Charakterisierung und Konvergenz mehrdimensionaler kontinuierlicher Verzweigungsprozesse

In dieser Arbeit wird eine Klasse von stochastischen Prozessen untersucht, die eine abstrakte Verzweigungseigenschaft besitzen. Die betrachteten Prozesse sind homogene Markov-Prozesse in stetiger Zeit mit Zuständen im mehrdimensionalen reellen Raum und dessen Ein-Punkt-Kompaktifizierung. Ausgehend von Minimalforderungen an die zugehörige Übergangsfunktion wird eine vollständige Charakterisierung der endlichdimensionalen Verteilungen mehrdimensionaler kontinuierlicher Verzweigungsprozesse vorgenommen. Mit Hilfe eines erweiterten Laplace-Kalküls wird gezeigt, dass jeder solche Prozess durch eine bestimmte spektral positive unendlich teilbare Verteilung eindeutig bestimmt ist. Umgekehrt wird nachgewiesen, dass zu jeder solchen unendlich teilbaren Verteilung ein zugehöriger Verzweigungsprozess konstruiert werden kann. Mit Hilfe der allgemeinen Theorie Markovscher Operatorhalbgruppen wird sichergestellt, dass jeder mehrdimensionale kontinuierliche Verzweigungsprozess eine Version mit Pfaden im Raum der cadlag-Funktionen besitzt. Ferner kann die (funktionale) schwache Konvergenz der Prozesse auf die vage Konvergenz der zugehörigen Charakterisierungen zurückgeführt werden. Hieraus folgen allgemeine Approximations- und Konvergenzsätze für die betrachtete Klasse von Prozessen. Diese allgemeinen Resultate werden auf die Unterklasse der sich verzweigenden Diffusionen angewendet. Es wird gezeigt, dass für diese Prozesse stets eine Version mit stetigen Pfaden existiert. Schließlich wird die allgemeinste Form der Fellerschen Diffusionsapproximation für mehrtypige Galton-Watson-Prozesse bewiesen.

Parameterschätzung in Verzweigungsprozessen mit Immigration

Im Verzweigungsprozess mit Immigration werden Schätzer für die erwartete Nachkommenzahl m eines Individuums und die erwartete Immigration λ pro Generation konstruiert. Sie sind nur aufgrund der beobachteten Populationsgröße einer jeden Generation konsistent, ohne Vorkenntnis darüber, ob der Prozess subkritisch (m<1), kritisch (m=1) oder superkritisch (m>1) ist. Im superkritischen Fall ist der Schätzer für λ jedoch nicht konsistent. Dies ist aber keine Einschränkung, denn es wird gezeigt, dass in diesem Fall kein konsistenter Schätzer für λ existiert. Des Weiteren werden Konvergenzgeschwindigkeit der Schätzer und asymptotische Verteilungen der Schätzfehler untersucht. Dabei werden die Fälle (m<1), (m>1) und (m=1) unterschieden, was gänzlich verschiedene Vorgehensweisen erfordert (Ergodizität, Martingalmethoden, Diffusionsapproximationen). Diese hier vorliegende Diplomarbeit orientiert sich an den Ideen und Ergebnissen von Wei und Winnicki (1989/90).

Partial reconstruction of the trajectories of a discretely observed branching diffusion with immigration and an application to inference

In this treatise we consider finite systems of branching particles where the particles move independently of each other according to d-dimensional diffusions. Particles are killed at a position dependent rate, leaving at their death position a random number of descendants according to a position dependent reproduction law. In addition particles immigrate at constant rate (one immigrant per immigration time). A process with above properties is called a branching diffusion withimmigration (BDI). In the first part we present the model in detail and discuss the properties of the BDI under our basic assumptions. In the second part we consider the problem of reconstruction of the trajectory of a BDI from discrete observations. We observe positions of the particles at discrete times; in particular we assume that we have no information about the pedigree of the particles. A natural question arises if we want to apply statistical procedures on the discrete observations: How can we find couples of particle positions which belong to the same particle? We give an easy to implement 'reconstruction scheme' which allows us to redraw or 'reconstruct' parts of the trajectory of the BDI with high accuracy. Moreover asymptotically the whole path can be reconstructed. Further we present simulations which show that our partial reconstruction rule is tractable in practice. In the third part we study how the partial reconstruction rule fits into statistical applications. As an extensive example we present a nonparametric estimator for the diffusion coefficient of a BDI where the particles move according to one-dimensional diffusions. This estimator is based on the Nadaraya-Watson estimator for the diffusion coefficient of one-dimensional diffusions and it uses the partial reconstruction rule developed in the second part above. We are able to prove a rate of convergence of this estimator and finally we present simulations which show that the estimator works well even if we leave our set of assumptions.

