This dissertation focuses on several issues pertaining to three-state progressive stochastic processes. Casting survival data within a three-state framework is an effective way to incorporate intermediate events into an analysis. These events can yield valuable insights into treatment interventions and the natural history of a process, especially when the right censoring is heavy. Exploiting the uni-directional nature of these processes allows for more effective modelling of the types of incomplete data commonly encountered in practice, as well as time-dependent explanatory variables and different time scales. In Chapter 2, we extend the model developed by Frydman (1995) by incorporating explanatory variables and by permitting interval censoring for the time to the terminal event. The resulting model is quite general and combines features of the models proposed by Frydman (1995) and Kim <i>et al</i>. (1993). The decomposition theorem of Gu (1996) is used to show that all of the estimating equations arising from Frydman's log likelihood function are self-consistent. An AIDS data set analyzed by these authors is used to illustrate our regression approach. Estimating the standard errors of our regression model parameters, by adopting a piecewise constant approach for the baseline intensity parameters, is the focus of Chapter 3. We also develop data-driven algorithms which select changepoints for the intervals of support, based on the Akaike and Schwarz Information Criteria. A sensitivity study is conducted to evaluate these algorithms. The AIDS example is considered here once more; standard errors are estimated for several piecewise constant regression models selected by the model criteria. Our results indicate that for both the example and the sensitivity study, the resulting estimated standard errors of certain model parameters can be quite large. Chapter 4 evaluates the goodness-of-link function for the transition intensity between states 2 and 3 in the regression model we introduced in chapter 2. By embedding this hazard function in a one-parameter family of hazard functions, we can assess its dependence on the specific parametric form adopted. In a simulation study, the goodness-of-link parameter is estimated and its impact on the regression parameters is assessed. The logistic specification of the hazard function from state 2 to state 3 is appropriate for the discrete, parametric-based data sets considered, as well as for the AIDS data. We also investigate the uniqueness and consistency of the maximum likelihood estimates based on our regression model for these AIDS data. In Chapter 5 we consider the possible efficiency gains realized in estimating the survivor function when an intermediate auxiliary variable is incorporated into a time-to-event analysis. Both Markov and hybrid time scale frameworks are adopted in the resulting progressive three-state model. We consider three cases for the amount of information available about the auxiliary variable: the observation is completely unknown, known exactly, or known to be within an interval of time. In the Markov framework, our results suggest that observing subjects at just two time points provides as much information about the survivor function as knowing the exact time of the intermediate event. There was generally a greater loss of efficiency in the hybrid time setting. The final chapter identifies some directions for future research.

This dissertation focuses on several issues pertaining to three-state progressive stochastic processes. Casting survival data within a three-state framework is an effective way to incorporate intermediate events into an analysis. These events can yield valuable insights into treatment interventions and the natural history of a process, especially when the right censoring is heavy. Exploiting the uni-directional nature of these processes allows for more effective modelling of the types of incomplete data commonly encountered in practice, as well as time-dependent explanatory variables and different time scales. In Chapter 2, we extend the model developed by Frydman (1995) by incorporating explanatory variables and by permitting interval censoring for the time to the terminal event. The resulting model is quite general and combines features of the models proposed by Frydman (1995) and Kim <i>et al</i>. (1993). The decomposition theorem of Gu (1996) is used to show that all of the estimating equations arising from Frydman's log likelihood function are self-consistent. An AIDS data set analyzed by these authors is used to illustrate our regression approach. Estimating the standard errors of our regression model parameters, by adopting a piecewise constant approach for the baseline intensity parameters, is the focus of Chapter 3. We also develop data-driven algorithms which select changepoints for the intervals of support, based on the Akaike and Schwarz Information Criteria. A sensitivity study is conducted to evaluate these algorithms. The AIDS example is considered here once more; standard errors are estimated for several piecewise constant regression models selected by the model criteria. Our results indicate that for both the example and the sensitivity study, the resulting estimated standard errors of certain model parameters can be quite large. Chapter 4 evaluates the goodness-of-link function for the transition intensity between states 2 and 3 in the regression model we introduced in chapter 2. By embedding this hazard function in a one-parameter family of hazard functions, we can assess its dependence on the specific parametric form adopted. In a simulation study, the goodness-of-link parameter is estimated and its impact on the regression parameters is assessed. The logistic specification of the hazard function from state 2 to state 3 is appropriate for the discrete, parametric-based data sets considered, as well as for the AIDS data. We also investigate the uniqueness and consistency of the maximum likelihood estimates based on our regression model for these AIDS data. In Chapter 5 we consider the possible efficiency gains realized in estimating the survivor function when an intermediate auxiliary variable is incorporated into a time-to-event analysis. Both Markov and hybrid time scale frameworks are adopted in the resulting progressive three-state model. We consider three cases for the amount of information available about the auxiliary variable: the observation is completely unknown, known exactly, or known to be within an interval of time. In the Markov framework, our results suggest that observing subjects at just two time points provides as much information about the survivor function as knowing the exact time of the intermediate event. There was generally a greater loss of efficiency in the hybrid time setting. The final chapter identifies some directions for future research.

