Simulations of ionization equilibria in weak polyelectrolyte solutions and gels

Theory and quantitative assessment of pH-responsive polyzwitterion-polyelectrolyte complexation

We introduce a theoretical framework to describe the pH-sensitive phase behavior of polyzwitterion-polyelectrolyte complex coacervates that reasonably captures the phenomenon from recent experimental observations. The polyzwitterion is described by a combinatorial sequence of the four states in which each zwitterionic monomer can occupy: dipolar, quasi-cationic, quasi-anionic, and fully neutralized. We explore the effects of various modifiable chemical and physical properties of the polymers-such as, pKa of the pH-active charged group on the zwitterion, equilibrium constant of salt condensation on the permanently charged group on the zwitterion, degrees of polymerization, hydrophobicity (via the Flory-Huggins interaction parameter), and dipole lengths-on the window of complexation across many stoichiometric mixing ratios of polyzwitterion and polyelectrolyte. The properties that determine the net charge of the polyzwitterion have the strongest effect on the pH range in which polyzwitterion-polyelectrolyte complexation occurs. We finish with general guidance for those interested in molecular design of polyzwitterion-polyelectrolyte complex coacervates and opportunities for future investigation.

Unexpected Two-Stage Swelling of Weak Polyelectrolyte Brushes with Divalent Counterions

We use particle-based, coarse-grained simulations to study the influence of divalent counterions on a weak polyelectrolyte brush. Our simulations show a profound influence of even small concentrations of divalent salt on the titration behavior of the brush, which is shown to be a combined effect of electrostatic interactions and the Donnan effect. Furthermore, we examine the partitioning of mono- and divalent counterions into the brush. We demonstrate the preferred uptake of divalent ions by the brush, which is further enhanced by electrostatic correlation effects. Finally, our simulations reveal a hitherto unobserved two-stage swelling of the brush as a function of the pH in the presence of divalent salt. This phenomenon arises as a consequence of charge regulation and ion partitioning.

Charge regulation of nanoparticles in the presence of multivalent electrolytes

We explore the charge regulation (CR) of spherical nanoparticles immersed in an asymmetric electrolyte of a specified pH. Using a recently developed reactive canonical Monte Carlo (MC) simulation method, titration isotherms are obtained for suspensions containing monovalent, divalent, and trivalent coions. A theory based on the modified Poisson-Boltzmann approximation, which incorporates the electrostatic ion solvation free energy and discrete surface charge effects, is used to compare with the simulation results. A remarkably good agreement is found without any fitting parameters, both for the ion distributions and titration curves, suggesting that ionic correlations between coions and hydronium ions at the nanoparticle surface play only a minor role in determining the association equilibrium between hydroniums and the functional sites on the nanoparticle surface. On the other hand, if suspension contains multivalent counterions, we observe a large deviation between theory and simulations, showing that the electrostatic correlations between counterions and hydronium ions at the nanoparticle surface are very significant and must be properly taken into account to correctly describe CR for such solutions.

The explicit bonding reaction ensemble Monte Carlo method

We present the explicit bonding Reaction ensemble Monte Carlo (eb-RxMC) method, designed to sample reversible bonding reactions in macromolecular systems in thermodynamic equilibrium. Our eb-RxMC method is based on the reaction ensemble method; however, its implementation differs from the latter by the representation of the reaction. In the eb-RxMC implementation, we are adding or deleting bonds between existing particles, instead of inserting or deleting particles with different chemical identities. This new implementation makes the eb-RxMC method suitable for simulating the formation of reversible linkages between macromolecules, which would not be feasible with the original implementation. To enable coupling of our eb-RxMC algorithm with molecular dynamics algorithm for the sampling of the configuration space, we biased the sampling of reactions only within a certain inclusion radius. We validated our algorithm using a set of ideally behaving systems undergoing dimerization and polycondensation reactions, for which analytical results are available. For dimerization reactions with various equilibrium constants and initial compositions, the degree of conversion measured in our simulations perfectly matched the reference values given by the analytical equations. We also showed that this agreement is not affected by the arbitrary choice of the inclusion radius or the stiffness of the harmonic bond potential. Next, we showed that our simulations can correctly match the analytical results for the distribution of the degree of polymerization and end-to-end distance of ideal chains in polycondensation reactions. Altogether, we demonstrated that our eb-RxMC simulations correctly sample both reaction and configuration spaces of these reference systems, opening the door to future simulations of more complex interacting macromolecular systems.

