Domain Decomposition Based Hybrid Methods of Finite Element and Finite Difference and Applications in Biomolecule Simulations

The dielectric continuum models, such as Poisson Boltzmann equation (PBE), size modified PBE (SMPBE), and nonlocal modified PBE (NMPBE), are important models in predicting the electrostatics of a biomolecule in an ionic solvent. To solve these dielectric continuum models efficiently, in this dissertation, new finite element and finite difference hybrid methods are constructed by Schwartz domain decomposition techniques based on a special seven-box partition of a cubic domain. As one important part of these methods, a finite difference optimal solver --- the preconditioned conjugate gradient method using a multigrid V-cycle preconditioner --- is described in details and proved to have a convergence rate independent of mesh size in solving a symmetric positive definite linear system. These new hybrid algorithms are programmed in Fortran, C, and Python based on the efficient finite element library DOLFIN from the FEniCS project, and are well validated by test models with known analytical solutions. Comparison numerical tests between the new hybrid solvers and the corresponding finite element solvers are done to show the improvement in efficiency. Finally, as applications, solvation free energy and binding free energy calculations are done and then compared to the experiment data

An Adapative Treecode-accelerated Boundary Integral Solver for Computing the Electrostatics of a Biomolecule

The Poisson-Boltzmann equation (PBE) is a widely-used model in the calculation of electrostatic potential for solvated biomolecules. PBE is an interface problem defined in the whole space with the interface being a molecular surface of a biomolecule, and has been solved numerically by finite difference, finite element, and boundary integral methods. Unlike the finite difference and finite element methods, the boundary integral method works directly over the whole space without approximating the whole space problem into an artificial boundary value problem. Hence, it is expected to solve PBE in higher accuracy. However, so far, it was only applied to a linear PBE model. Recently, a solution of PBE was split into three component functions. One of them, G, is a known function that collects all the singularity points of PBE so that the other two components become continuously twice differentiable within the protein and solvent regions. Such an approach has led to efficient PBE finite element solvers. This provided motivation to study the application of this solution decomposition to the development of a new boundary integral algorithm for solving PBE. Reformulating the interface problem of $\Psi$ into a boundary integral equation is nontrivial because the involved flux interface condition is discontinuous. Development of a fast numerical algorithm for solving the resulted boundary integral equation is an attractive research topic. In this masters thesis, we focus on one key step of our new boundary integral algorithm: how to solve for the second component function $\Psi$ of the PBE solution by a boundary integral method. This work becomes important by itself because the sum of $\Psi$ with $G$ gives the solution of the Poisson dielectric model for the case of a biomolecule in water. In this project, we obtain the new boundary integral equation and develop an adaptive treecode-accelerated boundary integral algorithm. We then program the new algorithm in Fortran and make various numerical tests to validate our new algorithm and program package. In particular, numerical tests performed against analytic models verify the effectiveness of the solver, and comparisons to experimental data verify its accuracy for real-world applications. In this way, it is demonstrated that this solver and solution decomposition can compute the electrostatics of a biomolecule in water with high numerical accuracy

