Nonlinear Poisson-Boltzmann Equation Research Articles

Effect of oppositely charged hydrophobic additives (alkanoates) on the stability of C14TAB foam films

The impact of the hydrophobicity of oppositely charged additives on the stability of C14TAB foam films has been studied using a thin film pressure balance (TFPB). Disjoining pressure isotherms of 10−4M C14TAB solutions containing negatively charged sodium alkanoates CnOONa with increasing chain length from n=4 to n=12 are measured. The results of disjoining pressure measurements are presented as maximum possible pressure Πmax applied to a stable film as a function of the alkanoate concentration. Furthermore, the surface properties of the mixed systems are described to gain a general understanding about the surface characteristics. Since C14TAB and alkanoates are oppositely charged, the question arises whether charge reversal at the film surfaces occurs and how it depends on the chain length of the alkanoate. The presented results indicate that C14TAB/alkanoate systems can be divided into three groups. Depending on the chain length, alkanoates act as organic salts (n=4), co-surfactant (n=6) or anionic surfactant (8≤n≤12). In case of alkanoates acting as anionic surfactant, the simulation of the isotherms by solving the non-linearized Poisson–Boltzmann equation under assumption of constant potential actually indicates that charge reversal occurs close to the nominal IEP. The film stabilities are markedly affected by the hydrophobicity and the amount of alkanoate.

Overlapping of the Diffuse Double Layer in the Vicinity of the Microporous Electrode

Introduction A microporous electrode with nm-sized micropores on the surface is an attractive subject in electrochemistry. Some experimental results show the physicochemical properties of microporous electrodes are not explained only in terms of the surface area. We have an interest in the electrical double layer in the narrow space. When the size of the micropore on the electrode is smaller than the diffuse part of the electrical double layer, the diffuse double layer show overlap across the pore. This overlapping of the diffuse double layer cannot be estimated on the one dimension model, such as Gouy–Chapman (GC) theory for planar electrodes. We propose a model of the diffuse double layers in the vicinity of micropores on the basis of the Poisson-Boltzmann equation without any complicated assumptions. This calculation shows essential features of the diffuse double layer on the electrode with micropores. The size dependence, the permittivity dependence, the potential dependence, and the electrolyte concentration dependence of the diffuse double layer and also the differential capacitance of the electrode with micropores are discussed.Model The model is described with a cylindrical coordinate. The electrode was assumed as a conductor with an equipotential surface. The diameter of a pore on the electrode was varied from 0.2 to 5 nm. To avoid singular points in the calculation, the radius of the corners was assumed to be 50 pm in the geometry of this model. The solution phase contains a 1:1 electrolyte. In the solution phase, the Poisson–Boltzmann equation was applied to obtain the profile of the electric potential. In this calculation, the relative permittivity was set to be from 10 to 78. The nonlinear Poisson–Boltzmann equation was numerically solved by the finite element method. The differential capacitance of the double layer was estimated from the dependence of the total charge on the electrode surface on the surface potential.Results and discussion The calculated potential profile showed the overlapping of the diffuse double layer in the micropore. The electric field in the micropore is weaker than that outside of the micropore, that is, the overlapped diffuse double layer provides the mild electric field. Assuming that the thickness of the diffuse double layer is the division of the surface potential by the gradient of the potential at the electrode surface, the degree of the overlapping can express qualitatively. When the size of the micropore on the electrode is smaller than the thickness of the diffuse double layer, the calculated capacitance agrees with the GC capacitance at the planner electrode. On the other hand, when the size of the micropore on the electrode is larger than the thickness of the diffuse double layer, the double layer capacitance converges on the GC capacitance at the microporous electrode. This means that the surface area of micropore becomes effective only at thinner diffuse double layers. The change of the thickness of the diffuse double layer in the micropore causes the strong dependence of the capacitance on the potential and the electrolyte concentration compared with the situation of planar electrode. The characteristic features of microporous electrodes would be originated from the heterogeneous electric field caused by the overlapping of the diffuse double layer.

