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Many diffusion processes in nature and society were found to be anomalous, in the sense of being fundamentally different from conventional Brownian motion. An important example is the migration of biological cells, which exhibits non-trivial temporal decay of velocity autocorrelation functions. This means that the corresponding dynamics is characterized by memory effects that slowly decay in time. Motivated by this we construct non-Markovian lattice-gas cellular automata models for moving agents with memory. For this purpose the reorientation probabilities are derived from velocity autocorrelation functions that are given a priori; in that respect our approach is “data-driven”. Particular examples we consider are velocity correlations that decay exponentially or as power laws, where the latter functions generate anomalous diffusion. The computational efficiency of cellular automata combined with our analytical results paves the way to explore the relevance of memory and anomalous diffusion for the dynamics of interacting cell populations, like confluent cell monolayers and cell clustering.

Das Ausmaß der permanenten klinischen Defizite bei Multiple-Sklerose-Patienten entsteht durch axonale Schädigung und axonalem Verlust. In dieser Studie wird eine entscheidende Rolle des niedrigaffinen Neurotrophinrezeptors p75NTR in Bezug auf axonale Schädigung in der experimentellen autoimmunen Encephalitomyelitis (EAE) gezeigt. Nach EAE Induktion per aktiver Immunisierung mit dem Myelin-Oligodendrozyten-Glykoprotein-Peptid MOG35-55 zeigten p75NTR KO Mäuse einen verschlechterten Krankheitsverlauf, stärkere Demyelinisierung und erhöhte axonale Schädigung. Um festzustellen, ob die erhöhten Defizite von einer aggressiveren Inflammation oder von einem vulnerableren zentralen Nervensystem (ZNS) stammen, wurden das Immunsystem während der peripheren Krankheitsentstehung und der darauffolgenden Krankheitsphase und Zellen des ZNS in vivo, ex vivo und in vitro untersucht. Es wurde kein Unterschied in der Qualität der Inflammation mit Hilfe von immunohistochemischen, durchfluss-zytometrischen, ELISA- und mRNA-Analysen gefunden, wodurch eine entscheidende Rolle der untersuchten Immunzellpopulationen ausgeschlossen werden kann. Notzdestrotrotz legt die konstitutive Expression von p75NTR auf B-Zellen eine Rolle für p75NTR während der Generation der Immunantwort innerhalb der Lymphknoten nahe, da p75NTR KO Mäuse von Anfang an einen erhöhten Krankheitsverlauf zeigten. Um die Effekte der p75NTR-Defizienz während des peripheren Primings zu umgehen, wurde die EAE auch durch adoptiven Transfer eines encephalitogenen MOG35-55-spezifischen T-Zell-Klones in p75NTR KO und Wildtyp (wt) Mäusen iduziert. Sie zeigten ähnliche Inzidenz, Beginn und Kinetik der Krankheit. Ein vergleichbares Ausmaß und eine vergleichbare Qualität der Inflammation wurde mit immunohistochemischen und mRNA-Analysen in beiden Mausstämmen am Höhepunkt der Krankheit gefunden. Nichtsdestotrotz leideten p75NTR KO Mäuse an signifikant erhöhten Krankheits-Scores in der chronischen Phase infolge von erhöhtem axonalen Schaden und Verlust. Dies deutet auf eine protektive Rolle von p75NTR im ZNS hin. In dieser Arbeit wird gezeigt, dass Astrozyten, aber nicht Mikroglia, p75NTR konstitutiv exprimieren. Jedoch wurde keine p75NTR-mediierte Regulation von Cytokinen/Chemokinen und der Produktion von reaktiven Sauerstoff-Spezies in vitro gefunden. Aktive Immunisierung von Knochenmark-Chimären, in denen nur Immun- oder ZNS-Zellen einen funktionierenden p75NTR tragen, bestätigen diese Resultate. Während p75NTR KO in wt- und wt in p75NTR KO-Chimären im Vergleich zu wt in wt-Chimären unter einem stärker ausgeprägten Krankheitsverlauf am Höhepunkt der Krankheit leideten, zeigten nur wt in p75NTR KO-Tiere erhöhten axonalen Schaden und Verlust. Zusammengefaßt deuten diese Daten darauf hin, dass p75NTR-Defizienz zu einer aggressiveren peripheren Immunantwort führt (namentlich durch B-Zellen). Desweiteren hat p75NTR auch neuroprotektive Eigenschaften innerhalb des ZNS (namentlich auf Neuronen) unter experimentellen inflammatorischen Bedingungen im ZNS.

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.

We study cellular automata as discrete dynamical systems and in particular investigate under which conditions two cellular automata are topologically conjugate. Based on work of McKinsey, Tarski, Pierce and Head we introduce derivative algebras to study the topological structure of sofic shifts in dimension one. This allows us to classify periodic cellular automata on sofic shifts up to topological conjugacy based on the structure of their periodic points. We also get new conjugacy invariants in the general case. Based on a construction by Hanf and Halmos, we construct a pair of non-homeomorphic subshifts whose disjoint sums with themselves are homeomorphic. From this we can construct two cellular automata on homeomorphic state spaces for which all points have minimal period two, which are, however, not topologically conjugate. We apply our methods to classify the 256 elementary cellular automata with radius one over the binary alphabet up to topological conjugacy. By means of linear algebra over the field with two elements and identities between Fibonacci-polynomials we show that every conjugacy between rule 90 and rule 150 cannot have only a finite number of local rules. Finally, we look at the sequences of finite dynamical systems obtained by restricting cellular automata to spatially periodic points. If these sequences are termwise conjugate, we call the cellular automata conjugate on all tori. We then study the invariants under this notion of isomorphism. By means of an appropriately defined entropy, we can show that surjectivity is such an invariant.