Anomalous mole fraction effect, electrostatics, and binding in ionic channels

Physics of Selective Conduction and Point Mutation in Biological Ion Channels

We introduce a statistical and linear response theory of selective conduction in biological ion channels with multiple binding sites and possible point mutation. We derive an effective grand-canonical ensemble and generalized Einstein relations for the selectivity filter, assuming strongly coordinated ionic motion, and allowing for ionic Coulomb blockade. The theory agrees well with data from the KcsA K^{+} channel and a mutant. We show that the Eisenman relations for thermodynamic selectivity follow from the condition for fast conduction and find that maximum conduction requires the binding sites to be nearly identical.

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.

Transport mechanisms of SARS-CoV-E viroporin in calcium solutions: Lipid-dependent Anomalous Mole Fraction Effect and regulation of pore conductance

The envelope protein E of the SARS-CoV coronavirus is an archetype of viroporin. It is a small hydrophobic protein displaying ion channel activity that has proven highly relevant in virus-host interaction and virulence. Ion transport through E channel was shown to alter Ca²⁺ homeostasis in the cell and trigger inflammation processes. Here, we study transport properties of the E viroporin in mixed solutions of potassium and calcium chloride that contain a fixed total concentration (mole fraction experiments). The channel is reconstituted in planar membranes of different lipid compositions, including a lipid mixture that mimics the endoplasmic reticulum-Golgi intermediate compartment (ERGIC) membrane where the virus localizes within the cell. We find that the E ion conductance changes non-monotonically with the total ionic concentration displaying an Anomalous Mole Fraction Effect (AMFE) only when charged lipids are present in the membrane. We also observe that E channel insertion in ERGIC-mimic membranes – including lipid with intrinsic negative curvature – enhances ion permeation at physiological concentrations of pure CaCl₂ or KCl solutions, with a preferential transport of Ca²⁺ in mixed KCl-CaCl₂ solutions. Altogether, our findings demonstrate that the presence of calcium modulates the transport properties of the E channel by interacting preferentially with charged lipids through different mechanisms including direct Coulombic interactions and possibly inducing changes in membrane morphology.

Investigating ion transport inside the pentameric ion channel encoded in COVID-19 E protein

Ion flow inside an ion channel can be described through continuum based Born-Poisson-Nernst-Planck (BPNP) equations in conjunction with the Lennard-Jones potential. Keeping in mind the ongoing pandemic, in this study, an attempt has been made to understand the selectivity and the current voltage relation of the COVID-19 E protein pentameric ion channel. Two ionic species, namely Na^{+} and Cl^{-}, have been considered here. E protein is one of the smallest structural protein which is embedded in the outer membrane of the virus. Once the virus is inside the host cell, this protein is expressed abundantly and is responsible for activities such as replication and budding of the virus. In the literature, we can find a few experimental studies focusing on understanding the activity of the channel formed by E proteins of different viruses. Here, we attempt the same study for the COVID-19 E protein ion channel through mathematical modeling. The channel geometry is calculated from the protein data bank file which was provided by NARLabs, Taiwan, using the hole program. Further, it was used to obtain the charge distribution using the pdbtopqr online program. The immersed boundary-lattice Boltzmann method (IB-LBM) has been implemented to numerically solve the system of equations in the channel generated by the protein data bank file. Further, an in-house code which operates on multiple GPUs and uses the cuda platform has been developed to achieve the goal of performing the current investigation.

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.

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.

A thermodynamic description for physiological transmembrane transport

A general formulation for both passive and active transmembrane transport is derived from basic thermodynamical principles. The derivation takes into account the energy required for the motion of molecules across membranes and includes the possibility of modeling asymmetric flow. Transmembrane currents can then be described by the general model in the case of electrogenic flow. As it is desirable in new models, it is possible to derive other well-known expressions for transmembrane currents as particular cases of the general formulation. For instance, the conductance-based formulation for current turns out to be a linear approximation of the general formula for current. Also, under suitable assumptions, other formulas for current based on electrodiffusion, like the constant field approximation by Goldman, can be recovered from the general formulation. The applicability of the general formulations is illustrated first with fits to existing data, and after, with models of transmembrane potential dynamics for pacemaking cardiocytes and neurons. The general formulations presented here provide a common ground for the biophysical study of physiological phenomena that depend on transmembrane transport.

