * * * * * * * * * * * * * * * * * * * * * * *

Stochastic

Moral of the story: State space is the same but discretized!

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

Consider the simple bacterial growth model: B

k

− → 2B.

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

Consider the simple bacterial growth model: B

k

− → 2B. Deterministic ODE: db(t) dt = kb(t), b(0) = b0 System has the solution b(t) = b0ekt

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

Consider the simple bacterial growth model: B

k

− → 2B. Deterministic ODE: db(t) dt = kb(t), b(0) = b0 System has the solution b(t) = b0ekt Stochastic CTMC: B(t) = B(0) + Y

• k

t B(s) ds

• .

What does it mean to have a “solution”?

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

There are two primary approaches to analyzing CTMCs.

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling 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 Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling 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 Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

1) Generate sample paths: B → 2B

1 2 3 4 5 10 20 30 40 50 60 70 Colony size Time

Three stochastic (coloured) realizations compared to the deterministic solution (black)

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

2) Evolve probability distribution:

10 12 14 16 18 20 0.2 0.4 0.6 0.8 1 n pn(0) 10 20 30 40 50 60 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 n pn(1) 10 20 30 40 50 60 0.02 0.04 0.06 0.08 0.1 n pn(2) 10 20 30 40 50 60 0.01 0.02 0.03 0.04 0.05 0.06 0.07 pn(3)

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

Of course, we should be careful... consider adding bacterial death to the model: B

2k

− → 2B.

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

Of course, we should be careful... consider adding bacterial death to the model: ∅

k

← − B

2k

− → 2B.

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

Of course, we should be careful... consider adding bacterial death to the model: ∅

k

← − B

2k

− → 2B. Deterministic model is (as before): db(t) dt = kb(t), b(0) = b0

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

Of course, we should be careful... consider adding bacterial death to the model: ∅

k

← − B

2k

− → 2B. Deterministic model is (as before): db(t) dt = kb(t), b(0) = b0

2 4 6 8 10 5 10 15 20 25 Time Colony Size Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

Of course, we should be careful... consider adding bacterial death to the model: ∅

k

← − B

2k

− → 2B. Deterministic model is (as before): db(t) dt = kb(t), b(0) = b0

2 4 6 8 10 5 10 15 20 25 Time Colony Size

More on . ← this later!

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling 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 Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling 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. Solutions stay at fixed points dx(t) dt = 0

• .

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling 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. Solutions stay at fixed points dx(t) dt = 0

• .

Stochastic model: Solutions do not stay at predicted fixed points! (Dynamic equilibrium...) Randomness destroys this notion of stability. :-(

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling Steady States vs. Stationary Distributions

To determine where trajectories of CTMC congregate, look for convergence in probability distribution!

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling 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 Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling 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) dt = 0 = ⇒ Prob(X(t) = X) = π(X) 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 Connections Between Approaches Future Work? Chemical Reaction Networks Stochastic vs. Deterministic Modelling 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) dt = 0 = ⇒ Prob(X(t) = X) = π(X) 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 Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour 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 Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

How are the deterministic and stochastic models related?

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour 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 Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour 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 Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour 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 Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour 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 Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

Intuitively, we expect the stochastic model to scale to the deterministic model as number of molecules is increased. Define XV (t) = X(t)/V (V scaling constant) and consider XV (t) ≈ 1 V X(0) +

r

• k=1

1 V Yk

• V |yk|

t ˆ κk XV (s)yk ds

• (y′

k − yk).

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

Intuitively, we expect the stochastic model to scale to the deterministic model as number of molecules is increased. Define XV (t) = X(t)/V (V scaling constant) and consider XV (t) ≈ 1 V X(0) +

r

• k=1

1 V Yk

• V |yk|

t ˆ κk XV (s)yk ds

• (y′

k − yk).

Applying law of large numbers as V → ∞ gives integral form: x(t) = x(0) +

r

• k=1

t κkx(s)yk ds

• · (y′

k − yk).

