Michael B. Partenskii

Theoretical perspectives on ion-channel electrostatics: continuum and microscopic approaches.

Peter Lauger introduced me (P.C.J.) to the field of ion-channel electrostatics while I was a sabbatical visitor at Konstanz in 1978–79. Lauger pointed out that the relative conductance of hydrophobic ions through phosphatidyl choline (PC) and glyceryl monooleate (GMO) membranes differed by a factor of about 100 (Hladky & Haydon, 1973), quite consistent with the difference in the water-membrane potential differences in the two systems (Pickar & Benz, 1978). However, cation conductance through gramicidin channels spanning these membranes only differs by a factor of 2–3 (Bamberg et al. 1976). Why? It is the pursuit of an answer to this question which led me into my researches in this field.

A semi-microscopic Monte Carlo study of permeation energetics in a gramicidin-like channel: the origin of cation selectivity

The influence of a gramicidin-like channel former on ion free energy barriers is studied using Monte Carlo simulation. The model explicitly describes the ion, the water dipoles, and the peptide carbonyls; the remaining degrees of freedom, bulk electrolyte, non-polar lipid and peptide regions, and electronic (high frequency) permittivity, are treated in continuum terms. Contributions of the channel waters and peptide COs are studied both separately and collectively. We found that if constrained to their original orientations, the COs substantially increase the cationic permeation free energy; with or without water present, CO reorientation is crucial for ion-CO interaction to lower cation free energy barriers; the translocation free energy profiles for potassium-, rubidium-, and cesium-like cations exhibit no broad barriers; the lipid-bound peptide interacts more effectively with anions than cations; anionic translocation free energy profiles exhibit well defined maxima. Using experimental data to estimate transfer free energies of ions and water from bulk electrolyte to a non-polar dielectric (continuum lipid), we found reasonable ion permeation profiles; cations bind and permeate, whereas anions cannot enter the channel. Cation selectivity arises because, for ions of the same size and charge, anions bind hydration water more strongly.

Membrane deformation and the elastic energy of insertion: Perturbation of membrane elastic constants due to peptide insertion

In peptide insertion, matching of the hydrophobic regions of both peptide and lipid molecules constrains the lipid molecules’ mobility and their ability to adjust position, orientation and conformation. This can be described as an insertion induced modification of membrane elastic constants close to the insertion. The perturbation’s correlation length (λc) must be comparable to the length of lipid molecules, ∼1.5 nm. We simulate this effect by introducing a “transition” function with decay length λc. The deformation profile u(r) and corresponding elastic free energy E are calculated using Euler-Lagrange equations. The proper choice of boundary conditions is discussed. Perturbation of the membrane’s compressional modulus is shown to have much greater influence than perturbation of the bending modulus. Experiments on gramicidin channels’ lifetime are discussed from this perspective. Possible implications of the nonuniformity of the elastic constants on the membrane-mediated interaction between the insertion...

Application of finite-difference methods to membrane-mediated protein interactions and to heat and magnetic field diffusion in plasmas

A robust finite-difference approach for solving physically distinct cross-disciplinary problems such as membrane-mediated protein-protein interactions and heat and magnetic field diffusion in plasmas is described for rectangular grids. Mathematical models representing these physical phenomena are fourth- and second-order partial differential equations with variable coefficients. The finite-difference coupled harmonic oscillators technique was developed to treat arbitrary aggregates of inclusions in membranes automatically accounting for their non-pairwise interactions. The method was applied to study the stabilization of ion channels in a cluster due to membrane-mediated interactions and to examine the effects of anisotropic membrane slope relaxation on the elastic free energy. To obtain contributions from heat and magnetic field diffusion, the splitting method for the physical processes has been used in the numerical solution of resistive magnetohydrodynamic equations. The fully implicit scheme is outlined, tested and applied to problems of the diffusive redistribution of magnetic field and heat in the plasma.

Extended dipolar chain model for ion channels: electrostriction effects and the translocational energy barrier.

We reinvestigate the dipolar chain model for an ion channel. Our goal is to account for the influence that ion-induced electrostriction of channel water has on the translocational energy barriers experienced by different ions in the channel. For this purpose, we refine our former model by relaxing the positional constraint on the ion and the water dipoles and by including Lennard-Jones contributions in addition to the electrostatic interactions. The positions of the ion and the waters are established by minimization of the free energy. As before, interaction with the external medium is described via the image forces. Application to alkali cations show that the short range interactions modulate the free energy profiles leading to a selectivity sequence for translocation. We study the influence of some structural parameters on this sequence and compare our theoretical predictions with observed results for gramicidin.

