Assessment of linear finite‐difference Poisson–Boltzmann solvers

CPU time and memory usage are two vital issues that any numerical solvers for the Poisson–Boltzmann equation have to face in biomolecular applications. In this study, we systematically analyzed the CPU time and memory usage of five commonly used finite‐difference solvers with a large and diversified set of biomolecular structures. Our comparative analysis shows that modified incomplete Cholesky conjugate gradient and geometric multigrid are the most efficient in the diversified test set. For the two efficient solvers, our test shows that their CPU times increase approximately linearly with the numbers of grids. Their CPU times also increase almost linearly with the negative logarithm of the convergence criterion at very similar rate. Our comparison further shows that geometric multigrid performs better in the large set of tested biomolecules. However, modified incomplete Cholesky conjugate gradient is superior to geometric multigrid in molecular dynamics simulations of tested molecules. We also investigated other significant components in numerical solutions of the Poisson–Boltzmann equation. It turns out that the time‐limiting step is the free boundary condition setup for the linear systems for the selected proteins if the electrostatic focusing is not used. Thus, development of future numerical solvers for the Poisson–Boltzmann equation should balance all aspects of the numerical procedures in realistic biomolecular applications. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

SMPBS: Web server for computing biomolecular electrostatics using finite element solvers of size modified Poisson‐Boltzmann equation

SMPBS (Size Modified Poisson‐Boltzmann Solvers) is a web server for computing biomolecular electrostatics using finite element solvers of the size modified Poisson‐Boltzmann equation (SMPBE). SMPBE not only reflects ionic size effects but also includes the classic Poisson‐Boltzmann equation (PBE) as a special case. Thus, its web server is expected to have a broader range of applications than a PBE web server. SMPBS is designed with a dynamic, mobile‐friendly user interface, and features easily accessible help text, asynchronous data submission, and an interactive, hardware‐accelerated molecular visualization viewer based on the 3Dmol.js library. In particular, the viewer allows computed electrostatics to be directly mapped onto an irregular triangular mesh of a molecular surface. Due to this functionality and the fast SMPBE finite element solvers, the web server is very efficient in the calculation and visualization of electrostatics. In addition, SMPBE is reconstructed using a new objective electrostatic free energy, clearly showing that the electrostatics and ionic concentrations predicted by SMPBE are optimal in the sense of minimizing the objective electrostatic free energy. SMPBS is available at the URL: smpbs.math.uwm.edu © 2017 Wiley Periodicals, Inc.     

Solving the finite-difference non-linear Poisson–Boltzmann equation

The Poisson–Boltzmann equation can be used to calculate the electrostatic potential field of a molecule surrounded by a solvent containing mobile ions. The Poisson–Boltzmann equation is a non-linear partial differential equation. Finite-difference methods of solving this equation have been restricted to the linearized form of the equation or a finite number of non-linear terms. Here we introduce a method based on a variational formulation of the electrostatic potential and standard multi-dimensional maximization methods that can be used to solve the full non-linear equation. © 1992 by John Wiley & Sons, Inc.     

Adaptive multilevel finite element solution of the Poisson–Boltzmann equation II. Refinement at solvent‐accessible surfaces in biomolecular systems

We apply the adaptive multilevel finite element techniques (Holst, Baker, and Wang 21 ) to the nonlinear Poisson–Boltzmann equation (PBE) in the context of biomolecules. Fast and accurate numerical solution of the PBE in this setting is usually difficult to accomplish due to presence of discontinuous coefficients, delta functions, three spatial dimensions, unbounded domains, and rapid (exponential) nonlinearity. However, these adaptive techniques have shown substantial improvement in solution time over conventional uniform‐mesh finite difference methods. One important aspect of the adaptive multilevel finite element method is the robust a posteriori error estimators necessary to drive the adaptive refinement routines. This article discusses the choice of solvent accessibility for a posteriori error estimation of PBE solutions and the implementation of such routines in the “Adaptive Poisson–Boltzmann Solver” (APBS) software package based on the “Manifold Code” (MC) libraries. Results are shown for the application of this method to several biomolecular systems. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 1343–1352, 2000     

