Hydrodynamic model of temperature change in open ionic channels

Joule heating involving ion currents through channel proteins

Ion currents associated with channel proteins in the presence of membrane potential are ubiquitous in cellular and organelle membranes. When an ion current occurs through a channel protein, Joule heating should occur. However, this Joule heating seems to have been largely overlooked in biology. Here we show theoretical investigation of Joule heating involving channel proteins in biological processes. We used electrochemical potential to derive the Joule's law for an ion current through an ion transport protein in the presence of membrane potential, and we suggest that heat production and absorption can occur. Simulation of temperature distribution around a single channel protein with the Joule heating revealed that the temperature increase was as small as <10-3 K, although an ensemble of channel proteins was suggested to exhibit a noticeable temperature increase. Thereby, we theoretically investigated the Joule heating of systems containing ensembles of channel proteins. Nerve is known to undergo rapid heat production followed by heat absorption during the action potential, and our simulation of Joule heating for a squid giant axon combined with the Hodgkin-Huxley model successfully reproduced the feature of the heat. Furthermore, we extended the theory of Joule heating to uncoupling protein 1 (UCP1), a solute carrier family transporter, which is important to the non-shivering thermogenesis in brown adipose tissue mitochondria (BATM). Our calculations showed that the Joule heat involving UCP1 was comparable to the literature calorimetry data of BATM. Joule heating of ion transport proteins is likely to be one of important mechanisms of cellular thermogenesis.

Emerging Diamond Quantum Sensing in Bio-Membranes

Bio-membranes exhibit complex but unique mechanical properties as communicative regulators in various physiological and pathological processes. Exposed to a dynamic micro-environment, bio-membranes can be seen as an intricate and delicate system. The systematical modeling and detection of their local physical properties are often difficult to achieve, both quantitatively and precisely. The recent emerging diamonds hosting quantum defects (i.e., nitrogen-vacancy (NV) center) demonstrate intriguing optical and spin properties, together with their outstanding photostability and biocompatibility, rendering them ideal candidates for biological applications. Notably, the extraordinary spin-based sensing enable the measurements of localized nanoscale physical quantities such as magnetic fields, electrical fields, temperature, and strain. These nanoscale signals can be optically read out precisely by simple optical microscopy systems. Given these exclusive properties, NV-center-based quantum sensors can be widely applied in exploring bio-membrane-related features and the communicative chemical reaction processes. This review mainly focuses on NV-based quantum sensing in bio-membrane fields. The attempts of applying NV-based quantum sensors in bio-membranes to investigate diverse physical and chemical events such as membrane elasticity, phase change, nanoscale bio-physical signals, and free radical formation are fully overviewed. We also discuss the challenges and future directions of this novel technology to be utilized in bio-membranes.

Towards understanding specific ion effects in aqueous media using thermodiffusion

Specific ion effects play an important role in scientific and technological processes. According to Hofmeister, the influence on the hydrogen bond network depends on the ion and leads to a specific order of the ions. Also thermodiffusion the mass transport caused by a temperature gradient is very sensitive to changes of the hydrogen bond network leading to a ranking according to hydrophilicity of the salt. Hence, we investigate various salt solutions in order to compare with the Hofmeister concept. We have studied three different sodium salts in water as a function of temperature (25-45[Formula: see text]C) and concentration (0.5-5 mol kg[Formula: see text]) using Thermal Diffusion Forced Rayleigh Scattering (TDFRS). The three anions studied, carbonate, acetate and thiocyanate, span the entire range of the Hofmeister series from hydrophilic to hydrophobic. We compare the results with the recent measurements of the corresponding potassium salts to see to what extent the cation changes the thermodiffusion of the salt.

