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This thesis deals with several aspects of non-perturbative calculations in low-dimensional quantum field theories. It is split into two main parts:
The first part focuses on method development and testing. Using exactly integrable QFTs in zero spacetime dimensions as toy models, the need for non-perturbative methods in QFT is demonstrated. In particular, we focus on the functional renormalization group (FRG) as a non-perturbative exact method and present a novel fluid-dynamic reformulation of certain FRG flow equations. This framework and the application of numerical schemes from the field of computational fluid dynamics (CFD) to the FRG is tested and benchmarked against exact results for correlation functions. We also draw several conclusions for the qualitative understanding and interpretation of renormalization group (RG) flows from this fluid-dynamic reformulation and discuss the generalization of our findings to realistic higher-dimensional QFTs.
The topics discussed in the second part are also manifold. In general, the second part of this thesis deals with the Gross-Neveu (GN) model, which is a prototype of a relativistic QFT. Even though being a model in two spacetime dimensions, it shares many features of realistic models and theories for high-energy particle physics, but also emerges as a limiting case from systems in solid state physics. Especially, it is interesting to study the model at non-vanishing temperatures and densities, thus, its thermodynamic properties and phase structure.
First, we use this model to test and apply our findings of the first part of this thesis in a realistic environment. We analyze how the fluid-dynamic aspects of the FRG realize themselves in the RG flow of a full-fledged QFT and how we profit from this numeric framework in actual calculations. Thereby, however, we also aim at answering a long-standing question: Is there still symmetry breaking and condensation at non-zero temperatures in the GN model, if one relaxes the commonly used approximation of an infinite number of fermion species and works with a finite number of fermions? In short: Is matter (in the GN model) in a single spatial dimension at non-zero temperature always gas-like?
In general, we also use the GN model to learn about the correct description of QFTs at non-zero temperatures and densities. This is of utmost relevance for model calculations in low-energy quan- tum chromodynamics (QCD) or other QFTs in medium and we draw several conclusions for the requirements for stable calculations at non-zero chemical potential.
Non-ribosomal peptide synthetase docking domains : structure, function and engineering strategies
(2021)
Non-ribosomal peptide synthetases (NRPSs) are known for their capability to produce a wide range of natural compounds and some of them possess interesting bioactivities relevant for clinical application like antibiotics, anticancer, and immunosuppressive drugs. The diverse bioactivity of non-ribosomal peptides (NRPs) originates from their structural diversity, which results not only from the incorporation of non-proteinogenic amino acids into the growing peptide chain, but also the formation of heterocycles or further peptide modifications like methylation, hydroxylation and acetylation.
The biosynthesis of NRPs is achieved via the orchestrated interplay of distinct catalytic domains, which are grouped to modules that are located on one or more polypeptide chains. Each cycle starts with the selection and activation of a specific amino acid by the adenylation (A) domain, which catalyzes the aminoacyl adenylate formation under ATP consumption. This activated amino acid is then bound via a thioester bond to the 4’-phosphopantetheine cofactor (PPant-arm) of the following thiolation (T) domain. Before substrate loading, the PPant-arm is post-translationally added to the T domain by a phosphopantetheinyl transferase (PPTase), which converts the inactive apo-T domain in its active holo-form. In the last step of the catalytic cycle, two T domain bound peptide building blocks are connected by the condensation (C) domain, resulting in peptide bond formation and transfer of the nascent peptide chain to the following module. Each catalytic cycle is performed by a C-A-T elongation module until the termination module with a C-terminal thioesterase (TE) domain is reached. Here, the peptide product is released by hydrolysis or intramolecular cyclisation.
In comparison to single-protein NRPSs, where all modules are encoded on a single polypeptide chain, multi-protein NRPS systems must also maintain a specific module order during the peptide biosynthesis. Therefore, small C-terminal and N-terminal communication-mediating (COM) domains/docking domains (DD) were identified in the C- and N-terminal regions of multi-protein NRPSs. It was shown that these domains mediate specific and selective non-covalent protein-protein interaction, even though DD interactions are generally characterized by low affinities.
The first publication of this work focuses on the Peptide-Antimicrobial-Xenorhabdus peptide-producing NRPS called PaxS, which consists of the three proteins PaxA, PaxB and PaxC. Here, in particular the trans DD interface between the C-terminal attached DD of PaxB and N-terminal attached DD of PaxC was structurally investigated and thermodynamically characterized by isothermal titration calorimetry (ITC), yielding a dissociation constant (KD) of ~25 µM, which is a DD typical affinity known from further characterized DD pairs. The artificial linking of the PaxB/C C/NDD pair via a glycine-serine (GS) linker facilitated the structure determination of the DD complex by solution nuclear magnetic resonance (NMR) spectroscopy. In comparison to known docking domain structures, this DD complex assembles in a completely new fold which is characterized by a central α-helix of PaxC NDD wrapped in two V-shaped α-helices of PaxB CDD.
