Moment closure inference for stochastic kinetic models

Moment closure inference forstochastic kinetic models

Colin Gillespie

School of Mathematics & Statistics

Talk outlineI An introduction to moment closure

I Case study: Aphids

I Conclusion

2/43

Birth-death process

Birth-death model

X −→ 2X and 2X −→ X

which has the propensity functions λX and µX .

Deterministic representationThe deterministic model is

dX (t)dt

= (λ− µ)X (t) ,

which can be solved to give X (t) = X (0) exp[(λ− µ)t ].

3/43

Stochastic representation

I In the stochastic framework, eachreaction has a probability of occurring

I The analogous version of thebirth-death process is the differenceequation

dpn

dt= λ(n− 1)pn−1 + µ(n + 1)pn+1

− (λ + µ)npn

Usually called the forward Kolmogorovequation or chemical master equation

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Moment equations

I Multiply the CME by enθ and sum over n, to obtain

∂M∂t

= [λ(eθ − 1) + µ(e−θ − 1)]∂M∂θ

where

M(θ; t) =∞

∑n=0

enθpn(t)

I 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

5/43

The mean equation

dE[N(t)]dt

= (λ− µ)E[N(t)]

I This ODE is solvable - the associated forward Kolmogorov equation isalso solvable

I The equation for the mean and deterministic ODE are identical

I When the rate laws are linear, the stochastic mean and deterministicsolution always correspond

6/43

The variance equation

I If we differentiate the p.d.e. w.r.t θ twice and set θ = 0, we get:

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.

I Differentiating three times gives an expression for the skewness, etc

7/43

Simple dimerisation model

Dimerisation

2X1 −→ X2 and X2 −→ 2X1

with propensities 0.5k1X1(X1 − 1) and k2X2.

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Dimerisation moment equations

I We formulate the dimer model in terms of moment equations

dE[X1]

dt= 0.5k1(E[X

21 ]− E[X1])− k2E[X1]

dE[X 21 ]

dt= k1(E[X

21 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 the variance

I The i th moment equation depends on the (i + 1)th equation

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Deterministic approximates stochastic

RewritingdE[X1]

dt= 0.5k1(E[X

21 ]− E[X1])− k2E[X1]

in terms of its variance, i.e. E[X 21 ] = Var[X1] + E[X1]2, we get

dE[X1]

dt= 0.5k1E [X1](E[X1]− 1) + 0.5k1Var[X1]− k2E[X1] (1)

I Setting Var[X1] = 0 in (1), recovers the deterministic equation

I So we can consider the deterministic models as an approximation tothe stochastic

I When we have polynomial rate laws, setting the variance to zeroresults in the deterministic equation

10/43

Simple dimerisation model

I To close the equations, we assume an underlying distribution

I The easiest option is to assume an underlying Normal distribution, i.e.

E[X 31 ] = 3E[X 2

1 ]E[X1]− 2E[X1]3

I But we could also use, the Poisson

E[X 31 ] = E[X1] + 3E[X1]

2 + E[X1]3

or the Log normal

E[X 31 ] =

(E[X 2

1 ]

E[X1]

)3

11/43

Heat shock modelI Proctor et al, 2005. Stochastic kinetic model of the heat shock system

I twenty-three reactionsI seventeen chemical species

I A single stochastic simulation up to t = 2000 takes about 35 minutes.

I If we convert the model to moment equations, we get 139 equationsADP Native Protein

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Gillespie, CS, 2009

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Density plots: heat shock model

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600 800 1000 1200 1400 600 800 1000 1200 1400ADP population

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P53-Mdm2 oscillation model

I Proctor and Grey, 2008I 16 chemical speciesI Around a dozen reactions

I The model contains an eventsI At t = 1, set X = 0

I If we convert the model to momentequations, we get 139 equations.

I However, in this case the momentclosure approximation doesn’t do towell!

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What went wrong?

