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The number density (“concentration”) of ions near a charged wall. The wall has charge density of . Charge is shown divided by . The diameter of the ions is 0.3 nm and they have charges of and −1e, where is the charge on a proton. Position is shown divided by nanometer. Dielectric coefficient was 78, temperature was 298 K, and the bulk concentration of ions was 1 M of the divalent cation and 2 M of the monovalent anion. The potential on the wall was set to −3.1 kT/e, i.e., −80 mV in accord with MC simulations (Ref. 202). Energy coupling coefficients in EnVarA were 0.5. The dotted circles show the size of the ions in the calculation. Ions are not allowed to overlap with the wall and so the densities are smooth functions until they reach the excluded zone produced by finite diameter of the ions. Calculations were done using (solid lines) the PNP-DFT and PNP-LJ in the form described in the text and Appendixes A and C. The form of the DFT differs in detail (but not spirit) from that in recent literature (Refs. 67, 68, and 203): we use DFT for uncharged interactions (following Refs. 66, 142, and 204) but we use EnVarA to deal with the electrostatics. EnVarA identically satisfies Gauss’ law (which is also one of the sum rules (Ref. 72 of statistical mechanics). This calculation is called PNP-DFT even though we report only equilibrium results: the variational method knows nothing of equilibrium and always computes a nonequilibrium transient response which may in special cases have no flows and a stationary solution. The results are qualitatively similar to the layering reported in MC simulations (Ref. 132). Quantitative differences are expected because the systems are not identical. Most importantly, simulations were of hard spheres (whereas we use Lennard-Jones or DFT in one dimension). There are many other small differences; e.g., MC uses an approximation to the solution of Poisson’s equation (Ref. 205) that produces results independent of system size, but without definite error bounds (Refs. 135 and 206).
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Binding of calcium to a DEEA channel. This figure shows the relative occupancy of the cylindrical space, that is to say, it shows the ratio of the spatial integral of the density of calcium (diameter of 1.98 Å) to the spatial integral of the density of sodium (diameter of 2.04 Å)—both within a cylindrical space of 3.5 Å radius and 3 Å length—of the stationary solution of the time dependent Euler–Lagrange equations. The concentration of sodium is maintained at 0.1 M on both sides of the channel. The concentration of calcium is varied and is shown on the horizontal axis. The dielectric constant was 80 and the temperature was 298 K. Geometrical setup is nearly the same as Figs. 1 and 2 of Ref. 15.
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Binding of sodium to a DEKA channel. This figure shows the relative occupancy of the cylindrical space, that is to say, it shows the ratio of the spatial integral of the density of calcium to the spatial integral of the density of sodium, both within a cylindrical space of 3 Å long and 3.5 Å radius of the stationary solution of the time dependent Euler–Lagrange equations. The concentration of sodium is maintained at 0.1 M on both sides of the channel. The concentration of calcium is varied and is shown on the horizontal axis. The dielectric constant was 80 and the temperature was 298 K. Geometrical setup is nearly the same as Figs. 1 and 2 of Ref. 15.
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The time response to a step function of voltage of a DEKA sodium channel (Glu-Asp-Lys-Ala). The voltage pulse started at −0.09 V and switched to the indicated voltage at and then back to −0.09 at . As shown in Fig. 5, the channel is 20 Å long and has 7 Å diameter. The concentration of NaCl was 0.9 M on one side and 0.1 M on the other. The sodium ion had diameter of 2.04 Å and chloride ion had diameter of 3.62 Å, diffusion coefficients were 1.68 and , respectively. The dielectric constant was 80 and temperature was 298 K. No units are shown for current because the number of channels being computed is arbitrary.
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The setup for the calculations of time dependent current shown in Fig. 4. The blue line shows the boundary of the one dimensional channel. The steep spread between the lines is a one dimensional representation of the baths used because it reduces the “resistance” to current flow or flux. That is to say, the greater cross sectional area to flow allows more flow for a given gradient of electrochemical potential than in the narrow 7 Å (diameter) channel through the bilayer itself. The dashed line represents the lipid bilayer membrane. The distribution of fixed charge along the channel wall is labeled “configuration of side chains.” The concentration of salts is shown in the baths. The units of the axes are divided by angstroms.
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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.
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