Ionic selectivity in L-type calcium channels by electrostatics and hard-core repulsion

Probing ion binding in the selectivity filter of the Cav1.1 channel with molecular dynamics

Cav1.1 is the voltage-gated calcium channel essential for the contraction of skeletal muscles upon membrane potential changes. Structural determination of the Cav1.1 channel opens the avenue toward understanding of the structure-function relationship of voltage-gated calcium channels. Here, we show that there exist two Ca2+-binding sites, termed S1 and S2, within the selectivity filter of Cav1.1 through extensive molecular dynamics simulations on various initial ion arrangement configurations. The formation of both binding sites is associated with the four Glu residues (Glu292/614/1014/1323) that constitute the so-called EEEE locus. At the S1 site near the extracellular side, the Ca2+ ion is coordinated with the negatively charged carboxylic groups of these Glu residues and of the Asp615 residue either in a direct way or via an intermediate water molecule. At the S2 site, Ca2+ binding shows two distinct states: an upper state involving two out of the four Glu residues in the EEEE locus and a lower state involving only one Glu residue. In addition, there exist two recruitment sites for Ca2+ above the entrance of the filter. These findings promote the understanding of mechanism for ion permeation and selectivity in calcium channels.

Setting Boundaries for Statistical Mechanics

Statistical mechanics has grown without bounds in space. Statistical mechanics of noninteracting point particles in an unbounded perfect gas is widely used to describe liquids like concentrated salt solutions of life and electrochemical technology, including batteries. Liquids are filled with interacting molecules. A perfect gas is a poor model of a liquid. Statistical mechanics without spatial bounds is impossible as well as imperfect, if molecules interact as charged particles, as nearly all atoms do. The behavior of charged particles is not defined until boundary structures and values are defined because charges are governed by Maxwell's partial differential equations. Partial differential equations require boundary structures and conditions. Boundary conditions cannot be defined uniquely 'at infinity' because the limiting process that defines 'infinity' includes such a wide variety of structures and behaviors, from elongated ellipses to circles, from light waves that never decay, to dipolar fields that decay steeply, to Coulomb fields that hardly decay at all. Boundaries and boundary conditions needed to describe matter are not prominent in classical statistical mechanics. Statistical mechanics of bounded systems is described in the EnVarA system of variational mechanics developed by Chun Liu, more than anyone else. EnVarA treatment does not yet include Maxwell equations.

Lithium administered to pregnant, lactating and neonatal rats: entry into developing brain

Background

Little is known about the extent of drug entry into developing brain, when administered to pregnant and lactating women. Lithium is commonly prescribed for bipolar disorder. Here we studied transfer of lithium given to dams, into blood, brain and cerebrospinal fluid (CSF) in embryonic and postnatal animals as well as adults.

Methods

Lithium chloride in a clinically relevant dose (3.2 mg/kg body weight) was injected intraperitoneally into pregnant (E15–18) and lactating dams (birth-P16/17) or directly into postnatal pups (P0–P16/17). Acute treatment involved a single injection; long-term treatment involved twice daily injections for the duration of the experiment. Following terminal anaesthesia blood plasma, CSF and brains were collected. Lithium levels and brain distribution were measured using Laser Ablation Inductively Coupled Plasma-Mass Spectrometry and total lithium levels were confirmed by Inductively Coupled Plasma-Mass Spectrometry.

Results

Lithium was detected in blood, CSF and brain of all fetal and postnatal pups following lithium treatment of dams. Its concentration in pups’ blood was consistently below that in maternal blood (30–35%) indicating significant protection by the placenta and breast tissue. However, much of the lithium that reached the fetus entered its brain. Levels of lithium in plasma fluctuated in different treatment groups but its concentration in CSF was stable at all ages, in agreement with known stable levels of endogenous ions in CSF. There was no significant increase of lithium transfer into CSF following application of Na⁺/K⁺ ATPase inhibitor (digoxin) in vivo, indicating that lithium transfer across choroid plexus epithelium is not likely to be via the Na⁺/K⁺ ATPase mechanism, at least early in development. Comparison with passive permeability markers suggested that in acute experiments lithium permeability was less than expected for diffusion but similar in long-term experiments at P2.

