Spatial arrangements are very important for many biological systems. This thesis presents several different studies of biological systems, which are related to spatial arrangements at two different levels: one is the growth of biological macromolecules, here related to protein aggregation, and the other is the spatial regulation of biological systems, here related to different aspects of the inflammatory response. All systems are studied using computational modelling and mathematical analysis.

The first part of the thesis explores different protein aggregation scenarios. In Chapter 1, we consider a previously studied and very general aggregation model describing frangible linear filaments. This model is especially relevant for the growth of amyloid fibres, that have been related to a number of serious human diseases, and which are known to grow in an accelerated self-enhanced manner.We derive an approximate analytical mathematical expression for the time evolution of the length distribution of the aggregate population, and we discuss the accuracy of the analytical expression. We also compare the model of frangible linear aggregation to experimentally obtained length distributions of growing insulin filaments.

In Chapter 2, we consider the aggregation of the protein p25α, influenced by the presence of the secondary chemical species heparin. Different concentrations of heparin present different environmental conditions, which cause the protein aggregates to form different structural shapes. We construct a mathematical model, which is fitted to experimental data for p25α aggregation at different heparin levels. The model incorporates a logistic-like growth assumption, which is motivated in the beginning of the chapter, and which represents an alternative model for accelerated growth of amyloid fibres. In Chapter 3, we consider the complex aggregation patterns of the whey proteins β-lactoglobulin (bLG) and α-lactalbumin (aLA), influenced by several different environmental conditions, which cause the aggregates to form the different structural shapes - here the varying environmental conditions are different pH and calcium concentrations.

We construct a mathematical model for the aggregation process, and fit the model to an array of experimental data. The model reproduces the dynamics of the aggregation process and predicts final size distributions of the aggregates, which agree well with the expectation based on experimental measurements. The second part of the thesis explores different spatial aspects of inflammatory response. In Chapter 4 we address the problem of cytokine signal transmission and the subsequent white blood cell recruitment during inflammatory response. We construct a simple model of the inflammatory response in a tissue cell, based on the regulatory network of the transcription factor NF-κB. We show that the simple model is able to produce either transient or continuous amplification of the cytokine signal depending on the external and internal conditions of the cell. We then construct a multicellular model of the tissue and show how coupled cells are able to function as an excitable medium and propagate waves of high cytokine concentration through the tissue. If the internal regulation in the cells is over-productive, the model predicts a continuous amplification of cytokines, which spans the entire system and resembles a situation of chronic inflammation in the tissue.

In Chapter 5 we consider inflammatory response in the islets of Langerhans, which are responsible for regulating the levels of blood sugar (by releasing insulin and glucagon) and which are located in the pancreas. Low-grade chronic inflammation and over-production of the cytokine IL-1β are characteristic features of islets in patients with type II diabetes. We expand the model of Chapter 4 in order to study the inflammatory response in islets of Langerhans, with a special focus on the influence imposed by the spatial conditions - namely the sizes and different possible shapes of the islets of Langerhans.

In agreement with experimental observations, we find that large islets are especially prone to transition into a state of chronic low-grade inflammation. Additionally, we find that different islet shapes may influence the risk of developing chronic inflammation - an observation, which implicates a connection between the distribution of different islet shapes and a protective function.