DNA block copolymers - synthesis, morphologies and applications

The last decades have witnessed significant and rapid progress in polymer chemistry and molecular biology. The invention of PCR and advances in automated solid phase synthesis of DNA have made this biological entity broadly available to all researchers across biological and chemical sciences. Thanks to the development of a variety of polymerization techniques, macromolecules can be synthesized with predetermined molecular weights and excellent structural control. In recent years these two exciting areas of research converged to generate a new type of nucleic acid hybrid material, consisting of oligodeoxynucleotides and organic polymers. By conjugating these two classes of materials, DNA block copolymers are generated exhibiting engineered material properties that cannot be realized with polymers or nucleic acids alone. Different synthetic strategies based on grafting onto routes in solution or on solid support were developed which afforded DNA block copolymers with hydrophilic, hydrophobic and thermoresponsive organic polymers in good yields. Beside the preparation of DNA block copolymers with a relative short DNA-segment, it was also demonstrated how these bioorganic polymers can be synthesized exhibiting large DNA blocks (>1000 bases) applying the polymerase chain reaction. Amphiphilic DNA block copolymers, which were synthesized fully automated in a DNA synthesizer, self-assemble into well-defined nanoparticles. Hybridization of spherical micelles with long DNA templates that encode several times the sequence of the micelle corona induced a transformation into rod-like micelles. The Watson-Crick motif aligned the hydrophobic polymer segments along the DNA double helix, which resulted in selective dimer formation. Even the length of the resulting nanostructures could be precisely adjusted by the number of nucleotides of the templates. In addition to changing the structural properties of DNA-b-PPO micelles, these materials were applied as 3D nanoscopic scaffolds for organic reactions. The DNA strands of the corona were organized by hydrophobic interactions of the organic polymer segments in such a fashion that several DNA-templated organic reactions proceeded in a sequence specific manner; either at the surface of the micelles or at the interface between the biological and the organic polymer blocks. The yields of reactions employing the micellar template were equivalent or better than existing template architectures. Aside from its physical properties and the morphologies achieved, an important requirement for a new biomaterial is its biocompatibility and interaction with living systems, i.e. human cells. The toxicity of the nanoparticles was analyzed by a cell proliferation assay. Motivated by the non-toxic nature of the amphiphilic DNA block copolymers, these nanoobjects were employed as drug delivery vehicles to target the anticancer drug to a tumor tissue. The micelles obtained from DNA block copolymers were easily functionalized with targeting units by hybridization. This facile route allowed studying the effect of the amount of targeting units on the targeting efficacy. By varying the site of functionalization, i.e. 5’ or 3’, the outcome of having the targeting unit at the periphery of the micelle or in the core of the micelle was studied. Additionally, these micelles were loaded with an anticancer drug, doxorubicin, and then applied to tumor cells. The viability of the cells was calculated in the presence and absence of targeting unit. It was demonstrated that the tumor cells bearing folate receptors showed a high mortality when the targeting unit was attached to the nanocarrier.

Investigations into the effect of nucleoside modifications on the physicochemical properties and biological function of DNA and RNA

This thesis explores the effect of chemical nucleoside modification on the physicochemical and biological properties of nucleic acids. Positional alteration on the Watson-Crick edge of purines and pyrimidines, the “C-H” edge of pyrimidines, as well as both the Hoogsteen and sugar edges of purines were attempted by means of copper catalyzed azide-alkyne cycloaddition. For this purpose, nucleic acid building blocks carrying terminal alkynes were synthesized and introduced into oligonucleotides by solid-phase oligonucleotide chemistry. rnOf particular interest was the effect of nucleoside modification on hydrogen bond formation with complementary nucleosides. The attachment of propargyl functionalities onto the N2 of guanosine and the N4 of 5-methylcytosine, respectively, followed by incorporation of the modified analogs into oligonucleotides, was successfully achieved. Temperature dependent UV-absorption melting measurements with duplexes formed between modified oligonucleotides and a variety of complementary strands resulted in melting temperatures for the respective duplexes. As a result, the effect that both the nature and the site of nucleoside modification have on base pairing properties could thus be assisted. rnTo further explore the enzymatic recognition of chemically modified nucleosides, the oligonucleotide containing the N2-modified guanosine derivative on the 5’-end, which was clicked to a fluorescent dye, was subjected to knockdown analyses of the eGFP reporter gene in the presence of increasing concentrations of siRNA duplexes. From these dose-dependent experiments, a clear effect of 5’-labeling on the knockdown efficiency could be seen. In contrast, 3’-labeling was found to be relatively insignificant.rn