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Qi R, Luo R. Robustness and Efficiency of Poisson-Boltzmann Modeling on Graphics Processing Units. J Chem Inf Model 2018; 59:409-420. [PMID: 30550277 DOI: 10.1021/acs.jcim.8b00761] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/15/2022]
Abstract
Poisson-Boltzmann equation (PBE) based continuum electrostatics models have been widely used in modeling electrostatic interactions in biochemical processes, particularly in estimating protein-ligand binding affinities. Fast convergence of PBE solvers is crucial in binding affinity computations as numerous snapshots need to be processed. Efforts have been reported to develop PBE solvers on graphics processing units (GPUs) for efficient modeling of biomolecules, though only relatively simple successive over-relaxation and conjugate gradient methods were implemented. However, neither convergence nor scaling properties of the two methods are optimal for large biomolecules. On the other hand, geometric multigrid (MG) has been shown to be an optimal solver on CPUs, though no MG have been reported for biomolecular applications on GPUs. This is not a surprise as it is a more complex method and depends on simpler but limited iterative methods such as Gauss-Seidel in its core relaxation procedure. The robustness and efficiency of MG on GPUs are also unclear. Here we present an implementation and a thorough analysis of MG on GPUs. Our analysis shows that robustness is a more pronounced issue than efficiency for both MG and other tested solvers when the single precision is used for complex biomolecules. We further show how to balance robustness and efficiency utilizing MG's overall efficiency and conjugate gradient's robustness, pointing to a hybrid GPU solver with a good balance of efficiency and accuracy. The new PBE solver will significantly improve the computational throughput for a range of biomolecular applications on the GPU platforms.
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Jadhao V, Solis FJ, de la Cruz MO. Free-energy functionals of the electrostatic potential for Poisson-Boltzmann theory. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:022305. [PMID: 24032831 DOI: 10.1103/physreve.88.022305] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/26/2013] [Indexed: 06/02/2023]
Abstract
In simulating charged systems, it is often useful to treat some ionic components of the system at the mean-field level and solve the Poisson-Boltzmann (PB) equation to get their respective density profiles. The numerically intensive task of solving the PB equation at each step of the simulation can be bypassed using variational methods that treat the electrostatic potential as a dynamic variable. But such approaches require the access to a true free-energy functional: a functional that not only provides the correct solution of the PB equation upon extremization, but also evaluates to the true free energy of the system at its minimum. Moreover, the numerical efficiency of such procedures is further enhanced if the free-energy functional is local and is expressed in terms of the electrostatic potential. Existing PB functionals of the electrostatic potential, while possessing the local structure, are not free-energy functionals. We present a variational formulation with a local free-energy functional of the potential. In addition, we also construct a nonlocal free-energy functional of the electrostatic potential. These functionals are suited for employment in simulation schemes based on the ideas of dynamical optimization.
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Affiliation(s)
- Vikram Jadhao
- Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, USA
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Zhou J, Schmitz R, Dünweg B, Schmid F. Dynamic and dielectric response of charged colloids in electrolyte solutions to external electric fields. J Chem Phys 2013; 139:024901. [DOI: 10.1063/1.4812692] [Citation(s) in RCA: 27] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/01/2023] Open
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Schmitz R, Dünweg B. Numerical electrokinetics. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2012; 24:464111. [PMID: 23113993 DOI: 10.1088/0953-8984/24/46/464111] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
A new lattice method is presented in order to efficiently solve the electrokinetic equations, which describe the structure and dynamics of the charge cloud and the flow field surrounding a single charged colloidal sphere, or a fixed array of such objects. We focus on calculating the electrophoretic mobility in the limit of small driving field, and systematically linearize the equations with respect to the latter. This gives rise to several subproblems, each of which is solved by a specialized numerical algorithm. For the total problem we combine these solvers in an iterative procedure. Applying this method, we study the effect of the screening mechanism (salt screening versus counterion screening) on the electrophoretic mobility, and find a weak non-trivial dependence, as expected from scaling theory. Furthermore, we find that the orientation of the charge cloud (i.e. its dipole moment) depends on the value of the colloid charge, as a result of a competition between electrostatic and hydrodynamic effects.
