Stochastic Chemical Reaction Networks Matthew Douglas Johnston - - PowerPoint PPT Presentation

Background Stochastic Models Interesting Systems

Stochastic Chemical Reaction Networks

Matthew Douglas Johnston University of Waterloo Fall 2010

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems

1 Background

Chemical Reactions Standard Model Stoichiometric Compatibility Classes

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems

1 Background

Chemical Reactions Standard Model Stoichiometric Compatibility Classes

2 Stochastic Models

Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems

1 Background

Chemical Reactions Standard Model Stoichiometric Compatibility Classes

2 Stochastic Models

Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

3 Interesting Systems

Lotka-Volterra System The Block

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

1 Background

Chemical Reactions Standard Model Stoichiometric Compatibility Classes

2 Stochastic Models

Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

3 Interesting Systems

Lotka-Volterra System The Block

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

An elementary reaction consists of a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O /

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

An elementary reaction consists of a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O Species/Reactants

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

An elementary reaction consists of a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O Reactant Complex/

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

An elementary reaction consists of a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O Product Complex/

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

An elementary reaction consists of a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O Reaction Constant/

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

An elementary reaction consists of a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O / Chemical kinetics is the study of the rates/dynamics resulting from systems of such reactions.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

An elementary reaction consists of a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O / Chemical kinetics is the study of the rates/dynamics resulting from systems of such reactions. To build a mathematical model, we need to make physical assumptions.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

An elementary reaction consists of a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O / Chemical kinetics is the study of the rates/dynamics resulting from systems of such reactions. To build a mathematical model, we need to make physical assumptions, e.g. Uniform distribution (well-mixed);

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

An elementary reaction consists of a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O / Chemical kinetics is the study of the rates/dynamics resulting from systems of such reactions. To build a mathematical model, we need to make physical assumptions, e.g. Uniform distribution (well-mixed); Temperature and volume are constant;

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

An elementary reaction consists of a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O / Chemical kinetics is the study of the rates/dynamics resulting from systems of such reactions. To build a mathematical model, we need to make physical assumptions, e.g. Uniform distribution (well-mixed); Temperature and volume are constant; Law of mass action applies.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

If many reactant molecules are involved (e.g. chemical reactor), we consider the reactant concentrations.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

If many reactant molecules are involved (e.g. chemical reactor), we consider the reactant concentrations. We will keep track of xi ≈ ni V = # of molecules of ith species Volume .

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

If many reactant molecules are involved (e.g. chemical reactor), we consider the reactant concentrations. We will keep track of xi ≈ ni V = # of molecules of ith species Volume . The concentrations are approximately continuous with respect to each occurrence of a reaction.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

If many reactant molecules are involved (e.g. chemical reactor), we consider the reactant concentrations. We will keep track of xi ≈ ni V = # of molecules of ith species Volume . The concentrations are approximately continuous with respect to each occurrence of a reaction. The reaction constant k represents the average occurrence rate of the reaction per time.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

Consider the general system Ci

ki

− → C′

i, i = 1, . . . , r.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

Consider the general system Ci

ki

− → C′

i, i = 1, . . . , r.

This system is governed by the system of autonomous, polynomial,

• rdinary differential equations

˙ x =

r

• i=1

ki(z′

i − zi)xzi .

(1)

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

Consider the general system Ci

ki

− → C′

i, i = 1, . . . , r.

This system is governed by the system of autonomous, polynomial,

• rdinary differential equations

˙ x =

r

• i=1

ki(z′

i − zi)xzi .

(1) We have the following important components: we sum over r reactions,

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

Consider the general system Ci

ki

− → C′

i, i = 1, . . . , r.

This system is governed by the system of autonomous, polynomial,

• rdinary differential equations

˙ x =

r

• i=1

ki(z′

i − zi)xzi .

(1) We have the following important components: we sum over r reactions, ki is the reaction rate,

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

Consider the general system Ci

ki

− → C′

i, i = 1, . . . , r.

