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88
Electrostatics of Nanosystems: Application to microtubules and the ribosome
 Proc. Natl. Acad. Sci. USA
, 2001
"... Evaluation of the electrostatic properties of biomolecules has become a standard practice in molecular biophysics. Foremost among the models used to elucidate the electrostatic potential is the PoissonBoltzmann equation, however, existing methods for solving this equation have limited the scope ..."
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Cited by 468 (23 self)
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Evaluation of the electrostatic properties of biomolecules has become a standard practice in molecular biophysics. Foremost among the models used to elucidate the electrostatic potential is the PoissonBoltzmann equation, however, existing methods for solving this equation have limited the scope of accurate electrostatic calculations to relatively small biomolecular systems.
Adaptive numerical treatment of elliptic systems on manifolds
 Advances in Computational Mathematics, 15(1):139
, 2001
"... ABSTRACT. Adaptive multilevel finite element methods are developed and analyzed for certain elliptic systems arising in geometric analysis and general relativity. This class of nonlinear elliptic systems of tensor equations on manifolds is first reviewed, and then adaptive multilevel finite element ..."
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Cited by 57 (26 self)
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ABSTRACT. Adaptive multilevel finite element methods are developed and analyzed for certain elliptic systems arising in geometric analysis and general relativity. This class of nonlinear elliptic systems of tensor equations on manifolds is first reviewed, and then adaptive multilevel finite element methods for approximating solutions to this class of problems are considered in some detail. Two a posteriori error indicators are derived, based on local residuals and on global linearized adjoint or dual problems. The design of Manifold Code (MC) is then discussed; MC is an adaptive multilevel finite element software package for 2 and 3manifolds developed over several years at Caltech and UC San Diego. It employs a posteriori error estimation, adaptive simplex subdivision, unstructured algebraic multilevel methods, global inexact Newton methods, and numerical continuation methods for the numerical solution of nonlinear covariant elliptic systems on 2 and 3manifolds. Some of the more interesting features of MC are described in detail, including some new ideas for topology and geometry representation in simplex meshes, and an unusual partition of unitybased method for exploiting parallel computers. A short example is then given which involves the Hamiltonian and momentum constraints in the Einstein equations, a representative nonlinear 4component covariant elliptic system on a Riemannian 3manifold which arises in general relativity. A number of operator properties and solvability results recently established are first summarized, making possible two quasioptimal a priori error estimates for Galerkin approximations which are then derived. These two results complete the theoretical framework for effective use of adaptive multilevel finite element methods. A sample calculation using the MC software is then presented.
A New Paradigm for Parallel Adaptive Meshing Algorithms
 SIAM J. Sci. Comput
, 2003
"... We present a new approach to the use of parallel computers with adaptive finite element methods. This approach addresses the load balancing problem in a new way, requiring far less communication than current approaches. It also allows existing sequential adaptive PDE codes such as PLTMG and MC to ru ..."
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Cited by 46 (9 self)
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We present a new approach to the use of parallel computers with adaptive finite element methods. This approach addresses the load balancing problem in a new way, requiring far less communication than current approaches. It also allows existing sequential adaptive PDE codes such as PLTMG and MC to run in a parallel environment without a large investment in recoding. In this new approach, the load balancing problem is reduced to the numerical solution of a small elliptic problem on a single processor, using a sequential adaptive solver, without requiring any modifications to the sequential solver. The small elliptic problem is used to produce a posteriori error estimates to predict future element densities in the mesh, which are then used in a weighted recursive spectral bisection of the initial mesh. The bulk of the calculation then takes place independently on each processor, with no communication, using possibly the same sequential adaptive solver. Each processor adapts its region of the mesh independently, and a nearly loadbalanced mesh distribution is usually obtained as a result of the initial weighted spectral bisection. Only the initial fanout of the mesh decomposition to the processors requires communication. Two additional steps requiring boundary exchange communication may be employed after the individual processors reach an adapted solution, namely, the construction of a global conforming mesh from the independent subproblems, followed by a final smoothing phase using the subdomain solutions as an initial guess. We present a series of convincing numerical experiments that illustrate the e#ectiveness of this approach. The justification of the initial refinement prediction step, as well as the justification of skipping the two communicationintensive steps, ...
The finite element approximation of the nonlinear poissonboltzmann equation
 SIAM Journal on Numerical Analysis
"... ABSTRACT. A widely used electrostatics model in the biomolecular modeling community, the nonlinear Poisson–Boltzmann equation, along with its finite element approximation, are analyzed in this paper. A regularized Poisson–Boltzmann equation is introduced as an auxiliary problem, making it possible t ..."
