Applications of integral equation theory to biological systems

Das „three-dimensional reference interaction site model“ (3D RISM) erlaubt es die Solvensverteilung, und somit die damit verbundenen thermodynamischen Eigenschaften, um ein gegebenes Solvat zu berechnen. Dies kann ein kleines, wirkstoffartiges Molekül sein oder ein Protein mit tausenden Atomen. Zusammen mit Methoden, wie Molekulardynamik- (MD) Simulationen und Kraftfeldern, ist es möglich, die Unterschiede in der freien Energie zwischen Konformeren, Molekülen und Komplexen in biologisch relevanten Systemen zu bestimmen. In dieser Arbeit werden durch Kombination von 3D RISM und MD Simulationen freie Energiedifferenzen zwischen zwei Konformeren eines Antikörpers berechnet und durch Tests mit verschiedenen Wassermodellen und Fehlerkorrekturen validiert. Allerdings entstehen durch starke strukturelle Fluktuationen während der Simulationen häufig große statistische Fehler, was die Anwendungsfelder solcher Methoden limitiert. Um das Problem abzuschwächen und um auf explizite Simulationen verzichten zu können, werden sogenannte „Localized Free Energies” (LFE) verwendet. Mit ihnen ist es möglich, die freie Energie auf ein atomweises Niveau herunter zu brechen, wo angenommen werden kann, dass besagte Fluktuationen einen geringeren Einfluss haben. Da eine solche Partitionierung rein virtuell ist, gibt es keinen experimentellen Weg, die LFEs zu validieren. Aus diesem Grund wird ihre Plausibilität durch Anwendung als Eingabeinformation für Methoden des maschinellen Lernens (ML) überprüft, indem der Verlust ihrer Vorhersagekraft durch ansteigende Störung der LFEs beobachtet wird. Mit bestätigter Plausibilität werden die LFEs beispielhaft auf eine Serie von Thrombin-Inhibitoren angwendet, um ihr Potential in der Medikamentenentwicklung zu zeigen. Darüberhinaus wird der Einfluss von experimentellen Unsicherheiten in den Kristallstrukturen sowie die Limitationen des Ansatzes selbst untersucht. Von der gleichen formalen Basis, wie sie auch bei den LFEs genutzt wurde, lassen sich auch die so genannten „Free Energy Derivatives” (FED) sehr effizient bestimmen. Diese beschreiben auf atomarer Ebene, wie sich die freie Energie in Abhängigkeit von Kraftfeldparametern verändert. Die LFEs werden ebenfalls anhand eines Thrombin Komplexes näher beleuchtet und ihr prädiktiver Einsatz wird anhand eines auf Literaturdaten basierenden in-silico Experiments demonstriert. The three-dimensional reference interaction site model (3D RISM) allows to compute the solvent distribution, and therefore the associated thermodynamic properties, around a given solute. This can be a small, drug-like molecule or a protein with several thousand atoms. Combined with other tools like molecular dynamics (MD) simulations and force fields, it is possible to study the differences in free energy of conformations, molecules, and complexes in biological relevant systems. By combining 3D RISM with MD simulations, the free energy difference between two structural conformers of an antibody is calculated, and the results are verified by tests with different water models and error corrections. However, due to strong structural fluctuation during the simulations, the statistical errors are often high, which limits the field of applications of such studies. To alleviate this problem and to be able to do without explicit simulations, so so-called localized free energies (LFE) are employed. With them it is possible to break down free energies to an atom-wise level, where said fluctuations can be assumed to have less influence on the results. Since such a partitioning is purely virtual, there is no experimental way to validate the LFEs. For this reason, their plausibility is checked by using them as input for machine learning (ML) models, analyzing the drop in predictive power upon increasing levels of perturbation in the LFE input. With the plausibility of the method established, the LFEs are applied to an exemplary series of thrombin inhibitors to illustrate their potential in a drug discovery context. Here they are used to identify the most relevant interactions between host and guest. Furthermore, the influence of experimental uncertainties in crystal structures and the limitations of the approach get explored. Coming from the same formal basis as it was used for the LFEs, it is possible to calculate so-called free energy derivatives (FED) very efficiently. They describe how the free energy changes with respect to the non-bonded force field parameters on an atomistic level. The FEDs are also applied to thrombin complex, exploring the capabilities of the approach and investigating the predictive applicability of the FEDs by performing an in-silico experiment on literature data

Development and Application of Numerical Methods in Biomolecular Solvation

This work addresses the development of fast summation methods for long range particle interactions and their application to problems in biomolecular solvation, which describes the interaction of proteins or other biomolecules with their solvent environment. At the core of this work are treecodes, tree-based fast summation methods which, for N particles, reduce the cost of computing particle interactions from O(N^2) to O(N log N). Background on fast summation methods and treecodes in particular, as well as several treecode improvements developed in the early stages of this work, are presented. Building on treecodes, dual tree traversal (DTT) methods are another class of tree-based fast summation methods which reduce the cost of computing particle interactions for N particles to O(N). The primary result of this work is the development of an O(N) dual tree traversal fast summation method based on barycentric Lagrange polynomial interpolation (BLDTT). This method is implemented to run across multiple GPU compute nodes in the software package BaryTree. Across different problem sizes, particle distributions, geometries, and interaction kernels, the BLDTT shows consistently better performance than the previously developed barycentric Lagrange treecode (BLTC). The first major biomolecular solvation application of fast summation methods presented is to the Poisson–Boltzmann implicit solvent model, and in particular, the treecode-accelerated boundary integral Poisson–Boltzmann solver (TABI-PB). The work on TABI-PB consists of three primary projects and an application. The first project investigates the impact of various biomolecular surface meshing codes on TABI-PB, and integrated the NanoShaper software into the package, resulting in significantly better performance. Second, a node patch method for discretizing the system of integral equations is introduced to replace the previous centroid collocation scheme, resulting in faster convergence of solvation energies. Third, a new version of TABI-PB with GPU acceleration based on the BLDTT is developed, resulting in even more scalability. An application investigating the binding of biomolecular complexes is undertaken using the previous Taylor treecode-based version of TABI-PB. In addition to these projects, work performed over the course of this thesis integrated TABI-PB into the popular Adaptive Poisson–Boltzmann Solver (APBS) developed at Pacific Northwest National Laboratory. The second major application of fast summation methods is to the 3D reference interaction site model (3D-RISM), a statistical-mechanics based continuum solvation model. This work applies cluster-particle Taylor expansion treecodes to treat long-range asymptotic Coulomb-like potentials in 3D-RISM, and results in significant speedups and improved scalability to the 3D-RISM package implemented in AmberTools. Additionally, preliminary work on specialized GPU-accelerated treecodes based on BaryTree for 3D-RISM long-range asymptotic functions is presented.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/168120/1/lwwilson_1.pd