This system is governed by the system of autonomous, polynomial,

• rdinary differential equations

˙ x =

r

• i=1

ki(z′

i − zi)xzi .

(1) We have the following important components: we sum over r reactions, ki is the reaction rate, (z′

i − zi) is the reaction vector,

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

Consider the general system Ci

ki

− → C′

i, i = 1, . . . , r.

This system is governed by the system of autonomous, polynomial,

• rdinary differential equations

˙ x =

r

• i=1

ki(z′

i − zi)xzi .

(1) We have the following important components: we sum over r reactions, ki is the reaction rate, (z′

i − zi) is the reaction vector, and

xzi = m

j=1(xj)zij is the mass-action term.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

Consider the (reversible) system A1

k1

⇄

k2

2A2.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

Consider the (reversible) system A1

k1

⇄

k2

2A2. This has the governing dynamics ˙ x1 ˙ x2

• = k1

−1 2

• x1 + k2
• 1

−2

• x2

2,

where x1 and x2 are the concentrations of A1 and A2 respectively.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

Consider the (reversible) system A1

k1

⇄

k2

2A2. This has the governing dynamics ˙ x1 ˙ x2

• = k1

−1 2

• x1 + k2
• 1

−2

• x2

2,

where x1 and x2 are the concentrations of A1 and A2 respectively.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

Consider the (reversible) system A1

k1

⇄

k2

2A2. This has the governing dynamics ˙ x1 ˙ x2

• = k1

−1 2

• x1 + k2
• 1

−2

• x2

2,

where x1 and x2 are the concentrations of A1 and A2 respectively.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

Consider the (reversible) system A1

k1

⇄

k2

2A2. This has the governing dynamics ˙ x1 ˙ x2

• = k1

−1 2

• x1 + k2
• 1

−2

• x2

2,

where x1 and x2 are the concentrations of A1 and A2 respectively.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

Consider the (reversible) system A1

k1

⇄

k2

2A2. This has the governing dynamics ˙ x1 ˙ x2

• = k1

−1 2

• x1 + k2
• 1

−2

• x2

2,

where x1 and x2 are the concentrations of A1 and A2 respectively.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

What kind of properties does this system have? ˙ x1 ˙ x2

• = k1

−1 2

• x1 + k2
• 1

−2

• x2

2

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

What kind of properties does this system have? ˙ x1 ˙ x2

• = k1

−1 2

• x1 + k2
• 1

−2

• x2

2

• =
• The (positive) equilibrium set is given by

E =

• x ∈ R2

>0 | x2 =

• k1

k2 x1

• .

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

What kind of properties does this system have? ˙ x1 ˙ x2

• = k1

−1 2

• x1 + k2
• 1

−2

• x2

2

The (positive) equilibrium set is given by E =

• x ∈ R2

>0 | x2 =

• k1

k2 x1

• .

For any k1, k2, x1, x2 we have f(x) ∈ S where S = span

• 1

−2

• .

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

x1 x2 E (x0+S)

Figure: Previous system with k1 = k2 = 1.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

x1 x2 E (x0+S)

Figure: Previous system with k1 = k2 = 1.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

x1 x2 E (x0+S)

Figure: Previous system with k1 = k2 = 1.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

x1 x2 E (x0+S)

Figure: Previous system with k1 = k2 = 1.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

The restriction of solutions is a general property.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

The restriction of solutions is a general property. Definition The stoichiometric subspace S ⊂ Rm is given by S = span

• z′

i − zi | i = 1, . . . , r

• .

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Chemical Reactions Standard Model Stoichiometric Compatibility Classes

The restriction of solutions is a general property. Definition The stoichiometric subspace S ⊂ Rm is given by S = span

• z′

i − zi | i = 1, . . . , r

• .

Theorem Solutions x(t) of (1) are restricted to stoichiometric compatibility classes such that x(t) ∈ (S + x0) ∩ Rm

+

∀t ≥ 0.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

1 Background

Chemical Reactions Standard Model Stoichiometric Compatibility Classes

2 Stochastic Models

Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

3 Interesting Systems

Lotka-Volterra System The Block

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

What about cases where the number of reactant molecules ni is small (e.g. biological cells)?