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Cited by 33 (13 self)
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ABSTRACT. A widely used electrostatics model in the biomolecular modeling community, the nonlinear Poisson–Boltzmann equation, along with its finite element approximation, are analyzed in this paper. A regularized Poisson–Boltzmann equation is introduced as an auxiliary problem, making it possible to study the original nonlinear equation with delta distribution sources. A priori error estimates for the finite element approximation are obtained for the regularized Poisson–Boltzmann equation based on certain quasiuniform grids in two and three dimensions. Adaptive finite element approximation through local refinement driven by an a posteriori error estimate is shown to converge. The Poisson–Boltzmann equation does not appear to have been previously studied in detail theoretically, and it is hoped that this paper will help provide molecular modelers with a better foundation for their analytical and computational work with the Poisson–Boltzmann equation. Note that this article apparently gives the first rigorous convergence result for a numerical discretization technique for the nonlinear Poisson– Boltzmann equation with delta distribution sources, and it also introduces the first provably convergent adaptive method for the equation. This last result is currently one of only a handful of existing convergence results of this type for nonlinear problems.
Quality meshing of implicit solvation models of biomolecular structures
 COMPUT. AIDED GEOM. DES
, 2006
"... This paper describes a comprehensive approach to construct quality meshes for implicit solvation models of biomolecular structures starting from atomic resolution data in the Protein Data Bank (PDB). First, a smooth volumetric electron density map is constructed from atomic data using weighted Gauss ..."
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Cited by 33 (8 self)
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This paper describes a comprehensive approach to construct quality meshes for implicit solvation models of biomolecular structures starting from atomic resolution data in the Protein Data Bank (PDB). First, a smooth volumetric electron density map is constructed from atomic data using weighted Gaussian isotropic kernel functions and a twolevel clustering technique. This enables the selection of a smooth implicit solvation surface approximation to the LeeRichards molecular surface. Next, a modified dual contouring method is used to extract triangular meshes for the surface, and tetrahedral meshes for the volume inside or outside the molecule within a bounding sphere/box of influence. Finally, geometric flow techniques are used to improve the surface and volume mesh quality. Several examples are presented, including generated meshes for biomolecules that have been successfully used in finite element simulations involving solvation energetics and rate binding constants.
Finite element solution of the steadystate Smoluchowski equation for rate constant calculations
 Biophysical Journal
, 2004
"... ABSTRACT This article describes the development and implementation of algorithms to study diffusion in biomolecular systems using continuum mechanics equations. Specifically, finite element methods have been developed to solve the steadystate Smoluchowski equation to calculate ligand binding rate co ..."
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Cited by 26 (14 self)
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ABSTRACT This article describes the development and implementation of algorithms to study diffusion in biomolecular systems using continuum mechanics equations. Specifically, finite element methods have been developed to solve the steadystate Smoluchowski equation to calculate ligand binding rate constants for large biomolecules. The resulting software has been validated and applied to mouse acetylcholinesterase. Rates for inhibitor binding to mAChE were calculated at various ionic strengths with several different reaction criteria. The calculated rates were compared with experimental data and show very good agreement when the correct reaction criterion is used. Additionally, these finite element methods require significantly less computational resources than existing particlebased Brownian dynamics methods.
Probabilistic interpretation and random walk on spheres algorithms for the PoissonBoltzmann equation in molecular dynamics
, 2010
"... Abstract. Motivated by the development of efficient Monte Carlo methods for PDE models in molecular dynamics, we establish a new probabilistic interpretation of a family of divergence form operators with discontinuous coefficients at the interface of two open subsets of R d. This family of operators ..."
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Cited by 19 (9 self)
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Abstract. Motivated by the development of efficient Monte Carlo methods for PDE models in molecular dynamics, we establish a new probabilistic interpretation of a family of divergence form operators with discontinuous coefficients at the interface of two open subsets of R d. This family of operators includes the case of the linearized PoissonBoltzmann equation used to compute the electrostatic free energy of a molecule. More precisely, we explicitly construct a Markov process whose infinitesimal generator belongs to this family, as the solution of a SDE including a non standard local time term related to the interface of discontinuity. We then prove an extended FeynmanKac formula for the PoissonBoltzmann equation. This formula allows us to justify various probabilistic numerical methods to approximate the free energy of a molecule. We analyse the convergence rate of these simulation procedures and numerically compare them on idealized molecules models. Résumé. Motivés par le développement de méthodes de MonteCarlo efficaces pour des équations aux dérivées partielles en dynamique moléculaire, nous établissons une nouvelle interprétation probabiliste d’une famille d’opérateurs sous forme divergence et à coefficients discontinus le long de l’interface de deux ouverts de R d. Cette famille d’opérateurs inclut le cas de l’équation de PoissonBoltzmann
ADAPTIVE FINITE ELEMENT MODELING TECHNIQUES FOR THE POISSONBOLTZMANN EQUATION
"... ABSTRACT. We consider the design of an effective and reliable adaptive finite element method (AFEM) for the nonlinear PoissonBoltzmann equation (PBE). We first examine the twoterm regularization technique for the continuous problem recently proposed by Chen, Holst, and Xu based on the removal of t ..."