Selective ion permeation involves complexation with carboxylates and lysine in a model human sodium channel

Bacterial and human voltage-gated sodium channels (Navs) exhibit similar cation selectivity, despite their distinct EEEE and DEKA selectivity filter signature sequences. Recent high-resolution structures for bacterial Navs have allowed us to learn about ion conduction mechanisms in these simpler homo-tetrameric channels, but our understanding of the function of their mammalian counterparts remains limited. To probe these conduction mechanisms, a model of the human Nav1.2 channel has been constructed by grafting residues of its selectivity filter and external vestibular region onto the bacterial NavRh channel with atomic-resolution structure. Multi-μs fully atomistic simulations capture long time-scale ion and protein movements associated with the permeation of Na+ and K+ ions, and their differences. We observe a Na+ ion knock-on conduction mechanism facilitated by low energy multi-carboxylate/multi-Na+ complexes, akin to the bacterial channels. These complexes involve both the DEKA and vestibular EEDD rings, acting to draw multiple Na+ into the selectivity filter and promote permeation. When the DEKA ring lysine is protonated, we observe that its ammonium group is actively participating in Na+ permeation, presuming the role of another ion. It participates in the formation of a stable complex involving carboxylates that collectively bind both Na+ and the Lys ammonium group in a high-field strength site, permitting pass-by translocation of Na+. In contrast, multiple K+ ion complexes with the DEKA and EEDD rings are disfavored by up to 8.3 kcal/mol, with the K+-lysine-carboxylate complex non-existent. As a result, lysine acts as an electrostatic plug that partially blocks the flow of K+ ions, which must instead wait for isomerization of lysine downward to clear the path for K+ passage. These distinct mechanisms give us insight into the nature of ion conduction and selectivity in human Nav channels, while uncovering high field strength carboxylate binding complexes that define the more general phenomenon of Na+-selective ion transport in nature.

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

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

Comparison of permeation mechanisms in sodium-selective ion channels

Voltage-gated sodium channels are the molecular components of electrical signaling in the body, yet the molecular origins of Na⁺-selective transport remain obscured by diverse protein chemistries within this family of ion channels. In particular, bacterial and mammalian sodium channels are known to exhibit similar relative ion permeabilities for Na⁺ over K⁺ ions, despite their distinct signature EEEE and DEKA sequences. Atomic-level molecular dynamics simulations using high-resolution bacterial channel structures and mammalian channel models have begun to describe how these sequences lead to analogous high field strength ion binding sites that drive Na⁺ conduction. Similar complexes have also been identified in unrelated acid sensing ion channels involving glutamate and aspartate side chains that control their selectivity. These studies suggest the possibility of a common origin for Na⁺ selective binding and transport.

Relative dielectric constants and selectivity ratios in open ionic channels

We investigate the effects of the relative dielectric coefficient on ionic flows in open ion channels, using mathematical analysis of reasonably general Poisson-Nernst-Planck type models that can include the finite sizes of ions. The value of the relative dielectric coefficient is of course a crucial parameter for ionic behavior in general. Using the powerful theory of singularly perturbed problems in applied mathematics, we show that some properties of open channels are quite insensitive to variation in the relative dielectric coefficient, thereby explaining such effects seen unexpectedly in simulations. The ratio between the total number of one ion species and that of another ion species, and the ratio between the flux of one ion species and that of another ion species do not depend significantly on the relative dielectric coefficient.

The enduring legacy of the "constant-field equation" in membrane ion transport

In 1943, David Goldman published a seminal paper in The Journal of General Physiology that reported a concise expression for the membrane current as a function of ion concentrations and voltage. This body of work was, and still is, the theoretical pillar used to interpret the relationship between a cell's membrane potential and its external and/or internal ionic composition. Here, we describe from an historical perspective the theory underlying the constant-field equation and its application to membrane ion transport.