Features of CPB: A Poisson–Boltzmann solver that uses an adaptive cartesian grid

The capabilities of an adaptive Cartesian grid (ACG)‐based Poisson–Boltzmann (PB) solver (CPB) are demonstrated. CPB solves various PB equations with an ACG, built from a hierarchical octree decomposition of the computational domain. This procedure decreases the number of points required, thereby reducing computational demands. Inside the molecule, CPB solves for the reaction‐field component (?_rf) of the electrostatic potential (?), eliminating the charge‐induced singularities in ?. CPB can also use a least‐squares reconstruction method to improve estimates of ? at the molecular surface. All surfaces, which include solvent excluded, Gaussians, and others, are created analytically, eliminating errors associated with triangulated surfaces. These features allow CPB to produce detailed surface maps of ? and compute polar solvation and binding free energies for large biomolecular assemblies, such as ribosomes and viruses, with reduced computational demands compared to other Poisson–Boltzmann equation solvers. The reader is referred to http://www.continuum‐dynamics.com/solution‐mm.html for how to obtain the CPB software. © 2014 Wiley Periodicals, Inc.     

Metabolomics approach based on ultra‐high‐performance liquid chromatography coupled with quadrupole‐time‐of‐flight mass spectrometry to identify the chemical constituents of the Traditional Chinese Er‐Zhi‐Pill

Er‐Zhi‐Pill, which consists of Ligustri lucidi fructus and Ecliptae prostratae herba , is a classical traditional Chinese medicinal formulation widely used as a liver‐nourishing and kidney‐enriching tonic. To identify the bioactive ingredients of Er‐Zhi‐Pill and characterize the variation of chemical constituents between co‐decoction and mix of individually decocted L. lucidi fructus and E. prostratae herba , a novel metabolomics approach based on ultra high performance liquid chromatography coupled with quadrupole time‐of‐flight mass spectrometry in both positive and negative ion modes, was established to comprehensively analyze chemical constituents and probe distinguishable chemical markers. In total, 68 constituents were unambiguously or tentatively identified through alignment of accurate molecular weights within an error margin of 5 ppm, elemental composition and fragmentation characteristics, including eight constituents, which were confirmed by comparing to reference standards. Furthermore, principal component analysis and partial least squares discriminant analysis using Simca‐p+ 12.0 software were applied to investigate chemical differences between formulations obtained by co‐decoction and a mixture of individual decoctions. Global chemical differences were found in samples of two different decoction methods, and 16 components, including salidroside, specneuzhenide and wedelolactone, contributed most to the observed differences. This study provides a basic chemical profile for the quality control and further mechanism research of Er‐Zhi‐Pill.     

Accurate discrimination of Gastrodia elata from different geographical origins using high‐performance liquid chromatography fingerprint combined with boosting partial least‐squares discriminant analysis

Gastrodia elata from different geographical origins varies in quality and pharmacological activity. This study focused on the classification and identification of Gastrodia elata from six producing areas using high‐performance liquid chromatography fingerprint combined with boosting partial least‐squares discriminant analysis. Before recognition analysis, a principal component analysis was applied to ascertain the discrimination possibility with high‐performance liquid chromatography fingerprints. And then, boosting partial least‐squares discriminant analysis and conventional partial least‐squares discriminant analysis were applied in this study. Experimental results indicated that the adaptive iteratively reweighted penalized least‐squares algorithm could eliminate the baseline drift of high‐performance liquid chromatography chromatograms effectively. And compared with partial least‐squares discriminant analysis, the total recognition rates using high‐performance liquid chromatography fingerprint combined with boosting partial least‐squares discriminant analysis for the calibration sets and prediction sets were improved from 94 to 100% and 86 to 97%, respectively. In conclusion, high‐performance liquid chromatography combined with boosting partial least‐squares discriminant analysis, which has such advantages as effective, specific, accurate, non‐polluting, has an edge for discrimination of traditional Chinese medicine from different geographical origins. And the proposed methodology is a useful tool to classify and identify Gastrodia elata from different geographical origins.     

