Charge regulating macro-ions in salt solutions: screening properties and electrostatic interactions

On the nature of screening in charge-regulated macroion solutions

We present a derivation of the screening length for a solution containing a charge-regulated macroion, e.g. protein, with its counterions. We show that it can be obtained directly from the second derivatives of the total free energy by taking recourse to the "uncertainty relation" of the Legendre transform, which connects the Hessians or the local curvatures of the free energy as a function of density and its Legendre transform, i.e., osmotic pressure, as a function of chemical potentials. Based on the Fowler-Guggenheim-Frumkin model of charge regulation, we then analyze the "screening resonance" and the "overscreening" of the screening properties of the charge-regulated macroion solution.

Electrostatic interactions between charge regulated spherical macroions

We study the interaction between two charge regulating spherical macroions with dielectric interior and dissociable surface groups immersed in a monovalent electrolyte solution. The charge dissociation is modelled via the Frumkin-Fowler-Guggenheim isotherm, which allows for multiple adsorption equilibrium states. The interactions are derived from the solutions of the mean-field Poisson-Boltzmann type theory with charge regulation boundary conditions. For a range of conditions we find symmetry breaking transitions from symmetric to asymmetric charge distribution exhibiting annealed charge patchiness, which results in like-charge attraction even in a univalent electrolyte-thus fundamentally modifying the nature of electrostatic interactions in charge-stabilized colloidal suspensions.

Curvature effects in interfacial acidity of amphiphilic vesicles

We analyze the changes in the vicinal acidity (pH) at a spherical amphiphilic membrane. The membrane is assumed to contain solvent accessible, embedded, dissociable, charge-regulated moieties. Basing our approach on the linear Debye-Hückel approximation, as well as on the nonlinear Poisson-Boltzmann theory, together with the general Frumkin-Fowler-Guggenheim adsorption isotherm model of the charge-regulation process, we analyze and review the dependence of the local pH on the position, as well as bulk electrolyte concentration, bulk pH, and curvature of the amphiphilic single membrane vesicle. With appropriately chosen adsorption parameters of the charge-regulation model, we find a good agreement with the available experimental data.

Charge fluctuations in charge-regulated systems: dependence on statistical ensemble

We investigate charge regulation of nanoparticles in concentrated suspensions, focusing on the effect of different statistical ensembles. We find that the choice of ensemble does not affect the mean charge of nanoparticles, but significantly alters the magnitude of its fluctuation. Specifically, we compared the behaviors of colloidal charge fluctuations in the semi-grand canonical and canonical ensembles and identified significant differences between the two. The choice of ensemble-whether the system is isolated or is in contact with a reservoir of acid and salt-will, therefore, affect the Kirkwood-Shumaker fluctuation-induced force inside concentrated suspensions. Our results emphasize the importance of selecting an appropriate ensemble that accurately reflects the experimental conditions when studying fluctuation-induced forces between polyelectrolytes, proteins, and colloidal particles in concentrated suspensions.

Unraveling fluctuation in gelatin and monovalent salt systems: coulombic starvation

Fluctuations play a key role in biological systems. Here, fluctuations in gelatin intensify with increasing salt concentration. We find a redistribution of hydrogen bonds in protein-salt systems due to unfulfilled hydration of the charges of gelatin and salt-ions, termed as coulombic starvation. This yielded three regions; no starvation, starvation of gelatin, and both gelatin-salt. The system reaches equilibrium with all charges being partially hydrated. This will aid in interpreting protein-metal ion interactions and designing biomaterials.

Theory of Charge Regulation of Colloidal Particles in Electrolyte Solutions

We present a theory that enables us to (i) calculate the effective surface charge of colloidal particles and (ii) efficiently obtain titration curves for different salt concentrations. The theory accounts for the shift of pH of solution due to the presence of 1:1 electrolyte. It also accounts self-consistently for the electrostatic potential produced by the deprotonated surface groups. To examine the accuracy of the theory, we have performed extensive reactive Monte Carlo simulations, which show excellent agreement between theory and simulations without any adjustable parameters.

