Moment Closure Based Parameter Inference of Stochastic Kinetic Models

Moment Closure Based ParameterInference of Stochastic Kinetic Models

Colin Gillespie

School of Mathematics & Statistics

Overview

Talk outlineI An introduction to moment closure

I Parameter inference

I Conclusion

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 ].

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 ].

Stochastic representation

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

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

dt= λ(n− 1)pn−1 + µ(n + 1)pn+1

− (λ + µ)npn

Usually called the forward Kolmogorovequation or chemical master equation

0 1 2 3 4Time

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

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

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

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

Simple dimerisation model

Dimerisation

2X1 −→ X2 and X2 −→ 2X1

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

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

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 model as an approximation tothe stochastic

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

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 model as an approximation tothe stochastic

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

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

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

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

5700000

5750000

5800000

5850000

5900000

5950000

6000000

0 500 1000 1500 2000 0 500 1000 1500 2000Time

Gillespie, CS, 2009

Density plots: heat shock model

Time t=200 Time t=2000

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

P53-Mdm2 oscillation model

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

I The model contains an eventI 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!

0 5 10 15 20 25 30Time

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

Parameter inference

0 10 20 30 40 50Time

0 2 4 6 8 10k1

I Simple immigration-deathprocess

I R1 : ∅k1−→ X

I R2 : Xk2−→ ∅

I The CME can be solved

I Discrete time course data

I The likelihood can be very flat

Parameter inference

● ●

● ● ●

● ●

0 10 20 30 40 50Time

0 2 4 6 8 10k1

I R1 : ∅k1−→ X

I R2 : Xk2−→ ∅

Parameter inference

● ●

● ● ●

● ●

0 10 20 30 40 50Time

0 2 4 6 8 10k1

I R1 : ∅k1−→ X

I R2 : Xk2−→ ∅

Lotka-Volterra model

The Lotka-Volterra predator prey system,describes the time evolution of twospecies, Y1 and Y2

I Prey birth: Y1 → 2Y1I Interaction: Y1 + Y2 → 2Y2I Predator death: Y2 → ∅I Since the Lotka-Volterra model

contains a non-linear rate law, the i th

moment equation depends on the(i + 1)th moment.

0 10 20 30 40

Species Predator Prey

0 10 20 30 40

Parameter estimation

I Let Y(tu) = (Y1(tu),Y2(tu))′ be the vector of the observed predatorand prey

I To infer c1, c2 and c3, we need to estimate

Pr[Y(tu)| Y(tu−1), c]

i.e. the solution of the forward Kolmogorov equation

I We will use moment closure to estimate this distribution:

Y(tu) |Y(tu−1), c ∼ N(ψu−1,Σu−1)

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

Parameter estimation

I Let Y(tu) = (Y1(tu),Y2(tu))′ be the vector of the observed predatorand prey

I To infer c1, c2 and c3, we need to estimate

Pr[Y(tu)| Y(tu−1), c]

i.e. the solution of the forward Kolmogorov equation

I We will use moment closure to estimate this distribution:

Y(tu) |Y(tu−1), c ∼ N(ψu−1,Σu−1)

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

Bayesian parameter inference

I Summarising our beliefs about c and the unobserved predatorpopulation Y2(0) via uninformative priors

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

p (y2, c | y1) ∝ p(c) p (y2(0))40

∏u=1

p (y(tu) | y(tu−1), c)

I For the results shown, we used a vanilla Metropolis-Hasting step toexplore the parameter and state spaces

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

Bayesian parameter inference

I Summarising our beliefs about c and the unobserved predatorpopulation Y2(0) via uninformative priors

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

p (y2, c | y1) ∝ p(c) p (y2(0))40

∏u=1

p (y(tu) | y(tu−1), c)

I For the results shown, we used a vanilla Metropolis-Hasting step toexplore the parameter and state spaces

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

Resultsc1 c2 c3

Mom. Clos.

Diffusion

Mom. Clos.

Diffusion

Fully O

artially Obs.

0.3 0.4 0.5 0.6 0.7 0.8 0.0015 0.0020 0.0025 0.0030 0.0035 0.2 0.3 0.4Parameter value

Auto regulation system

I This system contains twelve reactions and six species

I The species populations ranges from zero (for species i) to around65,000 for species G

I The moment closure approximation yields a closed set oftwenty-seven ODEs

I Six ODEs for the meansI Six ODEs for the variancesI Fifteen ODEs for the covariance terms

Stochastic realisation

0 10 20 30 40 50

Species

0 10 20 30 40 50

Species

0 10 20 30 40 50

Species

0 10 20 30 40 50

Parameter inferenceFully Obs. Partially Obs.

0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0Parameter value

I Posterior distributions for c1 toc8: mean ± 2 sd. True values inred

I Given information on allspecies, inference is reasonable

I For most of the parameters,fewer data points results inlarger credible regions

I But not in all cases!

0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0Parameter value

Future work

I Techniques for assessing the moment closure approximationI Better closure techniques

I Computer emulation for momentsI Using the moment closure approximation as a proposal distribution in

an MCMC algorithmI The proposal can be (almost) anything we wantI The likelihood can be calculated using anything we want

Acknowledgements

I Peter Milner I Darren Wilkinson

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.

Moment Closure Based Parameter Inference of Stochastic Kinetic Models

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