Multiscale modeling of a rectifying bipolar nanopore: Comparing Poisson-Nernst-Planck to Monte Carlo

In the framework of a multiscale modeling approach, we present a systematic study of a bipolar rectifying nanopore using a continuum and a particle simulation method. The common ground in the two methods is the application of the Nernst-Planck (NP) equation to compute ion transport in the framework of the implicit-water electrolytemodel. The difference is that the Poisson-Boltzmann theory is used in the Poisson-Nernst-Planck (PNP) approach, while the Local Equilibrium Monte Carlo (LEMC) method is used in the particle simulation approach (NP+LEMC) to relate the concentration profile to the electrochemical potential profile. Since we consider a bipolar pore which is short and narrow, we perform simulations using two-dimensional PNP. In addition, results of a non-linear version of PNP that takes crowding of ions into account are shown. We observe that the mean field approximation applied in PNP is appropriate to reproduce the basic behavior of the bipolar nanopore (e.g., rectification) for varying parameters of the system (voltage, surface charge,electrolyte concentration, and pore radius). We present current data that characterize the nanopore's behavior as a device, as well as concentration, electrical potential, and electrochemical potential profiles.

Permeation Mechanisms in the TMEM16B Calcium-Activated Chloride Channels

TMEM16A and TMEM16B encode for Ca²⁺-activated Cl⁻ channels (CaCC) and are expressed in many cell types and play a relevant role in many physiological processes. Here, I performed a site-directed mutagenesis study to understand the molecular mechanisms of ion permeation of TMEM16B. I mutated two positive charged residues R573 and K540, respectively located at the entrance and inside the putative channel pore and I measured the properties of wild-type and mutant TMEM16B channels expressed in HEK-293 cells using whole-cell and excised inside-out patch clamp experiments. I found evidence that R573 and K540 control the ion permeability of TMEM16B depending both on which side of the membrane the ion substitution occurs and on the level of channel activation. Moreover, these residues contribute to control blockage or activation by permeant anions. Finally, R573 mutation abolishes the anomalous mole fraction effect observed in the presence of a permeable anion and it alters the apparent Ca²⁺-sensitivity of the channel. These findings indicate that residues facing the putative channel pore are responsible both for controlling the ion selectivity and the gating of the channel, providing an initial understanding of molecular mechanism of ion permeation in TMEM16B.

Cation Selectivity in Biological Cation Channels Using Experimental Structural Information and Statistical Mechanical Simulation

Cation selective channels constitute the gate for ion currents through the cell membrane. Here we present an improved statistical mechanical model based on atomistic structural information, cation hydration state and without tuned parameters that reproduces the selectivity of biological Na+ and Ca2+ ion channels. The importance of the inclusion of step-wise cation hydration in these results confirms the essential role partial dehydration plays in the bacterial Na+ channels. The model, proven reliable against experimental data, could be straightforwardly used for designing Na+ and Ca2+ selective nanopores.

Voltage-gated calcium channels: Determinants of channel function and modulation by inorganic cations

