On the Physics of Size Selectivity

We demonstrate that two mechanisms used by biological ion channels to select particles by size are driven by entropy. With uncharged particles in an infinite cylinder, we show that a channel that attracts particles is small-particle selective and that a channel that repels water from the wall is large-particle selective. Comparing against extensive density-functional theory calculations of our model, we find that the main physics can be understood with surprisingly simple bulk models that neglect the confining geometry of the channel completely.Comment: 4 pages, 3 figures, Phys. Rev. Lett. (accepted

An Efficient Algorithm for Classical Density Functional Theory in Three Dimensions: Ionic Solutions

Classical density functional theory (DFT) of fluids is a valuable tool to analyze inhomogeneous fluids. However, few numerical solution algorithms for three-dimensional systems exist. Here we present an efficient numerical scheme for fluids of charged, hard spheres that uses $\mathcal{O}(N\log N)$ operations and $\mathcal{O}(N)$ memory, where $N$ is the number of grid points. This system-size scaling is significant because of the very large $N$ required for three-dimensional systems. The algorithm uses fast Fourier transforms (FFT) to evaluate the convolutions of the DFT Euler-Lagrange equations and Picard (iterative substitution) iteration with line search to solve the equations. The pros and cons of this FFT/Picard technique are compared to those of alternative solution methods that use real-space integration of the convolutions instead of FFTs and Newton iteration instead of Picard. For the hard-sphere DFT we use Fundamental Measure Theory. For the electrostatic DFT we present two algorithms. One is for the \textquotedblleft bulk-fluid\textquotedblright functional of Rosenfeld [Y. Rosenfeld. \textit{J. Chem. Phys.} 98, 8126 (1993)] that uses $\mathcal{O}(N\log N)$ operations. The other is for the \textquotedblleft reference fluid density\textquotedblright (RFD) functional [D. Gillespie et al., J. Phys.: Condens. Matter 14, 12129 (2002)]. This functional is significantly more accurate than the bulk-fluid functional, but the RFD algorithm requires $\mathcal{O}(N^{2})$ operations.Comment: 23 pages, 4 figure

Selecting Ions by Size in a Calcium Channel: The Ryanodine Receptor Case Study

AbstractMany calcium channels can distinguish between ions of the same charge but different size. For example, when cations are in direct competition with each other, the ryanodine receptor (RyR) calcium channel preferentially conducts smaller cations such as Li+ and Na+ over larger ones such as K+ and Cs+. Here, we analyze the physical basis for this preference using a previously established model of RyR permeation and selectivity. Like other calcium channels, RyR has four aspartate residues in its GGGIGDE selectivity filter. These aspartates have their terminal carboxyl group in the pore lumen, which take up much of the available space for permeating ions. We find that small ions are preferred by RyR because they can fit into this crowded environment more easily

Self-organized Models of Selectivity in Ca and Na Channels

A simple pillbox model with two adjustable parameters accounts for the selectivity of both DEEA Ca channels and DEKA Na channels in many ionic solutions of different composition and concentration. Only the side chains are different in the model of the Ca and Na channels. Parameters are the same for both channels in all solutions. 'Pauling' radii are used for ions. No information from crystal structures is used in the model. Side chains are grossly approximated as spheres. The predicted properties of the Na and Ca channels are very different. How can such a simple model give such powerful results when chemical intuition says that selectivity depends on the precise relation of ions and side chains? We use Monte Carlo simulations of this model that determine the most stable-lowest free energy-structure of the ions and side chains. Structure is the computed consequence of the forces in this model. The relationship of ions and side chains vary with ionic solution and are very different in simulations of the Na and Ca channels. Selectivity is a consequence of the 'induced fit' of side chains to ions and depends on the flexibility (entropy) of the side chains as well as their location. The model captures the relation of side chains and ions well enough to account for selectivity of both Na channels and Ca channels in the wide range of conditions measured in experiments. Evidently, the structures in the real Na and Ca channels responsible for selectivity are self-organized, at their free energy minimum. Oversimplified models are enough to account for selectivity if the models calculate the 'most stable' structure as it changes from solution to solution, and mutation to mutation.Comment: Version of http://www.ima.umn.edu/2008-2009/W12.8-12.08/abstracts.html, talk given at the Institute for Mathematics and its Applications, University of Minnesota, November 19, 2008. Abstract published in Biophysical Journal, Volume 96, Issue 3, 253

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 on 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 explicitwater 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 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

Distribution of ions between different dielectric media: direct simulation of the Donnan equilibrium in the grand canonical ensemble

A modification of the original Grand Canonical Monte Carlo (GCMC) method to handle Donnan equilibrium is proposed that provides an equilibrium between two implicit-solvent bath electrolytes with different dielectric constants. A solvation energy penalty (described by the Born theory) and an electrical potential difference (Donnan potential) exist between the two baths, the ‘system’ under investigation and the ‘reservoir’. These terms are deducted from the chemical potential of the ‘reservoir’, and the resulting chemical potential is used in the ‘system’. The simulation performed with this chemical potential in the acceptance probabilities of the ion insertion/deletions is called Donnan GCMC and provides a thermodynamic state in the ‘system’ that is in equilibrium with the electrolyte in the ‘reservoir’. The simulation provides the distribution of ions between the two baths (concentrations in both media) from a single run instead of a numerical procedure that requires several GCMC runs. Using individual ion insertion/deletions, the Donnan potential can be determined

Scaling Behavior of Bipolar Nanopore Rectification with Multivalent Ions

We present a scaling behavior of a rectifying bipolar nanopore as a function of the parameter ξ = R_P /(λz_if), where R_P is the radius of the pore, λ is the characteristic screening length of the electrolyte filling the pore, and z_if = \sqrt{z+|z− |} is a scaling factor that makes scaling work for electrolytes containing multivalent ions (z+ and z− are cation and the anion valences). By scaling we mean that the rectification of the pore (defined as the ratio of currents in the forward and reversed biased states) depends on pore radius, concentration, c, and ion valences via the parameter ξ implicitly. This feature is based on the fact that rectification depends on the voltage-sensitive appearance of depletion zones that, in turn, depend on the relation of RP to the rescaled screening length λz_if. In this modeling study, we use the Poisson-Nernst-Planck (PNP) theory and a particle simulation method, the Local Equilibrium Monte Carlo (LEMC). The latter can compute ion correlations that are ignored in the mean-field treatment of PNP and that are very important for multivalent ions (we show results for 1:1, 2:1, 3:1, and 2:2 electrolytes). In addition to the z_if factor, we show that one must choose a screening length appropriate to the system, in our case the Debye length for λ for PNP and the screening length given by the Mean Spherical Approximation for LEMC