Curvature effects in charge-regulated lipid bilayers

We formulate a theory of electrostatic interactions in lipid bilayer membranes where both monolayer leaflets contain dissociable moieties that are subject to charge regulation. We specifically investigate the coupling between membrane curvature and charge regulation of a lipid bilayer vesicle using both the linear Debye-Hückel (DH) and the non-linear Poisson-Boltzmann (PB) theory. We find that charge regulation of an otherwise symmetric bilayer membrane can induce charge symmetry breaking, non-linear flexoelectricity and anomalous curvature dependence of free energy. The pH effects investigated go beyond the paradigm of electrostatic renormalization of the mechano-elastic properties of membranes.

Where in the world are condensed counterions?

A scaling model of the concentration profiles of both condensed and free counterions is presented for solutions of spherical and cylindrical charged nanoparticles of different charge valences, nanoparticle sizes, and salt concentrations. The distribution of counterions for both spherical and cylindrical charged particles in salt-free solutions is determined by the condensation parameter γ₀ defined as the ratio of nanoparticle valence Z₀ to the number of Bjerrum lengths l_B = e²/(εkT) per nanoparticle size (γ₀ = Z₀l_B/(2r₀) for spherical nanoparticles with radii r₀ or γ₀ = Z₀l_B/L for cylindrical particles with length L), where ε is solution dielectric permittivity, e is elementary charge and kT is thermal energy. Depending on the magnitudes of the condensation parameter γ₀ and nanoparticle volume fraction ϕ, we find three qualitatively different regimes for the counterion distribution near charged particles: (i) weakly charged particles with no condensed counterions, (ii) regime of weak counterion condensation with less than half of the counterions condensed, and (iii) regime of strong counterion condensation with the majority of counterions condensed. The magnitude of electrostatic energy of a condensed counterion with respect to solution locations with zero electric field is larger than thermal energy kT, and the fraction of condensed counterions increases from less than half in the weak condensation regime to the majority of all counterions in the strong condensation regime. The condensed counterions are not bound to the nanoparticle surface but instead are localized within the condensed counterion zone near the charged particle. The thickness of the condensed counterion zone varies with the condensation parameter γ₀, the nanoparticle shape and volume fraction ϕ, and the salt concentration and can be as narrow as Bjerrum length (∼nm) or as large as the particle size (∼L the length of charged cylinder).

Dendritic polyelectrolytes with monovalent and divalent counterions: the charge regulation effect and counterion release

The charge regulation and the release of counterions are extremely important and substantial in determining the charge state of polyelectrolytes and the interaction between polyelectrolytes and proteins. Going beyond monovalent to multivalent cations, it is well-known that the effects of ions are qualitatively different. Therefore, the well-accepted descriptions of the charge regulation and the counterion release based on monovalent ions do not immediately apply to systems with multivalent ions. Here, we study the key structural and electrostatic features of charged dendrimers at hand of the pharmaceutically important dendritic polyglycerol sulfate (dPGS) macromolecule equilibrated with monovalent and divalent salts by molecular dynamics (MD) simulations. Following a simple but accurate scheme to determine its effective radius, the counterion condensed layer of the dPGS is determined with high accuracy and we observe the sequential replacement of condensed monovalent cations (MCs) to divalent cations (DCs) rendering a smaller dPGS effective charge versus the DC concentration. We resolve and track the release of counterions on the dPGS along its binding pathway with the plasma protein Human Serum Albumin (HSA). We find that the release of MCs remains favorable for the complexation leading to a considerable amount of release entropy as the driving force for complexation. The release of DCs only occurs above a certain DC concentration with a comparably smaller number of released ions than MCs. Its contribution to the binding free energy is small indicating a subtle cancellation between the entropy gain in releasing DCs and the enthalpy penalty from dissociating DCs from the dendrimer.

