Integrability of Moufang Foundations - A Contribution to the Classification of Twin Buildings

Buildings have been introduced by J. Tits in order to study semi-simple algebraic groups from a geometrical point of view. One of the most important results in the theory of buildings is the classification of irreducible spherical buildings of rank at least 3. About 25 years ago, M. Ronan and J. Tits defined the class of twin buildings, which generalize spherical buildings in a natural way. The motivation of their definition is provided by the theory of Kac-Moody groups. A 2-spherical twin building is uniquely determined by its local structure in almost all cases: The so-called foundation is the union of the rank 2 residues which contain an (arbitrary) chamber. Therefore, the classification of 2-spherical twin buildings reduces to the classification of all foundations which can be realized as the local structure of such a twin building. We call such a foundation "integrable". By a result of Tits, an integrable foundation is Moufang, which means that the rank 2 buildings in the foundation are Moufang polygons, and that the glueings are compatible with the Moufang structures induced on the rank 1 residues. As a consequence, the classification of Moufang polygons and the solution of the isomorphism problem for Moufang sets are essential to work out which Moufang polygons fit together in order to form a foundation. The present thesis contributes to establish complete lists of integrable foundations for certain types of diagrams, namely for simply laced diagrams and for 443 triangle diagrams. In this process, we closely follow the approach for the classification of spherical buildings. However, we have to refine the techniques used there, since in general, foundations don't only depend on the diagram and the defining field. Moreover, one of the main results in the context of Moufang sets is the solution of the isomorphism problem for Moufang sets of pseudo-quadratic spaces.Comment: PhD thesis, 221 page

Western European democracies and new Russian revisionism - Does alliance membership matter?

A steadily more aggressive course of action by Russia changed both the security situation in Europe and the security and defense policy of European states. Furthermore, in the second decade of the 21stcentury, NATO has shifted back its focus, away from stabilisation missions to collective defense. But not every member nation has kept pace with the adjustments. Particularly the obligation to invest at least 2% of the GDP into defense measures, which is also referred to as the NATO Defense Investment Pledge, has been a subject of disagreement among NATO members. This thesis takes the dissent as a starting point to highlight the reasons for the way and the intensity in which states react to a specific threat. It tries to determine if the reaction of a country to a new threat is predominantly influenced by the state’s alliance membership or simply the country’s geopolitical location. The countries Germany, Norway and Sweden, which differ in size and location and include NATO members as well as a non allied state, were chosen as comparison units. For all three states, both the defense expenditure and the procedure and intensity of changing the orientation of the armed forces within the period from the end of the Cold War until 2020 were analyzed. In order to achieve the objective of the thesis, the following research question shall be discussed and answered: Is membership in an alliance such as NATO the crucial factor that has determined western European states’ defense policy response to Russia’s new aggressive revisionism, or does geopolitical exposure to Russia offer a more convincing explanation? This thesis examines the aspects “defense expenditure” and “(re-)orientation of the armed forces” and reveals in both cases that geopolitics override the commitments that arise from the membership in an alliance. Regarding the first case, i.e. defense expenditure, it is notable that Sweden’s changes in defense investments keep up with those of the other states considered in this research. This applies to the time of rearmament in the 90’s as well as to the increase in investments after 2014 or, in the case of Sweden, especially after 2017. The second case, (re-)orientation of the armed forces, is even more distinct. It shows that Sweden reversed several measures taken in the age of stabilization missions at a good pace to regain mobilization abilities and high-intensity warfighting capabilities. Sweden’s effort was the decisive factor that led to the conclusion that geopolitics matter most

Phenotypes can be robust and evolvable if mutations have non-local effects on sequence constraints.

The mapping between biological genotypes and phenotypes plays an important role in evolution, and understanding the properties of this mapping is crucial to determine the outcome of evolutionary processes. One of the most striking properties observed in several genotype-phenotype (GP) maps is the positive correlation between the robustness and evolvability of phenotypes. This implies that a phenotype can be strongly robust against mutations and at the same time evolvable to a diverse range of alternative phenotypes. Here, we examine the causes for this positive correlation by introducing two analytically tractable GP map models that follow the principles of real biological GP maps. The first model is based on gene-like GP maps, reflecting the way in which genetic sequences are organized into protein-coding genes, and the second one is based on the GP map of RNA secondary structure. For both models, we find that a positive correlation between phenotype robustness and evolvability only emerges if mutations at one sequence position can have non-local effects on the sequence constraints at another position. This highlights that non-local effects of mutations are closely related to the coexistence of robustness and evolvability in phenotypes, and are likely to be an important feature of many biological GP maps

Using small samples to estimate neutral component size and robustness in the genotype-phenotype map of RNA secondary structure.