Molecular control via dynamic bonding enables material responsiveness in additively manufactured metallo-polyelectrolytes

Metallo-polyelectrolytes are versatile materials for applications like filtration, biomedical devices, and sensors, due to their metal-organic synergy. Their dynamic and reversible electrostatic interactions offer high ionic conductivity, self-healing, and tunable mechanical properties. However, the knowledge gap between molecular-level dynamic bonds and continuum-level material properties persists, largely due to limited fabrication methods and a lack of theoretical design frameworks. To address this critical gap, we present a framework, combining theoretical and experimental insights, highlighting the interplay of molecular parameters in governing material properties. Using stereolithography-based additive manufacturing, we produce durable metallo-polyelectrolytes gels with tunable mechanical properties based on metal ion valency and polymer charge sparsity. Our approach unveils mechanistic insights into how these interactions propagate to macroscale properties, where higher valency ions yield stiffer, tougher materials, and lower charge sparsity alters material phase behavior. This work enhances understanding of metallo-polyelectrolytes behavior, providing a foundation for designing advanced functional materials.

pyMBE: The Python-based molecule builder for ESPResSo

We present the Python-based Molecule Builder for ESPResSo (pyMBE), an open source software application to design custom coarse-grained (CG) models, as well as pre-defined models of polyelectrolytes, peptides, and globular proteins in the Extensible Simulation Package for Research on Soft Matter (ESPResSo). The Python interface of ESPResSo offers a flexible framework, capable of building custom CG models from scratch. As a downside, building CG models from scratch is prone to mistakes, especially for newcomers in the field of CG modeling, or for molecules with complex architectures. The pyMBE module builds CG models in ESPResSo using a hierarchical bottom-up approach, providing a robust tool to automate the setup of CG models and helping new users prevent common mistakes. ESPResSo features the constant pH (cpH) and grand-reaction (G-RxMC) methods, which have been designed to study chemical reaction equilibria in macromolecular systems with many reactive species. However, setting up these methods for systems, which contain several types of reactive groups, is an error-prone task, especially for beginners. The pyMBE module enables the automatic setup of cpH and G-RxMC simulations in ESPResSo, lowering the barrier for newcomers and opening the door to investigate complex systems not studied with these methods yet. To demonstrate some of the applications of pyMBE, we showcase several case studies where we successfully reproduce previously published simulations of charge-regulating peptides and globular proteins in bulk solution and weak polyelectrolytes in dialysis. The pyMBE module is publicly available as a GitHub repository (https://github.com/pyMBE-dev/pyMBE), which includes its source code and various sample and test scripts, including the ones that we used to generate the data presented in this article.

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.

Graphene oxide-DNA/graphene oxide-PDDA sandwiched membranes with neuromorphic function

The behavior of polyelectrolytes in confined spaces has direct relevance to the protein mediated ion transport in living organisms. In this paper, we govern lithium chloride transport by the interface provided by polyelectrolytes, polycation, poly(diallyldimethylammonium chloride) (PDDA) and, polyanion, double stranded deoxyribonucleic acid (dsDNA), in confined graphene oxide (GO) membranes. Polyelectrolyte-GO interfaces demonstrate neuromorphic functions that were successfully applied with nanochannel ion interactions contributed, resulting in ion memory effects. Excitatory and inhibitory post-synaptic currents were tuned continuously as the number of pulses applied increased accordingly, increasing decay times. Furthermore, we demonstrated the short-term memory of a trained vs untrained device in computation. On account of its simple and safe production along with its robustness and stability, we anticipate our device to be a low dimensional building block for arrays to embed artificial neural networks in hardware for neuromorphic computing. Additionally, incorporating such devices with sensing and actuating parts for a complete feedback loop produces robotics with its own ability to learn by modifying actuation based on sensing data.