Conclusions

Information obtained on the distribution of lithium in developing brain provides a basis for studying possible deleterious effects on brain development and behaviour in offspring of mothers undergoing lithium therapy.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12987-021-00285-w.

Maxwell Equations without a Polarization Field, Using a Paradigm from Biophysics

When forces are applied to matter, the distribution of mass changes. Similarly, when an electric field is applied to matter with charge, the distribution of charge changes. The change in the distribution of charge (when a local electric field is applied) might in general be called the induced charge. When the change in charge is simply related to the applied local electric field, the polarization field P is widely used to describe the induced charge. This approach does not allow electrical measurements (in themselves) to determine the structure of the polarization fields. Many polarization fields will produce the same electrical forces because only the divergence of polarization enters Maxwell's first equation, relating charge and electric forces and field. The curl of any function can be added to a polarization field P without changing the electric field at all. The divergence of the curl is always zero. Additional information is needed to specify the curl and thus the structure of the P field. When the structure of charge changes substantially with the local electric field, the induced charge is a nonlinear and time dependent function of the field and P is not a useful framework to describe either the electrical or structural basis-induced charge. In the nonlinear, time dependent case, models must describe the charge distribution and how it varies as the field changes. One class of models has been used widely in biophysics to describe field dependent charge, i.e., the phenomenon of nonlinear time dependent induced charge, called 'gating current' in the biophysical literature. The operational definition of gating current has worked well in biophysics for fifty years, where it has been found to makes neurons respond sensitively to voltage. Theoretical estimates of polarization computed with this definition fit experimental data. I propose that the operational definition of gating current be used to define voltage and time dependent induced charge, although other definitions may be needed as well, for example if the induced charge is fundamentally current dependent. Gating currents involve substantial changes in structure and so need to be computed from a combination of electrodynamics and mechanics because everything charged interacts with everything charged as well as most things mechanical. It may be useful to separate the classical polarization field as a component of the total induced charge, as it is in biophysics. When nothing is known about polarization, it is necessary to use an approximate representation of polarization with a dielectric constant that is a single real positive number. This approximation allows important results in some cases, e.g., design of integrated circuits in silicon semiconductors, but can be seriously misleading in other cases, e.g., ionic solutions.

Modeling the Device Behavior of Biological and Synthetic Nanopores with Reduced Models

Biological ion channels and synthetic nanopores are responsible for passive transport of ions through a membrane between two compartments. Modeling these ionic currents is especially amenable to reduced models because the device functions of these pores, the relation of input parameters (e.g., applied voltage, bath concentrations) and output parameters (e.g., current, rectification, selectivity), are well defined. Reduced models focus on the physics that produces the device functions (i.e., the physics of how inputs become outputs) rather than the atomic/molecular-scale physics inside the pore. Here, we propose four rules of thumb for constructing good reduced models of ion channels and nanopores. They are about (1) the importance of the axial concentration profiles, (2) the importance of the pore charges, (3) choosing the right explicit degrees of freedom, and (4) creating the proper response functions. We provide examples for how each rule of thumb helps in creating a reduced model of device behavior.

Rectification of bipolar nanopores in multivalent electrolytes: effect of charge inversion and strong ionic correlations

Bipolar nanopores have powerful rectification properties due to the asymmetry in the charge pattern on the wall of the nanopore. In particular, bipolar nanopores have positive and negative surface charges along the pore axis. Rectification is strong if the radius of the nanopore is small compared to the screening length of the electrolyte so that both cations and anions have depletion zones in the respective regions. The depths of these depletion zones is sensitive to sign of the external voltage. In this work, we are interested in the effect of the presence of strong ionic correlations (both between ions and between ions and surface charge) due to the presence of multivalent ions and large surface charges. We show that strong ionic correlations cause leakage of the coions, a phenomenon that is absent in mean field theories. In this modeling study, we use both the mean-field Poisson-Nernst-Planck (PNP) theory and a particle simulation method, Local Equilibrium Monte Carlo (LEMC), to show that phenomena such as overcharging and charge inversion cannot be reproduced with PNP, while LEMC is able to produce nonmonotonic dependence of currents and rectification as a function of surface charge strength.

