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Stochastic reaction-diffusion simulations has become an efficient approach for modelling spatial aspects of intracellular biochemical reaction networks. By accounting for intrinsic noise due to low copy number of chemical species, stochastic reaction-diffusion simulations have the ability to more accurately predict and model biological systems. As with many simulations software, exploration of the parameters associated with the model can be needed to yield new knowledge about the underlying system. The exploration can be conducted by executing parameter sweeps for a model. However, with little or no prior knowledge about the modelled system, the effort for practitioners to explore the parameter space can get overwhelming. To account for this problem we perform a feasibility study on an explorative behavioural analysis of stochastic reaction-diffusion simulations by applying spatial-temporal data mining to large parameter sweeps. By reducing individual simulation outputs into a feature space involving simple time series and distribution analytics, we were able to find similar behaving simulations after performing an agglomerative hierarchical clustering.

In this thesis, we study two stochastic reaction diffusion models - the diffusion limited reaction model of Smoluchowski, and a second approach popularized by Doi. Both models treat molecules as points undergoing Brownian motion. The former represents chemical reactions between two reactants through the use of reactive boundary conditions, with two molecules reacting instantly upon reaching the boundary of a properly embedded open set, termed the reaction region (or more generally some fixed lower dimensional sub-manifold). The Doi model uses reaction potentials, supported in the reaction region, whereby two molecules react with a fixed probability per unit time, λ, upon entering the reaction region. The problem considered is that of obtaining estimates for convergence rates, in λ, of the Doi model to the Smoluchowski model. This problem fits into the theory of singular perturbation or optimization, depending on which reactive boundary conditions one considers, and we solve it - at least for the bimolecular reaction with one stationary target.

In recent years, there has been increased awareness that stochasticity in chemical reactions and diffusion of molecules can have significant effects on the outcomes of intracellular processes, particularly given the low copy numbers of many proteins and mRNAs present in a cell. For such molecular species, the number and locations of molecules can provide a more accurate and detailed description than local concentration. In addition to diffusion, drift in the movements of molecules can play a key role in the dynamics of intracellular processes, and can often be modeled as arising from potential fields. Examples of sources of drift include active transport, variations in chemical potential, material heterogeneities in the cytoplasm, and local interactions with subcellular structures. This dissertation presents a new numerical method for simulating the stochastically varying numbers and locations of molecular species undergoing chemical reactions and drift-diffusion. The method combines elements of the First-Passage Kinetic Monte Carlo (FPKMC) method for reaction-diffusion systems and the Wang—Peskin—Elston lattice discretization of the Fokker—Planck equation that describes drift-diffusion processes in which the drift arises from potential fields. In the FPKMC method, each molecule is enclosed within a "protective domain," either by itself or with a small number of other molecules. To sample when a molecule leaves its protective domain or a reaction occurs, the original FPKMC method relies on analytic solutions of one- and two-body diffusion equations within the protective domains, and therefore cannot be used in situations with non-constant drift. To allow for such drift in our new method (hereafter Dynamic Lattice FPKMC or DL-FPKMC), each molecule undergoes a continuous-time random walk on a lattice within its protective domain, and the lattices change adaptively over time. One of the most commonly used spatial models for stochastic reaction-diffusion systems is the Smoluchowski diffusion-limited reaction (SDLR) model. The DL-FPKMC method generates convergent realizations of an extension of the SDLR model that includes drift from potentials. We present detailed numerical results demonstrating the convergence and accuracy of our method for various types of potentials (smooth, discontinuous, and constant). We also present several illustrative applications of DL-FPKMC, including examples motivated by cell biology.

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Wird der Äquator-Pol-Energietransfer als Wärmediffusion berücksichtigt, so gehen Energiebilanzmodelle in Reaktions-Diffusionsgleichungen über, deren Modellfall die (deterministische) Chafee-Infante-Gleichung darstellt. Ihre Lösung besitzt zwei stabile Zustände und mehrere instabile auf der separierenden Mannigfaltigkeit (Separatrix) der stabilen Anziehungsgebiete. Es wird bewiesen, dass die Lösung auf geeignet verkleinerten Anziehungsgebieten mit Minimalabstand zur Separatrix innerhalb von Zeitskalen relaxiert, die höchstens logarithmisch darin anwachsen. Motiviert durch statistische Belege aus grönländischen Zeitreihen wird diese partielle Differentialgleichung unter Störung mit unendlichdimensionalem, Hilbertraum-wertigen, regulär variierenden Lévy''schen reinen Sprungrauschen mit index alpha und Intensität epsilon untersucht. Ein kanonisches Beispiel dieses Rauschens ist alpha-stabiles Rauschen im Hilbertraum. Durch Erweiterung einer Methode von Imkeller und Pavlyukevich auf stochastische partielle Differentialgleichungen wird unter milden Bedingungen bewiesen, dass im Gegensatz zu Gauß''schem Rauschen die erwarteten Austritts- und übertrittszeiten zwischen Anziehungsgebieten polynomiell mit Ordnung in der inversen Intensität für kleine Rauschintensität anwachsen. In Kapitel 6 wird eine zusätzliche natürliche “Separatrixhypothese” über das Sprungmaß, eingeführt, die eine obere Schranke für die Austrittszeiten aus einer Umgebung der Separatrix impliziert. Dies ermöglicht den Nachweis einer oberen Schranke für die Austrittszeiten, welche gleichmäßig für Anfangsbedingungen in dem ganzen Anziehungsgebiet gilt. Es folgen zwei Lokalisierungsergebnisse. Schließlich wird gezeigt, dass die Lösung metastabiles Verhalten aufweist. Unter der “Separatrixhypothese” wird dies auf ein Ergebnis erweitert, welches gleichmäßig im Raum gilt. / If equator-to-pole energy transfer by heat diffusion is taken into account, Energy Balance Models turn into reaction-diffusion equations, whose prototype is the (deterministic) Chafee-Infante equation. Its solution has two stable states and several unstable ones on the separating manifold (separatrix) of the stable domains of attraction. We show, that on appropriately reduced domains of attraction of a minimal distance to the separatrix the solution relaxes in time scales increasing only logarithmically in it. Motivated by the statistical evidence from Greenland ice core time series, we consider this partial differential equation perturbed by an infinite-dimensional Hilbert space-valued regularly varying (pure jump) Lévy noise of index alpha and intensity epsilon. A proto-type of this noise is alpha-stable noise in the Hilbert space. Extending a method developed by Imkeller and Pavlyukevich to the SPDE setting we prove under mild conditions that in contrast to Gaussian perturbations the expected exit and transition times between the domains of attraction increase polynomially in the inverse intensity. In Chapter 6 we introduce an additional natural separatrix hypothesis on the jump measure that implies an upper bound on the exit time of a neighborhood of the separatrix. This allows to obtain an upper bound for the asymptotic exit time uniform for the initial positions inside the entire domain of attraction. It is followed by two localization results. Finally we prove that the solution exhibits metastable behavior. Under the separatrix hypothesis we can extend this to a result that holds uniformly in space.