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

What about cases where the number of reactant molecules ni is small (e.g. biological cells)? A few considerations: Differences between states is large - i.e. continuity of concentrations breaks down.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

What about cases where the number of reactant molecules ni is small (e.g. biological cells)? A few considerations: Differences between states is large - i.e. continuity of concentrations breaks down. Each occurrence of a reaction matters - i.e. we cannot average into a lump parameter ki.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

What about cases where the number of reactant molecules ni is small (e.g. biological cells)? A few considerations: Differences between states is large - i.e. continuity of concentrations breaks down. Each occurrence of a reaction matters - i.e. we cannot average into a lump parameter ki. We cannot tell when reactions will occur - i.e. the model is stochastic/probabilistic instead of deterministic.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

There are two general approaches to analyzing purely stochastic chemical kinetics systems:

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

There are two general approaches to analyzing purely stochastic chemical kinetics systems:

1 Evaluating sample trajectories/realizations.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

There are two general approaches to analyzing purely stochastic chemical kinetics systems:

1 Evaluating sample trajectories/realizations. 2 Analyzing the chemical master equation (models the

probability distribution over the admissible states as a function of time).

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

There are two general approaches to analyzing purely stochastic chemical kinetics systems:

1 Evaluating sample trajectories/realizations. 2 Analyzing the chemical master equation (models the

probability distribution over the admissible states as a function of time). Evaluating sample trajectories is simple to do numerically but not particularly insightful.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

There are two general approaches to analyzing purely stochastic chemical kinetics systems:

1 Evaluating sample trajectories/realizations. 2 Analyzing the chemical master equation (models the

probability distribution over the admissible states as a function of time). Evaluating sample trajectories is simple to do numerically but not particularly insightful. Solving the chemical master equation is typically several orders of magnitude beyond impossible.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

An algorithm for evaluating sample trajectories was developed by Doob (1945) and Gillespie (1977) [1]:

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

An algorithm for evaluating sample trajectories was developed by Doob (1945) and Gillespie (1977) [1]:

1 Initialize reactant numbers ni(0).

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

An algorithm for evaluating sample trajectories was developed by Doob (1945) and Gillespie (1977) [1]:

1 Initialize reactant numbers ni(0). 2 Determine time τ until next reaction.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

An algorithm for evaluating sample trajectories was developed by Doob (1945) and Gillespie (1977) [1]:

1 Initialize reactant numbers ni(0). 2 Determine time τ until next reaction. 3 Determine next reaction.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

An algorithm for evaluating sample trajectories was developed by Doob (1945) and Gillespie (1977) [1]:

1 Initialize reactant numbers ni(0). 2 Determine time τ until next reaction. 3 Determine next reaction. 4 Step forward τ, update system and return to step 2.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

An algorithm for evaluating sample trajectories was developed by Doob (1945) and Gillespie (1977) [1]:

1 Initialize reactant numbers ni(0). 2 Determine time τ until next reaction. 3 Determine next reaction. 4 Step forward τ, update system and return to step 2.

Typically carried out for a finite number of iterations or for a fixed amount of time.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

Consider the Michaelis-Menton enzyme reaction S + E

k+

1

⇄

k−

1

SE

k2

→ P + E

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

Consider the Michaelis-Menton enzyme reaction S + E

k+

1

⇄

k−

1

SE

k2

→ P + E Models the conversion of some substrate S into some product P via the enzyme E.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

Consider the Michaelis-Menton enzyme reaction S + E

k+

1

⇄

k−

1

SE

k2

→ P + E Models the conversion of some substrate S into some product P via the enzyme E. The deterministic model is a limiting case for ni → ∞ keeping ni/V constant.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

Figure: Comparison of deterministic and stochastic Michaelis-Menton enzyme mechanism.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

