Phase behavior of aqueous solutions containing dipolar proteins from second-order perturbation theory

Due to the interplay of Coulombic repulsion and attractive dipolar and van der Waals interactions, solutions of globular proteins display a rich variety of phase behavior featuring fluid-fluid and fluid-solid transitions that strongly depend on solution pH and salt concentration. Using a simple model for charge, dispersion and dipole-related contributions to the interprotein potential, we calculate phase diagrams for protein solutions within the framework of second-order perturbation theory. For each phase, we determine the Helmholtz energy as the sum of a hard-sphere reference term and a perturbation term that reflects both the electrostatic and dispersion interactions. Dipolar effects can induce fluid-fluid phase separation or crystallization even in the absence of any significant dispersion attraction. Because dissolved electrolytes screen the charge-charge repulsion more strongly than the dipolar attraction, the ionic strength dependence of the potential of mean force can feature a minimum at intermediate ionic strengths offering an explanation for the observed nonmonotonic dependence of the phase behavior on salt concentration. Inclusion of correlations between charge-dipole and dipole-dipole interactions is essential for a reliable calculation of phase diagrams for systems containing charged dipolar proteins and colloids.     

Toward the description of electrostatic interactions between globular proteins: potential of mean force in the primitive model

Monte Carlo simulations are used to calculate the exact potential of mean force between charged globular proteins in aqueous solution. The aim of the present paper is to study the influence of the ions of the added salt on the effective interaction between these nanoparticles. The charges of the model proteins, either identical or opposite, are either central or distributed on a discrete pattern. Contrarily to Poisson-Boltzmann predictions, attractive, and repulsive direct forces between proteins are not screened similarly. Moreover, it has been shown that the relative orientations of the charge patterns strongly influence salt-mediated interactions. More precisely, for short distances between the proteins, ions enhance the difference of the effective forces between (i) like-charged and oppositely charged proteins, (ii) attractive and repulsive relative orientations of the proteins, which may affect the selectivity of protein/protein recognition. Finally, such results observed with the simplest models are applied to a more elaborate one to demonstrate their generality.     

Contribution of charged groups to the enthalpic stabilization of the folded states of globular proteins

In this paper we use the results from all-atom molecular dynamics (MD) simulations of proteins and peptides to assess the individual contribution of charged atomic groups to the enthalpic stability of the native state of globular proteins and investigate how the distribution of charged atomic groups in terms of solvent accessibility relates to protein enthalpic stability. The contributions of charged groups is calculated using a comparison of nonbonded interaction energy terms from equilibrium simulations of charged amino acid dipeptides in water (the "unfolded state") and charged amino acids in globular proteins (the "folded state"). Contrary to expectation, the analysis shows that many buried, charged atomic groups contribute favorably to protein enthalpic stability. The strongest enthalpic contributions favoring the folded state come from the carboxylate (COO(-)) groups of either Glu or Asp. The contributions from Arg guanidinium groups are generally somewhat stabilizing, while N(+)(3) groups from Lys contribute little toward stabilizing the folded state. The average enthalpic gain due to the transfer of a methyl group in an apolar amino acid from solution to the protein interior is described for comparison. Notably, charged groups that are less exposed to solvent contribute more favorably to protein native-state enthalpic stability than charged groups that are solvent exposed. While solvent reorganization/release has favorable contributions to folding for all charged atomic groups, the variation in folded state stability among proteins comes mainly from the change in the nonbonded interaction energy of charged groups between the unfolded and folded states. A key outcome is that the calculated enthalpic stabilization is found to be inversely proportional to the excess charge density on the surface, in support of an hypothesis proposed previously.     

Interaction force between two charged plates immersed in a solution of charged particles. Coupling between double layer and depletion forces

When two parallel plates are immersed in a solution of small charged particles, the center of the particles is excluded from a region of thickness D/2 near the plate, where D is their diameter. The approach which Langmuir developed for the double layer repulsion in the presence of an electrolyte with ions of negligible size is extended to the case in which one of the "ions" is a charged particle of finite, relatively small size. A general expression for the force generated between the two charged plates immersed in an electrolyte solution containing relatively small charged particles is derived. In this expression, only the electrical potential at the middle distance between the plates is required to calculate the force. A Poisson-Boltzmann equation which accounts for the volume exclusion of the charged particles in the vicinity of the surface is solved to obtain the electrical potential at the middle between the two plates. Starting from this expression, some results obtained previously for the depletion force acting between two plates or two spheres are rederived. For charged plates immersed in a solution of an electrolyte and charged small particles, the effects of the particle charge, particle charge sign, particle size, and volume fraction of the particles on the force acting between the two plates are examined.     