Effects of nitric oxide on apoptosis and voltage-gated calcium channels in human cardiac myofibroblasts

We characterised the voltage-gated Ca²⁺ channels (VGCCs) in human cardiac fibroblasts (HCFs) and myofibroblasts (HCMFs) and investigated the effects of nitric oxide (NO) on apoptosis and on these channels. Western blotting and immunofluorescence analyses show that α-smooth muscle actin (a myofibroblast marker) was markedly expressed in passage (P) 12-15 but not in P4 HCF cells, whereas calponin (a fibroblast marker) was expressed only in P4 cells. Ca_V 1.2 (L-type) and Ca_V 3.3 (T-type) of VGCCs were highly expressed in P12-15 cells, but only weak Ca_V 2.3 (R-type) expression was identified in P4 cells using reverse transcription-polymerase chain reaction analysis. S-Nitroso-N-acetylpenicillamine (SNAP, an NO donor) decreased cell viability of HCMFs in a dose-dependent manner and induced apoptotic changes, and nifedipine (an L-type Ca²⁺ channel blocker) prevented apoptosis as shown with immunofluorescence staining and flow cytometry. Whole-cell mode patch-clamp recordings demonstrate the presence of L-type Ca²⁺ (I_{C a,L} ) and T-type Ca²⁺ (I_{C a,T} ) currents in HCMFs. SNAP inhibited I_{C a,L} of HCMFs, but pre-treatment with ODQ (a guanylate cyclase inhibitor) or KT5823 (a PKG inhibitor) prevented it. Pre-treating cells with KT5720 (a PKA inhibitor) or SQ22536 (an adenylate cyclase inhibitor) blocked SNAP-induced inhibition of I_{C a,L} . 8-Bromo-cyclic GMP or 8-bromo-cyclic AMP also inhibited I_{C a,L} . However, pre-treatment with N-ethylmaleimide (a thiol-alkylating reagent) did not block the SNAP effect, nor did DL-dithiothreitol (a reducing agent) reverse it. These data suggest that high concentrations of NO injure HCMFs and inhibit I_{C a,L} through the PKG and PKA signalling pathways but not through the S-nitrosylation pathway.

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.

Highly Selective Transport of Alkali Metal Ions by Nanochannels of Polyelectrolyte Threaded MIL-53 Metal Organic Framework

Conventional ion-exchange polymeric membranes have limited selectivity due to their nonuniform and unstable structures. The rigid, regular, high porosity of metal organic framework (MOF) generally provides MOF membrane with exclusion/sieving effect but lack of electrostatic screening. Here we report for the first time a nonbiological highly selective MOF membrane with polyelectrolyte threaded in the nanochannel of metal organic framework (polyelectrolyte∼MOF) and its selective transport of alkali metal cations. Poly(sodium vinyl sulfonated-co-acrylic acid)∼MIL-53(Al) is prepared on anodic aluminum oxide substrate via steps of MOF MIL-53(Al) growth followed by in situ polymerization. The poly(VS-co-AA)∼MIL-53(Al) membrane demonstrates highly specific selectivity in transport of alkali metal cations. Rate of ion transport correlates inversely with the hydrated diameter of the ion reaching a low limiting rate near 0.7 nm hydrated diameter. Charge exclusion is demonstrated with blockage of anion transport under a concentration gradient. The highly uniform porous nanostructure of MOF and ionic function of polyelectrolyte offers the MOF membrane with synergistic selectivity based on exclusion forces of the framework and Coulomb forces from fixed charges of polyelectrolytes in nanochannels.

