Ziel der Dissertation war die Schaffung einer Simulationskette, die auf Basis der Charakteristika eines Schaumes eine Vorhersage über die Verteilung der Eigenkreisfrequenzen dieser Schaumsorte ermöglicht. Zur Validierung der Simulationskette dienen an verschiedenen Schaumproben gemessene Eigenkreisfrequenzen für Längs- und Biegeschwingungen. Die Modellierung erfolgte als räumlicher stochastischer Prozess mithilfe der harmonischen Synthese. Notwendige Eingangsgrößen konnten anhand von CT-Scans der Proben bestimmt werden. Zur Bestimmung der Eigenkreisfrequenzen wurden eindimensionale Ansätze wie der Rayleigh-Quotient und die Finite Cell Method (FCM) als dreidimensionaler Ansatz getestet. Es konnte gezeigt werden, dass die FCM in Verbindung mit den modellierten räumlichen Prozessen die gemessenen Verteilungen der Eigenkreisfrequenzen gut abbilden kann. Der eindimensionale Berechnungsansatz eignet sich ebenfalls, jedoch nur für homogene und isotrope Schäume.:Einleitung 1.1 Einordnung 1.2 Charakterisierung von Schäumen 1.3 Motivation 1.4 Aufgaben und Aufbau der Arbeit 2 Wahrscheinlichkeitsrechnung und stochastische Prozesse 2.1 Wahrscheinlichkeitsrechnung und Zufallsvariablen 2.1.1 Wahrscheinlichkeitsrechnung 2.1.2 Zufallsvariablen 2.2 Stochastische Prozesse 2.2.1 Kenngrößen stochastischer Prozesse 2.2.2 Eigenschaften stochastischer Prozesse 2.2.3 Spektralanalyse stationärer stochastischer Prozesse 2.2.4 Schätztheorie 3 Analyse der Schaumproben 3.1 Probekörper 3.2 Auswertung der CT-Daten 3.2.1 Kalibrierung der Grauwerte 3.2.2 Verteilungsfunktion und Mittelwert 3.2.3 Varianz des Schwankungsanteils und mittlerer Füllgrad 3.2.4 Leistungsdichtespektrum und Autokorrelation 3.2.5 Porendurchmesser und Anisotropie 3.2.6 Flächeninhalt und Flächenträgheitsmoment 3.3 Messung des dynamischen Verhaltens 3.3.1 Theoretische Grundlagen zur Auswertung 3.3.2 Vorbereitung der Proben 3.3.3 Längseigenkreisfrequenzen 3.3.4 Biegeeigenkreisfrequenzen 3.3.5 Fazit 4 Simulationsmodelle zur Nachbildung von Schäumen 4.1 Vorüberlegungen zur Modellierung 4.2 Theoretische Grundlagen zur Erzeugung eindimensionaler stochastischer Prozesse 4.2.1 Karhunen-Loeve-Transformation . 4.2.2 Harmonische Synthese 4.2.3 Ergebnisse für Flächeninhaltsprozesse 4.3 Simulation mehrdimensionaler stochastischer Prozesse 4.3.1 Bewertungskriterien für die Qualität der Simulation 4.3.2 Ergebnisse für die virtuellen Schäume 4.3.3 Verbesserter Algorithmus 4.4 Vergleich von virtuellen Schäumen und CT-Daten 4.4.1 Keramikschäume 4.4.2 Metallschäume 4.4.3 Anmerkungen und Fazit 4.5 Erweiterung um den Mittelwert 4.6 Konzept zur Simulation von Prozessen größerer Abmessungen 4.7 Fazit 5 Bestimmung der Eigenkreisfrequenzen 5.1 Materialmodelle für die Betrachtung als eindimensionales Kontinuum 5.2 Eindimensionale Modelle 5.2.1 Modell mit konstantem Querschnitt 5.3 Eindimensionales Modell mit Berücksichtigung der Mikrostruktur 5.4 Dreidimensionales Modell mit der Finite Cell Method 5.4.1 Theoretische Grundlagen 5.4.2 Anpassung und Optimierung der verwendeten Toolbox 5.4.3 Konvergenz und Festlegung der Zellgröße 5.5 Diskussion der Ergebnisse 6 Zusammenfassung und Ausblick Literatur A Daten zu Geometrie und Material der verwendeten Probekörper B Sensoren und Parameter für die Messung / The aim of this thesis was to create a simulation chain which, on the basis of the characteristics of a foam, enables a prediction of the distribution of the eigenfrequencies for this type of foam. Eigenfrequencies measured on different foam samples for longitudinal and flexural vibrations were used to validate the simulation chain. The modeling was done as a spatial stochastic