Poisson-Boltzmann-Nernst-Planck model

Bridging Innovations of Phase Change Heat Transfer to Electrochemical Gas Evolution Reactions

Bubbles play a ubiquitous role in electrochemical gas evolution reactions. However, a mechanistic understanding of how bubbles affect the energy efficiency of electrochemical processes remains limited to date, impeding effective approaches to further boost the performance of gas evolution systems. From a perspective of the analogy between heat and mass transfer, bubbles in electrochemical gas evolution reactions exhibit highly similar dynamic behaviors to them in the liquid-vapor phase change. Recent developments of liquid-vapor phase change systems have substantially advanced the fundamental knowledge of bubbles, leading to unprecedented enhancement of heat transfer performance. In this Review, we aim to elucidate a promising opportunity of understanding bubble dynamics in electrochemical gas evolution reactions through a lens of phase change heat transfer. We first provide a background about key parallels between electrochemical gas evolution reactions and phase change heat transfer. Then, we discuss bubble dynamics in gas evolution systems across multiple length scales, with an emphasis on exciting research problems inspired by new insights gained from liquid-vapor phase change systems. Lastly, we review advances in engineered surfaces for manipulating bubbles to enhance heat and mass transfer, providing an outlook on the design of high-performance gas evolving electrodes.

Ordered patterns in electroactive polymer ionic liquid blends: effect of long range interactions

Designing multifunctional soft materials via pattern formation has been a major challenge for scientists and engineers. Soft materials based on polymers are the perfect candidates for designing such materials as they are not only easy to handle, but also offer diverse combinations of mechanical and chemical properties. Here, we present a polymer-based ternary system and reveal, using modelling and simulations, the mechanisms for creating patterned surfaces. Specifically, we consider polymer ionic liquid (PIL) blends and demonstrate that exposure to a uniform electric field results in the formation of ordered patterns through phase separation. Our approach is based on reaction-diffusion phenomena and utilizes Poisson-Boltzmann-Nernst-Planck equations to capture the long-range interactions of ionic liquids in both weak and strong segregation limits. Furthermore, we elucidate that the ordered patterns in our PIL blend can be tuned by changing the direction of the electric field. From the structural characterization point of view, we reveal that the presence of the electric field significantly enhances the domain growth rate and their respective ordering in a remarkable fashion. We believe this non-invasive technique is a significant step towards the development of ordered structures at microscopic length scales and can be utilized for micro-scale fabrication from soft materials.

Exploring ssDNA translocation through α-hemolysin using coarse-grained steered molecular dynamics

Protein nanopores have proven to be effective for single-molecule studies, particularly for single-stranded DNA (ssDNA) translocation. Previous experiments demonstrated their ability to distinguish differences in purine and pyrimidine bases and in the orientation of the ssDNA molecule inside nanopores. Unfortunately, the microscopic details of ssDNA translocation over experimental time scales, which are not accessible through all-atom molecular dynamics (MD), have yet to be examined. However, coarse-grained (CG) MD simulations enable systems to be simulated over longer characteristic times closer to experiments than all-atom MD. This paper studies ssDNA translocation through α-hemolysin nanopores exploiting steered MD using the MARTINI CG force field. The impacts of the sequence length, orientation inside the nanopore and DNA charges on translocation dynamics as well as the conformational dynamics of ssDNA during the translocation are explored. Our results highlight the efficacy of CG molecular dynamics in capturing the experimental properties of ssDNA translocation, including a wide distribution in translocation times per base. In particular, the phosphate charges of the DNA molecule are crucial in the translocation dynamics and impact the translocation rate. Additionally, the influence of the ssDNA molecule orientation on the translocation rate is explained by the conformational differences of ssDNA inside the nanopore during its translocation. Our study emphasizes the significance of obtaining sufficient statistics via CG MD, which can elucidate the great variety of translocation processes.

A meshless stochastic method for Poisson-Nernst-Planck equations

A plethora of biological, physical, and chemical phenomena involve transport of charged particles (ions). Its continuum-scale description relies on the Poisson-Nernst-Planck (PNP) system, which encapsulates the conservation of mass and charge. The numerical solution of these coupled partial differential equations is challenging and suffers from both the curse of dimensionality and difficulty in efficiently parallelizing. We present a novel particle-based framework to solve the full PNP system by simulating a drift-diffusion process with time- and space-varying drift. We leverage Green's functions, kernel-independent fast multipole methods, and kernel density estimation to solve the PNP system in a meshless manner, capable of handling discontinuous initial states. The method is embarrassingly parallel, and the computational cost scales linearly with the number of particles and dimension. We use a series of numerical experiments to demonstrate both the method's convergence with respect to the number of particles and computational cost vis-à-vis a traditional partial differential equation solver.

Significantly Increased Specific Discharge Capacitance at Carbon Fibers Created via Architected Ultramicropores

Carbon is commonly used as an electrode material for supercapacitors operating on an electrical double-layer energy storage mechanism. However, the low specific capacitance limits its application. Increasing the specific surface area is by far the most common expansion method, and surprisingly, they are not always positively correlated. The overmuch specific surface will show the characteristics of nanoconfinement, and the potential synergistic enhancement mechanism of various key parameters is still controversial. In this work, carbon fiber electrodes with different ultramicropore structures were designed in order to improve the utilization rate and the discharge capacitance. It has been found that when the ultramicropore entrance's surface is too small, it will lead to the decrease of the external charge of the pore transport channel, and then, the selectivity of the opposite ions will decrease. The numerical simulation based on Poisson and Nernst-Planck equations also indicates that ions have difficulty diffusing into the micropores when their entrance surface decreases. Surface properties within the nanocontainment space become critical factors influencing ion transport and adsorption. The specific discharge capacitance of carbon fiber is increased from 3 to 1430 mF cm-2.

