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The analysis of time-delayed dynamics on networks may help to understand many systems from physics, biology, and engineering, such as coupled laser arrays, gene-regulatory networks and complex ecosystems. Beside the complexity due to the network structure, the analysis is further complicated by the presence of the delays. Delay systems are in general infinite dimensional and thus can display complex dynamics as oscillations and chaos. The mathematical difficulties related to the delays hinders the analysis of delay networks. Thus, little is known yet about basic relations between network structure and delay dynamics. It has been shown that networks without delays can be studied efficiently with the generalized modeling approach, which analyzes the stability of an assumed steady state by a direct parametrization of the Jacobian matrix. In this thesis, I demonstrate the extension of the generalized modeling approach to delay networks and analyze networks of delay-coupled delay oscillators, with delayed auto-catalytic growth on the nodes and delayed transport between nodes. For degree-homogeneous networks (DHONs), in which each node has the same number of links, the bifurcation lines that border the stable areas can be calculated analytically, where the topology of the network is described only by the eigenvalues of the adjacency matrix. For undirected networks, the stability pattern in the parameter space of growth and transport delay is governed by two periodic sets of tongues of instability, which depend on the largest positive and the smallest negative eigenvalue. The direct relation between the eigenvalue and the bifurcation lines allows us to predict stability patterns for networks with certain topological properties. Thus, bipartite networks display a characteristic periodicity of tongues. In order to analyze the stability of degree-heterogeneous networks (DHENs), I apply a numerical sampling method based on Cauchy\'s Argument Principle. The stability patterns of these networks resembles the pattern of DHONs, which is governed by the two periodic sets. For networks with sufficiently many links, one set disappears, and the stability of DHENs can be approximates by the stability of a fully-connected network with the same average degree. However, random DHENs tend to be more stable than DHONs, and DHENs with a broad degree-distribution tend to be more stable than DHENs with a narrow distribution. Thus, such networks are more likely to give rise to amplitude death, i.e. the stabilization of an unstable steady state through diffusive coupling. The stability pattern of DHENs can be qualitatively different than the pattern in DHONs. However, for small growth delays, close to the critical delay of the single node system, the bifurcation lines of all DHENs with the same average degree coincide. This, is particularly interesting, because there the stability depends on a global property of the network, which suggests a diverging interaction length. In summary, the extension of generalized modeling to time-delay networks reveals basic relations between the delay dynamics and the topology. The generality of our model should allow to apply these results to a large class of real-world systems.

In dieser Dissertation modellieren und analysieren wir scRNA-Seq-Daten, um Mechanismen, die biologischen Prozessen zugrunde liegen, zu verstehen In scRNA-Seq-Experimenten wird biologisches Rauschen mit technischem Rauschen vermischt. Mittels eines vereinfachten scRNA-Seq-Modells leiten wir eine analytische Verteilungsfunktion für die beobachtete Verteilung unter Kenntnis einer Ausgangsverteilung her. Charakteristiken und sogar ein allgemeines Moment der Ausgangsverteilung können aus der beobachteten Verteilung berechnet werden. Unsere Formeln stellen den Ausgangspunkt zur Quantifizierung von Zellvariabilität dar. Wir haben eine vollständig lineare Analyse von Transkriptomdaten entwickelt, die zeigt, dass sich Zellen während des Zellzyklus auf einer ebenen zirkulären Trajektorie im Transkriptomraum bewegen. In immortalisierten Zelllinien stellen wir fest, dass die Transkriptomdynamiken des Zellzyklus hauptsächlich unabhängig von den Dynamiken anderer Zellprozesse stattfinden. Unser Algorithmus (“Revelio”) bringt eine einfache Methode mit sich, um unsynchronisierte Zellen nach der Zeit zu ordnen und ermöglicht das exakte Entfernen von Zellzykluseffekten. Die Form der Zellzyklus-Trajektorie zeigt, dass der Zellzyklus sich dazu entwickelt hat, Änderungen der transkriptionellen Aktivitäten und der damit verbundenen regulativen Anstrengungen zu minimieren. Dieses Konstruktionsprinzip könnte auch für andere Prozesse relevant sein. Durch die Verwendung von metabolischer Molekülmarkierung erweitern wir Modelle zur mRNA-Kinetik, um dynamische mRNA-Ratenparameter für Transkription, Splicing und Degradation zu erhalten und die Lösungen auf den Zellzyklus anzuwenden. Wir zeigen, dass unser Modell zwischen Genen mit ähnlicher Genexpression aber unterschiedlicher Genregulation unterscheiden kann. Zwar enthalten scRNA-Seq-Daten aktuell noch zu viel technisches Rauschen, unser Modell wird jedoch für das zukünftige Errechnen von dynamischen mRNA-Ratenparametern von großem Nutzen sein. / In this dissertation, we model and analyse scRNA-seq data to understand mechanisms underlying biological processes. In scRNA-seq experiments, biological noise gets convoluted with various sources of technical noise. With the help of a simplified scRNA-seq model, we derive an analytical probability distribution function for the observed output distribution given a true input distribution. We find that characteristics and even general moments of the input distribution can be calculated from the output distribution. Our formulas are a starting point for the quantification of cell-to-cell variability. We developed a fully linear analysis of transcriptome data which reveals that cells move along a planar circular trajectory in transcriptome space during the cell cycle. Additionally, we find in immortalized cell lines that cell cycle transcriptome dynamics occur largely independently from other cellular processes. Our algorithm (“Revelio”) offers a simple method to order unsynchronized cells in time and enables the precise removal of cell cycle effects from the data. The shape of the cell cycle trajectory indicates that the cell cycle has evolved to minimize changes of transcriptional activity and their related regulatory efforts. This design principle may be of relevance to other cellular processes. By considering metabolic labelling, we extend existing mRNA kinetic models to obtain dynamic mRNA rate parameters for transcription, splicing and degradation and apply our solutions to the cell cycle. We can distinguish genes with similar expression values but different gene regulation strategies. While current scRNA-seq data contains too much technical noise, the model will be of great value for inferring dynamic mRNA rate parameters in future research.