A new approach to precise mapping of local temperature fields in submicrometer aqueous volumes

Nanodiamonds hosting temperature-sensing centers constitute a closed thermodynamic system. Such a system prevents direct contact of the temperature sensors with the environment making it an ideal environmental insensitive nanosized thermometer. A new design of a nanodiamond thermometer, based on a 500-nm luminescent nanodiamond embedded into the inner channel of a glass submicron pipette is reported. All-optical detection of temperature, based on spectral changes of the emission of "silicon-vacancy" centers with temperature, is used. We demonstrate the applicability of the thermometric tool to the study of temperature distribution near a local heater, placed in an aqueous medium. The calculated and experimental values of temperatures are shown to coincide within measurement error at gradients up to 20 °C/μm. Until now, temperature measurements on the submicron scale at such high gradients have not been performed. The new thermometric tool opens up unique opportunities to answer the urgent paradigm-shifting questions of cell physiology thermodynamics.

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.

Nonequilibrium dynamic structure factor of a lipid bilayer in the presence of an in-plane temperature gradient

There is rapidly increasing evidence that nanoscale temperature heterogeneities are involved in important biological processes. Combining nanoheating and nanoscale thermosensors forms the basis of emerging unique methods of cell therapy, tissue engineering, and regenerative medicine. Understanding corresponding phenomena seems to require a mesoscopic nonequilibrium hydrodynamic theory. In this paper, a Langevin-type model of dynamics of phonon modes propagating along a bilayer lipid membrane in the presence of an in-plane temperature gradient is proposed. Corresponding quantitative estimates for the Brillouin components of the nonequilibrium dynamic structure factor and the equal-time longitudinal momentum-density correlation function for a lipid bilayer are obtained. The analysis reveals that for typical values of parameters of lipid bilayer, the longitudinal temperature gradient of the order of 5qK for wave numbers q from 0.01 to 1nm^{-1} induces significant asymmetry of the Brillouin components of the dynamic structure factor and long-range spatial correlations in the plane of the bilayer. The corresponding membrane temperature gradients seem to be typical or achievable for cellular processes responsible for intracellular temperature variations and such external physical impacts as high-intensity electromagnetic pulses or heating of membrane-associated nanoparticles.

Non-Neurotoxic Nanodiamond Probes for Intraneuronal Temperature Mapping

Optical biomarkers have been used extensively for intracellular imaging with high spatial and temporal resolution. Extending the modality of these probes is a key driver in cell biology. In recent years, the nitrogen-vacancy (NV) center in nanodiamond has emerged as a promising candidate for bioimaging and biosensing with low cytotoxicity and stable photoluminescence. Here we study the electrophysiological effects of this quantum probe in primary cortical neurons. Multielectrode array recordings across five replicate studies showed no statistically significant difference in 25 network parameters when nanodiamonds are added at varying concentrations over various time periods, 12-36 h. The physiological validation motivates the second part of the study, which demonstrates how the quantum properties of these biomarkers can be used to report intracellular information beyond their location and movement. Using the optically detected magnetic resonance from the nitrogen-vacancy defects within the nanodiamonds we demonstrate enhanced signal-to-noise imaging and temperature mapping from thousands of nanodiamond probes simultaneously. This work establishes nanodiamonds as viable multifunctional intraneuronal sensors with nanoscale resolution, which may ultimately be used to detect magnetic and electrical activity at the membrane level in excitable cellular systems.

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

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

Multiscale Multiphysics and Multidomain Models I: Basic Theory

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

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.