Na+-Mg2+ ion effects on conformation and translocation dynamics of single-stranded RNA: Cooperation and competition

The nanopore-based translocation of a single-stranded RNA (ssRNA) in mixed salt solution has garnered increasing interest for its biological and technological significance. However, it is challenging to comprehensively understand the effects of the mixed ion species on the translocation dynamics due to their cooperation and competition, which can be directly reflected by the ion screening and neutralizing effects, respectively. In this study, Langevin dynamics simulation is employed to investigate the properties of ssRNA conformation and translocation in mixed Na+-Mg2+ ion environments. Simulation results reveal that the ion screening effect dominates the change in the ssRNA conformational size, the ion neutralizing effect controls the capture rate of the ssRNA by the nanopore, and both of them take charge of the different changes in translocation time of the ssRNA under various mixed ion environments. Under high Na+ ion concentration, as Mg2+ concentration increases, the ion neutralizing effect strengthens, weakening the driving force inside the nanopore, leading to longer translocation time. Conversely, at low Na+ concentration, an increase in Mg2+ concentration enhances the ion screening effect, aiding in faster translocation. Furthermore, these simulation results will be explained by quantitative analysis, advancing a deeper understanding of the complicated effects of the mixed Na+-Mg2+ ions.

Partial mean-field model for neurotransmission dynamics

This article addresses reaction networks in which spatial and stochastic effects are of crucial importance. For such systems, particle-based models allow us to describe all microscopic details with high accuracy. However, they suffer from computational inefficiency if particle numbers and density get too large. Alternative coarse-grained-resolution models reduce computational effort tremendously, e.g., by replacing the particle distribution by a continuous concentration field governed by reaction-diffusion PDEs. We demonstrate how models on the different resolution levels can be combined into hybrid models that seamlessly combine the best of both worlds, describing molecular species with large copy numbers by macroscopic equations with spatial resolution while keeping the spatial-stochastic particle-based resolution level for the species with low copy numbers. To this end, we introduce a simple particle-based model for the binding dynamics of ions and vesicles at the heart of the neurotransmission process. Within this framework, we derive a novel hybrid model and present results from numerical experiments which demonstrate that the hybrid model allows for an accurate approximation of the full particle-based model in realistic scenarios.

Electric-field-induced oscillations in ionic fluids: a unified formulation of modified Poisson-Nernst-Planck models and its relevance to correlation function analysis

We theoretically investigate an electric-field-driven system of charged spheres as a primitive model of concentrated electrolytes under an applied electric field. First, we provide a unified formulation for the stochastic charge and density dynamics of the electric-field-driven primitive model using the stochastic density functional theory (DFT). The stochastic DFT integrates the four frameworks (the equilibrium and dynamic DFTs, the liquid state theory and the field-theoretic approach), which allows us to justify in a unified manner various modifications previously made for the Poisson-Nernst-Planck model. Next, we consider stationary density-density and charge-charge correlation functions of the primitive model with a static electric field. We predict an electric-field-induced synchronization between emergences of density and charge oscillations. We are mainly concerned with the emergence of stripe states formed by segregation bands transverse to the external field, thereby demonstrating the following: (i) the electric-field-induced crossover occurs prior to the conventional Kirkwood crossover without an applied electric field, and (ii) the ion concentration dependence of the decay lengths at the onset of oscillations bears a similarity to the underscreening behavior found by recent simulation and theoretical studies on equilibrium electrolytes. Also, the 2D inverse Fourier transform of the correlation function illustrates the existence of stripe states beyond the electric-field-induced Kirkwood crossover.

Design and simulation of a millifluidic device for differential detection of SARS-CoV-2 and H1N1 based on triboelectricity

Differential diagnosis of pathogenic diseases, presently coronavirus disease 2019 (COVID-19) and influenza, is crucial with due attention to their superspreading events, presumably long incubation period, particular complications, and treatments. In this paper, a label-free, self-powered, and ultrafast immunosensor device working based on triboelectric effect is proposed. Equilibrium constants of specific antibody-antigen reactions are accompanied by IEP-relevant electric charges of antigens to recognize SARS-CoV-2 and H1N1. Simulation attributes including fluid flow and geometrical parameters are optimized so that the maximum capture efficiency of 85.63% is achieved. Accordingly, antibody-antigen complexes form electric double layers (EDLs) across the channel interfaces. The resultant built-in electric field affects the following external electric field derived from triboelectricity, leading to the variation of open-circuit voltage as a sensing metric. The device is flexible to operate in conductor-to-dielectric single-electrode and contact-separation modes simultaneously. While the detection limit is reduced utilizing the single-electrode mode compared to the latter one, surface treatment of the triboelectric pair contributes to the sensitivity enhancement. A threshold value equal to −4.113 V is featured to discriminate these two viruses in a vast detectable region; however, further surface engineering can allow the on-site detection of any electrically-charged pathogen applying the emerging triboelectric immunosensor enjoying a lower detection limit.

Geometric algebra generation of molecular surfaces

Geometric algebra is a powerful framework that unifies mathematics and physics. Since its revival in the 1960s, it has attracted great attention and has been exploited in fields like physics, computer science and engineering. This work introduces a geometric algebra method for the molecular surface generation that uses the Clifford-Fourier transform (CFT) which is a generalization of the classical Fourier transform. Notably, the classical Fourier transform and CFT differ in the derivative property in [Formula: see text] for k even. This distinction is due to the non-commutativity of geometric product of pseudoscalars with multivectors and has significant consequences in applications. We use the CFT in [Formula: see text] to benefit from the derivative property in solving partial differential equations (PDEs). The CFT is used to solve the mode decomposition process in PDE transform. Two different initial cases are proposed to make the initial shapes in the present method. The proposed method is applied first to small molecules and proteins. To validate the method, the molecular surfaces generated are compared to surfaces of other definitions. Applications are considered to protein electrostatic surface potentials and solvation free energy. This work opens the door for further applications of geometric algebra and CFT in biological sciences.

Biomimetic Nanochannels: From Fabrication Principles to Theoretical Insights

Biological nanochannels which can regulate ionic transport across cell membranes intelligently play a significant role in physiological functions. Inspired by these nanochannels, numerous artificial nanochannels have been developed during recent years. The exploration of smart solid-state nanochannels can lay a solid foundation, not only for fundamental studies of biological systems but also practical applications in various fields. The basic fabrication principles, functional materials, and diverse applications based on artificial nanochannels are summarized in this review. In addition, theoretical insights into transport mechanisms and structure-function relationships are discussed. Meanwhile, it is believed that improvements will be made via computer-guided strategy in designing more efficient devices with upgrading accuracy. Finally, some remaining challenges and perspectives for developments in both novel conceptions and technology of this inspiring research field are stated.