Multiscale analysis of the effect of surface charge pattern on a nanopore's rectification and selectivity properties: From all-atom model to Poisson-Nernst-Planck

We report a multiscale modeling study for charged cylindrical nanopores using three modeling levels that include (1) an all-atom explicit-water model studied with molecular dynamics, and reduced models with implicit water containing (2) hard-sphere ions studied with the Local Equilibrium Monte Carlo simulation method (computing ionic correlations accurately), and (3) point ions studied with Poisson-Nernst-Planck theory (mean-field approximation). We show that reduced models are able to reproduce device functions (rectification and selectivity) for a wide variety of charge patterns, that is, reduced models are useful in understanding the mesoscale physics of the device (i.e., how the current is produced). We also analyze the relationship of the reduced implicit-water models with the explicit-water model and show that diffusion coefficients in the reduced models can be used as adjustable parameters with which the results of the explicit- and implicit-water models can be related. We find that the values of the diffusion coefficients are sensitive to the net charge of the pore but are relatively transferable to different voltages and charge patterns with the same total charge.

On the selectivity of the NaChBac channel: an integrated computational and experimental analysis of sodium and calcium permeation

Ion channel selectivity is essential for their function, yet the molecular basis of a channel's ability to select between ions is still rather controversial. In this work, using a combination of molecular dynamics simulations and electrophysiological current measurements we analyze the ability of the NaChBac channel to discriminate between calcium and sodium. Our simulations show that a single calcium ion can access the Selectivity Filter (SF) interacting so strongly with the glutamate ring so as to remain blocked inside. This is consistent with the tiny calcium currents recorded in our patch-clamp experiments. Two reasons explain this scenario. The first is the higher free energy of ion/SF binding of Ca²⁺ with respect to Na⁺. The second is the strong electrostatic repulsion exerted by the resident ion that turns back a second potentially incoming Ca²⁺, preventing the knock-on permeation mechanism. Finally, we analyzed the possibility of the Anomalous Mole Fraction Effect (AMFE), i.e. the ability of micromolar Ca²⁺ concentrations to block Na⁺ currents. Current measurements in Na⁺/Ca²⁺ mixed solutions excluded the AMFE, in agreement with metadynamics simulations showing the ability of a sodium ion to by-pass and partially displace the resident calcium. Our work supports a new scenario for Na⁺/Ca²⁺ selectivity in the bacterial sodium channel, challenging the traditional notion of an exclusion mechanism strictly confining Ca²⁺ ions outside the channel.

Simulation of a model nanopore sensor: Ion competition underlies device behavior

We study a model nanopore sensor with which a very low concentration of analyte molecules can be detected on the basis of the selective binding of the analyte molecules to the binding sites on the pore wall. The bound analyte ions partially replace the current-carrier cations in a thermodynamic competition. This competition depends both on the properties of the nanopore and the concentrations of the competing ions (through their chemical potentials). The output signal given by the device is the current reduction caused by the presence of the analyte ions. The concentration of the analyte ions can be determined through calibration curves. We model the binding site with the square-well potential and the electrolyte as charged hard spheres in an implicit background solvent. We study the system with a hybrid method in which we compute the ion flux with the Nernst-Planck (NP) equation coupled with the Local Equilibrium Monte Carlo (LEMC) simulation technique. The resulting NP+LEMC method is able to handle both strong ionic correlations inside the pore (including finite size of ions) and bulk concentrations as low as micromolar. We analyze the effect of bulk ion concentrations, pore parameters, binding site parameters, electrolyte properties, and voltage on the behavior of the device.

