Department of Physics • University of Cologne

Improving the understanding of pedestrian crowd dynamics is a growing field with essential contributions from many areas of science utilizing empirical data and mathematical models. Pedestrians systems exhibit a number of self-organization phenomena like lane formation and temporal clogging which are interesting from a physics perspective. Simulations of these scenarios can improve the understanding of these phenomena by focusing on minimal models that can reproduce the observations. Many of these phenomena in pedestrian dynamics are not unique to the system and are observed in related fields like granular materials, colloids, and active matter. The presented work introduces methods from solid-state physics and granular matter to quantify the spatial order of pedestrian systems. The focus is on bottleneck flow when a crowd has to pass a spatially restricted area (e.g., a door leading to a hall), a crucial scenario in pedestrian dynamics and the flow of granular matter.

The genetic heterogeneity found within a tumor also extends to the spatial dimension of clonal architecture, but only recently has high-depth multi-region sequencing data become available. We use spatially resolved genetic data to study a tumor’s mode of evolution and growth as well as population genetic theory of mutant type extinction to infer cancer cell turnover rates and mutation rates.

Abstract In the context of tumor growth, niches can play an important role in the evolutionary process. In this thesis we will investigate the possible consequences of niches on the allele frequency distribution of mutations which constitutes a fundamental quantity in standard genetics. Somatic mutations in tumor tis- sue are cell–intrinsic properties whose variety constitutes to the intra–tumor heterogeneity. This can result in different phenotypes throughout the disease, that decrease the success of homogeneous therapy strategies. Intra–tumor heterogeneity can change during the outgrowth of a tumor. Here we will describe the dynamics of this evolution with a statistical model including a niche environment and will later investigate the heterogeneity that occurs within the spatial expansion of a tumor. Spatial patterns of allele frequency distributions can teach us about the underlying evolutionary processes and moreover show rules and limitations for the interpretation on cancer sequencing data.

Granular materials consist of macroscopic, athermal packings of particles which only interact via dissipative contact forces. The systems can adopt different states of aggregation depending on the external driving conditions. The viscosity dependence of a granular fluid will be probed by an active microrheology setup. We demonstrate access to three viscosity regimes depending on the shear rate. In a static granular system we measure the contact forces on grain scale and evaluate the local mechanical heterogeneities in an amorphous and crystalline packing.

We show that various models in statistical modelling can be derived from the maximum entropy principle. As a particular example, we discuss the Ising model in the context of the inverse Ising problem. Applications of maximum entropy models in disentangling direct interaction from indirect correlation in cancer data will be introduced. Another application of maximum entropy models in semi-supervised cancer classification will also be discussed.

The Plefka expansion is a method for approximately analyzing complicated probabilistic systems such as those consisting of many interacting particles. In principle, it consists of an expansion in the interaction strength around the non-interacting case. Important applications are the calculation of expectation values of observables as well as the problem of statistical inference encountered for example in the inverse Ising problem. In order to assess the applicability of the expansion to a specific system, it is important to know the radius of convergence. Unfortunately, this radius is only known for very few systems, the Sherrington-Kirkpatrick model of a spin glass being the most prominent one. This talk will introduce the general concept of the Plefka expansion applied to the Ising model as well as explicitly derive the radius of convergence for the 1D ferromagnet.

The structure of fitness landscapes, which determine the behaviour of evolutionary processes through mutation and natural selection, is an current issue of theoretical biology. In this colloquium one aspect of the NK-model, which is a biologically motivated construction for complex correlated fitness landscapes, is analyzed with regard to some typical characteristics of fitness landscapes. This aspect is tthe choice of the neighborhood describing the interaction between mutations at different genetic loci. It is shown by simulation that, although the fitness correlation does not depend on the choice of neighborhood, some global measures like the number of maxima and the number of evolutionary accessible paths to the global maximum depend, sometimes strongly, on that choice. Tested examples for choice of neighborhood are random, adjacent and block neighborhood. Also, the distribution of evolutionary accessible paths in the block model is derived from the distribution in the random energy model (House-of-Cards model).