Variational multiscale models for charge transport

This work presents a few variational multiscale models for charge transport in complex physical, chemical and biological systems and engineering devices, such as fuel cells, solar cells, battery cells, nanofluidics, transistors and ion channels. An essential ingredient of the present models, introduced in an earlier paper (Bulletin of Mathematical Biology, 72, 1562-1622, 2010), is the use of differential geometry theory of surfaces as a natural means to geometrically separate the macroscopic domain from the microscopic domain, meanwhile, dynamically couple discrete and continuum descriptions. Our main strategy is to construct the total energy functional of a charge transport system to encompass the polar and nonpolar free energies of solvation, and chemical potential related energy. By using the Euler-Lagrange variation, coupled Laplace-Beltrami and Poisson-Nernst-Planck (LB-PNP) equations are derived. The solution of the LB-PNP equations leads to the minimization of the total free energy, and explicit profiles of electrostatic potential and densities of charge species. To further reduce the computational complexity, the Boltzmann distribution obtained from the Poisson-Boltzmann (PB) equation is utilized to represent the densities of certain charge species so as to avoid the computationally expensive solution of some Nernst-Planck (NP) equations. Consequently, the coupled Laplace-Beltrami and Poisson-Boltzmann-Nernst-Planck (LB-PBNP) equations are proposed for charge transport in heterogeneous systems. A major emphasis of the present formulation is the consistency between equilibrium LB-PB theory and non-equilibrium LB-PNP theory at equilibrium. Another major emphasis is the capability of the reduced LB-PBNP model to fully recover the prediction of the LB-PNP model at non-equilibrium settings. To account for the fluid impact on the charge transport, we derive coupled Laplace-Beltrami, Poisson-Nernst-Planck and Navier-Stokes equations from the variational principle for chemo-electro-fluid systems. A number of computational algorithms is developed to implement the proposed new variational multiscale models in an efficient manner. A set of ten protein molecules and a realistic ion channel, Gramicidin A, are employed to confirm the consistency and verify the capability. Extensive numerical experiment is designed to validate the proposed variational multiscale models. A good quantitative agreement between our model prediction and the experimental measurement of current-voltage curves is observed for the Gramicidin A channel transport. This paper also provides a brief review of the field.

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.

Differential geometry based multiscale models

Large chemical and biological systems such as fuel cells, ion channels, molecular motors, and viruses are of great importance to the scientific community and public health. Typically, these complex systems in conjunction with their aquatic environment pose a fabulous challenge to theoretical description, simulation, and prediction. In this work, we propose a differential geometry based multiscale paradigm to model complex macromolecular systems, and to put macroscopic and microscopic descriptions on an equal footing. In our approach, the differential geometry theory of surfaces and geometric measure theory are employed as a natural means to couple the macroscopic continuum mechanical description of the aquatic environment with the microscopic discrete atomistic description of the macromolecule. Multiscale free energy functionals, or multiscale action functionals are constructed as a unified framework to derive the governing equations for the dynamics of different scales and different descriptions. Two types of aqueous macromolecular complexes, ones that are near equilibrium and others that are far from equilibrium, are considered in our formulations. We show that generalized Navier-Stokes equations for the fluid dynamics, generalized Poisson equations or generalized Poisson-Boltzmann equations for electrostatic interactions, and Newton's equation for the molecular dynamics can be derived by the least action principle. These equations are coupled through the continuum-discrete interface whose dynamics is governed by potential driven geometric flows. Comparison is given to classical descriptions of the fluid and electrostatic interactions without geometric flow based micro-macro interfaces. The detailed balance of forces is emphasized in the present work. We further extend the proposed multiscale paradigm to micro-macro analysis of electrohydrodynamics, electrophoresis, fuel cells, and ion channels. We derive generalized Poisson-Nernst-Planck equations that are coupled to generalized Navier-Stokes equations for fluid dynamics, Newton's equation for molecular dynamics, and potential and surface driving geometric flows for the micro-macro interface. For excessively large aqueous macromolecular complexes in chemistry and biology, we further develop differential geometry based multiscale fluid-electro-elastic models to replace the expensive molecular dynamics description with an alternative elasticity formulation.

