Mónika Valiskó

The effect of protein dielectric coefficient on the ionic selectivity of a calcium channel.

Calcium-selective ion channels are known to have carboxylate-rich selectivity filters, a common motif that is primarily responsible for their high Ca(2+) affinity. Different Ca(2+) affinities ranging from micromolar (the L-type Ca channel) to millimolar (the ryanodine receptor channel) are closely related to the different physiological functions of these channels. To understand the physical mechanism for this range of affinities given similar amino acids in their selectivity filters, we use grand canonical Monte Carlo simulations to assess the binding of monovalent and divalent ions in the selectivity filter of a model Ca channel. We use a reduced model where the electolyte is modeled by hard-sphere ions embedded in a continuum dielectric solvent, while the interior of protein surrounding the channel is allowed to have a dielectric coefficient different from that of the electrolyte. The induced charges that appear on the protein/lumen interface are calculated by the induced charge computation method [Boda et al., Phys. Rev. E 69, 046702 (2004)]. It is shown that decreasing the dielectric coefficient of the protein attracts more cations into the pore because the proteins carboxyl groups induce negative charges on the dielectric boundary. As the density of the hard-sphere ions increases in the filter, Ca(2+) is absorbed into the filter with higher probability than Na(+) because Ca(2+) provides twice the charge to neutralize the negative charge of the pore (both structural carboxylate oxygens and induced charges) than Na(+) while occupying about the same space (the charge/space competition mechanism). As a result, Ca(2+) affinity is improved an order of magnitude by decreasing the protein dielectric coefficient from 80 to 5. Our results indicate that adjusting the dielectric properties of the protein surrounding the permeation pathway is a possible way for evolution to regulate the Ca(2+) affinity of the common four-carboxylate motif.

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 Ca2+ and predicts the blockade of alkali metal ion currents by micromolar Ca2+. Conductance patterns observed in varied mixtures containing Na+ and Li+, or Ba2+ and Ca2+, are predicted. Ca2+ is substantially more potent in blocking Na+ current than Ba2+. In apparent contrast to experiments using buffered Ca2+ solutions, the predicted potency of Ca2+ 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 Ca2+ for the pore. These experiments depend on the problematic estimation of Ca2+ activity in solutions buffered for Ca2+ and pH in a varying background of bulk salt. Simulations of Ca2+ 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 Ca2+ 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.

Simulations of calcium channel block by trivalent cations: Gd3+ 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.

Anomalous temperature dependence of the differential capacitance in valence asymmetric electrolytes. Comparison of Monte Carlo simulation results and the field theoretical approach

In this paper, we consider the anomalous behaviour of the differential capacitance as a function of the reduced temperature for the case of valence asymmetric systems. Monte Carlo simulation results of these systems are discussed in view of the predictions of a field theoretical approach. We show that a simple field theory for point ions is sufficient to capture the main features of the phenomenon. In particular, the existence of scaling of the curves related to ionic strength is discussed as well as the increase of the differential capacitance peak as a function of the asymmetry between ions. This indicates that important features of the phenomenon are essentially related to the electrical properties of the system.

Unraveling the behavior of the individual ionic activity coefficients on the basis of the balance of ion-ion and ion-water interactions.

We investigate the individual activity coefficients of pure 1:1 and 2:1 electrolytes using our theory that is based on the competition of ion-ion (II) and ion-water (IW) interactions (Vincze et al. J. Chem. Phys. 2010, 133, 154507). The II term is computed from grand canonical Monte Carlo simulations on the basis of the implicit solvent model of electrolytes using hard sphere ions with Pauling radii. The IW term is computed on the basis of Borns treatment of solvation using experimental hydration free energies. The two terms are coupled through the concentration-dependent dielectric constant of the electrolyte. We show that the theory can provide valuable insight into the nonmonotonic concentration dependence of individual activity coefficients. We compare our theoretical predictions against experimental data measured by electrochemical cells containing ion-specific electrodes. We find good agreement for 2:1 electrolytes, while the accuracy is worse for 1:1 systems. This deviation in accuracy is explained by the fact that the two competing terms (II and IW) are much larger in the 2:1 case, resulting in smaller relative errors. The difference of the excess chemical potentials of cations and anions is determined by asymmetries in the properties of the two ions: charge, radius, and hydration free energies.

Ions and Inhibitors in the Binding Site of HIV Protease: Comparison of Monte Carlo Simulations and the Linearized Poisson-Boltzmann Theory

Proteins can be influenced strongly by the electrolyte in which they are dissolved, and we wish to model, understand, and ultimately control such ionic effects. Relatively detailed Monte Carlo (MC) ion simulations are needed to capture biologically important properties of ion channels, but a simpler treatment of ions, the linearized Poisson-Boltzmann (LPB) theory, is often used to model processes such as binding and folding, even in settings where the LPB theory is expected to be inaccurate. This study uses MC simulations to assess the reliability of the LPB theory for such a system, the constrained, anionic active site of HIV protease. We study the distributions of ions in and around the active site, as well as the energetics of displacing ions when a protease inhibitor is inserted into the active site. The LPB theory substantially underestimates the density of counterions in the active site when divalent cations are present. It also underestimates the energy cost of displacing these counterions, but the error is not consequential because the energy cost is less than kBT, according to the MC calculations. Thus, the LPB approach will often be suitable for studying energetics, but the more detailed MC approach is critical when ionic distributions and fluxes are at issue.

