An inverse averaging finite element method for solving three-dimensional Poisson-Nernst-Planck equations in nanopore system simulations.

The Poisson-Nernst-Planck (PNP) model plays an important role in simulating nanopore systems. In nanopore simulations, the large-size nanopore system and convection-domination Nernst-Planck (NP) equations will bring convergence difficulties and numerical instability problems. Therefore, we propose an improved finite element method (FEM) with an inverse averaging technique to solve the three-dimensional PNP model, named inverse averaging FEM (IAFEM). At first, the Slotboom variables are introduced aiming at transforming non-symmetric NP equations into self-adjoint second-order elliptic equations with exponentially behaved coefficients. Then, these exponential coefficients are approximated with their harmonic averages, which are calculated with an inverse averaging technique on every edge of each tetrahedral element in the grid. Our scheme shows good convergence when simulating single or porous nanopore systems. In addition, it is still stable when the NP equations are convection domination. Our method can also guarantee the conservation of computed currents well, which is the advantage that many stabilization schemes do not possess. Our numerical experiments on benchmark problems verify the accuracy and robustness of our scheme. The numerical results also show that the method performs better than the standard FEM when dealing with convection-domination problems. A successful simulation combined with realistic chemical experiments is also presented to illustrate that the IAFEM is still effective for three-dimensional interconnected nanopore systems.

Improved Ion Transport and High Energy Conversion through Hydrogel Membrane with 3D Interconnected Nanopores.

To mimic and use the functions of the ion transport system that are central to biological processes, bioinspired ion-selective membranes are developed and show great potential in a variety of fields. However, the practical applications of them are now limited due to low pore density, low conductivity, or scale-up difficulty. Herein, we demonstrate a 2-hydroxyethyl methacrylate phosphate (HEMAP) hydrogel membrane with 3D interconnected nanopores and space charged through simple photopolymerization. The HEMAP hydrogel membrane exhibits high conductance and outstanding ion selectivity, and the membrane-based osmotic power generator shows the excellent output power density up to 5.38 W/m2. Both experimentally and theoretically, the 3D interconnected structure is revealed to play a key role in enhancing charge-governed ion transport and energy conversion. This work highlights the advantages of 3D interconnected nanopores in ion diffusion and shows the potential of our designed hydrogel membrane in osmotic energy conversion, water desalination, and sensors.

Molecular Sparse Representation by a 3D Ellipsoid Radial Basis Function Neural Network via L1 Regularization.

The three-dimensional structures and shapes of biomolecules provide essential information about their interactions and functions. Unfortunately, the computational cost of biomolecular shape representation is an active challenge which increases rapidly as the number of atoms increase. Recent developments in sparse representation and deep learning have shown significant improvements in terms of time and space. A sparse representation of molecular shape is also useful in various other applications, such as molecular structure alignment, docking, and coarse-grained molecular modeling. We have developed an ellipsoid radial basis function neural network (ERBFNN) and an algorithm for sparsely representing molecular shape. To evaluate a sparse representation model of molecular shape, the Gaussian density map of the molecule is approximated using ERBFNN with a relatively small number of neurons. The deep learning models were trained by optimizing a nonlinear loss function with L1 regularization. Experimental results reveal that our algorithm can represent the original molecular shape with a relatively higher accuracy and fewer scale of ERBFNN. Our network in principle is applicable to the multiresolution sparse representation of molecular shape and coarse-grained molecular modeling. Executable files are available at https://github.com/SGUI-LSEC/SparseGaussianMolecule. The program was implemented in PyTorch and was run on Linux.

Molecular Surface Remeshing with Local Region Refinement.

Molecular surface mesh generation is a prerequisite for using the boundary element method (BEM) and finite element method (FEM) in implicit-solvent modeling. Molecular surface meshes typically have small angles, redundant vertices, and low-quality elements. In the implicit-solvent modeling of biomolecular systems it is usually required to improve the mesh quality and eliminate low-quality elements. Existing methods often fail to efficiently remove low-quality elements, especially in complex molecular meshes. In this paper, we propose a mesh refinement method that smooths the meshes, eliminates invalid regions in a cut-and-fill strategy, and improves the minimal angle. We compared our method with four different state-of-the-art methods and found that our method showed a significant improvement over state-of-the-art methods in minimal angle, aspect ratio, and other meshing quality measurements. In addition, our method showed satisfactory results in terms of the ratio of regular vertices and the preservation of area and volume.