Molecular Simulations to Elucidate Transport Phenomena in Polymeric Membranes

Despite decades of dominance in separation technology, progress in the design and development of high-performance polymer-based membranes has been incremental. Recent advances in materials science and chemical synthesis provide opportunities for molecular-level design of next-generation membrane materials. Such designs necessitate a fundamental understanding of transport and separation mechanisms at the molecular scale. Molecular simulations are important tools that could lead to the development of fundamental structure-property-performance relationships for advancing membrane design. In this Perspective, we assess the application and capability of molecular simulations to understand the mechanisms of ion and water transport across polymeric membranes. Additionally, we discuss the reliability of molecular models in mimicking the structure and chemistry of nanochannels and transport pathways in polymeric membranes. We conclude by providing research directions for resolving key knowledge gaps related to transport phenomena in polymeric membranes and for the construction of structure-property-performance relationships for the design of next-generation membranes.

Can ionic concentration changes due to mechanical deformation be responsible for the neurostimulation caused by focused ultrasound? A simulation study

Objective.Ultrasound stimulation is an emerging neuromodulation technique, for which the exact mechanism of action is still unknown. Despite the number of hypotheses such as mechanosensitive ion channels and intermembrane cavitation, they fail to explain all of the observed experimental effects. Here we are investigating the ionic concentration change as a prime mechanism for the neurostimulation by the ultrasound.Approach.We derive the direct analytical relationship between the mechanical deformations in the tissue and the electric boundary conditions for the cable theory equations and solve them for two types of neuronal axon models: Hodgkin-Huxley and C-fibre. We detect the activation thresholds for a variety of ultrasound stimulation cases including continuous and pulsed ultrasound and estimate the mechanical deformations required for reaching the thresholds and generating action potentials (APs).Main results.We note that the proposed mechanism strongly depends on the mechanical properties of the neural tissues, which at the moment cannot be located in literature with the required certainty. We conclude that given certain common linear assumptions, this mechanism alone cannot cause significant effects and be responsible for neurostimulation. However, we also conclude that if the lower estimation of mechanical properties of neural tissues in literature is true, or if the normal cavitation occurs during the ultrasound stimulation, the proposed mechanism can be a prime cause for the generation of APs.Significance.The approach allows prediction and modelling of most observed experimental effects, including the probabilistic ones, without the need for any extra physical effects or additional parameters.

Computational modeling of protein conformational changes - Application to the opening SARS-CoV-2 spike

We present a new approach to compute and analyze the dynamical electro-geometric properties of proteins undergoing conformational changes. The molecular trajectory is obtained from Markov state models, and the electrostatic potential is calculated using the continuum Poisson-Boltzmann equation. The numerical electric potential is constructed using a parallel sharp numerical solver implemented on adaptive Octree grids. We introduce novel a posteriori error estimates to quantify the solution's accuracy on the molecular surface. To illustrate the approach, we consider the opening of the SARS-CoV-2 spike protein using the recent molecular trajectory simulated through the Folding@home initiative. We analyze our results, focusing on the characteristics of the receptor-binding domain and its vicinity. This work lays the foundation for a new class of hybrid computational approaches, producing high-fidelity dynamical computational measurements serving as a basis for protein bio-mechanism investigations.

A continuum PNP model of double species ion transport between graphene

In this paper, we employ Poisson-Nernst-Planck equations in conjunction with the Navier-Stokes equations and the mean-field theory to investigate the charge transport of double species monovalent ions in double-layered graphene sheets, driven by an external electric field. Unlike most classical models, we develop a simple mechanical model by incorporating the microscopic effects of the physical systems so that the ionic interactions between ions and the host material, and the steric repulsions among ions are considered. Taking Li⁺-PF6^-1monovalent ions as an example, we find that the transport pattern for the present double species ionic systems and that for the pure lithium ions are dramatically different. Due to the mutual attractive ionic forces between ions and their counter-ions, such double species systems turn out to be more stable than that of the single species systems. In addition, the storage patterns of the former systems are richer, which depend heavily on initial ionic states, however less on the strength of applied electric fields and external temperatures when the graphene separation is not too large. The fluid flows, the electric conductivities and the stability of such double species systems are subsequently scrutinized. The current study could form a theoretical basis to help designing high-performance lithium batteries and explaining ion transport in biological channels.

Transport-Based Modeling of Bubble Nucleation on Gas Evolving Electrodes

Bubble nucleation is ubiquitous in gas evolving reactions that are instrumental for a variety of electrochemical systems. Fundamental understanding of the nucleation process, which is critical to system optimization, remains limited as prior works generally focused on the thermodynamics and have not considered the coupling between surface geometries and different forms of transport in the electrolytes. Here, we establish a comprehensive transport-based model framework to identify the underlying mechanism for bubble nucleation on gas evolving electrodes. We account for the complex effects on the electrical field, ion migration, ion diffusion, and gas diffusion arising from surface heterogeneities and gas pockets initiated from surface crevices. As a result, we show that neglecting these effects leads to significant underprediction of the energy needed for nucleation. Our model provides a non-monotonic relationship between the surface cavity size and the overpotential required for nucleation, which is physically more consistent than the monotonic relationship suggested by a traditional thermodynamics-based model. We also identify the significance of the gas diffuse layer thickness, a parameter controlled by external flow fields and overall electrode geometries, which has been largely overlooked in previous models. Our model framework offers guidelines for practical electrochemical systems whereby, without changing the surface chemistry, nucleation on electrodes can be tuned by engineering the cavity size and the gas diffuse layer thickness.

Effects of Weak Electrolytes on Electric Double Layer Ion Distributions

Many common experimental systems have electric double layers containing weak electrolytes, including systems with buffers. The pH at the boundary of the diffuse layer is an important parameter for determining the physicochemical state of the system, including surface charge density. We show that the Boltzmann equilibrium relation can be used as an exact solution for weak electrolyte electric double layers. Using these results, we provide a closed-form relation for the maximum pH change in a buffered electric double layer, in terms of the boundary potential. Importantly, our results suggest that equilibrium electric double layer concepts developed for strong electrolytes can be expanded to include weak electrolytes.

