Charge Regulation Triggers Condensation of Short Oligopeptides to Polyelectrolytes.

Electrostatic interactions between charged macromolecules are ubiquitous in biological systems, and they are important also in materials design. Attraction between oppositely charged molecules is often interpreted as if the molecules had a fixed charge, which is not affected by their interaction. Less commonly, charge regulation is invoked to interpret such interactions, i.e., a change of the charge state in response to a change of the local environment. Although some theoretical and simulation studies suggest that charge regulation plays an important role in intermolecular interactions, experimental evidence supporting such a view is very scarce. In the current study, we used a model system, composed of a long polyanion interacting with cationic oligolysines, containing up to 8 lysine residues. We showed using both simulations and experiments that while these lysines are only weakly charged in the absence of the polyanion, they charge up and condense on the polycations if the pH is close to the pKa of the lysine side chains. We show that the lysines coexist in two distinct populations within the same solution: (1) practically nonionized and free in solution; (2) highly ionized and condensed on the polyanion. Using this model system, we demonstrate under what conditions charge regulation plays a significant role in the interactions of oppositely charged macromolecules and generalize our findings beyond the specific system used here.

Sequestration of Small Ions and Weak Acids and Bases by a Polyelectrolyte Complex Studied by Simulation and Experiment.

Mixing of oppositely charged polyelectrolytes can result in phase separation into a polymer-poor supernatant and a polymer-rich polyelectrolyte complex (PEC). We present a new coarse-grained model for the Grand-reaction method that enables us to determine the composition of the coexisting phases in a broad range of pH and salt concentrations. We validate the model by comparing it to recent simulations and experimental studies, as well as our own experiments on poly(acrylic acid)/poly(allylamine hydrochloride) complexes. The simulations using our model predict that monovalent ions partition approximately equally between both phases, whereas divalent ones accumulate in the PEC phase. On a semiquantitative level, these results agree with our own experiments, as well as with other experiments and simulations in the literature. In the sequel, we use the model to study the partitioning of a weak diprotic acid at various pH values of the supernatant. Our results show that the ionization of the acid is enhanced in the PEC phase, resulting in its preferential accumulation in this phase, which monotonically increases with the pH. Currently, this effect is still waiting to be confirmed experimentally. We explore how the model parameters (particle size, charge density, permittivity, and solvent quality) affect the measured partition coefficients, showing that fine-tuning of these parameters can make the agreement with the experiments almost quantitative. Nevertheless, our results show that charge regulation in multivalent solutes can potentially be exploited in engineering the partitioning of charged molecules in PEC-based systems at various pH values.

Explaining Giant Apparent pK_{a} Shifts in Weak Polyelectrolyte Brushes.

Recent experiments on weak polyelectrolyte brushes found marked shifts in the effective pK_{a} that are linear in the logarithm of the salt concentration. Comparing explicit-particle simulations with mean-field calculations we show that for high grafting densities the salt concentration effect can be explained using the ideal Donnan theory, but for low grafting densities the full shift is due to a combination of the Donnan effect and the polyelectrolyte effect. The latter originates from electrostatic correlations that are neglected in the Donnan picture and that are only approximately included in the mean-field theory. Moreover, we demonstrate that the magnitude of the polyelectrolyte effect is almost invariant with respect to salt concentration but depends on the grafting density of the brush. This invariance is due to a complex cancellation of multiple effects. Based on our results, we show how the experimentally determined pK_{a} shifts may be used to infer the grafting density of brushes, a parameter that is difficult to measure directly.

Reply to the 'Comment on "Simulations of ionization equilibria in weak polyelectrolyte solutions and gels"' by J. Landsgesell, L. Nová, O. Rud, F. Uhlík, D. Sean, P. Hebbeker, C. Holm and P. Kosovan, Soft Matter, 2019, 15, 1155-1185.

Levin and Bakhshandeh suggested in their comment that (1), we stated in our recent review that pH-pKA is a universal parameter for titrating systems, that (2), we omitted to mention in our review the broken symmetry of the constant pH algorithm, and that (3), a constant pH simulation must include a grand-canonical exchange of ions with the reservoir. As a reply to (1), we point out that Levin and Bakhshandeh misquoted and hence invalidated our original statement. We therefore explain in detail under which circumstances pH-pKA can be a universal parameter, and also demonstrate why their numerical example is not in contradiction to our statement. Moreover, the fact that pH-pKA is not a universal parameter for titrating systems is well known in the pertinent literature. Regarding (2), we admit that the symmetry-breaking feature of the constant pH algorithm has escaped our attention at the time of writing the review. We added some clarifying remarks to this behavior. Concerning (3), we point out that the grand-canonical coupling and the resultant Donnan potential are not features of single-phase systems, but are essential for two-phase systems, as was shown in a recent paper by some of us, see J. Landsgesell et al., Macromolecules, 2020, 53, 3007-3020.

