Download Stochastic Chemical Reaction Networks

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.
k
2H2 + O2 −→ 2H2 O
/
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.
k
2H2 + O2 −→ 2H2 O
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.
k
2H2 + O2 −→ 2H2 O
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.
k
2H2 + O2 −→ 2H2 O
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.
k
2H2 + O2 −→ 2H2 O
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.
k
2H2 + O2 −→ 2H2 O
/
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.
k
2H2 + O2 −→ 2H2 O
/
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.
k
2H2 + O2 −→ 2H2 O
/
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.
k
2H2 + O2 −→ 2H2 O
/
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.
k
2H2 + O2 −→ 2H2 O
/
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 ≈
# of molecules of i th species
ni
=
.
V
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 ≈
# of molecules of i th species
ni
=
.
V
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 ≈
# of molecules of i th species
ni
=
.
V
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
k
i
Ci −→
Ci0 , 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
k
i
Ci −→
Ci0 , i = 1, . . . , r .
This system is governed by the system of autonomous, polynomial,
ordinary differential equations
ẋ =
r
X
ki (z0i − zi )xzi .
i =1
Matthew Douglas Johnston
Stochastic Chemical Reaction Networks
(1)
Background
Stochastic Models
Interesting Systems
Chemical Reactions
Standard Model
Stoichiometric Compatibility Classes
Consider the general system
k
i
Ci −→
Ci0 , i = 1, . . . , r .
This system is governed by the system of autonomous, polynomial,
ordinary differential equations
ẋ =
r
X
ki (z0i − zi )xzi .
i =1
We have the following important components:
we sum over r reactions,
Matthew Douglas Johnston
Stochastic Chemical Reaction Networks
(1)
Background
Stochastic Models
Interesting Systems
Chemical Reactions
Standard Model
Stoichiometric Compatibility Classes
Consider the general system
k
i
Ci −→
Ci0 , i = 1, . . . , r .
This system is governed by the system of autonomous, polynomial,
ordinary differential equations
ẋ =
r
X
ki (z0i − zi )xzi .
i =1
We have the following important components:
we sum over r reactions,
ki is the reaction rate,
Matthew Douglas Johnston
Stochastic Chemical Reaction Networks
(1)
Background
Stochastic Models
Interesting Systems
Chemical Reactions
Standard Model
Stoichiometric Compatibility Classes
Consider the general system
k
i
Ci −→
Ci0 , i = 1, . . . , r .
This system is governed by the system of autonomous, polynomial,
ordinary differential equations
ẋ =
r
X
ki (z0i − zi )xzi .
i =1
We have the following important components:
we sum over r reactions,
ki is the reaction rate,
(z0i − zi ) is the reaction vector,
Matthew Douglas Johnston
Stochastic Chemical Reaction Networks
(1)
Background
Stochastic Models
Interesting Systems
Chemical Reactions
Standard Model
Stoichiometric Compatibility Classes
Consider the general system
k
i
Ci −→
Ci0 , i = 1, . . . , r .
This system is governed by the system of autonomous, polynomial,
ordinary differential equations
ẋ =
r
X
ki (z0i − zi )xzi .
i =1
We have the following important components:
we sum over r reactions,
ki is the reaction rate,
(z0i − zi ) is the reaction vector, and
Q
zij is the mass-action term.
xzi = m
j=1 (xj )
Matthew Douglas Johnston
Stochastic Chemical Reaction Networks
(1)
Background
Stochastic Models
Interesting Systems
Chemical Reactions
Standard Model
Stoichiometric Compatibility Classes
Consider the (reversible) system
k1
A1 2A2 .
k2
Matthew Douglas Johnston
Stochastic Chemical Reaction Networks
Background
Stochastic Models
Interesting Systems
Chemical Reactions
Standard Model
Stoichiometric Compatibility Classes
Consider the (reversible) system
k1
A1 2A2 .
k2
This has the governing dynamics
x˙1
−1
1
= k1
x1 + k2
x22 ,
x˙2
2
−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
k1
A1 2A2 .
k2
This has the governing dynamics
−1
x˙1
1
x1 + k2
= k1
x22 ,
2
x˙2
−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
k1
A1 2A2 .
k2
This has the governing dynamics
1
−1
x˙1
x1 + k2
x22 ,
= k1
−2
2
x˙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
k1
A1 2A2 .
k2
This has the governing dynamics
1
x˙1
−1
x1 + k2
x22 ,
= k1
−2
x˙2
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
k1
A1 2A2 .
k2
This has the governing dynamics
x˙1
−1
1
x22 ,
= k1
x1 + k2
x˙2
2
−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?
−1
x˙1
1
= k1
x1 + k2
x22
x˙2
2
−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?
0
−1
x˙1
1
2
=
= k1
x1 + k2
x2
0
x˙2
2
−2
The (positive) equilibrium set is given by
(
r
E=
x∈
R2>0
Matthew Douglas Johnston
| x2 =
k1
x1
k2
)
.
Stochastic Chemical Reaction Networks
Background
Stochastic Models
Interesting Systems
Chemical Reactions
Standard Model
Stoichiometric Compatibility Classes
What kind of properties does this system have?
−1
1
x˙1
x1 + k2
x22
= k1
2
−2
x˙2
The (positive) equilibrium set is given by
(
r
E=
x∈
R2>0
| x2 =
k1
x1
k2
)
.