Dilute polyelectrolyte solutions: recent progress and open questions

Polyelectrolytes are a class of polymers possessing ionic groups on their repeating units. Since counterions can dissociate from the polymer backbone, polyelectrolyte chains are strongly influenced by electrostatic interactions. As a result, the physical properties of polyelectrolyte solutions are significantly different from those of electrically neutral polymers. The aim of this article is to highlight key results and some outstanding questions in the polyelectrolyte research from recent literature. We focus on the influence of electrostatics on conformational and hydrodynamic properties of polyelectrolyte chains. A compilation of experimental results from the literature reveals significant disparities with theoretical predictions. We also discuss a new class of polyelectrolytes called poly(ionic liquid)s that exhibit unique physical properties in comparison to ordinary polyelectrolytes. We conclude this review by listing some key research challenges in order to fully understand the conformation and dynamics of polyelectrolytes in solutions.

On the Validity of Constant pH Simulations

Constant pH (cpH) simulations are now a standard tool for investigating charge regulation in coarse-grained models of polyelectrolytes and colloidal systems. Originally developed for studying solutions with implicit ions, extending this method to systems with explicit ions or solvents presents several challenges. Ensuring proper charge neutrality within the simulation cell requires performing titration moves in sync with the insertion or deletion of ions, a crucial aspect often overlooked in the literature. Contrary to the prevailing views, cpH simulations are inherently grand-canonical, meaning that the controlled pH is that of the reservoir. The presence of the Donnan potential between the implicit reservoir and the simulation cell introduces significant differences between titration curves calculated for open and closed systems; the pH of an isolated (closed) system is different from the pH of the reservoir for the same protonation state of the polyelectrolyte. To underscore this point, in this paper, we will compare the titration curves calculated using the usual cpH algorithm with those from the exact canonical simulation algorithm. In the latter case, titration moves adhere to the correct detailed balance condition, and pH is calculated using the recently introduced surface Widom insertion algorithm. Our findings reveal a very significant difference between the titration isotherms obtained using the standard cpH algorithm and the canonical titration algorithm, emphasizing the importance of using the correct simulation approach when studying charge regulation of polyelectrolytes, proteins, and colloidal particles.

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.

Monte Carlo simulation of the ionization and uptake behavior of cationic oligomers into pH-responsive polyelectrolyte microgels of opposite charge - a model for oligopeptide uptake and release

External stimuli can tune the uptake and release of guest molecules in microgels. Especially their pH responsiveness makes microgels exciting candidates for drug delivery systems. When both microgel and guest molecules are pH-responsive, predicting the electrostatically driven uptake can be complex since the ionization depends on many parameters. In this work, we performed Metropolis Monte Carlo simulations while systematically varying the pK of the monomers, the concentrations of microgel and guest molecules to obtain a better understanding of the uptake of weak cationic oligomers as a model for oligopeptides into a weak anionic polyelectrolyte microgel. Further, we varied the chain length of the oligomers. The polyelectrolyte networks can take up oligomers when both the network and the oligomers are charged. The presence of both species in the system leads to a mutual enhancement of their ionization. The uptake induces a release of counterions and results in complex formation between the oligomers and the network, leading to the collapse of the networks. Longer oligomers enhance the ionization of the network and, therefore, the complexation. A higher microgel concentration increases the uptake only around the isoelectric point but prevents the uptake due to lower entropy gain at counterion release at higher pH. The results give an insight into the uptake of cationic oligomers into oppositely charged polyelectrolyte microgels and provide hints for the design of anionic microgels as carriers for guest molecules e.g. antimicrobial peptides.

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.

Induced phase transformation in ionizable colloidal nanoparticles

Acid-base equilibria directly influence the functionality and behavior of particles in a system. Due to the ionizing effects of acid-base functional groups, particles will undergo charge exchange. The degree of ionization and their intermolecular and electrostatic interactions are controlled by varying the pH and salt concentration of the solution in a system. Although the pH can be tuned in experiments, it is hard to model this effect using simulations or theoretical approaches. This is due to the difficulty in treating charge regulation and capturing the cooperative effects in a colloidal suspension with Coulombic interaction. In this work, we analyze a suspension of ionizable colloidal particles via molecular dynamics (MD) simulations, along with Monte Carlo simulations for charge regulation (MC-CR) and derive a phase diagram of the system as a function of pH. It is observed that as pH increases, particles functionalized with acid groups change their arrangement from face-centered cubic (FCC) packing to a disordered state. We attribute these transitions to an increase in the degree of charge polydispersity arising from an increase in pH. Our work shows that charge regulation leads to amorphous solids in colloids when the mean nanoparticle charge is sufficiently high.

