Ion Permeation and Selectivity of OmpF Porin: A Theoretical Study Based on Molecular Dynamics, Brownian Dynamics, and Continuum Electrodiffusion Theory

Supralinear scaling behavior of ionic transport in membrane nanochannels regulated by outer-surface charges

The peculiarity of ion transport at the nanoscale is revealed through electrophysiological studies of two biological ion channels: the cation-selective bacterial porin-OmpF and the mitochondrial voltage-dependent anion channel (VDAC). We provide evidence of an unprecedented scaling behavior in the power-law relationship between conductivity and concentration G ∼ c α with α > 1 when functional groups attached to the pore inner wall have opposite charges to those located in the nanochannel's outer surface. Indeed, we find α ∼ 1.4 both for OmpF in positively charged membranes and for VDAC in negatively charged ones. The experiments are analyzed using different levels of theoretical models, starting with an equivalent circuit where total electrical current is described as the sum of ionic currents. Subsequently, we show that electrical circuits incorporating simplifying assumptions such as local electroneutrality and Donnan equilibrium consistently account for the measured G-c relationships yielding extremely similar results to the numerical results of structure-based Poisson-Nernst-Planck equations computed without these assumptions. We demonstrate that unexpected scaling exponents do not correspond to deviations from these classical equilibrium/electroneutrality assumptions, but rather to the structural features of the pore that are not included in oversimplified models in terms of shape and/or charge distribution. In contrast to the predictions of widely accepted models, we demonstrate both experimentally and theoretically that the conductance of ion-selective nanochannels can be drastically reduced in dilute solutions through a mechanism in which membrane charges and pore charges do not compensate for each other but act as interacting sites of opposite charge. Our insights into the critical role of external surface charges aim to open new conceptual avenues for developing nanofluidic devices with enhanced capabilities for energy conversion and sensing properties.

Molecular permeation through large pore channels: computational approaches and insights

Computational methods such as molecular dynamics (MD) have illuminated how single-atom ions permeate membrane channels and how selectivity among them is achieved. Much less is understood about molecular permeation through eukaryotic channels that mediate the flux of small molecules (e.g. connexins, pannexins, LRRC8s, CALHMs). Here we describe computational methods that have been profitably employed to explore the movements of molecules through wide pores, revealing mechanistic insights, guiding experiments, and suggesting testable hypotheses. This review illustrates MD techniques such as voltage-driven flux, potential of mean force, and mean first-passage-time calculations, as applied to molecular permeation through wide pores. These techniques have enabled detailed and quantitative modeling of molecular interactions and movement of permeants at the atomic level. We highlight novel contributors to the transit of molecules through these wide pathways. In particular, the flexibility and anisotropic nature of permeant molecules, coupled with the dynamics of pore-lining residues, lead to bespoke permeation dynamics. As more eukaryotic large-pore channel structures and functional data become available, these insights and approaches will be important for understanding the physical principles underlying molecular permeation and as guides for experimental design.

Quantitative insights into the mechanism of proton conduction and selectivity for the human voltage-gated proton channel Hv1

Human voltage-gated proton (hHv1) channels are crucial for regulating essential biological processes such as immune cell respiratory burst, sperm capacitation, and cancer cell migration. Despite the significant concentration difference between protons and other ions in physiological conditions, hHv1 demonstrates remarkable proton selectivity. Our calculations of single-proton, cation, and anion permeation free energy profiles quantitatively demonstrate that the proton selectivity of the wild-type channel originates from its strong proton affinity via the titration of the key residues D112 and D174, although the channel imposes similar kinetic blocking effects for protons compared to other ions. A two-proton knock-on model is proposed to mathematically explain the electrophysiological measurements of the pH-dependent proton conductance in the conductive state. Moreover, it is shown that the anion selectivity of the D112N mutant channel is tied to impaired proton transport and substantial anion leakage.

Molecular Mechanism of Ciprofloxacin Translocation Through the Major Diffusion Channels of the ESKAPE Pathogens Klebsiella pneumoniae and Enterobacter cloacae

Experimental studies on the translocation and accumulation of antibiotics in Gram-negative bacteria have revealed details of the properties that allow efficient permeation through bacterial outer membrane porins. Among the major outer membrane diffusion channels, OmpF has been extensively studied to understand the antibiotic translocation process. In a few cases, this knowledge has also helped to improve the efficacy of existing antibacterial molecules. However, the extension of these strategies to enhance the efficacy of other existing and novel drugs require comprehensive molecular insight into the permeation process and an understanding of how antibiotic and channel properties influence the effective permeation rates. Previous studies have investigated how differences in antibiotic charge distribution can influence the observed permeation pathways through the OmpF channel, and have shown that the dynamics of the L3 loop can play a dominant role in the permeation process. Here, we perform all-atom simulations of the OmpF orthologs, OmpE35 from Enterobacter cloacae and OmpK35 from Klebsiella pneumoniae. Unbiased simulations of the porins and biased simulations of the ciprofloxacin permeation processes through these channels provide insight into the differences in the permeation pathway and energetics. In addition, we show that similar to the OmpF channel, antibiotic-induced dynamics of the L3 loop are also operative in the orthologs. However, the sequence and structural differences, influence the extent of the L3 loop fluctuations with OmpK35 showing greater stability in unbiased runs and subdued fluctuations in simulations with ciprofloxacin.

Detection of helical water flows in sub-nanometer channels

Nanoscale flows of liquids can be revealed in various biological processes and underlie a wide range of nanofluidic applications. Though the integral characteristics of these systems, such as permeability and effective diffusion coefficient, can be measured in experiments, the behaviour of the flows within nanochannels is still a matter of speculation. Herein, we used a combination of quadrupolar solid-state NMR spectroscopy, computer simulation, and dynamic vapour sorption measurements to analyse water diffusion inside peptide nanochannels. We detected a helical water flow coexisting with a conventional axial flow that are independent of each other, immiscible, and associated with diffusion coefficients that may differ up to 3 orders of magnitude. The trajectory of the helical flow is dictated by the screw-like distribution of ionic groups within the channel walls, while its flux is governed by external water vapour pressure. Similar flows may occur in other types of nanochannels containing helicoidally distributed ionic groups and be exploited in various nanofluidic lab-on-a-chip devices.