Response to “Comment on ‘The nonmonotonic concentration dependence of the mean activity coefficient of electrolytes is a result of a balance between solvation and ion–ion correlations’” [J. Chem. Phys.134, 157101 (2011)]

Fraenkel commented1 on our paper2 in which we offered a different view of the activity coefficient (or, equivalently, the excess chemical potential) of electrolytes. Fraenkel summarized the main points of our theory nicely. Also, he offered his smaller-ion-shell (SiS) model3, 4 as an alternative to our approach. In this response, we categorize our thoughts along a few main issues. (1) The most important issue for us is the question whether the (concentration-dependent) dielectric constant, (c), of the solution should be used in the model, or the dielectric constant of the pure solvent (that is equal to the dielectric constant of the solution at infinite dilution). We argue that is a macroscopic physical parameter whose value is well-defined and measurable for a given thermodynamic state point (different concentration corresponds to a different state point). In the implicit solvent model of electrolytes the ions are modeled on the molecular level, therefore, they are characterized by microscopic parameters such as the ionsize parameters (ISP). The solvent, on the other hand, is modeled on the basis of its average dielectric response, which is characterized by a macroscopic physical parameter: the dielectric constant. In our view, physical variables that enter the calculations as external parameters, such as the dielectric constant, should have the experimental values for the given state point. The structure of the solvent is influenced by the presence of ions and decreases with increasing concentration (dielectric saturation). Unfortunately, measurements of the static dielectric constant of an electrolyte have several technical difficulties. The extrapolation to zero frequency based on dielectric relaxation data over a frequency range requires special instrumentation to determine the dielectric loss. For example, newer experiments by Buchner et al.5 for NaCl gave (c) values that are larger than those reported by Barthel et al. previously.6 The absence of experimental data for (c) makes the applicability of our approach limited and, at the same time, from a practical point of view, necessarily requires the use of theories applying the dielectric constant of the solvent. The absence or availability of experimental data, however, does not influence the correctness of the physical picture behind our model. Our opinion is that solvation should be taken into account by a concentration-dependent dielectric constant rather than by an increased, solvated ionic radius. (2) Our paper might give the (false) impression that the question whether adjustable parameters are used or not is a central issue for us. It is not. It depends on the intention of the investigator: if one wants to adjust molecular parameters to fit with experiments, it is legitimate to do so. Actually, that is what we do in our ion channel studies, for example.7 There are parameters, however, whose values are well-defined so they should not be adjusted. Even if we adjust a parameter, the resulting value should have a sensible physical meaning. In general, we can shed a light on our standpoint regarding modeling if we cite Occam’s razor that “admonishes us to choose from a set of otherwise equivalent models of a given phenomenon the simplest one. In any given model, Occam’s razor helps us to ‘shave off’ those concepts, variables, or constructs that are not really needed to explain the phenomenon. By doing that, developing the model will become much easier, and there is less chance of introducing inconsistencies, ambiguities, and redundancies.”8 Paraphrasing this idea for the case of adjustable parameters, we can offer a less strong statement: “In a model, we should minimize the number of adjustable parameters.” (3) The third issue is the question of the statistical– mechanical method with which the results for the activity coefficient [and the ion–ion (II) term in our approach] are calculated. Our opinion is that the best method should be used even if it is computationally expensive (especially in the age of fast computers). Technically simpler methods might be justified by computational convenience, but when scientific knowledge is sought, the most accurate method is favorable. According to the complexity of the studied many-body systems, more complex methods usually provide more accurate results. (Alas, Occam’s razor does not help here.) For a well-defined molecular model of a many-particle system, computer simulations, if performed properly, provide exact results apart from statistical uncertainties and system-size effects. Theories necessarily use approximations. Their results, therefore, should be tested against simulation data. The simulation results contain only the error due to the imperfectness of the model, while the theoretical results also contain the error due to the imperfectness of the approximation to derive the theory. Separation of these two errors is important for the activity coefficient of electrolytes as demonstrated in our paper using grand canonical Monte Carlo (MC) simulations9 and the mean spherical

A systematic Monte Carlo simulation and renormalized perturbation theoretical study of the dielectric constant of the polarizable Stockmayer fluid

A systematic Monte Carlo (MC) simulation and perturbation theory (PT) study is reported for the dielectric constant of the polarizable Stockmayer fluid. Our MC simulations apply the ‘pair approximation for polarization interaction’ procedure suggested by Předota et al. The theoretical approach is based on our newly introduced equation (Valiskó et al., 2002, Molec. Phys., 100, 559) which is a density expansion for the dielectric constant using Wertheims renormalized PT method. The agreement between our MC and PT results is excellent for low to moderate dipole moments and polarizabilities. At stronger dipolar interactions ergodicity problems and anisotropic behaviour appear where simulation results become uncertain and the theoretical approach becomes invalid.

The dielectric constant of polarizable fluids from the renormalized perturbation theory

A perturbation theoretical equation for the dielectric constant of polarizable dipolar fluids is proposed. For the fluctuation of the dipole moment, namely for the Kirkwood g-factor, a formula is given on the basis of Wertheims renormalized perturbation theory. Using this formula, a series expansion for ε(p) is suggested on the basis of the Kirkwood equation, which gives an implicit function for ε as a function of ¶. The same series expansion can be derived from the Clausius-Mosotti equation—thus it proves to be independent of the boundary conditions. The resulting equation gives excellent results for the dielectric constant of the polarizable Stockmayer fluid producing good agreement with computer simulation data. The series expansion gives better results than the Kirkwood equation itself.

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 nanopores behavior as a device, as well as concentration, electrical potential, and electrochemical potential profiles.