Single-chain simulation of Ising density functional theory for weak polyelectrolytes

Conventional theories of weak polyelectrolytes are either computationally prohibitive to account for the multidimensional inhomogeneity of polymer ionization in a liquid environment or oversimplistic in describing the coupling effects of ion-explicit electrostatic interactions and long-range intrachain correlations. To bridge this gap, we implement the Ising density functional theory (iDFT) for ionizable polymer systems using the single-chain-in-mean-field algorithm. The single-chain-in-iDFT (sc-iDFT) shows significant improvements over conventional mean-field methods in describing segment-level dissociation equilibrium, specific ion effects, and long-range intrachain correlations. With an explicit consideration of the fluctuations of polymer configurations and the position-dependent ionization of individual polymer segments, sc-iDFT provides a faithful description of the structure and thermodynamic properties of inhomogeneous weak polyelectrolyte systems across multiple length scales.

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.

Titration in Canonical and Grand-Canonical Ensembles

We discuss problems associated with the notion of pH in heterogeneous systems. For homogeneous systems, standardization protocols lead to a well-defined quantity, which, although different from Sørensen's original idea of pH, is well reproducible and has become accepted as the measure of the "hydrogen potential". On the other hand, for heterogeneous systems, pH defined in terms of the chemical part of the electrochemical activity is thermodynamically inconsistent and runs afoul of the Gibbs-Guggenheim principle that forbids splitting of the electrochemical potential into separate chemical and electrostatic parts, since only the sum of two has any thermodynamic meaning. The problem is particularly relevant for modern simulation methods which involve charge regulation of proteins, polyelectrolytes, nanoparticles, colloidal suspensions, and so forth. In this paper, we show that titration isotherms calculated using semigrand canonical simulations can be very different from the ones obtained using canonical reactive Monte Carlo simulations.

A new method for reactive constant pH simulations

We present a simulation method that allows us to calculate the titration curves for systems undergoing protonation/deprotonation reactions-such as charged colloidal suspensions with acidic/basic surface groups, polyelectrolytes, polyampholytes, and proteins. The new approach allows us to simultaneously obtain titration curves both for systems in contact with salt and acid reservoir (semi-grand canonical ensemble) and for isolated suspensions (canonical ensemble). To treat the electrostatic interactions, we present a new method based on Ewald summation-which accounts for the existence of both Bethe and Donnan potentials within the simulation cell. We show that the Donnan potential dramatically affects the pH of a suspension. Counterintuitively, we find that in suspensions with a large volume fraction of nanoparticles and low ionic strength, the number of deprotonated groups can be 100% larger in an isolated system, compared to a system connected to a reservoir by a semi-permeable membrane-both systems being at exactly the same pH.

Renormalized charge and dielectric effects in colloidal interactions: a numerical solution of the nonlinear Poisson-Boltzmann equation for unknown boundary conditions

 The Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, introduced more than 70 years ago, is a hallmark of colloidal particle modeling. For highly charged particles in the dilute regime, it is often supplemented by Alexander's prescription (Alexander et al. in J Chem Phys 80:5776, 1984) for using a renormalized charge. Here, we solve the problem of the interaction between two charged colloids at finite ionic strength, including dielectric mismatch effects, using an efficient numerical scheme to solve the nonlinear Poisson-Boltzmann (NPB) equation with unknown boundary conditions. Our results perfectly match the analytical predictions for the renormalized charge by Trizac and coworkers (Aubouy et al. in J Phys A 36:5835, 2003). Moreover, they allow us to reinterpret previous molecular dynamics (MD) simulation results by Kreer et al. (Phys Rev E 74:021401, 2006), rendering them now in agreement with the expected behavior. We furthermore find that the influence of polarization becomes important only when the Debye layers overlap significantly.

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.