Voltage-gated calcium channels (VGCCs) represent a key link between electrical signals and non-electrical processes, such as contraction, secretion and transcription. Evolved to achieve high rates of Ca(2+)-selective flux, they possess an elaborate mechanism for selection of Ca(2+) over foreign ions. It has been convincingly linked to competitive binding in the pore, but the fundamental question of how this is reconcilable with high rates of Ca(2+) transfer remains unanswered. By virtue of their similarity to Ca(2+), polyvalent cations can interfere with the function of VGCCs and have proven instrumental in probing the mechanisms underlying selective permeation. Recent emergence of crystallographic data on a set of Ca(2+)-selective model channels provides a structural framework for permeation in VGCCs, and warrants a reconsideration of their diverse modulation by polyvalent cations, which can be roughly separated into three general mechanisms: (I) long-range interactions with charged regions on the surface, affecting the local potential sensed by the channel or influencing voltage-sensor movement by repulsive forces (electrostatic effects), (II) short-range interactions with sites in the ion-conducting pathway, leading to physical obstruction of the channel (pore block), and in some cases (III) short-range interactions with extracellular binding sites, leading to non-electrostatic modifications of channel gating (allosteric effects). These effects, together with the underlying molecular modifications, provide valuable insights into the function of VGCCs, and have important physiological and pathophysiological implications. Allosteric suppression of some of the pore-forming Cavα1-subunits (Cav2.3, Cav3.2) by Zn(2+) and Cu(2+) may play a major role for the regulation of excitability by endogenous transition metal ions. The fact that these ions can often traverse VGCCs can contribute to the detrimental intracellular accumulation of metal ions following excessive release of endogenous Cu(2+) and Zn(2+) or exposure to non-physiological toxic metal ions.

Sodium channel selectivity and conduction: prokaryotes have devised their own molecular strategy

The molecular strategy for alkali cation selectivity by a bacterial sodium channel resembles those of eukaryotic calcium and potassium channels, rather than those of eukaryotic sodium channels.

Striking structural differences between voltage-gated sodium (Nav) channels from prokaryotes (homotetramers) and eukaryotes (asymmetric, four-domain proteins) suggest the likelihood of different molecular mechanisms for common functions. For these two channel families, our data show similar selectivity sequences among alkali cations (relative permeability, Pion/PNa) and asymmetric, bi-ionic reversal potentials when the Na/K gradient is reversed. We performed coordinated experimental and computational studies, respectively, on the prokaryotic Nav channels NaChBac and NavAb. NaChBac shows an “anomalous,” nonmonotonic mole-fraction dependence in the presence of certain sodium–potassium mixtures; to our knowledge, no comparable observation has been reported for eukaryotic Nav channels. NaChBac’s preferential selectivity for sodium is reduced either by partial titration of its highly charged selectivity filter, when extracellular pH is lowered from 7.4 to 5.8, or by perturbation—likely steric—associated with a nominally electro-neutral substitution in the selectivity filter (E191D). Although no single molecular feature or energetic parameter appears to dominate, our atomistic simulations, based on the published NavAb crystal structure, revealed factors that may contribute to the normally observed selectivity for Na over K. These include: (a) a thermodynamic penalty to exchange one K⁺ for one Na⁺ in the wild-type (WT) channel, increasing the relative likelihood of Na⁺ occupying the binding site; (b) a small tendency toward weaker ion binding to the selectivity filter in Na–K mixtures, consistent with the higher conductance observed with both sodium and potassium present; and (c) integrated 1-D potentials of mean force for sodium or potassium movement that show less separation for the less selective E/D mutant than for WT. Overall, tight binding of a single favored ion to the selectivity filter, together with crucial inter-ion interactions within the pore, suggests that prokaryotic Nav channels use a selective strategy more akin to those of eukaryotic calcium and potassium channels than that of eukaryotic Nav channels.

Paradoxical one-ion pore behavior of the long β-helical peptide of marine cytotoxic polytheonamide B

The cytotoxic 48-mer peptide, polytheonamide B (pTB), from a marine sponge forms a β^6.3-helix with an inner diameter of 4 Å and a length of 45 Å, features that allow the selective permeation of monovalent cations across targeted cell membranes. To characterize this long, narrow pore, electrophysiological examination using a planar lipid bilayer method was performed. The single-channel current amplitude exhibited saturation for concentrated Cs⁺ or K⁺ solution, and the reversal potential in mixed solutions did not exhibit any anomalous mole-fraction behavior. These results suggest the one-ion permeation mechanism. This is in contrast to the short (26 Å) β^6.3-helical gramicidin channel, which holds two ions simultaneously. The paradoxical one-ion permeation through the long pTB channel was modeled with a discrete-state Markov model. Ions permeate through the channel by stepping between two binding sites in the pore, but never occupy these sites simultaneously in either pure or mixed ion solution.