New insights into the effects of small permanent charge on ionic flows: A higher order analysis

This study investigated how permanent charges influence the dynamics of ionic channels. Using a quasi-one-dimensional classical Poisson-Nernst-Planck (PNP) model, we investigated the behavior of two distinct ion species-one positively charged and the other negatively charged. The spatial distribution of permanent charges was characterized by zero values at the channel ends and a constant charge $ Q_0 $ within the central region. By treating the classical PNP model as a boundary value problem (BVP) for a singularly perturbed system, the singular orbit of the BVP depended on $ Q_0 $ in a regular way. We therefore explored the solution space in the presence of a small permanent charge, uncovering a systematic dependence on this parameter. Our analysis employed a rigorous perturbation approach to reveal higher-order effects originating from the permanent charges. Through this investigation, we shed light on the intricate interplay among boundary conditions and permanent charges, providing insights into their impact on the behavior of ionic current, fluxes, and flux ratios. We derived the quadratic solutions in terms of permanent charge, which were notably more intricate compared to the linear solutions. Through computational tools, we investigated the impact of these quadratic solutions on fluxes, current-voltage relations, and flux ratios, conducting a thorough analysis of the results. These novel findings contributed to a deeper comprehension of ionic flow dynamics and hold potential implications for enhancing the design and optimization of ion channel-based technologies.

Dynamics accelerate the kinetics of ion diffusion through channels: Continuous-time random walk models beyond the mean field approximation

Ion channels are proteins that play a significant role in physiological processes, including neuronal excitability and signal transduction. However, the precise mechanisms by which these proteins facilitate ion diffusion through cell membranes are not well understood. This is because experimental techniques to characterize ion channel activity operate on a time scale too large to understand the role of the various protein conformations on diffusion. Meanwhile, computational approaches operate on a time scale too short to rationalize the observed behavior at the microscopic scale. In this paper, we present a continuous-time random walk model that aims to bridge the scales between the atomistic models of ion channels and the experimental measurement of their conductance. We show how diffusion slows down in complex systems by using 3D lattices that map out the pore geometry of two channels: Nav1.7 and gramicidin. We also introduce spatial and dynamic site disorder to account for system heterogeneity beyond the mean field approximation. Computed diffusion coefficients show that an increase in spatial disorder slows down diffusion kinetics, while dynamic disorder has the opposite effect. Our results imply that microscopic or phenomenological models based on the potential of mean force data overlook the functional importance of protein dynamics on ion diffusion through channels.

Electro-osmotic flow in nanoconfinement: Solid-state and protein nanopores

Electro-osmotic flow (EOF) is a phenomenon where fluid motion occurs in porous materials or micro/nano-channels when an external electric field is applied. In the particular example of single-molecule electrophoresis using single nanopores, the role of EOF on the translocation velocity of the analyte molecule through the nanopore is not fully understood. The complexity arises from a combination of effects from hydrodynamics in restricted environments, electrostatics emanating from charge decorations and geometry of the pores. We address this fundamental issue using the Poisson-Nernst-Planck and Navier-Stokes (PNP-NS) equations for cylindrical solid-state nanopores and three representative protein nanopores (α-hemolysin, MspA, and CsgG). We present the velocity profiles inside the nanopores as a function of charge decoration and geometry of the pore and applied electric field. We report several unexpected results: (a) The apparent charges of the protein nanopores are different from their net charge and the surface charge of the whole protein geometry, and the net charge of inner surface is consistent with the apparent charge. (b) The fluid velocity depends non-monotonically on voltage. The three protein nanopores exhibit unique EOF and velocity-voltage relations, which cannot be simply deduced from their net charge. Furthermore, effective point mutations can significantly change both the direction and the magnitude of EOF. The present computational analysis offers an opportunity to further understand the origins of the speed of transport of charged macromolecules in restricted space and to design desirable nanopores for tuning the speed of macromolecules through nanopores.

Finite ion size effects on I-V relations via Poisson-Nernst-Planck systems with two cations: A case study

We consider a quasi-one-dimensional Poisson-Nernst-Planck model with two cations having the same valances and one anion. Bikerman's local hard-sphere potential is included to account for ion size effects. Under some further restrictions on the boundary conditions of the two cations, we obtain approximations of the I-V (current-voltage) relations by treating the ion sizes as small parameters. Critical potentials are identified, which play critical roles in characterizing finite ion size effects on ionic flows. Nonlinear interplays between system parameters, such as boundary concentrations and diffusion coefficients, are analyzed. To provide more intuitive illustrations of our analytical results and better understanding of the dynamics of ionic flows through membrane channels, numerical simulations are performed.

The mechanism of an electrostatic nanofilter: overcoming entropy with electrostatics

General porins are nature's sieving machinery in the outer membrane of Gram-negative bacteria. Their unique hourglass-shaped architecture is highly conserved among different bacterial membrane proteins and other biological channels. These biological nanopores have been designed to protect the interior of the bacterial cell from leakage of toxic compounds while selectively allowing the entry of the molecules needed for cell growth and function. The mechanism of transport through porins is of utmost and direct interest for drug discovery, extending toward nanotechnology applications for blue energy, separations, and sequencing. Here we present a theoretical framework for analysing the filter of general porins in relation to translocating molecules with the aid of enhanced molecular simulations quantitatively. Using different electrostatic probes in the form of a series of related molecules, we describe the nature of this filter and how to finely tune permeability by exploiting electrostatic interactions between the pore and the translocating molecule. Eventually, we show how enhanced simulations constitute today a valid tool for characterising the mechanism and quantifying energetically the transport of molecules through nanopores.

Fast prediction of antibiotic permeability through membrane channels using Brownian dynamics

The efficient permeation across the Gram-negative bacterial membrane is an important step in the overall process of antibacterial action of a molecule and the one that has posed a significant hurdle on the way toward approved antibiotics. Predicting the permeability for a large library of molecules and assessing the effect of different molecular transformations on permeation rates of a given molecule is critical to the development of effective antibiotics. We present a computational approach for obtaining estimates of molecular permeability through a porin channel in a matter of hours using a Brownian dynamics approach. The fast sampling using a temperature acceleration scheme enables the approximate estimation of permeability using the inhomogeneous solubility diffusion model. Although the method is a significant approximation to similar all-atom approaches tested previously, we show that the present approach predicts permeabilities that correlate fairly well with the respective experimental permeation rates from liposome swelling experiments and accumulation rates from antibiotic accumulation assays, and is significantly, i.e., about 14 times, faster compared with a previously reported approach. The possible applications of the scheme in high-throughput screening for fast permeators are discussed.

