Implications of a high dielectric constant in proteins

The electrostatic potential inside and around α-lactalbumin: Fluctuations and mean-field models

The electrostatic potential (EP) generated by the protein α-lactoalbumin in the presence of added salt is computed as a thermal average at a given point in space. With this aim, constant pH Monte Carlo (MC) simulations are performed within the primitive model, namely, the solvent is treated as a continuum dielectric. The study of the thermal and spatial fluctuations of the EP reveals that they are remarkably high inside the protein. The calculations indicate that fluctuations inside the protein are mainly due to the asymmetric distribution of the charge groups, while the charge fluctuations of the titratable groups play a minor role. The computed EP matches very well with the one obtained from the Poisson equation for the average charge density in spherical symmetry. The Tanford–Kirkwood multipole expansion reproduces the simulated angular-averaged potential rather accurately. Surprisingly, two of the simplest mean-field models, the linear Poisson–Boltzmann (PB) equation and Donnan potential, provide good estimations of the average EP in the effective protein surface (surface EP). The linear PB equation predicts a linear relationship between charge and surface EP, which is numerically reproduced only if the small ions within the protein are taken into account. On the other hand, the partition coefficients of the small ions inside and outside the protein predicted by Donnan theory reproduce reasonably well the simulation results.

Electrokinetic behavior of a pH-regulated dielectric cylindrical nanopore

A continuum model is adopted to describe the electrokinetic behavior of a pH-regulated cylindrical nanopore, the surface of which has charge-regulated carboxyl groups. We focus on the influences of the permittivity of the nanopore material, nanopore size, salt concentration, and solution pH on this behavior, and the underlying mechanisms. The influence of the nanopore permittivity becomes significant when a nanopore is shorter than ca. 50 nm. It is interesting to observe that if it is longer than ca. 100 nm, the nanopore conductance decreases with increasing permittivity. If it is sufficiently short, the conductance increases with increasing permittivity. If the nanopore length takes a medium level, the conductance is insensitive to the variation in the permittivity. For a short nanopore (~20 nm), the conductivity increases with increasing permittivity. However, if pH is sufficiently high, it becomes insensitive to permittivity. Although the larger the permittivity the greater the conductivity, in general, this effect becomes insignificant when the bulk salt concentration is sufficiently high, implying that the effect of membrane polarization is important only if the bulk salt concentration is sufficiently low.

A semi-GCMC simulation study of electrolytic capacitors with adsorbed titrating peptides

We use semi-grand canonical Monte Carlo simulations to study an electrolytic capacitor with an adsorbed peptide on the electrode surfaces. Only homogeneous peptides are considered, consisting of only a single residue type. We find that the classical double-hump camel-shaped differential capacitance in such systems is augmented by the addition of a third peak, due to the capacitance contribution of the peptide, essentially superimposed on the salt contribution. This mechanistic picture is justified using a simple mean-field analysis. We find that the position of this third peak can be tuned to various surface potential values by adjusting the ambient pH of the electrolyte solution. We investigate the effect of changing the residue type and the concentration of the adsorbed peptide and of the supporting electrolyte. Varying the residue species and pH allows one to modify the capacitance profile as a function of surface potential, facilitating the design of varying discharging patterns for the capacitor.

Many-body interactions between charged particles in a polymer solution: the protein regime

We study the phase behavior of charged particles in electrolyte solutions wherein non-adsorbing polymers are added to provide an attractive depletion interaction. The polymer has a radius of gyration similar to that of the particle radius, which causes significant many-body effects in the effective polymer mediated interaction between particles. We use a recently developed analytical theory, which gives a closed expression for the full depletion interaction, accounting for all orders of many-body terms in the potential of mean force. We compare with simulations of an explicit polymer model and show that the potential of mean force provides an accurate and computationally efficient description for the charged particle/polymer mixture, over a range of electrolyte concentrations. Furthermore, we demonstrate that the usual pair potential approach is highly inaccurate for these systems. A simple simulation method is used to estimate the limits of stability of the mixture. The pair approximation is shown to predict a much greater region of instability compared with the many-body treatment, due to its overestimation of the polymer depletion effect.