The first manuscript of this work focuses on the application of synthetic zippers (SZ) to mimic natural docking domains, enabling the easy assembly of NRPS building blocks encoded on different plasmids in a functional way. Here, the high-affinity interaction of SZs unambiguously defines the order of the synthetases derived from single-protein NRPSs in the engineered NRPS system and allows the recombination in a plug-and-play manner. Notably, the SZ engineering strategy even facilitates the functional assembly of NRPSs derived from Gram-positive and Gram-negative bacteria. Furthermore, the functional incorporation of SZs into NRPS modules is not limited to a specific linker region, so we could introduce them within all native NRPS linker regions (A-T, T-C, C-A).
The second publication and the second manuscript of this thesis again focus on the multi-protein PaxS, in particular on the trans interface between the proteins PaxA and PaxB on a molecular level by solution NMR. Therefore, the PaxA CDD adjacent T domain was included into the structural investigation besides the native interaction partner PaxB NDD. Before a three-dimensional structure could be obtained from NMR data, the NH groups located in the peptide bonds had to be assigned to the respective amino acids of the proteins (backbone assignment). Based on these backbone assignments, the secondary structure of PaxA T1-CDD and PaxB NDD in the absence and presence of the respective interaction partner were predicted.
The structural and functional characterization of the PaxA T1-CDD:PaxB NDD complex is summarized in manuscript two. The thermodynamic analysis of this complex by ITC determined a KD value of ~250 nM, whereas the discrete DDs did not interact at all. The high-affinity interaction allowed to determine the solution NMR structure of the PaxA T1-CDD:PaxB NDD complex without the covalent linkage of the interaction partners and an extended docking domain interface could be determined. This interface comprises on the one hand α-helix 4 of the PaxA T1 domain together with the α-helical CDD, and on the other hand the PaxB NDD, which is composed of two α-helices separated by a sharp bend.
...
Non-ribosomal peptide synthetases (NRPSs) are modular biosynthetic megaenzymes producing many important natural products and refer to a specific set of peptides in bacteria’s and fungi’s secondary metabolism. With the actual purpose of providing advantages within their respective ecological niche, the bioactivity of the structurally highly diverse products ranges from, e.g., antibiotic (e.g., vancomycin) to immunosuppressive (e.g., cyclosporin A) to cytostatic (e.g., echinomycin or thiocoralin) activity.
An NRPS module consists of at least three core domains that are essential for the incorporation of specific substrates with the 'multiple carrier thiotemplate mechanism' into a growing peptide chain: an adenylation (A) domain selects and activates a cognate amino acid; a thiolation (T) domain shuffles the activated amino acid and the growing peptide chain, which are attached at its post-translationally 4ʹ-phosphopantetheine (4'-PPant) group, between the active sites; a condensation (C) domain links the upstream and downstream substrates. NRPS synthesis is finished with the transfer of the assembled peptide to the C-terminal chain-terminating domain. Accordingly, the intermediate is either released by hydrolysis as a linear peptide chain or by an intramolecular nucleophilic attack as a cyclic peptide.
The NRPS’s modular character seems to imply straightforward engineering to take advantage of their features but appears to be more challenging. Since the pioneering NRPS engineering approaches focused on the reprogramming and replacement of A domains, several working groups developed advanced methods to perform a complete replacement of subdomains or single or multiple catalytic domains.
The first part of this work focusses parts of the publication with the title 'De novo design and engineering of non-ribosomal peptide synthetases', which follows up assembly line engineering with the development of a new guideline. Thereby, the pseudodimeric V-shaped structure of the C domain is exploited to separate the N-terminal (CDSub) and C-terminal (CASub) subdomains alongside a four-AA-long linker. This results in the creation of self-contained, catalytically active CASub-A-T-CDSub (XUC) building blocks. As an advantage over the previous XU concept, the characteristics (substrate- and stereoselectivity) assigned to the C domain subunits are likewise exchanged, and thus, no longer represent a barrier. Furthermore, with the XUC concept, no important interdomain interfaces are disrupted during the catalytic cycle of NRPS, allow to expect much higher production titers. Moreover, the XUC concept shows a more flexible application within its genus origin of building blocks to create peptide libraries. Additionally, with this concept only 80 different XUC building blocks are needed to cover the entire proteinogenic amino acid spectrum.