Anionically Functionalized Glycogen Encapsulates Melittin by Multivalent Interaction.

We developed acid-functionalized glycogen conjugates as supramolecular carriers for efficient encapsulation and inhibition of a model cationic peptide melittin─the main component of honeybee venom. For this purpose, we synthesized and characterized a set of glycogens, functionalized to various degrees by several different acid groups. These conjugates encapsulate melittin up to a certain threshold amount, beyond which they precipitate. Computer simulations showed that sufficiently functionalized conjugates electrostatically attract melittin, resulting in its efficient encapsulation in a broad pH range around the physiological pH. Hemolytic assays confirmed in vitro that the effective inhibition of melittin's hemolytic activity occurs for highly functionalized samples, whereas no inhibition is observed when using low-functionalized conjugates. It can be concluded that functional glycogens are promising carriers for cationic molecular cargos or antidotes against animal venoms under conditions, in which suitable properties such as biodegradability and biocompatibility are crucial.

Both Charge-Regulation and Charge-Patch Distribution Can Drive Adsorption on the Wrong Side of the Isoelectric Point.

The mechanism of protein-polyelectrolyte complexation on the wrong side of the isoelectric point has long puzzled researchers. Two alternative explanations have been proposed in the literature: (a) the charge-patch (CP) mechanism, based on the inhomogeneous distribution of charges on the protein, and (b) the charge-regulation (CR) mechanism, based on the variable charge of weak acid and base groups, which may invert the protein charge in the presence of another highly charged object. To discern these two mechanisms, we simulated artificially constructed short peptides, containing acidic and basic residues, arranged in a blocklike or alternating sequence. Our simulations of these peptides, interacting with polyelectrolytes, showed that charge patch and charge regulation alone can both lead to adsorption on the wrong side of the pI value. Their simultaneous presence enhances adsorption, whereas their absence prevents adsorption. Our simulation results were rationalized by following the variation of the charge regulation capacity and dipole moments of these peptides with the pH. Specifically for lysozyme, we found that charge patch prevails at physiological pH, whereas charge regulation prevails near the pI, thereby explaining seemingly contradicting conclusions in the literature. By applying the same approach to other proteins, we developed a general framework for assessing the role of the CP and CR mechanisms in existing case studies and for predicting how various proteins interact with polyelectrolytes at different pH values.

Reversible multilayered vesicle-like structures with fluid hydrophobic and interpolyelectrolyte layers.

Hydrophobic blocks of amphiphilic block copolymers often form glassy micellar cores at room temperature with a rigid structure that limits their applications as nanocapsules for targeted delivery. Nevertheless, we prepared and analyzed core/shell micelles with a soft core, formed by a self-assembled block copolymer consisting of a hydrophobic block and a polycation block, poly(lauryl acrylate)-block-poly(trimethyl-aminoethyl acrylate) (PLA-QPDMAEA), in aqueous solution. By light and small-angle neutron scattering, by transmission electron microscopy and by fluorescence spectroscopy, we showed that these core/shell micelles are spherical and cylindrical with a fluid-like PLA core and a positively charged outer shell and that they can encapsulate and release hydrophobic solutes. Moreover, after mixing these PLA-QPDMAEA core/shell micelles with another diblock copolymer, consisting of a hydrophilic block and a polyanion block, namely poly(ethylene oxide)-block-poly(methacrylic acid) (PEO-PMAA), we observed the formation of novel vesicle-like multicompartment structures containing both soft hydrophobic and interpolyelectrolyte (IPEC) layers. By combining small-angle neutron scattering with self-consistent field modeling, we confirmed the formation of these complex vesicle-like structures with a swollen PEO core, an IPEC inner layer, a PLA soft layer, an IPEC outer layer and a loose PEO corona. Thus, these multicompartment micelles with fluid and IPEC layers and a hydrophilic corona may be used as nanocapsules with several tunable properties, including the ability to control the thickness of each layer, the charge of the IPEC layers and the stability of the micelles, to deliver both hydrophobic and multivalent solutes.

Role of pKA in Charge Regulation and Conformation of Various Peptide Sequences.