I The Moment closure (tends) to fail when there is a large differencebetween the deterministic and stochastic formulations

I In this particular case, strongly correlated species

I Typically when the MC approximation fails, it gives a negativevariance

I The MC approximation does work well for other parameter values forthe p53 model

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Part II

Cotton aphids

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Cotton aphids

Aphid infestation (G & Golightly, 2010)A cotton aphid infestation of a cotton plant can result in:

I leaves that curl and pucker

I seedling plants become stunted and may die

I a late season infestation can result in stained cotton

I cotton aphids have developed resistance to many chemicaltreatments and so can be difficult to treat

I Basically it costs someone a lot of money

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Cotton aphids

The data consists of

I five observations at each plot

I the sampling times are t=0, 1.14, 2.29, 3.57 and 4.57 weeks (i.e.every 7 to 8 days)

I three blocks, each being in a distinct area

I three irrigation treatments (low, medium and high)

I three nitrogen levels (blanket, variable and none)

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The data

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The dataZero Variable Block

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Some notation

Let

I n(t) to be the size of the aphid population at time tI c(t) to be the cumulative aphid population at time t

1. We observe n(t) at discrete time points2. We don’t observe c(t)3. c(t) ≥ n(t)

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The model

We assume, based on previous modelling (Matis et al., 2004)

I An aphid birth rate of λn(t)

I An aphid death rate of µn(t)c(t)

I So extinction is certain, as eventually µnc > λn for large t

21/43

The model

Deterministic representationPrevious modelling efforts have focused on deterministic models:

dN(t)dt

= λN(t)− µC(t)N(t)

dC(t)dt

= λN(t)

Some problemsI Initial and final aphid populations are quite small

I No allowance for ‘natural’ random variation

I Solution: use a stochastic model

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The model

Stochastic representation

Let pn,c(t) denote the probability:

I there are n aphids in the population at time t

I a cumulative population size of c at time t

This gives the forward Kolmogorov equation

dpn,c(t)dt

= λ(n− 1)pn−1,c−1(t) + µc(n + 1)pn+1,c(t)

− n(λ + µc)pn,c(t)

Even though this equation is fairly simple, it still can’t be solved exactly.

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Some simulations

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Parameters: n(0) = c(0) = 1, λ = 1.7 and µ = 0.001 24/43

Stochastic parameter estimation

I Let X(tu) = (n(tu), c(tu))′ be the vector of observed aphid countsand unobserved cumulative population size at time tu;

I To infer λ and µ, we need to estimate

Pr[X(tu)| X(tu−1),λ, µ]

i.e. the solution of the forward Kolmogorov equation

I We will use moment closure to estimate this distribution

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Moment equations for the means

dE[n(t)]dt

= λE[n(t)]− µ(E[n(t)]E[c(t)] + Cov[n(t), c(t)])

dE[c(t)]dt

= λE[n(t)]

I The equation for the E[n(t)] depends on the Cov[n(t), c(t)]

I Setting Cov[n(t), c(t)]=0 gives the deterministic model

I We obtain similar equations for higher-order moments

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Parameter inference

Given

I the parameters: {λ, µ}I the initial states: X(tu−1) = (n(tu−1), c(tu−1));

We haveX(tu) |X(tu−1),λ, µ ∼ N(ψu−1,Σu−1)

where ψu−1 and Σu−1 are calculated using the moment closureapproximation

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Parameter inference

I Summarising our beliefs about {λ, µ} and the unobservedcumulative population c(t0) via priors p(λ, µ) and p(c(t0))

I The joint posterior for parameters and unobserved states (for a singledata set) is

p (λ, µ, c | n) ∝ p(λ, µ) p (c(t0))4

∏u=1

p (x(tu) | x(tu−1),λ, µ)

I For the results shown, we used a simple random walk MH step toexplore the parameter and state spaces

I For more complicated models, we can use a Durham & Gallant stylebridge (Milner, G & Wilkinson, 2012).