Figure: Comparison of deterministic and stochastic Michaelis-Menton enzyme mechanism.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

Figure: Comparison of deterministic and stochastic Michaelis-Menton enzyme mechanism.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

Figure: Comparison of deterministic and stochastic Michaelis-Menton enzyme mechanism.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

Evaluating sample trajectories is illustrative but not particularly enlightening.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

Evaluating sample trajectories is illustrative but not particularly enlightening. We are typically interested in questions such as:

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

Evaluating sample trajectories is illustrative but not particularly enlightening. We are typically interested in questions such as:

1 What is the probability that Xi = ni, Xi ≥ ni, etc., at time t?

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

Evaluating sample trajectories is illustrative but not particularly enlightening. We are typically interested in questions such as:

1 What is the probability that Xi = ni, Xi ≥ ni, etc., at time t? 2 Does the system have steady states?

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

Evaluating sample trajectories is illustrative but not particularly enlightening. We are typically interested in questions such as:

1 What is the probability that Xi = ni, Xi ≥ ni, etc., at time t? 2 Does the system have steady states?

If we sample enough trajectories we can build curves of best fit and confidence intervals but we will still miss many details.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

Evaluating sample trajectories is illustrative but not particularly enlightening. We are typically interested in questions such as:

1 What is the probability that Xi = ni, Xi ≥ ni, etc., at time t? 2 Does the system have steady states?

If we sample enough trajectories we can build curves of best fit and confidence intervals but we will still miss many details. We can do better - in fact, we can model the evolution of these probabilities explicitly!

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

We will let P(n; t) = P(X1 = n1, . . . , Xm = nm; t) and λi(n) = ki V |zi|−1

m

• j=1

nj! (nj − zij)! denote the transition probability from one state to another.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

We will let P(n; t) = P(X1 = n1, . . . , Xm = nm; t) and λi(n) = ki V |zi|−1

m

• j=1

nj! (nj − zij)! denote the transition probability from one state to another. The chemical master equation is given by dP(n; t) dt =

• i∈I

λi(n + zi − z′

i)P(n + zi − z′ i; t) − P(n; t)

• i∈O

λi(n)

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

We will let P(n; t) = P(X1 = n1, . . . , Xm = nm; t) and λi(n) = ki V |zi|−1

m

• j=1

nj! (nj − zij)! denote the transition probability from one state to another. The chemical master equation is given by dP(n; t) dt =

• i∈I

λi(n + zi − z′

i)P(n + zi − z′ i; t) − P(n; t)

• i∈O

λi(n) where I are the reactions which lead into a given state...

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

We will let P(n; t) = P(X1 = n1, . . . , Xm = nm; t) and λi(n) = ki V |zi|−1

m

• j=1

nj! (nj − zij)! denote the transition probability from one state to another. The chemical master equation is given by dP(n; t) dt =

• i∈I

λi(n + zi − z′

i)P(n + zi − z′ i; t) − P(n; t)

• i∈O

λi(n) where I are the reactions which lead into a given state and O are the reactions which lead from a given state.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

Reconsider the earlier system A1

k1

⇄

k2

2A2.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

Reconsider the earlier system A1

k1

⇄

k2

2A2. Consider the states (2, 0), (1, 2), and (0, 4)...

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

Reconsider the earlier system A1

k1

⇄

k2

2A2. Consider the states (2, 0), (1, 2), and (0, 4), for which we have dP(2, 0; t) dt = 2k2 V P(1, 2; t) − 2k1P(2, 0; t) dP(1, 2; t) dt = 12k2 V P(0, 4; t) + 2k1P(2, 0; t) −

• k1 + 2k2

V

• P(1, 2; t)

dP(0, 4; t) dt = k1P(1, 2; t) − 12k2 V P(0, 4; t).