Multiscale modeling of a rectifying bipolar nanopore: explicit-water versus implicit-water simulations

In a multiscale modeling approach, we present computer simulation results for a rectifying bipolar nanopore at two modeling levels. In an all-atom model, we use explicit water to simulate ion transport directly with the molecular dynamics technique. In a reduced model, we use implicit water and apply the Local Equilibrium Monte Carlo method together with the Nernst-Planck transport equation. This hybrid method makes the fast calculation of ion transport possible at the price of lost details. We show that the implicit-water model is an appropriate representation of the explicit-water model when we look at the system at the device (i.e., input vs. output) level. The two models produce qualitatively similar behavior of the electrical current for different voltages and model parameters. Looking at the details of concentration and potential profiles, we find profound differences between the two models. These differences, however, do not influence the basic behavior of the model as a device because they do not influence the z-dependence of the concentration profiles which are the main determinants of current. These results then address an old paradox: how do reduced models, whose assumptions should break down in a nanoscale device, predict experimental data? Our simulations show that reduced models can still capture the overall device physics correctly, even though they get some important aspects of the molecular-scale physics quite wrong; reduced models work because they include the physics that is necessary from the point of view of device function. Therefore, reduced models can suffice for general device understanding and device design, but more detailed models might be needed for molecular level understanding.

Understanding Sodium Channel Function and Modulation Using Atomistic Simulations of Bacterial Channel Structures

Sodium channels are chief proteins involved in electrical signaling in the nervous system, enabling critical functions like heartbeat and brain activity. New high-resolution X-ray structures for bacterial sodium channels have created an opportunity to see how these proteins operate at the molecular level. An important challenge to overcome is establishing relationships between the structures and functions of mammalian and bacterial channels. Bacterial sodium channels are known to exhibit the main structural features of their mammalian counterparts, as well as several key functional characteristics, including selective ion conduction, voltage-dependent gating, pore-based inactivation and modulation by local anesthetic, antiarrhythmic and antiepileptic drugs. Simulations have begun to shed light on each of these features in the past few years. Despite deviations in selectivity signatures for bacterial and mammalian channels, simulations have uncovered the nature of the multiion conduction mechanism associated with Na(+) binding to a high-field strength site established by charged glutamate side chains. Simulations demonstrated a surprising level of flexibility of the protein, showing that these side chains are active participants in the permeation process. They have also uncovered changes in protein structure, leading to asymmetrical collapses of the activation gate that have been proposed to correspond to inactivated structures. These observations offer the potential to examine the mechanisms of state-dependent drug activity, focusing on pore-blocking and pore-based slow inactivation in bacterial channels, without the complexities of inactivation on multiple timescales seen in eukaryotic channels. Simulations have provided molecular views of the interactions of drugs, consistent with sites predicted in mammalian channels, as well as a wealth of other sites as potential new drug targets. In this chapter, we survey the new insights into sodium channel function that have emerged from studies of simpler bacterial channels, which provide an excellent learning platform, and promising avenues for mechanistic discovery and pharmacological development.

Numerical methods for a Poisson-Nernst-Planck-Fermi model of biological ion channels

Numerical methods are proposed for an advanced Poisson-Nernst-Planck-Fermi (PNPF) model for studying ion transport through biological ion channels. PNPF contains many more correlations than most models and simulations of channels, because it includes water and calculates dielectric properties consistently as outputs. This model accounts for the steric effect of ions and water molecules with different sizes and interstitial voids, the correlation effect of crowded ions with different valences, and the screening effect of polarized water molecules in an inhomogeneous aqueous electrolyte. The steric energy is shown to be comparable to the electrical energy under physiological conditions, demonstrating the crucial role of the excluded volume of particles and the voids in the natural function of channel proteins. Water is shown to play a critical role in both correlation and steric effects in the model. We extend the classical Scharfetter-Gummel (SG) method for semiconductor devices to include the steric potential for ion channels, which is a fundamental physical property not present in semiconductors. Together with a simplified matched interface and boundary (SMIB) method for treating molecular surfaces and singular charges of channel proteins, the extended SG method is shown to exhibit important features in flow simulations such as optimal convergence, efficient nonlinear iterations, and physical conservation. The generalized SG stability condition shows why the standard discretization (without SG exponential fitting) of NP equations may fail and that divalent Ca(2+) may cause more unstable discrete Ca(2+) fluxes than that of monovalent Na(+). Two different methods-called the SMIB and multiscale methods-are proposed for two different types of channels, namely, the gramicidin A channel and an L-type calcium channel, depending on whether water is allowed to pass through the channel. Numerical methods are first validated with constructed models whose exact solutions are known. The experimental data of both channels are then used to verify and explain novel features of PNPF as compared with previous PNP models. The PNPF currents are in accord with the experimental I-V (V for applied voltages) data of the gramicidin A channel and I-C (C for bath concentrations) data of the calcium channel with 10(-8)-fold bath concentrations that pose severe challenges in theoretical simulations.