Peptides containing amino acids with ionisable side chains represent a typical example of weak ampholytes, that is, molecules with multiple titratable acid and base groups, which generally exhibit charge regulating properties upon changes in pH. Charged groups on an ampholyte interact electrostatically with each other, and their interaction is coupled to conformation of the (macro)molecule, resulting in a complex feedback loop. Their charge-regulating properties are primarily determined by the pKA of individual ionisable side-chains, modulated by electrostatic interactions between the charged groups. The latter is determined by the amino acid sequence in the peptide chain. In our previous work we introduced a simple coarse-grained model of a flexible peptide. We validated it against experiments, demonstrating its ability to quantitatively predict charge on various peptides in a broad range of pH. In the current work, we investigated two types of peptide sequences: diblock and alternating, each of them consisting of an equal number of amino acids with acid and base side-chains. We showed that changing the sequence while keeping the same overall composition has a profound effect on the conformation, whereas it practically does not affect total charge on the peptide. Nevertheless, the sequence significantly affects the charge state of individual groups, showing that the zero net effect on the total charge is a consequence of unexpected cancellation of effects. Furthermore, we investigated how the difference between the pKA of acid and base side chains affects the charge and conformation of the peptide, showing that it is possible to tune the charge-regulating properties by following simple guiding principles based on the pKA and on the amino acid sequence. Our current results provide a theoretical basis for understanding of the complex coupling between the ionisation and conformation in flexible polyampholytes, including synthetic polymers, biomimetic materials and biological molecules, such as intrinsically disordered proteins, whose function can be regulated by changes in the pH.

Computer modeling of polymer stars in variable solvent conditions: a comparison of MD simulations, self-consistent field (SCF) modeling and novel hybrid Monte Carlo SCF approach.

Computer-aided modeling is a systematic approach to grasp the physics of macromolecules, but it remains essential to know when to trust the results and when not. For a polymer star, we consider three approaches: (i) Molecular Dynamics (MD) simulations and implementing a coarse-grained model, (ii) the self-consistent field approach based on a mean-field approximation and implementing the lattice model due to Scheutjens and Fleer (SF-SCF) and (iii) novel hybrid Monte Carlo self-consistent field (MC-SCF) method, which combines a coarse-grained model driven by a Monte Carlo method and a mean-field representation driven by SF-SCF. We compare the performance of these approaches under a wide range of solvent qualities. The MD approach is formally the most exact but suffers from reasonable convergence. The mean-field approach works similarly in all solvent qualities but is quantitatively least accurate. The MC-SCF hybrid allows us to combine the benefits of the simulation route and the effective performance of SCF. We consider the center-to-end distance Rce, the radius of gyration Rg2 of the star and the polymer density profiles φ(r) of polymer-segments in it. All three methods show a good qualitative agreement one to another. The MC-SCF method is in good agreement with the scaling predictions in the whole range of solvent quality values showing that it grasps the essential physics while remaining computationally in bounds.

Multivalent counterions accumulate in star-like polyelectrolytes and collapse the polymer in spite of increasing its ionization.

We used computer simulations to explore the dissociative and conformational behaviour of branched weak polyelectrolytes with multivalent counterions. We compared simulated titration curves and chain sizes in the presence of added salt of various valencies, keeping the total charge of salt constant. We showed that multivalent counterions enhance ionization of the weak polyelectrolytes, in spite of collapsing of the chains. We provided evidence that such an effect is absent in systems with only monovalent counterions at the same ionic strength, and thus cannot be attributed to electrostatic screening. We attributed it to strong ion-ion correlations that we quantified by comparing potentials of mean force with the mean electrostatic potentials. Finally, we used the partition coefficient to quantify the ability of star-like polyelectrolytes to capture multivalent ions, that is important for water-treatment applications. Our work provides fundamental understanding of the mechanism of polyelectrolyte collapse and ionization response upon addition of multivalent ions.

Interactions of star-like polyelectrolyte micelles with hydrophobic counterions.

Hydrophobicity of a counterion has a profound effect on the interaction with polyelectrolytes similar to that of multivalency. Specifically, understanding this interaction in weak polyelectrolyte micelles might assist in developing nanocarriers for pH-controlled encapsulation and release. We used star-like weak polyelectrolyte micelles of polystyrene-block-poly(2-vinyl pyridine) (PS-P2VP) with fixed aggregation number as a model polyelectrolyte, and cobalt bis(1,2-dicarbollide) (COSAN) as a model hydrophobic anion. We used NMR to assess the mobility of the polymer segments in the presence of varying amounts of COSAN, and at varying protonation degrees of the polyelectrolyte. Same experiments with indifferent electrolyte (NaCl) were used as a control. Furthermore, we used coarse-grained simulations to obtain a detailed picture of the effect of hydrophobic counterions on the conformation of the micelles. A small amount of hydrophobic counterions causes morphological changes within the micelles, whereas a bigger amount causes precipitation. This was confirmed both in simulations and in experiments. Furthermore, adsorption of the counterions induces ionization of the collapsed segments of the polyelectrolyte. Although the COSAN/P2VP system is rather specific, the generic model used in the coarse-grained simulations shows that the observed behavior is a consequence of synergy of hydrophobic and electrostatic attraction between polyelectrolytes and hydrophobic counterions. Our study provides general insights into the molecular mechanisms of these interactions.