Coarse-Grained Modeling of the Titration and Conductance Behavior of Aqueous Fullerene Hexa Malonic Acid (FHMA) Solutions

The coarse-grained continuum primitive model is developed and used to characterize the titration and electrical conductance behavior of aqueous solutions of fullerene hexa malonic acid (FHMA). The spherical FHMA molecule, a highly charged electrolyte with an absolute valence charge as large as 12, is modeled as a dielectric sphere in Newtonian fluid, and electrostatics are treated numerically at the level of the non-linear Poisson-Boltzmann equation. Transport properties (electrophoretic mobilities and conductances) of the various charge states of FHMA are numerically computed using established numerical algorithms. For reasonable choices of the model parameters, good agreement between experiment (published literature) and modeling is achieved. In order to accomplish this, however, a moderate degree of specific binding of principal counterion and FHMA must be included in the modeling. It should be emphasized, however, that alternative explanations are possible. This comparison is made at 25 °C for both Na(+) and Ca(2+) principal counterions. The model is also used to characterize the different charge states and degree of counterion binding to those charge states as a function of pH.

ELECTROSTATICS STUDY OF A SINGLE-STRANDED DNA: A PROSPECTIVE FOR SINGLE MOLECULE SEQUENCING

Single molecule DNA sequencing requires new approaches to identify nucleotide bases. Using molecular dynamics simulations, we investigate the intrinsic electrostatics of single-stranded DNA by solving the nonlinear Poisson–Boltzmann equation. The results show variations of the molecular electrostatic potential (MEP) within 3 nm from the center of the sugar backbone, with suitably differentiable variations at 1.4 nm distance. MEP variations among four nucleotide bases are the most significant near ~ 33.7° and ~ 326.3° angular orientation, while the influences of the neighboring bases on MEPs become insignificant after the 3rd-nearest neighbors. This analysis shows potential for direct electronic sequencing of individual DNA molecules. [Formula: see text]Special Issue Comment: This paper about DNA sequencing based on molecular electrostatic potential maps of the DNA in the channel is related to the Special Issue articles about: measuring enzymes,32 and solving single molecules' trajectories with the RDF approach33 and with the QuB software.34

Extensions to Study Electrochemical Interfaces - A Contribution to the Theory of Ions

In the present study an alternative model allows the extension of the Debye-Huckel Theory (DHT) considering time dependence explicitly. From the Electro-Quasistatic approach (EQS) done in earlier studies time dependent potentials are suitable to describe several phenomena especially conducting media as well as the behaviour of charged particles in arbitrary solutions acting as electrolytes. This leads to a new formulation of the meaning of the nonlinear Poisson-Boltzmann Equation (PBE). If a concentration and/or flux gradient of particles is considered the original structure of the mPBE will be modified leading to a new nonlinear partial differential equation (nPDE) of the third order. It is shown how one can derive classes of solutions for the potential function analytically by application of an algebraic method. The benefit of the mathematical tools used here is the fact that closed-form solutions can be calculated without any numerical approximations.

Sensitivities to parameterization in the size-modified Poisson-Boltzmann equation

Experimental results have demonstrated that the numbers of counterions surrounding nucleic acids differ from those predicted by the nonlinear Poisson-Boltzmann equation, NLPBE. Some studies have fit these data against the ion size in the size-modified Poisson-Boltzmann equation, SMPBE, but the present study demonstrates that other parameters, such as the Stern layer thickness and the molecular surface definition, can change the number of bound ions by amounts comparable to varying the ion size. These parameters will therefore have to be fit simultaneously against experimental data. In addition, the data presented here demonstrate that the derivative, SK, of the electrostatic binding free energy, ΔGel, with respect to the logarithm of the salt concentration is sensitive to these parameters, and experimental measurements of SK could be used to parameterize the model. However, although better values for the Stern layer thickness and ion size and better molecular surface definitions could improve the model's predictions of the numbers of ions around biomolecules and SK, ΔGel itself is more sensitive to parameters, such as the interior dielectric constant, which in turn do not significantly affect the distributions of ions around biomolecules. Therefore, improved estimates of the ion size and Stern layer thickness to use in the SMPBE will not necessarily improve the model's predictions of ΔGel.