Low-Frequency Impedance-Based Cell Discrimination Considering Ion Transport Model in Cell Suspension

Low-frequency impedance-based (LFI) cell discrimination as a novel non-destructive and non-invasive cell discrimination is proposed. LFI cell discrimination discriminates the cell type by considering an ion transport model in cell suspension. Ion transport model in cell suspension is constructed on the basis of Fick's laws of diffusion in the extracellular region under ion permeability P which represents the characteristics of cell type. P is achieved using the ion transport model equation through an iterative curve fitting to an ion concentration in extracellular region obtained from low-frequency impedance which is assumed to be linearly related to the ion concentration in extracellular region. In experiment, the electrical impedance spectra from the frequency of 200 kHz to 2.0 MHz are measured over time during producing ions from intracellular region to extracellular one in cell suspension using an impedance analyzer and an interdigitated array electrode system. As a target cell type, two different cell types based on Medical Research Council 5 (MRC-5), which are different in intracellular component are used. The curve fitting is performed for the low-frequency impedance at 200 kHz at which impedance reflects the ion concentration in extracellular region in order to obtain P of each cell type. As a result, each cell type has its own P. The proposed LFI cell discrimination successfully discriminates the cell type.

Dynamics of ionic flows via Poisson-Nernst-Planck systems with local hard-sphere potentials: Competition between cations

We study a quasi-one-dimensional steady-state Poisson-Nernst-Planck type model for ionic flows through a membrane channel with three ion species, two positively charged with the same valence and one negatively charged. Bikerman's local hard-sphere potential is included in the model to account for ion sizes. The problem is treated as a boundary value problem of a singularly perturbed differential system. Under the framework of a geometric singular perturbation theory, together with specific structures of this concrete model, the existence and uniqueness of solutions to the boundary value problem for small ion sizes is established. Furthermore, treating the ion sizes as small parameters, we derive an approximation of individual fluxes, from which one can further study the qualitative properties of ionic flows and extract concrete information directly related to biological measurements. Of particular interest is the competition between two cations due to the nonlinear interplay between finite ion sizes, diffusion coefficients and boundary conditions, which is closely related to selectivity phenomena of open ion channels with given protein structures. Furthermore, we are able to characterize the distinct effects of the nonlinear interplays between these physical parameters. Numerical simulations are performed to identify some critical potentials which play critical roles in examining properties of ionic flows in our analysis.

Resonance-driven ion transport and selectivity in prokaryotic ion channels

Ion channels exhibit a remarkably high accuracy in selecting uniquely its associated type of ion. The mechanisms behind ion selectivity are not well understood. Current explanations build mainly on molecular biology and bioinformatics. Here we propose a simple physical model for ion selectivity based on the driven damped harmonic oscillator (DDHO). The driving force for this oscillator is provided by self-organizing harmonic turbulent structures in the dehydrating ionic flow through the ion channel, namely, oscillating pressure waves in one dimension, and toroidal vortices in two and three dimensions. Density fluctuations caused by these turbulences efficiently transmit their energy to aqua ions that resonate with the driving frequency. Consequently, these release their hydration shell and exit the ion channel as free ions. Existing modeling frameworks do not express the required complex spatiotemporal dynamics, hence we introduce a macroscopic continuum model for ionic dehydration and transport, based on the hydrodynamics of a dissipative ionic flow through an ion channel, subject to electrostatic and amphiphilic interactions. This model combines three classical physical fields: Navier-Stokes equations from hydrodynamics, Gauss's law from Maxwell theory, and the convection-diffusion equation from continuum physics. Numerical experiments with mixtures of chemical species of ions in various degrees of hydration indeed reveal the emergence of strong oscillations in the ionic flow that are instrumental in the efficient dehydration and cause a strong ionic jet into the cell. As such, they provide a powerful engine for the DDHO mechanism. Theoretical predictions of the modeling framework match significantly with empirical patch-clamp data. The DDHO standard response curve defines a unique resonance frequency that depends on the mass and charge of the ion. In this way, the driving oscillations act as a selection mechanism that filters out one specific ion. Application of the DDHO model to real ion data shows that this mechanism indeed clearly distinguishes between chemical species and between aqua and bare ions with a large Mahalanobis distance and high oscillator quality. The DDHO framework helps to understand how SNP mutations can cause severe genetic pathologies as they destroy the geometry of the channel and so alter the resonance peaks of the required ion type.

Boundary Layer Effects on Ionic Flows Via Classical Poisson-Nernst-Planck Systems

A quasi-one-dimensional steady-state Poisson-Nernst-Planck model of two oppositely charged ion species through a membrane channel is analyzed. The model problem is treated as a boundary value problem of a singularly perturbed differential system. Our analysis is based on the geometric singular perturbation theory but, most importantly, on specific structures of this concrete model. The existence and (local ) uniqueness of solutions to the boundary value problem is established. In particular, an approximation of both the individual flux and the I-V (current-voltage) relation are derived explicitly from the zeroth order approximation (in ") solutions, from which the boundary layer effects on ionic flows are studied in great details.

A Flux Ratio and a Universal Property of Permanent Charges Effects on Fluxes

In this work, we consider ionic flow through ion channels for an ionic mixture of a cation species (positively charged ions) and an anion species (negatively charged ions), and examine effects of a positive permanent charge on fluxes of the cation species and the anion species. For an ion species, and for any given boundary conditions and channel geometry,we introduce a ratio _(Q) = J(Q)/J(0) between the flux J(Q) of the ion species associated with a permanent charge Q and the flux J(0) associated with zero permanent charge. The flux ratio _(Q) is a suitable quantity for measuring an effect of the permanent charge Q: if _(Q) > 1, then the flux is enhanced by Q; if _ < 1, then the flux is reduced by Q. Based on analysis of Poisson-Nernst-Planck models for ionic flows, a universal property of permanent charge effects is obtained: for a positive permanent charge Q, if _1(Q) is the flux ratio for the cation species and _2(Q) is the flux ratio for the anion species, then _1(Q) < _2(Q), independent of boundary conditions and channel geometry. The statement is sharp in the sense that, at least for a given small positive Q, depending on boundary conditions and channel geometry, each of the followings indeed occurs: (i) _1(Q) < 1 < _2(Q); (ii) 1 < _1(Q) < _2(Q); (iii) _1(Q) < _2(Q) < 1. Analogous statements hold true for negative permanent charges with the inequalities reversed. It is also shown that the quantity _(Q) = |J(Q) − J(0)| may not be suitable for comparing the effects of permanent charges on cation flux and on anion flux. More precisely, for some positive permanent charge Q, if _1(Q) is associated with the cation species and _2(Q) is associated with the anion species, then, depending on boundary conditions and channel geometry, each of the followings is possible: (a) _1(Q) > _2(Q); (b) _1(Q) < _2(Q).