Ion conduction and conformational flexibility of a bacterial voltage-gated sodium channel

Voltage-gated Na(+) channels play an essential role in electrical signaling in the nervous system and are key pharmacological targets for a range of disorders. The recent solution of X-ray structures for the bacterial channel NavAb has provided an opportunity to study functional mechanisms at the atomic level. This channel's selectivity filter exhibits an EEEE ring sequence, characteristic of mammalian Ca(2+), not Na(+), channels. This raises the fundamentally important question: just what makes a Na(+) channel conduct Na(+) ions? Here we explore ion permeation on multimicrosecond timescales using the purpose-built Anton supercomputer. We isolate the likely protonation states of the EEEE ring and observe a striking flexibility of the filter that demonstrates the necessity for extended simulations to study conduction in this channel. We construct free energy maps to reveal complex multi-ion conduction via knock-on and "pass-by" mechanisms, involving concerted ion and glutamate side chain movements. Simulations in mixed ionic solutions reveal relative energetics for Na(+), K(+), and Ca(2+) within the pore that are consistent with the modest selectivity seen experimentally. We have observed conformational changes in the pore domain leading to asymmetrical collapses of the activation gate, similar to proposed inactivated structures of NavAb, with helix bending involving conserved residues that are critical for slow inactivation. These structural changes are shown to regulate access to fenestrations suggested to be pathways for lipophilic drugs and provide deeper insight into the molecular mechanisms connecting drug activity and slow inactivation.

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.

Na(+)/Ca(2+) selectivity in the bacterial voltage-gated sodium channel NavAb

The recent publication of a number of high resolution bacterial voltage-gated sodium channel structures has opened the door for the mechanisms employed by these channels to distinguish between ions to be elucidated. The way these channels select between Na⁺ and K⁺ has been investigated in computational studies, but the selectivity for Na⁺ over Ca²⁺ has not yet been studied in this way. Here we use molecular dynamics simulations to calculate the energetics of Na⁺ and Ca²⁺ transport through the channel. Single ion profiles show that Ca²⁺ experiences a large barrier midway through the selectivity filter that is not seen by Na⁺. This barrier is caused by the need for Ca²⁺ to partly dehydrate to pass through this region and the lack of compensating interactions with the protein. Multi-ion profiles show that ions can pass each other in the channel, which is why the presence of Ca²⁺ does not block Na⁺ conduction despite binding more strongly in the pore.

Ion correlations in nanofluidic channels: effects of ion size, valence, and concentration on voltage- and pressure-driven currents

The effects of ion-ion and ion-wall correlations in nanochannels are explored, specifically how they influence voltage- and pressure-driven currents and pressure-to-voltage energy conversion. Cations of different diameters (0.15, 0.3, and 0.9 nm) and different valences (+1, +2, and +3) at concentrations ranging from 10(-6) M to 1 M are considered in 50-nm- and 100-nm-wide nanoslits with wall surface charges ranging from 0 C/m(2) to -0.3 C/m(2). These parameters are typical of nanofluidic devices. Ion correlations have significant effects on device properties over large parts of this parameter space. These effects are the result of ion layering (oscillatory concentration profiles) for large monovalent cations and charge inversion (more cations in the first layer near the wall than necessary to neutralize the surface charge) for the multivalent cations. The ions were modeled as charged, hard spheres using density functional theory of fluids, and current was computed with the Navier-Stokes equations with two different no-slip conditions.

