Bayesian inference for stochastic kinetic models using data on proportions of cell death

Abstract

The PolyQ model is a large stochastic kinetic model that describes protein aggregation within human cells as they undergo ageing. The presence of protein aggregates in cells is a known feature in many age-related diseases, such as Huntington's. Experimental data are available consisting of the proportions of cell death over time. This thesis is motivated by the need to make inference for the rate parameters of the PolyQ model. Ideally observations would be obtained on all chemical species, observed continuously in time. More realistically, it would be hoped that partial observations were available on the chemical species observed discretely in time. However, current experimental techniques only allow noisy observations on the proportions of cell death at a few discrete time points. This presents an ambitious inference problem. The model has a large state space and it is not possible to evaluate the data likelihood analytically. However, realisations from the model can be obtained using a stochastic simulator such as the Gillespie algorithm. The time evolution of a cell can be repeatedly simulated, giving an estimate of the proportion of cell death. Various MCMC schemes can be constructed targeting the posterior distribution of the rate parameters. Although evaluating the marginal likelihood is challenging, a pseudo-marginal approach can be used to replace the marginal likelihood with an easy to construct unbiased estimate. Another alternative which allows for the sampling error in the simulated proportions is also considered. Unfortunately, in practice, simulation from the model is too slow to be used in an MCMC inference scheme. A fast Gaussian process emulator is used to approximate the simulator. This emulator produces fully probabilistic predictions of the simulator output and can be embedded into inference schemes for the rate parameters. The methods developed are illustrated in two smaller models; the birth-death model and a medium sized model of mitochondrial DNA. Finally, inference on the large PolyQ model is considered.