A new Green's function Monte Carlo algorithm for the solution of the three-dimensional nonlinear Poisson–Boltzmann equation: Application to the modeling of plasma sheath layers

Abstract. The objective of this paper is the exposition of a recently-developed Green's function Monte Carlo (GFMC) algorithm for the solution of nonlinear partial differential equations and its application to the modeling of the plasma sheath region around a spherical conducting object, carrying a potential and moving at low speeds through a partially ionized medium. The plasma sheath is modeled in equilibrium through the GFMC solution of the nonlinear Poisson–Boltzmann (NPB) equation. The traditional GFMC-based approaches for the solution of nonlinear equations involve the iterative solution of a series of linear problems. Over the last several years, one of the authors of this paper, K. Chatterjee, has been developing a philosophically different approach, where the linearization of the equation of interest is not required and hence there is no need for iteration. Instead, an approximate expression for the Green's function is obtained using perturbation theory, which is used to formulate the random walk equations within the problem sub-domains, where the random-walker makes its walks. On the other hand, as a trade-off, the dimensions of these sub-domains have to be restricted by the limitations imposed by perturbation theory. The greatest advantage of this approach is the ease and simplicity of parallelization stemming from the lack of the need for iteration, as a result of which the parallelization procedure is identical to the parallelization procedure for the GFMC solution of a linear problem. However, the premise of the algorithm is novel and rigorous mathematical justification has to be established in the future. The application areas of interest include the communication blackout problem during a space vehicle's re-entry into the atmosphere and electromagnetic propagation through the atmosphere/ionosphere in UHF/GPS applications.

High-Performance Computing and Big Data in Omics-Based Medicine

High-Performance Computing and Big Data in Omics-Based Medicine

A combined MPI-CUDA parallel solution of linear and nonlinear Poisson-Boltzmann equation.

The Poisson-Boltzmann equation models the electrostatic potential generated by fixed charges on a polarizable solute immersed in an ionic solution. This approach is often used in computational structural biology to estimate the electrostatic energetic component of the assembly of molecular biological systems. In the last decades, the amount of data concerning proteins and other biological macromolecules has remarkably increased. To fruitfully exploit these data, a huge computational power is needed as well as software tools capable of exploiting it. It is therefore necessary to move towards high performance computing and to develop proper parallel implementations of already existing and of novel algorithms. Nowadays, workstations can provide an amazing computational power: up to 10 TFLOPS on a single machine equipped with multiple CPUs and accelerators such as Intel Xeon Phi or GPU devices. The actual obstacle to the full exploitation of modern heterogeneous resources is efficient parallel coding and porting of software on such architectures. In this paper, we propose the implementation of a full Poisson-Boltzmann solver based on a finite-difference scheme using different and combined parallel schemes and in particular a mixed MPI-CUDA implementation. Results show great speedups when using the two schemes, achieving an 18.9x speedup using three GPUs.

Hot-Spots Detection - Application to a Variety of Different Protein-Based Systems

Protein-based complexes are the basis of many key systems in nature and have been the subject of intense research in the last decades [1]. Computational Alanine Scanning Mutagenesis approaches have been extensively used in the determination of the most important residues for complex formation, the Hot-spots [1-3]. However, this approach is usually applied to the study of protein-protein interfaces. In our work we focused on study of protein-DNA interfaces, which are also crucial in nature, but have not been the subject of as much research. We carry out MD simulations of several protein-DNA complexes and tested the influence of the variation of different parameters on the determination of the binding free energy terms of 78 mutations: solvent representation, internal dielectric constant, Linear and Nonlinear Poisson-Boltzmann equation, simulation time, number of structures analyzed and energetic terms involved [4]. In a different perspective, we have also explored the machine-learning approaches as these are faster while maintaining accuracy. We tested a group of twelve SASA-based features for their ability to correlate and differentiate hot-spots. The features tested were shown to be capable of discerning between hot- and null-spots, while presenting low correlations. A new method using support machine learning algorithms was developed: SBHD (Sasa-Based Hot-spot Detection) [5]. With these advances in the field druggable sites can be easier found in a variety of protein-based systems.[1] Moreira IS, Fernandes PA, Ramos MJ, Proteins, 2007, 68: 803-812.[2] Moreira IS, Fernandes PA, Ramos MJ, J. Comput. Chem., 2007, 28:644-654.[3] Pimenta AC, Martins JM, Fernandes R, Moreira IS, 2013, J. Chem. Info. Model- Online.[4] Ramos RM, Moreira IS, 2013, J. Chem. Theor. Comput., 9, 4243-4256.[5] Martins JM, Ramos RM, Pimenta AC, Moreira IS, 2013, Proteins- Online.

