Der erste Teil dieser Arbeit konzentriert sich auf die Mechanismen hormoninduzierter Ca2+-oszillationen, und wie diese von Konzentrationsschwankungen des Ca2+-freisetzenden Botenstoffes Inositol-1,4,5-trisphosphat (IP3) beinflusst werden. Wir konnten zeigen, dass IP3-Oszillationen die Frequenzkodierung des äußeren Stimulus durch Ca2+-Ozillationen deutlich verstärken. Zwei Mechanismen für das Entstehen der IP3-Oszillationen wurden untersucht: es zeigte sich, dass die Aktivierung der Phospholipase C durch Ca2+ der wahrscheinlichste Mechanismus ist. Um die Rolle der IP3-Oszillationen genauer zu verstehen, wurde ein Modell für den Stoffwechsel des IP3-Vorläufers Phosphatidylinositol-4,5-bisphosphat (PIP2) entwickelt. Es zeigt sich, dass die scheinbar nutzlosen Phosphorylierungs/Dephosphorylierungszyklen eine wichtige Rolle für den PIP2-Haushalt spielen. Durch Nachliefern von PIP2 während der Stimulierung ermöglichen sie anhaltende Ca2+-signale. Der zweite Teil der Arbeit beschäftigt sich mit einem DNS-Reparaturweg, der Reparatur mittels Entfernung von Nukleotiden (NER). Dieser Reparaturmechanismus ist äußerst vielseitig und entfernt Pyrimidinpaare, die durch UV-Strahlung erzeugt wurden, oder Schäden, die durch chemische Agentien erzeugt wurden. Es wurde ein mathematisches Model erarbeitet, das die Grundeigenschaften der NER beschreiben soll. Erstens wurde untersucht, wie die Bindungs- und Freisetzungskinetik der Reparaturfaktoren mit den strukturellen Eigenschaften des Systems, beispielsweise der Bindungsreihenfolge, zusammenhängt. Zweitens wurden anhand von in vivo gemessenen Rekrutierungskinetiken dreier Proteinfaktoren die Modellparameter bestimmt. Das so angepasste Modell sagt unter anderem eine Sättigung der NER durch den Verbrauch des Erkennungsfaktors vorher. Die theoretischen Untersuchungen deuten darauf hin, dass ein sequentieller Anlagerungsmechanismus im Hinblick auf Effizienz und auf Spezifität gegenüber den beschädigten Substrat große Vorteile bringen kann. / The first part of this thesis focuses on the mechanisms of hormone induced Ca2+ oscillations and how these depend on fluctuations in the concentration of the Ca2+-releasing messenger, inositol 1,4,5-trisphosphate (IP3). We were able to show that IP3 oscillations greatly enhances the ability to frequency encode the hormone stimulus by Ca2+ oscillations. Two mechanisms for the generation of IP3-oscillations have been investigated, we could show that Ca2+-activation of phospholipase C is the most probable mechanism. To better understand the role of IP3-oscillations a detailed model for the phosphoinositide pathway has been developed. The model illustrates the importance of futile (de)phosphorylation cycles for regenerating phosphatidylinositol-4,5-bisphophat during stimulation, an essential property to support long-lasting Ca2+ signals. The second part of the thesis is devoted to nucleotide excision repair (NER). It is a versatile DNA repair mechanism that can remove lesions such as UV light induced pyrimidine dimers and bulky adducts caused by chemical agents. To understand the mechanisms underlying the protein assembly during NER and the performance of repair, a mathematical model, delineating hallmarks and general characteristics of NER, has been developed. First, the binding and dissociation kinetics of repair factors are related to the structural properties of the system, such as the sequential order in which the factors enter repair. Second, using in vivo kinetic data for the recruitment of three different proteins at local damaged nuclei, the model parameters are determined and the dynamic behavior of the repair process is scrutinized in detail. The observed saturation of NER is predicted to rely on the high engagement of the recognition factor in repair. The theoretical analysis of repair performance indicates that a sequential assembly process is remarkably advantageous in terms of repair efficiency and can show a marked selectivity for the damaged substrate.