Adaptive multilevel finite element solution of the Poisson–Boltzmann equation I. Algorithms and examples

This article is the first of two articles on the adaptive multilevel finite element treatment of the nonlinear Poisson–Boltzmann equation (PBE), a nonlinear eliptic equation arising in biomolecular modeling. Fast and accurate numerical solution of the PBE is usually difficult to accomplish, due to the presence of discontinuous coefficients, delta functions, three spatial dimensions, unbounded domain, and rapid (exponential) nonlinearity. In this first article, we explain how adaptive multilevel finite element methods can be used to obtain extremely accurate solutions to the PBE with very modest computational resources, and we present some illustrative examples using two well‐known test problems. The PBE is first discretized with piece‐wise linear finite elements over a very coarse simplex triangulation of the domain. The resulting nonlinear algebraic equations are solved with global inexact Newton methods, which we have described in an article appearing previously in this journal. A posteriori error estimates are then computed from this discrete solution, which then drives a simplex subdivision algorithm for performing adaptive mesh refinement. The discretize–solve–estimate–refine procedure is then repeated, until a nearly uniform solution quality is obtained. The sequence of unstructured meshes is used to apply multilevel methods in conjunction with global inexact Newton methods, so that the cost of solving the nonlinear algebraic equations at each step approaches optimal O(N) linear complexity. All of the numerical procedures are implemented in MANIFOLD CODE (MC), a computer program designed and built by the first author over several years at Caltech and UC San Diego. MC is designed to solve a very general class of nonlinear elliptic equations on complicated domains in two and three dimensions. We describe some of the key features of MC, and give a detailed analysis of its performance for two model PBE problems, with comparisons to the alternative methods. It is shown that the best available uniform mesh‐based finite difference or box‐method algorithms, including multilevel methods, require substantially more time to reach a target PBE solution accuracy than the adaptive multilevel methods in MC. In the second article, we develop an error estimator based on geometric solvent accessibility, and present a series of detailed numerical experiments for several complex biomolecules. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 1319–1342, 2000     

The Calculation of Surface Potential for Spherical Ionic Micelles

In this article, an approximate expression for the calculation of surface potential for spherical ionic micelles has been presented. This simple analytic form overcomes the complexity of original theory. The calculated values of surface potential of spherical ionic micelles are in quite good agreement with the exact numerical values of nonlinear Poisson‐Boltzmann equation.     

A solvation‐free‐energy functional: A reference‐modified density functional formulation

The three‐dimensional reference interaction site model (3D‐RISM) theory, which is one of the most applicable integral equation theories for molecular liquids, overestimates the absolute values of solvation‐free‐energy (SFE) for large solute molecules in water. To improve the free‐energy density functional for the SFE of solute molecules, we propose a reference‐modified density functional theory (RMDFT) that is a general theoretical approach to construct the free‐energy density functional systematically. In the RMDFT formulation, hard‐sphere (HS) fluids are introduced as the reference system instead of an ideal polyatomic molecular gas, which has been regarded as the appropriate reference system of the interaction‐site‐model density functional theory for polyatomic molecular fluids. We show that using RMDFT with a reference HS system can significantly improve the absolute values of the SFE for a set of neutral amino acid side‐chain analogues as well as for 504 small organic molecules. © 2015 Wiley Periodicals, Inc.     

Numerical study of electroosmotic slip flow of fractional Oldroyd-B fluids at high zeta potentials

In this paper, an investigation of the electroosmotic flow of fractional Oldroyd-B fluids in a narrow circular tube with high zeta potential is presented. The Navier linear slip law at the walls is considered. The potential field is applied along the walls described by the nonlinear Poisson–Boltzmann equation. It's worth noting here that the linear Debye–Hückel approximation can't be used at the condition of high zeta potential and the exact solution of potential in cylindrical coordinates can't be obtained. Therefore, the Matlab bvp4c solver method and the finite difference method are employed to numerically solve the nonlinear Poisson–Boltzmann equation and the governing equations of the velocity distribution, respectively. To verify the validity of our numerical approach, a comparison has been made with the previous work in the case of low zeta potential and the excellent agreement between the solutions is clear. Then, in view of the obtained numerical solution for the velocity distribution, the numerical solutions of the flow rate and the shear stress are derived. Furthermore, based on numerical analysis, the influence of pertinent parameters on the potential distribution and the generation of flow is presented graphically.     