Golden aspect ratio for ion transport simulation in nanopores

Access resistance indicates how well current carriers from a bulk medium can converge to a pore or opening and is an important concept in nanofluidic devices and in cell physiology. In simplified scenarios, when the bulk dimensions are infinite in all directions, it depends only on the resistivity and pore radius. These conditions are not valid in all-atom molecular dynamics simulations of transport, due to the computational cost of large simulation cells, and can even break down in micro- and nanoscale systems due to strong confinement. Here, we examine a scaling theory for the access resistance that predicts a special simulation cell aspect ratio-the golden aspect ratio-where finite-size effects are eliminated. Using both continuum and all-atom simulations, we demonstrate that this golden aspect ratio exists and that it takes on a universal value in linear response and moderate concentrations. Outside of linear response, it gains an apparent dependence on characteristics of the transport scenario (concentration, voltages, etc.) for small simulation cells, but this dependence vanishes at larger length scales. These results will enable the use of all-atom molecular dynamics simulations to study contextual properties of access resistance-its dependence on protein and molecular-scale fluctuations, the presence of charges, and other functional groups-and yield the opportunity to quantitatively compare computed and measured resistances.

Numerical Simulation of Ion Transport in a Nano-Electrospray Ion Source at Atmospheric Pressure

Understanding ion transport properties from the ion source to the mass spectrometer (MS) is essential for optimizing device performance. Numerical simulation helps in understanding of ion transport properties and, furthermore, facilitates instrument design. In contrast to previously reported numerical studies, ion transport simulations in a continuous injection mode whilst considering realistic space-charge effects have been carried out. The flow field was solved using Reynolds-averaged Navier-Stokes (RANS) equations, and a particle-in-cell (PIC) method was applied to solve a time-dependent electric field with local charge density. A series of ion transport simulations were carried out at different cone gas flow rates, ion source currents, and capillary voltages. A force evaluation analysis reveals that the electric force, the drag force, and the Brownian force are the three dominant forces acting on the ions. Both the experimental and simulation results indicate that cone gas flow rates of ≤250 slph (standard liter per hour) are important for high ion transmission efficiency, as higher cone gas flow rates reduce the ion signal significantly. The simulation results also show that the ion transmission efficiency reduces exponentially with an increased ion source current. Additionally, the ion loss due to space-charge effects has been found to be predominant at a higher ion source current, a lower capillary voltage, and a stronger cone gas counterflow. The interaction of the ion driving force, ion opposing force, and ion dispersion is discussed to illustrate ion transport mechanism in the ion source at atmospheric pressure. Graphical Abstract.

Anomalies in the equilibrium and nonequilibrium properties of correlated ions in complex molecular environments

Emergent statistical attributes, and therefore the equations of state, of an assembly of interacting charge carriers embedded within a complex molecular environment frequently exhibit a variety of anomalies, particularly in the high-density (equivalently, the concentration) regime, which are not well understood, because they do not fall under the low-concentration phenomenologies of Debye-Hückel-Onsager and Poisson-Nernst-Planck, including their variants. To go beyond, we here use physical concepts and mathematical tools from quantum scattering theory, transport theory with the Stosszahlansatz of Boltzmann, and classical electrodynamics (Lorentz gauge) and obtain analytical expressions both for the average and the frequency-wave vector-dependent longitudinal and transverse current densities, diffusion coefficient, and the charge density, and therefore the analytical expressions for (a) the chemical potential, activity coefficient, and the equivalent conductivity for strong electrolytes and (b) the current-voltage characteristics for ion-transport processes in complex molecular environments. Using a method analogous to the notion of Debye length and thence the electrical double layer, we here identify a pair of characteristic length scales (longitudinal and the transverse), which, being wave vector and frequency dependent, manifestly exhibit nontrivial fluctuations in space-time. As a unifying theme, we advance a quantity (inverse length dimension), g_{scat}^{(a)}, which embodies all dynamical interactions, through various quantum scattering lengths, relevant to molecular species a, and the analytical behavior which helps us to rationalize the properties of strong electrolytes, including anomalies, in all concentration regimes. As an example, the behavior of g_{scat}^{(a)} in the high-concentration regime explains the anomalous increase of the Debye length with concentration, as seen in a recent experiment on electrolyte solutions. We also put forth an extension of the standard diffusion equation, which manifestly incorporates the effects arising from the underlying microscopic collisions among constituent molecular species. Furthermore, we show a nontrivial connection between the current-voltage characteristics of electrolyte solutions and the Landauer's approach to electrical conduction in mesoscopic solids and thereby establish a definite conceptual bridge between the two disjoint subjects. For numerical insight, we present results on the aqueous solution of KCl as an example of strong electrolyte, and the transport (conduction as well as diffusion) of K^{+} ions in water, as an example of ion transport across the voltage-gated channels in biological cells.

Electrokinetic Lattice Boltzmann Solver Coupled to Molecular Dynamics: Application to Polymer Translocation

We have developed a theoretical and computational approach to deal with systems that involve a disparate range of spatiotemporal scales, such as those composed of colloidal particles or polymers moving in a fluidic molecular environment. Our approach is based on a multiscale modeling that combines the slow dynamics of the large particles with the fast dynamics of the solvent into a unique framework. The former is numerically solved via Molecular Dynamics and the latter via a multicomponent Lattice Boltzmann. The two techniques are coupled together to allow for a seamless exchange of information between the descriptions. Being based on a kinetic multicomponent description of the fluid species, the scheme is flexible in modeling charge flow within complex geometries and ranging from large to vanishing salt concentration. The details of the scheme are presented and the method is applied to the problem of translocation of a charged polymer through a nanopores. Lastly, we discuss the advantages and complexities of the approach.

Fractional Poisson-Nernst-Planck Model for Ion Channels I: Basic Formulations and Algorithms

In this work, we propose a fractional Poisson-Nernst-Planck model to describe ion permeation in gated ion channels. Due to the intrinsic conformational changes, crowdedness in narrow channel pores, binding and trapping introduced by functioning units of channel proteins, ionic transport in the channel exhibits a power-law-like anomalous diffusion dynamics. We start from continuous-time random walk model for a single ion and use a long-tailed density distribution function for the particle jump waiting time, to derive the fractional Fokker-Planck equation. Then, it is generalized to the macroscopic fractional Poisson-Nernst-Planck model for ionic concentrations. Necessary computational algorithms are designed to implement numerical simulations for the proposed model, and the dynamics of gating current is investigated. Numerical simulations show that the fractional PNP model provides a more qualitatively reasonable match to the profile of gating currents from experimental observations. Meanwhile, the proposed model motivates new challenges in terms of mathematical modeling and computations.