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

Reconsider the earlier system A1

k1

⇄

k2

2A2. Consider the states (2, 0), (1, 2), and (0, 4), for which we have dP(2, 0; t) dt = 2k2 V P(1, 2; t) − 2k1P(2, 0; t) dP(1, 2; t) dt = 12k2 V P(0, 4; t) + 2k1P(2, 0; t) −

• k1 + 2k2

V

• P(1, 2; t)

dP(0, 4; t) dt = k1P(1, 2; t) − 12k2 V P(0, 4; t). This can be solved explicitly!

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

Nice features about the CME:

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

Nice features about the CME:

1 It is linear!

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

Nice features about the CME:

1 It is linear! 2 If it can be solved, it completely describes every aspect of the

mechanism.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

Nice features about the CME:

1 It is linear! 2 If it can be solved, it completely describes every aspect of the

mechanism. Less-than-nice features about the CME:

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

Nice features about the CME:

1 It is linear! 2 If it can be solved, it completely describes every aspect of the

mechanism. Less-than-nice features about the CME:

1 It is typically massive (for unbounded systems, it is

infinite-dimensional).

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

Nice features about the CME:

1 It is linear! 2 If it can be solved, it completely describes every aspect of the

mechanism. Less-than-nice features about the CME:

1 It is typically massive (for unbounded systems, it is

infinite-dimensional).

2 Mass-action term must be computed for each state.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

Nice features about the CME:

1 It is linear! 2 If it can be solved, it completely describes every aspect of the

mechanism. Less-than-nice features about the CME:

1 It is typically massive (for unbounded systems, it is

infinite-dimensional).

2 Mass-action term must be computed for each state. 3 Connections between states can be complicated near the

boundary.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Lotka-Volterra System The Block

1 Background

Chemical Reactions Standard Model Stoichiometric Compatibility Classes

2 Stochastic Models

Small-Scale Considerations Gillespie Algorithm Chemical Master Equation

3 Interesting Systems

Lotka-Volterra System The Block

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Lotka-Volterra System The Block

Modelling chemical kinetics systems stochastically can qualitatively change the dynamics of a system.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Lotka-Volterra System The Block

Modelling chemical kinetics systems stochastically can qualitatively change the dynamics of a system. Consider the Lotka-Volterra predator-prey system (A1 is the prey, A2 is the predator) A1

k1

− → 2A1 A1 + A2

k2

− → 2A2 A2

k3

− → O.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Lotka-Volterra System The Block

Modelling chemical kinetics systems stochastically can qualitatively change the dynamics of a system. Consider the Lotka-Volterra predator-prey system (A1 is the prey, A2 is the predator) A1

k1

− → 2A1 A1 + A2

k2

− → 2A2 A2

k3

− → O. For the rate constant values k1 = k2 = k3 = 1 the large-scale deterministic system has a unique positive equilibrium x∗

1 = x∗ 2 = 1

which is a centre, however...

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

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

Background Stochastic Models Interesting Systems Lotka-Volterra System The Block

The stable equilibrium concentration is no longer stable! (In fact, none of the stable periodic orbits are stable.)

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Lotka-Volterra System The Block

The stable equilibrium concentration is no longer stable! (In fact, none of the stable periodic orbits are stable.) Oscillatory behaviour remains but appears almost chaotic.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Lotka-Volterra System The Block

The stable equilibrium concentration is no longer stable! (In fact, none of the stable periodic orbits are stable.) Oscillatory behaviour remains but appears almost chaotic. Furthermore, extinction events which were not possible in the continuous, deterministic system are now possible.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Lotka-Volterra System The Block

The stable equilibrium concentration is no longer stable! (In fact, none of the stable periodic orbits are stable.) Oscillatory behaviour remains but appears almost chaotic. Furthermore, extinction events which were not possible in the continuous, deterministic system are now possible. Although it is unlikely for either the predator or the prey to go extinct, it is irreversible — carried over a long enough time scale, extinction is the inevitable outcome of the system!