Simulations of ionization equilibria in weak polyelectrolyte solutions and gels.

This article recapitulates the state of the art regarding simulations of ionization equilibria of weak polyelectrolyte solutions and gels. We start out by reviewing the essential thermodynamics of ionization and show how the weak polyelectrolyte ionization differs from the ionization of simple weak acids and bases. Next, we describe simulation methods for ionization reactions, focusing on two methods: the constant-pH ensemble and the reaction ensemble. After discussing the advantages and limitations of both methods, we review the existing simulation literature. We discuss coarse-grained simulations of weak polyelectrolytes with respect to ionization equilibria, conformational properties, and the effects of salt, both in good and poor solvent conditions. This is followed by a discussion of branched star-like weak polyelectrolytes and weak polyelectrolyte gels. At the end we touch upon the interactions of weak polyelectrolytes with other polymers, surfaces, nanoparticles and proteins. Although proteins are an important class of weak polyelectrolytes, we explicitly exclude simulations of protein ionization equilibria, unless they involve protein-polyelectrolyte interactions. Finally, we try to identify gaps and open problems in the existing simulation literature, and propose challenges for future development.

Local pH and Effective pK of a Polyelectrolyte Chain: Two Names for One Quantity?

In recent experiments, the "local pH" near polyelectrolyte chains was determined from the shift in the effective acidity constant of fluorescent pH indicators attached to the macromolecules. This indirect determination raises the question if the analyzed quantity was indeed the "local pH" and what this term actually means. In this study, we combined experiments and simulations to demonstrate that the shift in ionization constant is slightly lower than the difference between the pH and the "local pH". This offset is caused by correlations between fluctuations in chain conformation, small-ion distribution, and fluorophore ionization.

Local pH and effective pKA of weak polyelectrolytes - insights from computer simulations.

In this work we study the titration behavior of weak polyelectrolytes by computer simulations. We analyze the local pH near the chains at various conditions and provide molecular-level insight which complements the recent experimental determination of this quantity. Next, we analyze the non-ideal titration behaviour of weak polyelectrolytes in solution, calculate the effective ionization constant and compare the simulation results with theoretical predictions. In contrast with the universal behaviour with respect to chain length, we find non-universality and deviations from theory with respect to polymer concentration and permittivity of the solvent. The latter we explain in terms of counterion condensation and ion correlation effects, which lead to reversal of the non-ideal titration behaviour at very low permittivities. We discuss the impact of these findings on the interpretation of experimental results.

A self-consistent mean-field model for polyelectrolyte gels.

We present a novel approach to modeling polyelectrolyte gels, exploiting the analogy between star-branched polymers and polymer networks as a computationally inexpensive yet reliable alternative to full-scale simulations. In the numerical mean-field model of a star-like polymer we modify the boundary conditions to represent an infinite network. We validate the predictions of our new model against a coarse-grained simulation model. We also validate it against a phenomenological analytical model which has been previously shown to agree with simulations in a limited range of parameters. The mean-field model explicitly considers local density gradients and agrees with the simulation results in a broad range of parameters, beyond that of the analytical model. Finally, we use the mean-field model for predictions of the swelling behaviour of weak polyelectrolyte gels under different pH conditions. We demonstrate that the local density gradients are important and that the ionization of the weak polyelectrolyte gel is significantly suppressed. Under the studied conditions the effective pKA is about one unit higher than that of the free monomer. This shift in the effective pKA stems from the different pH values inside and outside the gel.

Charge-controlled nano-structuring in partially collapsed star-shaped macromolecules.