Treating ion distribution with Gaussian‐based smooth dielectric function in DelPhi

The standard treatment of ions in the framework of the Poisson–Boltzmann equation relies on molecular surfaces, which are commonly constructed along with the Stern layer. The molecular surface determines where ions can be present. In the Gaussian‐based smooth dielectric function in DelPhi, smooth boundaries between the solute and solvent take the place of molecular surface. Therefore, this invokes the question of how to model mobile ions in the water phase without a definite solute‐solvent boundary. This article reports a natural extension of the Gaussian‐based smooth dielectric function approach that treats mobile ions via Boltzmann distribution with an added desolvation penalty. Thus, ion concentration near macromolecules is governed by the local electrostatic potential and the desolvation penalty (from being partially desolvated). The approach is tested against the experimental salt dependence of binding free energy on 7 protein–protein complexes and 12 DNA–protein complexes, resulting in Pearson correlations of 0.95 and 0.88, respectively. © 2017 Wiley Periodicals, Inc.     

Modeling and simulation of morphology variation during the solidification of polymer melts with amorphous and semi‐crystalline phases

The solidification of polymer melts in practical processing such as extrusion, injection molding and blow molding can significantly influence the inner structure and performance of final products. The investigation of its mechanism has both scientific and industrial interests. In the study, the three‐dimensional mathematical model is developed for the simulation of morphology variation in the solidification of polymer melts with amorphous and semi‐crystalline phases. The amorphous phase is simulated as the finite extensible nonlinear elastic dumbbell with a peterlin closure approximation (FENE‐P) fluid and the semi‐crystalline phase is approximated as rigid rods that grow and oriented in the flow field. The model of amorphous phase and semi‐crystalline phase are coupled through the stress and momentum balance and the feedback of crystallinity to the system relaxation time. The evolution of crystallization kinetics process are described by using a set of Schneider equation that discriminating the relative roles of the thermal and the flow effect on the crystallization behavior. With the standard Galerkin formulation adopted as basic computational framework, the discrete elastic viscous stress splitting algorithm in cooperating with the streamline upwinding approach serves as a relatively robust numerical scheme by using penalty finite element–finite difference simulation with a decoupled solving algorithm. The proposed mathematical model and numerical method have been successfully applied to the investigation of solidification of polymer melts in the extrusion process. The variations of orientation and crystallization morphology during the solidification process are further discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

A rigorous approach to the formulation of extended Born‐Oppenheimer equation for a three‐state system

If a coupled three‐state electronic manifold forms a sub‐Hilbert space, it is possible to express the non‐adiabatic coupling (NAC) elements in terms of adiabatic–diabatic transformation (ADT) angles. Consequently, we demonstrate: (a) Those explicit forms of the NAC terms satisfy the Curl conditions with non‐zero Divergences; (b) The formulation of extended Born‐Oppenheimer (EBO) equation for any three‐state BO system is possible only when there exists coordinate independent ratio of the gradients for each pair of ADT angles leading to zero Curls at and around the conical intersection(s). With these analytic advancements, we formulate a rigorous EBO equation and explore its validity as well as necessity with respect to the approximate one (Sarkar and Adhikari, J Chem Phys 2006, 124, 074101) by performing numerical calculations on two different models constructed with different chosen forms of the NAC elements. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Determination of Solvatochromic Regression Coefficients for the Molybdenum(VI) Complex with Ethylenediamine‐N,N′‐diacetic Acid by Using Kamlet‐Abboud‐Taft Equation

The Solver, Microsoft Excel 2000 powerful optimization package, has been used to perform non‐linear least‐squares curve fitting on the basis of Gauss‐Newton method for the calculation of solvatochromic regression coefficients for the complexation of molybdenum(VI) with ethylenediamine‐N,N′‐diacetic acid and dissociation constants at 25°C and constant ionic strength 0.1 mol·L⁻¹ sodium perchlorate in different aqueous solutions of methanol. A combination of potentiometric and UV spectrophotometric methods have been used for experimental studies. Non specific and specific solute‐solvent interactions were interpreted by correlating the equilibrium data with solvent parameters using the Kamlet‐Abboud‐Taft solvatochromic equation. Finally the influence of the solvent on the stability of the complex was discussed on the basis of the correlation results and the contribution of α (hydrogen‐bond donor acidity), β (hydrogen‐bond acceptor basicity) and π* (dipolarity/polarizability) parameters.     