Chromatographic measurement of interactions between unlike proteins

A chromatographic method is used to measure ovalbumin-lysozyme and BSA-lysozyme interactions in aqueous salt solutions as a function of solution conditions (pH, ionic strength, salt type). In this method, one protein is immobilized on the support surface, and the other, dissolved in a buffer/electrolyte solution, flows over that surface. The retention time provides a measure of immobile/mobile protein–protein interactions. Trends in ovalbumin-lysozyme interactions suggest that they are primarily electrostatic. The identity of the electrolyte has a strong influence on the magnitude of the interaction. Assuming a potential of mean force that contains a hard sphere, electrostatic, and square-well potential, experimental results are used to fit the square-well depth. For BSA-lysozyme interactions, the square-well depth depends on which protein is immobilized on the solid phase.     

Particle Interactions in Diffusiophoresis and Electrophoresis of Colloidal Spheres with Thin but Polarized Double Layers

The diffusiophoretic and electrophoretic motions of two colloidal spheres in the solution of a symmetrically charged electrolyte are analyzed using a method of reflections. The particles are oriented arbitrarily with respect to the electrolyte gradient or the electric field, and they are allowed to differ in radius and in zeta potential. The thickness of the electric double layers surrounding the particles is assumed to be small relative to the radius of each particle and to the gap width between the particles, but the effect of polarization of the mobile ions in the diffuse layer is taken into account. A slip velocity of fluid and normal fluxes of solute ions at the outer edge of the thin double layer are used as the boundary conditions for the fluid phase outside the double layers. The method of reflections is based on an analysis of the electrochemical potential and fluid velocity disturbances produced by a single dielectric sphere placed in an arbitrarily varying electrolyte gradient or electric field. The solution for two-sphere interactions is obtained in expansion form correct to O(r(12)(-7)), where r(12) is the distance between the particle centers. Our analytical results are found to be in good agreement with the available numerical solutions obtained using a boundary collocation method. On the basis of a model of statistical mechanics, the results of two-sphere interactions are used to analytically determine the first-order effect of the volume fraction of particles of each type on the mean diffusiophoretic and eletrophoretic velocities in a bounded suspension. For a suspension of identical spheres, the mean diffusiophoretic velocity can be decreased or increased as the volume fraction of the particles is increased, while the mean electrophoretic velocity is reduced with the increase in the particle concentration. Generally speaking, the particle interaction effects can be quite significant in typical situations. Copyright 2000 Academic Press.     

Revealing Conformational Variants of Solution‐Phase Intrinsically Disordered Tau Protein at the Single‐Molecule Level

Intrinsically disordered proteins, such as tau protein, adopt a variety of conformations in solution, complicating solution‐phase structural studies. We employed an anti‐Brownian electrokinetic (ABEL) trap to prolong measurements of single tau proteins in solution. Once trapped, we recorded the fluorescence anisotropy to investigate the diversity of conformations sampled by the single molecules. A distribution of anisotropy values obtained from trapped tau protein is conspicuously bimodal while those obtained by trapping a globular protein or individual fluorophores are not. Time‐resolved fluorescence anisotropy measurements were used to provide an explanation of the bimodal distribution as originating from a shift in the compaction of the two different families of conformations.     

Specific ion effects in solutions of globular proteins: comparison between analytical models and simulation

Monte Carlo simulations have been performed for ion distributions outside a single globular macroion and for a pair of macroions, in different salt solutions. The model that we use includes both electrostatic and van der Waals interactions between ions and between ions and macroions. Simulation results are compared with the predictions of the Ornstein-Zernike equation with the hypernetted chain closure approximation and the nonlinear Poisson-Boltzmann equation, both augmented by pertinent van der Waals terms. Ion distributions from analytical approximations are generally very close to the simulation results. This demonstrates that properties that are related to ion distributions in the double layer outside a single interface can to a good approximation be obtained from the Poisson-Boltzmann equation. We also present simulation and integral equation results for the mean force between two globular macroions (with properties corresponding to those of hen-egg-white lysozyme protein at pH 4.3) in different salt solutions. The mean force and potential of mean force between the macroions become more attractive upon increasing the polarizability of the counterions (anions), in qualitative agreement with experiments. We finally show that the deduced second virial coefficients agree quite well with experimental results.     

New local composition model for electrolyte solutions

A new model for representation of the excess Gibbs energy of electrolyte solutions is proposed. The excess Gibbs energy of an electrolyte solution is expressed as a sum of contributions of a long-range and a short-range excess Gibbs energy term. The Pitzer–Debye–Hückel model is used as a long-range contribution to the excess Gibbs energy. A new expression based on the local composition concept, which is the non-random factor (NRF)–Wilson model, is developed to account for the short-range contribution to the excess Gibbs energy. The main difference between this model and the electrolyte-NRF model available in the literature is the assumption that the short-range energy parameter between species in a local cell has an enthalpic rather than Gibbs energy nature. The utility of the model is demonstrated with the successful representation of the mean ionic activity coefficient of several aqueous electrolyte solutions. The results show that with only two adjustable parameters per electrolyte, the model is valid for the whole range of electrolyte concentration, from dilute solution up to saturation. The results are compared with those obtained from the NRTL, NRF and Wilson models. The model presented in this work produces better results.     