The second part of this work addresses the influence of the C domain on activity and specificity of A domains. In a comprehensive analysis, a clear influence of different C domains on the in vitro activation rate and the in vivo substrate spectrum could be observed. Further in situ and in silico characterizations indicate that these influences are neither the result of the respective A domains promiscuity nor the C domain’s proofreading, but due to an 'extended gatekeeping' function of the C domain. This novel term of an 'extended gatekeeping' function describes the very nature of interfaces that C domains can form with an A domain of interest. Therefore, the C-A interface is assumed to have a more significant contribution to a selectivity filter function.
The third part of this work combines the NRPS engineering with phylogenetic/evolutionary perspectives. At first, the C-A interface could be precisely defined and further identified to encode equivalent information corresponding to the complete C-A didomain. Moreover, the comparison of NRPSs topology reveals hints for a co-evolutionary relatedness of the C-A didomain and could be shown to reassemble even after separation. In this regard, based on a designed CAopt.py algorithm, the reassembling-compatibility of hybrid interfaces could be determined by scoring of the co-expressed NRPS hybrids. This algorithm also enables the randomization of the interface sequences, thus, leading to the identification of more functional interface variant, which cause significantly higher peptide production and could even be applied to other native and hybrid interfaces.
The present work deals with photoionization in the realm of the absorption of one single photon. The formal treatment of one-photon ionization usually employs a semi-classical approach, where the electron’s initial and final states are described as quantum-mechanical wave functions but the photon is treated as a classical electromagnetic wave. In the calculation of photoionization cross sections with this semi-classical method, there is an often used approximation which is called the electric dipole approximation. Mathematically, the application of the dipole approximation corresponds to truncating the series expansion of an exponential after the leading term. Physically, this means neglecting the linear photon momentum and the spatial dependence of the light field. The dipole approximation is valid if the wavelength of the light is much larger than the spatial extent of the target and if the photon momentum is small compared to the momenta of the reaction products, which is generally the case for photon energies short above the electron binding energy.
For the present work, we experimentally investigated nondipolar photoionization, i.e., one-photon ionization at high photon energies where the dipole approximation breaks down. In our experiments, we irradiated single atoms and molecules with such high-energetic photons and measured the three-dimensional momentum distributions of the reaction fragments to uncover the effects of the linear photon momentum and the spatially-dependent light field on photoionization. Our observations allow the first profound insight into photoionization that reveals all photon properties, i.e., photon energy, spin, linear momentum, and the speed of light. Hopefully, our efforts make a constructive contribution to the understanding and the further exploration of light-matter interaction.
Nonequilibrium phase transitions in chiral fluid dynamics including dissipation and fluctuation
(2011)
Chiral fluid dynamics combines the fluid dynamic expansion of a hot and dense plasma created in a heavy-ion collision with the explicit propagation of fluctuations at the chiral phase transition of quantum chromodynamics. From systems in equilibrium long-range fluctuations are expected at a conjectured critical point. Heavy-ion collisions are, however, finite in size and time and very dynamic. It is thus likely that nonequilibrium effects diminish the signal of a critical point. They can, however, stimulate phenomena at a first order phase transitions, like nucleation and spinodal decomposition. Both of phase transition scenarios are investigated in this work. Based on the linear sigma model with constituent quarks a consistent quantum field theoretical approach using the two-particle irreducible effective action is developed to derive both, the local equilibrium properties of the expanding quark fluid and the damping and noise terms in the Langevin equation of the order parameter of the phase transition, the sigma field. Within this formalism it is possible to obtain a conserved energy-momentum tensor of the coupled system. It describes the energy dissipation from the sigma field to the heat bath during relaxation. Within this model we investigate nonequilibrium phenomena in a scenario with a critical point and a first order phase transition. We observe long relaxation times at the phase transition, phase coexistence at the first order phase transition and critical slowing down at the critical point. We find a substantial supercooling in a first order phase transition in our model and due to the energy-momentum exchange also reheating is present. While at the critical point the correlation length increases slightly we find an enhanced intensity of nonequilibrium fluctuations at the first order phase transition, which leads to an increased production of sigma mesons.
In this work a nonlinear evolution of pure states of a finite dimensional quantum system is introduced, in particular a Riccati evolution equation.
It is shown how this class of dynamics is actually a Hamiltonian dynamics in the complex projective space.