Phase Separation of Polyelectrolytes: The Effect of Charge Regulation

Complex coacervation, known as the liquid–liquid phase separation of solutions with oppositely charged polyelectrolytes, has attracted substantial interest in recent years. We study the effect of the charge regulation (CR) mechanism on the complex coacervation by including short-range interactions between the charged sites on the polymer chains as well as an association–dissociation energy parameter in the CR mechanism. We investigate the phase diagrams of two CR models: (i) the hopping CR model (HCR) and (ii) the asymmetric CR model (ACR). It is shown that during the phase separation that the polymers in the condensed phase are more charged than those in the dilute phase, in accordance with Le Chatelier’s principle. In addition, secondary CR effects also influence the change in the volume fraction of the two phases. The latter can cause the charge difference between the two phases to change nonmonotonically as a function of the CR parameters.

Accurate Determination of the Quantity and Spatial Distribution of Counterions around a Spherical Macroion

The accurate distribution of countercations (Rb⁺ and Sr²⁺ ) around a rigid, spherical, 2.9-nm size polyoxometalate cluster, {Mo₁₃₂ }^42- , is determined by anomalous small-angle X-ray scattering. Both Rb⁺ and Sr²⁺ ions lead to shorter diffuse lengths for {Mo₁₃₂ } than prediction. Most Rb⁺ ions are closely associated with {Mo₁₃₂ } by staying near the skeleton of {Mo₁₃₂ } or in the Stern layer, whereas more Sr²⁺ ions loosely associate with {Mo₁₃₂ } in the diffuse layer. The stronger affinity of Rb⁺ ions towards {Mo₁₃₂ } than that of Sr²⁺ ions explains the anomalous lower critical coagulation concentration of {Mo₁₃₂ } with Rb⁺ compared to Sr²⁺ . The anomalous behavior of {Mo₁₃₂ } can be attributed to majority of negative charges being located at the inner surface of its cavity. The longer anion-cation distance weakens the Coulomb interaction, making the enthalpy change owing to the breakage of hydration layers of cations more important in regulating the counterion-{Mo₁₃₂ } interaction.

Lattice-gas model of a charge regulated planar surface

In this work, we consider a lattice-gas model of charge regulation with electrostatic interactions within the Debye-Hückel level of approximation. In addition to long-range electrostatic interactions, the model incorporates the nearest-neighbor interactions for representing non-electrostatic forces between adsorbed ions. The Frumkin-Fowler-Guggenheim isotherm obtained from the mean-field analysis accurately reproduces the simulation data points.

Charge regulation of colloidal particles in aqueous solutions

We study the charge regulation of colloidal particles inside aqueous electrolyte solutions. To stabilize a colloidal suspension against precipitation, colloidal particles are synthesized with either acidic or basic groups on their surface. On contact with water, these surface groups undergo proton transfer reactions, resulting in colloidal surface charge. The charge is determined by the condition of local chemical equilibrium between hydronium ions inside the solution and at the colloidal surface. We use a model of Baxter sticky spheres to explicitly calculate the equilibrium dissociation constants and to construct a theory which is able to quantitatively predict the effective charge of colloidal particles with either acidic or basic surface groups. The predictions of the theory for the model are found to be in excellent agreement with the results of Monte Carlo simulations. This theory is further extended to treat colloidal particles with a mixture of both acidic and basic surface groups.

Critical behavior of charge-regulated macro-ions

Based on a collective description of electrolytes composed of charge-regulated macro-ions and simple salt ions, we analyze their equilibrium charge state in the bulk and their behavior in the vicinity of an external electrified surface. The mean-field formulation of mobile macro-ions in an electrolyte bathing solution is extended to include interactions between association and dissociation sites. We demonstrate that above a critical concentration of salt, similar to the critical micelle concentration, a non-trivial distribution of charge states sets in. Such a charge state can eventually lead to a liquid-liquid phase separation based on charge regulation.

Charge regulation radically modifies electrostatics in membrane stacks

Motivated by biological membrane-containing organelles in plants and photosynthetic bacteria, we study charge regulation in a model membrane stack. Considering (de)protonation as the simplest mechanism of charge equilibration between the membranes and with the bathing environment, we uncover a symmetry-broken charge state in the stack with a quasiperiodic effective charge sequence. In the case of a monovalent bathing salt solution our model predicts complex, inhomogeneous charge equilibria depending on the strength of the (de)protonation reaction, salt concentration, and membrane size. Our results shed light on the basic reorganization mechanism of thylakoid membrane stacks.