Conformational Dynamics of Loop L3 in OmpF: Implications toward Antibiotic Translocation and Voltage Gating

In the present work, we delineate the molecular mechanism of a bulky antibiotic permeating through a bacterial channel and uncover the role of conformational dynamics of the constriction loop in this process. Using the temperature accelerated sliced sampling approach, we shed light onto the dynamics of the L3 loop, in particular the F118 to S125 segment, at the constriction regions of the OmpF porin. We complement the findings with single channel electrophysiology experiments and applied-field simulations, and we demonstrate the role of hydrogen-bond stabilization in the conformational dynamics of the L3 loop. A molecular mechanism of permeation is put forward wherein charged antibiotics perturb the network of stabilizing hydrogen-bond interactions and induce conformational changes in the L3 segment, thereby aiding the accommodation and permeation of bulky antibiotic molecules across the constriction region. We complement the findings with single channel electrophysiology experiments and demonstrate the importance of the hydrogen-bond stabilization in the conformational dynamics of the L3 loop. The generality of the present observations and experimental results regarding the L3 dynamics enables us to identify this L3 segment as the source of gating. We propose a mechanism of OmpF gating that is in agreement with previous experimental data that showed the noninfluence of cysteine double mutants that tethered the L3 tip to the barrel wall on the OmpF gating behavior. The presence of similar loop stabilization networks in porins of other clinically relevant pathogens suggests that the conformational dynamics of the constriction loop is possibly of general importance in the context of antibiotic permeation through porins.

Local Molecular Field Theory for Coulomb Interactions in Aqueous Solutions

Coulomb interactions play a crucial role in a wide array of processes in aqueous solutions but present conceptual and computational challenges to both theory and simulations. We review recent developments in an approach addressing these challenges─local molecular field (LMF) theory. LMF theory exploits an exact and physically suggestive separation of intermolecular Coulomb interactions into strong short-range and uniformly slowly varying long-range components. This allows us to accurately determine the averaged effects of the long-range components on the short-range structure using effective single particle fields and analytical corrections, greatly reducing the need for complex lattice summation techniques used in most standard approaches. The simplest use of these ideas in aqueous solutions leads to the short solvent (SS) model, where both solvent-solvent and solute-solvent Coulomb interactions have only short-range components. Here we use the SS model to give a simple description of pairing of nucleobases and biologically relevant ions in water.

Mathematical Analysis on Current-Voltage Relations via Classical Poisson-Nernst-Planck Systems with Nonzero Permanent Charges under Relaxed Electroneutrality Boundary Conditions

We focus on a quasi-one-dimensional Poisson-Nernst-Planck model with small permanent charges for ionic flows of two oppositely charged ion species through an ion channel. Of particular interest is to examine the dynamics of ionic flows in terms of I-V (current-voltage) relations with boundary layers due to the relaxation of neutral conditions on boundary concentrations. This is achieved by employing the regular perturbation analysis on the solutions established through geometric singular perturbation analysis. Rich dynamics are observed, particularly, the nonlinear interplays among different physical parameters are characterized. Critical potentials are identified, which play critical roles in the study of ionic flows and can be estimated experimentally. Numerical simulations are performed to further illustrate and provide more intuitive understandings of our analytical results.

Structure and Computational Electrophysiology of Ac-LS3, a Synthetic Ion Channel

Computer simulations are reported on Ac-LS3, a synthetic ion channel, containing 21 residues with a Leu-Ser-Ser-Leu-Leu-Ser-Leu heptad repeat, which forms ions channels upon application of voltage. A hexameric, coiled-coil bundle initially positioned perpendicular to the membrane settled into a stable, tilted structure after 1.5 μs, most likely to improve contacts between the non-polar exterior of the channel and the hydrophobic core of the membrane. Once tilted, the bundle remained in this state during subsequent simulations of nearly 10 μs at voltages ranging from 200 to -100 mV. In contrast, attempts to identify a stable pentameric structure failed, thus supporting the hypothesis that the channel is a hexamer. Results at 100 mV were used to reconstruct the free energy profiles for K⁺ and Cl^- in the channel. This was done by way of several methods in which results of molecular dynamics (MD) simulations were combined with the electrodiffusion model. Two of them developed recently do not require knowledge of the diffusivity. Instead, they utilize one-sided density profiles and committor probabilities. The consistency between different methods is very good, supporting the utility of the newly developed methods for reconstructing free energies of ions in channels. The flux of K⁺, which accounts for most of the current through the channel, calculated directly from MD matches well the total measured current. However, the current of Cl^- is somewhat overestimated, possibly due to a slightly unbalanced force field involving chloride. The current-voltage dependence was also reconstructed by way of a recently developed, efficient method that requires simulations only at a single voltage, yielding good agreement with the experiment. Taken together, the results demonstrate that computational electrophysiology has become a reliable tool for studying how channels mediate ion transport through membranes.

DNA-Based Optical Quantification of Ion Transport across Giant Vesicles

Accurate measurements of ion permeability through cellular membranes remains challenging due to the lack of suitable ion-selective probes. Here we use giant unilamellar vesicles (GUVs) as membrane models for the direct visualization of mass translocation at the single-vesicle level. Ion transport is indicated with a fluorescently adjustable DNA-based sensor that accurately detects sub-millimolar variations in K⁺ concentration. In combination with microfluidics, we employed our DNA-based K⁺ sensor for extraction of the permeation coefficient of potassium ions. We measured K⁺ permeability coefficients at least 1 order of magnitude larger than previously reported values from bulk experiments and show that permeation rates across the lipid bilayer increase in the presence of octanol. In addition, an analysis of the K⁺ flux in different concentration gradients allows us to estimate the complementary H⁺ flux that dissipates the charge imbalance across the GUV membrane. Subsequently, we show that our sensor can quantify the K⁺ transport across prototypical cation-selective ion channels, gramicidin A and OmpF, revealing their relative H⁺/K⁺ selectivity. Our results show that gramicidin A is much more selective to protons than OmpF with a H⁺/K⁺ permeability ratio of ∼10⁴.

Charged Nanochannels in Covalent Organic Framework Membranes Enabling Efficient Ion Exclusion

Controllable ion transport through nanochannels is crucial for biological and artificial membrane systems. Covalent organic frameworks (COFs) with regular and tunable nanochannels are emerging as an ideal material platform to develop synthetic membranes for ion transport. However, ion exclusion by COF membranes remains challenging because most COF materials have large-sized nanochannels leading to nonselective transport of small ions. Here we develop ionic COF membranes (iCOFMs) to control ion transport through charged framework nanochannels, the interior surfaces of which are covered with arrayed sulfonate groups to render superior charge density. The overlap of an electrical double layer in charged nanochannels blocks the entry of co-ions, narrows their passageways, and concomitantly restrains the permeation of counterions via the charge balance. These highly charged large-sized nanochannels within the iCOFM enable ion exclusion while maintaining intrinsically high water permeability. Our results reveal possibilities for controllable ion transport based on COF membranes for water purification, ionic separation, sensing, and energy conversion.