In this projective space it is shown that there is a nonlinear superposition rule, consistent with its linear counterpart in the Hilbert space. As an example, the developed nonlinear formalism is applied to the semiclassical Jaynes–Cummings model.
Later, it is shown that there is an inherent nonlinear evolution in the dynamics of the so-called generalized coherent states.
To show this, the fact that in quantum mechanics it is possible to immerse a ''classical'' manifold into the Hilbert space is employed, such that one may parametrize the time-dependence of the wave function through the variation of parameters in the classical manifold.
The immersion allows to consider the so-called principle of analogy, i.e. using the procedures and structures available from the classical setting to employ them in the quantum setting.
Finally, it is introduced the contact Hamiltonian mechanics, an extension of symplectic Hamiltonian mechanics, and it is showed that it is a natural candidate for a geometric description of non-dissipative and dissipative systems.
In the context of information theory, the term Mutual Information has first been formulated by Claude Elwood Shannon. Information theory is the consistent mathematical description of technical communication systems. To this day, it is the basis of numerous applications in modern communications engineering and yet became indispensable in this field. This work is concerned with the development of a concept for nonlinear feature selection from scalar, multivariate data on the basis of the mutual information. From the viewpoint of modelling, the successful construction of a realistic model depends highly on the quality of the employed data. In the ideal case, high quality data simply consists of the relevant features for deriving the model. In this context, it is important to possess a suitable method for measuring the degree of the, mostly nonlinear, dependencies between input- and output variables. By means of such a measure, the relevant features could be specifically selected. During the course of this work, it will become evident that the mutual information is a valuable and feasible measure for this task and hence the method of choice for practical applications. Basically and without the claim of being exhaustive, there are two possible constellations that recommend the application of feature selection. On the one hand, feature selection plays an important role, if the computability of a derived system model cannot be guaranteed, due to a multitude of available features. On the other hand, the existence of very few data points with a significant number of features also recommends the employment of feature selection. The latter constellation is closely related to the so called "Curse of Dimensionality". The actual statement behind this is the necessity to reduce the dimensionality to obtain an adequate coverage of the data space. In other word, it is important to reduce the dimensionality of the data, since the coverage of the data space exponentially decreases, for a constant number of data points, with the dimensionality of the available data. In the context of mapping between input- and output space, this goal is ideally reached by selecting only the relevant features from the available data set. The basic idea for this work has its origin in the rather practical field of automotive engineering. It was motivated by the goals of a complex research project in which the nonlinear, dynamic dependencies among a multitude of sensor signals should be identified. The final goal of such activities was to derive so called virtual sensors from identified dependencies among the installed automotive sensors. This enables the real-time computability of the required variable without the expenses of additional hardware. The prospect of doing without additional computing hardware is a strong motive force in particular in automotive engineering. In this context, the major problem was to find a feasible method to capture the linear- as well as the nonlinear dependencies. As mentioned before, the goal of this work is the development of a flexibly applicable system for nonlinear feature selection. The important point here is to guarantee the practicable computability of the developed method even for high dimensional data spaces, which are rather realistic in technical environments. The employed measure for the feature selection process is based on the sophisticated concept of mutual information. The property of the mutual information, regarding its high sensitivity and specificity to linear- and nonlinear statistical dependencies, makes it the method of choice for the development of a highly flexible, nonlinear feature selection framework. In addition to the mere selection of relevant features, the developed framework is also applicable for the nonlinear analysis of the temporal influences of the selected features. Hence, a subsequent dynamic modelling can be performed more efficiently, since the proposed feature selection algorithm additionally provides information about the temporal dependencies between input- and output variables. In contrast to feature extraction techniques, the developed feature selection algorithm in this work has another considerable advantage. In the case of cost intensive measurements, the variables with the highest information content can be selected in a prior feasibility study. Hence, the developed method can also be employed to avoid redundance in the acquired data and thus prevent for additional costs.
The cones of nonnegative polynomials and sums of squares arise as central objects in convex algebraic geometry and have their origin in the seminal work of Hilbert ([Hil88]). Depending on the number of variables n and the degree d of the polynomials, Hilbert famously characterizes all cases of equality between the cone of nonnegative polynomials and the cone of sums of squares. This equality precisely holds for bivariate forms, quadratic forms and ternary quartics ([Hil88]). Since then, a lot of work has been done in understanding the difference between these two cones, which has major consequences for many practical applications such as for polynomial optimization problems. Roughly speaking, minimizing polynomial functions (constrained as well as unconstrained) can be done efficiently whenever certain nonnegative polynomials can be written as sums of squares (see Section 2.3 for the precise relationship). The underlying reason is the fundamental difference that checking nonnegativity of polynomials is an NP-hard problem whenever the degree is greater or equal than four ([BCSS98]), whereas checking whether a polynomial can be written as a sum of squares is a semidefinite feasibility problem (see Section 2.2). Although the complexity status of the semidefinite feasibility problem is still an open problem, it is polynomial for fixed number of variables. Hence, understanding the difference between nonnegative polynomials and sums of squares is highly desirable both from a theoretical and a practical viewpoint.