A Finite Element Solution of Lateral Periodic Poisson-Boltzmann Model for Membrane Channel Proteins.

Membrane channel proteins control the diffusion of ions across biological membranes. They are closely related to the processes of various organizational mechanisms, such as: cardiac impulse, muscle contraction and hormone secretion. Introducing a membrane region into implicit solvation models extends the ability of the Poisson-Boltzmann (PB) equation to handle membrane proteins. The use of lateral periodic boundary conditions can properly simulate the discrete distribution of membrane proteins on the membrane plane and avoid boundary effects, which are caused by the finite box size in the traditional PB calculations. In this work, we: (1) develop a first finite element solver (FEPB) to solve the PB equation with a two-dimensional periodicity for membrane channel proteins, with different numerical treatments of the singular charges distributions in the channel protein; (2) add the membrane as a dielectric slab in the PB model, and use an improved mesh construction method to automatically identify the membrane channel/pore region even with a tilt angle relative to the z-axis; and (3) add a non-polar solvation energy term to complete the estimation of the total solvation energy of a membrane protein. A mesh resolution of about 0.25 Å (cubic grid space)/0.36 Å (tetrahedron edge length) is found to be most accurate in linear finite element calculation of the PB solvation energy. Computational studies are performed on a few exemplary molecules. The results indicate that all factors, the membrane thickness, the length of periodic box, membrane dielectric constant, pore region dielectric constant, and ionic strength, have individually considerable influence on the solvation energy of a channel protein. This demonstrates the necessity to treat all of those effects in the PB model for membrane protein simulations.

An ionic concentration and size dependent dielectric permittivity Poisson-Boltzmann model for biomolecular solvation studies.

By considering the influence of volume exclusion on the solvent dielectric, a variable dielectric Poisson-Boltzmann (VDPB) model is explored for molecular solvation studies by using a dielectric as an explicit function of ionic sizes and concentrations. A finite element method is adopted and an iterative strategy is introduced to numerically solve the VDPB equation. According to our computations, the current dielectric model can result in considerable differences compared with the traditional Poisson-Boltzmann (PB) solutions, especially for those systems with highly charged biomolecule and/or under high salt concentration condition. The model to certain extent captures the fact of dielectric decrement of electrolyte solutions, which is especially remarkable in the vicinity of molecules. Counter-ion concentration very near the molecular surface in VDPB calculation is found higher than that in PB. The new dielectric model may also influence the charge compensation behavior near biomolecular surface. For a spherical cavity solvated in a concentrated ionic solution, charge inversion is observed in VDPB, which does not occur with the traditional PB model. Besides, the solvation energy predicted by VDPB will always be greater than that by PB. Moreover, differing from PB, the VDPB also allows non-monotonous dependencies of solvation energy on ionic strength.

Ionic size effects to molecular solvation energy and to ion current across a channel resulted from the nonuniform size-modified PNP equations.

Ionic finite size can impose considerable effects to both the equilibrium and non-equilibrium properties of a solvated molecular system, such as the solvation energy, ionic concentration, and transport in a channel. As discussed in our former work [B. Lu and Y. C. Zhou, Biophys. J. 100, 2475 (2011)], a class of size-modified Poisson-Boltzmann (PB)/Poisson-Nernst-Planck (PNP) models can be uniformly studied through the general nonuniform size-modified PNP (SMPNP) equations deduced from the extended free energy functional of Borukhov et al. [I. Borukhov, D. Andelman, and H. Orland, Phys. Rev. Lett. 79, 435 (1997)] This work focuses on the nonuniform size effects to molecular solvation energy and to ion current across a channel for real biomolecular systems. The main contributions are: (1) we prove that for solvation energy calculation with nonuniform size effects (through equilibrium SMPNP simulation), there exists a simplified approximation formulation which is the same as the widely used one in PB community. This approximate form avoids integration over the whole domain and makes energy calculations convenient. (2) Numerical calculations show that ionic size effects tend to negate the solvation effects, which indicates that a higher molecular solvation energy (lower absolute value) is to be predicted when ionic size effects are considered. For both calculations on a protein and a DNA fragment systems in a 0.5M 1:1 ionic solution, a difference about 10 kcal/mol in solvation energies is found between the PB and the SMPNP predictions. Moreover, it is observed that the solvation energy decreases as ionic strength increases, which behavior is similar as those predicted by the traditional PB equation (without size effect) and by the uniform size-modified Poisson-Boltzmann equation. (3) Nonequilibrium SMPNP simulations of ion permeation through a gramicidin A channel show that the ionic size effects lead to reduced ion current inside the channel compared with the results without considering size effects. As a component of the current, the drift term is the main contribution to the total current. The ionic size effects to the total current almost come through the drift term, and have little influence on the diffusion terms in SMPNP.