Operator splitting ADI schemes for pseudo-time coupled nonlinear solvation simulations

This work introduces novel operator splitting alternating direction implicit (ADI) schemes to overcome numerical difficulties in solving pseudo-time coupled nonlinear partial differential equations (PDEs) for biomolecular solvation analysis. Based on the variational formulation, a pseudo-transient continuation model has been previously formulated to couple a nonlinear Poisson–Boltzmann (NPB) equation for the electrostatic potential with a generalized Laplace–Beltrami equation defining the biomolecular surface. However, the standard numerical integration of the time dependent NPB equation is known to be very inefficient. Moreover, it encounters instability issues for smoothly varying solute–solvent interfaces so that a filtering process has to be conducted. In the present work, we propose to solve the unsteady NPB equation in an operator splitting framework so that the nonlinear instability can be completely avoided through an analytical integration. Central finite differences are employed to discretize the nonhomogeneous diffusion term of the generalized NPB equation to form tridiagonal matrices in the Douglas and Douglas–Rachford type ADI schemes. The proposed time splitting ADI schemes are found to be unconditionally stable for solving the NPB equation in benchmark examples with analytical solutions. For the solvation analysis involving two pseudo-time coupled nonlinear PDEs, the time stability of the NPB equation can be maintained by using very large time increments, so that without sacrificing the accuracy, the present biomolecular simulation becomes over ten times faster.

Design parameters for voltage-controllable directed assembly of single nanoparticles

Techniques to reliably pick-and-place single nanoparticles into functional assemblies are required to incorporate exotic nanoparticles into standard electronic circuits. In this paper we explore the use of electric fields to drive and direct the assembly process, which has the advantage of being able to control the nano-assembly process at the single nanoparticle level. To achieve this, we design an electrostatic gating system, thus enabling a voltage-controllable nanoparticle picking technique. Simulating this system with the nonlinear Poisson–Boltzmann equation, we can successfully characterize the parameters required for single particle placement, the key being single particle selectivity, in effect designing a system that can achieve this controllably. We then present the optimum design parameters required for successful single nanoparticle placement at ambient temperature, an important requirement for nanomanufacturing processes.

Electrostatic forces in the Poisson-Boltzmann systems.

Continuum modeling of electrostatic interactions based upon numerical solutions of the Poisson-Boltzmann equation has been widely used in structural and functional analyses of biomolecules. A limitation of the numerical strategies is that it is conceptually difficult to incorporate these types of models into molecular mechanics simulations, mainly because of the issue in assigning atomic forces. In this theoretical study, we first derived the Maxwell stress tensor for molecular systems obeying the full nonlinear Poisson-Boltzmann equation. We further derived formulations of analytical electrostatic forces given the Maxwell stress tensor and discussed the relations of the formulations with those published in the literature. We showed that the formulations derived from the Maxwell stress tensor require a weaker condition for its validity, applicable to nonlinear Poisson-Boltzmann systems with a finite number of singularities such as atomic point charges and the existence of discontinuous dielectric as in the widely used classical piece-wise constant dielectric models.

Computational Alanine Scanning Mutagenesis-An Improved Methodological Approach for Protein-DNA Complexes.

Proteins and protein-based complexes are the basis of many key systems in nature and have been the subject of intense research in the last decades, in an attempt to acquire comprehensive knowledge of reactions that take place in nature. Computational Alanine Scanning Mutagenesis approaches have been extensively used in the study of protein interfaces and in the determination of the most important residues for complex formation, the Hot-spots. However, as it is usually applied to the study of protein-protein interfaces, we tried to modify and apply it to the study of protein-DNA interfaces, which are also crucial in nature but have not been the subject of as much research. In this work, we carry out MD simulations of seven protein-DNA complexes and tested the influence of the variation of different parameters on the determination of the binding free energy terms (ΔΔGbinding) of 78 mutations: solvent representation, internal dielectric constant, Linear and Nonlinear Poisson-Boltzmann equation, Generalized Born model, simulation time, number of structures analyzed, number of MD trajectories, force field used, and energetic terms involved. Overall, this new approach gave an average error of 1.55 kcal/mol, and P, R, F1, accuracy, and specificity values of 0.78, 0.50, 0.61, 0.77, and 0.92, respectively. This improved computational alanine scanning mutagenesis approach may serve as a tool to explore the behavior of this important class of complexes.

