Weihua Geng, Ph.D. - US grants

Current biomolecular simulations are unable to reach the long time scales needed to study conformation changes such as protein folding. One of the main obstacles is the high cost of computing the electrostatic forces among the solvent water molecules surrounding the protein. To address this issue, this project adopts an implicit solvent model in which the electrostatic potential satisfies the Poisson-Boltzmann (PB) equation. Numerical solution of the PB equation poses a challenge due to the geometric complexity of the molecular surface, the discontinuity in the dielectric function, and the unbounded computational domain. The investigators will overcome these difficulties by developing a boundary integral PB solver using a new Cartesian treecode algorithm for screened Coulomb interactions. The treecode-accelerated PB solver will be tested on benchmark examples such as Kirkwood's solution for a spherical surface, and the results will be compared with those obtained using other PB solvers. In addition to the electrostatic potential, the code will be extended to compute other important quantities such as the solvation free energy and solvation forces needed for dynamics.


One obstacle facing current biomolecular simulations is the expense of computing the self-induced electrostatic forces among the molecules in the system. Advances in computer hardware alone won't achieve the improvements necesssary for studying long time molecular dynamics. This project therefore focuses on improving the mathematical algorithms used in these computer simulations. In addition to enabling more accurate and efficient biomolecular simulations, the algorithms developed will be potentially useful in other applications where electrostatic forces play a role, for example in modeling charge transport in fuel cells. The project will contribute to training the scientific workforce by supporting the research of a postdoc who will be mentored by the PI.

The goal of the proposed project is to develop second order interface methods embedded in the alternating direction implicit (ADI) framework for solving the 3D nonlinear Poisson-Boltzmann (PB) equation with complex dielectric interfaces. Efficiency and accuracy are known to be the two major difficulties for solving the nonlinear PB equation numerically. The efficiency concern stems from the needs for solving the PB equation in demanding applications, such as one-time solution to systems with large spatial degrees of freedom, and/or million-time solutions in dynamical simulations. The accuracy concern is due to various challenging features of the PB model, including piecewisely-defined dielectric constants, a strong nonlinearity, singular point charges, and complex dielectric interfaces. Without addressing these features, fine meshes have to be used for a reliable simulation, which in turn impairs efficiency. In this project, a new pseudo-transient continuation formulation will be constructed based on a suitable regularization setting so that the singular charges are represented analytically. The nonlinear term of the PB equation will be integrated exactly with time splitting techniques. To deal with piecewise dielectric constants, a tensor product decomposition of 3D interface conditions will be carried out to derive essentially 1D jump conditions so that the dielectric interface can be accommodated along each Cartesian direction in an alternating manner. Fast algebraic solvers will be developed for solving matrices of each Cartesian direction. Consequently, the proposed matched ADI approaches not only maintain both the simplicity of Cartesian grids and the efficiency of the Thomas algorithm, but also achieve spatially second order accuracy in resolving complex dielectric interfaces.

The electrostatic interactions are vital not only for the study of biological and chemical systems and processes at the molecular level, but also for the design of semiconductor devices at the nanoscale. The PB model, in which the electrostatic interactions are computed implicitly via a mean force approach, can surprisingly well describe the electrostatics of a charged system. This model finds broad applications in science and engineering, such as modeling the charged polymers and surfactants in interface and colloid science, studying transistors on very large scale integration (VLSI) semiconductor devices in nanotechnology, and analyzing structure, function, and dynamics of solvated biomolecules including proteins and DNAs in molecular biology. The proposed mathematical modeling, algorithm development, and numerical computations will address key scientific challenges in interdisciplinary fields involving computational mathematics, chemistry, biology, and electrical engineering. The planned research activities will bring new advances to computational mathematics and lead to reliable simulation tools for the electrostatic analysis of various physical, chemical, and biological systems/devices. In addition, this project will provide interdisciplinary research and training opportunities for students pursing careers in science and engineering.

