Stochastic Models in Biology: Methods and Applications to Intracellular Transport

Mammalian cells regulate their glucose levels by redistributing glucose transporter proteins within the cell. Glucose Transporter 4 (GLUT4) is the main insulin-regulated glucose transporter in mammalian cells. Insulin stimulates the release of sequestered GLUT4 from intracellular compartments and the subsequent translocation to the plasma membrane. The breakdown of this process is implicated in type-2 diabetes. However, the mechanisms by which GLUT4 is released and trafficked remain unclear. This thesis utilises a queuing network to test a possible mechanism of GLUT4 sequestration and release in response to insulin. The contribution of this thesis is twofold. Firstly, it develops a new model for intracellular GLUT4 trafficking. A closed queueing network is developed as a novel approach to study this system. This approach allows mechanisms to be modelled directly -- something not possible with mean-field models. The second contribution is a series of studies related to the inference of stochastic systems. These studies investigate the effects of measurement error on parameter inference, develop a distance measure to enable parameter inference of simulated stochastic models using small datasets, and the development of a surrogate model as an initial investigation into efficient parameter inference for systems which are expensive to simulate. The first part of this thesis focuses on deterministic models. The effects of measurement error for parameter inference in the biological sciences, using deterministic models, are investigated. It is common in mathematical biology to consider measurement error in the measurement of the dependent variable. However, less attention has been given to errors in the independent variable. Through a series of synthetic data studies, the effects of various error models are investigated, with a particular focus given to error in the time a measurement is taken. It is shown that in a wide range of scenarios, parameter inference is robust to these errors without explicitly accounting for the errors. However, the investigations indicate that some systems, such as oscillating systems, are particularly susceptible to these errors and parameter estimates become biased. To aid researchers in the biological sciences, methods to correct for measurement error taken from the statistics literature are reviewed, and the applicability of these methods assessed in a biological context by considering data availability and necessary assumptions on the error. Mean-field models have previously been used to model GLUT4 translocation. Mean-field models smooth out and average the contributions of multiple microscale structures and stochastic molecule-molecule interactions and individual processes, suggesting the need for a more complex, stochastic modelling approach. Queuing networks have been extensively used to model computer systems, manufacturing systems, road traffic routing, electric vehicle charging, etc., and could be amenable to modelling cellular processes. Evaluation of the results of these models are often limited to steady state average measures, such as average queue length and wait time. However, these measures do not always fully describe the behaviour of the system; for example, they do not capture start up or transient effects. In cellular processes, these start up and transient effects are critical to the operation of the system. In this case, repeated stochastic simulations are required to compute trajectories for analysis and comparison with experimental observations. In order to infer model parameters, a distance measure to compare stochastic model outputs to time-series data is required. A measure suitable the problem of GLUT4 translocation is developed and characterised. In this thesis a novel model for intracellular GLUT4 trafficking is developed. The model is a closed queuing network consisting of four stations: the endosome store, the microtubules, the fusion sites, and the plasma membrane. In this model the customers represent GLUT4 packaged as vesicles. Here, a biologically plausible mechanism for the release and subsequent translocation of sequestered GLUT4 is presented -- that insulin action occurs only at the fusion sites. The hypothesis is that insulin stimulates the activation of the fusion sites, and inactive fusion sites refuse service to customers on the preceding microtubule. The refusal of service by inactive fusion servers sequesters the GLUT4 on the associated microtubules and only releases them when the site becomes active post-insulin stimulation. To test this hypothesis the queuing model needed to be fit with experimental data. The experimental data consists of time-series with small sample sizes and destructive measurements. Previously available methods were unsuitable for inference with this model and data. In this study, a distance measure that compares stochastic model outputs to time-series data was developed. This distance measure compares data to model outputs across three scales: first at individual time points, then across time-series, and finally across multiple experiments. This distance utilises comparisons of the empirical cumulative distribution functions of the data and model output at the time point scale. This allows comparisons without making assumptions about the distribution of either the data or the model outputs and allows inference with simulated model outputs. Two broad categories of comparators at each point were considered, based on the empirical cumulative distribution (ECDF) of the data and of the model outputs: discrete based measures such as the Kolmogorov-Smirnov distance, and integrated measures such as the Wasserstein-1 distance between the ECDFs. It was found that the discrete based measures were highly sensitive to parameter changes near the synthetic data parameters but were largely insensitive otherwise, whereas the integrated distances had smoother transitions as the parameters approached the true values. The integrated measures were also found to be robust to noise added to the synthetic data, replicating experimental error. The characteristics of the identified distances provides the basis for the design of an algorithm suitable for fitting stochastic models to real world stochastic data. Using this distance the queuing model is fit to experimental data to test the hypothesis that insulin stimulated activation of the fusion sites alone is sufficient to capture the system dynamics. The results indicate that it is sufficient and suggest that the activation of fusion sites in response to insulin could be a primary determinant of the system dynamics. This work identifies a biologically plausible mechanism by which GLUT4 can be sequestered and then released and trafficked to the plasma membrane in response to insulin stimulation. Investigations into alternative or secondary mechanisms of insulin action were prohibited by the computational cost of the queuing model. As a first step towards circumventing this restriction, a surrogate model that encodes the features of the queuing model was developed. This model uses feedback terms in a system of differential equations to approximate the blocking mechanisms seen in the queuing model. A sensitivity analysis of the surrogate model was performed and the correspondence of the surrogate model to the queuing model was assessed. The results of the sensitivity analyses can then be used in the development of an efficient inference scheme for the queuing model. A method for efficient inference will allow the GLUT4 trafficking system to be further studied with the queuing model developed in this thesis, and enable a wider search of the existing parameter space of the model. Indeed, extra complexity could then be added to the model to incorporate and assess the effects of additional processes, alternative mechanisms, and secondary actions of insulin on the system. The models and methods developed in this thesis provide insight into the dynamics of GLUT4 trafficking and lay the foundations for the wider use of stochastic models constrained with experimental data.