Ion Competition in Condensed DNA Arrays in the Attractive Regime

Physical origin of DNA condensation by multivalent cations remains unsettled. Here, we report quantitative studies of how one DNA-condensing ion (Cobalt3+ Hexammine, or Co3+Hex) and one nonDNA-condensing ion (Mg2+) compete within the interstitial space in spontaneously condensed DNA arrays. As the ion concentrations in the bath solution are systematically varied, the ion contents and DNA-DNA spacings of the DNA arrays are determined by atomic emission spectroscopy and x-ray diffraction, respectively. To gain quantitative insights, we first compare the experimentally determined ion contents with predictions from exact numerical calculations based on nonlinear Poisson-Boltzmann equations. Such calculations are shown to significantly underestimate the number of Co3+Hex ions, consistent with the deficiencies of nonlinear Poisson-Boltzmann approaches in describing multivalent cations. Upon increasing the concentration of Mg2+, the Co3+Hex-condensed DNA array expands and eventually redissolves as a result of ion competition weakening DNA-DNA attraction. Although the DNA-DNA spacing depends on both Mg2+ and Co3+Hex concentrations in the bath solution, it is observed that the spacing is largely determined by a single parameter of the DNA array, the fraction of DNA charges neutralized by Co3+Hex. It is also observed that only ∼20% DNA charge neutralization by Co3+Hex is necessary for spontaneous DNA condensation. We then show that the bath ion conditions can be reduced to one variable with a simplistic ion binding model, which is able to describe the variations of both ion contents and DNA-DNA spacings reasonably well. Finally, we discuss the implications on the nature of interstitial ions and cation-mediated DNA-DNA interactions.

Separation of Ions Using Polyelectrolyte-Modified Nanoporous Track-Etched Membranes

Selective ion exclusion from charged nanopores in track-etched membranes allows separation of ions with different charges or mobilities. This study examines pressure-driven transport of dissolved ions through track-etched membranes modified by adsorption of poly(styrene sulfonate) (PSS)/protonated poly(allylamine) (PAH) films. For nominal 30 nm pores modified with a single layer of PSS, Br(-)/SO4(2-) selectivities are ∼3.4 with SO4(2-) rejections around 85% due to selective electrostatic exclusion of the divalent anion from the negatively charged pore. Corresponding membranes containing an adsorbed PSS/PAH bilayer are positively charged and exhibit average K(+)/Mg(2+) selectivities >10 at 8 mM ionic strength, and Mg(2+) rejections are >97.5% at ionic strengths <5 mM. The high rejection of Mg(2+) compared to SO4(2-) likely results from both a smaller pore size after deposition of the PAH layer and higher surface charge because of Mg(2+) adsorption. Simultaneous modeling of K(+) and Mg(2+) rejections using the nonlinearized Poisson-Boltzmann equation gives an average modified pore diameter of 8.4 ± 2.1 nm, which does not vary significantly with ionic strength. This diameter is smaller than that calculated from hydraulic permeabilities and estimated pore densities, suggesting that narrow regions near the pore entrance control ion transport. In addition to simple electrostatic exclusion, streaming potentials lead to differing rejections of Br(-) and acetate in PSS/PAH-modified pores, and of Li(+) and Cs(+) in PSS-modified pores. For these cases, electrical migration of ions toward the feed solution results in higher rejection of the more mobile ion.

A New Approach to Implement Absorbing Boundary Condition in Biomolecular Electrostatics

This paper discusses a novel approach to employ the absorbing boundary condition in conjunction with the finite-element method (FEM) in biomolecular electrostatics. The introduction of Bayliss-Turkel absorbing boundary operators in electromagnetic scattering problem has been incorporated by few researchers. However, in the area of biomolecular electrostatics, this boundary condition has not been investigated yet. The objective of this paper is twofold. First, to solve nonlinear Poisson-Boltzmann equation using Newton's method and second, to find an efficient and acceptable solution with minimum number of unknowns. In this work, a Galerkin finite-element formulation is used along with a Bayliss-Turkel absorbing boundary operator that explicitly accounts for the open field problem by mapping the Sommerfeld radiation condition from the far field to near field. While the Bayliss-Turkel condition works well when the artificial boundary is far from the scatterer, an acceptable tolerance of error can be achieved with the second order operator. Numerical results on test case with simple sphere show that the treatment is able to reach the same level of accuracy achieved by the analytical method while using a lower grid density. Bayliss-Turkel absorbing boundary condition (BTABC) combined with the FEM converges to the exact solution of scattering problems to within discretization error.