Surging footprints of mathematical modeling for prediction of transdermal permeability

In vivo skin permeation studies are considered gold standard but are difficult to perform and evaluate due to ethical issues and complexity of process involved. In recent past, a useful tool has been developed by combining the computational modeling and experimental data for expounding biological complexity. Modeling of percutaneous permeation studies provides an ethical and viable alternative to laboratory experimentation. Scientists are exploring complex models in magnificent details with advancement in computational power and technology. Mathematical models of skin permeability are highly relevant with respect to transdermal drug delivery, assessment of dermal exposure to industrial and environmental hazards as well as in developing fundamental understanding of biotransport processes. Present review focuses on various mathematical models developed till now for the transdermal drug delivery along with their applications.

ESES: Software for Eulerian solvent excluded surface

Solvent excluded surface (SES) is one of the most popular surface definitions in biophysics and molecular biology. In addition to its usage in biomolecular visualization, it has been widely used in implicit solvent models, in which SES is usually immersed in a Cartesian mesh. Therefore, it is important to construct SESs in the Eulerian representation for biophysical modeling and computation. This work describes a software package called Eulerian solvent excluded surface (ESES) for the generation of accurate SESs in Cartesian grids. ESES offers the description of the solvent and solute domains by specifying all the intersection points between the SES and the Cartesian grid lines. Additionally, the interface normal at each intersection point is evaluated. Furthermore, for a given biomolecule, the ESES software not only provides the whole surface area, but also partitions the surface area according to atomic types. Homology theory is utilized to detect topological features, such as loops and cavities, on the complex formed by the SES. The sizes of loops and cavities are measured based on persistent homology with an evolutionary partial differential equation-based filtration. ESES is extensively validated by surface visualization, electrostatic solvation free energy computation, surface area and volume calculations, and loop and cavity detection and their size estimation. We used the Amber PBSA test set in our electrostatic solvation energy, area, and volume validations. Our results are either calibrated by analytical values or compared with those from the MSMS software. © 2017 Wiley Periodicals, Inc.

Object-oriented Persistent Homology

Persistent homology provides a new approach for the topological simplification of big data via measuring the life time of intrinsic topological features in a filtration process and has found its success in scientific and engineering applications. However, such a success is essentially limited to qualitative data classification and analysis. Indeed, persistent homology has rarely been employed for quantitative modeling and prediction. Additionally, the present persistent homology is a passive tool, rather than a proactive technique, for classification and analysis. In this work, we outline a general protocol to construct object-oriented persistent homology methods. By means of differential geometry theory of surfaces, we construct an objective functional, namely, a surface free energy defined on the data of interest. The minimization of the objective functional leads to a Laplace-Beltrami operator which generates a multiscale representation of the initial data and offers an objective oriented filtration process. The resulting differential geometry based object-oriented persistent homology is able to preserve desirable geometric features in the evolutionary filtration and enhances the corresponding topological persistence. The cubical complex based homology algorithm is employed in the present work to be compatible with the Cartesian representation of the Laplace-Beltrami flow. The proposed Laplace-Beltrami flow based persistent homology method is extensively validated. The consistence between Laplace-Beltrami flow based filtration and Euclidean distance based filtration is confirmed on the Vietoris-Rips complex for a large amount of numerical tests. The convergence and reliability of the present Laplace-Beltrami flow based cubical complex filtration approach are analyzed over various spatial and temporal mesh sizes. The Laplace-Beltrami flow based persistent homology approach is utilized to study the intrinsic topology of proteins and fullerene molecules. Based on a quantitative model which correlates the topological persistence of fullerene central cavity with the total curvature energy of the fullerene structure, the proposed method is used for the prediction of fullerene isomer stability. The efficiency and robustness of the present method are verified by more than 500 fullerene molecules. It is shown that the proposed persistent homology based quantitative model offers good predictions of total curvature energies for ten types of fullerene isomers. The present work offers the first example to design object-oriented persistent homology to enhance or preserve desirable features in the original data during the filtration process and then automatically detect or extract the corresponding topological traits from the data.

Matched Interface and Boundary Method for Elasticity Interface Problems

Elasticity theory is an important component of continuum mechanics and has had widely spread applications in science and engineering. Material interfaces are ubiquity in nature and man-made devices, and often give rise to discontinuous coefficients in the governing elasticity equations. In this work, the matched interface and boundary (MIB) method is developed to address elasticity interface problems. Linear elasticity theory for both isotropic homogeneous and inhomogeneous media is employed. In our approach, Lamé's parameters can have jumps across the interface and are allowed to be position dependent in modeling isotropic inhomogeneous material. Both strong discontinuity, i.e., discontinuous solution, and weak discontinuity, namely, discontinuous derivatives of the solution, are considered in the present study. In the proposed method, fictitious values are utilized so that the standard central finite different schemes can be employed regardless of the interface. Interface jump conditions are enforced on the interface, which in turn, accurately determines fictitious values. We design new MIB schemes to account for complex interface geometries. In particular, the cross derivatives in the elasticity equations are difficult to handle for complex interface geometries. We propose secondary fictitious values and construct geometry based interpolation schemes to overcome this difficulty. Numerous analytical examples are used to validate the accuracy, convergence and robustness of the present MIB method for elasticity interface problems with both small and large curvatures, strong and weak discontinuities, and constant and variable coefficients. Numerical tests indicate second order accuracy in both L∞ and L2 norms.

