Symmetric cell division in gram-negative bacteria is essential for generating two equal-sized daughter cells, each containing cellular material crucial for growth and future replication. The Min system, comprised of proteins MinC, MinD and MinE, is particularly important for this process since its deletion leads to minicells incapable of further replication. This thesis focuses on the interactions involving MinE that are important for allowing cell division at the mid-cell and for directing the dynamic localization of MinD that is observed in vivo. Previous experiments have shown that the MinE protein contains an N-terminal region that is required to stimulate MinD-catalyzed ATP hydrolysis in the Min protein interaction cycle. However, MinD-binding residues in MinE identified by in vitro MinD ATPase assays were subsequently found to be buried in the hydrophobic dimeric interface in the MinE structure, raising the possibility that these residues are not directly involved in the interaction. To address this issue, the ability of N-terminal MinE peptides to stimulate MinD activity was studied to determine the role of these residues in MinD activation. Our results implied that MinE likely undergoes a change in conformation or oligomerization state before binding MinD. In addition we performed circular dichroism spectroscopy of MinE. The data suggest that direct interactions between MinE and the lipid membrane can lead to conformational changes in MinE. Using NMR spectroscopy in an attempt to observe this structure change, different membrane-mimetic environments were tested. However the results strongly suggest that structural studies on the membrane-bound state of MinE will pose significant challenges. Taken together, the results in this thesis open the door for further exploration of the interactions involving MinE in order to gain a better understanding of the dynamic localization patterns formed by these proteins in vivo.

Symmetric cell division in gram-negative bacteria is essential for generating two equal-sized daughter cells, each containing cellular material crucial for growth and future replication. The Min system, comprised of proteins MinC, MinD and MinE, is particularly important for this process since its deletion leads to minicells incapable of further replication. This thesis focuses on the interactions involving MinE that are important for allowing cell division at the mid-cell and for directing the dynamic localization of MinD that is observed in vivo. Previous experiments have shown that the MinE protein contains an N-terminal region that is required to stimulate MinD-catalyzed ATP hydrolysis in the Min protein interaction cycle. However, MinD-binding residues in MinE identified by in vitro MinD ATPase assays were subsequently found to be buried in the hydrophobic dimeric interface in the MinE structure, raising the possibility that these residues are not directly involved in the interaction. To address this issue, the ability of N-terminal MinE peptides to stimulate MinD activity was studied to determine the role of these residues in MinD activation. Our results implied that MinE likely undergoes a change in conformation or oligomerization state before binding MinD. In addition we performed circular dichroism spectroscopy of MinE. The data suggest that direct interactions between MinE and the lipid membrane can lead to conformational changes in MinE. Using NMR spectroscopy in an attempt to observe this structure change, different membrane-mimetic environments were tested. However the results strongly suggest that structural studies on the membrane-bound state of MinE will pose significant challenges. Taken together, the results in this thesis open the door for further exploration of the interactions involving MinE in order to gain a better understanding of the dynamic localization patterns formed by these proteins in vivo.

Often predictor variables in regression models are measured with errors. This is known as an errors-in-variables (EIV) problem. The statistical analysis of the data ignoring the EIV is called naive analysis. As a result, the variance of the errors is underestimated. This affects any statistical inference that may subsequently be made about the model parameter estimates or the response prediction. In some cases (e.g. quadratic polynomial models) the parameter estimates and the model prediction is biased. The errors can occur in different ways. These errors are mainly classified into classical (i.e. occur in observational studies) or Berkson type (i.e. occur in designed experiments). This thesis addresses the problem of the Berkson EIV and their effect on the statistical analysis of data fitted using linear and nonlinear models. In particular, the case when the errors are dependent and have heterogeneous variance is studied. Both analytical and empirical tools have been used to develop new approaches for dealing with this type of errors. Two different scenarios are considered: mixture experiments where the model to be estimated is linear in the parameters and the EIV are correlated; and bioassay dose-response studies where the model to be estimated is nonlinear. EIV following Gaussian distribution, as well as the much less investigated non-Gaussian distribution are examined. When the errors occur in mixture experiments both analytical and empirical results showed that the naive analysis produces biased and inefficient estimators for the model parameters. The magnitude of the bias depends on the variances of the EIV for the mixture components, the model and its parameters. First and second Scheffé polynomials are used to fit the response. To adjust for the EIV, four different approaches of corrections are proposed. The statistical properties of the estimators are investigated, and compared with the naive analysis estimators. Analytical and empirical weighted regression calibration methods are found to give the most accurate and efficient results. The approaches require the error variance to be known prior to the analysis. The robustness of the adjusted approaches for misspecified variance was also examined. Different error scenarios of EIV in the settings of concentrations in bioassay dose-response studies are studied (i.e. dependent and independent errors). The scenarios are motivated by real-life examples. Comparisons between the effects of the errors are illustrated using the 4-prameter Hill model. The results show that when the errors are non-Gaussian, the nonlinear least squares approach produces biased and inefficient estimators. An extension of the well-known simulation-extrapolation (SIMEX) method is developed for the case when the EIV lead to biased model parameters estimators, and is called Berkson simulation-extrapolation (BSIMEX). BSIMEX requires the error variance to be known. The robustness of the adjusted approach for misspecified variance is examined. Moreover, it is shown that BSIMEX performs better than the regression calibration methods when the EIV are dependent, while the regression calibration methods are preferable when the EIV are independent.