Complete Charge Regulation by a Redox Enzyme Upon Single Electron Transfer

The degree by which metalloproteins partially regulate net charge (Z) upon electron transfer (ET) was recently measured for the first time using "protein charge ladders" of azurin, cytochrome c, and myoglobin [Angew. Chem. Int. Ed. 2018, 57(19), 5364-5368; Angew. Chem. 2018, 130, 5462-5466]. Here, we show that Cu, Zn superoxide dismutase (SOD1) is unique among proteins in its ability to resist changes in net charge upon single ET (e.g., ΔZ_ET(SOD1) =0.05±0.08 per electron, compared to ΔZ_ET(Cyt-c) =1.19±0.02). This total regulation of net charge by SOD1 is attributed to the protonation of the bridging histidine upon copper reduction, yielding redox centers that are isoelectric at both copper oxidation states. Charge regulation by SOD1 would prevent long range coulombic perturbations to residue pK_a 's upon ET at copper, allowing SOD1's "electrostatic loop" to attract superoxide with equal affinity (at both redox states of copper) during diffusion-limited reduction and oxidation of superoxide.

Active volume regulation in adhered cells

Recent experiments reveal that the volume of adhered cells is reduced as their basal area is increased. During spreading, the cell volume decreases by several thousand cubic micrometers, corresponding to large pressure changes of the order of megapascals. We show theoretically that the volume regulation of adhered cells is determined by two concurrent conditions: mechanical equilibrium with the extracellular environment and a generalization of Donnan (electrostatic) equilibrium that accounts for active ion transport. Spreading affects the structure and hence activity of ion channels and pumps, and indirectly changes the ionic content in the cell. We predict that more ions are released from the cell with increasing basal area, resulting in the observed volume-area dependence. Our theory is based on a minimal model and describes the experimental findings in terms of measurable, mesoscale quantities. We demonstrate that two independent experiments on adhered cells of different types fall on the same master volume-area curve. Our theory also captures the measured osmotic pressure of adhered cells, which is shown to depend on the number of proteins confined to the cell, their charge, and their volume, as well as the ionic content. This result can be used to predict the osmotic pressure of cells in suspension.

Correlation Length in Concentrated Electrolytes: Insights from All-Atom Molecular Dynamics Simulations

We study the correlation length of the charge-charge pair correlations in concentrated electrolyte solutions by means of all-atom, explicit-solvent molecular dynamics simulations. We investigate LiCl and NaI in water, which constitute highly soluble, prototypical salts for experiments, as well as two more complex, molecular electrolyte systems of lithium bis(trifluoromethane)sulfonimide (LiTFSI), a salt commonly employed in electrochemical storage systems, in water, and in an organic solvent mixture of dimethoxyethane and dioxolane. Our simulations support the recent experimental observations as well as theoretical predictions of a nonmonotonic behavior of the correlation length with increasing salt concentration. We observe a Debye-Hückel like regime at low concentration, followed by a minimum reached when d/λ_D ≃ 1, where λ_D is the Debye correlation length and d is the effective ionic diameter, and an increasing correlation length with salt concentration in very concentrated electrolytes. As in the experiments, we find that the screening length in the concentrated regime follows a universal scaling law as a function d/λ_D for all studied salts. However, the scaling exponent is significantly lower than the experimentally measured one and lies in the range of the theoretical predictions based on much simpler electrolyte models.

One-dimensional Coulomb system in a sticky wall confinement: Exact results

This work investigates a one-component one-dimensional Coulomb system in sticky wall confinement. Sticky wall is introduced as an alternative and intuitive depiction of charge regulation, the notion that a surface charge is not a fixed but a fluctuating quantity in dynamic equilibrium with its immediate environment. Emphasis is placed on intuitive derivation and expressions are obtained by observing that the partition function of a charge regulated system can be decomposed into a collection of independent equilibriums with different fixed surface charges. Adsorbed particles behave as ideal-gas particles in a one-dimensional box whose length corresponds to the parameter of stickiness. Among various scenarios considered are a single- and two-wall confinement as well as the case of sticky counterions capable of associating into pairs. Exact solutions provide a view of the role and behavior of surface charge fluctuations, which is an important step in the "beyond-mean-field" analysis. Consequently, the model serves as a simple paradigm of the mechanism that gives rise to the Kirkwood-Shumaker interactions detected in real systems.

