Jaydeep P. Bardhan

Simulated x-ray scattering of protein solutions using explicit-solvent models

X-ray solution scattering shows new promise for the study of protein structures, complementing crystallography and nuclear magnetic resonance. In order to realize the full potential of solution scattering, it is necessary to not only improve experimental techniques but also develop accurate and efficient computational schemes to relate atomistic models to measurements. Previous computational methods, based on continuum models of water, have been unable to calculate scattering patterns accurately, especially in the wide-angle regime which contains most of the information on the secondary, tertiary, and quaternary structures. Here we present a novel formulation based on the atomistic description of water, in which scattering patterns are calculated from atomic coordinates of protein and water. Without any empirical adjustments, this method produces scattering patterns of unprecedented accuracy in the length scale between 5 and 100 A, as we demonstrate by comparing simulated and observed scattering patterns for myoglobin and lysozyme.

Biomolecular electrostatics using a fast multipole BEM on up to 512 gpus and a billion unknowns

Abstract We present teraflop-scale calculations of biomolecular electrostatics enabled by the combination of algorithmic and hardware acceleration. The algorithmic acceleration is achieved with the fast multipole method ( fmm ) in conjunction with a boundary element method ( bem ) formulation of the continuum electrostatic model, as well as the bibee approximation to bem . The hardware acceleration is achieved through graphics processors, gpu s. We demonstrate the power of our algorithms and software for the calculation of the electrostatic interactions between biological molecules in solution. The applications demonstrated include the electrostatics of protein–drug binding and several multi-million atom systems consisting of hundreds to thousands of copies of lysozyme molecules. The parallel scalability of the software was studied in a cluster at the Nagasaki Advanced Computing Center, using 128 nodes, each with 4 gpu s. Delicate tuning has resulted in strong scaling with parallel efficiency of 0.8 for 256 and 0.5 for 512 gpu s. The largest application run, with over 20 million atoms and one billion unknowns, required only one minute on 512 gpu s. We are currently adapting our bem software to solve the linearized Poisson–Boltzmann equation for dilute ionic solutions, and it is also designed to be flexible enough to be extended for a variety of integral equation problems, ranging from Poisson problems to Helmholtz problems in electromagnetics and acoustics to high Reynolds number flow.

Accurate Solution of Multi-Region Continuum Biomolecule Electrostatic Problems Using the Linearized Poisson-Boltzmann Equation with Curved Boundary Elements

We present a boundary‐element method (BEM) implementation for accurately solving problems in biomolecular electrostatics using the linearized Poisson–Boltzmann equation. Motivating this implementation is the desire to create a solver capable of precisely describing the geometries and topologies prevalent in continuum models of biological molecules. This implementation is enabled by the synthesis of four technologies developed or implemented specifically for this work. First, molecular and accessible surfaces used to describe dielectric and ion‐exclusion boundaries were discretized with curved boundary elements that faithfully reproduce molecular geometries. Second, we avoided explicitly forming the dense BEM matrices and instead solved the linear systems with a preconditioned iterative method (GMRES), using a matrix compression algorithm (FFTSVD) to accelerate matrix–vector multiplication. Third, robust numerical integration methods were employed to accurately evaluate singular and near‐singular integrals over the curved boundary elements. Fourth, we present a general boundary‐integral approach capable of modeling an arbitrary number of embedded homogeneous dielectric regions with differing dielectric constants, possible salt treatment, and point charges. A comparison of the presented BEM implementation and standard finite‐difference techniques demonstrates that for certain classes of electrostatic calculations, such as determining absolute electrostatic solvation and rigid‐binding free energies, the improved convergence properties of the BEM approach can have a significant impact on computed energetics. We also demonstrate that the improved accuracy offered by the curved‐element BEM is important when more sophisticated techniques, such as nonrigid‐binding models, are used to compute the relative electrostatic effects of molecular modifications. In addition, we show that electrostatic calculations requiring multiple solves using the same molecular geometry, such as charge optimization or component analysis, can be computed to high accuracy using the presented BEM approach, in compute times comparable to traditional finite‐difference methods.

SoftWAXS: a computational tool for modeling wide-angle X-ray solution scattering from biomolecules

This paper describes a computational approach to estimating wide-angle X-ray solution scattering (WAXS) from proteins, which has been implemented in a computer program called SoftWAXS. The accuracy and efficiency of SoftWAXS are analyzed for analytically solvable model problems as well as for proteins. Key features of the approach include a numerical procedure for performing the required spherical averaging and explicit representation of the solute-solvent boundary and the surface of the hydration layer. These features allow the Fourier transform of the excluded volume and hydration layer to be computed directly and with high accuracy. This approach will allow future investigation of different treatments of the electron density in the hydration shell. Numerical results illustrate the differences between this approach to modeling the excluded volume and a widely used model that treats the excluded-volume function as a sum of Gaussians representing the individual atomic excluded volumes. Comparison of the results obtained here with those from explicit-solvent molecular dynamics clarifies shortcomings inherent to the representation of solvent as a time-averaged electron-density profile. In addition, an assessment is made of how the calculated scattering patterns depend on input parameters such as the solute-atom radii, the width of the hydration shell and the hydration-layer contrast. These results suggest that obtaining predictive calculations of high-resolution WAXS patterns may require sophisticated treatments of solvent.

