Statistical modelling of Ca2+ oscillations in the presence of single cell heterogeneity

Intracellular calcium oscillations are a versatile signalling mechanism responsible for many biological phenomena including immune responses and insulin secretion. There is now compelling evidence that whole-cell calcium oscillations are stochastic, arising from random molecular interactions at the subcellular level. This poses a significant challenge for modelling. In this thesis, we develop a probabilistic approach that treats calcium oscillations as a stochastic point process. By employing an intensity function — a one dimension function over time which corresponds to the mean calcium spiking rate — we capture intrinsic cellular heterogeneity as well as inhomogeneous extracellular conditions, such as time-dependent stimulation. We adopt a Bayesian approach to infer the model parameters from calcium oscillations. Under this approach we need to be able to infer the intensity function. One method is to use a parametric model for the intensity function. For example we could assume the intensity function has the linear form x(t) = at + b. Then the intensity function is reduced to only needing to infer the two parameters a and b. However, parametric models suffer from strict assumptions, in this case, for the shape of the intensity function. Therefore, to lessen such assumptions, we utilise a non-parametric approach. This requires a prior distribution over the space of functions. We use two such priors, namely Gaussian processes and piecewise constant functions. We use Markov chain Monte Carlo (MCMC) techniques to sample from the posterior distribution to obtain estimates for our model parameters. Although advan- tageous — due to sampling from the true posterior distribution — MCMC algorithms can experience issues relating to their computational cost and imprecise samplers. We discuss the issues arising for our particular model and data and develop methods to improve the functionality of the MCMC algorithms in this case. For example we discuss the difficulty of inferring the length scale of the Gaussian process when fitted from calcium oscillations. An important mechanism of calcium oscillations is the refractory period, the min- imum amount of time before the next calcium oscillation. Thus, it may be beneficial to explicitly include the refractory period as part of the model. We investigate the advantages and disadvantages of including the refractory period. We fit the model to HEK293 cells and astrocytes challenged under a variety of stimulation protocols. We find that our model can accurately generate surrogate spike sequences with similar properties to those the model is fitted from. Therefore, the model can be used to cheaply create spike sequences that are synonymous to those found experimentally. Moreover, our model captures the similarity between calcium spike sequences obtained from step-change stimulus protocols and constant stimulus protocols. Combining intensity functions inferred from constant stimulus experiments closely follow the intensity function from a step change experiment. This implies it may be possible to build surrogate spike sequences for complex time-dependent stimulation protocols by combining results from simpler experiments. Of particular interest are patterns found in the intensity function which describes the heterogeneity in the calcium oscillations over time. Common patterns could help to understand the different time scales of the calcium response. Standard approaches often fail in grouping intensity functions with similar shape. Therefore, we develop an approach to cluster intensity functions based on their shape alone by utilising the Haar basis. In summary, we have developed novel statistical approaches based on the concept of stochastic point processes and non-standard MCMC techniques. We have successfully applied these new methodologies to gain a deeper understanding into the stochastic nature of intracellular calcium oscillations, in particular how different cell types respond to a variety of stimulation protocols. In turn, this brings us one step closer to unravel the complex dynamics of this pivotal intracellular messenger which controls life from its very beginning to its end