process using harmonic synthesis. Necessary input parameters were determined from CT scans of the specimens. One-dimensional approaches such as the Rayleigh quotient, and the Finite Cell Method (FCM) as a three-dimensional approach were tested in order to determine the eigenfrequencies. It could be shown that the FCM, in conjunction with the modeled spatial processes, is able to reproduce the measured distributions of the eigenfrequencies. The one-dimensional calculation approach is also suitable, but only for homogeneous and isotropic foams.:Einleitung 1.1 Einordnung 1.2 Charakterisierung von Schäumen 1.3 Motivation 1.4 Aufgaben und Aufbau der Arbeit 2 Wahrscheinlichkeitsrechnung und stochastische Prozesse 2.1 Wahrscheinlichkeitsrechnung und Zufallsvariablen 2.1.1 Wahrscheinlichkeitsrechnung 2.1.2 Zufallsvariablen 2.2 Stochastische Prozesse 2.2.1 Kenngrößen stochastischer Prozesse 2.2.2 Eigenschaften stochastischer Prozesse 2.2.3 Spektralanalyse stationärer stochastischer Prozesse 2.2.4 Schätztheorie 3 Analyse der Schaumproben 3.1 Probekörper 3.2 Auswertung der CT-Daten 3.2.1 Kalibrierung der Grauwerte 3.2.2 Verteilungsfunktion und Mittelwert 3.2.3 Varianz des Schwankungsanteils und mittlerer Füllgrad 3.2.4 Leistungsdichtespektrum und Autokorrelation 3.2.5 Porendurchmesser und Anisotropie 3.2.6 Flächeninhalt und Flächenträgheitsmoment 3.3 Messung des dynamischen Verhaltens 3.3.1 Theoretische Grundlagen zur Auswertung 3.3.2 Vorbereitung der Proben 3.3.3 Längseigenkreisfrequenzen 3.3.4 Biegeeigenkreisfrequenzen 3.3.5 Fazit 4 Simulationsmodelle zur Nachbildung von Schäumen 4.1 Vorüberlegungen zur Modellierung 4.2 Theoretische Grundlagen zur Erzeugung eindimensionaler stochastischer Prozesse 4.2.1 Karhunen-Loeve-Transformation . 4.2.2 Harmonische Synthese 4.2.3 Ergebnisse für Flächeninhaltsprozesse 4.3 Simulation mehrdimensionaler stochastischer Prozesse 4.3.1 Bewertungskriterien für die Qualität der Simulation 4.3.2 Ergebnisse für die virtuellen Schäume 4.3.3 Verbesserter Algorithmus 4.4 Vergleich von virtuellen Schäumen und CT-Daten 4.4.1 Keramikschäume 4.4.2 Metallschäume 4.4.3 Anmerkungen und Fazit 4.5 Erweiterung um den Mittelwert 4.6 Konzept zur Simulation von Prozessen größerer Abmessungen 4.7 Fazit 5 Bestimmung der Eigenkreisfrequenzen 5.1 Materialmodelle für die Betrachtung als eindimensionales Kontinuum 5.2 Eindimensionale Modelle 5.2.1 Modell mit konstantem Querschnitt 5.3 Eindimensionales Modell mit Berücksichtigung der Mikrostruktur 5.4 Dreidimensionales Modell mit der Finite Cell Method 5.4.1 Theoretische Grundlagen 5.4.2 Anpassung und Optimierung der verwendeten Toolbox 5.4.3 Konvergenz und Festlegung der Zellgröße 5.5 Diskussion der Ergebnisse 6 Zusammenfassung und Ausblick Literatur A Daten zu Geometrie und Material der verwendeten Probekörper B Sensoren und Parameter für die Messung

All cells maintain essential metal nutrients at optimal levels by metal homeostasis. P-type ATPases, a crucial superfamily of integral membrane proteins, are involved in the active transport of metal ions across biological membranes driven by the motive force of ATP- hydrolysis. The PIB-type ATPase subfamily, also called CPx-ATPases, fulfills a key role in heavy metal homoeostasis among the most widespread species from bacteria to human. In humans, the defect in copper transporters is the direct cause of severe neurological and hepatic disorders such as Wilson and Menkes diseases, therefore, understanding the molecular function of these pumps is of paramount importance in human health. Cu+-ATPases have two transmembrane metal binding sites (TM-MBS) and three cytosolic domains, namely the