Colloid vibration potential in a suspension of spherical colloidal particles

When a sound wave is applied to a suspension of colloidal particles in an electrolyte solution, the colloid vibration potential (CVP) is produced in the suspension. The CVP is proportional to the difference between the mass density of the particles and that of the electrolyte solution. For a suspension of biological colloids such as cells, whose mass density is only slightly different from the electrolyte solution, its CVP becomes very small so that the magnitude of the ion vibration potential (IVP) of the electrolyte solution exceeds that of CVP. This causes difficulty in analyzing the CVP in biological systems. In the present paper, we show that even in such cases the phase of CVP becomes much greater than that of IVP.     

Molecular dynamics simulation of nanocolloidal amorphous silica particles: Part I

Explicit molecular dynamics simulations were applied to a pair of amorphous silica nanoparticles in aqueous solution, with diameter of 4.4 nm and with four different background electrolyte concentrations, to extract the mean force acting between the two silica nanoparticles. Dependences of the interparticle forces on the separation and the background electrolyte concentration were demonstrated. The nature of the interaction of the counterions with charged silica surface sites (deprotonated silanols) was investigated. A "patchy" double layer of adsorbed sodium counterions was observed. Dependences of the interparticle potential of mean force on the separation and the background electrolyte concentration were demonstrated. Direct evidence of the solvation forces is presented in terms of changes of the water ordering at the surfaces of the isolated and double nanoparticles. The nature of the interaction of the counterions with charged silica surface sites (deprotonated silanols) was investigated in terms of quantifying the effects of the number of water molecules separately inside each pair of nanoparticles by defining an impermeability measure. A direct correlation was found between the impermeability (related to the silica surface "hairiness") and the disruption of water ordering. Differences in the impermeability between the two nanoparticles are attributed to differences in the calculated electric dipole moment.     

Ado and additional comments on the interfacial inhomogeneous regions near a biological membrane

The non-linear differential equations, relating concentration and electric potential profiles in inhomogeneous interfacial regions, resulting from the inclusion of the contributions from ion-ion interaction energy terms to the limiting expression for the chemical potential of an ion are presented. The analysis of these equations is shown to imply the necessary existence of a dielectric profile in the inhomogeneous region. The resulting relations between the surface charge density and the electric potential difference for the system are presented with illustrative numerical calculations. The magnitude of the inhomogeneous region is shown to extend up to 23 Å for a uni-normal electrolyte solution and to 75 Å for a 0.1 N solution of electrolytes, when the surface potential is about −50 mV with respect to the bulk solution. The calculation of the electric potential profile, the charge density profile and the concentration profile based on the assumed validity of an expression for dielectric profile is included.     

Surface tension of aqueous electrolyte solutions. Thermodynamics

A thermodynamic theory is developed for obtaining the enthalpic and entropic contributions to the surface excess Gibbs energy of electrolyte solutions from the dependence of the surface tension on concentration and temperature. For elaboration, accurate activity coefficients in solution as functions of concentration and temperature are required. The theory is elaborated for (1-1) electrolytes and applied to HClO(4), HNO(3), NaCl, NaBr, and LiCl, of which the first two adsorb positively and the other three negatively. One of the conspicuous outcomes is that in all cases, the surface excess entropies slightly decrease with electrolyte activity but remain close to that of pure water, whereas the enthalpy is different from that. The implication is that the driving force for positive or negative adsorption must have an enthalpic origin. This finding can be useful in developing and evaluating theoretical models for the interpretation of surface tensions of electrolyte solutions.     

Modified Protein-Water Interactions in CHARMM36m for Thermodynamics and Kinetics of Proteins in Dilute and Crowded Solutions

Proper balance between protein-protein and protein-water interactions is vital for atomistic molecular dynamics (MD) simulations of globular proteins as well as intrinsically disordered proteins (IDPs). The overestimation of protein-protein interactions tends to make IDPs more compact than those in experiments. Likewise, multiple proteins in crowded solutions are aggregated with each other too strongly. To optimize the balance, Lennard-Jones (LJ) interactions between protein and water are often increased about 10% (with a scaling parameter, λ = 1.1) from the existing force fields. Here, we explore the optimal scaling parameter of protein-water LJ interactions for CHARMM36m in conjunction with the modified TIP3P water model, by performing enhanced sampling MD simulations of several peptides in dilute solutions and conventional MD simulations of globular proteins in dilute and crowded solutions. In our simulations, 10% increase of protein-water LJ interaction for the CHARMM36m cannot maintain stability of a small helical peptide, (AAQAA)₃ in a dilute solution and only a small modification of protein-water LJ interaction up to the 3% increase (λ = 1.03) is allowed. The modified protein-water interactions are applicable to other peptides and globular proteins in dilute solutions without changing thermodynamic properties from the original CHARMM36m. However, it has a great impact on the diffusive properties of proteins in crowded solutions, avoiding the formation of too sticky protein-protein interactions.     

