N-body problems encompass a variety of fields such as electrostatics, molecularbiology and astrophysics. If there are N particles in the system, the brute force algorithmfor these problems based on particle-particle interaction takes O(N2), whichis clearly expensive for large values of N. There have been some approximation algorithmslike the Barnes-Hut Method and the Fast Multipole Method (FMM) proposedfor these problems to reduce the complexity. However, the applicability of these algorithmsare limited to operators with analytic multipole expansions or restricted tosimulations involving low accuracy. The shortcoming of N-body treecodes are moreevident for particles in motion where the movement of the particles are not consideredwhen evaluating the potential. If the displacement of the particles are small, thenupdating the multipole coefficients for all the nodes in the tree may not be requiredfor computing the potential to a reasonable accuracy. This study focuses on some ofthe limitations of the existing approximation schemes and presents new algorithmsthat can be used for N-body simulations to efficiently compute potentials and forces.In the case of electrostatics, existing algorithms use Cartesian coordinates to evaluatethe potentials of the form r−, where 1. The use of such coordinates toseparate the variables results in cumbersome expressions and does not exploit the inherent spherical symmetry found in these kernels. For such potentials, we providea new multipole expansion series and construct a method which is asymptoticallysuperior than the current treecodes. The advantage of this expansion series is furtherdemonstrated by an algorithm that can compute the forces to the desired accuracy.For particles in motion, we introduce a new method in which we retain the multipolecoefficients when performing multipole updates (to the parent nodes) at every timestep. This results in considerable savings in time while maintaining the accuracy. Wefurther illustrate the efficiency of our algorithms through numerical experiments.

N-body problems encompass a variety of fields such as electrostatics, molecularbiology and astrophysics. If there are N particles in the system, the brute force algorithmfor these problems based on particle-particle interaction takes O(N2), whichis clearly expensive for large values of N. There have been some approximation algorithmslike the Barnes-Hut Method and the Fast Multipole Method (FMM) proposedfor these problems to reduce the complexity. However, the applicability of these algorithmsare limited to operators with analytic multipole expansions or restricted tosimulations involving low accuracy. The shortcoming of N-body treecodes are moreevident for particles in motion where the movement of the particles are not consideredwhen evaluating the potential. If the displacement of the particles are small, thenupdating the multipole coefficients for all the nodes in the tree may not be requiredfor computing the potential to a reasonable accuracy. This study focuses on some ofthe limitations of the existing approximation schemes and presents new algorithmsthat can be used for N-body simulations to efficiently compute potentials and forces.In the case of electrostatics, existing algorithms use Cartesian coordinates to evaluatethe potentials of the form r−, where 1. The use of such coordinates toseparate the variables results in cumbersome expressions and does not exploit the inherent spherical symmetry found in these kernels. For such potentials, we providea new multipole expansion series and construct a method which is asymptoticallysuperior than the current treecodes. The advantage of this expansion series is furtherdemonstrated by an algorithm that can compute the forces to the desired accuracy.For particles in motion, we introduce a new method in which we retain the multipolecoefficients when performing multipole updates (to the parent nodes) at every timestep. This results in considerable savings in time while maintaining the accuracy. Wefurther illustrate the efficiency of our algorithms through numerical experiments.

The implicit-solvent model uses continuum electrostatic theory to represent the salt solution around dissolved biomolecules, leading to a coupled system of the Poisson-Boltzmann and Poisson equations. This thesis uses the implicit-solvent model to study solvation, binding and adsorption of proteins. We developed an implicit-solvent model solver that uses the boundary element method (BEM), called PyGBe. BEM numerically solves integral equations along the biomolecule-solvent interface only, therefore, it does not need to discretize the entire domain. PyGBe accelerates the BEM with a treecode algorithm and runs on graphic processing units. We performed extensive verification and validation of the code, comparing it with experimental observations, analytical solutions, and other numerical tools. Our results suggest that a BEM approach is more appropriate than volumetric based methods, like finite-difference or finite-element, for high accuracy calculations. We also discussed the effect of features like solvent-filled cavities and Stern layers in the implicit-solvent model, and realized that they become relevant in binding energy calculations. The application that drove this work was nano-scale biosensors-- devices designed to detect biomolecules. Biosensors are built with a functionalized layer of ligand molecules, to which the target molecule binds when it is detected. With our code, we performed a study of the orientation of proteins near charged surfaces, and investigated the ideal conditions for ligand molecule adsorption. Using immunoglobulin G as a test case, we found out that low salt concentration in the solvent and high positive surface charge density leads to favorable orientations of the ligand molecule for biosensing applications. We also studied the plasmonic response of localized surface plasmon resonance (LSPR) biosensors. LSPR biosensors monitor the plasmon resonance frequency of metallic nanoparticles, which shifts when a target molecule binds to a ligand molecule. Electrostatics is a valid approximation to the LSPR biosensor optical phenomenon in the long-wavelength limit, and BEM was able to reproduce the shift in the plasmon resonance frequency as proteins approach the nanoparticle.