Highly efficient and exact method for parallelization of grid‐based algorithms and its implementation in DelPhi

The Gauss–Seidel (GS) method is a standard iterative numerical method widely used to solve a system of equations and, in general, is more efficient comparing to other iterative methods, such as the Jacobi method. However, standard implementation of the GS method restricts its utilization in parallel computing due to its requirement of using updated neighboring values (i.e., in current iteration) as soon as they are available. Here, we report an efficient and exact (not requiring assumptions) method to parallelize iterations and to reduce the computational time as a linear/nearly linear function of the number of processes or computing units. In contrast to other existing solutions, our method does not require any assumptions and is equally applicable for solving linear and nonlinear equations. This approach is implemented in the DelPhi program, which is a finite difference Poisson–Boltzmann equation solver to model electrostatics in molecular biology. This development makes the iterative procedure on obtaining the electrostatic potential distribution in the parallelized DelPhi several folds faster than that in the serial code. Further, we demonstrate the advantages of the new parallelized DelPhi by computing the electrostatic potential and the corresponding energies of large supramolecular structures. © 2012 Wiley Periodicals, Inc.     

Quantitative analysis of tea using ytterbium‐based internal standard near‐infrared spectroscopy coupled with boosting least‐squares support vector regression

The present study demonstrated the possibility of utilizing the ytterbium (Yb)‐based internal standard near‐infrared (NIR) spectroscopic measurement technique coupled with multivariate calibration for quantitative analysis of tea, including total free amino acids and total polyphenols in tea. Yb is a rare earth element aimed to compensate for the spectral variation induced by the alteration of sample quantity during the spectral measurement of the powdered samples. Boosting was invoked to be combined with least‐squares support vector regression (LS‐SVR), forming boosting least‐squares support vector regression (BLS‐SVR) for the multivariate calibration task. The results showed that the tea quality could be accurately and rapidly determined via the Yb‐based internal standard NIR spectroscopy combined with BLS‐SVR method. Moreover, the introduction of boosting drastically enhanced the performance of individual LS‐SVR, and BLS‐SVR compared favorably with partial least‐squares regression. Copyright © 2013 John Wiley & Sons, Ltd.     

Chemometrics comparison of gas chromatography with mass spectrometry and comprehensive two‐dimensional gas chromatography with time‐of‐flight mass spectrometry Daphnia magna metabolic profiles exposed to salinity

The performances of gas chromatography with mass spectrometry and of comprehensive two‐dimensional gas chromatography with time‐of‐flight mass spectrometry are examined through the comparison of Daphnia magna metabolic profiles. Gas chromatography with mass spectrometry and comprehensive two‐dimensional gas chromatography with mass spectrometry were used to compare the concentration changes of metabolites under saline conditions. In this regard, a chemometric strategy based on wavelet compression and multivariate curve resolution–alternating least squares is used to compare the performances of gas chromatography with mass spectrometry and comprehensive two‐dimensional gas chromatography with time‐of‐flight mass spectrometry for the untargeted metabolic profiling of Daphnia magna in control and salinity‐exposed samples. Examination of the results confirmed the outperformance of comprehensive two‐dimensional gas chromatography with time‐of‐flight mass spectrometry over gas chromatography with mass spectrometry for the detection of metabolites in D. magna samples. The peak areas of multivariate curve resolution–alternating least squares resolved elution profiles in every sample analyzed by comprehensive two‐dimensional gas chromatography with time‐of‐flight mass spectrometry were arranged in a new data matrix that was then modeled by partial least squares discriminant analysis. The control and salt‐exposed daphnids samples were discriminated and the most relevant metabolites were estimated using variable importance in projection and selectivity ratio values. Salinity de‐regulated 18 metabolites from metabolic pathways involved in protein translation, transmembrane cell transport, carbon metabolism, secondary metabolism, glycolysis, and osmoregulation.