Gene transcription in individual living cells is inevitably a stochastic and dynamic process. Little is known about how cells and organisms learn to balance the fidelity of transcriptional control and the stochasticity of transcription dynamics. In an effort to elucidate the contribution of environmental signals to this intricate balance, a Three State Model was recently proposed, and the transcription system was assumed to transit among three different functional states randomly. In this work, we employ this model to demonstrate how the stochastic dynamics of gene transcription can be characterized by the three transition parameters. We compute the probability distribution of a zero transcript event and its conjugate, the distribution of the time durations in gene on or gene off periods, the transition frequency between system states, and the transcriptional bursting frequency. We also exemplify the mathematical results by the experimental data on prokaryotic and eukaryotic transcription. The analysis reveals that no promoters will be definitely turned on to transcribe within a finite time period, no matter how strong the induction signals are applied, and how abundant the activators are available. Although stronger extrinsic signals could enhance promoter activation rate, the promoter creates an intrinsic ceiling that no signals could cross over in a finite time. Consequently, among a large population of isogenic cells, only a portion of the cells, but not the whole population, could be induced by environmental signals to express a particular gene within a finite time period. We prove that the gene on duration follows an exponential distribution, and the gene off intervals show a local maximum that is best described by assuming two sequential exponential process. The transition frequencies are determined by a system of stochastic differential equations, or equivalently, an iterative scheme of integral operators. We prove that for each positive integer n , there associates a unique time, called the peak instant, at which the nth transcript synthesis cycle since time zero proceeds most likely. These moments constitute a time series preserving the nature order of n.

Pektatlyase (Pel-15) aus dem alkalophilen Bodenbakterium Bacillus spec. KSM-P15 ist mit 197 Aminosäuren eines der kleinsten, bekannten β-3-Solenoidproteine. Sie spaltet Polygalakturonsäurederivate in einem Ca2+-abhängigen β-Eliminierungsprozess. Wie bei allen Proteinen dieser Enzymfamilie ist auch die Polypeptidkette von Pel-15 zu einer einsträngigen, rechtsgängigen, parallelen β-Helix aufgewunden. In diesem Strukturmotiv enthält jede Windung drei β-Stränge, die jeweils durch flexible Schleifenbereiche miteinander verbunden sind. Insgesamt acht Windungen stapeln sich in Pel-15 übereinander und bilden entlang der Helixachse flächige, parallele β-Faltblätter aus. Im Bereich dieser β-Faltblätter existiert ein ausgedehntes Netzwerk von Wasserstoffbrückenbindungen, durch das der hydrophobe Kern, der sich im Inneren der β-Helix befindet, vom umgebenden Lösungsmittel abgeschirmt wird. Besondere Abschlussstrukturen an beiden Enden der β-Helix, wie sie typischerweise bei anderen Ver-tretern dieser Strukturklasse ausgeprägt werden, sind in Pel-15 nicht zu beobachten. Stattdessen sind die terminalen Bereiche der β-Helix über Salzbrücken und hydrophobe Seitenkettenkontakte stabilisiert. In der vorliegenden Dissertation wurde die Pektatlyase Pel-15 hinsichtlich ihres Faltungsgleichgewichtes, ihrer enzymatischen Aktivität und der Kinetik ihrer Strukturbildung charakterisiert. In eine evolutionär konservierte Helixwindung wurden destabilisierende Mutationen eingeführt, und deren Auswirkungen mittels spektroskopischer Methoden analysiert. Die