An Unfitted Finite Element Poisson-Boltzmann Solver with Automatic Resolving of Curved Molecular Surface.

So far, the existing Poisson-Boltzmann (PB) solvers that accurately take into account the interface jump conditions need a pregenerated body-fitted mesh (molecular surface mesh). However, qualified biomolecular surface meshing and its implementation into numerical methods remains a challenging and laborious issue, which practically hinders the progress of further developments and applications of a bunch of numerical methods in this field. In addition, even with a molecular surface mesh, it is only a low-order approximation of the original curved surface. In this article, an interface-penalty finite element method (IPFEM), which is a typical unfitted finite element method, is proposed to solve the Poisson-Boltzmann equation (PBE) without requiring the user to generate a molecular surface mesh. The Gaussian molecular surface is used to represent the molecular surface and can be automatically resolved with a high-order approximation within our method. Theoretical convergence rates of the IPFEM for the linear PB equation have been provided and are well validated on a benchmark problem with an analytical solution (we also noticed from numerical examples that the IPFEM has similar convergence rates for the nonlinear PBE). Numerical results on a set of different-sized biomolecules demonstrate that the IPFEM is numerically stable and accurate in the calculation of biomolecular electrostatic solvation energy.

A parallel finite element simulator for ion transport through three-dimensional ion channel systems.

A parallel finite element simulator, ichannel, is developed for ion transport through three-dimensional ion channel systems that consist of protein and membrane. The coordinates of heavy atoms of the protein are taken from the Protein Data Bank and the membrane is represented as a slab. The simulator contains two components: a parallel adaptive finite element solver for a set of Poisson-Nernst-Planck (PNP) equations that describe the electrodiffusion process of ion transport, and a mesh generation tool chain for ion channel systems, which is an essential component for the finite element computations. The finite element method has advantages in modeling irregular geometries and complex boundary conditions. We have built a tool chain to get the surface and volume mesh for ion channel systems, which consists of a set of mesh generation tools. The adaptive finite element solver in our simulator is implemented using the parallel adaptive finite element package Parallel Hierarchical Grid (PHG) developed by one of the authors, which provides the capability of doing large scale parallel computations with high parallel efficiency and the flexibility of choosing high order elements to achieve high order accuracy. The simulator is applied to a real transmembrane protein, the gramicidin A (gA) channel protein, to calculate the electrostatic potential, ion concentrations and I - V curve, with which both primitive and transformed PNP equations are studied and their numerical performances are compared. To further validate the method, we also apply the simulator to two other ion channel systems, the voltage dependent anion channel (VDAC) and α-Hemolysin (α-HL). The simulation results agree well with Brownian dynamics (BD) simulation results and experimental results. Moreover, because ionic finite size effects can be included in PNP model now, we also perform simulations using a size-modified PNP (SMPNP) model on VDAC and α-HL. It is shown that the size effects in SMPNP can effectively lead to reduced current in the channel, and the results are closer to BD simulation results.

Parameter-Efficient Densely Connected Dual Attention Network for Phonocardiogram Classification.