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Simulation study

I Three treatments & two blocks

I Baseline birth and death rates: {λ = 1.75, µ = 0.00095}I Treatment 2 increases µ by 0.0004

I Treatment 3 increases λ by 0.35

I The block effect reduces µ by 0.0003

Treatment 1 Treatment 2 Treatment 3

Block 1 {1.75, 0.00095} {1.75, 0.00135} {2.1, 0.00095}Block 2 {1.75, 0.00065} {1.75, 0.00105} {2.1, 0.00065}

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Simulated data

Treament 1 Treatment 2 Treatment 3

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Parameter structure

I Let i, k represent the block and treatments level, i ∈ {1, 2} andk ∈ {1, 2, 3}

I For each data set, we assume birth rates of the form:

λik = λ + αi + βk

where α1 = β1 = 0

I So for block 1, treatment 1 we have:

λ11 = λ

and for block 2, treatment 1 we have:

λ21 = λ + α2

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MCMC scheme

I Using the MCMC scheme described previously, we generated 2Miterates and thinned by 1K

I This took a few hours and convergence was fairly quick

I We used independent proper uniform priors for the parameters

I For the initial unobserved cumulative population, we had

c(t0) = n(t0) + ε

where ε has a Gamma distribution with shape 1 and scale 10.

I This set up mirrors the scheme that we used for the real data set

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Marginal posterior distributions forλ and µ

Birth Rate

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Death Rate

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Marginal posterior distributions for birthrates

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I We obtained similar densities for the death rates.

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Application to the cotton aphid data set

Recall that the data consists of

I five observations on twenty randomly chosen leaves in each plot;

I three blocks, each being in a distinct area;

I three irrigation treatments (low, medium and high);

I three nitrogen levels (blanket, variable and none);

I the sampling times are t=0, 1.14, 2.29, 3.57 and 4.57 weeks (i.e.every 7 to 8 days).

Following in the same vein as the simulated data, we are estimating 38parameters (including interaction terms) and the latent cumulative aphidpopulation.

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Cotton aphid dataMarginal posterior distributions

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Does the model fit the data?

I We simulate predictive distributions from the MCMC output, i.e. werandomly sample parameter values (λ, µ) and the unobserved statec and simulate forward

I We simulate forward using the Gillespie simulatorI not the moment closure approximation

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Does the model fit the data?

Predictive distributions for 6 of the 27 Aphid data sets

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Summarising the results

I Consider the additional number of aphids per treatment combination

I Set c(0) = n(0) = 1 and tmax = 6

I We now calculate the number of aphids we would see for eachparameter combination in addition to the baseline

I For example, the effect due to medium water:

λ211 = λ + αWater (M) and µ211 = µ + α∗Water (M)

I SoAdditional aphids = c i

Water (M) − c ibaseline

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Aphids over baselineMain Effects

Aphids

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Block 3 Block 2

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Water (H)

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Aphids over baselineInteractions

Aphids

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Conclusions

I The 95% credible intervals for the baseline birth and death rates are(1.64, 1.86) and (0.00090, 0.00099).

I Main effects have little effect by themselves

I However block 2 appears to have a very strong interaction withnitrogen

I Moment closure parameter inference is a very useful technique forestimating parameters in stochastic population models

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Future work

Aphid modelI Other data sets suggest that there is aphid immigration in the early

stages

I Model selection for stochastic models

I Incorporate measurement error

Moment closureI Better closure techniques

I Assessing the fit

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Acknowledgements

I Andrew GolightlyI Peter MilnerI Darren Wilkinson

I Richard Boys

I Jim Matis (Texas A & M)

ReferencesI Gillespie, CS Moment closure approximations for mass-action models. IET Systems Biology 2009.

I Gillespie, CS, Golightly, A Bayesian inference for generalized stochastic population growth models with application to aphids.Journal of the Royal Statistical Society, Series C 2010.

I Milner, P, Gillespie, CS, Wilkinson, DJ Moment closure approximations for stochastic kinetic models with rational rate laws.Mathematical Biosciences 2011.

I Milner, P, Gillespie, CS and Wilkinson, DJ Moment closure based parameter inference of stochastic kinetic models.Statistics and Computing 2012.

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Moment closure inference for stochastic kinetic models

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Transcript of Moment closure inference for stochastic kinetic models