Solution Of Poisson-Boltzmann Equation Research Articles

Physically constrained learning of MOS capacitor electrostatics

In recent years, neural networks have achieved phenomenal success across a wide range of applications. They have also proven useful for solving differential equations. The focus of this work is on the Poisson–Boltzmann equation (PBE) that governs the electrostatics of a metal–oxide–semiconductor capacitor. We were motivated by the question of whether a neural network can effectively learn the solution of PBE using the methodology pioneered by Lagaris et al. [IEEE Trans. Neural Netw. 9 (1998)]. In this method, a neural network is used to generate a set of trial solutions that adhere to the boundary conditions, which are then optimized using the governing equation. However, the challenge with this method is the lack of a generic procedure for creating trial solutions for intricate boundary conditions. We introduce a novel method for generating trial solutions that adhere to the Robin and Dirichlet boundary conditions associated with the PBE. Remarkably, by optimizing the network parameters, we can learn an optimal trial solution that accurately captures essential physical insights, such as the depletion width, the threshold voltage, and the inversion charge. Furthermore, we show that our functional solution can extend beyond the sampling domain.

A Deep Neural Network Based on ResNet for Predicting Solutions of Poisson–Boltzmann Equation

The Poisson–Boltzmann equation (PBE) arises in various disciplines including biophysics, electrochemistry, and colloid chemistry, leading to the need for efficient and accurate simulations of PBE. However, most of the finite difference/element methods developed so far are rather complicated to implement. In this study, we develop a ResNet-based artificial neural network (ANN) to predict solutions of PBE. Our networks are robust with respect to the locations of charges and shapes of solvent–solute interfaces. To generate train and test sets, we have solved PBE using immersed finite element method (IFEM) proposed in (Kwon, I.; Kwak, D. Y. Discontinuous bubble immersed finite element method for Poisson–Boltzmann equation. Communications in Computational Physics 2019, 25, pp. 928–946). Once the proposed ANNs are trained, one can predict solutions of PBE in almost real time by a simple substitution of information of charges/interfaces into the networks. Thus, our algorithms can be used effectively in various biomolecular simulations including ion-channeling simulations and calculations of diffusion-controlled enzyme reaction rate. The performance of the ANN is reported in the result section. The comparison between IFEM-generated solutions and network-generated solutions shows that root mean squared error are below 5·10−7. Additionally, blow-ups of electrostatic potentials near the singular charge region and abrupt decreases near the interfaces are represented in a reasonable way.

A flux-jump preserved gradient recovery technique for accurately predicting the electrostatic field of an immersed biomolecule

Poisson-Boltzmann equation (PBE) and its variants are important implicit continuum models for predicting the electrostatics of solvated biomolecules. In this paper, in order to accurately predict the gradient of electrostatics, we propose a new flux-jump preserved gradient recovery method and then fulfill it in the program using Python and Fortran. Two numerical tests with available analytical solutions are presented to well validate the new program. Then the new proposed method is applied to recover the gradient of a simple dipole case as well as two protein cases using the numerical solutions of both PBE and SMPBE.

Nano-colloid electrophoretic transport: Fully explicit modelling via dissipative particle dynamics

Abstract In present study, a novel fully explicit approach using dissipative particle dynamics (DPD) method is introduced for modelling electrophoretic transport of nano-colloids in an electrolyte solution. Slater type charge smearing function included in 3D Ewald summation method is employed to treat electrostatic interaction. Moreover, capability of different thermostats are challenged to control the system temperature and study the dynamic response of colloidal electrophoretic mobility under practical ranges of external electric field in nano scale application ( 0.072 E 0.361 v / n m ) covering non-linear response regime, and ionic salt concentration (0.049 S C 0.69 [M]) covering weak to strong Debye screening of the colloid. The effect of different colloidal repulsions are then studied on temperature, reduced mobility and zeta potential which is computed based on charge distribution within the spherical colloidal EDL. System temperature and electrophoretic mobility both show a direct and inverse relationship respectively with electric field and colloidal repulsion. Mobility declining with colloidal repulsion reaches a plateau which is a relatively constant value at each electrolyte salinity for A i i > 600 in DPD units regardless of electric field intensity. Nose–Hoover–Lowe–Andersen and Lowe-Andersen thermostats are found to function more effectively under high electric fields ( E > 0.145 [ v / n m ] ) while thermal equilibrium is maintained. Reasonable agreements are achieved by benchmarking the radial distribution function with available electrolyte structure modellings, as well as comparing reduced mobility against conventional Smoluchowski and Huckel theories, and numerical solution of Poisson-Boltzmann equation.

