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.