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.

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.

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 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.

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.

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.

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.

Conic shapes have higher sensitivity than cylindrical ones in nanopore DNA sequencing.

Nanopores have emerged as helpful research tools for single molecule detection. Through continuum modeling, we investigated the effects of membrane thickness, nanopore size, and pore shape on current signal characteristics of DNA. The simulation results showed that, when reducing the pore diameter, the amplitudes of current signals of DNA increase. Moreover, we found that, compared to cylindrically shaped nanopores, conical-shaped nanopores produce greater signal amplitudes from biomolecules translocation. Finally, we demonstrated that continuum model simulations for the discrimination of DNA and RNA yield current characteristics approximately consistent with experimental measurements and that A-T and G-C base pairs can be distinguished using thin conical solid-state nanopores. Our study not only suggests that computational approaches in this work can be used to guide the designs of nanopore for single molecule detection, but it also provides several possible ways to improve the current amplitudes of nanopores for better resolution.

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.

Frontiers in biomolecular mesh generation and molecular visualization systems.

With the development of biomolecular modeling and simulation, especially implicit solvent modeling, higher requirements are set for the stability, efficiency and mesh quality of molecular mesh generation software. In this review, we summarize the recent works in biomolecular mesh generation and molecular visualization. First, we introduce various definitions of molecular surface and corresponding meshing software. Second, as the mesh quality significantly influences biomolecular simulation, we investigate some remeshing methods in the fields of computer graphics and molecular modeling. Then, we show the application of biomolecular mesh in the boundary element method (BEM) and the finite element method (FEM). Finally, to conveniently visualize the numerical results based on the mesh, we present two types of molecular visualization systems.

Quality improvement of surface triangular mesh using a modified Laplacian smoothing approach avoiding intersection.

We present a systematic procedure to improve the qualities of triangular molecular surface meshes and at the same time preserve the manifoldness. The procedure utilizes an algorithm to remove redundant points having three or four valences and another algorithm to smooth the mesh using a modified version of Laplacian method without causing intersecting triangles. This approach can be effectively applied to any manifold surface meshes with arbitrary complex geometry. In this paper, the tested meshes are biomolecular surface meshes exhibiting typically highly irregular geometry. The results show that the qualities of the surface meshes are greatly improved and the manifoldness of the surface meshes are preserved. Compared with the original meshes, these improved molecular surface meshes can be directly applied to boundary element simulations and generation of body-fitted volume meshes using Tetgen. The procedure has been incorporated into our triangular molecular surface mesh generator, TMSmesh 2.0. It can be also used as a standalone program and works together with any other surface triangular mesh generator to obtain qualified manifold mesh. The package is downloadable at https://doi.org/10.6084/m9.figshare.5346169.v1 and can be run online at http://www.xyzgate.com.

Incorporating Born solvation energy into the three-dimensional Poisson-Nernst-Planck model to study ion selectivity in KcsA K

Potassium channels are much more permeable to potassium than sodium ions, although potassium ions are larger and both carry the same positive charge. This puzzle cannot be solved based on the traditional Poisson-Nernst-Planck (PNP) theory of electrodiffusion because the PNP model treats all ions as point charges, does not incorporate ion size information, and therefore cannot discriminate potassium from sodium ions. The PNP model can qualitatively capture some macroscopic properties of certain channel systems such as current-voltage characteristics, conductance rectification, and inverse membrane potential. However, the traditional PNP model is a continuum mean-field model and has no or underestimates the discrete ion effects, in particular the ion solvation or self-energy (which can be described by Born model). It is known that the dehydration effect (closely related to ion size) is crucial to selective permeation in potassium channels. Therefore, we incorporated Born solvation energy into the PNP model to account for ion hydration and dehydration effects when passing through inhomogeneous dielectric channel environments. A variational approach was adopted to derive a Born-energy-modified PNP (BPNP) model. The model was applied to study a cylindrical nanopore and a realistic KcsA channel, and three-dimensional finite element simulations were performed. The BPNP model can distinguish different ion species by ion radius and predict selectivity for K^{+} over Na^{+} in KcsA channels. Furthermore, ion current rectification in the KcsA channel was observed by both the PNP and BPNP models. The I-V curve of the BPNP model for the KcsA channel indicated an inward rectifier effect for K^{+} (rectification ratio of ∼3/2) but indicated an outward rectifier effect for Na^{+} (rectification ratio of ∼1/6).

An effective sequence-alignment-free superpositioning of pairwise or multiple structures with missing data.