FFTSVD: A Fast Multiscale Boundary-Element Method Solver Suitable for Bio-MEMS and Biomolecule Simulation

This paper presents a fast boundary-element method (BEM) algorithm that is well suited for solving electrostatics problems that arise in traditional and bio-microelectromechanical systems (bio-MEMS) design. The algorithm, FFTSVD, is Greens-function-independent for low-frequency kernels and efficient for inhomogeneous problems. FFTSVD is a multiscale algorithm that decomposes the problem domain using an octree and uses sampling to calculate low-rank approximations to dominant source distributions and responses. Long-range interactions at each length scale are computed using the FFT. Computational results illustrate that the FFTSVD algorithm performs better than precorrected-FFT (pFFT)-style algorithms or the multipole-style algorithms in FastCap.

Fast methods for simulation of biomolecule electrostatics

Computer simulation is an important tool for improving our understanding of biomolecule electrostatics, in part to aid in drug design. However, the numerical techniques used in these simulation tools do not exploit fast solver approaches widely used in analyzing integrated circuit interconnects. In this paper we describe one popular formulation used to analyze biomolecule electrostatics, present an integral formulation of the problem, and apply the precorrected-FFT method to accelerate the solution of the integral equations.

Biomolecular electrostatics—I want your solvation (model)*

We review the mathematical and computational foundations for implicit-solvent models in theoretical chemistry and molecular biophysics. These models are valuable theoretical tools for studying the influence of a solvent, often water or an aqueous electrolyte, on a molecular solute such as a protein. Detailed chemical and physical aspects of implicit-solvent models have been addressed in numerous exhaustive reviews, as have numerical algorithms for simulating the most popular models. This work highlights several important conceptual developments, focusing on selected works that spotlight the need for research at the intersections between chemical, biological, mathematical, and computational physics. To introduce the field to computational scientists, we begin by describing the basic theoretical ideas of implicit-solvent models and numerical implementations. We then address practical and philosophical challenges in parameterization, and major advances that speed up calculations (covering continuum theories based on Poisson as well as faster approximate theories such as generalized Born). We briefly describe the main shortcomings of existing models, and survey promising developments that deliver improved realism in a computationally tractable way, i.e. without increasing simulation time significantly. The review concludes with a discussion of ongoing modeling challenges and relevant trends in high-performance computing and computational science.

Streaming current and wall dissolution over 48 h in silica nanochannels

We present theoretical and experimental studies of the streaming current induced by a pressure-driven flow in long, straight, electrolyte-filled nanochannels. The theoretical work builds on our recent one-dimensional model of electro-osmotic and capillary flow, which self-consistently treats both the ion concentration profiles, via the nonlinear Poisson-Boltzmann equation, and the chemical reactions in the bulk electrolyte and at the solid-liquid interface. We extend this model to two dimensions and validate it against experimental data for electro-osmosis and pressure-driven flows, using eight 1-μm-wide nanochannels of heights varying from 40 nm to 2000 nm. We furthermore vary the electrolyte composition using KCl and borate salts, and the wall coating using 3-cyanopropyldimethylchlorosilane. We find good agreement between prediction and experiment using literature values for all parameters of the model, i.e., chemical reaction constants and Stern-layer capacitances. Finally, by combining model predictions with measurements over 48 h of the streaming currents, we develop a method to estimate the dissolution rate of the silica walls, typically around 0.01 mg/m(2)/h, equal to 45 pm/h or 40 nm/yr, under controlled experimental conditions.

Numerical solution of boundary-integral equations for molecular electrostatics.

Numerous molecular processes, such as ion permeation through channel proteins, are governed by relatively small changes in energetics. As a result, theoretical investigations of these processes require accurate numerical methods. In the present paper, we evaluate the accuracy of two approaches to simulating boundary-integral equations for continuum models of the electrostatics of solvation. The analysis emphasizes boundary-element method simulations of the integral-equation formulation known as the apparent-surface-charge (ASC) method or polarizable-continuum model (PCM). In many numerical implementations of the ASC/PCM model, one forces the integral equation to be satisfied exactly at a set of discrete points on the boundary. We demonstrate in this paper that this approach to discretization, known as point collocation, is significantly less accurate than an alternative approach known as qualocation. Furthermore, the qualocation method offers this improvement in accuracy without increasing simulation time. Numerical examples demonstrate that electrostatic part of the solvation free energy, when calculated using the collocation and qualocation methods, can differ significantly; for a polypeptide, the answers can differ by as much as 10 kcal/mol (approximately 4% of the total electrostatic contribution to solvation). The applicability of the qualocation discretization to other integral-equation formulations is also discussed, and two equivalences between integral-equation methods are derived.

Efficiently accounting for ion correlations in electrokinetic nanofluidic devices using density functional theory

The electrokinetic behavior of nanofluidic devices is dominated by the electrical double layers at the device walls. Therefore, accurate, predictive models of double layers are essential for device design and optimization. In this paper, we demonstrate that density functional theory (DFT) of electrolytes is an accurate and computationally efficient method for computing finite ion size effects and the resulting ion-ion correlations that are neglected in classical double layer theories such as Poisson-Boltzmann. Because DFT is derived from liquid-theory thermodynamic principles, it is ideal for nanofluidic systems with small spatial dimensions, high surface charge densities, high ion concentrations, and/or large ions. Ion-ion correlations are expected to be important in these regimes, leading to nonlinear phenomena such as charge inversion, wherein more counterions adsorb at the wall than is necessary to neutralize its surface charge, leading to a second layer of co-ions. We show that DFT, unlike other theories that do not include ion-ion correlations, can predict charge inversion and other nonlinear phenomena that lead to qualitatively different current densities and ion velocities for both pressure-driven and electro-osmotic flows. We therefore propose that DFT can be a valuable modeling and design tool for nanofluidic devices as they become smaller and more highly charged.