Symmetric cell division in gram-negative bacteria is essential for generating two equal-sized daughter cells, each containing cellular material crucial for growth and future replication. The Min system, comprised of proteins MinC, MinD and MinE, is particularly important for this process since its deletion leads to minicells incapable of further replication. This thesis focuses on the interactions involving MinE that are important for allowing cell division at the mid-cell and for directing the dynamic localization of MinD that is observed in vivo. Previous experiments have shown that the MinE protein contains an N-terminal region that is required to stimulate MinD-catalyzed ATP hydrolysis in the Min protein interaction cycle. However, MinD-binding residues in MinE identified by in vitro MinD ATPase assays were subsequently found to be buried in the hydrophobic dimeric interface in the MinE structure, raising the possibility that these residues are not directly involved in the interaction. To address this issue, the ability of N-terminal MinE peptides to stimulate MinD activity was studied to determine the role of these residues in MinD activation. Our results implied that MinE likely undergoes a change in conformation or oligomerization state before binding MinD. In addition we performed circular dichroism spectroscopy of MinE. The data suggest that direct interactions between MinE and the lipid membrane can lead to conformational changes in MinE. Using NMR spectroscopy in an attempt to observe this structure change, different membrane-mimetic environments were tested. However the results strongly suggest that structural studies on the membrane-bound state of MinE will pose significant challenges. Taken together, the results in this thesis open the door for further exploration of the interactions involving MinE in order to gain a better understanding of the dynamic localization patterns formed by these proteins in vivo.

[ES] Esta tesis trata sobre la integración numérica de sistemas hamiltonianos con potenciales explícitamente dependientes del tiempo. Los problemas de este tipo son comunes en la física matemática, porque provienen de la mecánica cuántica, clásica y celestial. La meta de la tesis es construir integradores para unos problemas relevantes no autónomos: la ecuación de Schrödinger, que es el fundamento de la mecánica cuántica; las ecuaciones de Hill y de onda, que describen sistemas oscilatorios; el problema de Kepler con la masa variante en el tiempo. El Capítulo 1 describe la motivación y los objetivos de la obra en el contexto histórico de la integración numérica. En el Capítulo 2 se introducen los conceptos esenciales y unas herramientas fundamentales utilizadas a lo largo de la tesis. El diseño de los integradores propuestos se basa en los métodos de composición y escisión y en el desarrollo de Magnus. En el Capítulo 3 se describe el primero. Su idea principal consta de una recombinación de unos integradores sencillos para obtener la solución del problema. El concepto importante de las condiciones de orden se describe en ese capítulo. En el Capítulo 4 se hace un resumen de las álgebras de Lie y del desarrollo de Magnus que son las herramientas algebraicas que permiten expresar la solución de ecuaciones diferenciales dependientes del tiempo. La ecuación lineal de Schrödinger con potencial dependiente del tiempo está examinada en el Capítulo 5. Dado su estructura particular, nuevos métodos casi sin conmutadores, basados en el desarrollo de Magnus, son construidos. Su eficiencia es demostrada en unos experimentos numéricos con el modelo de Walker-Preston de una molécula dentro de un campo electromagnético. En el Capítulo 6, se diseñan los métodos de Magnus-escisión para las ecuaciones de onda y de Hill. Su eficiencia está demostrada en los experimentos numéricos con varios sistemas oscilatorios: con la ecuación de Mathieu, la ec. de Hill matricial, las ecuaciones de onda y de Klein-Gordon-Fock. El Capítulo 7 explica cómo el enfoque algebraico y el desarrollo de Magnus pueden generalizarse a los problemas no lineales. El ejemplo utilizado es el problema de Kepler con masa decreciente. El Capítulo 8 concluye la tesis, reseña los resultados y traza las posibles direcciones de la investigación futura. / [CA] Aquesta tesi tracta de la integració numèrica de sistemes hamiltonians amb potencials explícitament dependents