Anti-polyelectrolyte and polyelectrolyte effects on conformations of polyzwitterionic chains in dilute aqueous solutions

Polyzwitterions (PZs) are considered as model synthetic analogs of intrinsically disordered proteins. Based on this analogy, PZs in dilute aqueous solutions are expected to attain either globular (i.e. molten, compact) or random coil conformations. Addition of salt is expected to open these conformations. To the best of our knowledge, these hypotheses about conformations of PZs have never been verified. In this study, we test these hypotheses by studying effects of added salt [potassium bromide (KBr)] on gyration and hydrodynamic radii of poly(sulfobetaine methacrylate) in dilute aqueous solutions using dynamic light scattering and small-angle X-ray scattering, respectively. Effects of zwitteration are revealed by direct comparisons of the PZs with the polymers of the same backbone but containing (1) no explicit charges on side groups such as poly(2-dimethylaminoethyl methacrylate)s and (2) explicit cationic side groups with tertiary amino bromide pendants. Zeta-potential measurements, transmission electron microscopy, and ab initio molecular dynamics simulations reveal that the PZs acquire net positive charge in near salt-free conditions due to protonation but retain coiled conformations. Added KBr leads to nonmonotonic changes exhibiting an increase followed by a decrease in radius of gyration (and hydrodynamic radius), which are called antipolyelectrolyte and polyelectrolyte effects, respectively. Charge regulation and screening of charge-charge interactions are discussed in relation to the antipolyelectrolyte and polyelectrolyte effects, respectively, which highlight the importance of salt in affecting net charge and conformations of PZs.

pH-Sensitive Poly(acrylic acid)-g-poly(L-lysine) Charge-Driven Self-Assembling Hydrogels with 3D-Printability and Self-Healing Properties

We report the rheological behavior of aqueous solutions of a graft copolymer polyampholyte, constituted of polyacrylic acid (PAA) backbone grafted by Poly(L-lysine) (PAA-b-PLL). The graft copolymer self-assembles in aqueous media, forming a three-dimensional (3D) network through polyelectrolyte complexation of the oppositely charged PAA and PLL segments. Rheological investigations showed that the hydrogel exhibits interesting properties, namely, relatively low critical gel concentration, elastic response with slow dynamics, remarkable extended critical strain to flow, shear responsiveness, injectability, 3D printability and self-healing. Due to the weak nature of the involved polyelectrolyte segments, the hydrogel properties display pH-dependency, and they are affected by the presence of salt. Especially upon varying pH, the PLL secondary structure changes from random coil to α-helix, affecting the crosslinking structural mode and, in turn, the overall network structure as reflected in the rheological properties. Thanks to the biocompatibility of the copolymer constituents and the biodegradability of PLL, the designed gelator seems to exhibit potential for bioapplications.

Unusual Aspects of Charge Regulation in Flexible Weak Polyelectrolytes

This article reviews the state of the art of the studies on charge regulation (CR) effects in flexible weak polyelectrolytes (FWPE). The characteristic of FWPE is the strong coupling of ionization and conformational degrees of freedom. After introducing the necessary fundamental concepts, some unconventional aspects of the the physical chemistry of FWPE are discussed. These aspects are: (i) the extension of statistical mechanics techniques to include ionization equilibria and, in particular, the use of the recently proposed Site Binding-Rotational Isomeric State (SBRIS) model, which allows the calculation of ionization and conformational properties on the same foot; (ii) the recent progresses in the inclusion of proton equilibria in computer simulations; (iii) the possibility of mechanically induced CR in the stretching of FWPE; (iv) the non-trivial adsorption of FWPE on ionized surfaces with the same charge sign as the PE (the so-called "wrong side" of the isoelectric point); (v) the influence of macromolecular crowding on CR.

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.

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.

Chain-Shattering Polymers as Degradable Microdispersive Solid-Phase Extraction Sorbents

A chain-shattering polymer (CSP) has been proposed as a microdispersive solid-phase extraction (μdSPE) sorbent in a proof-of-concept study of degradable materials for analytical purposes. The responsive CSP was synthesized from 1,3,5-tris(bromomethyl)-2-nitrobenzene acting as the self-immolative trigger responsive unit and 2,6-naphthalenedicarboxylic acid as aromatic linker to enhance noncovalent aromatic interactions with the analytes. The CSP was characterized and applied as a μdSPE sorbent of a group of plasticizers, which were selected as model analytes, from different types of environmental water samples (tap, waste, and spring waters). Gas chromatography coupled to mass spectrometry detection was used for analyte determination. Mean recovery values were in the range of 80%–118% with RSD values below 22%. After the extraction, the polymer could be efficiently degraded by UV irradiation or by chemical reduction, recovering the aromatic linker. This work has proved the potential of CSPs as recyclable sorbents, paving the way to more environmentally benign analytical procedures.

