A meshless, high-order integral equation method for smooth surfaces, with application to biomolecular electrostatics

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.Includes bibliographical references (p. 87-97).In this thesis, we develop methods for efficient simulation of biomolecular electrostatics based on Poisson-Boltzmann equation. Current techniques using finite-difference solution of differential formulation have many drawbacks. We present an integral formulation that resolves these difficulties and enables an efficient implementation using a recently developed fast solver. The new approach can solve practical engineering problems with good accuracy, but only with an aid of a high quality mesh generator, and sometimes require a large number of panels to discretize a surface. To this end, a novel approach to discretize singular integral equations is proposed. Unlike the traditional boundary element method using panel discretization, the new method is meshless and capable of achieving spectral convergence: numerical errors decrease exponentially fast with increasing size of basis set. We will describe a number of techniques in our approach, including the use of global, high order basis, quadrature-based panel integration, and innovative surface representation. The biomolecular problem is particularly suited for this method because molecular surfaces are typically smooth and can be represented globally using spherical harmonics.(cont.) The use of flat panels in the traditional approach would incur significant geometrical distortion, in addition to much slower convergence rate. Computational results demonstrate that for a practical problem at engineering accuracy (a tolerance of 10¡3) this new approach requires one to two orders of magnitude fewer unknowns than a flat panel method. For a more stringent tolerance of 10¡6, a comparison to an analytically solvable problem reveals that an improvement more than three orders of magnitude has been achieved.by Shih-Hsien Kuo.Ph.D