Correlation induced electrostatic effects in biomolecular systems

An understanding of electrostatic interactions in biomolecular systems is crucial for many applications in molecular biology. This thesis focuses on the theoretical modeling of two effects: first, the change in the dielectric properties of water due to hydrogen bond formation and second, the reentrant condensation of proteins induced by protein-metal ion complexation. A nonlocal response theory is necessary to describe the dielectric effects of hydrogen bond formation. Correctly formulating this theory for a solvated biomolecule is challenging, because the biomolecule\u27s cavity poses an obstacle for the water network. We develop a theory explicitly incorporating boundary conditions to describe the water network on the molecular surface. We implement an accurate and efficient finite difference solver, which offers the possibility to easily investigate different physically motivated boundary effects. A detailed analysis of different nonlocal models reveals that, for the macroscopic behavior, the boundary conditions are of minor importance, while for a detailed understanding of the electrostatics near the molecular surface the correct modeling of the hydrogen bond formation is crucial. Recent experimental findings describe a reentrant condensation of proteins in solutions of varying metal ion concentration. We present a heuristic model to account for the metal ion binding on the molecular surface which qualitatively and quantitatively explains the phase diagram of this condensation effect.In der vorliegenden Arbeit konzentrieren wir uns auf die Beschreibung elektrostatischer Phänomene in biomolekularen Systemen. Zuerst untersuchen wir den Einfluss von Wasserstoffbrückenbindungen auf die dielektrischen Eigenschaften von Wasser. Dafür ist die Einführung eines nichtlokalen dielektrischen Operators notwendig. Die nichtlokale Reaktion des Wassers wird durch das gelöste Protein und der damit entstandenen Kavität maßgeblich beeinflusst.Wir entwickeln ein Differentialgleichungssystem, welches Veränderungen der dielektrischen Eigenschaften an der Moleküloberfläche explizit berücksichtigt. Um diese Randeffekte genauer zu analysieren und um unsere Modellgleichungen auf ionische Lösungen zu erweitern, implementieren wir ein modifiziertes Finite-Differenzen-Verfahren, welches sich, neben Effizienz, durch hohe Genauigkeit auszeichnet. Mit diesem Lösungsverfahren untersuchen wir erstmals verschiedene Wassermodelle. Die Analyse zeigt, dass die Veränderungen der Randbedingung an der Moleküloberfläche auf makroskopische Größen von untergeordneter Bedeutung sind, jedoch einen signifikanten Einfluss auf das elektrostatische Potential in der Nähe des Moleküls hat. Des Weiteren betrachten wir einen kürzlich entdeckten Effekt in Proteinlösungen: die Bindungsaffinität von gelösten Metallionen induziert die Bildung von Protein-Metallionen-Komplexen. Diese können in Abhängigkeit der gelösten Ionenkonzentration kondensieren und wieder in Lösung gehen. In Analogie zu Protonierungsmodellen entwickeln wir eine Theorie zur Beschreibung der Komplexbildung. Erste Vergleiche mit Experimenten zeigen, dass das vorgeschlagene Modell den Kondensationseffekt qualitativ und quantitativ erklären kann

Biomolecules in a structured solvent : a novel formulation of nonlocal electrostatics and its numerical solution

The accurate modeling of the dielectric properties of water is crucial for many applications in physics, computational chemistry, and molecular biology. In principle this becomes possible in the framework of nonlocal electrostatics, but since the complexity of the underlying equations seemed overwhelming, the approach was considered unfeasible for biomolecular purposes. In this work, we propose a novel formulation of nonlocal electrostatics which for the first time allows for numerical solutions for the nontrivial molecular geometries arising in the applications mentioned before. The approach is illustrated by its application to simple geometries, and its usefulness for the computation of solvation free energies is demonstrated for the case of monoatomic ions. In order to extend the applicability of nonlocal electrostatics to nontrivial systems like large biomolecules, a boundary element method for its numerical solution is developed and implemented. The resulting solver is then used to predict the free energies of solvation of polyatomic molecules with high accuracy. Finally, the nonlocal electrostatic potential of the protein trypsin is computed and interpreted qualitatively.Die präzise Modellierung der dielektrischen Eigenschaften des Wassers ist für viele Anwendungen in Physik, Computational Chemistry und Molekularbiologie von entscheidender Bedeutung. Theoretisch ist eine solche Modellierung im Rahmen der sogenannten nichtlokalen Elektrostatik möglich, doch da die dabei auftretenden Gleichungssysteme bislang als beinahe unlösbar schwierig galten, schien dieser Zugang für biomolekulare Problemstellungen ungeeignet. In dieser Arbeit präsentieren wir eine neuartige Formulierung der nichtlokalen Elektrostatik, die zum ersten Mal die Entwicklung numerischer Methoden erlaubt, die auf die nichttrivialen molekularen Geometrien, wie sie in den oben genannten Forschungsgebieten auftreten, anwendbar sind. Wir demonstrieren unseren Zugang zunächst durch die Anwendung auf einfache Modellgeometrien und zeigen seine Nützlichkeit für die Berechnung freier Solvatationsenergien einatomiger Ionen. Um die Anwendbarkeit der nichtlokalen Elektrostatik auf nichttriviale Systeme, wie z.B. große Biomoleküle zu erweitern, wird eine Randelementmethode zur numerischen Lösung der präsentierten Gleichungen entwickelt und implementiert. Der resultierende Randelementl öser wird daraufhin zur genauen Vorhersage der freien Solvatationsenergien kleiner Moleküle verwendet. Schließlich wird das nichtlokale elektrostatische Potential des Proteins Trypsin berechnet und qualitativ interpretiert