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Affiliation(s)
- R Schmitz
- Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany
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Zimmer MJ, Geyer T. Do we have to explicitly model the ions in brownian dynamics simulations of proteins? J Chem Phys 2012; 136:125102. [PMID: 22462897 DOI: 10.1063/1.3698593] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/16/2023] Open
Abstract
Brownian dynamics (BD) is a very efficient coarse-grained simulation technique which is based on Einstein's explanation of the diffusion of colloidal particles. On these length scales well beyond the solvent granularity, a treatment of the electrostatic interactions on a Debye-Hückel (DH) level with its continuous ion densities is consistent with the implicit solvent of BD. On the other hand, since many years BD is being used as a workhorse simulation technique for the much smaller biological proteins. Here, the assumption of a continuous ion density, and therefore the validity of the DH electrostatics, becomes questionable. We therefore investigated for a few simple cases how far the efficient DH electrostatics with point charges can be used and when the ions should be included explicitly in the BD simulation. We find that for large many-protein scenarios or for binary association rates, the conventional continuum methods work well and that the ions should be included explicitly when detailed association trajectories or protein folding are investigated.
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Affiliation(s)
- Melanie J Zimmer
- Zentrum für Bioinformatik, Universität des Saarlandes, D-66041 Saarbrücken, Germany
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Wen J, Zhou S, Xu Z, Li B. Competitive adsorption and ordered packing of counterions near highly charged surfaces: From mean-field theory to Monte Carlo simulations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:041406. [PMID: 22680474 PMCID: PMC3725615 DOI: 10.1103/physreve.85.041406] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/19/2012] [Indexed: 06/01/2023]
Abstract
Competitive adsorption of counterions of multiple species to charged surfaces is studied by a size-effect-included mean-field theory and Monte Carlo (MC) simulations. The mean-field electrostatic free-energy functional of ionic concentrations, constrained by Poisson's equation, is numerically minimized by an augmented Lagrangian multiplier method. Unrestricted primitive models and canonical ensemble MC simulations with the Metropolis criterion are used to predict the ionic distributions around a charged surface. It is found that, for a low surface charge density, the adsorption of ions with a higher valence is preferable, agreeing with existing studies. For a highly charged surface, both the mean-field theory and the MC simulations demonstrate that the counterions bind tightly around the charged surface, resulting in a stratification of counterions of different species. The competition between mixed entropy and electrostatic energetics leads to a compromise that the ionic species with a higher valence-to-volume ratio has a larger probability to form the first layer of stratification. In particular, the MC simulations confirm the crucial role of ionic valence-to-volume ratios in the competitive adsorption to charged surfaces that had been previously predicted by the mean-field theory. The charge inversion for ionic systems with salt is predicted by the MC simulations but not by the mean-field theory. This work provides a better understanding of competitive adsorption of counterions to charged surfaces and calls for further studies on the ionic size effect with application to large-scale biomolecular modeling.
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Affiliation(s)
- Jiayi Wen
- Department of Mathematics, and the NSF Center for Theoretical Biological Physics, University of California, San Diego, 9500 Gilman Drive, Mail code: 0112, La Jolla, CA 92093-0112, USA
| | - Shenggao Zhou
- Department of Mathematics, Zhejiang University, No. 38 Zheda Road, Hangzhou, 310027, P. R. China, and Department of Mathematics and the NSF Center for Theoretical Biological Physics, University of California, San Diego, 9500 Gilman Drive, Mail code: 0112, La Jolla, CA 92093-0112, USA
| | - Zhenli Xu
- Department of Mathematics and Institute of Natural Sciences, Shanghai Jiao Tong University, 800 Dongchuan Rd., Shanghai, 200240, P. R. China
| | - Bo Li
- Department of Mathematics and the NSF Center for Theoretical Biological Physics, University of California, San Diego, 9500 Gilman Drive, Mail code: 0112, La Jolla, CA 92093-0112, USA
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Zhou S, Wang Z, Li B. Mean-field description of ionic size effects with nonuniform ionic sizes: a numerical approach. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:021901. [PMID: 21929014 PMCID: PMC3727298 DOI: 10.1103/physreve.84.021901] [Citation(s) in RCA: 36] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/17/2011] [Indexed: 05/14/2023]
Abstract
Ionic size effects are significant in many biological systems. Mean-field descriptions of such effects can be efficient but also challenging. When ionic sizes are different, explicit formulas in such descriptions are not available for the dependence of the ionic concentrations on the electrostatic potential, that is, there is no explicit Boltzmann-type distributions. This work begins with a variational formulation of the continuum electrostatics of an ionic solution with such nonuniform ionic sizes as well as multiple ionic valences. An augmented Lagrange multiplier method is then developed and implemented to numerically solve the underlying constrained optimization problem. The method is shown to be accurate and efficient, and is applied to ionic systems with nonuniform ionic sizes such as the sodium chloride solution. Extensive numerical tests demonstrate that the mean-field model and numerical method capture qualitatively some significant ionic size effects, particularly those for multivalent ionic solutions, such as the stratification of multivalent counterions near a charged surface. The ionic valence-to-volume ratio is found to be the key physical parameter in the stratification of concentrations. All these are not well described by the classical Poisson-Boltzmann theory, or the generalized Poisson-Boltzmann theory that treats uniform ionic sizes. Finally, various issues such as the close packing, limitation of the continuum model, and generalization of this work to molecular solvation are discussed.