Cell migration, the active translocation of cells is involved in various biological processes, e.g. development of tissues and organs, tumor invasion and wound healing. Cell migration behavior can be divided into two distinct classes: single cell migration and collective cell migration. Single cell migration describes the migration of cells without interaction with other cells in their environment. Collective cell migration is the joint, active movement of multiple cells, e.g. in the form of strands, cohorts or sheets which emerge as the result of individual cell-cell interactions. Collective cell migration can be observed during branching morphogenesis, vascular sprouting and embryogenesis. Experimental studies of single cell migration have been extensive. Collective cell migration is less well investigated due to more difficult experimental conditions than for single cell migration. Especially, experimentally identifying the impact of individual differences in cell phenotypes on individual cell migration behavior inside cell populations is challenging because the tracking of individual cell trajectories is required. In this thesis, a novel mathematical modeling approach, individual-based lattice-gas cellular automata (IB-LGCA), that allows to investigate the migratory behavior of individual cells inside migrating cell populations by enabling the tracking of individual cells is introduced. Additionally, stochastic differential equation (SDE) approximations of individual cell trajectories for IB-LGCA models are constructed. Such SDE approximations allow the analytical description of the trajectories of individual cells during single cell migration. For a complete analytical description of the trajectories of individual cell during collective cell migration the aforementioned SDE approximations alone are not sufficient. Analytical approximations of the time development of selected observables for the cell population have to be added. What observables have to be considered depends on the specific cell migration mechanisms that is to be modeled. Here, partial integro-differential equations (PIDE) that approximate the time evolution of the expected cell density distribution in IB-LGCA are constructed and coupled to SDE approximations of individual cell trajectories. Such coupled PIDE and SDE approximations provide an analytical description of the trajectories of individual cells in IB-LGCA with density-dependent cell-cell interactions. Finally, an IB-LGCA model and corresponding analytical approximations were applied to investigate the impact of changes in cell-cell and cell-ECM forces on the migration behavior of an individual, labeled cell inside a population of epithelial cells. Specifically, individual cell migration during the epithelial-mesenchymal transition (EMT) was considered. EMT is a change from epithelial to mesenchymal cell phenotype which is characterized by cells breaking adhesive bonds with surrounding epithelial cells and initiating individual migration along the extracellular matrix (ECM). During the EMT, a transition from collective to single cell migration occurs. EMT plays an important role during cancer progression, where it is believed to be linked to metastasis development. In the IB-LGCA model epithelial cells are characterized by balanced cell-cell and cell-ECM forces. The IB-LGCA model predicts that the balance between cell-cell and cell-ECM forces can be disturbed to some degree without being accompanied by a change in individual cell migration behavior. Only after the cell force balance has been strongly interrupted mesenchymal migration behavior is possible. The force threshold which separates epithelial and mesenchymal migration behavior in the IB-LGCA has been identified from the corresponding analytical approximation. The IB-LGCA model allows to obtain quantitative predictions about the role of cell forces during EMT which in the context of mathematical modeling of EMT is a novel approach.

Cell migration, the active translocation of cells is involved in various biological processes, e.g. development of tissues and organs, tumor invasion and wound healing. Cell migration behavior can be divided into two distinct classes: single cell migration and collective cell migration. Single cell migration describes the migration of cells without interaction with other cells in their environment. Collective cell migration is the joint, active movement of multiple cells, e.g. in the form of strands, cohorts or sheets which emerge as the result of individual cell-cell interactions. Collective cell migration can be observed during branching morphogenesis, vascular sprouting and embryogenesis. Experimental studies of single cell migration have been extensive. Collective cell migration is less well investigated due to more difficult experimental conditions than for single cell migration. Especially, experimentally identifying the impact of individual differences in cell phenotypes on individual cell migration behavior inside cell populations is challenging because the tracking of individual cell trajectories is required. In this thesis, a novel mathematical modeling approach, individual-based lattice-gas cellular automata (IB-LGCA), that allows to investigate the migratory behavior of individual cells inside migrating cell populations by enabling the tracking of individual cells is introduced. Additionally, stochastic differential equation (SDE) approximations of individual cell trajectories for IB-LGCA models are constructed. Such SDE approximations allow the analytical description of the trajectories of individual cells during single cell migration. For a complete analytical description of the trajectories of individual cell during collective cell migration the aforementioned SDE approximations alone are not sufficient. Analytical approximations of the time development of selected observables for the cell population have to be added. What observables have to be considered depends on the specific cell migration mechanisms that is to be modeled. Here, partial integro-differential equations (PIDE) that approximate the time evolution of the expected cell density distribution in IB-LGCA are constructed and coupled to SDE approximations of individual cell trajectories. Such coupled PIDE and SDE approximations provide an analytical description of the trajectories of individual cells in IB-LGCA with density-dependent cell-cell interactions. Finally, an IB-LGCA model and corresponding analytical approximations were applied to investigate the impact of changes in cell-cell and cell-ECM forces on the migration behavior of an individual, labeled cell inside a population of epithelial cells. Specifically, individual cell migration during the epithelial-mesenchymal transition (EMT) was considered. EMT is a change from epithelial to mesenchymal cell phenotype which is characterized by cells breaking adhesive bonds with surrounding epithelial cells and initiating individual migration along the extracellular matrix (ECM). During the EMT, a transition from collective to single cell migration occurs. EMT plays an important role during cancer progression, where it is believed to be linked to metastasis development. In the IB-LGCA model epithelial cells are characterized by balanced cell-cell and cell-ECM forces. The IB-LGCA model predicts that the balance between cell-cell and cell-ECM forces can be disturbed to some degree without being accompanied by a change in individual cell migration behavior. Only after the cell force balance has been strongly interrupted mesenchymal migration behavior is possible. The force threshold which separates epithelial and mesenchymal migration behavior in the IB-LGCA has been identified from the corresponding analytical approximation. The IB-LGCA model allows to obtain quantitative predictions about the role of cell forces during EMT which in the context of mathematical modeling of EMT is a novel approach.