Ion permeation in the NanC porin from Escherichia coli: free energy calculations along pathways identified by coarse-grain simulations

Using the X-ray structure of a recently discovered bacterial protein, the N-acetylneuraminic acid-inducible channel (NanC), we investigate computationally K(+) and Cl(-) ions' permeation. We identify ion permeation pathways that are likely to be populated using coarse-grain Monte Carlo simulations. Next, we use these pathways as reaction coordinates for umbrella sampling-based free energy simulations. We find distinct tubelike pathways connecting specific binding sites for K(+) and, more pronounced, for Cl(-) ions. Both ions permeate the porin preserving almost all of their first hydration shell. The calculated free energy barriers are G(#) ≈ 4 kJ/mol and G(#) ≈ 8 kJ/mol for Cl(-) and K(+), respectively. Within the approximations associated with these values, discussed in detail in this work, we suggest that the porin is slightly selective for Cl(-) versus K(+). Our suggestion is consistent with the experimentally observed weak Cl(-) over K(+) selectivity. A rationale for the latter is suggested by a comparison with previous calculations on strongly anion selective porins.

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.

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.

Electrodiffusion model simulation of the potassium channel

The drift-diffusion (Poisson-Nernst-Planck) model is applied to the potassium channel in a biological membrane plus surrounding solution baths. Two-dimensional cylindrically symmetric simulations of the K channel in KCl solutions are presented which show significant boundary layers at the ends of the channel and display the spreading of charge into the bath regions. The computed current-voltage curve shows excellent agreement with experimental measurements. In addition, the response of the K channel to time-dependent applied voltages is investigated.

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.

Insights on the permeability of wide protein channels: measurement and interpretation of ion selectivity

Ion channels are hollow proteins that have evolved to exhibit discrimination between charged solutes. This property, known as ion selectivity is critical for several biological functions. By using the bacterial porin OmpF as a model system of wide protein channels, we demonstrate that significant insights can be gained when selectivity measurements are combined with electrodiffusion continuum models and simulations based on the atomic structure. A correct interpretation of the mechanisms ruling the many sources of channel discrimination is a first, indispensable step for the understanding of the controlled movement of ions into or out of cells characteristic of many physiological processes. We conclude that the scattered information gathered from several independent approaches should be appropriately merged to provide a unified and coherent picture of the channel selectivity.

Multiple Scales in the Simulation of Ion Channels and Proteins

Computation of living processes creates great promise for the everyday life of mankind and great challenges for physical scientists. Simulations molecular dynamics have great appeal to biologists as a natural extension of structural biology. Once a biologist sees a structure, she/he wants to see it move. Molecular biology has shown that a small number of atoms, sometimes even one messenger ion, like Ca(2+), can control biological function on the scale of cells, organs, tissues, and organisms. Enormously concentrated ions-at number densities of ~20 M-in protein channels and enzymes are responsible for many of the characteristics of living systems, just as highly concentrated ions near electrodes are responsible for many of the characteristics of electrochemical systems. Here we confront the reality of the scale differences of ions. We show that the scale differences needed to simulate all the atoms of biological cells are 10(7) in linear dimension, 10(21) in three dimensions, 10(9) in resolution, 10(11) in time, and 10(13) in particle number (to deal with concentrations of Ca(2+)). These scales must be dealt with simultaneously if the simulation is to deal with most biological functions. Biological function extends across all of them, all at once in most cases. We suggest a computational approach using explicit multiscale analysis instead of implicit simulation of all scales. The approach is based on an energy variational principle EnVarA introduced by Chun Liu to deal with complex fluids. Variational methods deal automatically with multiple interacting components and scales. When an additional component is added to the system, the resulting Euler Lagrange field equations change form automatically-by algebra alone-without additional unknown parameters. Multifaceted interactions are solutions of the resulting equations. We suggest that ionic solutions should be viewed as complex fluids with simple components. Highly concentrated solutions-dominated by interactions of components-are easily computed by EnVarA. Successful computation of ions concentrated in special places may be a significant step to understanding the defining characteristics of biological and electrochemical systems. Indeed, computing ions near proteins and nucleic acids may prove as important to molecular biology and chemical technology as computing holes and electrons has been to our semiconductor and digital technology.