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Affiliation(s)
- Shenggao Zhou
- Department of Mathematics, Zhejiang University, No. 38 Zheda Road, Hangzhou, 310027, P. R. China, and Department of Mathematics and the NSF Center for Theoretical Biological Physics, University of California, San Diego, 9500 Gilman Drive, Mail code: 0112, La Jolla, CA 92093-0112, USA.
| | - Zhongming Wang
- Department of Mathematics, Department of Chemistry and Biochemistry, and the NSF Center for Theoretical Biological Physics, University of California, San Diego, 9500 Gilman Drive, Mail code: 0112, La Jolla, CA 92093-0112, USA.
| | - Bo Li
- Department of Mathematics and the NSF Center for Theoretical Biological Physics, University of California, San Diego, 9500 Gilman Drive, Mail code: 0112, La Jolla, CA 92093-0112, USA.
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Golovnev A, Trimper S. Analytical solution of the Poisson–Nernst–Planck equations in the linear regime at an applied dc-voltage. J Chem Phys 2011; 134:154902. [DOI: 10.1063/1.3580288] [Citation(s) in RCA: 27] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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Xu Z. Treecode algorithm for pairwise electrostatic interactions with solvent-solute polarization. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:020902. [PMID: 20365522 DOI: 10.1103/physreve.81.020902] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/26/2009] [Revised: 01/18/2010] [Indexed: 05/29/2023]
Abstract
An O(N log N) treecode algorithm is presented for computing pairwise interactions of electrostatic free energy for reaction potentials with polarization effects due to the macroscopic solvent. A multipole expansion for a cluster is used to account for particles inside the cluster, where a spatial difference is applied to obtain the expansion coefficients of the polarization function. Numerical tests are performed to illustrate the accuracy and efficiency of the approach. The algorithm is significant in speeding up generalized Born methods for biomolecular simulations under the framework of macroscopically treating solvents.
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Affiliation(s)
- Zhenli Xu
- Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China and Department of Mathematics and Statistics, University of North Carolina, Charlotte, North Carolina 28223, USA.
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Wang H, Junghans C, Kremer K. Comparative atomistic and coarse-grained study of water: what do we lose by coarse-graining? THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2009; 28:221-229. [PMID: 19198906 DOI: 10.1140/epje/i2008-10413-5] [Citation(s) in RCA: 188] [Impact Index Per Article: 12.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/05/2008] [Revised: 12/05/2008] [Indexed: 05/27/2023]
Abstract
We employ the inverse Boltzmann method to coarse-grain three commonly used three-site water models (TIP3P, SPC and SPC/E) where one molecule is replaced with one coarse-grained particle with isotropic two-body interactions only. The shape of the coarse-grained potentials is dominated by the ratio of two lengths, which can be rationalized by the geometric constraints of the water clusters. It is shown that for simple two-body potentials either the radial distribution function or the geometrical packing can be optimized. In a similar way, as needed for multiscale methods, either the pressure or the compressibility can be fitted to the all atom liquid. In total, a speed-up by a factor of about 50 in computational time can be reached by this coarse-graining procedure.
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Affiliation(s)
- Han Wang
- Max Planck Institute for Polymer Research, Ackermannweg 10, D-55128 Mainz, Germany
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