actuator (A-domain) and phosphorylation and nucleotide-binding domain (PN), and regulatory N-terminal heavy metal binding domain (HMBD). Here, we have studied the Legionella pneumophila CopA (LpCopA) and its isolated cytosolic domains to improve our understanding of the functional interaction of the protein domains during metal transport relate this to the known structure of this ATPase. To elucidate how cytosolic ligands (Cu+ and nucleotide) stimulate the interactions among the cytosolic domains and may transmit conformational changes to the TM-MBS, the interactions among recombinant isolated cytosolic domains were first examined biochemically by co-purification and spectroscopically by circular dichroism, time-resolved fluorescence and site-directed fluorescent labeling assays. The Cu+-dependent interaction between the A-domain and HMBD has been postulated as a mechanism for activating the ATPase cycle. This question was addressed here by studying copper-dependent interactions between the isolated expressed domains. Spectroscopic evidence is provided that an HMBD-A complex is formed in the presence of Cu+ which binds with 100-200 nM affinity to the recombinant HMBD. In contrast, the A-domain interacts with the PN domain in a nucleotide-dependent fashion. This molecular recognition is required for the dephosphorylation step in the catalytic cycle. The interaction was investigated in more detail by the use of a decameric peptide derived from the PN-binding interface of the A-domain and carrying the conserved TGE-motif involved in dephosphorylation. Its binding to the isolated PN domain in a weakly nucleotide-dependent manner, is demonstrated here by stopped-flow fluorescence spectroscopy. Several ATPase assays were modified to assess the functionality of the PN-domain and full length LpCopA. The peptide was found to reduce the catalytic turnover of full length LpCopA. This agrees with the expected slowing down of the reformation of the PN-A-domain interaction since the peptide occupies their binding interface. Thus, the synthetic peptide provides a means to study specifically the influence of PN-A-domain interactions on the structure and function of LpCopA. This was done by time-correlated single photon counting (TCSPC) method. The time-dependent Stokes shift of the environmentally sensitive fluorophore BADAN which was covalently attached to the conserved CPC-motif in the TM-MBS was measured. The data indicate that the interior of the ATPase is hydrated and the mobility of the intra-protein water varies from high to low at C382 at the “luminal side” and C384 at the “cytosolic side” of the TM-MBS, respectively. This finding is consistent with the recent MD simulation of LpCopA, bringing the first experimental evidence on a luminal-open conformation of E2~P state. The A-domain-derived decapeptide, although binding to the cytosolic head piece, induces structural changes also at the TM-MBS. The peptide-stabilized state (with a disrupted PN-A interface) renders the C384 environment more hydrophobic as evidenced by TCSPC. Taken together, the data from cytosolic domain interactions, ATPase assays and of time-dependent Stoke shift analyses of BADAN-labeled LpCopA reveal the presence of hydrated intramembraneous sites whose degree of hydration is regulated by the rearrangement of cytosolic domains, particularly during the association and dissociation of the PN-A domains. Copper affects this arrangement by inducing the linkage of the A-domain to the HMBD. The latter appears to play not only an autoinhibitory but also a chaperone-like role in transferring Cu+ to the TM-MBS during catalytic turnover.