Fully implicit ADI schemes for solving the nonlinear Poisson-Boltzmann equation

Abstract The Poisson-Boltzmann (PB) model is an effective approach for the electrostatics analysis of solvated biomolecules. The nonlinearity associated with the PB equation is critical when the underlying electrostatic potential is strong, but is extremely difficult to solve numerically. In this paper, we construct two operator splitting alternating direction implicit (ADI) schemes to efficiently and stably solve the nonlinear PB equation in a pseudo-transient continuation approach. The operator splitting framework enables an analytical integration of the nonlinear term that suppresses the nonlinear instability. A standard finite difference scheme weighted by piecewise dielectric constants varying across the molecular surface is employed to discretize the nonhomogeneous diffusion term of the nonlinear PB equation, and yields tridiagonal matrices in the Douglas and Douglas-Rachford type ADI schemes. The proposed time splitting ADI schemes are different from all existing pseudo-transient continuation approaches for solving the classical nonlinear PB equation in the sense that they are fully implicit. In a numerical benchmark example, the steady state solutions of the fully-implicit ADI schemes based on different initial values all converge to the time invariant analytical solution, while those of the explicit Euler and semi-implicit ADI schemes blow up when the magnitude of the initial solution is large. For the solvation analysis in applications to real biomolecules with various sizes, the time stability of the proposed ADI schemes can be maintained even using very large time increments, demonstrating the efficiency and stability of the present methods for biomolecular simulation.

Electroosmotic Augmentation in Flexural Plate Wave Micropumps

We investigate the effectiveness and applicability of electroosmotic augmentation in flexural plate wave (FPW) micropumps for enhanced capabilities. Flow rates generated in FPW micro-scale flow systems are restricted particularly when the channel height is greater than the acoustic wave length. The proposed concept can be exploited to integrate micropumps into complex microfluidic chips improving the portability of micro-total-analysis systems along with the capabilities of actively controlling acoustics and electrokinetics for micro-mixer applications. A computational study of electroosmotic augmentation in FPW micropumps is presented where FPWs are considered by a moving wall model. A transient analysis of compressible flows of water is performed for microchannels. An isothermal equation of state for water is employed. The nonlinear Poisson-Boltzmann and Laplace equations are used to model the induced electric double layer (EDL) potential and the applied electric potential. Coupled electroosmotic and acoustics cases are investigated for two channel heights while the electric field intensity of the electrokinetic body forces and actuation frequency of acoustic excitations are varied. For deeper microchannels, increasing the actuation frequency of the wall motion does not improve the generated flow rate significantly. Inclusion of electroosmotic effects is more efficient than increasing the intensity of acoustic perturbations whenever high flow rates are required in micro-mixer applications.

Electrostatic free energy for a confined nanoscale object in a fluid

We present numerical calculations of electrostatic free energies, based on the nonlinear Poisson-Boltzmann (PB) equation, for the case of an isolated spherical nano-object in an aqueous suspension, interacting with charged bounding walls. We focus on systems with a low concentration of monovalent ions (≲10(-4) M), where the range of electrostatic interactions is long (~30 nm) and comparable to the system and object dimensions (~100 nm). Locally tailoring the geometry of the boundaries creates a modulation in the object-wall interaction, which for appropriately chosen system dimensions can be strong enough to result in stable spatial trapping of a nanoscale entity. A detailed view of the underlying mechanism of the trap shows that the physics depends predominantly on counterion entropy and the depth of the potential well is effectively independent of the object's dielectric function; we further note an appreciable trap depth even for an uncharged object in the fluid. These calculations not only provide a quantitative framework for understanding geometry-driven electrostatic effects at the nanoscale, but will also aid in identifying contributions from phenomena beyond mean field PB electrostatics, e.g., Casimir and other fluctuation-driven forces.