Iso-energy cutoff for the calculation of interionic potential of mean force in water

An iso-energy cutoff scheme is introduced for the calculation of the potential of mean force between two ions in water. The cutoff criterion is based on the optimal interaction of the water dipole with the ion pair, for which analytical expressions are derived. Formulas are also derived to characterize the solvent reorganization contribution to the potential of mean force. Treatment of the contributions from waters outside the cutoff is also discussed. © 1994 John Wiley & Sons, Inc.     

The Molten Globule State of a Globular Protein in a Cell Is More or Less Frequent Case Rather than an Exception

Quite a long time ago, Oleg B. Ptitsyn put forward a hypothesis about the possible functional significance of the molten globule (MG) state for the functioning of proteins. MG is an intermediate between the unfolded and the native state of a protein. Its experimental detection and investigation in a cell are extremely difficult. In the last decades, intensive studies have demonstrated that the MG-like state of some globular proteins arises from either their modifications or interactions with protein partners or other cell components. This review summarizes such reports. In many cases, MG was evidenced to be functionally important. Thus, the MG state is quite common for functional cellular proteins. This supports Ptitsyn’s hypothesis that some globular proteins may switch between two active states, rigid (N) and soft (MG), to work in solution or interact with partners.     

Electrolyte distribution around two like-charged rods: their effective attractive interaction and angular dependent charge reversal

A simple model for two like-charged parallel rods immersed in an electrolyte solution is considered. We derived the three point extension (TPE) of the hypernetted chain/mean spherical approximation (TPE-HNC/MSA) and Poisson-Boltzmann (TPE-PB) integral equations. We numerically solve these equations and compare them to our results of Monte Carlo (MC) simulations. The effective interaction force, F(T), the charge distribution profiles, rho(el)(x,y), and the angular dependent integrated charge function, P(theta), are calculated for this system. The analysis of F(T) is carried out in terms of the electrostatic and entropic (depletion) contributions, F(E) and F(C). We studied several cases of monovalent and divalent electrolytes, for which the ionic size and concentration are varied. We find good qualitative agreement between TPE-HNC/MSA and MC in all the cases studied. The rod-rod force is found to be attractive when immersed in large size, monovalent or divalent electrolytes. In general, the TPE-PB has poor agreement with the MC. For large monovalent and divalent electrolytes, we find angular dependent charge reversal charge inversion and polarizability. We discuss the intimate relationship between this angular dependent charge reversal and rod-rod attraction.     

Molecular dynamics investigation of ferrous-ferric electron transfer in a hydrolyzing aqueous solution: calculation of the pH dependence of the diabatic transfer barrier and the potential of mean force

We present a molecular model for ferrous-ferric electron transfer in an aqueous solution that accounts for electronic polarizability and exhibits spontaneous cation hydrolysis. An extended Lagrangian technique is introduced for carrying out calculations of electron-transfer barriers in polarizable systems. The model predicts that the diabatic barrier to electron transfer increases with increasing pH, due to stabilization of the Fe3+ by fluctuations in the number of hydroxide ions in its first coordination sphere, in much the same way as the barrier would increase with increasing dielectric constant in the Marcus theory. We have also calculated the effect of pH on the potential of mean force between two hydrolyzing ions in aqueous solution. As expected, increasing pH reduces the potential of mean force between the ferrous and ferric ions in the model system. The magnitudes of the predicted increase in diabatic transfer barrier and the predicted decrease in the potential of mean force nearly cancel each other at the canonical transfer distance of 0.55 nm. Even though hydrolysis is allowed in our calculations, the distribution of reorganization energies has only one maximum and is Gaussian to an excellent approximation, giving a harmonic free energy surface in the reorganization energy F(DeltaE) with a single minimum. There is thus a surprising amount of overlap in electron-transfer reorganization energies for Fe(2+)-Fe(H2O)6(3+), Fe(2+)-Fe(OH)(H2O)5(2+), and Fe(2+)-Fe(OH)2(H2O)+ couples, indicating that fluctuations in hydrolysis state can be viewed on a continuum with other solvent contributions to the reorganization energy. There appears to be little justification for thinking of the transfer rate as arising from the contributions of different hydrolysis states. Electronic structure calculations indicate that Fe(H2O)6(2+)-Fe(OH)n(H2O)(6-n)(3-n)+ complexes interacting through H3O2- bridges do not have large electronic couplings.