BACKGROUND: Superpositioning is an important problem in structural biology. Determining an optimal superposition requires a one-to-one correspondence between the atoms of two proteins structures. However, in practice, some atoms are missing from their original structures. Current superposition implementations address the missing data crudely by ignoring such atoms from their structures. RESULTS: In this paper, we propose an effective method for superpositioning pairwise and multiple structures without sequence alignment. It is a two-stage procedure including data reduction and data registration. CONCLUSIONS: Numerical experiments demonstrated that our method is effective and efficient. The code package of protein structure superposition method for addressing the cases with missing data is implemented by MATLAB, and it is freely available from: http://sourceforge.net/projects/pssm123/files/?source=navbar.

Charged Substrate and Product Together Contribute Like a Nonreactive Species to the Overall Electrostatic Steering in Diffusion-Reaction Processes.

The Debye-Hückel limiting law is used to study the binding kinetics of substrate-enzyme system as well as to estimate the reaction rate of a electrostatically steered diffusion-controlled reaction process. It is based on a linearized Poisson-Boltzmann model and known for its accurate predictions in dilute solutions. However, the substrate and product particles are in nonequilibrium states and are possibly charged, and their contributions to the total electrostatic field cannot be explicitly studied in the Poisson-Boltzmann model. Hence the influences of substrate and product on reaction rate coefficient were not known. In this work, we consider all the charged species, including the charged substrate, product, and mobile salt ions in a Poisson-Nernst-Planck model, and then compare the results with previous work. The results indicate that both the charged substrate and product can significantly influence the reaction rate coefficient with different behaviors under different setups of computational conditions. It is interesting to find that when substrate and product are both considered, under an overall neutral boundary condition for all the bulk charged species, the computed reaction rate kinetics recovers a similar Debye-Hückel limiting law again. This phenomenon implies that the charged product counteracts the influence of charged substrate on reaction rate coefficient. Our analysis discloses the fact that the total charge concentration of substrate and product, though in a nonequilibrium state individually, obeys an equilibrium Boltzmann distribution, and therefore contributes as a normal charged ion species to ionic strength. This explains why the Debye-Hückel limiting law still works in a considerable range of conditions even though the effects of charged substrate and product particles are not specifically and explicitly considered in the theory.

Parameterization for molecular Gaussian surface and a comparison study of surface mesh generation.

The molecular Gaussian surface has been frequently used in the field of molecular modeling and simulation. Typically, the Gaussian surface is defined using two controlling parameters; the decay rate and isovalue. Currently, there is a lack of studies in which a systematic approach in the determination of optimal parameterization according to the geometric features has been done. In this paper, surface area, volume enclosed by the surface and Hausdorff distance are used as three criteria for the parameterization to make the Gaussian surface approximate the solvent excluded surface (SES) well. For each of these three criteria, a search of the parameter space is carried out in order to determine the optimal parameter values. The resulted parameters are close to each other and result in similar calculated molecular properties. Approximation of the VDW surface is also done by analyzing the explicit expressions of the Gaussian surface and VDW surface, which analysis and parameters can be similarly applied to the solvent accessible surface (SAS) due to its geometric similarity to the VDW surface. Once the optimal parameters are obtained, we compare the performance of our Gaussian surface generation software TMSmesh with other commonly used software programs, focusing primarily on mesh quality and fidelity. Additionally, the Poisson-Boltzmann solvation energies based on the surface meshes generated by TMSmesh and those generated by other software programs are calculated and compared for a set of molecules with different sizes. The results of these comparisons validate both the accuracy and the applicability of the parameterized Gaussian surface.

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.

VCMM: a visual tool for continuum molecular modeling.

This paper describes the design and function of a visualization tool, VCMM, for visualizing and analyzing data, and interfacing solvers for generic continuum molecular modeling. In particular, an emphasis of the program is to treat the data set based on unstructured mesh as used in finite/boundary element simulations, which largely enhances the capabilities of current visualization tools in this area that only support structured mesh. VCMM is segmented into molecular, meshing and numerical modules. The capabilities of molecular module include molecular visualization and force field assignment. Meshing module contains mesh generation, analysis and visualization tools. Numerical module currently provides a few finite/boundary element solvers of continuum molecular modeling, and contains several common visualization tools for the numerical result such as line and plane interpolations, surface probing, volume rendering and stream rendering. Three modules can exchange data with each other and carry out a complete process of modeling. Interfaces are also designed in order to facilitate usage of other mesh generation tools and numerical solvers. We develop a technique to accelerate data retrieval and have combined many graphical techniques in visualization. VCMM is highly extensible, and users can obtain more powerful functions by introducing relevant plug-ins. VCMM can also be useful in other fields such as computational quantum chemistry, image processing, and material science.

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.