Cardiac auscultation, exhibited by phonocardiogram (PCG), is a non-invasive and low-cost diagnostic method for cardiovascular diseases (CVDs). However, deploying it in practice is quite challenging, due to the inherent murmurs and a limited number of supervised samples in heart sound data. To solve these problems, not only heart sound analysis based on handcrafted features, but also computer-aided heart sound analysis based on deep learning have been extensively studied in recent years. Though with elaborate design, most of these methods still use additional pre-processing to improve classification performance, which heavily relies on time-consuming experienced engineering. In this article, we propose a parameter-efficient densely connected dual attention network (DDA) for heart sound classification. It combines two advantages simultaneously of the purely end-to-end architecture and enriched contextual representations of the self-attention mechanism. Specifically, the densely connected structure can automatically extract the information flow of heart sound features hierarchically. Alongside, improving contextual modeling capabilities, the dual attention mechanism adaptively aggregates local features with global dependencies via a self-attention mechanism, which captures the semantic interdependencies across position and channel axes respectively. Extensive experiments across stratified 10-fold cross-validation strongly evidence that our proposed DDA model surpasses current 1D deep models on the challenging Cinc2016 benchmark with significant computational efficiency.

Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processes II: size effects on ionic distributions and diffusion-reaction rates.

The effects of finite particle size on electrostatics, density profiles, and diffusion have been a long existing topic in the study of ionic solution. The previous size-modified Poisson-Boltzmann and Poisson-Nernst-Planck models are revisited in this article. In contrast to many previous works that can only treat particle species with a single uniform size or two sizes, we generalize the Borukhov model to obtain a size-modified Poisson-Nernst-Planck (SMPNP) model that is able to treat nonuniform particle sizes. The numerical tractability of the model is demonstrated as well. The main contributions of this study are as follows. 1), We show that an (arbitrarily) size-modified PB model is indeed implied by the SMPNP equations under certain boundary/interface conditions, and can be reproduced through numerical solutions of the SMPNP. 2), The size effects in the SMPNP effectively reduce the densities of highly concentrated counterions around the biomolecule. 3), The SMPNP is applied to the diffusion-reaction process for the first time, to our knowledge. In the case of low substrate density near the enzyme reactive site, it is observed that the rate coefficients predicted by SMPNP model are considerably larger than those by the PNP model, suggesting both ions and substrates are subject to finite size effects. 4), An accurate finite element method and a convergent Gummel iteration are developed for the numerical solution of the completely coupled nonlinear system of SMPNP equations.

AFMPB: An Adaptive Fast Multipole Poisson-Boltzmann Solver for Calculating Electrostatics in Biomolecular Systems.

A Fortran program package is introduced for rapid evaluation of the electrostatic potentials and forces in biomolecular systems modeled by the linearized Poisson-Boltzmann equation. The numerical solver utilizes a well-conditioned boundary integral equation (BIE) formulation, a node-patch discretization scheme, a Krylov subspace iterative solver package with reverse communication protocols, and an adaptive new version of fast multipole method in which the exponential expansions are used to diagonalize the multipole to local translations. The program and its full description, as well as several closely related libraries and utility tools are available at http://lsec.cc.ac.cn/lubz/afmpb.html and a mirror site at http://mccammon.ucsd.edu/. This paper is a brief summary of the program: the algorithms, the implementation and the usage.

Solutions to a reduced Poisson-Nernst-Planck system and determination of reaction rates.

We study a reduced Poisson-Nernst-Planck (PNP) system for a charged spherical solute immersed in a solvent with multiple ionic or molecular species that are electrostatically neutralized in the far field. Some of these species are assumed to be in equilibrium. The concentrations of such species are described by the Boltzmann distributions that are further linearized. Others are assumed to be reactive, meaning that their concentrations vanish when in contact with the charged solute. We present both semi-analytical solutions and numerical iterative solutions to the underlying reduced PNP system, and calculate the reaction rate for the reactive species. We give a rigorous analysis on the convergence of our simple iteration algorithm. Our numerical results show the strong dependence of the reaction rates of the reactive species on the magnitude of its far field concentration as well as on the ionic strength of all the chemical species. We also find non-monotonicity of electrostatic potential in certain parameter regimes. The results for the reactive system and those for the non-reactive system are compared to show the significant differences between the two cases. Our approach provides a means of solving a PNP system which in general does not have a closed-form solution even with a special geometrical symmetry. Our findings can also be used to test other numerical methods in large-scale computational modeling of electro-diffusion in biological systems.