PB-SAM, a Novel Solution to the Poison-Boltzmann Equation for Applications in Coarse Grain Dynamics

While molecular dynamics (MD) simulations routinely and beneficially address questions focused on detailed atomistic chemistry, structure, and kinetics over fast timescales, there exists another set of problems on the mesoscale, where limitations of size and timescales in MD are reached. These problems may be overcome by the use of coarse-grain (CG) representations, e.g. the Poisson-Boltzmann equation (PBE), of microscopic systems integrated into simulations utilizing stochastic dynamics. At supramolecular distances, electrostatic forces dominate many first interaction events. Applications of these interactions range from recognition events for biomolecular complex formation to proton transport in membranes.A realistic simulation must include an accurate treatment of bulk electrolytes for these large-scale systems. This can be achieved using continuum mean-field theory with the PB equation, which under low field condition be further simplified to the linearized Poisson-Boltzmann equation (LPBE). A variety of analytical and numerical techniques have been developed to solve the PB equation. Our group has achieved a fundamental result in deriving the first completely general analytical solution to the linearized Poisson Boltzmann equation (LPBE) for computing the screened (salty) electrostatic interaction between arbitrary numbers of nanoscale spheres of arbitrarily complex charge distributions, separated by arbitrary distance (or concentration), adapted from the generalized Kirkwood PBE solution. This analytical solution in turn serves as the foundation of a semi-analytical approach to solving the LPBE for any nanoscale shape (PB-SAM). We have recently incorporated PB-SAM into a Brownian Dynamic approach to create a robust coarse-grained simulation tool to study a range of biological systems, including a recent study of Barnase/Barstar association under crowded conditions. We have developed a highly parallelized version of PB-SAM that is competitive with current PBE software, and are exploring 3-body implementations to further increase PB-SAM efficiency.

Theory of lubrication due to polyelectrolyte hydrogels with arbitrary salt concentration and degree of compression

A Poisson-Boltzmann equation solution is used to determine the thickness of a thin fluid lubricating layer predicted to separate two polyelectrolyte hydrogels in contact for arbitrary salt concentration as a function of applied load and fixed charge and salt concentration. We consider loads ranging from 1 Pa, at which the thin fluid layer thickness is of the order of micron, up to loads of the order of a MPa, at which it is estimated to be of the order of an angstrom. This allows us to predict the thickness of this layer over the wide range of loads that can occur in various applications of hydrogels.

Analytical solutions of the nonlinear Poisson–Boltzmann equation in mixture of electrolytes

The Poisson–Boltzmann theory is still considered as an exact theory under low or even moderate electrolyte concentrations. As of today, however, just the analytical solutions in a single electrolyte were obtained, while people often meet mixed electrolytes in practical applications. Thus the analytical solutions of Poisson–Boltzmann equation (PBE) in different mixed electrolytes are still an important issue. The analytical solutions of PBE in mixed electrolytes with monovalent and bivalent counterions were firstly derived in the present study. The potential distribution and concentration profile curves in EDL were obtained with one dimensional space by the analytical solutions of PBE. The potential distribution curves of different mixtures showed that the power of the screen surface charge for bivalent counterion was stronger than that for monovalent counterion, thus the concentration of bivalent counterion in EDL was larger than that of monovalent counterion.

A New and Efficient Poisson-Boltzmann Solver for Interaction of Multiple Proteins.

We derive a new numerical approach to solving the linearized Poisson Boltzmann equation (PBE) by representing the protein surface as a collection of spheres in which the surface charges can then be iteratively solved by new analytical multipole methods previously introduced by us [Lotan & Head-Gordon, 2006]. We show that our Poisson Boltzmann semi-analytical method, PB-SAM, realizes better accuracy, more flexible memory management, and at reduced cost relative to either finite difference or boundary element method PBE solvers. We provide two new benchmarks of PBE solution accuracy to test the numerical PBE solutions based on (1) arrays of up to hundreds of spherical low dielectric geometries with asymmetric charges in which mutual polarization is treated exactly, and (2) two overlapping spheres with increasing charge asymmetry by solving the PB-SAM method to very high pole order. We illustrate the strength of the PB-SAM approach by computing the potential profile of an array of 60 T1-particle forming monomers of the bromine mosaic virus.

Modeling of the Mass Transfer Effect in Biphasic Enzyme Membrane Reactor for Hydrolysis of Olive Oil

This article presents a model to explain the mass transfer phenomenon of the ions (OH-, Na+ and RCOO-) formed during the hydrolysis reaction on the aqueous side of the reaction sites in the biphasic enzyme membrane reactor (EMR) using lipase immobilized on modified poly (methyl methacrylate-ethylene glycol dimethacrylate) (PMMA-EGDM) clay composite membrane. Our model divides the aqueous side of the system into two regions, an unstirred film (aqueous side) adjacent to the pore end (region I) and charged capillaries of the membrane (region II). The process involves transport of Na+, OH- and RCOO- ions in region I whereas the film is considered as an electro-neutral region. Space charge model (consisting of Nernst-Planck equation for ion transport and nonlinear Poisson-Boltzmann (PB) equation for charge distribution) is used to describe transport of ions inside charged capillaries (pores) in region II. The solution of the nonlinear PB equation is computationally intensive and to overcome this difficulty, a series solution of the PB equation for the charge distribution inside the capillaries is developed. The suggested solution is used to determine the rate of formation of oleic acid using the information on pore size of the membrane (determined from molecular weight cut-off experiment) and diffusivity data from the literature. The result shows that the concentration of RCOO- ions at the boundary of region I of the aqueous phase increases with increasing the duration of reaction. The pore wall charge determined using the model is found to be in the range of 0.715-0.735 mV.