Proteins are large biomolecules each consisting of a unique sequence of amino acids with a complex three-dimensional structure. Proteins perform many essential functions in living organisms, and some diseases are associated with improper protein structure. Hence there is great interest among biomedical researchers in understanding the structure, dynamics, and function of proteins. In their natural environment proteins are surrounded by water with dissolved salt; the protein/solvent interactions are critical to proper function in the organism. Laboratory experiments are used to study these protein/solvent interactions, but computer simulations are also increasingly employed to complement the experiments. The investigators will use their expertise in computational mathematics to develop improved numerical algorithms and software for computing protein/solvent interactions, with potential impact on areas such as protein folding and synthetic drug design. Several applications will be studied in collaboration with bioscientists. The software developed will be posted in open source format on a public website and will be installed in a widely distributed molecular simulation software package for use by bio-computational researchers. The project will train a postdoc and a graduate student in this important branch of scientific research.

The project will develop improved numerical algorithms and software for computing electrostatic solvent effects which play a key role in determining protein structure, dynamics, and function. Computing these effects is challenging, and implicit solvent models based on the Poisson-Boltzmann (PB) equation for the electrostatic potential are a popular approach to reducing the cost. However, grid-based PB simulations encounter difficulties due to the singular point charges representing the protein, the complex geometry and discontinuous dielectric constant across the molecular surface, and the unbounded computational domain. In previous NSF-supported research, the investigators developed a new treecode-accelerated boundary integral (TABI) potential solver with improved accuracy and efficiency, low memory usage, and straightforward parallelization. The current project has the following components. 1. (algorithm development) The investigators will extend the current TABI potential solver to compute the electrostatic solvation forces needed for molecular dynamics simulations. This requires careful discretization of singular integrals representing the induced charge on the molecular surface separating the low-dielectric protein domain from the high-dielectric solvent domain. 2. (parallel computing) The investigators will develop a new parallel TABI solver for graphics processing units (GPUs), taking advantage of the treecode's low memory and communication requirements. 3. (biological applications) The investigators will apply the new TABI potential solver and force driver to study solvent effects in proteins. Applications to be studied in collaboration with bioscientists include: (a) pH-dependent properties of a bactericidal lectin protein; (b) structural changes of a neurotransmitter receptor in an ionic environment which is relevant to an autoimmune disease; (c) extension of TABI to incorporate polarizable atomic multipole solutes.

This award supports the 2019 NSF-CBMS conference "Mathematical Molecular Bioscience and Biophysics," hosted by University of Alabama, Tuscaloosa, during May 13-17, 2019. The conference will feature Professor Guowei Wei of Michigan State University as the Principal Lecturer. Mathematical molecular bioscience and biophysics has been emerging as a promising interdisciplinary research area at the interface of mathematics and biology, driven by the trends of contemporary life sciences that transform biosciences from macroscopic to microscopic or molecular, and from qualitative and phenomenological to quantitative and predictive. The conference will promote biological studies for solving cutting edge problems at molecular level, so that mathematics can play a more important role in addressing fundamental challenges in molecular biosciences and biophysics. This conference consists of ten principal lectures, together with supplemental presentations by other experts and round table discussions. The conference aims to attract many junior mathematicians, including undergraduate and graduate students, postdoctoral fellows, and young faculty, to enter this new interdisciplinary field. In addition, the meeting will benefit the hosting university by enhancing its research program and raising its visibility to peer institutes in the southeastern region.

Mathematical Molecular Bioscience and Biophysics (MMBB) concerns the development of mathematical theories, models, methods, schemes, and algorithms for elucidating molecular mechanisms and for solving open problems at the forefront of molecular biosciences and biophysics. The lecture series will provide a thorough overview of the MMBB literature to mathematical and biological societies. Numerous areas of mathematics, including differential equations, functional analysis, harmonic analysis, Lie groups, Lie algebras, geometry, graph theory, topology, combinatorics, multiscale modeling, inverse problems, optimization, machine learning, stochastic analysis, uncertainty quantification, fuzzy logic, statistical inference, and nonparametric regression, have found important applications in MMBB and many successful applications will be illustrated in this conference. Biological open problems at the forefront of the MMBB, such as those associated with drug design and discovery, which will stimulate new research directions in modeling and computation of biomolecular structure, function, dynamics and transport, will be identified. For more information, please refer to the conference webpage: http://cbms.ua.edu