Improved Ion Transport in Hydrogel-Based Nanofluidics for Osmotic Energy Conversion.

In nature, ultrafast signal transfer based on ion transport, which is the foundation of biological processes, commonly works in a hydrogel-water mixed mechanism. Inspired by organisms' hydrogel-based system, we introduce hydrogel into nanofluidics to prepare a hydrogel hybrid membrane. The introduction of a space charged hydrogel improves the ion selectivity evidently. Also, a power generator based on the hydrogel hybrid membrane shows an excellent energy conversion property; a maximum power density up to 11.72 W/m2 is achieved at a 500-fold salinity gradient. Furthermore, the membrane shows excellent mechanical properties. These values are achievable, which indicates our membrane's huge potential applications in osmotic energy conversion.

Enzymatic activity versus structural dynamics: the case of acetylcholinesterase tetramer.

The function of many proteins, such as enzymes, is modulated by structural fluctuations. This is especially the case in gated diffusion-controlled reactions (where the rates of the initial diffusional encounter and of structural fluctuations determine the overall rate of the reaction) and in oligomeric proteins (where function often requires a coordinated movement of individual subunits). A classic example of a diffusion-controlled biological reaction catalyzed by an oligomeric enzyme is the hydrolysis of synaptic acetylcholine (ACh) by tetrameric acetylcholinesterase (AChEt). Despite decades of efforts, the extent to which enzymatic efficiency of AChEt (or any other enzyme) is modulated by flexibility is not fully determined. This article attempts to determine the correlation between the dynamics of AChEt and the rate of reaction between AChEt and ACh. We employed equilibrium and nonequilibrium electro-diffusion models to compute rate coefficients for an ensemble of structures generated by molecular dynamics simulation. We found that, for the static initial model, the average reaction rate per active site is approximately 22-30% slower in the tetramer than in the monomer. However, this effect of tetramerization is modulated by the intersubunit motions in the tetramer such that a complex interplay of steric and electrostatic effects either guides or blocks the substrate into or from each of the four active sites. As a result, the rate per active site calculated for some of the tetramer structures is only approximately 15% smaller than the rate in the monomer. We conclude that structural dynamics minimizes the adverse effect of tetramerization, allowing the enzyme to maintain similar enzymatic efficiency in different oligomerization states.

Free energy for the permeation of Na(+) and Cl(-) ions and their ion-pair through a zwitterionic dimyristoyl phosphatidylcholine lipid bilayer by umbrella integration with harmonic fourier beads.

Understanding the mechanism of ion permeation across lipid bilayers is key to controlling osmotic pressure and developing new ways of delivering charged, drug-like molecules inside cells. Recent reports suggest ion-pairing as the mechanism to lower the free energy barrier for the ion permeation in disagreement with predictions from the simple electrostatic models. In this paper we quantify the effect of ion-pairing or charge quenching on the permeation of Na(+) and Cl(-) ions across DMPC lipid bilayer by computing the corresponding potentials of mean force (PMFs) using fully atomistic molecular dynamics simulations. We find that the free energy barrier to permeation reduces in the order Na(+)-Cl(-) ion-pair (27.6 kcal/mol) > Cl(-) (23.6 kcal/mol) > Na(+) (21.9 kcal/mol). Furthermore, with the help of these PMFs we derive the change in the binding free energy between the Na(+) and Cl(-) with respect to that in water as a function of the bilayer permeation depth. Despite the fact that the bilayer boosts the Na(+)-Cl(-) ion binding free energy by as high as 17.9 kcal/mol near its center, ion-pairing between such hydrophilic ions as Na(+) and Cl(-) does not assist their permeation. However, based on a simple thermodynamic cycle, we suggest that ion-pairing between ions of opposite charge and solvent philicity could enhance ion permeation. Comparison of the computed permeation barriers for Na(+) and Cl(-) ions with available experimental data supports this notion. This work establishes general computational methodology to address ion-pairing in fluid anisotropic media and details the ion permeation mechanism on atomic level.