Synthetic nanopores as a test case for ion channel theories: the anomalous mole fraction effect without single filing

The predictions of a theory for the anomalous mole fraction effect (AMFE) are tested experimentally with synthetic nanopores in plastic. The negatively charged synthetic nanopores under consideration are highly cation selective and 50 A in diameter at their smallest point. These pores exhibit an AMFE in mixtures of Ca(2+) and monovalent cations. An AMFE occurs when the conductance through a pore is lower in a mixture of salts than in the pure salts at the same concentration. For ion channels, the textbook interpretation of the AMFE is that multiple ions move through the pore in coordinated, single-file motion. However, because the synthetic nanopores are so wide, their AMFE shows that single filing is not necessary for the AMFE. It is shown that the AMFE in the synthetic nanopores is explained by a theory of preferential ion selectivity. The unique properties of the synthetic nanopores allow us to experimentally confirm several predictions of this theory. These same properties make synthetic nanopores an interesting new platform to test theories of ion channel permeation and selectivity in general.

Energetics of divalent selectivity in a calcium channel: the ryanodine receptor case study

A model of the ryanodine receptor (RyR) calcium channel is used to study the energetics of binding selectivity of Ca(2+) versus monovalent cations. RyR is a calcium-selective channel with a DDDD locus in the selectivity filter, similar to the EEEE locus of the L-type calcium channel. While the affinity of RyR for Ca(2+) is in the millimolar range (as opposed to the micromolar range of the L-type channel), the ease of single-channel measurements compared to L-type and its similar selectivity filter make RyR an excellent candidate for studying calcium selectivity. A Poisson-Nernst-Planck/density functional theory model of RyR is used to calculate the energetics of selectivity. Ca(2+) versus monovalent selectivity is driven by the charge/space competition mechanism in which selectivity arises from a balance of electrostatics and the excluded volume of ions in the crowded selectivity filter. While electrostatic terms dominate the selectivity, the much smaller excluded-volume term also plays a substantial role. In the D4899N and D4938N mutations of RyR that are analyzed, substantial changes in specific components of the chemical potential profiles are found far from the mutation site. These changes result in the significant reduction of Ca(2+) selectivity found in both theory and experiments.

Microscopic detection of thermogenesis in a single HeLa cell

We report here the technique for detection and measurement of the temperature changes in single cells using a recently devised microthermometer (a glass micropipette filled with the thermosensitive fluorescent dye Europium (III) thenoyltrifluoroacetonate trihydrate). We found that the heat production in a single HeLa cell occurred with some time delay after the ionomycin-induced Ca(2+) influx from the extracellular space. The time delay inversely depended on extracellular [Ca(2+)], and the increase in temperature was suppressed when Ca(2+)-ATPases were blocked by thapsigargin. These observations strongly suggest that the enzymatic activity of Ca(2+)-ATPases in endoplasmic reticulum leads to the heat production. This study has therefore paved the way for studying the thermogenesis at the single-cell level.

Accelerated diffusion of Na+ in a hydrophobic region revealed by molecular dynamics simulations of a synthetic ion channel

To get insight into the significance of the hydrophobic lining on the ion permeation, we performed molecular dynamics simulations on a Na(+) permeation through a de novo synthetic hydrophobic channel. Electrophysiological study has suggested that the channel is formed from a tail-to-tail associated dimer of a cyclic octa-peptide coupled with hydrophobic acyl chains. The acyl chains line the channel pore while the cyclic peptide forms the channel entrance [Z. Qi, M. Sokabe, K. Donowaki, H. Ishida, Biophys. J. 76 (1999) 631]. Molecular dynamics simulation of water in the channel indicated that the inferred structure is physically reasonable [Z. Qi, M. Sokabe, Biophys. Chem. 71 (1998) 35]. In the present study, the potential energy profile of the Na(+) and the energy contributions from each component of the system at different positions along the channel axis were calculated. An energy well instead of a peak is located at the central hydrophobic cavity of the channel, due to its ability of accommodating at least five water molecules to hydrate the ion. Interestingly, the ion diffuses much faster in the hydrophobic acyl chain region, particularly in the central hydrophobic cavity, than it does in the peptide ring region and even surprisingly faster than that in the bulk phase. These results provide a physical basis for an idea that the hydrophobic lining of the K(+) channel [D.A. Doyle, J.M. Cabral, R.A. Pfuetzner, A. Kuo, J.M. Gulbis, S.L. Cohen, B.T. Chait, R. MacKinnon, Science 280 (1998) 69] plays an active role to facilitate the ion permeation through the channel pore.