Diffusion of molecules through nanopores under confinement: Time-scale bridging and crowding effects via Markov state model

Passive transport of molecules through nanopores is characterized by the interaction of molecules with pore internal walls and by a general crowding effect due to the constricted size of the nanopore itself, which limits the presence of molecules in its interior. The molecule-pore interaction is treated within the diffusion approximation by introducing the potential of mean force and the local diffusion coefficient for a correct statistical description. The crowding effect can be handled within the Markov state model approximation. By combining the two methods, one can deal with complex free energy surfaces taking into account crowding effects. We recapitulate the equations bridging the two models to calculate passive currents assuming a limited occupancy of the nanopore in a wide range of molecular concentrations. Several simple models are analyzed to clarify the consequences of the model. Eventually, a biologically relevant case of transport of an antibiotic molecule through a bacterial porin is used to draw conclusions (i) on the effects of crowding on transport of small molecules through biological channels, and (ii) to demonstrate its importance for modelling of cellular transport.

What have molecular simulations contributed to understanding of Gram-negative bacterial cell envelopes?

Bacterial cell envelopes are compositionally complex and crowded and while highly dynamic in some areas, their molecular motion is very limited, to the point of being almost static in others. Therefore, it is no real surprise that studying them at high resolution across a range of temporal and spatial scales requires a number of different techniques. Details at atomistic to molecular scales for up to tens of microseconds are now within range for molecular dynamics simulations. Here we review how such simulations have contributed to our current understanding of the cell envelopes of Gram-negative bacteria.

Role of internal loop dynamics in antibiotic permeability of outer membrane porins

Antibiotic resistance in Gram-negative pathogens has been identified as an urgent threat to human health by the World Health Organization. The major challenge with treating infections by these pathogens is developing antibiotics that can traverse the dense bacterial outer membrane (OM) formed by a mesh of lipopolysaccharides. Effective antibiotics permeate through OM porins, which have evolved for nutrient diffusion; however, the conformational states of these porins regulating permeation are still unclear. Here, we used molecular dynamics simulations, free energy calculations, Markov-state modeling, and whole-cell accumulation assays to provide mechanistic insight on how a porin shifts between open and closed states. We provide a mechanism of how Gram-negative bacteria confer resistance to antibiotics.

Gram-negative bacteria pose a serious public health concern due to resistance to many antibiotics, caused by the low permeability of their outer membrane (OM). Effective antibiotics use porins in the OM to reach the interior of the cell; thus, understanding permeation properties of OM porins is instrumental to rationally develop broad-spectrum antibiotics. A functionally important feature of OM porins is undergoing open–closed transitions that modulate their transport properties. To characterize the molecular basis of these transitions, we performed an extensive set of molecular dynamics (MD) simulations of Escherichia coli OM porin OmpF. Markov-state analysis revealed that large-scale motion of an internal loop, L3, underlies the transition between energetically stable open and closed states. The conformation of L3 is controlled by H bonds between highly conserved acidic residues on the loop and basic residues on the OmpF β-barrel. Mutation of key residues important for the loop’s conformation shifts the equilibrium between open and closed states and regulates translocation of permeants (ions and antibiotics), as observed in the simulations and validated by our whole-cell accumulation assay. Notably, one mutant system G119D, which we find to favor the closed state, has been reported in clinically resistant bacterial strains. Overall, our accumulated ∼200 µs of simulation data (the wild type and mutants) along with experimental assays suggest the involvement of internal loop dynamics in permeability of OM porins and antibiotic resistance in Gram-negative bacteria.

A Deep Dive into VDAC1 Conformational Diversity Using All-Atom Simulations Provides New Insights into the Structural Origin of the Closed States

The voltage-dependent anion channel 1 (VDAC1) is a crucial mitochondrial transporter that controls the flow of ions and respiratory metabolites entering or exiting mitochondria. As a voltage-gated channel, VDAC1 can switch between a high-conducting “open” state and a low-conducting “closed” state emerging at high transmembrane (TM) potentials. Although cell homeostasis depends on channel gating to regulate the transport of ions and metabolites, structural hallmarks characterizing the closed states remain unknown. Here, we performed microsecond accelerated molecular dynamics to highlight a vast region of VDAC1 conformational landscape accessible at typical voltages known to promote closure. Conformers exhibiting durable subconducting properties inherent to closed states were identified. In all cases, the low conductance was due to the particular positioning of an unfolded part of the N-terminus, which obstructed the channel pore. While the N-terminal tail was found to be sensitive to voltage orientation, our models suggest that stable low-conducting states of VDAC1 predominantly take place from disordered events and do not result from the displacement of a voltage sensor or a significant change in the pore. In addition, our results were consistent with conductance jumps observed experimentally and corroborated a recent study describing entropy as a key factor for VDAC gating.

Computational methods and theory for ion channel research

Ion channels are fundamental biological devices that act as gates in order to ensure selective ion transport across cellular membranes; their operation constitutes the molecular mechanism through which basic biological functions, such as nerve signal transmission and muscle contraction, are carried out. Here, we review recent results in the field of computational research on ion channels, covering theoretical advances, state-of-the-art simulation approaches, and frontline modeling techniques. We also report on few selected applications of continuum and atomistic methods to characterize the mechanisms of permeation, selectivity, and gating in biological and model channels.

Unveiling the Gating Mechanism of CRAC Channel: A Computational Study

CRAC channel is ubiquitous and its importance in the regulation of the immune system is testified by the severe immunodeficiencies caused by its mutations. In this work we took advantage of the availability of open and closed structures of this channel to run for the first time simulations of the whole gating process reaching the relevant time-scale with an enhanced sampling technique, Targeted Molecular Dynamics. Our simulations highlighted a complex allosteric propagation of the conformational change from peripheral helices, where the activator STIM1 binds, to the central pore helices. In agreement with mutagenesis data, our simulations revealed the key role of residue H206 whose displacement creates an empty space behind the hydrophobic region of the pore, thus releasing a steric brake and allowing the opening of the channel. Conversely, the process of pore closing culminates with the formation of a bubble that occludes the pore even in the absence of steric block. This mechanism, known as "hydrophobic gating", has been observed in an increasing number of biological ion channels and also in artificial nanopores. Our study therefore shows promise not only to better understand the molecular origin of diseases caused by disrupted calcium signaling, but also to clarify the mode of action of hydrophobically gated ion channels, possibly even suggesting strategies for the biomimetic design of synthetic nanopores.