INFLUENCE OF A CHANNEL-FORMING PEPTIDE ON ENERGY BARRIERS TO ION PERMEATION, VIEWED FROM A CONTINUUM DIELECTRIC PERSPECTIVE

The continuum three-dielectric model for an aqueous ion channel pore-forming peptide-membrane system is extended to account for the finite length of the channel. We focus on the electrostatic influence that a channel-forming peptide may exert on energy barriers to ion permeation. The nonlinear dielectric behavior of channel water caused by dielectric saturation in the presence of an ion is explicitly modeled by assigning channel water a mean dielectric constant much less than that of bulk water. An exact solution of the continuum problem is formulated by approximating the dielectric behavior of bulk water, assigning it a dielectric constant of infinity. The validity of this approximation is verified by comparison with a Poisson-Boltzmann description of the electrolyte. The formal equivalence of high ionic strength and high electrolyte dielectric constant is demonstrated. We estimate limits on the reduction of the electrostatic free energy caused by ionic interaction with the channel-forming peptide. We find that even assigning this region an epsilon of 100, its influence is insufficient to lower permeation free energy barriers to values consistent with observed channel conductances. We provide estimates of the effective dielectric constant of this highly polarizable region, by comparing energy barriers computed using the continuum approach with those found from a semi-microscopic analysis of a simplified model of a gramicidin-like charge distribution. Possible ways of improving both models are discussed.

Membrane stability under electrical stress: A nonlocal electroelastic treatment

Existing models of membrane instability and breakdown under an applied voltage are critically examined. An alternative, speculative treatment of the electroelastic model is suggested, based on the assumption that spatial dispersion of the elastic moduli leads to their effective softening at short wave lengths. The model parameters that account for these effects are chosen to ensure that short wave length thickness fluctuations become unstable at moderate applied voltages, ∼1–1.5 V. With these parameters we treat the membrane stretching diagram and membrane thickness fluctuations. The stretching diagram agrees with experimental findings and earlier calculations. Computed thickness fluctuations are consistent with previous investigations.

The admissible sign of the differential capacity, instabilities, and phase transitions at electrified interfaces

The issue of the allowed sign of the differential capacity C of electrochemical interfaces has a long history dating back to the so‐called ‘‘Cooper–Harrison catastrophe.’’ Previously suggested ‘‘electromechanical models’’ are modified to include entropic contributions; the possibility of C<0 for an isolated electric cell is supported by rigorous solution of the model. We also provide new evidence that for an electric cell in contact with a potentiostat (‘‘extended system’’) the overall C of the cell must be positive; the contribution of an individual double layer may still be negative. The previous statistical mechanical derivation of the upper boundary for C−1 is generalized for a quite general model of an electrolyte in contact with hard charged walls. We also discuss the possibility of electric instabilities and phase transitions in an extended system when an isolated prototype possesses a negative capacity branch.

A dipolar chain model for the electrostatics of transmembrane ion channels

Abstract Many ion selective channels contain a narrow region in which water molecules cannot pass one another. We model this “single file” domain by a linear chain of molecular dipoles embedded in continuum, low dielectric (ϵm = 2 ) slab surrounded by high dielectric (ϵ = ∞) continua. We present exact results for the free energy, polarization and susceptibility of the molecular dipoles as functions of an applied electric field and of the location of a cation in the chain. For parameters typical of the gramicidin channel we find that for an ion-free channel, the dipolar susceptibility is high at low electric fields. When an ion replaces one of the water dipoles in the chain the susceptibility is reduced 100-fold.

Stabilization of ion channels due to membrane-mediated elastic interaction

Recent work shows that linked gramicidin channels may have much longer lifetimes than single channels. We establish that the stabilization of the individual channels can be caused by membrane-mediated elastic interactions between such inclusions. In linear elastic theory, interaction can be rigorously described in terms of coupled harmonic oscillators. We determine the “effective spring constants” for various assemblies using the smectic bilayer model. We consider a range of aggregates; in clusters, channel lifetimes may increase by several orders of magnitude, an effect that is especially pronounced for a channel with many near neighbors.