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This thesis deals with thickness optimization of shells. The overall task is to find an optimal thickness distribution in order to minimize the deformation of a loaded shell with prescribed volume. In addition, lower and upper bounds for the thickness are given. The shell is made of elastic, isotropic, homogeneous material. The deformation is modeled using equations from Linear Elasticity. Here, a basic shell model based on the Reissner-Mindlin assumption is used. Both the stationary and the dynamic case are considered. The continuity and the Gâteaux-differentiability of the control-to-state operator is investigated. These results are applied to the reduced objective with help of adjoint theory. In addition, techniques from shape optimization are compared to the optimal control approach. In the following, the theoretical results are applied to cylindrical shells and an efficient numerical implementation is presented. Finally, numerical results are shown and analyzed for different examples.
A slice is an intersection of a hyperplane and a self-similar set. The main purpose of this work is the mathematical description of slices. A suitable tool to describe slices are branching dynamical systems. Such systems are a generalisation of ordinary discrete dynamical systems for multivalued maps. Simple examples are systems arising from Bernoulli convolutions and beta-representations. The connection between orbits of branching dynamical systems and slices is demsonstrated and conditions are derived under which the geometry of a slice can be computed. A number of interesting 2-d and 3-d slices through 3-d and 4-d fractals is discussed.
We present classical and hybrid modeling approaches for genetic regulatory networks focusing on promoter analysis for negatively and positively autoregulated networks. The main aim of this thesis is to introduce an alternative mathematical approach to model gene regulatory networks based on piecewise deterministic Markov processes (PDMP). During somitogenesis, a process describing the early segmentation in vertebrates, molecular oscillators play a crucial role as part of a segmentation clock. In mice, these oscillators are called Hes1 and Hes7 and are commonly modeled by a system of two delay differential equations including a Hill function, which describes gene repression by their own gene products. The Hill coefficient, which is a measure of nonlinearity of the binding processes in the promoter, is assumed to be equal to two, based on the fact that Hes1 and Hes7 form dimers.However, by standard arguments applied to binding analysis, we show that a higher Hill coefficient is reasonable. This leads to results different from those in literature which requires a more sophisticated model. For the Hes7 oscillator we present a system of ordinary differential equations including a Michaelis-Menten term describing a nonlinear degradation of the proteins by the ubiquitinpathway. As demonstrated by the Hes1 and Hes7 oscillator, promoter behavior can have strong influence on the dynamical behavior of genetic networks. Since purely deterministic systems cannot reveal phenomenons caused by the inherent random fluctuations, we propose a novel approach based on PDMPs. Such models allow to model binding processes of transcription factors to binding sites in a promoter as random processes, where all other processes like synthesis, degradation or dimerization of the gene products are modeled in deterministic manner. We present and discuss a simulation algorithm for PDMPs and apply it to three types of genetic networks: an unregulated gene, a toggle switch, and a positively autoregulated network. The different regulation characteristics are analyzed and compared by numerical means. Furthermore, we determine analytical solutions of the stationary distributions of one negatively, and three positively autoregulated networks. Based on these results, we analyze attenuation of noise in a negative feedback loop, and the question of graded or binary response in autocatalytic networks.
Die vorliegende Arbeit ist im Bereich der parameterfreien Statistik anzusiedeln und beschäftigt sich mit der Anwendung von ordinalen Verfahren auf Zeitreihen und Bilddaten. Die Basis bilden dabei die sogenannten ordinalen Muster in ein bzw. zwei Dimensionen. Der erste Hauptteil der Arbeit gibt einen Überblick über die breiten Einsatzmöglichkeiten ordinaler Muster in der Zeitreihenanalyse. Mit ihrer Hilfe wird bei simulierten gebrochenen Brownschen Bewegungen der Hurst-Exponenten geschätzt und anhand von EEG-Daten eine Klassifikationsaufgabe gelöst. Des Weiteren wird die auf der Verteilung der ordinalen Muster beruhende Permutationsentropie eingesetzt, um in Magnetresonanztomographie (MRT)-Ruhedaten Kopfbewegungen der Probanden zu detektieren. Der zweite Hauptteil der Arbeit befasst sich mit der Erweiterung der ordinalen Muster auf zwei Dimensionen, um sie für Bilddaten nutzbar zu machen. Nach einigen Betrachtungen an fraktalen Oberflächen steht eine automatisierte und robuste Einschätzung der Qualität struktureller MRT-Daten im Vordergrund.
