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An internal seminar introducing the moment closure technique for stochastic kinetic models
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An introduction to moment closure techniques
Colin Gillespie
School of Mathematics & StatisticsNewcastle University
July 30, 2008
Colin Gillespie An introduction to moment closure techniques
Modelling
Let’s start with a simple birth-death model.
Birth-death model
X −→ X − 1 and X −→ X + 1
which has the propensity functions µX and λX .
The deterministic model is
dX (t)dt
= (λ− µ)X (t) ,
which can be solved to give X (t) = X (0) exp[(λ− µ)t ].
Colin Gillespie An introduction to moment closure techniques
Deterministic Solution: λ < µ
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Colin Gillespie An introduction to moment closure techniques
Stochastic Simulation
It’s very easy to simulate the birth-death process usingGillespie’s method:
1 Update reaction clock;2 Choose a reaction to occur;3 Repeat.
Colin Gillespie An introduction to moment closure techniques
Four Stochastic Simulations
4 Simulations of a birth-death process
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Simulation 1 Simulation 2
Simulation 3
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Simulation 4
Colin Gillespie An introduction to moment closure techniques
Stochastic Mean and Variance
If we simulated the process a large number of times (say109), then we could calculate the population mean andvariance.We could construct an approximate 95% tolerance interval
Mean ± 2√
Variance
Colin Gillespie An introduction to moment closure techniques
Four Stochastic SimulationsMean (Green), tolerance interval (red), simulation(blue)
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Colin Gillespie An introduction to moment closure techniques
Mean and Variance
In this talk we will look at a quick method for estimating themean and variance, without using stochastic simulation
Colin Gillespie An introduction to moment closure techniques
Moment generating function
Let pn(t) be the probability that the population is of size nat time t .The moment generating function is defined as
M(θ; t) ≡∞∑
n=0
pn(t)enθ .
If we differentiate M(θ; t) w.r.t θ and set θ = 0, we getE[N(t)], i.e. the mean.If we differentiate M(θ; t) w.r.t θ twice, and set θ = 0, weget E[N(t)2] and hence
Var[N(t)] = E[N(t)2]− E[N(t)]2 .
Colin Gillespie An introduction to moment closure techniques
General idea
The birth-death process has the following CME:
dpn
dt= λ(n − 1)pn−1 + µ(n + 1)pn+1 − (λ + µ)npn
After multiplying the CME by enθ and summing over n, weobtain
∂M∂t
= [λ(eθ − 1) + µ(e−θ − 1)]∂M∂θ
Colin Gillespie An introduction to moment closure techniques
Moment Equations
If we differentiate this p.d.e. w.r.t θ and set θ = 0, we get:
dE[N(t)]dt
= (λ− µ)E[N(t)]
where E[N(t)] is the mean. This is a single ODE that wecan solve to obtain a value for the mean.If we differentiate the p.d.e. w.r.t θ twice and set θ = 0, weget:
dE[N(t)2]
dt= (λ− µ)E[N(t)] + 2(λ− µ)E[N(t)2]
and hence the variance Var[N(t)] = E[N(t)2]− E[N(t)]2
So instead of simulating the process 109 to estimate themean and variance, we can simply solve two ODEs.
Colin Gillespie An introduction to moment closure techniques
Moment Equations
If we differentiate this p.d.e. w.r.t θ and set θ = 0, we get:
dE[N(t)]dt
= (λ− µ)E[N(t)]
where E[N(t)] is the mean. This is a single ODE that wecan solve to obtain a value for the mean.If we differentiate the p.d.e. w.r.t θ twice and set θ = 0, weget:
dE[N(t)2]
dt= (λ− µ)E[N(t)] + 2(λ− µ)E[N(t)2]
and hence the variance Var[N(t)] = E[N(t)2]− E[N(t)]2
So instead of simulating the process 109 to estimate themean and variance, we can simply solve two ODEs.
Colin Gillespie An introduction to moment closure techniques
Part I
Examples
Colin Gillespie An introduction to moment closure techniques
Simple Dimerisation model
The dimerisation model has the following biochemicalreactions:
Dimerisation
2X1 −→ X2 and X2 −→ 2X1
We can formulate the dimer model in terms of momentequations, namely,
dE[X1]
dt= 0.5k1(E[X 2
1 ]− E[X1])− k2E[X1]
dE[X 21 ]
dt= k1(E[X 2
1 X2]− E[X1X2]) + 0.5k1(E[X 21 ]− E[X1])
+ k2(E[X1]− 2E[X 21 ])
where E[X1] is the mean of X1 and E[X 21 ]− E[X1]
2 is thevariance of X1.