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

Biological molecules are basic components of life and electrostatic forces play a key role in determining their properties. In order to complement physical experiments, computer simulation of these forces is critical to advance the understanding of how biological molecules function. This project will contribute new computational tools for electrostatics of solvated biomolecules. The research will focus on charge transport in ion channel proteins with the goal of assisting in the study of neuron-related autoimmune disease. The software developed will be made available in open source format to the scientific community and will be installed in public software packages. Elements of the project research will be included in a mathematical biology course taught by one of the PIs. The results will be disseminated at scientific conferences and academic seminars, and will be published in scientific journals. The project will contribute to the national scientific workforce by training a postdoctoral scholar, as well as a graduate student and several undergraduate students.

The proposed research develops improved computational tools for electrostatics of solvated biomolecules. The project has two components, (1) improving existing computational tools for the Poisson-Boltzmann (PB) and 3D-RISM (Reference Interaction Site Model) implicit solvent models, and (2) developing a boundary element method for the Poisson-Nernst-Planck (PNP) model. In the first component the PIs will upgrade their previous treecode-accelerated boundary integral (TABI) PB solver with more efficient techniques including node patch discretization, NanoShaper molecular surface triangulation, a new preconditioning strategy, and the GPU-accelerated barycentric Lagrange dual tree traversal (BLDTT) fast multipole method. The improved TABI-PB solver will be applied to accelerate computation of the electrostatic free energy of solvated viruses and the long-range asymptotic correlation functions in 3D-RISM. In the second component, the PIs will develop a novel integral equation based method for the PNP model applied to solvated ion channel proteins embedded in a membrane. The method will combine a boundary/volume element approach for the electrostatic potential with a particle method for the drift-diffusion of dissolved ions. The goal is to apply the new computational PNP tool to study the Acetylcholine receptor (AChR), an ion channel protein that plays a significant role in neuron-related autoimmune diseases such as myasthenia gravis and possibly also the Covid-19 coronavirus.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

This collaborative research project aims to improve the implicit solvent modeling for studying electrostatic interaction between solutes, such as proteins, DNA, and RNA, and their surrounding solvent environment. The research will improve on current approaches and will formulate a new polarizable multipole implicit solvent model with improved and enhanced modeling accuracy. Furthermore, efficient and accurate numerical algorithms will be developed to meet computational challenges of the new model. This research will provide biophysicists a new tool for analyzing electrostatic interactions of solvated biomolecules in the form of models and algorithms implemented in a freely available software package. In addition, this project will offer interdisciplinary research and training opportunities for undergraduate and graduate students in biological modeling, computation, and mathematical analysis.

The project will address limitations in the existing implicit solvent models for studying electrostatic interaction between solutes. These include the facts that the solute charge sources are often modeled as point charges located at atomic centers, and this rough approximation to the quantum mechanical charge density is known to be a major source of the modeling errors. Moreover, polarization, an important physical phenomenon account for the redistribution of the electron density in the presence of an external electric field is missing in this point charge model. The project will develop a novel nonlinear Poisson-Boltzmann (PB) model associated with an atomic polarizable multipole (PM) force field to describe the self-consistent polarization process and study electrostatic interactions among permanent multipoles, induced dipoles, and reaction-field potential. The coupling of PM source with the nonlinear PB (NPB) equation, as opposed to the linearized PB, is challenging in many aspects involving modeling and numerical difficulties such as charge singularities, geometric complexity, interface jumps, nonlinearity, polarization, as well as high computational cost. To overcome such difficulties, a set of efficient, accurate, and seamlessly coupled numerical methods will be developed to resolve numerical challenges associated with the PM-NPB model. In particular, multipole charge singularities are regularized using Green’s function based decomposition; self-consistent polarization, which involves repeatedly solving an NPB equation across the molecular interface, is efficiently realized by a linearized iterative algorithm coupled with a fast 3D Augmented Matched Interface and Boundary (AMIB) method. Finally, model benchmarking and biological applications will be carried out to ensure that the research results provide a robust tool for simulating electrostatic interactions.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.