Acid-Base Equilibrium and Dielectric Environment Regulate Charge in Supramolecular Nanofibers

Peptide amphiphiles are a class of molecules that can self-assemble into a variety of supramolecular structures, including high-aspect-ratio nanofibers. It is challenging to model and predict the charges in these supramolecular nanofibers because the ionization state of the peptides are not fixed but liable to change due to the acid-base equilibrium that is coupled to the structural organization of the peptide amphiphile molecules. Here, we have developed a theoretical model to describe and predict the amount of charge found on self-assembled peptide amphiphiles as a function of pH and ion concentration. In particular, we computed the amount of charge of peptide amphiphiles nanofibers with the sequence C₁₆ − V₂A₂E₂. In our theoretical formulation, we consider charge regulation of the carboxylic acid groups, which involves the acid-base chemical equilibrium of the glutamic acid residues and the possibility of ion condensation. The charge regulation is coupled with the local dielectric environment by allowing for a varying dielectric constant that also includes a position-dependent electrostatic solvation energy for the charged species. We find that the charges on the glutamic acid residues of the peptide amphiphile nanofiber are much lower than the same functional group in aqueous solution. There is a strong coupling between the charging via the acid-base equilibrium and the local dielectric environment. Our model predicts a much lower degree of deprotonation for a position-dependent relative dielectric constant compared to a constant dielectric background. Furthermore, the shape and size of the electrostatic potential as well as the counterion distribution are quantitatively and qualitatively different. These results indicate that an accurate model of peptide amphiphile self-assembly must take into account charge regulation of acidic groups through acid–base equilibria and ion condensation, as well as coupling to the local dielectric environment.

Accelerated simulation method for charge regulation effects

The net charge of solvated entities, ranging from polyelectrolytes and biomolecules to charged nanoparticles and membranes, depends on the local dissociation equilibrium of individual ionizable groups. Incorporation of this phenomenon, charge regulation (CR), in theoretical and computational models requires dynamic, configuration-dependent recalculation of surface charges and is therefore typically approximated by assuming constant net charge on particles. Various computational methods exist that address this. We present an alternative, particularly efficient CR Monte Carlo method (CR-MC), which explicitly models the redistribution of individual charges and accurately samples the correct grand-canonical charge distribution. In addition, we provide an open-source implementation in the large-scale Atomic/Molecular Massively Parallel Simulator molecular dynamics (MD) simulation package, resulting in a hybrid MD/CR-MC simulation method. This implementation is designed to handle a wide range of implicit-solvent systems that model discreet ionizable groups or surface sites. The computational cost of the method scales linearly with the number of ionizable groups, thereby allowing accurate simulations of systems containing thousands of individual ionizable sites. By matter of illustration, we use the CR-MC method to quantify the effects of CR on the nature of the polyelectrolyte coil-globule transition and on the effective interaction between oppositely charged nanoparticles.

DPD Modelling of the Self- and Co-Assembly of Polymers and Polyelectrolytes in Aqueous Media: Impact on Polymer Science

This review article is addressed to a broad community of polymer scientists. We outline and analyse the fundamentals of the dissipative particle dynamics (DPD) simulation method from the point of view of polymer physics and review the articles on polymer systems published in approximately the last two decades, focusing on their impact on macromolecular science. Special attention is devoted to polymer and polyelectrolyte self- and co-assembly and self-organisation and to the problems connected with the implementation of explicit electrostatics in DPD numerical machinery. Critical analysis of the results of a number of successful DPD studies of complex polymer systems published recently documents the importance and suitability of this coarse-grained method for studying polymer systems.

Inducing pH control over the critical micelle concentration of zwitterionic surfactants via polyacids adsorption: Effect of chain length and structure

HYPOTHESIS

The critical concentration above which micelles form from zwitterionic surfactant solutions and their thermodynamic stability is affected by the interaction with weak Brønsted polyacid chains (A_n) via the formation of charged hydrogen bonds between the latter and anionic moieties.