Rationalizing the generation of broad spectrum antibiotics with the addition of a positive charge

Antibiotic resistance of Gram-negative bacteria is largely attributed to the low permeability of their outer membrane (OM). Recently, we disclosed the eNTRy rules, a key lesson of which is that the introduction of a primary amine enhances OM permeation in certain contexts. To understand the molecular basis for this finding, we perform an extensive set of molecular dynamics (MD) simulations and free energy calculations comparing the permeation of aminated and amine-free antibiotic derivatives through the most abundant OM porin of E. coli, OmpF. To improve sampling of conformationally flexible drugs in MD simulations, we developed a novel, Monte Carlo and graph theory based algorithm to probe more efficiently the rotational and translational degrees of freedom visited during the permeation of the antibiotic molecule through OmpF. The resulting pathways were then used for free-energy calculations, revealing a lower barrier against the permeation of the aminated compound, substantiating its greater OM permeability. Further analysis revealed that the amine facilitates permeation by enabling the antibiotic to align its dipole to the luminal electric field of the porin and form favorable electrostatic interactions with specific, highly-conserved charged residues. The importance of these interactions in permeation was further validated with experimental mutagenesis and whole cell accumulation assays. Overall, this study provides insights on the importance of the primary amine for antibiotic permeation into Gram-negative pathogens that could help the design of future antibiotics. We also offer a new computational approach for calculating free-energy of processes where relevant molecular conformations cannot be efficiently captured.

Outer Membrane Protein F Is Involved in Biofilm Formation, Virulence and Antibiotic Resistance in Cronobacter sakazakii

In some Gram-negative bacteria, ompF encodes outer membrane protein F (OmpF), which is a cation-selective porin and is responsible for the passive transport of small molecules across the outer membrane. However, there are few reports about the functions of this gene in Cronobacter sakazakii. To investigate the role of ompF in detail, an ompF disruption strain (ΔompF) and a complementation strain (cpompF) were successfully obtained. We find that OmpF can affect the ability of biofilm formation in C. sakazakii. In addition, the variations in biofilm composition of C. sakazakii were examined using Raman spectroscopy analyses caused by knocking out ompF, and the result indicated that the levels of certain biofilm components, including lipopolysaccharide (LPS), were significantly decreased in the mutant (ΔompF). Then, SDS-PAGE was used to further analyze the LPS content, and the result showed that the LPS levels were significantly reduced in the absence of ompF. Therefore, we conclude that OmpF affects biofilm formation in C. sakazakii by reducing the amount of LPS. Furthermore, the ΔompF mutant showed decreased (2.7-fold) adhesion to and invasion of HCT-8 cells. In an antibiotic susceptibility analysis, the ΔompF mutant showed significantly smaller inhibition zones than the WT, indicating that OmpF had a positive effect on the influx of antibiotics into the cells. In summary, ompF plays a positive regulatory role in the biofilm formation and adhesion/invasion, which is achieved by regulating the amount of LPS, but is a negative regulator of antibiotic resistance in C. sakazakii.

Improved Sampling and Free Energy Estimates for Antibiotic Permeation through Bacterial Porins

Antibiotics enter into bacterial cells via protein channels that serve as low-energy pathways through the outer membrane, which is otherwise impenetrable. Insights into the molecular mechanisms underlying the transport processes are vital for the development of effective antibacterials. A much-desired prerequisite is an accurate and reproducible determination of free energy surfaces for antibiotic translocation, enabling quantitative and meaningful comparisons of permeation mechanisms for different classes of antibiotics. Inefficient sampling along the orthogonal degrees of freedom, for example, in umbrella sampling and metadynamics approaches, is however a key limitation affecting the accuracy and the convergence of free energy estimates. To overcome this limitation, two sampling methods have been employed in the present study that, respectively, combine umbrella sampling and metadynamics-style biasing schemes with temperature acceleration for improved sampling along orthogonal degrees of freedom. As a model for the transport of bulky solutes, the ciprofloxacin-OmpF system has been selected. The well-tempered metadynamics approach with multiple walkers is compared with its "temperature-accelerated" variant in terms of improvements in sampling and convergence of free energy estimates. We find that the inclusion of collective variables governing solute degrees of freedom and solute-water interactions within the sampling scheme largely alleviates sampling issues. Concerning improved sampling and convergence of free energy estimates from independent simulations, the temperature-accelerated sliced sampling approach that combines umbrella sampling with temperature-accelerated molecular dynamics performs even better as shown for the ciprofloxacin-OmpF system.

An antibiotic-resistance conferring mutation in a neisserial porin: Structure, ion flux, and ampicillin binding

Gram-negative bacteria cause the majority of highly drug-resistant bacterial infections. To cross the outer membrane of the complex Gram-negative cell envelope, antibiotics permeate through porins, trimeric channel proteins that enable the exchange of small polar molecules. Mutations in porins contribute to the development of drug-resistant phenotypes. In this work, we show that a single point mutation in the porin PorB from Neisseria meningitidis, the causative agent of bacterial meningitis, can strongly affect the binding and permeation of beta-lactam antibiotics. Using X-ray crystallography, high-resolution electrophysiology, atomistic biomolecular simulation, and liposome swelling experiments, we demonstrate differences in drug binding affinity, ion selectivity and drug permeability of PorB. Our work further reveals distinct interactions between the transversal electric field in the porin eyelet and the zwitterionic drugs, which manifest themselves under applied electric fields in electrophysiology and are altered by the mutation. These observations may apply more broadly to drug-porin interactions in other channels. Our results improve the molecular understanding of porin-based drug-resistance in Gram-negative bacteria.

The Whole Is Bigger than the Sum of Its Parts: Drug Transport in the Context of Two Membranes with Active Efflux

Cell envelope plays a dual role in the life of bacteria by simultaneously protecting it from a hostile environment and facilitating access to beneficial molecules. At the heart of this ability lie the restrictive properties of the cellular membrane augmented by efflux transporters, which preclude intracellular penetration of most molecules except with the help of specialized uptake mediators. Recently, kinetic properties of the cell envelope came into focus driven on one hand by the urgent need in new antibiotics and, on the other hand, by experimental and theoretical advances in studies of transmembrane transport. A notable result from these studies is the development of a kinetic formalism that integrates the Michaelis-Menten behavior of individual transporters with transmembrane diffusion and offers a quantitative basis for the analysis of intracellular penetration of bioactive compounds. This review surveys key experimental and computational approaches to the investigation of transport by individual translocators and in whole cells, summarizes key findings from these studies and outlines implications for antibiotic discovery. Special emphasis is placed on Gram-negative bacteria, whose envelope contains two separate membranes. This feature sets these organisms apart from Gram-positive bacteria and eukaryotic cells by providing them with full benefits of the synergy between slow transmembrane diffusion and active efflux.

