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Interest lies in inference for the rate parameters in a complex stochastic biological model describing the aggregation of proteins within human cells. Protein aggregation is a factor in many age-related diseases such as Alzheimer's disease. Ideally time-course measurements on all chemical species in the model would be available. However, current experimental techniques only allow noisy observations on the proportions of cell death at a few time points. Although the model has a large state space and is analytically intractable, realisations from the model can be obtained using a stochastic simulator. The time evolution of a cell can be repeatedly simulated giving an estimate of the proportion of cell death. Unfortunately, simulation from the model is too slow to be used in an MCMC inference scheme. A Gaussian process emulator, which is very fast, can be used to approximate the simulator. An MCMC scheme can be constructed targeting the posterior distribution of interest, however evaluating the marginal likelihood is challenging. A pseudo-marginal approach replaces the marginal likelihood with an easy to construct unbiased estimate while still targeting the true posterior. The methods will be illustrated using a toy birth-death model, allowing comparison with the exact model.
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Inferring parameters in a large stochastic model using only proportions of celldeath: insights from the birth-death model
Holly Ainsworth, Richard Boys & Colin Gillespie∗Newcastle University, UK
REFERENCES[1] Tang, M.Y., Proctor, C.J., Woulfe, J., Gray, D.A.
Experimental and Computational Analysis ofPolyglutamine-Mediated Cytotoxicity In PLoS Com-put Biol., 2010 .
INTRODUCTION →• Expanded polyglutamine (PolyQ) proteins
are known to be the causative agents in anumber of neurodegenerative diseases, butthey are still poorly understood.
• Aggregation of specific proteins are part ofthe normal process of ageing in the brainas well as in many age-related diseases.
• A causative link between aggregation anddisease is not universally accepted.
STOCHASTIC MODEL ↓Stochastic kinetic model with uncertain pa-rameters:
• 27 (chemical) species
• 70 reactions
• 40 rate constants, denoted θ
The model aims to explore the relationshipbetween PolyQ, p38MAPK activation, genera-tion of reactive oxygen species (ROS), protea-some inhibition and inclusion body formation.
DATA ←Data are proportions (of cell death) - not quan-titative trait measurements
Scenario 24hrs 36hrs 48hrs
GFP 15.03 14.55 26.08H25 18.97 18.07 22.50H103 21.68 23.44 36.44
SIMULATION ↓Denote the probability of cell death at time t, pt(θ). Note the probability of cell death dependson parameters θ. Given θ, the Gillespie algorithm can be used to simulate the time evolution ofa particular cell:
• The simulation gives us a binary time series for the cell, with 1 = death and 0 = no death.
• Repeating the above for n cells gives us a handle on pt(θ) via the observed proportion ofcell death p̂t(θ), where
p̂t(θ) ∼1
nBin(n, pt(θ)).
BIRTH-DEATH MODEL →Let x denote the number of individualspresent in the population. In chemical kineticnotation, this system is represented as
R1 : xλ−→ 2x (birth)
R2 : 2xµ−→ x (death)
Example simulations from the model:
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0 4 8 12Time
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of e
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Simulator n = 10 n = 100 n = 1000 n = 10000
Comparisons with PolyQ model:
• Compare a single population governed bya birth-death process with a cell governedby the PolyQ model.
• A cell becoming extinct in the birth-deathprocess can be likened to a cell dying inthe PolyQ model.
• Proportions of extinction from the birth-death model are comparible to propor-tions of cell death from the PolyQ model.
An analytic expression for the probability ofextinction in the birth-death process for givent, λ, µ and initial population level is available.
INFERENCE ↓Work with proportions on the logit scale and assume data model:
yt = logitxt = logit pt(θ) + σεt, t = 1, . . . , T
where εt ∼ N(0, 1) independently. The posterior of interest is
π(θ, σ|y) ∝ π(θ)π(σ)π(y|θ, σ).
Approaches to inference:
• Vanilla MCMC - approximate the distribution of p̂t(θ) and account for the uncertainty using
elogit p̂t(θ) ∼ N(
logit pt(θ),1
npt(θ)[1− pt(θ)]
)approximately.
• Pseudo marginal 1 - construct a Monte Carlo estimate of the marginal likelihood.
• Pseudo marginal 2 - at each iteration of the MCMC scheme, use a (SIR) particle filter toconstruct a SMC approximation to the marginal likelihood.
RESULTS
log(λ) log(µ) log(σ)
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n=
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−3 −2 −1 0 1 −0.5 0.0 0.5 1.0 −3 −2 −1 0 1Parameter value
Den
sity
Vanilla MCMC Pseudo−marginal MCMC 1 Pseudo−marginal MCMC 2 Inference using exact probability of death
• Both pseudo-marginal schemes involvemore work. They use n×#particles runsof the simulator at each iteration, com-pared to n runs for the original scheme.
• The pseudo-marginal approach has theadvantage that it performs exact infer-ence (for a particular choice of n).
• This will be useful particularly whenthe asymptotic distributional result forelogit p̂t(θ) is poor for small n.
• The SMC approach to estimating themarginal likelihood appears to mix betterthan the MC approach.