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.

Competition among Ca2+, Mg2+, and Na+ for model ion channel selectivity filters: determinants of ion selectivity

Because voltage-gated ion channels play critical biological roles, understanding how they can discriminate the native metal ion from rival cations in the milieu is of great interest. Although Ca(2+), Mg(2+), and Na(+) are present in comparable concentrations outside the cell, the factors governing the competition among these cations for the selectivity filter of voltage-gated Ca(2+) ion channel remain unclear. Using density functional theory combined with continuum dielectric methods, we evaluate the effect of (1) the number, chemical type, and charge of the ligands lining the pore, (2) the pore's rigidity, size, symmetry, and solvent accessibility, and (3) the Ca(2+) hydration number outside the selectivity filter on the competition among Ca(2+), Mg(2+), and Na(+) in model selectivity filters. The calculations show how the outcome of this competition depends on the interplay between electronic and solvation effects. Selectivity for monovalent Na(+) over divalent Ca(2+)/Mg(2+) is achieved when solvation effects outweigh electrostatic effects; thus filters comprising a few weak charge-donating groups such as Ser/Thr side chains, where electrostatic effects are relatively weak and are easily overcome by solvation effects, are Na(+)-selective. In contrast, selectivity for divalent Ca(2+)/Mg(2+) over monovalent Na(+) is achieved when metal-ligand electrostatic effects outweigh solvation effects. The key differences in selectivity between Mg(2+) and Ca(2+) lie in the pore size, oligomericity, and solvent accessibility. The results, which are consistent with available experimental data, reveal how the structure and composition of the ion channel selectivity pore had adapted to the specific physicochemical properties of the native metal ion to enhance the competitiveness of the native metal toward rival cations.

Ionizable side chains at catalytic active sites of enzymes

Catalytic active sites of enzymes of known structure can be well defined by a modern program of computational geometry. The CASTp program was used to define and measure the volume of the catalytic active sites of 573 enzymes in the Catalytic Site Atlas database. The active sites are identified as catalytic because the amino acids they contain are known to participate in the chemical reaction catalyzed by the enzyme. Acid and base side chains are reliable markers of catalytic active sites. The catalytic active sites have 4 acid and 5 base side chains, in an average volume of 1,072 Å(3). The number density of acid side chains is 8.3 M (in chemical units); the number density of basic side chains is 10.6 M. The catalytic active site of these enzymes is an unusual electrostatic and steric environment in which side chains and reactants are crowded together in a mixture more like an ionic liquid than an ideal infinitely dilute solution. The electrostatics and crowding of reactants and side chains seems likely to be important for catalytic function. In three types of analogous ion channels, simulation of crowded charges accounts for the main properties of selectivity measured in a wide range of solutions and concentrations. It seems wise to use mathematics designed to study interacting complex fluids when making models of the catalytic active sites of enzymes.

Current and selectivity in a model sodium channel under physiological conditions: Dynamic Monte Carlo simulations

A reduced model of a sodium channel is analyzed using Dynamic Monte Carlo simulations. These include the first simulations of ionic current under approximately physiological ionic conditions through a model sodium channel and an analysis of how mutations of the sodium channel's DEKA selectivity filter motif transform the channel from being Na(+) selective to being Ca(2+) selective. Even though the model of the pore, amino acids, and permeant ions is simplified, the model reproduces the fundamental properties of a sodium channel (e.g., 10 to 1 Na(+) over K(+) selectivity, Ca(2+) exclusion, and Ca(2+) selectivity after several point mutations). In this model pore, ions move through the pore one at a time by simple diffusion and Na(+) versus K(+) selectivity is due to both the larger K(+) not fitting well into the selectivity filter that contains amino acid terminal groups and K(+) moving more slowly (compared to Na(+)) when it is in the selectivity filter.