Electrodiffusion Phenomena in Neuroscience and the Nernst–Planck–Poisson Equations

This work is aimed to give an electrochemical insight into the ionic transport phenomena in the cellular environment of organized brain tissue. The Nernst–Planck–Poisson (NPP) model is presented, and its applications in the description of electrodiffusion phenomena relevant in nanoscale neurophysiology are reviewed. These phenomena include: the signal propagation in neurons, the liquid junction potential in extracellular space, electrochemical transport in ion channels, the electrical potential distortions invisible to patch-clamp technique, and calcium transport through mitochondrial membrane. The limitations, as well as the extensions of the NPP model that allow us to overcome these limitations, are also discussed.

Competition between Cations via Classical Poisson-Nernst-Planck Models with Nonzero but Small Permanent Charges

We study a one-dimensional Poisson–Nernst–Planck system for ionic flow through a membrane channel. Nonzero but small permanent charge, the major structural quantity of an ion channel, is included in the model. Two cations with the same valences and one anion are included in the model, which provides more rich and complicated correlations/interactions between ions. The cross-section area of the channel is included in the system, and it provides certain information of the geometry of the three-dimensional channel, which is critical for our analysis. Geometric singular perturbation analysis is employed to establish the existence and local uniqueness of solutions to the system for small permanent charges. Treating the permanent charge as a small parameter, through regular perturbation analysis, we are able to derive approximations of the individual fluxes explicitly, and this allows us to study the competition between two cations, which is related to the selectivity phenomena of ion channels. Numerical simulations are performed to provide a more intuitive illustration of our analytical results, and they are consistent.

How to Enter a Bacterium: Bacterial Porins and the Permeation of Antibiotics

Despite tremendous successes in the field of antibiotic discovery seen in the previous century, infectious diseases have remained a leading cause of death. More specifically, pathogenic Gram-negative bacteria have become a global threat due to their extraordinary ability to acquire resistance against any clinically available antibiotic, thus urging for the discovery of novel antibacterial agents. One major challenge is to design new antibiotics molecules able to rapidly penetrate Gram-negative bacteria in order to achieve a lethal intracellular drug accumulation. Protein channels in the outer membrane are known to form an entry route for many antibiotics into bacterial cells. Up until today, there has been a lack of simple experimental techniques to measure the antibiotic uptake and the local concentration in subcellular compartments. Hence, rules for translocation directly into the various Gram-negative bacteria via the outer membrane or via channels have remained elusive, hindering the design of new or the improvement of existing antibiotics. In this review, we will discuss the recent progress, both experimentally as well as computationally, in understanding the structure-function relationship of outer-membrane channels of Gram-negative pathogens, mainly focusing on the transport of antibiotics.

The intrinsically disordered N-terminus of the voltage-dependent anion channel

The voltage-dependent anion channel (VDAC) is a critical β-barrel membrane protein of the mitochondrial outer membrane, which regulates the transport of ions and ATP between mitochondria and the cytoplasm. In addition, VDAC plays a central role in the control of apoptosis and is therefore of great interest in both cancer and neurodegenerative diseases. Although not fully understood, it is presumed that the gating mechanism of VDAC is governed by its N-terminal region which, in the open state of the channel, exhibits an α-helical structure positioned midway inside the pore and strongly interacting with the β-barrel wall. In the present work, we performed molecular simulations with a recently developed force field for disordered systems to shed new light on known experimental results, showing that the N-terminus of VDAC is an intrinsically disordered region (IDR). First, simulation of the N-terminal segment as a free peptide highlighted its disordered nature and the importance of using an IDR-specific force field to properly sample its conformational landscape. Secondly, accelerated dynamics simulation of a double cysteine VDAC mutant under applied voltage revealed metastable low conducting states of the channel representative of closed states observed experimentally. Related structures were characterized by partial unfolding and rearrangement of the N-terminal tail, that led to steric hindrance of the pore. Our results indicate that the disordered properties of the N-terminus are crucial to properly account for the gating mechanism of VDAC.

The voltage-dependent anion channel (VDAC) is a membrane protein playing a pivotal role in the transport of ions or ATP across the mitochondrial outer membrane as well as in the induction of apoptosis. At high enough membrane potential, VDAC is known to transition from an open state to multiple closed states, reducing the flow of ions through the channel and blocking the passage of large metabolites. While the structure of the open state was resolved more than a decade ago, a molecular description of the gating mechanism of the channel is still missing. Here we show that the N-terminus of VDAC is an intrinsically disordered region and that such a property has a profound impact on its dynamics either as a free peptide or as part of the channel. By taking disordered properties of the N-terminus into account, we managed to generate long-lived closed conformations of the channel at experimental values of the membrane potential. Our results provide new insights into the molecular mechanism driving the gating of VDAC.

Specific adsorption of trivalent cations in biological nanopores determines conductance dynamics and reverses ionic selectivity

Adsorption processes are central to ionic transport in industrial and biological membrane systems. Multivalent cations modulate the conductive properties of nanofluidic devices through interactions with charged surfaces that depend principally on the ion charge number. Considering that ion channels are specialized valves that demand a sharp specificity in ion discrimination, we investigate the adsorption dynamics of trace amounts of different salts of trivalent cations in biological nanopores. We consider here OmpF from Escherichia coli, an archetypical protein nanopore, to probe the specificity of biological nanopores to multivalent cations. We systematically compare the effect of three trivalent electrolytes on OmpF current-voltage relationships and characterize the degree of rectification induced by each ion. We also analyze the open channel current noise to determine the existence of equilibrium/non-equilibrium mechanisms of ion adsorption and evaluate the extent of charge inversion through selectivity measurements. We show that the interaction of trivalent electrolytes with biological nanopores occurs via ion-specific adsorption yielding differential modulation of ion conduction and selectivity inversion. We also demonstrate the existence of non-equilibrium fluctuations likely related to ion-dependent trapping-detrapping processes. Our study provides fundamental information relevant to different biological and electrochemical systems where transport phenomena involve ion adsorption in charged surfaces under nanoscale confinement.

Millisecond-Long Simulations of Antibiotics Transport through Outer Membrane Channels

To reach their target site inside Gram-negative bacteria, almost all antibiotics need to cross the outer membrane. Computational modeling of such processes can be numerically demanding due to the size of the systems and especially due to the timescales involved. Recently, a hybrid Brownian and molecular dynamics approach, i.e., Brownian dynamics including explicit atoms (BRODEA), has been developed and evaluated for studying the transport of monoatomic ions through membrane channels. Later on, this numerically efficient scheme has been applied to determine the free energy surfaces of the ciprofloxacin and enrofloxacin translocation through the porin OmpC using temperature-accelerated simulations. To improve the usability and accuracy of the approach, schemes to approximate the position-dependent diffusion constant of the molecule while traversing the pore had to be established. To this end, we have studied the translocation of the charged phosphonic acid antibiotic fosfomycin through the porin OmpF from Escherichia coli devising and benchmarking several diffusion models. To test the efficiency and sensitivity of these models, the effect of OmpF mutations on the permeation of fosfomycin was analyzed. Permeation events have been recorded over millisecond-long biased and unbiased simulations, from which thermodynamics and kinetics quantities of the translocation processes were determined. As a result, the use of the BRODEA approach, together with the appropriate diffusion model, was seen to accurately reproduce the findings observed in electrophysiology experiments and all-atom molecular dynamics simulations. These results suggest that the BRODEA approach can become a valuable tool for screening numerous compounds to evaluate their outer membrane permeability, a property important in the development of new antibiotics.