A computational study of ion conductance in the KcsA K(+) channel using a Nernst-Planck model with explicit resident ions.

The biophysical mechanisms underlying the relationship between the structure and function of the KcsA K(+) channel are described. Because of the conciseness of electrodiffusion theory and the computational advantages of a continuum approach, the Nernst-Planck (NP) type models, such as the Goldman-Hodgkin-Katz and Poisson-NP (PNP) models, have been used to describe currents in ion channels. However, the standard PNP (SPNP) model is known to be inapplicable to narrow ion channels because it cannot handle discrete ion properties. To overcome this weakness, the explicit resident ions NP (ERINP) model was formulated, which applies a local explicit model where the continuum model fails. Then, the effects of the ERI Coulomb potential, the ERI induced potential, and the ERI dielectric constant for ion conductance were tested in the ERINP model. The current-voltage (I-V) and current-concentration (I-C) relationships determined in the ERINP model provided biologically significant information that the traditional continuum model could not, explicitly taking into account the effects of resident ions inside the KcsA K(+) channel. In addition, a mathematical analysis of the K(+) ion dynamics established a tight structure-function system with a shallow well, a deep well, and two K(+) ions resident in the selectivity filter. Furthermore, the ERINP model not only reproduced the experimental results with a realistic set of parameters, but it also reduced CPU costs.

Toward a Model for Activation of Orai Channel.

Store-operated calcium release-activated calcium (CRAC) channels mediate a variety of cellular signaling functions. The CRAC channel pore-forming protein, Orai1, is a hexamer arranged with 3-fold symmetry. Despite its importance in moving Ca2+ ions into cells, a detailed mechanistic understanding of Orai1 activation is lacking. Herein, a working model is proposed for the putative open state of Orai from Drosophila melanogaster (dOrai), which involves a "twist-to-open" gating mechanism. The proposed model is supported by energetic, structural, and experimental evidence. Fluorescent imaging demonstrates that each subunit on the intracellular side of the pore is inherently strongly cross-linked, which is important for coupling to STIM1, the pore activator, and graded activation of the Orai1 channel. The proposed model thus paves the way for understanding key aspects of calcium signaling at a molecular level.

Continuum simulations of acetylcholine consumption by acetylcholinesterase: a Poisson-Nernst-Planck approach.

The Poisson-Nernst-Planck (PNP) equation provides a continuum description of electrostatic-driven diffusion and is used here to model the diffusion and reaction of acetylcholine (ACh) with acetylcholinesterase (AChE) enzymes. This study focuses on the effects of ion and substrate concentrations on the reaction rate and rate coefficient. To this end, the PNP equations are numerically solved with a hybrid finite element and boundary element method at a wide range of ion and substrate concentrations, and the results are compared with the partially coupled Smoluchowski-Poisson-Boltzmann model. The reaction rate is found to depend strongly on the concentrations of both the substrate and ions; this is explained by the competition between the intersubstrate repulsion and the ionic screening effects. The reaction rate coefficient is independent of the substrate concentration only at very high ion concentrations, whereas at low ion concentrations the behavior of the rate depends strongly on the substrate concentration. Moreover, at physiological ion concentrations, variations in substrate concentration significantly affect the transient behavior of the reaction. Our results offer a reliable estimate of reaction rates at various conditions and imply that the concentrations of charged substrates must be coupled with the electrostatic computation to provide a more realistic description of neurotransmission and other electrodiffusion and reaction processes.

Molecular surface-free continuum model for electrodiffusion processes.

Incorporation of van der Waals interactions enables the continuum model of electrodiffusion in biomolecular system to avoid the artifacts of introducing a molecular surface and the painful task of the surface mesh generation. Calculation examples show that the electrostatics, diffusion-reaction kinetics, and molecular surface defined as an isosurface of a certain density distribution can be extracted from the solution of the Poisson-Nernst-Planck equations using this model. The molecular surface-free model enables a wider usage of some modern numerical methodologies such as finite element methods for biomolecular modeling, and sheds light on a new paradigm of continuum modeling for biomolecular systems.