Second order Method for Solving 3D Elasticity Equations with Complex Interfaces

Elastic materials are ubiquitous in nature and indispensable components in man-made devices and equipments. When a device or equipment involves composite or multiple elastic materials, elasticity interface problems come into play. The solution of three dimensional (3D) elasticity interface problems is significantly more difficult than that of elliptic counterparts due to the coupled vector components and cross derivatives in the governing elasticity equation. This work introduces the matched interface and boundary (MIB) method for solving 3D elasticity interface problems. The proposed MIB elasticity interface scheme utilizes fictitious values on irregular grid points near the material interface to replace function values in the discretization so that the elasticity equation can be discretized using the standard finite difference schemes as if there were no material interface. The interface jump conditions are rigorously enforced on the intersecting points between the interface and the mesh lines. Such an enforcement determines the fictitious values. A number of new techniques has been developed to construct efficient MIB elasticity interface schemes for dealing with cross derivative in coupled governing equations. The proposed method is extensively validated over both weak and strong discontinuity of the solution, both piecewise constant and position-dependent material parameters, both smooth and nonsmooth interface geometries, and both small and large contrasts in the Poisson's ratio and shear modulus across the interface. Numerical experiments indicate that the present MIB method is of second order convergence in both L∞ and L2 error norms for handling arbitrarily complex interfaces, including biomolecular surfaces. To our best knowledge, this is the first elasticity interface method that is able to deliver the second convergence for the molecular surfaces of proteins..

Modeling the electric potential across neuronal membranes: the effect of fixed charges on spinal ganglion neurons and neuroblastoma cells

We present a model for the electric potential profile across the membranes of neuronal cells. We considered the resting and action potential states, and analyzed the influence of fixed charges of the membrane on its electric potential, based on experimental values of membrane properties of the spinal ganglion neuron and the neuroblastoma cell. The spinal ganglion neuron represents a healthy neuron, and the neuroblastoma cell, which is tumorous, represents a pathological neuron. We numerically solved the non-linear Poisson-Boltzmann equation for the regions of the membrane model we have adopted, by considering the densities of charges dissolved in an electrolytic solution and fixed on both glycocalyx and cytoplasmic proteins. Our model predicts that there is a difference in the behavior of the electric potential profiles of the two types of cells, in response to changes in charge concentrations in the membrane. Our results also describe an insensitivity of the neuroblastoma cell membrane, as observed in some biological experiments. This electrical property may be responsible for the low pharmacological response of the neuroblastoma to certain chemotherapeutic treatments.

Singular perturbation solutions of steady-state Poisson-Nernst-Planck systems

We study the Poisson-Nernst-Planck (PNP) system with an arbitrary number of ion species with arbitrary valences in the absence of fixed charges. Assuming point charges and that the Debye length is small relative to the domain size, we derive an asymptotic formula for the steady-state solution by matching outer and boundary layer solutions. The case of two ionic species has been extensively studied, the uniqueness of the solution has been proved, and an explicit expression for the solution has been obtained. However, the case of three or more ions has received significantly less attention. Previous work has indicated that the solution may be nonunique and that even obtaining numerical solutions is a difficult task since one must solve complicated systems of nonlinear equations. By adopting a methodology that preserves the symmetries of the PNP system, we show that determining the outer solution effectively reduces to solving a single scalar transcendental equation. Due to the simple form of the transcendental equation, it can be solved numerically in a straightforward manner. Our methodology thus provides a standard procedure for solving the PNP system and we illustrate this by solving some practical examples. Despite the fact that for three ions, previous studies have indicated that multiple solutions may exist, we show that all except for one of these solutions are unphysical and thereby prove the existence and uniqueness for the three-ion case.

Multiscale geometric modeling of macromolecules I: Cartesian representation

This paper focuses on the geometric modeling and computational algorithm development of biomolecular structures from two data sources: Protein Data Bank (PDB) and Electron Microscopy Data Bank (EMDB) in the Eulerian (or Cartesian) representation. Molecular surface (MS) contains non-smooth geometric singularities, such as cusps, tips and self-intersecting facets, which often lead to computational instabilities in molecular simulations, and violate the physical principle of surface free energy minimization. Variational multiscale surface definitions are proposed based on geometric flows and solvation analysis of biomolecular systems. Our approach leads to geometric and potential driven Laplace-Beltrami flows for biomolecular surface evolution and formation. The resulting surfaces are free of geometric singularities and minimize the total free energy of the biomolecular system. High order partial differential equation (PDE)-based nonlinear filters are employed for EMDB data processing. We show the efficacy of this approach in feature-preserving noise reduction. After the construction of protein multiresolution surfaces, we explore the analysis and characterization of surface morphology by using a variety of curvature definitions. Apart from the classical Gaussian curvature and mean curvature, maximum curvature, minimum curvature, shape index, and curvedness are also applied to macromolecular surface analysis for the first time. Our curvature analysis is uniquely coupled to the analysis of electrostatic surface potential, which is a by-product of our variational multiscale solvation models. As an expository investigation, we particularly emphasize the numerical algorithms and computational protocols for practical applications of the above multiscale geometric models. Such information may otherwise be scattered over the vast literature on this topic. Based on the curvature and electrostatic analysis from our multiresolution surfaces, we introduce a new concept, the polarized curvature, for the prediction of protein binding sites.

Multiscale Multiphysics and Multidomain Models I: Basic Theory

This work extends our earlier two-domain formulation of a differential geometry based multiscale paradigm into a multidomain theory, which endows us the ability to simultaneously accommodate multiphysical descriptions of aqueous chemical, physical and biological systems, such as fuel cells, solar cells, nanofluidics, ion channels, viruses, RNA polymerases, molecular motors and large macromolecular complexes. The essential idea is to make use of the differential geometry theory of surfaces as a natural means to geometrically separate the macroscopic domain of solvent from the microscopic domain of solute, and dynamically couple continuum and discrete descriptions. Our main strategy is to construct energy functionals to put on an equal footing of multiphysics, including polar (i.e., electrostatic) solvation, nonpolar solvation, chemical potential, quantum mechanics, fluid mechanics, molecular mechanics, coarse grained dynamics and elastic dynamics. The variational principle is applied to the energy functionals to derive desirable governing equations, such as multidomain Laplace-Beltrami (LB) equations for macromolecular morphologies, multidomain Poisson-Boltzmann (PB) equation or Poisson equation for electrostatic potential, generalized Nernst-Planck (NP) equations for the dynamics of charged solvent species, generalized Navier-Stokes (NS) equation for fluid dynamics, generalized Newton's equations for molecular dynamics (MD) or coarse-grained dynamics and equation of motion for elastic dynamics. Unlike the classical PB equation, our PB equation is an integral-differential equation due to solvent-solute interactions. To illustrate the proposed formalism, we have explicitly constructed three models, a multidomain solvation model, a multidomain charge transport model and a multidomain chemo-electro-fluid-MD-elastic model. Each solute domain is equipped with distinct surface tension, pressure, dielectric function, and charge density distribution. In addition to long-range Coulombic interactions, various non-electrostatic solvent-solute interactions are considered in the present modeling. We demonstrate the consistency between the non-equilibrium charge transport model and the equilibrium solvation model by showing the systematical reduction of the former to the latter at equilibrium. This paper also offers a brief review of the field.