Guardians of the Cell: State-of-the-Art of Membrane Proteins from a Computational Point-of-View

Membrane proteins (MPs) encompass a large family of proteins with distinct cellular functions, and although representing over 50% of existing pharmaceutical drug targets, their structural and functional information is still very scarce. Over the last years, in silico analysis and algorithm development were essential to characterize MPs and overcome some limitations of experimental approaches. The optimization and improvement of these methods remain an ongoing process, with key advances in MPs' structure, folding, and interface prediction being continuously tackled. Herein, we discuss the latest trends in computational methods toward a deeper understanding of the atomistic and mechanistic details of MPs.

VDAC Gating Thermodynamics, but Not Gating Kinetics, Are Virtually Temperature Independent

The voltage-dependent anion channel (VDAC) is the most abundant protein in the mitochondrial outer membrane and an archetypical β-barrel channel. Here, we study the effects of temperature on VDAC channels reconstituted in planar lipid membranes at the single- and multichannel levels within the 20°C to 40°C range. The temperature dependence of conductance measured on a single channel in 1 M KCl shows an increase characterized by a 10°C temperature coefficient Q₁₀ = 1.22 ± 0.02, which exceeds that of the bathing electrolyte solution conductivity, Q₁₀ = 1.17 ± 0.01. The rates of voltage-induced channel transition between the open and closed states measured on multichannel membranes also show statistically significant increases, with temperatures that are consistent with activation energy barriers of ∼10 ± 3 kcal/mol. At the same time, the gating thermodynamics, as characterized by the gating charge and voltage of equipartitioning, does not display any measurable temperature dependence. The two parameters stay within 3.2 ± 0.2 elementary charges and 30 ± 2 mV, respectively. Thus, whereas the channel kinetics, specifically its conductance and rates of gating response to voltage steps, demonstrates a clear increase with temperature, the conformational voltage-dependent equilibria are virtually insensitive to temperature. These results, which may be a general feature of β-barrel channel gating, suggest either an entropy-driven gating mechanism or a role for enthalpy-entropy compensation.

Dynamic interaction of fluoroquinolones with magnesium ions monitored using bacterial outer membrane nanopores

Divalent ions are known to have a severe effect on the translocation of several antibiotic molecules into (pathogenic) bacteria. In the present study we have investigated the effect of divalent ions on the permeability of norfloxacin across the major outer membrane channels from E. coli (OmpF and OmpC) and E. aerogenes (Omp35 and Omp36) at the single channel level. To understand the rate limiting steps in permeation, we reconstituted single porins into planar lipid bilayers and analyzed the ion current fluctuations caused in the presence of norfloxacin. Moreover, to obtain an atomistic view, we complemented the experiments with millisecond-long free energy calculations based on temperature-accelerated Brownian dynamics simulations to identify the most probable permeation pathways of the antibiotics through the respective pores. Both, the experimental analysis and the computational modelling, suggest that norfloxacin is able to permeate through the larger porins, i.e., OmpF, OmpC, and Omp35, whereas it only binds to the slightly narrower porin Omp36. Moreover, divalent ions can bind to negatively charged residues inside the porin, reversing the ion selectivity of the pore. In addition, the divalent ions can chelate with the fluoroquinolone molecules and alter their physicochemical properties. The results suggest that the conjugation with either pores or molecules must break when the antibiotic molecules pass the lumen of the porin, with the conjugation to the antibiotic being more stable than that to the respective pore. In general, the permeation or binding process of fluoroquinolones in porins occurs irrespective of the presence of divalent ions, but the presence of divalent ions can vary the kinetics significantly. Thus, a detailed investigation of the interplay of divalent ions with antibiotics and pores is of key importance in developing new antimicrobial drugs.

Accurate modeling of a biological nanopore with an extended continuum framework

Despite the broad success of biological nanopores as powerful instruments for the analysis of proteins and nucleic acids at the single-molecule level, a fast simulation methodology to accurately model their nanofluidic properties is currently unavailable. This limits the rational engineering of nanopore traits and makes the unambiguous interpretation of experimental results challenging. Here, we present a continuum approach that can faithfully reproduce the experimentally measured ionic conductance of the biological nanopore Cytolysin A (ClyA) over a wide range of ionic strengths and bias potentials. Our model consists of the extended Poisson-Nernst-Planck and Navier-Stokes (ePNP-NS) equations and a computationally efficient 2D-axisymmetric representation for the geometry and charge distribution of the nanopore. Importantly, the ePNP-NS equations achieve this accuracy by self-consistently considering the finite size of the ions and the influence of both the ionic strength and the nanoscopic scale of the pore on the local properties of the electrolyte. These comprise the mobility and diffusivity of the ions, and the density, viscosity and relative permittivity of the solvent. Crucially, by applying our methodology to ClyA, a biological nanopore used for single-molecule enzymology studies, we could directly quantify several nanofluidic characteristics difficult to determine experimentally. These include the ion selectivity, the ion concentration distributions, the electrostatic potential landscape, the magnitude of the electro-osmotic flow field, and the internal pressure distribution. Hence, this work provides a means to obtain fundamental new insights into the nanofluidic properties of biological nanopores and paves the way towards their rational engineering.

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.

Molecular Mean-Field Theory of Ionic Solutions: A Poisson-Nernst-Planck-Bikerman Model

We have developed a molecular mean-field theory—fourth-order Poisson–Nernst–Planck–Bikerman theory—for modeling ionic and water flows in biological ion channels by treating ions and water molecules of any volume and shape with interstitial voids, polarization of water, and ion-ion and ion-water correlations. The theory can also be used to study thermodynamic and electrokinetic properties of electrolyte solutions in batteries, fuel cells, nanopores, porous media including cement, geothermal brines, the oceanic system, etc. The theory can compute electric and steric energies from all atoms in a protein and all ions and water molecules in a channel pore while keeping electrolyte solutions in the extra- and intracellular baths as a continuum dielectric medium with complex properties that mimic experimental data. The theory has been verified with experiments and molecular dynamics data from the gramicidin A channel, L-type calcium channel, potassium channel, and sodium/calcium exchanger with real structures from the Protein Data Bank. It was also verified with the experimental or Monte Carlo data of electric double-layer differential capacitance and ion activities in aqueous electrolyte solutions. We give an in-depth review of the literature about the most novel properties of the theory, namely Fermi distributions of water and ions as classical particles with excluded volumes and dynamic correlations that depend on salt concentration, composition, temperature, pressure, far-field boundary conditions etc. in a complex and complicated way as reported in a wide range of experiments. The dynamic correlations are self-consistent output functions from a fourth-order differential operator that describes ion-ion and ion-water correlations, the dielectric response (permittivity) of ionic solutions, and the polarization of water molecules with a single correlation length parameter.