Colin Gillespie An introduction to moment closure techniques
Simple Dimerisation model
The dimerisation model has the following biochemicalreactions:
Dimerisation
2X1 −→ X2 and X2 −→ 2X1
We can formulate the dimer model in terms of momentequations, namely,
dE[X1]
dt= 0.5k1(E[X 2
1 ]− E[X1])− k2E[X1]
dE[X 21 ]
dt= k1(E[X 2
1 X2]− E[X1X2]) + 0.5k1(E[X 21 ]− E[X1])
+ k2(E[X1]− 2E[X 21 ])
where E[X1] is the mean of X1 and E[X 21 ]− E[X1]
2 is thevariance of X1.The i th moment equation depends on the (i + 1)th
equation.Colin Gillespie An introduction to moment closure techniques
Simple Dimerisation model
The dimerisation model has the following biochemicalreactions:
Dimerisation
2X1 −→ X2 and X2 −→ 2X1
We can formulate the dimer model in terms of momentequations, namely,
dE[X1]
dt= 0.5k1E [X1](E[X1]− 1) + 0.5k1Var[X1]− k2E[X1]
where E[X1] is the mean of X1 and E[X 21 ]− E[X1]
2 is thevariance of X1.The deterministic equation is an approximation to thestochastic mean.
Colin Gillespie An introduction to moment closure techniques
Simple Dimerisation model
To close the equations, we usually assume that theunderlying distribution is Normal or Lognormal.The easiest option is to assume an underlying Normaldistribution, i.e.
E[X 31 ] = 3E[X 2
1 ]E[X1]− 2E[X1]3
But we could also use, the Poisson
E[X 31 ] = E[X1] + 3E[X1]
2 + E[X1]3
or the Lognormal
E[X 31 ] =
(E[X 2
1 ]
E[X1]
)3
Colin Gillespie An introduction to moment closure techniques
Simple Dimerisation model
To close the equations, we usually assume that theunderlying distribution is Normal or Lognormal.The easiest option is to assume an underlying Normaldistribution, i.e.
E[X 31 ] = 3E[X 2
1 ]E[X1]− 2E[X1]3
But we could also use, the Poisson
E[X 31 ] = E[X1] + 3E[X1]
2 + E[X1]3
or the Lognormal
E[X 31 ] =
(E[X 2
1 ]
E[X1]
)3
Colin Gillespie An introduction to moment closure techniques
Simple Dimerisation model
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Colin Gillespie An introduction to moment closure techniques
Heat Shock Model
Proctor et al, 2005 - 23 reactions, 17 chemical speciesA single stochastic simulation up to t = 2000 takes about35 minutes.If we convert the model to moment equations, we get 139equations.
A python script automatically generates the ODEs from anSBML file
These can be solved in less than 5 minutes using MapleHopefully I’ll start outputting in sundials, so this should beeven quicker
Colin Gillespie An introduction to moment closure techniques
Heat Shock Model
Time
AD
P
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800
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1200
0 500 1000 1500 2000
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000’
s)
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Colin Gillespie An introduction to moment closure techniques
Univariate Distributions
ADP
Den
sity
0.000
0.002
0.004
0.006
600 800 1000 1200 1400
Time t=200
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Time t=2000
Colin Gillespie An introduction to moment closure techniques
Bivariate Distributions at time t = 2000
ADP
Nat
P
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4e+0
65e
+06
6e+0
67e
+06
Colin Gillespie An introduction to moment closure techniques
P53-Mdm2 Oscillations model
Proctor and Grey, 2008 - 16 chemical species and about adozen reactions.The model contains two events.If we convert the model to moment equations, we get 139equations.However, in this case the moment closure approximationdoesn’t do to well!
Colin Gillespie An introduction to moment closure techniques
P53-Mdm2 Oscillations model
Proctor and Grey, 2008 - 16 chemical species and about adozen reactions.The model contains two events.If we convert the model to moment equations, we get 139equations.However, in this case the moment closure approximationdoesn’t do to well!
Colin Gillespie An introduction to moment closure techniques
P53 MeanMC(black), True (red)
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Colin Gillespie An introduction to moment closure techniques
P53 MeanMC(black), True (red), Deterministic(green)
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Colin Gillespie An introduction to moment closure techniques
What went wrong?
The Moment closure (tends) to fail when there is a largedifference between the deterministic and stochasticformulations.I believe it failed because of strongly correlated speciesTypically when the MC approximation fails, it gives anegative varianceThe MC approximation does work well for other parametervalues for the p53 model.
Colin Gillespie An introduction to moment closure techniques
Software
Python script that takes in a SBML file and outputs themoment equations.Currently outputs as a Maple file (University has a sitelicence)Hopefully it will soon output as a sundials/GSL C file(Sort of) supports events.Currently only handles polynomial rate laws, but could beupgrade to handle more complicated rate laws.
Colin Gillespie An introduction to moment closure techniques
References
For an introduction to Moment closure see papers by Matiset al over the last 20 years.Gillespie, C.S. Moment closure approximations formass-action models. IET Systems Biology, in press
Colin Gillespie An introduction to moment closure techniques