Charge reversal at a planar boundary between two dielectrics

Despite the ubiquitous character and relevance of the electric double layer in the entire realm of interface and colloid science, very little is known of the effect that surface heterogeneity exerts on the underlying mechanisms of ion adsorption. Herein, computer simulations offer a perspective that, in sharp contrast to the homogeneously charged surface, discrete groups promote multivalent counterion binding, leading to charge reversal but possibly having not a sign change of the electrophoretic mobility. Counterintuitively, the introduction of dielectric images yields a significantly greater accumulation of counterions, which further facilitates the magnitude of charge reversal. The reported results are very sensitive to both the degree of ion hydration and the representation of surface charges. Our findings shed light on the mechanism for charge reversal over a broad range of coupling regimes operating the adsorption of counterions through surface group bridging attraction with their own images and provide opportunities for experimental studies and theoretical development.

Hypothesis: The unfolding power of protein dielectricity

A hypothesis is proposed on a potential role of protein dielectricity as an unfolding factor in protein-protein interactions. It is suggested that large protein complexes and aggregation seeds can unfold target proteins by virtue of their effect on the dielectric properties of water at the protein-solvent interface. Here, similar to the effect of membrane surfaces, protein surface can cause decrease in the local dielectric constant of solvent and thereby induce structural changes in a target protein approaching this surface. Some potential implementations of this hypothetical mechanism are also discussed.

Do we have to explicitly model the ions in brownian dynamics simulations of proteins?

Brownian dynamics (BD) is a very efficient coarse-grained simulation technique which is based on Einstein's explanation of the diffusion of colloidal particles. On these length scales well beyond the solvent granularity, a treatment of the electrostatic interactions on a Debye-Hückel (DH) level with its continuous ion densities is consistent with the implicit solvent of BD. On the other hand, since many years BD is being used as a workhorse simulation technique for the much smaller biological proteins. Here, the assumption of a continuous ion density, and therefore the validity of the DH electrostatics, becomes questionable. We therefore investigated for a few simple cases how far the efficient DH electrostatics with point charges can be used and when the ions should be included explicitly in the BD simulation. We find that for large many-protein scenarios or for binary association rates, the conventional continuum methods work well and that the ions should be included explicitly when detailed association trajectories or protein folding are investigated.

pK(a) values for the unfolded state under native conditions explain the pH-dependent stability of PGB1

Understanding the role of electrostatics in protein stability requires knowledge of these interactions in both the folded and unfolded states. Electrostatic interactions can be probed experimentally by characterizing ionization equilibria of titrating groups, parameterized as pK(a) values. However, pK(a) values of the unfolded state are rarely accessible under native conditions, where the unfolded state has a very low population. Here, we report pK(a) values under nondenaturing conditions for two unfolded fragments of the protein G B1 domain that mimic the unfolded state of the intact protein. pK(a) values were determined for carboxyl groups by monitoring their pH-dependent (13)C chemical shifts. Monte Carlo simulations using a Gaussian chain model provide corrections for changes in electrostatic interactions that arise from fragmentation of the protein. Most pK(a) values for the unfolded state agree well with model values, but some residues show significant perturbations that can be rationalized by local electrostatic interactions. The pH-dependent stability was calculated from the experimental pK(a) values of the folded and unfolded states and compared to experimental stability data. The use of experimental pK(a) values for the unfolded state results in significantly improved agreement with experimental data, as compared to calculations based on model data alone.

Electrostatic chameleons in biological systems

Due to large equilibrium fluctuations of protons at physiological pH, the orthophosphate ion as well as the imidazole group on histidine substantially regulate their charge upon approaching charged interfaces. This implies that these-and comparable-ions function as electostatic "proximity switches" when interacting with lipid membranes, DNA, proteins, etc. Using straightforward statistical thermodynamics as well as mesoscopic computer simulations we quantify the charge regulation mechanism and argue that it is important in a range of biological as well as technical processes.

Chemistry of Hofmeister anions and osmolytes

The study of the interactions of salts and osmolytes with macromolecules in aqueous solution originated with experiments concerning protein precipitation more than 100 years ago. Today, these solutes are known to display recurring behavior for myriad biological and chemical processes. Such behavior depends both on the nature and concentration of the species in solution. Despite the generality of these effects, our understanding of the molecular-level details of ion and osmolyte specificity is still quite limited. Here, we review recent studies of the interactions between anions and urea with model macromolecular systems. A mechanism for specific ion effects is elucidated for aqueous systems containing charged and uncharged polymers, polypeptides, and proteins. The results clearly show that the effects of the anions are local and involve direct interactions with macromolecules and their first hydration shell. Also, a hydrogen-bonding mechanism is tested for the urea denaturation of proteins with some of these same systems. In that case, direct hydrogen bonding can be largely discounted as the key mechanism for urea stabilization of uncollapsed and/or unfolded structures.