Simulations of the molecular mechanisms in P-type ATPases

Die P-Typ-ATPasen finden sich in allen Domänen des Lebens und stellen die größte Gruppe aktiver Ionentransporter in Zellen dar. Es handelt sich bei den P-Typ-ATPasen um integrale Membranproteine, die eine große Anzahl verschiedenster Ionen aktiv über eine biologische Membran transportieren. Die für diesen Ionentransport notwendige Energie wird durch Bindung und Hydrolyse von Adenosintriphosphat (ATP) und durch Phosphorylierung des Enzyms gewonnen. Diese, im cytoplasmatischen Teil gewonnene Energie, muss für den Ionentransport von der Phosphorylierungsstelle zur räumlich entfernten transmembranen Ionenbindungsstelle übertragen werden, bei dem das Protein einem Reaktionszyklus mit zwei Hauptkonformationszuständen E1 und E2 unterliegt. Zwischen diesen beiden Zuständen finden große strukturelle Änderungen statt, durch die die Ionenaffintät und die Zugänglichkeit der Ionenbindungsstelle reguliert wird. Da dieser Mechanismus der Energiegewinnung für alle Ionenpumpen dieser Art ähnlich ist, wurde die Ca2+-ATPase und die Na+/K+-ATPase als Modellproteine für die Untersuchung molekularer Mechanismen in P-Typ-ATPasen ausgewählt. Im Rahmen der vorliegenden Arbeit soll die Energietransduktion in P-Typ-ATPasen im Allgemeinen und der Protonengegentransport bzw. ein potentieller Protonentransportweg in der Ca2+-ATPase im Speziellen untersucht werden. Die beiden oben genannten Mechanismen sollen mittels computergestützter Methoden analysiert werden. Vor allem die Ca2+-ATPase ist prädestiniert für computergestützte Untersuchungen, da für diese sehr viele hochaufgelöste Röntgenstrukturdaten vorliegen, wenn auch bisher aufgrund der Größe und Komplexität des Systems nur sehr wenige theoretische Arbeiten durchgeführt wurden. Um den Energietransduktionsmechanismus in P-Typ-ATPasen zu untersuchen, wurde mittels Elektrostatik-Rechnungen der Einfluss eines elektrischen Feldes auf die verschiedenen Transmembranhelices untersucht. Dazu wurde ein Simulationssystem entwickelt, welches aus einem molekularen Kondensator besteht, der im Modell das Anlegen eines homogenen elektrischen Feldes über den Transmembranbereich simuliert. Da es sich bei dem Energietransduktionsmechanismus um einen dynamischen Prozess handelt, wurden die Elektrostatik-Rechnungen um Molekulardynamik-Simulationen erweitert. Mit diesen kann die konformelle Dynamik der P-Typ-ATPasen während der Energietransduktion in die Elektrostatik-Rechnungen einbezogen werden. Aus Spannungsklemmen-Fluorometrie-Experimenten, bei denen eine Spannung über eine Membran angelegt wird, kann geschlossen werden, dass die Helix M5 für die Energietransduktion verantwortlich ist. Mit den in dieser Arbeit durchgeführten Elektrostatik-Rechnungen konnte für verschiedene Enzymzustände der Ca2+-ATPase und für die Na+/K+-ATPase gezeigt werden, dass die Helix M5 die größten Konformeränderungen aufgrund des elektrischen Feldes aufweist. Durch die Erweiterung der Elektrostatik-Rechnungen um die Methode der Molekulardynamik-Simulation konnte zusätzlich die elektrische Feldstärke reduziert werden. Auch dabei zeigte sich, dass auf der Helix M5 die meisten Rotameränderungen durch das elektrische Feld induziert werden. Die aus Experimenten vermutete Rolle der Helix M5 als wichtiges Energietransduktionselement ließ sich mit diesen Simulationsrechnungen bestätigen. Um einen möglichen Protonenweg durch den Transmembranbereich der Ca2+-ATPase aufzuklären, wurden explizite Wassermoleküle in sechs verschiedene Enzymzustände der Ca2+-ATPase eingefügt. Aus Experimenten ist bekannt, dass in der Ca2+-ATPase ein Protonengegentransport stattfindet. Deshalb wurden für verschiedene Enzymzustände der Ca2+-ATPase mittels Elektrostatik-Rechnungen die Protonierungen der eingefügten Wassermoleküle sowie der titrierbaren Aminosäuren bestimmt. Aus den Ergebnissen dieser Rechnungen kann geschlossen werden, dass es sich bei dem Protonentransfer nicht um einen linearen Transport der Protonen handelt. Die Untersuchungen zeigen einen mehrstufigen Prozess, an dem Protonen in verschiedenen Transmembranbereichen der Ca2+-ATPase beteiligt sind. Anhand der berechneten Protonierungszustände der eingefügten Wassermoleküle und der pK-Werte der Aminosäuren im Transmembranbereich konnte weiterhin ein möglicher Protonenweg identifiziert werden.P-type ATPases are present in all domains of life and they represent a large group of active ion transporters in cells. All P-type ATPases are integral membrane proteins which transport a number of different ions across biological membranes. The necessary energy for that ion transport is obtained by binding and hydrolysis of adenosine triphosphate (ATP) and phosphorylation of the protein. This energy has to be transferred from the phosphorylation site to the ion binding site which is approx. 50 Å apart. The protein underlies a reaction cycle which can be described by two main conformational states E1 and E2. The conformational change of these two enzyme states regulates the ion affinity and accessibility of the binding sites. The mechanism is similar for all members of the P-type-ATPases. The Ca2+-ATPase and the Na+/K+-ATPase were chosen as model proteins to study the molecular mechanisms in P-type ATPases. In this study, the energy transduction mechanism of P-type ATPases and the proton countertransport and its potential proton pathways in the Ca2+-ATPase were analyzed by computational approaches. Mainly the Ca2+-ATPase is predestinated for computational studies, because X-ray structures with high atomic resolution are available for almost all enzyme states of the reaction cycle. However, only few theoretical work has been done so far due to the dimension and the complexity of the molecular system. To study the energy transduction mechanism in P-type ATPases, electrostatic calculations were performed to analyze the impact of an electric field on the different transmembrane helices. Therefore a simulation system was developed which consists of a molecular capacitor that allows to apply a homogenous electric field to the transmembrane region. Since the energy transduction mechanism is a dynamic process, the electrostatic calculations have been combined with molecular dynamics simulations in order to facilitate a certain protein backbone dynamics. From voltage clamp fluorometry experiments it was concluded that the transmembrane helix M5 participates as an energy transduction element. The analysis of the electrostatic calculations reveal that most conformational and ionization changes occur on helix M5 for different enzyme states of the Ca2+-ATPase and of the Na+/K+-ATPase. The combination with the molecular dynamics simulations allowed to reduce the applied electric field strength. It was possible to confirm the role of the transmembrane helix M5 as an energy transduction element. To study a possible proton pathway in the transmembrane region of the Ca2+-ATPase, explicit water molecules were inserted in six different enzyme states of the Ca2+-ATPase. From experimental data it is known, that a proton countertransport takes place in the Ca2+-ATPase, likely along titratable amino acids and water molecules inside the Ca2+-ATPase. Due to this reason the ionization state of the inserted water molecules and the titratable amino acids were calculated for various enzyme states of the Ca2+-ATPase. From electrostatic calculations it can be deduced, that the proton transfer is not a linear transport from the lumen to the cytoplasm. Instead it is a multi-step process, in which protons at different sites inside the transmembrane region of the Ca2+-ATPase participate. Due to the calculated pK values of the amino acids and the explicit water molecules it was possible to identify a potential proton path