The study of sow reproduction traits is important in livestock science and production to increase animal survival and economic efficiency. This work deals with the detection of different effects on within-litter variance of birth weight by applying different statistical models with different distributional assumptions. The piglets within one litter were separated by sex. The trait of sow was formed from the sample variances of birth weights within litter separated by sex to consider the sex effect on mean birth weight. A linear mixed model (LMM) approach was fitted to the logarithmized sample variance and the sample standard deviation. A generalized linear mixed model with gamma distributed residuals and log-link function was applied to the untransformed sample variance. Appropriate weights were constructed to account for individual litter sizes. Models were compared by analysing data from Landrace and Large White. The estimates of heritability for the different traits ranged from 6-14%. The LMM for the weighted standard deviation of birth weights was identified as most suitable in terms of residual normality. Furthermore, the impact of piglets´ sex on birth weight variability was tested, but it was only proved for one practical dataset. Additionally, we analysed the influence of including or not including birth weights of stillborn piglets on the estimates of variance components of birth weight variability. With omitted stillborns the estimates of heritability resulted in about 2% higher values than in investigations of total born piglets. We were interested in the presence of the random boar effect on birth weight variability. The corresponding variance component was tested via restricted likelihood ratio test. Among others, the null distribution of the test statistic was approximated by parametric bootstrap simulations which were computational intensive. We picked up a two-parametric approach from literature and proposed a three-parametric approach to approximate the null distribution of the test statistic. We have analysed correlated data in balanced (simulated data) and unbalanced (empirical data) designs. The two-parametric approach using a scaled mixture of chisquare-distributions as well as a three-parametric approach, that uses a mixture of the point mass at zero and a gamma distribution, behaved most solid in all investigations and were most powerful in the simulation study.
In the PhD-thesis a conditional random field approach and its implementation is presented to predict the interaction sites of protein homo- and heterodimers using the spatial structure of one protein partner from a complex. The method includes a substantially simple edge feature model. A novel node feature class is introduced that is called -change in free energy-. The Online Large-Margin algorithm is adapted in order to train the model parameters given a classified reference set of proteins. A significantly higher prediction accuracy is achieved by combining our new node feature class with the standard node feature class relative accessible surface area. The quality of the predictions is measured by computing the area under the receiver operating characteristic.
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.
Simple Summary
Paratuberculosis is a disease which affects ruminants worldwide. Many countries have implemented certification and monitoring systems to control the disease, particularly in dairy herds. Monitoring herds certified as paratuberculosis non-suspect is an important component of paratuberculosis herd certification programs. The challenge is to detect the introduction or reintroduction of the infectious agent as early as possible with reasonable efforts but high certainty. In our study, we evaluated different low-cost testing schemes in herds where the share of infected animals was low, resulting in a low within-herd prevalence of animals shedding the bacteria that causes paratuberculosis in their feces. The test methods used were repeated pooled milk samples and fecal samples from the barn environment. Our study showed that numerous repetitions of different samples are necessary to monitor such herds with sufficiently high certainty. In the case of herds with a very low prevalence, our study showed that a combination of different sampling approaches is required.
Abstract
An easy-to-use and affordable surveillance system is crucial for paratuberculosis control. The use of environmental samples and milk pools has been proven to be effective for the detection of Mycobacterium avium subsp. paratuberculosis (MAP)-infected herds, but not for monitoring dairy herds certified as MAP non-suspect. We aimed to evaluate methods for the repeated testing of large dairy herds with a very low prevalence of MAP shedders, using different sets of environmental samples or pooled milk samples, collected monthly over a period of one year in 36 herds with known MAP shedder prevalence. Environmental samples were analyzed by bacterial culture and fecal PCR, and pools of 25 and 50 individual milk samples were analyzed by ELISA for MAP-specific antibodies. We estimated the cumulative sensitivity and specificity for up to twelve sampling events by adapting a Bayesian latent class model and taking into account the between- and within-test correlation. Our study revealed that at least seven repeated samplings of feces from the barn environment are necessary to achieve a sensitivity of 95% in herds with a within-herd shedder prevalence of at least 2%. The detection of herds with a prevalence of less than 2% is more challenging and, in addition to numerous repetitions, requires a combination of different samples.