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Lotka-Volterra System The Block

Horn and Jackson consider the following system in their seminal paper “General Mass Action Kinetics” [2]: 2A1 + A2

1

− → 3A1

ǫ ↑

↓ ǫ 3A2

1

← − A1 + 2A2.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Lotka-Volterra System The Block

Horn and Jackson consider the following system in their seminal paper “General Mass Action Kinetics” [2]: 2A1 + A2

1

− → 3A1

ǫ ↑

↓ ǫ 3A2

1

← − A1 + 2A2. The system exhibits varying behaviour depending on the value of ǫ:

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Lotka-Volterra System The Block

Horn and Jackson consider the following system in their seminal paper “General Mass Action Kinetics” [2]: 2A1 + A2

1

− → 3A1

ǫ ↑

↓ ǫ 3A2

1

← − A1 + 2A2. The system exhibits varying behaviour depending on the value of ǫ:

1 ǫ ≥ 1/6: one stable equilibrium.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Lotka-Volterra System The Block

Horn and Jackson consider the following system in their seminal paper “General Mass Action Kinetics” [2]: 2A1 + A2

1

− → 3A1

ǫ ↑

↓ ǫ 3A2

1

← − A1 + 2A2. The system exhibits varying behaviour depending on the value of ǫ:

1 ǫ ≥ 1/6: one stable equilibrium. 2 0 < ǫ < 1/6: two stable and one unstable equilibria.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Lotka-Volterra System The Block

Horn and Jackson consider the following system in their seminal paper “General Mass Action Kinetics” [2]: 2A1 + A2

1

− → 3A1

ǫ ↑

↓ ǫ 3A2

1

← − A1 + 2A2. The system exhibits varying behaviour depending on the value of ǫ:

1 ǫ ≥ 1/6: one stable equilibrium. 2 0 < ǫ < 1/6: two stable and one unstable equilibria. 3 ǫ = 0: two stable boundary equilibria.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Lotka-Volterra System The Block

(a) (b) (c)

Figure: Block system with (a) ǫ ≥ 1/6, (b) 0 < ǫ < 1/6, and (c) ǫ = 0.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Lotka-Volterra System The Block

Let’s restrict our attention to the case 0 < ǫ < 1/6.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Lotka-Volterra System The Block

Let’s restrict our attention to the case 0 < ǫ < 1/6. For the deterministic system, each compatibility class is divided into two regions by the unstable equilibrium.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Lotka-Volterra System The Block

Let’s restrict our attention to the case 0 < ǫ < 1/6. For the deterministic system, each compatibility class is divided into two regions by the unstable equilibrium. Trajectories originating on one side or the other collapse to their respective stability equilibrium concentrations.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Lotka-Volterra System The Block

Let’s restrict our attention to the case 0 < ǫ < 1/6. For the deterministic system, each compatibility class is divided into two regions by the unstable equilibrium. Trajectories originating on one side or the other collapse to their respective stability equilibrium concentrations. What would happen if we modelled the system stochastically?

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Lotka-Volterra System The Block

Let’s restrict our attention to the case 0 < ǫ < 1/6. For the deterministic system, each compatibility class is divided into two regions by the unstable equilibrium. Trajectories originating on one side or the other collapse to their respective stability equilibrium concentrations. What would happen if we modelled the system stochastically? Trajectories can jump from one side to the other!

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Lotka-Volterra System The Block

Figure: Block system with ǫ = 0.12.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Lotka-Volterra System The Block

Thanks for coming out!

Matthew Douglas Johnston Stochastic Chemical Reaction Networks

Background Stochastic Models Interesting Systems Lotka-Volterra System The Block

• D. Gillespie. Exact Stochastic Simulation of Coupled Chemical
• Reactions. The Journal of Physical Chemistry,

81(25):2340-2361, 1977.

• F. Horn and R. Jackson. General mass action kinetics. Archive

for Rational Mechanics and Analysis, 47:187-194, 1972.

• D. Wilkinson. Stochastic Modelling for Systems Biology.

Chapman & Hall, 2006.

Matthew Douglas Johnston Stochastic Chemical Reaction Networks