The application of natural products (NPs) as drugs and lead compounds has greatly improved human health over the past few decades. Despite their success, we still need to find new NPs that can be used as drugs to combat increasing drug resistance via new modes of action and to develop safer treatments with less side effects.
Entomopathogenic bacteria of Xenorhabdus and Photorhabdus that live in mutualistic symbiosis with nematodes are considered as promising producers of NPs, since more than 6.5% of their genomes are assigned to biosynthetic gene clusters (BGCs) responsible for production of secondary metabolites. The investigation on NPs from Xenorhabdus and Photorhabdus can not only provide new compounds for drug discovery but also help to understand the biochemical basis involved in mutualistic and pathogenic symbiosis of bacteria, nematode host and insect prey.
Nonribosomal peptides (NRPs) are a large class of NPs that are mainly found in bacteria and fungi. They are biosynthesized by nonribosomal peptide synthetases (NRPSs) and display diverse functions, representing more than 20 clinically used drugs. Although a large number of NRPs have been identified in Xenorhabdus and Photorhabdus, the advanced genome sequencing and bioinformatic analysis indicate that these bacteria still have many unknown NRPS-encoding gene clusters for NRP production that are worth to explore. Therefore, this thesis focuses on the discovery, biosynthesis, structure identification, and biological functions of new NRPs from Xenorhabdus and Photorhabdus.
The first publication describes the isolation and structure elucidation of seven new rhabdopeptide/xenortide-like peptides (RXPs) from X. innexi, incorporating putrescine or ammonia as the C-terminal amines. Bioactivity testing of these RXPs revealed potent antiprotozoal activity against the causative agents of sleeping sickness (Trypanosoma brucei rhodesiense) and malaria (Plasmodium falciparum), making them the most active RXP derivatives known to date. Biosynthetically, the initial NRPS module InxA might act iteratively with a flexible methyltransferase activity to catalyze the incorporation of the first five or six N-methylvaline/valine to these peptides.
The second publication focuses on the structure elucidation of seven unusual methionine-containing RXPs that were found as minor products in E. coli carrying the BGC kj12ABC from Xenorhabdus KJ12.1. To confirm the proposed structures from detailed HPLC-MS analysis, a solid-phase peptide synthesis (SPPS) method was developed for the synthesis of these partially methylated RXPs. These RXPs also exhibited good effects against T. brucei rhodesiense and P. falciparum, suggesting RXPs might play a role in protecting insect cadaver from soil-living protozoa to support the symbiosis with nematodes.
The third publication presents the identification of a new peptide library, named photohexapeptide library, which occurred after the biosynthetic gene phpS was activated in P. asymbiotica PB68.1 via promoter exchange. The chemical diversity of the photohexapeptides results from unusual promiscuous specificity of five out of six adenylation (A) domains being an excellent example of how to create compound libraries in nature. Furthermore, photohexapeptides enrich the family of the rare linear D-/L-peptide NPs.
The fourth publication concentrates on the structure elucidation of a new cyclohexapeptide, termed photoditritide, which was produced by P. temperata Meg1 after the biosynthetic gene pdtS was activated via promoter exchange. Photoditritide so far is the only example of a peptide from entomopathogenic bacteria that contains the uncommon amino acid homoarginine. The potent antimicrobial activity of photoditritide against Micrococcus luteus implies that photoditritide can protect the insect cadaver from food competitor bacteria in the complex life cycle of nematode and bacteria.
The last publication reports a new family of cyclic lipopeptides (CLPs), named phototemtides, which were obtained after the BGC pttABC from P. temperata Meg1 was heterologously expressed in E. coli. The gene pttA encodes an MbtH protein that was required for the biosynthesis of phototemtides in E. coli. To determine the absolute configurations of the hydroxy fatty acids, a total synthesis of the major compound phototemtide A was performed. Although the antimalarial activity of phototemtide A is only weak, it might be a starting point towards a selective P. falciparum compound, as it shows no activity against any other tested organisms.