Die dem Leben zugrundeliegenden Prozesse sind hochkomplex. Sie werden zu einem Großteil durch Proteine umgesetzt. Diese spielen eine tragende Rolle für die morphologische Struktur und Vielfalt sowie Spezifität der Fähigkeiten der verschiedenen Zelltypen. Jedoch wirken Proteine nicht isoliert für sich allein sondern indem sie miteinander oder mit anderen Molekülen in der Zelle (DNA, Metabolite, Signalstoffe etc.) wechselwirken. Gerät dieses Geflecht von aufeinander abgestimmten Wechselwirkungen aus dem Gleichgewicht, kann das eine Ursache für Erkrankungen sein. Die Kenntnis über fehlregulierte Interaktionen kann dabei helfen, die betreffende Krankheit besser zu verstehen und gegen sie zu intervenieren. Die vorliegende Dissertation beschäftigt sich mit der Identifizierung von solch differentiell regulierten Interaktionen. Im Rahmen der Arbeit wurde eine Methode mit dem Namen ExprEssence entwickelt, welche diejenigen Interaktionen in einem Protein-Protein-Interaktionsnetzwerk identifiziert, die sich zwischen zwei verglichenen Zuständen (z.B. krank versus gesund) am stärksten unterscheiden. Ziel ist es, das Netzwerk auf die wesentlichen Unterschiede zwischen den zwei untersuchten Zuständen zu reduzieren. Hierzu werden Genexpressions- oder Proteomdaten der beiden Zustände in das bereits bestehende Netzwerk integriert. Aus diesen Daten wird die Stärke/Häufigkeit des Auftretens der einzelnen Interaktionen des Netzwerks geschätzt. Die Interaktionen, deren Interaktionsstärken sich zwischen den betrachteten Zuständen am stärksten unterscheiden, werden beibehalten – die restlichen Interaktionen werden verworfen. Dies ergibt ein verkleinertes Subnetzwerk, das aus jenen Interaktionen besteht, die am stärksten differentiell reguliert sind. Diese Interaktionen und ihre Proteine sind Kandidaten für eine Erklärung der biologischen Unterschiede der betrachteten Zustände auf molekularem Niveau. Die Methode wurde auf verschiedene biologische Fragestellungen angewandt und mit anderen ähnlichen Methoden verglichen. Bei der Untersuchung der Unterschiede zwischen Erfolg und Misserfolg einer chemotherapeutischen Brustkrebstherapie konnte beispielsweise gezeigt werden, dass das mit ExprEssence erstellte Subnetzwerk einen stärkeren Bezug zu den bereits bekannten Therapieerfolg-relevanten Mechanismen aufweist als die Methoden, mit denen ExprEssence verglichen wurde. Weiterhin wurde im Subnetzwerk eine möglicherweise für den Therapieerfolg relevante Interaktion identifiziert, die in diesem Zusammenhang bisher nicht betrachtet wurde. Deren Bedeutung konnte in der experimentellen Nachverfolgung weiter untermauert werden. Einen weiteren Schwerpunkt der Arbeit bildete die Untersuchung des Interaktoms eines spezialisierten Zelltyps der Niere – des Podozyten. Dieser Zelltyp ist essentiell für die Filtrationskompetenz der Niere. Ein Interaktionsnetzwerk mit spezifisch für den Podozyten relevanten Interaktion gib es bisher nicht. Daher wurde ein Podozyten-spezifisches Protein-Protein-Interaktionsnetzwerk aus wissenschaftlichen Veröffentlichungen zusammengestellt und öffentlich verfügbar gemacht. Genexpressionsdaten vielfältiger Art, beispielsweise von Podozyten in verschiedenen Entwicklungsstadien oder in Zellkultur, wurden in das Netzwerk integriert und mit ExprEssence analysiert. So konnte beispielsweise gezeigt werden, dass die Dedifferenzierung von in Kultur gehaltenen Podozyten nicht dem Umkehrweg der zuvor durchlaufenen Differenzierung entspricht. Neben ExprEssence wurde weitere Software entwickelt, die die Anwendbarkeit von ExprEssence erweitert – MovieMaker und ExprEsSector. Mit MovieMaker werden die Übergänge zwischen den betrachteten Zuständen nachvollziehbarer visualisiert. ExprEsSector bildet die Vereinigungs- und Schnittmengen-Netzwerke von ExprEssence-Subnetzwerken. So können beispielsweise verschiedenen Krankheiten gemeinsame Veränderungen vom Normalzustand identifiziert werden. Ist für eine Krankheit bereits ein Therapieansatz vorhanden, der auf eine fehlregulierte Interaktion einwirkt, und ist diese Interaktion auch in der anderen Krankheit gleichartig differentiell reguliert, kann geprüft werden, ob diese Therapie auf die zweite Krankheit übertragen werden kann. Neben der Vorstellung und Diskussion der erzielten Ergebnisse, wird auch auf methodisch bedingte Nachteile eingegangen. Es werden Strategien aufgezeigt, wie die negativen Einflüsse möglichst minimiert werden können oder wie sie bei der Bewertung der Ergebnisse zu berücksichtigen sind. In Anbetracht der immer schneller ansteigenden Menge biologischer Daten ist es eine wesentliche Herausforderung geworden, aus diesen die essentiellen Informationen zu extrahieren. Der integrative Ansatz der Verknüpfung von Informationen verschiedener Quellen wurde mit ExprEssence und den Erweiterungen MovieMaker und ExprEsSector in einem Konzept zur Identifizierung zustandsrelevanter molekularer Mechanismen in intuitiv leicht erfassbarer Form umgesetzt.