Energy variational analysis of ions in water and channels: Field theory for primitive models of complex ionic fluids

Ionic solutions are mixtures of interacting anions and cations. They hardly resemble dilute gases of uncharged noninteracting point particles described in elementary textbooks. Biological and electrochemical solutions have many components that interact strongly as they flow in concentrated environments near electrodes, ion channels, or active sites of enzymes. Interactions in concentrated environments help determine the characteristic properties of electrodes, enzymes, and ion channels. Flows are driven by a combination of electrical and chemical potentials that depend on the charges, concentrations, and sizes of all ions, not just the same type of ion. We use a variational method EnVarA (energy variational analysis) that combines Hamilton's least action and Rayleigh's dissipation principles to create a variational field theory that includes flow, friction, and complex structure with physical boundary conditions. EnVarA optimizes both the action integral functional of classical mechanics and the dissipation functional. These functionals can include entropy and dissipation as well as potential energy. The stationary point of the action is determined with respect to the trajectory of particles. The stationary point of the dissipation is determined with respect to rate functions (such as velocity). Both variations are written in one Eulerian (laboratory) framework. In variational analysis, an "extra layer" of mathematics is used to derive partial differential equations. Energies and dissipations of different components are combined in EnVarA and Euler-Lagrange equations are then derived. These partial differential equations are the unique consequence of the contributions of individual components. The form and parameters of the partial differential equations are determined by algebra without additional physical content or assumptions. The partial differential equations of mixtures automatically combine physical properties of individual (unmixed) components. If a new component is added to the energy or dissipation, the Euler-Lagrange equations change form and interaction terms appear without additional adjustable parameters. EnVarA has previously been used to compute properties of liquid crystals, polymer fluids, and electrorheological fluids containing solid balls and charged oil droplets that fission and fuse. Here we apply EnVarA to the primitive model of electrolytes in which ions are spheres in a frictional dielectric. The resulting Euler-Lagrange equations include electrostatics and diffusion and friction. They are a time dependent generalization of the Poisson-Nernst-Planck equations of semiconductors, electrochemistry, and molecular biophysics. They include the finite diameter of ions. The EnVarA treatment is applied to ions next to a charged wall, where layering is observed. Applied to an ion channel, EnVarA calculates a quick transient pile-up of electric charge, transient and steady flow through the channel, stationary "binding" in the channel, and the eventual accumulation of salts in "unstirred layers" near channels. EnVarA treats electrolytes in a unified way as complex rather than simple fluids. Ad hoc descriptions of interactions and flow have been used in many areas of science to deal with the nonideal properties of electrolytes. It seems likely that the variational treatment can simplify, unify, and perhaps derive and improve those descriptions.

Simulations of calcium channel block by trivalent cations: Gd(3+) competes with permeant ions for the selectivity filter

Current through L-type calcium channels (Ca(V)1.2 or dihydropyridine receptor) can be blocked by micromolar concentrations of trivalent cations like the lanthanide gadolinium (Gd(3+)). It has been proposed that trivalent block is due to ions competing for a binding site in both the open and closed configuration, but possibly with different trivalent affinities. Here, we corroborate this general view of trivalent block by computing conductance of a model L-type calcium channel. The model qualitatively reproduces the Gd(3+) concentration dependence and the effect that substantially more Gd(3+) is required to produce similar block in the presence of Sr(2+) (compared to Ba(2+)) and even more in the presence of Ca(2+). Trivalent block is explained in this model by cations binding in the selectivity filter with the charge/space competition mechanism. This is the same mechanism that in the model channel governs other selectivity properties. Specifically, selectivity is determined by the combination of ions that most effectively screen the negative glutamates of the protein while finding space in the midst of the closely packed carboxylate groups of the glutamate residues.