del temps. Els problemes d'aquest tipus són comuns en la física matemàtica, perquè provenen de la mecànica quàntica, clàssica i celest. L'objectiu de la tesi és construir integradors per a uns problemes rellevants no autònoms: l'equació de Schrödinger, que és el fonament de la mecànica quàntica; les equacions de Hill i d'ona, que descriuen sistemes oscil·latoris; el problema de Kepler amb la massa variant en el temps. El Capítol 1 descriu la motivació i els objectius de l'obra en el context històric de la integració numèrica. En Capítol 2 s'introdueixen els conceptes essencials i unes ferramentes fonamentals utilitzades al llarg de la tesi. El disseny dels integradors proposats es basa en els mètodes de composició i escissió i en el desenvolupament de Magnus. En el Capítol 3, es descriu el primer. La seua idea principal consta d'una recombinació d'uns integradors senzills per a obtenir la solució del problema. El concepte important de les condicions d'orde es descriu en eixe capítol. El Capítol 4 fa un resum de les àlgebres de Lie i del desenvolupament de Magnus que són les ferramentes algebraiques que permeten expressar la solució d'equacions diferencials dependents del temps. L'equació lineal de Schrödinger amb potencial dependent del temps està examinada en el Capítol 5. Donat la seua estructura particular, nous mètodes quasi sense commutadors, basats en el desenvolupament de Magnus, són construïts. La seua eficiència és demostrada en uns experiments numèrics amb el model de Walker-Preston d'una molècula dins d'un camp electromagnètic. En el Capítol 6 es dissenyen els mètodes de Magnus-escissió per a les equacions d'onda i de Hill. El seu rendiment està demostrat en els experiments numèrics amb diversos sistemes oscil·latoris: amb l'equació de Mathieu, l'ec. de Hill matricial, les equacions d'onda i de Klein-Gordon-Fock. El Capítol 7 explica com l'enfocament algebraic i el desenvolupament de Magnus poden generalitzar-se als problemes no lineals. L'exemple utilitzat és el problema de Kepler amb massa decreixent. El Capítol 8 conclou la tesi, ressenya els resultats i traça les possibles direccions de la investigació futura. / [EN] The present thesis addresses the numerical integration of Hamiltonian systems with explicitly time-dependent potentials. These problems are common in mathematical physics because they come from quantum, classical and celestial mechanics. The goal of the thesis is to construct integrators for several import ant non-autonomous problems: the Schrödinger equation, which is the cornerstone of quantum mechanics; the Hill and the wave equations, that describe oscillating systems; the Kepler problem with time-variant mass. Chapter 1 describes the motivation and the aims of the work in the historical context of numerical integration. In Chapter 2 essential concepts and some fundamental tools used throughout the thesis are introduced. The design of the proposed integrators is based on the composition and splitting methods and the Magnus expansion. In Chapter 3, the former is described. Their main idea is to recombine some simpler integrators to obtain the solution. The salient concept of order conditions is described in that chapter. Chapter 4 summarises Lie algebras and the Magnus expansion ¿ algebraic tools that help to express the solution of time-dependent differential equations. The linear Schrödinger equation with time-dependent potential is considered in Chapter 5. Given its particular structure, new, Magnus-based quasi-commutator-free integrators are build. Their efficiency is shown in numerical experiments with the Walker-Preston model of a molecule in an electromagnetic field. In Chapter 6, Magnus-splitting methods for the wave and the Hill equations are designed. Their performance is demonstrated in numerical experiments with various oscillatory systems: the Mathieu equation, the matrix Hill eq., the wave and the Klein-Gordon-Fock eq. Chapter 7 shows how the algebraic approach and the Magnus expansion can be generalised to non-linear problems. The example used is the Kepler problem with decreasing mass. The thesis is concluded by Chapter 8, in which the results are reviewed and possible directions of future work are outlined. / Kopylov, N. (2019). Magnus-based geometric integrators for dynamical systems with time-dependent potentials [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/118798