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

Intuitively, we expect the stochastic model to scale to the deterministic model as number of molecules is increased. Define XV (t) = X(t)/V (V scaling constant) and consider XV (t) ≈ 1 V X(0) +

r

• k=1

1 V Yk

• V |yk|

t ˆ κk XV (s)yk ds

• (y′

k − yk).

Applying law of large numbers as V → ∞ gives integral form: x(t) = x(0) +

r

• k=1

t κkx(s)yk ds

• · (y′

k − yk).

Convergence holds on compact time intervals [0, T]. [7]

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

Consider the stochastic model A + B

2/V

− → 2B B

1

− → A and initial conditions A(0) = 3V and B(0) = V so that AV (0) = 3 and BV (0) = 1.

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

Consider the stochastic model A + B

2/V

− → 2B B

1

− → A and initial conditions A(0) = 3V and B(0) = V so that AV (0) = 3 and BV (0) = 1. Corresponding ODE model is A + B

2

− → 2B B

1

− → A with initial conditions a(0) = 3 and b(0) = 1.

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

1 2 3 4 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5

V=1

A B 1 2 3 4 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5

V=10

A B 1 2 3 4 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5

V=100 A B

1 2 3 4 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5

V=1000

A B

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

Also consider the Lotka-Volterra model: Stochastic Deterministic A

100

− → 2A A

100

− → 2A A + B

10/V

− → 2B A + B

10

− → 2B B

100

− → ∅ B

100

− → ∅. where A = prey and B = predator.

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

Also consider the Lotka-Volterra model: Stochastic Deterministic A

100

− → 2A A

100

− → 2A A + B

10/V

− → 2B A + B

10

− → 2B B

100

− → ∅ B

100

− → ∅. 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 Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

5 10 15 20 25 30 5 10 15 20 25 V=1 50 100 150 200 250 300 50 100 150 200 250 300 V=10 400 600 800 1000 1200 1400 1600 1800 2000 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 V=100 500 1000 1500 2000 2500 3000 3500 4000 500 1000 1500 2000 2500 3000 3500 4000 V=200

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

So stochastic models are basically deterministic plus noise.

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

So stochastic models are basically deterministic plus noise. Not so fast! Consider toy model more closely: A + B − → 2B B − → A

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

So stochastic models are basically deterministic plus noise. Not so fast! Consider toy model more closely: A + B − → 2B B − → A State space is:

A B 1 2 3 4 1 2 3 4

* * * * * * * * * * * * * * * * * * * * * * * *

Stochastic

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

So stochastic models are basically deterministic plus noise. Not so fast! Consider toy model more closely: A + B − → 2B B − → A State space is:

A B 1 2 3 4 1 2 3 4

* * * * * * * * * * * * * * * * * * * * * * * *

Stochastic

← − Irreversible transition!

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

Similar phenomenon occurs for Lotka-Volterra model: A − → 2A A + B − → 2B B − → ∅.

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

Similar phenomenon occurs for Lotka-Volterra model: A − → 2A A + B − → 2B 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 Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

Similar phenomenon occurs for Lotka-Volterra model: A − → 2A A + B − → 2B 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 Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

How do we reconcile this with the previous convergence results?

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour 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 Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour 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!

1 2 3 4 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5

V=1 B A

5 10 15 20 25 30 35 5 10 15 20 25 30 35 40

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

We seek network conditions which guarantee models behave analogously/disparately on the interval [0, ∞).

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour 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 Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

First, we introduce some elements from Chemical Reaction Network Theory (CRNT).