Reinterpreting the anomalous mole fraction effect: the ryanodine receptor case study

The origin of the anomalous mole fraction effect (AMFE) in calcium channels is explored with a model of the ryanodine receptor. This model predicted and experiments verified new AMFEs in the cardiac isoform. In mole fraction experiments, conductance is measured in mixtures of ion species X and Y as their relative amounts (mole fractions) vary. This curve can have a minimum (an AMFE). The traditional interpretation of the AMFE is that multiple interacting ions move through the pore in a single file. Mole fraction curves without minima (no AMFEs) are generally interpreted as X displacing Y from the pore in a proportion larger than its bath mole fraction (preferential selectivity). We find that the AMFE is also caused by preferential selectivity of X over Y, if X and Y have similar conductances. This is a prediction applicable to any channel and provides a fundamentally different explanation of the AMFE that does not require single filing or multiple occupancy: preferential selectivity causes the resistances to current flow in the baths, channel vestibules, and selectivity filter to change differently with mole fraction, and produce the AMFE.

Protein structure and ionic selectivity in calcium channels: selectivity filter size, not shape, matters

Calcium channels have highly charged selectivity filters (4 COO(-) groups) that attract cations in to balance this charge and minimize free energy, forcing the cations (Na(+) and Ca(2+)) to compete for space in the filter. A reduced model was developed to better understand the mechanism of ion selectivity in calcium channels. The charge/space competition (CSC) mechanism implies that Ca(2+) is more efficient in balancing the charge of the filter because it provides twice the charge as Na(+) while occupying the same space. The CSC mechanism further implies that the main determinant of Ca(2+) versus Na(+) selectivity is the density of charged particles in the selectivity filter, i.e., the volume of the filter (after fixing the number of charged groups in the filter). In this paper we test this hypothesis by changing filter length and/or radius (shape) of the cylindrical selectivity filter of our reduced model. We show that varying volume and shape together has substantially stronger effects than varying shape alone with volume fixed. Our simulations show the importance of depletion zones of ions in determining channel conductance calculated with the integrated Nernst-Planck equation. We show that confining the protein side chains with soft or hard walls does not influence selectivity.

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

A physical model of selective "ion binding" in the L-type calcium channel is constructed, and consequences of the model are compared with experimental data. This reduced model treats only ions and the carboxylate oxygens of the EEEE locus explicitly and restricts interactions to hard-core repulsion and ion-ion and ion-dielectric electrostatic forces. The structural atoms provide a flexible environment for passing cations, thus resulting in a self-organized induced-fit model of the selectivity filter. Experimental conditions involving binary mixtures of alkali and/or alkaline earth metal ions are computed using equilibrium Monte Carlo simulations in the grand canonical ensemble. The model pore rejects alkali metal ions in the presence of biological concentrations of Ca(2+) and predicts the blockade of alkali metal ion currents by micromolar Ca(2+). Conductance patterns observed in varied mixtures containing Na(+) and Li(+), or Ba(2+) and Ca(2+), are predicted. Ca(2+) is substantially more potent in blocking Na(+) current than Ba(2+). In apparent contrast to experiments using buffered Ca(2+) solutions, the predicted potency of Ca(2+) in blocking alkali metal ion currents depends on the species and concentration of the alkali metal ion, as is expected if these ions compete with Ca(2+) for the pore. These experiments depend on the problematic estimation of Ca(2+) activity in solutions buffered for Ca(2+) and pH in a varying background of bulk salt. Simulations of Ca(2+) distribution with the model pore bathed in solutions containing a varied amount of Li(+) reveal a "barrier and well" pattern. The entry/exit barrier for Ca(2+) is strongly modulated by the Li(+) concentration of the bath, suggesting a physical explanation for observed kinetic phenomena. Our simulations show that the selectivity of L-type calcium channels can arise from an interplay of electrostatic and hard-core repulsion forces among ions and a few crucial channel atoms. The reduced system selects for the cation that delivers the largest charge in the smallest ion volume.