The scope of this project was to implement a quorum sensing model capable of synchronised oscillations from the article ”A synchronized quorum of genetic clocks” [1] in the software framework URDME [2]. The model consists of a system of partial differential equations describing time delayed and coupled biochemical reactions. In URDME, the time delay system was formed using a cascade of reactions in which the rate of each reaction was set so that the expected total time for all reactions in the cascade corresponds to a certain delay time. One reason for this cascade delay model is that it might better capture the inherently stochastic nature of the delay mechanism in the quorum sensing network, as opposed to a model using explicit delays.Another reason is simplicity of implementation, as delays are not explicitly supported in URDME. After initial tests suggested that the cascade delay model gave satisfying results, it was incorporated into the quorum sensing model from the article, which was implemented by rewriting the differential equations as a system of biochemical reactions. Simulations in one and two dimensions were then done, with both stochastic and deterministic solution methods. The one dimensional and two dimensional simulations yielded distinct synchronised oscillations with a cascade delay containing five sub-reactions. Several results from the simulations of the original article could be reproduced. From the results, it was concluded that the proposed cascade delay model was successful in modelling the delayed reactions in the quorum sensing network. In future studies, it is suggested that the individual cells, in which most of the reactions in the quorum sensing network happen, are modelled with greater resolution.

Quantitative descriptions of reaction kinetics formulated at the stochastic mesoscopic level are frequently used to study various aspects of regulation and control in models of cellular control systems. For this type of systems, numerical simulation offers a variety of challenges caused by the high dimensionality of the problem and the multiscale properties often displayed by the biochemical model. In this thesis I have studied several aspects of stochastic simulation of both well-stirred and spatially heterogenous systems. In the well-stirred case, a hybrid method is proposed that reduces the dimension and stiffness of a model. We also demonstrate how both a high performance implementation and a variance reduction technique based on quasi-Monte Carlo can reduce the computational cost to estimate the probability density of the system. In the spatially dependent case, the use of unstructured, tetrahedral meshes to sample realizations of the stochastic process is proposed. Using such meshes, we then extend the reaction-diffusion framework to incorporate active transport of cellular cargo in a seamless manner. Finally, two multilevel methods for spatial stochastic simulation are considered. One of them is a space-time adaptive method combining exact stochastic, approximate stochastic and macroscopic modeling levels to reduce the simualation cost. The other method blends together mesoscale and microscale simulation methods to locally increase modeling resolution. / eSSENCE

Self-organizing patterns arise in a variety of ways in nature, the complex patterning observed on animal coats is such an example. It is already known that the mechanisms responsible for pattern formation starts at the developmental stage of an embryo. However, the actual process determining cell fate has been, and still is, unknown. The mathematical interest for pattern formation emerged from the theories formulated by the mathematician and computer scientist Alan Turing in 1952. He attempted to explain the mechanisms behind morphogenesis and how the process of spatial cell differentiation from homogeneous cells lead to organisms with different complexities and shapes. Turing formulated a mathematical theory and proposed a reaction-diffusion system where morphogens, a postulated chemically active substance, moderated the whole mechanism. He concluded that this process was stable as long as diffusion was neglected; otherwise this would lead to a diffusion-driven instability, which is the fundamental part of pattern formation. The mathematical theory describing this process consists of solving partial differential equations and Turing considered deterministic reaction-diffusion systems.   This thesis will start with introducing the reader to the problem and then gradually build up the mathematical theory needed to get an understanding of the stochastic reaction-diffusion systems that is the focus of the thesis. This study will to a large extent simulate stochastic systems using numerical computations and in order to be computationally feasible a compartment-based model will be used. Noise is an inherent part of such systems, so the study will also discuss the effects of noise and morphogen kinetics on different geometries with boundaries of different complexities from one-dimensional cases up to three-dimensions.