All cells maintain essential metal nutrients at optimal levels by metal homeostasis. P-type ATPases, a crucial superfamily of integral membrane proteins, are involved in the active transport of metal ions across biological membranes driven by the motive force of ATP- hydrolysis. The PIB-type ATPase subfamily, also called CPx-ATPases, fulfills a key role in heavy metal homoeostasis among the most widespread species from bacteria to human. In humans, the defect in copper transporters is the direct cause of severe neurological and hepatic disorders such as Wilson and Menkes diseases, therefore, understanding the molecular function of these pumps is of paramount importance in human health. Cu+-ATPases have two transmembrane metal binding sites (TM-MBS) and three cytosolic domains, namely the actuator (A-domain) and phosphorylation and nucleotide-binding domain (PN), and regulatory N-terminal heavy metal binding domain (HMBD). Here, we have studied the Legionella pneumophila CopA (LpCopA) and its isolated cytosolic domains to improve our understanding of the functional interaction of the protein domains during metal transport relate this to the known structure of this ATPase. To elucidate how cytosolic ligands (Cu+ and nucleotide) stimulate the interactions among the cytosolic domains and may transmit conformational changes to the TM-MBS, the interactions among recombinant isolated cytosolic domains were first examined biochemically by co-purification and spectroscopically by circular dichroism, time-resolved fluorescence and site-directed fluorescent labeling assays. The Cu+-dependent interaction between the A-domain and HMBD has been postulated as a mechanism for activating the ATPase cycle. This question was addressed here by studying copper-dependent interactions between the isolated expressed domains. Spectroscopic evidence is provided that an HMBD-A complex is formed in the presence of Cu+ which binds with 100-200 nM affinity to the recombinant HMBD. In contrast, the A-domain interacts with the PN domain in a nucleotide-dependent fashion. This molecular recognition is required for the dephosphorylation step in the catalytic cycle. The interaction was investigated in more detail by the use of a decameric peptide derived from the PN-binding interface of the A-domain and carrying the conserved TGE-motif involved in dephosphorylation. Its binding to the isolated PN domain in a weakly nucleotide-dependent manner, is demonstrated here by stopped-flow fluorescence spectroscopy. Several ATPase assays were modified to assess the functionality of the PN-domain and full length LpCopA. The peptide was found to reduce the catalytic turnover of full length LpCopA. This agrees with the expected slowing down of the reformation of the PN-A-domain interaction since the peptide occupies their binding interface. Thus, the synthetic peptide provides a means to study specifically the influence of PN-A-domain interactions on the structure and function of LpCopA. This was done by time-correlated single photon counting (TCSPC) method. The time-dependent Stokes shift of the environmentally sensitive fluorophore BADAN which was covalently attached to the conserved CPC-motif in the TM-MBS was measured. The data indicate that the interior of the ATPase is hydrated and the mobility of the intra-protein water varies from high to low at C382 at the “luminal side” and C384 at the “cytosolic side” of the TM-MBS, respectively. This finding is consistent with the recent MD simulation of LpCopA, bringing the first experimental evidence on a luminal-open conformation of E2~P state. The A-domain-derived decapeptide, although binding to the cytosolic head piece, induces structural changes also at the TM-MBS. The peptide-stabilized state (with a disrupted PN-A interface) renders the C384 environment more hydrophobic as evidenced by TCSPC. Taken together, the data from cytosolic domain interactions, ATPase assays and of time-dependent Stoke shift analyses of BADAN-labeled LpCopA reveal the presence of hydrated intramembraneous sites whose degree of hydration is regulated by the rearrangement of cytosolic domains, particularly during the association and dissociation of the PN-A domains. Copper affects this arrangement by inducing the linkage of the A-domain to the HMBD. The latter appears to play not only an autoinhibitory but also a chaperone-like role in transferring Cu+ to the TM-MBS during catalytic turnover.