Permeation through the calcium release channel of cardiac muscle

Current voltage (I-V) relations were measured from the calcium release channel (CRC) of the sarcoplasmic reticulum of cardiac muscle in 12 KCl solutions, symmetrical and asymmetrical, from 25 mM to 2 M. I-V curves are nearly linear, in the voltage range +/- 150 mV approximately 12kT/e, even in asymmetrical solutions, e.g., 2 M // 100 mM. It is awkward to describe straight lines as sums of exponentials in a wide range of solutions and potentials, and so traditional barrier models have difficulty fitting this data. Diffusion theories with constant fields predict curvilinear I-V relations, and so they are also unsatisfactory. The Poisson and Nernst-Planck equations (PNP) form a diffusion theory with variable fields. They fit the data by using adjustable parameters for the diffusion constant of each ion and for the effective density of fixed (i.e., permanent) charge P(x) along the channel's "filter" (7-A diameter, 10 A long). If P(x) is described by just one parameter, independent of x (i.e., P(x) = P0 = -4.2 M), the fits are satisfactory (RMS error/RMS current = 6.4/67), and the estimates of diffusion coefficients are reasonable D(K) = 1.3 x 10(-6) cm2/s, D(Cl) = 3.9 x 10(-6) cm2/s. The CRC seems to have a small selectivity filter with a very high density of permanent charge. This may be a design principle of channels specialized for large flux. The Appendix derives barrier models, and their prefactor, from diffusion theories (with variable fields) and argues that barrier models are poor descriptions of CRCs in particular and open channels in general.

Permeation through an open channel: Poisson-Nernst-Planck theory of a synthetic ionic channel

The synthetic channel [acetyl-(LeuSerSerLeuLeuSerLeu)3-CONH2]6 (pore diameter approximately 8 A, length approximately 30 A) is a bundle of six alpha-helices with blocked termini. This simple channel has complex properties, which are difficult to explain, even qualitatively, by traditional theories: its single-channel currents rectify in symmetrical solutions and its selectivity (defined by reversal potential) is a sensitive function of bathing solution. These complex properties can be fit quantitatively if the channel has fixed charge at its ends, forming a kind of macrodipole, bracketing a central charged region, and the shielding of the fixed charges is described by the Poisson-Nernst-Planck (PNP) equations. PNP fits current voltage relations measured in 15 solutions with an r.m.s. error of 3.6% using four adjustable parameters: the diffusion coefficients in the channel's pore DK = 2.1 x 10(-6) and DCl = 2.6 x 10(-7) cm2/s; and the fixed charge at the ends of the channel of +/- 0.12e (with unequal densities 0.71 M = 0.021e/A on the N-side and -1.9 M = -0.058e/A on the C-side). The fixed charge in the central region is 0.31e (with density P2 = 0.47 M = 0.014e/A). In contrast to traditional theories, PNP computes the electric field in the open channel from all of the charges in the system, by a rapid and accurate numerical procedure. In essence, PNP is a theory of the shielding of fixed (i.e., permanent) charge of the channel by mobile charge and by the ionic atmosphere in and near the channel's pore. The theory fits a wide range of data because the ionic contents and potential profile in the channel change significantly with experimental conditions, as they must, if the channel simultaneously satisfies the Poisson and Nernst-Planck equations and boundary conditions. Qualitatively speaking, the theory shows that small changes in the ionic atmosphere of the channel (i.e., shielding) make big changes in the potential profile and even bigger changes in flux, because potential is a sensitive function of charge and shielding, and flux is an exponential function of potential.