Multiscale multiphysics and multidomain models--flexibility and rigidity

The emerging complexity of large macromolecules has led to challenges in their full scale theoretical description and computer simulation. Multiscale multiphysics and multidomain models have been introduced to reduce the number of degrees of freedom while maintaining modeling accuracy and achieving computational efficiency. A total energy functional is constructed to put energies for polar and nonpolar solvation, chemical potential, fluid flow, molecular mechanics, and elastic dynamics on an equal footing. The variational principle is utilized to derive coupled governing equations for the above mentioned multiphysical descriptions. Among these governing equations is the Poisson-Boltzmann equation which describes continuum electrostatics with atomic charges. The present work introduces the theory of continuum elasticity with atomic rigidity (CEWAR). The essence of CEWAR is to formulate the shear modulus as a continuous function of atomic rigidity. As a result, the dynamics complexity of a macromolecular system is separated from its static complexity so that the more time-consuming dynamics is handled with continuum elasticity theory, while the less time-consuming static analysis is pursued with atomic approaches. We propose a simple method, flexibility-rigidity index (FRI), to analyze macromolecular flexibility and rigidity in atomic detail. The construction of FRI relies on the fundamental assumption that protein functions, such as flexibility, rigidity, and energy, are entirely determined by the structure of the protein and its environment, although the structure is in turn determined by all the interactions. As such, the FRI measures the topological connectivity of protein atoms or residues and characterizes the geometric compactness of the protein structure. As a consequence, the FRI does not resort to the interaction Hamiltonian and bypasses matrix diagonalization, which underpins most other flexibility analysis methods. FRI's computational complexity is of O(N(2)) at most, where N is the number of atoms or residues, in contrast to O(N(3)) for Hamiltonian based methods. We demonstrate that the proposed FRI gives rise to accurate prediction of protein B-Factor for a set of 263 proteins. We show that a parameter free FRI is able to achieve about 95% accuracy of the parameter optimized FRI. An interpolation algorithm is developed to construct continuous atomic flexibility functions for visualization and use with CEWAR.

Multiscale geometric modeling of macromolecules II: Lagrangian representation

Geometric modeling of biomolecules plays an essential role in the conceptualization of biolmolecular structure, function, dynamics, and transport. Qualitatively, geometric modeling offers a basis for molecular visualization, which is crucial for the understanding of molecular structure and interactions. Quantitatively, geometric modeling bridges the gap between molecular information, such as that from X-ray, NMR, and cryo-electron microscopy, and theoretical/mathematical models, such as molecular dynamics, the Poisson-Boltzmann equation, and the Nernst-Planck equation. In this work, we present a family of variational multiscale geometric models for macromolecular systems. Our models are able to combine multiresolution geometric modeling with multiscale electrostatic modeling in a unified variational framework. We discuss a suite of techniques for molecular surface generation, molecular surface meshing, molecular volumetric meshing, and the estimation of Hadwiger's functionals. Emphasis is given to the multiresolution representations of biomolecules and the associated multiscale electrostatic analyses as well as multiresolution curvature characterizations. The resulting fine resolution representations of a biomolecular system enable the detailed analysis of solvent-solute interaction, and ion channel dynamics, whereas our coarse resolution representations highlight the compatibility of protein-ligand bindings and possibility of protein-protein interactions.

Interacting ions in biophysics: real is not ideal

Ions in water are important throughout biology, from molecules to organs. Classically, ions in water were treated as ideal noninteracting particles in a perfect gas. Excess free energy of each ion was zero. Mathematics was not available to deal consistently with flows, or interactions with other ions or boundaries. Nonclassical approaches are needed because ions in biological conditions flow and interact. The concentration gradient of one ion can drive the flow of another, even in a bulk solution. A variational multiscale approach is needed to deal with interactions and flow. The recently developed energetic variational approach to dissipative systems allows mathematically consistent treatment of the bio-ions Na(+), K(+), Ca(2+), and Cl(-) as they interact and flow. Interactions produce large excess free energy that dominate the properties of the high concentration of ions in and near protein active sites, ion channels, and nucleic acids: the number density of ions is often >10 M. Ions in such crowded quarters interact strongly with each other as well as with the surrounding protein. Nonideal behavior found in many experiments has classically been ascribed to allosteric interactions mediated by the protein and its conformation changes. The ion-ion interactions present in crowded solutions-independent of conformation changes of the protein-are likely to change the interpretation of many allosteric phenomena. Computation of all atoms is a popular alternative to the multiscale approach. Such computations involve formidable challenges. Biological systems exist on very different scales from atomic motion. Biological systems exist in ionic mixtures (like extracellular and intracellular solutions), and usually involve flow and trace concentrations of messenger ions (e.g., 10(-7) M Ca(2+)). Energetic variational methods can deal with these characteristic properties of biological systems as we await the maturation and calibration of all-atom simulations of ionic mixtures and divalents.

A molecular level prototype for mechanoelectrical transducer in mammalian hair cells

The mechanoelectrical transducer (MET) is a crucial component of mammalian auditory system. The gating mechanism of the MET channel remains a puzzling issue, though there are many speculations, due to the lack of essential molecular building blocks. To understand the working principle of mammalian MET, we propose a molecular level prototype which constitutes a charged blocker, a realistic ion channel and its surrounding membrane. To validate the proposed prototype, we make use of a well-established ion channel theory, the Poisson-Nernst-Planck equations, for three-dimensional (3D) numerical simulations. A wide variety of model parameters, including bulk ion concentration, applied external voltage, blocker charge and blocker displacement, are explored to understand the basic function of the proposed MET prototype. We show that our prototype prediction of channel open probability in response to blocker relative displacement is in remarkable accordance with experimental observation of rat cochlea outer hair cells. Our results appear to suggest that tip links which connect hair bundles gate MET channels.

Ionic Interactions Are Everywhere

Ionic solutions are dominated by interactions because they must be electrically neutral, but classical theory assumes no interactions. Biological solutions are rather like seawater, concentrated enough so that the diameter of ions also produces important interactions. In my view, the theory of complex fluids is needed to deal with the interacting reality of biological solutions.