The inverse and direct Hofmeister series for lysozyme

Anion effects on the cloud-point temperature for the liquid-liquid phase transition of lysozyme were investigated by temperature gradient microfluidics under a dark field microscope. It was found that protein aggregation in salt solutions followed 2 distinct Hofmeister series depending on salt concentration. Namely, under low salt conditions the association of anions with the positively charged lysozyme surface dominated the process and the phase transition temperature followed an inverse Hofmeister series. This inverse series could be directly correlated to the size and hydration thermodynamics of the anions. At higher salt concentrations, the liquid-liquid phase transition displayed a direct Hofmeister series that correlated with the polarizability of the anions. A simple model was derived to take both charge screening and surface tension effects into account at the protein/water interface. Fitting the thermodynamic data to this model equation demonstrated its validity in both the high and low salt regimes. These results suggest that in general positively charged macromolecular systems should show inverse Hofmeister behavior only at relatively low salt concentrations, but revert to a direct Hofmeister series as the salt concentration is increased.

Association and electrostatic steering of alpha-lactalbumin-lysozyme heterodimers

The salt and pH dependent association of hen egg white lysozyme with alpha-lactalbumin whey proteins has been studied using molecular level Monte Carlo simulations. A highly uneven charge distribution of alpha-lactalbumin leads to strongly ordered heterodimers that may facilitate the formation of structured, mesoscopic aggregates. This electrostatic steering gives rise to 80% alignment at 5 mM 1 : 1 salt which, due to screening, diminishes to 60% at 100 mM salt. The free energy of interaction minima, dominated by electrostatics, ranges between -9 kT at 1 mM salt to -2 kT at 100 mM (neutral pH). Calculated osmotic second virial cross coefficients indicate complexation in the pH interval 6-10. Multivalent ions are found to effectively destabilize the protein complex and, at constant ionic strength, the order is La(3+) > Ca(2+) > Mg(2+) > Na(+). Upon binding of calcium to alpha-lactalbumin both the interaction and orientational alignment with lysozyme are reduced due to induced changes in the whey protein charge distribution. This potentially explains the experimentally observed absence of supramolecular structuring for the calcium loaded holo alpha-lactalbumin. Where available, good agreement is found with experimental data.

Identifying stabilizing key residues in proteins using interresidue interaction energy matrix

We are proposing an interresidue interaction energy map (IEM)--a new tool for protein structure analysis and protein bioinformatics. This approach employs the sum of pair-wise interaction energies of a particular residue as a measure of its structural importance. We will show that the IEM can serve as a means for identifying key residues responsible for the stability of a protein. Our method can be compared with the interresidue contact map but has the advantage of weighting the contacts by the stabilization energy content which they bring to the protein structure. For the theoretical adjustment of the proposed method, we chose the Trp-cage mini protein as a model system to compare a spectrum of computational methods ranging from the ab initio MP2 level through the DFT method to empirical force-field methods. The IEM method correctly identifies Tryptophane 6 as the key residue in the Trp-cage. The other residues with the highest stabilizing contributions correspond to the structurally important positions in the protein. We have further tested our method on the Trp2Cage miniprotein--a P12W mutant of the Trp-cage and on two proteins from the rubredoxin family that differ in their thermostability. Our method correctly identified the thermodynamically more stable variants in both cases and therefore can also be used as a tool for the relative measurement of protein stability. Finally, we will point out the important role played by dispersion energy, which contributes significantly to the total stabilization energy and whose role in aromatic pairs is clearly dominant. Surprisingly, the dispersion energy plays an even more important role in the interaction of prolines with aromatic systems.

Faunus: An object oriented framework for molecular simulation

Background

We present a C++ class library for Monte Carlo simulation of molecular systems, including proteins in solution. The design is generic and highly modular, enabling multiple developers to easily implement additional features. The statistical mechanical methods are documented by extensive use of code comments that – subsequently – are collected to automatically build a web-based manual.

Results

We show how an object oriented design can be used to create an intuitively appealing coding framework for molecular simulation. This is exemplified in a minimalistic C++ program that can calculate protein protonation states. We further discuss performance issues related to high level coding abstraction.

Conclusion

C++ and the Standard Template Library (STL) provide a high-performance platform for generic molecular modeling. Automatic generation of code documentation from inline comments has proven particularly useful in that no separate manual needs to be maintained.