Efficient numerical algorithms for surface formulations of mathematical models for biomolecule analysis and design

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.Includes bibliographical references (p. 179-183).This thesis presents a set of numerical techniques that extend and improve computational modeling approaches for biomolecule analysis and design. The presented research focuses on surface formulations of modeling problems related to the estimation of the energetic cost to transfer a biomolecule from the gas phase to aqueous solution. The thesis discusses four contributions to modeling biomolecular interactions. First, the thesis presents an approach to allow accurate discretization of the most prevalent mathematical definitions of the biomolecule-solvent interface; also presented are a number of accurate techniques for numerically integrating possibly singular functions over the discretized surfaces. Such techniques are essential for solving surface formulations numerically. The second part of the thesis presents a fast multiscale numerical algorithm, FFTSVD, that efficiently solves large boundary-element method problems in biomolecule electrostatics. The algorithm synthesizes elements of other popular fast algorithms to achieve excellent efficiency and flexibility. The third thesis component describes an integral-equation formulation and boundary-element method implementation for biomolecule electrostatic analysis.(cont.) The formulation and implementation allow the solution of complicated molecular topologies and physical models. Furthermore, by applying the methods developed in the first half of the thesis, the implementation can deliver superior accuracy for competitive performance. Finally, the thesis describes a highly efficient numerical method for calculating a biomolecular charge distribution that minimizes the free energy' change of binding to another molecule. The approach, which represents a novel PDE-constrained methodology, builds on well-developed physical theory. Computational results illustrate not only the method's improved performance but also its application to realistic biomolecule problems.by Jaydeep Porter Bardhan.Ph.D

Computational ligand design and analysis in protein complexes using inverse methods, combinatorial search, and accurate solvation modeling

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemistry, 2006.Vita.Includes bibliographical references (p. 207-230).This thesis presents the development and application of several computational techniques to aid in the design and analysis of small molecules and peptides that bind to protein targets. First, an inverse small-molecule design algorithm is presented that can explore the space of ligands compatible with binding to a target protein using fast combinatorial search methods. The inverse design method was applied to design inhibitors of HIV-1 protease that should be less likely to induce resistance mutations because they fit inside a consensus substrate envelope. Fifteen designed inhibitors were chemically synthesized, and four of the tightest binding compounds to the wild-type protease exhibited broad specificity against a panel of drug resistance mutant proteases in experimental tests. Inverse protein design methods and charge optimization were also applied to improve the binding affinity of a substrate peptide for an inactivated mutant of HIV-1 protease, in an effort to learn more about the thermodynamics and mechanisms of peptide binding. A single mutant peptide calculated to have improved binding electrostatics exhibited greater than 10-fold improved affinity experimentally.(cont.) The second half of this thesis presents an accurate method for evaluating the electrostatic component of solvation and binding in molecular systems, based on curved boundary-element method solutions of the linearized Poisson-Boltzmann equation. Using the presented FFTSVD matrix compression algorithm and other techniques, a full linearized Poisson-Boltzmann equation solver is described that is capable of solving multi-region problems in molecular continuum electrostatics to high precision.Michael Darren Altman.Ph.D