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

First, we introduce some elements from Chemical Reaction Network Theory (CRNT). Connectivity of (S, C, R):

C3 C2 C1

Reversible

C3 C2 C1

Weakly Reversible

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

We have the following definitions: Linkage class: Connected component

C1 C2 C3 C4 C5 C6 C7 C8

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

We have the following definitions: Linkage class: Connected component

C1 C2 C3 C4 C5 C6 C7 C8

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

We have the following definitions: Linkage class: Connected component Strong linkage class: Maximal weakly reversible component (subset of nodes)

C1 C2 C3 C4 C5 C6 C7 C8

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

We have the following definitions: Linkage class: Connected component Strong linkage class: Maximal weakly reversible component (subset of nodes)

C1 C2 C3 C4 C5 C6 C7 C8

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour 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

C1 C2 C3 C4 C5 C6 C7 C8

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour 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

C1 C2 C3 C4 C5 C6 C7 C8

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour 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 Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour 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). Example 1: A + B ⇄ 2B B ⇄ A n = 4, ℓ = 2, s = 1 = ⇒ δ = 1

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour 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). Example 1: A + B ⇄ 2B B ⇄ A n = 4, ℓ = 2, s = 1 = ⇒ δ = 1 Example 2: A + B ⇄ C B ⇄ A n = 4, ℓ = 2, s = 2 = ⇒ δ = 0

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour 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 Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour 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 ∈ Rm

>0 is guaranteed to be complex

balanced, i.e. for every y∗ ∈ C

r

• k=1

yk=y∗

κk(c)yk =

r

• k=1

y′

k=y∗

κk(c)yk.

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour 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 ci, i.e. π(X) = M

m

• i=1

cXi

i

Xi!, X ∈ Γ where Γ a closed irreducible component and M is a normalizing constant.

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

The stochastic network really is just deterministic plus noise!

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

The stochastic network really is just deterministic plus noise! Example: S + E ⇄ SE ⇄ P + E S ⇄ ∅ ⇄ E.

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

The stochastic network really is just deterministic plus noise! Example: S + E ⇄ SE ⇄ P + E 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 Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

Now re-consider the example A + B

α

− → 2B B

β

− → A.

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

Now re-consider the example A + B

α

− → 2B B

β

− → A. Deterministic model: da(t) dt = −db(t) dt = βb(t) − αa(t)b(t) = b(t) (β − αa(t)) . Model has absolute concentration robustness since a∗ = β/α (stable steady state) regardless of initial conditions.

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

Now re-consider the example A + B

α

− → 2B B

β

− → A. Deterministic model: da(t) dt = −db(t) dt = βb(t) − αa(t)b(t) = b(t) (β − αa(t)) . 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 Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour 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 Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour 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 Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

Example 1: Re-consider A + B

α

− → 2B B

β

− → A.

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

Example 1: Re-consider A + B

α

− → 2B B

β

− → A. Can check deficiency is one and there is a positive steady state.

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

Example 1: Re-consider A + B

α

− → 2B B

β

− → A. Can check deficiency is one and there is a positive steady state. Non-terminal complexes are: {A + B, B} which differ only in A = ⇒ ACR in A.

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

Example 2: Consider the following [8]:

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

XD X XT Xp Xp+Y XpY X+Yp XD+Y

p

XDY

p

XD+Y

k1 k2[ ] k3 k4 k5 k6 k7 k8 k9 k10 k11 [ ] T D Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

XD X XT Xp Xp+Y XpY X+Yp XD+Y

p

XDY

p

XD+Y

k1 k2[ ] k3 k4 k5 k6 k7 k8 k9 k10 k11 [ ] T D

We have that: network is deficiency of one; permits positive steady states; and non-terminal complexes (blue) XD and XD + Yp differ in Yp. It follows that the network is ACR in Yp (also stable).

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

What can we say about the corresponding stochastically modelled systems?

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

What can we say about the corresponding stochastically modelled systems? Theorem (Stochastic ACR (Anderson, Enciso, Johnston, 2014 [2])) Suppose that (S, C, R) satisfies the following: The deficiency is one (i.e. δ = 1); System admits a positive steady state; There are non-terminal complexes which differ only in S; and Network is conservative Then, with probability one, there will be an extinction event.