Ergebnisse zeigen, dass Pel-15 in Gegenwart des Denaturierungsmittels Guanidiniumhydrochlorid einen hyperfluoreszenten Gleichgewichtsustand (HF) populiert, der nach Messungen von Faltungs- und Entfaltungskinetiken ein konformationelles Ensemble aus den Zuständen HFslow und HFfast darstellt. Diese HF-Zustände sind durch eine hohe Aktivierungsbarriere voneinander getrennt. In Rückfaltungsexperimenten populieren nur etwa 80 % der faltenden Moleküle den Zwischenzustand HFslow, der mit einer Zeitkonstante von ca. 100 s zu HFfast weiterreagiert. Die Denaturierungsmittelabhängigkeit dieser Reaktion ist sehr gering, was eine trans-/cis-Prolylisomerisierung als geschwindigkeitslimitierenden Schritt nahelegt. Die Existenz eines cis-Peptides in der nativen Struktur macht es erforderlich, den denaturierten Zustand als ein Ensemble kinetisch separierter Konformationen, kurz: DSE, zu betrachten, das durch die Spezies Ufast und Uslow populiert wird. Nach dem in dieser Arbeit aufgestellten „Minimalmodell der Pel-15 Faltung“ stehen die HF-Spezies (HFslow, HFfast) mit den Konformationen des DSE in einem thermodynamischen Kreisprozess. Das Modell positioniert HFfast und die native Konformation N auf die „native Seite“ der Aktivierungsbarriere und trägt damit der Tatsache Rechnung, dass die Gleichgewichtseinstellung zwischen diesen Spezies zu schnell ist, um mit manuellen Techniken erfasst zu werden. Die hochaffine Bindung von Ca2+ (Kd = 10 μM) verschiebt sich das Faltungsgleichgewicht bereits in Gegenwart von 1 mM CaCl2 soweit auf die Seite des nativen Zustandes, das HFfast nicht länger nachweisbar ist. Entgegen anfänglicher Vermutungen kommt einer lokalen, evolutionär konservierten Disulfidbrücke im Zentrum der β-Helix eine wichtige Stabilisierungsfunktion zu. Die Disulfidbrücke befindet sich in einem kurzen Schleifenbereich der β-Helix nahe dem aktiven Zentrum. Obwohl ihr Austausch gegen die Reste Val und Ala die freie Stabilisierungsenthalpie des Proteins um ca. 10 kJ/mol reduziert, lässt die Struktur im Bereich der Mutationsstelle keine gravierende Veränderung erkennen. Auch die katalytisch relevante Ca2+-Bindungsaffinität bleibt unbeeinflusst; dennoch zeigen Enzymaktivitätstests für VA-Mutanten eine Reduktion der enzymatischen Aktivität um fast 50 % an. Die evolutionär konservierte Helixwindung im Allgemeinen und die in ihr enthaltene Disulfidbrücke im Besonderen müssen nach den vorliegenden Ergebnissen also eine zentrale Funktion sowohl für die Struktur des katalytischen Zentrums als auch für die Strukturbildung der β-Helix während der Faltungsreaktion besitzen. Die Ergebnisse dieser Arbeit finden in mehreren Punkten Anklang an Faltungseigenschaften, die für andere β -Helixproteine beschrieben wurden. Vor allem aber prädestinieren sie Pel-15 als ein neues, β-helikales Modellprotein. Aufgrund seiner einfachen Topologie, seiner niedrigen Windungszahl und seiner hohen thermodynamischen Stabilität ist Pel-15 sehr gut geeignet, die Determinanten von Stabilität und Strukturbildung des parallelen β-Helix-Motivs in einer Auflösung zu studieren, die aufgrund der Komplexität bestehender β-helikaler Modellsysteme bislang nicht zur Verfügung stand. / Pectate lyase Pel-15 was isolated from alcaliphlic Bacillus spec. strain KSM-P15. Like all pectate lyases Pel-15 binds and subsequently cleaves polygalacturonic acid, the main pectic compound in plant cell walls and middle lamellae, in a Ca2+ dependent beta-elimination reaction. With 197 amino acids and a molecular mass of only 21 kDa the protein is one of the smallest right-handed parallel beta-helical proteins known today. Polypeptide chains that are classified into this structural family adopt super-helical folds in which each “solenoid stack” consists