Hydrophobic polyelectrolytes exhibit intra-molecular nano-scale self-organization instead of macroscopic phase separation because of the interplay between short-range hydrophobic attraction and long-range electrostatic repulsion. We aim to unravel how the morphology of the intra-molecular nanostructures can be controlled through the topology of the macromolecule on one hand and by adjustable ionization on the other hand. Specifically, we focus on hydrophobic star-branched polyelectrolytes, composed of either strong or weak acidic monomers. While both collapse in a globule when uncharged, and expand to full stretching of arms at high ionization, they exhibit quite different intermediate scenarios. For the strong ones, we observe the formation of bundles of arms as the main structural motif, and for the weak ones the intramolecular micelle-like structure is found at the same overall charge of the macromolecule. Here intramolecular disproportionation leaves some arms in a collapsed virtually neutral core, while others are substantially ionized and stretched in the corona.

Mean squared displacement from fluorescence correlation spectroscopy.

Under certain conditions, the mean squared displacement (MSD) can be retrieved from fluorescence correlation spectroscopy (FCS) measurements. However, in the general case this procedure is not valid, and the apparent MSD obtained from FCS data may substantially differ from the true one. In this work we discuss under which conditions this procedure can be applied. Furthermore, we use computer simulations to obtain the MSD and the apparent MSD for the diffusion of a single polymer chain under various approximations. Based on the simulation results we discuss the reliability of the apparent MSD obtained from FCS, showing that it systematically deviates from the true MSD. We also propose a general procedure to verify the reliability of the apparent MSD by measurements at various focal spot sizes.

The drag of the tails: Diffusion of sticky nanoparticles in dilute polymer solutions.

We devise a simple coarse-grained model of a nanoparticle exhibiting attractive interactions with a polymer in good solvent and use Monte Carlo simulations to study how adsorption of the polymer affects the hydrodynamic properties of the nanoparticle. We show that the hydrodynamic radius of the polymer-decorated nanoparticle increases with increasing polymer chain length or concentration. The slowdown of diffusion of the sticky nanoparticles is predicted to occur at polymer concentrations many orders of magnitude below the overlap concentration, in contrast with the case of non-sticky nanoparticles. To rationalize our findings, we employ the concept of trains, loops, and tails, which has been used in earlier theoretical studies of polymer adsorption at interfaces. We show that dominant contribution to the increase of the hydrodynamic radius of the polymer-decorated nanoparticles comes from the tails, which stretch far from the surface. On the contrary, the much more numerous but shorter loops and tails play only a minor role.

Coarse-grained simulations of an ionic liquid-based capacitor: II. Asymmetry in ion shape and charge localization.

In this work, which is a continuation of part I, we introduce a primitive model for an ionic liquid (IL) that can account for the planar shape of cations typical for ILs like imidazolium. The model consists of a spherical anion and a triangular cation consisting of three spheres, where one or all three vertices of the triangle can carry electric charge. We use molecular dynamics simulations to study the differential capacitance Cd of an ionic liquid confined between two planar electrodes. Our goal is to elucidate the complex dependence of Cd on the electrode potential U in terms of simple entities such as the shape and charge distribution of the ions. For this purpose, we compare the results from the current model to the results based on the models with spherical cations that possess asymmetry in ion valence and shape that were analyzed in detail in part I of this work. We show that the various possible stackings of the triangles near the cathode lead to noticeable new features in Cd(U) as compared to the spherical models. Different distributions of charges on the triangle lead to different preferred orientations of the cations near the cathode that are moreover potential dependent.

Coarse-grained simulations of an ionic liquid-based capacitor: I. Density, ion size, and valency effects.

We introduce a hierarchy of generic coarse-grained models of ionic liquids of increasing complexity. We use them in molecular dynamics simulations to study the differential capacitance of a capacitor consisting of an ionic liquid between two planar electrodes. The primary goal is to explain the complex dependence of the differential capacitance Cd on the electrode potential U in simple terms, e.g. in terms of the size and valency of the ions. For this purpose we introduce the symmetric model A, which qualitatively reproduces the Cd(U) dependence predicted by the mean-field theory but also reveals strong quantitative deviations. We further introduce size asymmetry in model A by increasing the cation size. In model B we vary the cation valency, keeping the sizes of both ions constant. We show that simultaneous increases in size and valency may compensate for each other, leading to a Cd(U) very similar to that for the symmetric case. We interpret distinct features in Cd(U) on the basis of the density profiles of the ions and charge density profiles. We focus on the first two ion layers at the electrode, and demonstrate that the polarization of the ionic liquid proceeds through replacement of one ion type by the other, in contrast to the simple increase in ion concentrations typical for dilute systems. The understanding gained for the simple models serves as a reference for interpretation of complex effects of ion size, valency and shape. This is carried through in part II (a separate article) where we show how the planar shape of ions in model C brings new features to the Cd(U) curve and also to the polarization mechanism.