Charge Regulation of Colloidal Particles: Theory and Simulations

To explore charge regulation (CR) in physicochemical and biophysical systems, we present a model of colloidal particles with sticky adsorption sites which account for the formation of covalent bonds between the hydronium ions and the surface functional groups. Using this model and Monte Carlo simulations, we find that the standard Ninham and Parsegian (NP) theory of CR leads to results which deviate significantly from computer simulations. The problem with the NP approach is traced back to the use of a bulk equilibrium constant to account for surface chemical reactions. To resolve this difficulty we present a new theory of CR. The fundamental ingredient of the new approach is the sticky length, which is nontrivially related to the bulk equilibrium constant. The theory is found to be in excellent agreement with computer simulations, without any adjustable parameters. As an application of the theory we calculate the effective charge of colloidal particles containing carboxyl groups, as a function of pH and salt concentration.

What Are We Missing by Not Measuring the Net Charge of Proteins?

The net electrostatic charge (Z) of a folded protein in solution represents a bird's eye view of its surface potentials-including contributions from tightly bound metal, solvent, buffer, and cosolvent ions-and remains one of its most enigmatic properties. Few tools are available to the average biochemist to rapidly and accurately measure Z at pH≠pI. Tools that have been developed more recently seem to go unnoticed. Most scientists are content with this void and estimate the net charge of a protein from its amino acid sequence, using textbook values of pK_a . Thus, Z remains unmeasured for nearly all folded proteins at pH≠pI. When marveling at all that has been learned from accurately measuring the other fundamental property of a protein-its mass-one wonders: what are we missing by not measuring the net charge of folded, solvated proteins? A few big questions immediately emerge in bioinorganic chemistry. When a single electron is transferred to a metalloprotein, does the net charge of the protein change by approximately one elementary unit of charge or does charge regulation dominate, that is, do the pK_a values of most ionizable residues (or just a few residues) adjust in response to (or in concert with) electron transfer? Would the free energy of charge regulation (ΔΔG_z ) account for most of the outer sphere reorganization energy associated with electron transfer? Or would ΔΔG_z contribute more to the redox potential? And what about metal binding itself? When an apo-metalloprotein, bearing minimal net negative charge (e.g., Z=-2.0) binds one or more metal cations, is the net charge abolished or inverted to positive? Or do metalloproteins regulate net charge when coordinating metal ions? The author's group has recently dusted off a relatively obscure tool-the "protein charge ladder"-and used it to begin to answer these basic questions.

Role of metallic core for the stability of virus-like particles in strongly coupled electrostatics

Electrostatic interactions play important roles in the formation and stability of viruses and virus-like particles (VLPs) through processes that often involve added, or naturally occurring, multivalent ions. Here, we investigate the electrostatic or osmotic pressure acting on the proteinaceous shell of a generic model of VLPs, comprising a charged outer shell and a metallic nanoparticle core, coated by a charged layer and bathed in an aqueous electrolyte solution. Motivated by the recent studies accentuating the role of multivalent ions for the stability of VLPs, we focus on the effects of multivalent cations and anions in an otherwise monovalent ionic solution. We perform extensive Monte-Carlo simulations based on appropriate Coulombic interactions that consistently take into account the effects of salt screening, the dielectric polarization of the metallic core, and the strong-coupling electrostatics due to multivalent ions. We specifically study the intricate roles these factors play in the electrostatic stability of the model VLPs. It is shown that while the insertion of a metallic nanoparticle by itself can produce negative, inward-directed, pressure on the outer shell, addition of only a small amount of multivalent counterions can robustly engender negative pressures, enhancing the VLP stability across a wide range of values for the system parameters.