For any k1 , k2 , x1 , x2 we have f(x) ∈ S where
1
S = span
.
−2
Matthew Douglas Johnston
Stochastic Chemical Reaction Networks
Background
Stochastic Models
Interesting Systems
Chemical Reactions
Standard Model
Stoichiometric Compatibility Classes
(x0+S)
E
x2
x1
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
(x0+S)
E
x2
x1
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
(x0+S)
E
x2
x1
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
(x0+S)
E
x2
x1
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 z0i − 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 z0i − zi | i = 1, . . . , r .
Theorem
Solutions x(t) of (1) are restricted to stoichiometric
compatibility classes such that
x(t) ∈ (S + x0 ) ∩ Rm
+
Matthew Douglas Johnston
∀t ≥ 0.
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
k1+
k
S + E SE →2 P + E
k1−
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
k1+
k
S + E SE →2 P + E
k1−
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
k1+
k
S + E SE →2 P + E
k1−
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
Y
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
Y
j=1
nj !
(nj − zij )!
denote the transition probability from one state to another.
The chemical master equation is given by
X
X
dP(n; t)
=
λi (n + zi − z0i )P(n + zi − z0i ; t) − P(n; t)
λi (n)
dt
i ∈I
Matthew Douglas Johnston
i ∈O
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
Y
j=1
nj !
(nj − zij )!
denote the transition probability from one state to another.
The chemical master equation is given by
X
X
dP(n; t)
λi (n + zi − z0i )P(n + zi − z0i ; t) − P(n; t)
=
λi (n)
dt
i ∈I
i ∈O
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
Y
j=1
nj !
(nj − zij )!
denote the transition probability from one state to another.
The chemical master equation is given by
X
X
dP(n; t)
=
λi (n + zi − z0i )P(n + zi − z0i ; t) − P(n; t)
λi (n)
dt
i ∈I
i ∈O
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
k1
A1 2A2 .
k2
Matthew Douglas Johnston
Stochastic Chemical Reaction Networks
Background
Stochastic Models
Interesting Systems
Small-Scale Considerations
Gillespie Algorithm
Chemical Master Equation
Reconsider the earlier system
k1
A1 2A2 .
k2
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
k1
A1 2A2 .
k2
Consider the states (2, 0), (1, 2), and (0, 4), for which we have
dP(2, 0; t)
2k2
=
P(1, 2; t) − 2k1 P(2, 0; t)
dt
V
dP(1, 2; t)
12k2
2k2
=
P(0, 4; t) + 2k1 P(2, 0; t) − k1 +
P(1, 2; t)
dt
V
V
12k2
dP(0, 4; t)
= k1 P(1, 2; t) −
P(0, 4; t).
dt
V
Matthew Douglas Johnston
Stochastic Chemical Reaction Networks
Background
Stochastic Models
Interesting Systems
Small-Scale Considerations
Gillespie Algorithm
Chemical Master Equation
Reconsider the earlier system
k1
A1 2A2 .
k2
Consider the states (2, 0), (1, 2), and (0, 4), for which we have
dP(2, 0; t)
2k2
=
P(1, 2; t) − 2k1 P(2, 0; t)
dt
V
dP(1, 2; t)
12k2
2k2
=
P(0, 4; t) + 2k1 P(2, 0; t) − k1 +
P(1, 2; t)
dt
V
V
12k2
dP(0, 4; t)
= k1 P(1, 2; t) −
P(0, 4; t).
dt
V
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)
k
1
A1 −→
2A1
k
2
A1 + A2 −→
2A2
k
3
A2 −→
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)
k
1
A1 −→
2A1
k
2
A1 + A2 −→
2A2
k
3
A2 −→
O.
For the rate constant values k1 = k2 = k3 = 1 the large-scale
deterministic system has a unique positive equilibrium x1∗ = x2∗ = 1
which is a centre, however...
Matthew Douglas Johnston
Stochastic Chemical Reaction Networks
Background
Stochastic Models
Interesting Systems
Matthew Douglas Johnston
Lotka-Volterra System
The Block
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]:
1
2A1 + A2 −→
↑
3A2
3A1
↓
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]:
1
2A1 + A2 −→
↑
3A2
3A1
↓
1
←− A1 + 2A2 .
The system exhibits varying behaviour depending on the value of :
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]:
1
2A1 + A2 −→
↑
3A2
3A1
↓
1
←− A1 + 2A2 .
The system exhibits varying behaviour depending on the value of :
1
≥ 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]:
1
2A1 + A2 −→
↑
3A2
3A1
↓
1
←− A1 + 2A2 .
The system exhibits varying behaviour depending on the value of :
1
≥ 1/6: one stable equilibrium.
2
0 < < 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]:
1
2A1 + A2 −→
↑
3A2
3A1
↓
1
←− A1 + 2A2 .
The system exhibits varying behaviour depending on the value of :
1
≥ 1/6: one stable equilibrium.
2
0 < < 1/6: two stable and one unstable equilibria.
3
= 0: two stable boundary equilibria.
Matthew Douglas Johnston
Stochastic Chemical Reaction Networks
Background
Stochastic Models
Interesting Systems
(a)
Lotka-Volterra System
The Block
(b)
(c)
Figure: Block system with (a) ≥ 1/6, (b) 0 < < 1/6, and (c) = 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 < < 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 < < 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 < < 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 < < 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 < < 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 = 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