of three beta-structured regions that are connected by flexible turn segments. Along its longitudinal axis the right-handed parallel beta-helix thus comprises three elongated parallel beta-sheets that are stabilized by an extensive network of hydrogen bonds wrapping around the densely packed hydrophobic core. Together with the shield-like arrangement of hydrogen bonds this hydrophobic core is considered as the main contributor to an exceptionally high stability that is a common feature of all beta-helical proteins. In contrast to most right-handed parallel beta-helices, Pel-15 is devoid of any terminal capping domains and laterally associated secondary structure. Therefore, this protein is considered to be a promising model protein of a pure beta-helix which will help to understand the determinants of both parallel beta-sheet formation and stability. In the dissertation at hand optical spectroscopic methods were used to assess the enzymatic activity, the folding/unfolding equilibrium and the kinetic mechanism of structure formation in neutral buffered solutions. Results indicate that Pel-15 populates a hyper-fluorescent equilibrium intermediate (HF) that is effectively populated in presence of the denaturing agent guanidinium hydrochloride (GdmCl). According to kinetic folding and unfolding experiments HF is not only an essential on-pathway intermediate but has to be considered as a conformational ensemble in which several hyperfluorescent states are in thermodynamic equilibrium with each other. According to their existence in kinetic folding trajectories these different HF-species were termed HFslow and HFfast. The activation energy between both states is remarkably high leading to a time constant of about 100 seconds for the reaction HFslow ⇆ HFfast. Since native Pel-15 contains an energetically disfavoured cis-prolyl peptide between A59 and P60 it is proposed that HFslow and HFfast differ in their prolyl peptide conformations. Two main results emerge from this dissertation. First, an extensive study of the Pel-15 folding- and unfolding behaviour facilitated the proposal of a “minimal folding model”. According to this model the HF-states and the according denatured species Uslow and Ufast are aligned into a thermodynamic circle. This implies that unfolded polypeptide chains reach the HF-ensemble via parallel folding trajectories. Since the native conformation N together with HFfast are on the same side of the activation barrier, it is the reaction HFslow ⇆ HFfast that is the rate limiting step in the folding reaction of Pel-15. Second, the importance of an evolutionarily conserved disulfide bond in the central region of Pel-15 was tested by site directed mutagenesis and subsequent spectroscopic characterization. The exchange of the disulfide against a hydrophobic pair of alanine and valine decreases the folding free energy by about 10 kJ/mol. Although this value is unexpectedly high, structural perturbations around both mutational positions are small as was deduced from X-Ray crystallography. Interestingly, the stability decrease is accompanied by a major loss of enzymatic activity while the Ca2+ binding affinity is not significantly affected. It is therefore concluded that the allosterically relevant disulfide bond stabilizes long-range interactions that stabilize several adjacent solenoid turns near the N-terminus of the protein. Indeed, planar stacking interactions are perturbed and flexibility of N-terminal loops is increased once the disulfide bond is removed. This dissertation establishes Pel-15 as a novel beta-helical model protein and – even more important – smoothes the way for a generally accepted perspective on the formation and stability of parallel beta-sheet proteins.