Geometry-guided computation of 3D electrostatics for large biomolecules

Electrostatic interactions play a central role in biological processes. Development of fast computational methods to solve the underlying Poisson–Boltzmann equation (PBE) is vital for biomolecular modeling and simulation package. In this paper, we propose new methods for efficiently computing the electrostatic potentials for large molecules by using the geometry of the molecular shapes to guide the computation. The accuracy and stability of the solution to the PBE is quite sensitive to the boundary layer between the solvent and the solute which defines the molecular surface. In this paper, we present a new interface-layer-focused PBE solver. First, we analytically construct the molecular surface of the molecule and compute a distance field from the surface. We then construct nested iso-surface layers outwards and inwards from the surface using the distance field. We have developed a volume simplification algorithm to adaptively adjust the density of the irregular grid based on the importance to the PBE solution. We have generalized the finite difference methods using Taylor series expansion on the irregular grids. Our algorithm achieves about three times speedup in the iterative solution process of PBE, with more accurate results on an analytical solvable testing case, compared with the popular optimized DelPhi program.

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 solution of the poisson-boltzmann equation between two spheres: Modified iterative method

We present a modification on the successive overrelaxation (SOR) method and the iteration of the Green's function integral representation for the solution of the (nonlinear) Poisson-Boltzmann equation between two spheres. In comparison with other attempts, which approximate the geometry or the nonlinearity, the computations here are done for the full problem and compared with those done by the finite element method as a typical method for such problems. For the parameters of general interest, while the SOR method does not work, and the iteration of the integral representation is limited in its convergence, our modification to these iterative schemes converge. The modified SOR surpasses both methods in simplicity and speed; it is about 100 times faster than the modified iteration of the integral representation, with the latter being still simpler and faster than the finite element method. These two examples further illustrate the advantage of our recent modification to iterative methods, which is based on an analytical fixed point argument.

Selective Counterion Binding to Linear Polyelectrolytes in Aqueous Solutions

The selective counterion condensation induced by a charged cylindrical polyelectrolyte in aqueous solution containing different mono- and di-valent ions was investigated by means of the numerical solutions of the non linearized cylindrical Poisson-Boltzmann equation. The critical charge fraction fc, above which the condensation predicted by the Manning theory occurs, has been evaluated in some selected cases of relevance in biopolyelectrolyte systems and a selective binding of counterions has been shown to occur. To take into account the effects of different types of mono- and di-valent salts, an extension of the Manning theory recently proposed by Paoletti et al. and based on Gibbs free energy minimization, was considered. Discrepancies between the charge distribution of bound ions calculated by means of numerical solutions of Poisson-Boltzmann equation and those predicted by the extended Manning theory, were found.

Studies on Photo-Effects in Semiconductor Surfaces

Photo-voltage at the semiconductor surface has been investigated by using a new solution of Poisson-Boltzmann equation. Brattain and Bardeen modex for surface traps has been taken in the formulation. Dependence of normalised surface photo-voltage on surface potential under different carrier densities has been estimated and the results are presented graphically.

Asymptotic behavior of the solutions of the poisson-boltzmann equation and its physical consequences

Expressions are derived describing the asymptotic behavior of the solutions of the Poisson-Boltzmann equation (PBE) with increase of the surface potential or surface charge. In contrast to the solutions of the linearized PBE, the exact solutions remain finite everywhere in the aqueous medium at infinite surface charge density. This peculiarity gives rise to some notable physical effects: (a) the electrostatic disjoining pressure P el in a liquid film of fixed width cannot exceed a finite limiting value at any surface charge density; (b) at constant pressure, the width of a liquid film tends to a finite value at infinite increase of its surface charge; (c) the ζ-potential of an interface is limited from above and becomes insensitive to the value of the surface charge if the latter is sufficiently high. Numerical estimates show that the specific asymptotic behavior of the exact solutions of PBE manifests itself at surface charge densities which are readily accessible in experimental measurements. A comparison with the data available about foam films and lyotropic lamellar lipid phases provides evidence that P el and the width of a liquid film may become indistinguishably close to their asymptotic values. Thus, the asymptotic effects may be used also as a criterion for the extent of validity of the standard Gouy-Chapman theory.

Lower Bounds on Solutions of the Poisson-Boltzmann Equation near the Limit of Infinite Dilution

Rigorous existence, uniqueness and lower bound results for solutions of the nonlinear Poisson-Boltzmann equation are obtained. The relationship of these bounds with certain assumptions in Manning's limiting law theory for polyelectrolytes is discussed.

Bounds on Solutions of the Poisson-Boltzmann Equation near Infinite Dilution-The Moderately Charged Case

Upper and lower bounds on solutions of the nonlinear Poisson-Boltzmann equation are obtained for the case where the so-called surface charge parameter is less than unity (the moderately charged case). The relationship of these bounds with Manning's limiting law theory for polyelectrolytes is discussed. In particular, we are able to provide rigorous justification (different from Manning's own justification which is based on cluster theory) for Manning's use of the Debye-Hückel approximation.