## Institut fĂĽr Mathematik und Informatik

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Trade of cattle between farms forms a complex trade network. We investigate partitions of this network for cattle trade in Germany. These partitions are groups of farms with similar properties and they are inferred directly from the trade pattern between farms. We make use of a rather new method known as stochastic block modeling (SBM) in order to divide the network into smaller units. SBM turns out to outperform the more established community detection method in the context of disease control in terms of trade restriction. Moreover, SBM is also superior to geographical based trade restrictions and could be a promising approach for disease control.

A common task in natural sciences is to
describe, characterize, and infer relations between discrete
objects. A set of relations E on a set of objects V can
naturally be expressed as a graph G = (V, E). It is
therefore often convenient to formalize problems in natural
sciences as graph theoretical problems.
In this thesis we will examine a number of problems found in
life sciences in particular, and show how to use graph theoretical
concepts to formalize and solve the presented problems. The
content of the thesis is a collection of papers all
solving separate problems that are relevant to biology
or biochemistry.
The first paper examines problems found in self-assembling
protein design. Designing polypeptides, composed of concatenated
coiled coil units, to fold into polyhedra turns out
to be intimately related to the concept of 1-face embeddings in
graph topology. We show that 1-face embeddings can be
canonicalized in linear time and present algorithms to enumerate
pairwise non-isomorphic 1-face embeddings in orientable surfaces.
The second and third paper examine problems found in evolutionary
biology. In particular, they focus on
inferring gene and species trees directly from sequence data
without any a priori knowledge of the trees topology. The second
paper characterize when gene trees can be inferred from
estimates of orthology, paralogy and xenology relations when only
partial information is available. Using this characterization an
algorithm is presented that constructs a gene tree consistent
with the estimates in polynomial time, if one exists. The
shown algorithm is used to experimentally show that gene trees
can be accurately inferred even in the case that only 20$\%$ of
the relations are known. The third paper explores how to
reconcile a gene tree with a species tree in a biologically
feasible way, when the events of the gene tree are known.
Biologically feasible reconciliations are characterized using
only the topology of the gene and species tree. Using this
characterization an algorithm is shown that constructs a
biologically feasible reconciliation in polynomial time, if one
exists.
The fourth and fifth paper are concerned with with the analysis
of automatically generated reaction networks. The fourth paper
introduces an algorithm to predict thermodynamic properties of
compounds in a chemistry. The algorithm is based on
the well known group contribution methods and will automatically
infer functional groups based on common structural motifs found
in a set of sampled compounds. It is shown experimentally that
the algorithm can be used to accurately
predict a variety of molecular properties such as normal boiling
point, Gibbs free energy, and the minimum free energy of RNA
secondary structures. The fifth and final paper presents a
framework to track atoms through reaction networks generated by a
graph grammar. Using concepts found in semigroup theory, the
paper defines the characteristic monoid of a reaction network. It
goes on to show how natural subsystems of a reaction network organically
emerge from the right Cayley graph of said monoid. The
applicability of the framework is proven by applying it to the
design of isotopic labeling experiments as well as to the
analysis of the TCA cycle.

Mathematical phylogenetics provides the theoretical framework for the reconstruction and analysis of phylogenetic trees and networks. The underlying theory is based on various mathematical disciplines, ranging from graph theory to probability theory.
In this thesis, we take a mostly combinatorial and graph-theoretical position and study different problems concerning phylogenetic trees and networks.
We start by considering phylogenetic diversity indices that rank species for conservation. Two such indices for rooted trees are the Fair Proportion index and the Equal Splits index, and we analyze how different they can be from each other and under which circumstances they coincide. Moreover, we define and investigate analogues of these indices for unrooted trees.
Subsequently, we study the Shapley value of unrooted trees, another popular phylogenetic diversity index. We show that it may fail as a prioritization criterion in biodiversity conservation and is outcompeted by an existing greedy approach. Afterwards, we leave the biodiversity setting and consider the Shapley value as a tree reconstruction tool. Here, we show that non-isomorphic trees may have permutation-equivalent Shapley transformation matrices and identical Shapley values, implying that the Shapley value cannot reliably be employed in tree reconstruction.
In addition to phylogenetic diversity indices, another class of indices frequently discussed in mathematical phylogenetics, is the class of balance indices. In this thesis, we study one of the oldest and most popular of them, namely the Colless index for rooted binary trees. We focus on its extremal values and analyze both its maximum and minimum values as well as the trees that achieve them.
Having analyzed various questions regarding phylogenetic trees, we finally turn to phylogenetic networks. We focus on a certain class of phylogenetic networks, namely tree-based networks, and consider this class both in a rooted and in an unrooted setting.
First, we prove the existence of a rooted non-binary universal tree-based network with n leaves for all positive integers n, that is, we show that there exists a rooted non-binary tree-based network with $n$ leaves that has every non-binary phylogenetic tree on the same leaf set as a base tree.
Finally, we study unrooted tree-based networks and introduce a class of networks that are necessarily tree-based, namely edge-based networks. We show that edge-based networks are closely related to a family of graphs in classical graph theory, so-called generalized series-parallel graphs, and explore this relationship in full detail.
In summary, we add new insights into existing concepts in mathematical phylogenetics, answer open questions in the literature, and introduce new concepts and approaches. In doing so, we make a small but relevant contribution to current research in mathematical phylogenetics.

In this thesis, we elaborate upon Bayesian changepoint analysis, whereby our focus is on three big topics: approximate sampling via MCMC, exact inference and uncertainty quantification. Besides, modeling matters are discussed in an ongoing fashion. Our findings are underpinned through several changepoint examples with a focus on a well-log drilling data.

Tuberculosis (TB) has tremendous public health relevance. It most frequently affects the lung and is characterized by the development of unique tissue lesions, termed granulomas. These lesions encompass various immune populations, with macrophages being most extensively investigated. Myeloid derived suppressor cells (MDSCs) have been recently identified in TB patients, both in the circulation and at the site of infection, however their interactions with Mycobacterium tuberculosis (Mtb) and their impact on granulomas remain undefined. We generated human monocytic MDSCs and observed that their suppressive capacities are retained upon Mtb infection. We employed an in vitro granuloma model, which mimics human TB lesions to some extent, with the aim of analyzing the roles of MDSCs within granulomas. MDSCs altered the structure of and affected bacterial containment within granuloma-like structures. These effects were partly controlled through highly abundant secreted IL-10. Compared to macrophages, MDSCs activated primarily the NF-ÎşB and MAPK pathways and the latter largely contributed to the release of IL-10 and replication of bacteria within in vitro generated granulomas. Moreover, MDSCs upregulated PD-L1 and suppressed proliferation of lymphocytes, albeit with negligible effects on Mtb replication. Further comprehensive characterization of MDSCs in TB will contribute to a better understanding of disease pathogenesis and facilitate the design of novel immune-based interventions for this deadly infection.