In genotype-phenotype (GP) maps, the genotypes that map to the same phenotype are usually not randomly distributed across the space of genotypes, but instead are predominantly connected through one-point mutations, forming network components that are commonly referred to as neutral components (NCs). Because of their impact on evolutionary processes, the characteristics of these NCs, like their size or robustness, have been studied extensively. Here, we introduce a framework that allows the estimation of NC size and robustness in the GP map of RNA secondary structure. The advantage of this framework is that it only requires small samples of genotypes and their local environment, which also allows experimental realizations. We verify our framework by applying it to the exhaustively analysable GP map of RNA sequence length L = 15, and benchmark it against an existing method by applying it to longer, naturally occurring functional non-coding RNA sequences. Although it is specific to the RNA secondary structure GP map in the first place, our framework can probably be transferred and adapted to other sequence-to-structure GP maps.MW was supported by the EPSRC and the Gatsby Charitable Foundation. SEA was supported by the Gatsby Charitable Foundation

Neutral components show a hierarchical community structure in the genotype-phenotype map of RNA secondary structure.

Genotype-phenotype (GP) maps describe the relationship between biological sequences and structural or functional outcomes. They can be represented as networks in which genotypes are the nodes, and one-point mutations between them are the edges. The genotypes that map to the same phenotype form subnetworks consisting of one or multiple disjoint connected components-so-called neutral components (NCs). For the GP map of RNA secondary structure, the NCs have been found to exhibit distinctive network features that can affect the dynamical processes taking place on them. Here, we focus on the community structure of RNA secondary structure NCs. Building on previous findings, we introduce a method to reveal the hierarchical community structure solely from the sequence constraints and composition of the genotypes that form a given NC. Thereby, we obtain modularity values similar to common community detection algorithms, which are much more complex. From this knowledge, we endorse a sampling method that allows a fast exploration of the different communities of a given NC. Furthermore, we introduce a way to estimate the community structure from genotype samples, which is useful when an exhaustive analysis of the NC is not feasible, as is the case for longer sequence lengths.MW was supported by the EPSRC and the Gatsby Charitable Foundation. SEA was supported by the Gatsby Charitable Foundation and the Alan Turing Institute

Radiosynthesis of L-[18F]fluorotryptophan by isotopic exchange on carbonyl-activated precursors

In the past a variety of 18F-labeled aromatic amino acids have been developed, primarily for tumor diagnostics with positron-emission-tomography. Recently tryptophan got high attention, since evidence came up that some tumors exhibit an elevated consumption of it. So far, this amino acid could only be radiofluorinated by unsatisfactory approaches. In the work here, a simpler, 3-step method for a nucleophilic radiosynthesis of L-4-[18F]fluorotryptophan was developed. For this a carbonyl activated precursor was radiofluorinated by isotopic exchange, followed by removal of the activating formyl group by reductive decarbonylation and subsequent hydrolysis of the protecting groups under acidic conditions. First, the influence of positions of fluorine and of the formyl group on the isotopic exchange was examined in several fluoro-1H-indolecarbaldehydes where different protecting groups were attached to the indole nitrogen. Further, a decarbonylation reaction with Rh(PPh3)3 on those molecules was carried out and optimized. The best results regarding radiochemical yield and chemical stability were obtained with 1-benzyl-4-fluoro-1H-indole-5-carbaldehyde. Based on these results a concept for the synthesis of precursors for L-6-[18F]fluorotryptophan and L-4-[18F]fluorotryptophan was developed. Hereby the compounds benzyl (2S,5S)-2-tert-butyl-5-[(1-benzyl-4-fluoro-5-formyl-1H–indol–3-yl)methyl]–3–methyl–4–oxoimidazolidine-1-carboxylate and benzyl (2S,5S)–2–tert–butyl–5-[(1–Boc–4–fluoro–5–formyl-1H–indol–3 yl) methyl]-3-methyl-4-oxoimidazolidine-1-carboxylate were prepared following 11-step linear synthetic pathways with overall yields of about 8 % and an enantiomeric purity of > 99 %. While the radiosynthesis of L-6-[18F]fluorotryptophan was not successful due to the failing hydrolysis of the benzyl group, L-4-[18F]fluorotryptophan was prepared by the three step radiosynthesis, consisting of an isotopic exchange, a reductive decarbonylation with Rh(PPh3)3 and the hydrolysis of the protecting groups with HCl, yielding an enantiomeric purity of > 99 %. After optimization of this procedure L-4-[18F]fluorotryptophan was isolated in a radiochemical yield of ca. 13 % and a molar activity of > 70 MBq/mmol within about 115 min. Hence, a new and more efficient nucleophilic radiosynthesis of L-4-[18F]fluoro-tryptophan was developed which is now available for preclinical evaluation