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

What can we say about the corresponding stochastically modelled systems? Theorem (Stochastic ACR (Anderson, Enciso, Johnston, 2014 [2])) Suppose that (S, C, R) satisfies the following: The deficiency is one (i.e. δ = 1); System admits a positive steady state; There are non-terminal complexes which differ only in S; and Network is conservative (NEW) Then, with probability one, there will be an extinction event.

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

Different long-term behaviour from deterministic prediction is guaranteed for wide class of networks!

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

Different long-term behaviour from deterministic prediction is guaranteed for wide class of networks! XD X XT Xp Xp+Y XpY X+Yp XD+Y

p

XDY

p

XD+Y

k1 k2[ ] k3 k4 k5 k6 k7 k8 k9 k10 k11 [ ] T D

Network is conservative = ⇒ stochastic system inevitably converges to boundary, not (deterministically-stable) ACR value!

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work? Scaling Limit Classes with Similar Behaviour Classes with Different Behaviour

Different long-term behaviour from deterministic prediction is guaranteed for wide class of networks! XD X XT Xp Xp+Y XpY X+Yp XD+Y

p

XDY

p

XD+Y

k1 k2[ ] k3 k4 k5 k6 k7 k8 k9 k10 k11 [ ] T D

Network is conservative = ⇒ stochastic system inevitably converges to boundary, not (deterministically-stable) ACR value! Absorbing boundary: Xp = Xtot, Yp = Ytot, rest = 0

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work?

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 Connections Between Approaches Future Work?

Many questions remain:

1 What are general network conditions for the existence of a

stationary distribution (positive recurrence of CMTC)?

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work?

Many questions remain:

1 What are general network conditions for the existence of a

stationary distribution (positive recurrence of CMTC)?

2 What does the state space / extinction set look like?

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work?

Many questions remain:

1 What are general network conditions for the existence of a

stationary distribution (positive recurrence of CMTC)?

2 What does the state space / extinction set look like? 3 If extinction is rare, what does the quasi-stationary

distribution look like?

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work?

Selected Bibliography

David F. Anderson, Gheorghe Craciun, and Thomas G. Kurtz. Product-form stationary distributions for deficiency zero chemical reaction networks. Bull. Math. Biol., 72(8):1947–1970, 2011. David F. Anderson, German Enciso, and Matthew D. Johnston. Stochastic analysis of chemical reaction networks with absolute concentration robustness. J. R. Soc. Interface, 11(93):20130943, 2014. Martin Feinberg. Complex balancing in general kinetic systems. Arch. Ration. Mech. Anal., 49:187–194, 1972. Daniel Gillespie. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys., 22(4):403–434, 1976. Fritz Horn. Necessary and sufficient conditions for complex balancing in chemical kinetics. Arch. Ration.

• Mech. Anal., 49:172–186, 1972.

Fritz Horn and Roy Jackson. General mass action kinetics. Arch. Ration. Mech. Anal., 47:81–116, 1972. Tom G. Kurtz. The relationship between stochastic and deterministic models for chemical reactions. J.

• Chem. Phys., 57:2976–2978, 1972.

Guy Shinar and Martin Feinberg. Structural sources of robustness in biochemical reaction networks. Science, 327(5971):1389–1391, 2010. Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work?

The Chemical Master Equation is given by dP(X(t) = X) dt =

• k∈I

λk(X + yk − y′

k)P(X(t) = X + yk − y′ k)

− P(X(t) = X)

• k∈O

λk(X) where I are the reactions which lead into a given state in one reaction step and O are the reactions which lead from a given state. Deterministic mass action: λk(X) = ˆ κk

m

• i=1

Xi! yki!(Xi − yki)!1Xi≥yki

Matthew Douglas Johnston Stochastic CRNs

Background Connections Between Approaches Future Work?

10 20 30 40 50 0.02 0.04 0.06 0.08

Molecules of Y

p

Quasi−stationary probabilities

Xtot = 100 Ytot = 3500 Xtot = 1000 Ytot = 35000 Xtot = 10000 Ytot = 350000 Poisson

Matthew Douglas Johnston Stochastic CRNs