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

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

A method for treating the passage of a charged hard sphere ion as it passes through a sharp dielectric boundary

In the implicit solvent models of electrolytes (such as the primitive model (PM)), the ions are modeled as point charges in the centers of spheres (hard spheres in the case of the PM). The surfaces of the spheres are not polarizable which makes these models appropriate to use in computer simulations of electrolyte systems where these ions do not leave their host dielectrics. The same assumption makes them inappropriate in simulations where these ions cross dielectric boundaries because the interaction energy of the point charge with the polarization charge induced on the dielectric boundary diverges. In this paper, we propose a procedure to treat the passage of such ions through dielectric interfaces with an interpolation method. Inspired by the "bubble ion" model (in which the ion's surface is polarizable), we define a space-dependent effective dielectric coefficient, ε(eff)(r), for the ion that overlaps with the dielectric boundary. Then, we replace the "bubble ion" with a point charge that has an effective charge q/ε(eff)(r) and remove the portion of the dielectric boundary where the ion overlaps with it. We implement the interpolation procedure using the induced charge computation method [D. Boda, D. Gillespie, W. Nonner, D. Henderson, and B. Eisenberg, Phys. Rev. E 69, 046702 (2004)]. We analyze the various energy terms using a spherical ion passing through an infinite flat dielectric boundary as an example.

Analyzing the components of the free-energy landscape in a calcium selective ion channel by Widom's particle insertion method

The selectivity filter of the L-type calcium channel works as a Ca(2+) binding site with a very large affinity for Ca(2+) versus Na(+). Ca(2+) replaces half of the Na(+) ions in the filter even when these ions are present in 1 μM and 30 mM concentrations in the bath, respectively. The energetics of this strong selectivity is analyzed in this paper. We use Widom's particle insertion method to compute the space-dependent profiles of excess chemical potential in our grand canonical Monte Carlo simulations. These profiles define the free-energy landscape for the various ions. Following Gillespie [Biophys. J. 94, 1169 (2008)], the difference of the excess chemical potentials for the two competing ions defines the advantage that one of the ions has over the other in the competition for space in the crowded selectivity filter. These advantages depend on ionic bath concentrations: the ion that is present in the bath in larger quantity (Na(+)) has the "number" advantage which is balanced by the free-energy advantage of the other ion (Ca(2+)). The excess chemical potentials are decomposed into hard sphere exclusion and electrostatic components. The electrostatic terms correspond to interactions with the mean electric field produced by ions and induced charges as well to ionic correlations beyond the mean field description. Dielectrics are needed to produce micromolar Ca(2+) versus Na(+) selectivity in the L-type channel. We study the behavior of these terms with changes in bath concentrations of ions, charges, and diameters of ions, as well as geometrical parameters such as radius of the pore and the dielectric constant of the protein. Ion selectivity in calcium binding proteins probably has a similar mechanism.

Selectivity sequences in a model calcium channel: role of electrostatic field strength

The energetics that give rise to selectivity sequences of ionic binding selectivity of Li(+), Na(+), K(+), Rb(+), and Cs(+) in a model of a calcium channel are considered. This work generalizes Eisenman's classic treatment (Biophys J 2(Suppl. 2):259, 1962) by including multiple, mobile binding site oxygens that coordinate many permeating ions (all modeled as charged, hard spheres). The selectivity filter of the model calcium channel allows the carboxyl terminal groups of glutamate and aspartate side chains to directly interact with and coordinate the permeating ions. Ion dehydration effects are represented with a Born energy between the dielectric coefficients of the selectivity filter and the bath. High oxygen concentration creates a high field strength site that prefers small ions, as in Eisenman's model. On the other hand, a low filter dielectric constant also creates a high field strength site, but this site prefers large ions, contrary to Eisenman's model. These results indicate that field strength does not have a unique effect on ionic binding selectivity sequences once entropic, electrostatic, and dehydration forces are included in the model. Thus, Eisenman's classic relationship between field strength and selectivity sequences must be supplemented with additional information about selectivity filters such as the calcium channel that has amino acid side chains mixing with ions to make a crowded permeation pathway.