Effects of Diffusion Coefficients and Permanent Charge on Reversal Potentials in Ionic Channels

In this work, the dependence of reversal potentials and zero-current fluxes on diffusion coefficients are examined for ionic flows through membrane channels. The study is conducted for the setup of a simple structure defined by the profile of permanent charges with two mobile ion species, one positively charged (cation) and one negatively charged (anion). Numerical observations are obtained from analytical results established using geometric singular perturbation analysis of classical Poisson–Nernst–Planck models. For 1:1 ionic mixtures with arbitrary diffusion constants, Mofidi and Liu (arXiv:1909.01192) conducted a rigorous mathematical analysis and derived an equation for reversal potentials. We summarize and extend these results with numerical observations for biological relevant situations. The numerical investigations on profiles of the electrochemical potentials, ion concentrations, and electrical potential across ion channels are also presented for the zero-current case. Moreover, the dependence of current and fluxes on voltages and permanent charges is investigated. In the opinion of the authors, many results in the paper are not intuitive, and it is difficult, if not impossible, to reveal all cases without investigations of this type.

Short solvent model for ion correlations and hydrophobic association

Coulomb interactions play a major role in determining the thermodynamics, structure, and dynamics of condensed-phase systems, but often present significant challenges. Computer simulations usually use periodic boundary conditions to minimize corrections from finite cell boundaries but the long range of the Coulomb interactions generates significant contributions from distant periodic images of the simulation cell, usually calculated by Ewald sum techniques. This can add significant overhead to computer simulations and hampers the development of intuitive local pictures and simple analytic theory. In this paper, we present a general framework based on local molecular field theory to accurately determine the contributions from long-ranged Coulomb interactions to the potential of mean force between ionic or apolar hydrophobic solutes in dilute aqueous solutions described by standard classical point charge water models. The simplest approximation leads to a short solvent (SS) model, with truncated solvent-solvent and solute-solvent Coulomb interactions and long-ranged but screened Coulomb interactions only between charged solutes. The SS model accurately describes the interplay between strong short-ranged solute core interactions, local hydrogen-bond configurations, and long-ranged dielectric screening of distant charges, competing effects that are difficult to capture in standard implicit solvent models.

Solution of Ion Channel Flow Using Immersed Boundary-Lattice Boltzmann Methods

Poisson-Nernst-Planck (PNP) model has been extensively used for the study of channel flow under the influence of electrochemical gradients. PNP theory is a continuum description of ion flow where ionic distributions are described in terms of concentrations. Nonionic interparticle interactions are not considered in this theory as in continuum framework, their impact on the solution is minimal. This theory holds true for dilute flows or flows where channel radius is significantly larger than ion radius. However, for ion channel flows, where channel dimensions and ionic radius are of similar magnitude, nonionic interactions, particularly related to the size of the ions (steric effect), play an important role in defining the selectivity of the channel, concentration distribution of ionic species, and current across the channel, etc. To account for the effect of size of ions, several modifications to PNP equations have been proposed. One such approach is the introduction of Lennard-Jones potential to the energy variational formulation of PNP system. This study focuses on understanding the role of steric effect on flow properties. To discretize the system, Lattice Boltzmann method has been used. The system is defined by modified PNP equations where the steric effect is described by Lennard-Jones potential. In addition, boundary conditions for the complex channel geometry have been treated using immersed boundary method.

Selectivity of the KcsA potassium channel: Analysis and computation

Ion channels regulate the flux of ions through cell membranes and play significant roles in many physiological functions. Most of the existing literature focuses on computational approaches based on molecular dynamics simulation or numerical solution of the modified Poisson-Nernst-Planck (PNP) system. In this paper, we present an analytical and computational study of a mathematical model of the KcsA potassium channel, including the effects of ion size (Bikerman model) and solvation energy (Born model). Under equilibrium conditions, we obtain an analytical solution of our modified PNP system, which is used to explain selectivity of KcsA of various ions (K^{+}, Na^{+}, Cl^{-}, Ca^{2+}, and Ba^{2+}) due to negative permanent charges inside the filter region and the effect of ion sizes. Our results show that K^{+} is always selected over Na^{+}, as smaller Na^{+} ions have larger solvation energy. As the amount of negative charges in the filter exceeds a critical value, divalent ions (Ca^{2+} and Ba^{2+}) can enter the filter region and block the KcsA channel. For the nonequilibrium cases, due to difficulties associated with a pure analytical or numerical approach, we use a hybrid analytical-numerical method to solve the modified PNP system. Our predictions of selectivity of KcsA channels and saturation phenomenon of the current-voltage (I-V) curve agree with experimental observations.

Manipulation of charge distribution in the arginine and glutamate clusters of the OmpG pore alters sugar specificity and ion selectivity

OmpG is a general diffusion pore in the E. coli outer membrane with a molecular architecture comprising a 14-stranded β-barrel scaffold and unique structural features. In contrast to other non-specific porins, OmpG lacks a central constriction zone and has an exceptionally wide pore diameter of about 13 Å. The equatorial plane of OmpG harbors an annulus of four alternating basic and acidic patches whose function is only poorly characterized. We have investigated the role of charge distribution for ion selectivity and sugar transport with the help of OmpG variants mutated in the annulus. Substituting the glutamate residues of the annulus for histidines or alanines led to a strong reduction in cation selectivity. Replacement of the glutamates in the annulus by histidine residues also disfavored the passage of pentoses and hexoses relative to disaccharides. Our results demonstrate that despite the wide pore diameter, an annulus only consisting of two opposing basic patches confers reduced cation and monosaccharide transport compared to OmpG wild type. Furthermore, randomization of charged residues in the annulus had the potential to abolish pH-dependency of sugar transport. Our results indicate that E¹⁵, E³¹, R⁹², R¹¹¹ and R²¹¹ in the annulus form electrostatic interactions with R²²⁸, E²²⁹ and D²³² in loop L6 that influence pH-dependency of sugar transport.