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Cited by 15 (9 self)
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ABSTRACT. We consider the design of an effective and reliable adaptive finite element method (AFEM) for the nonlinear PoissonBoltzmann equation (PBE). We first examine the twoterm regularization technique for the continuous problem recently proposed by Chen, Holst, and Xu based on the removal of the singular electrostatic potential inside biomolecules; this technique made possible the development of the first complete solution and approximation theory for the PoissonBoltzmann equation, the first provably convergent discretization, and also allowed for the development of a provably convergent AFEM. However, in practical implementation, this twoterm regularization exhibits numerical instability. Therefore, we examine a variation of this regularization technique which can be shown to be less susceptible to such instability. We establish a priori estimates and other basic results for the continuous regularized problem, as well as for Galerkin finite element approximations. We show that the new approach produces regularized continuous and discrete problems with the same mathematical advantages of the original regularization. We then design an AFEM scheme for the new regularized problem, and show that the resulting AFEM scheme is accurate and reliable, by proving a contraction result for the error. This result, which is one of the first results of this type for nonlinear elliptic problems, is based on using continuous and discrete a priori L ∞ estimates to establish quasiorthogonality. To provide a highquality geometric model as input to the AFEM algorithm, we also describe a class of featurepreserving adaptive mesh generation algorithms designed specifically for constructing meshes of biomolecular structures, based on the intrinsic local structure tensor of the molecular surface. All of the algorithms described in the article are implemented in the Finite Element Toolkit (FETK), developed and maintained at UCSD. The stability advantages of the new regularization scheme are demonstrated with FETK through comparisons with the original regularization approach for a model problem. The convergence and accuracy of the overall AFEM algorithm is also illustrated by numerical approximation of electrostatic solvation energy for an insulin protein.
FFTSVD: A fast multiscale boundaryelement method solver suitable for biomems and biomolecule simulation
 IEEE Transactions on ComputerAided Design of Integrated Circuits and Systems
, 2006
"... Abstract—This paper presents a fast boundaryelement method (BEM) algorithm that is well suited for solving electrostatics problems that arise in traditional and biomicroelectromechanical systems (bioMEMS) design. The algorithm, FFTSVD, is Green’sfunctionindependent for lowfrequency kernels an ..."
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Cited by 14 (5 self)
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Abstract—This paper presents a fast boundaryelement method (BEM) algorithm that is well suited for solving electrostatics problems that arise in traditional and biomicroelectromechanical systems (bioMEMS) design. The algorithm, FFTSVD, is Green’sfunctionindependent for lowfrequency kernels and efficient for inhomogeneous problems. FFTSVD is a multiscale algorithm that decomposes the problem domain using an octree and uses sampling to calculate lowrank approximations to dominant source distributions and responses. Longrange interactions at each length scale are computed using the FFT. Computational results illustrate that the FFTSVD algorithm performs better than precorrectedFFT (pFFT)style algorithms or the multipolestyle algorithms in FastCap. Index Terms—BioMEMS, biomolecule, boundary element, electrostatic, fast solver, FFTSVD. I.
Accurate evaluation of electrostatics for macromolecules in solution
 Methods Appl. Anal
"... Abstract. Most biochemical processes involve macromolecules in solution. The corresponding electrostatics is of central importance for understanding their structures and functions. An accurate and efficient numerical scheme is introduced to evaluate the corresponding electrostatic potential and forc ..."
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Cited by 12 (0 self)
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Abstract. Most biochemical processes involve macromolecules in solution. The corresponding electrostatics is of central importance for understanding their structures and functions. An accurate and efficient numerical scheme is introduced to evaluate the corresponding electrostatic potential and force by solving the governing PoissonBoltzmann equation. This paper focuses on the following issues: (i) the point charge singularity problem, (ii) the dielectric discontinuity problem across a molecular surface, and (iii) the infinite domain problem. Green’s function associated with the point charges plus a harmonic function is introduced as the zeroth order approximation to the solution to solve the point charge singularity problem. A jump condition capturing finite difference scheme is adopted to solve the discontinuity problem across molecule surfaces, where a bodyfitting grid is used. The infinite domain problem is solved by mapping the outer infinite domain into a finite domain. The corresponding stiffness matrix is symmetric and positive definite, therefore, fast algorithm such as preconditioned conjugate gradient method can be applied for inner iteration. Finally, the resulting scheme is second order accurate for both the potential and its gradient. 1. Introduction. Most