Diese Dissertation untersucht Aspekte des Zusammenspiels von ergodischem Langzeitver- halten und der Glättungseigenschaft dynamischer Systeme, die von stochastischen Differen- tialgleichungen (SDEs) mit Sprüngen erzeugt sind. Im Speziellen werden SDEs getrieben von Lévy-Prozessen und der Marcusschen kanonischen Gleichung untersucht. Ein vari- ationeller Ansatz für den Malliavin-Kalkül liefert eine partielle Integration, sodass eine Variation im Raum in eine Variation im Wahrscheinlichkeitsmaß überführt werden kann. Damit lässt sich die starke Feller-Eigenschaft und die Existenz glatter Dichten der zuge- hörigen Markov-Halbgruppe aus einer nichtstandard Elliptizitätsbedingung an eine Kom- bination aus Gaußscher und Sprung-Kovarianz ableiten. Resultate für Sprungdiffusionen auf Untermannigfaltigkeiten werden aus dem umgebenden Euklidischen Raum hergeleitet. Diese Resultate werden dann auf zufällige dynamische Systeme angewandt, die von lin- earen stochastischen Differentialgleichungen erzeugt sind. Ruelles Integrierbarkeitsbedin- gung entspricht einer Integrierbarkeitsbedingung an das Lévy-Maß und gewährleistet die Gültigkeit von Oseledets multiplikativem Ergodentheorem. Damit folgt die Existenz eines Lyapunov-Spektrums. Schließlich wird der top Lyapunov-Exponent über eine Formel der Art von Furstenberg–Khasminsikii als ein ergodisches Mittel der infinitesimalen Wachs- tumsrate über die Einheitssphäre dargestellt. / The present thesis investigates certain aspects of the interplay between the ergodic long time behavior and the smoothing property of dynamical systems generated by stochastic differential equations (SDEs) with jumps, in particular SDEs driven by Lévy processes and the Marcus’ canonical equation. A variational approach to the Malliavin calculus generates an integration-by-parts formula that allows to transfer spatial variation to variation in the probability measure. The strong Feller property of the associated Markov semigroup and the existence of smooth transition densities are deduced from a non-standard ellipticity condition on a combination of the Gaussian and a jump covariance. Similar results on submanifolds are inferred from the ambient Euclidean space. These results are then applied to random dynamical systems generated by linear stochas- tic differential equations. Ruelle’s integrability condition translates into an integrability condition for the Lévy measure and ensures the validity of the multiplicative ergodic theo- rem (MET) of Oseledets. Hence the exponential growth rate is governed by the Lyapunov spectrum. Finally the top Lyapunov exponent is represented by a formula of Furstenberg– Khasminskii–type as an ergodic average of the infinitesimal growth rate over the unit sphere.

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