EXPERIMENTS

The interaction between zwitterionic micelles and polyacids capable of forming hydrogen bonds, and its dependence on the environmental pH and polymer structure, has been studied with constant-pH simulations and a restricted primitive model for all electrolytes.

FINDINGS

At low pH, the formation of polyacid/micelle complexes is witnessed independently of the polymer size or structure, so that the concentration above which micelles form is substantially decreased compared to polyacid-free cases. Upon rising pH, polymer desorption takes place within a narrow range of pH values, its location markedly depending on the size and structure of polyacids, and on the relative disposition between headgroup charged moieties. Thus, the desorption onset for long linear polyacids (A₆₀) interacting with sulphobetainic headgroups is roughly two pH units higher than for six decameric chains (6A₁₀) adsorbed onto micelles bearing phosphorylcholinic headgroups. This effect, together with the preferential desorption of chain ends at intermediate pH, may be exploited for drug delivery purposes or building advanced metamaterials.

Ising density functional theory for weak polyelectrolytes with strong coupling of ionization and intrachain correlations

We report a theoretical framework for weak polyelectrolytes by combining the polymer density functional theory with the Ising model for charge regulation. The so-called Ising density functional theory provides an accurate description of the effects of polymer conformation on the ionization of individual segments and is able to account for both the intra- and interchain correlations due to the excluded-volume effects, chain connectivity, and electrostatic interactions. Theoretical predictions of the titration behavior and microscopic structure of ionizable polymers are found to be in excellent agreement with the experiment.

Thermodynamic non-ideality in charge regulation of weak polyelectrolytes

Polymer ionization differs from that for their monomeric counterparts due to intramolecular correlations. Such effects are conventionally described in terms of the site-binding model that accounts for short-range interactions between neighboring sites. With an apparent equilibrium constant for each ionizable group and the nearest-neighbor energy as adjustable parameters, the site-binding method is useful to correlate experimental titration curves when the site-site interactions are insignificant at long ranges. This work aims to describe the electrostatic behavior of weak polyelectrolytes in aqueous solutions on the basis of the intrinsic equilibrium constants of the individual ionizable groups and solution conditions underlying the thermodynamic non-ideality. A molecular thermodynamic model is proposed for the protonation of weak polyelectrolytes by incorporating classical density functional theory into the site-binding model to account for the effects of the local ionic environment on both inter-chain and intra-chain correlations. By an extensive comparison of theoretical predictions with experimental titration curves, we demonstrate that the thermodynamic model is able to quantify the ionization behavior of weak polyelectrolytes over a broad range of molecular architectures and solution conditions.

Conformation and Ionization Behavior of Charge-Regulating Polyelectrolyte Brushes in a Poor Solvent

Understanding the structural response of weak polyelectrolyte brushes upon external stimuli is crucial for their applications ranging from modifying surface properties to the development of smart and intelligent materials. In this work, coarse-grained molecular dynamics simulations were carried out to investigate the conformation and ionization behavior of charge-regulating polyelectrolyte brushes under poor solvent conditions, using an implicit solvent model. The results show that, while the thickness of a sparse polyelectrolyte brush shows a similar behavior to that of a single chain, namely, a monotonic change as a function of solvent quality (modeled by an effective segment-segment attraction strength parameter) and solution pH, a dense polyelectrolyte brush exhibits more complex behavior. An unexpected reexpansion is observed when the effective segment-segment attraction strength is further increased, especially in the case of a high pH. In the latter case, strong attraction in polymer segments promotes the formation of large, interchain, cylindrical aggregates, leading to an increase in brush thickness.

Monte Carlo simulations of weak polyampholyte microgels: pH-dependence of conformation and ionization

We performed Metropolis Monte Carlo simulations to investigate the impact of varying acid and base dissociation constants on the pH-dependent ionization and conformation of weak polyampholyte microgels under salt-free conditions and under explicit consideration of the chemical ionization equilibria of the acidic and basic groups and their electrostatic interaction. Irrespective of their relative acid and base dissociation constant, all of the microgels undergo a pH-dependent charge reversal from positive to negative with a neutral charge at the isoelectric point. This charge reversal is accompanied by a U-shaped swelling transition of the microgels with a minimum of their size at the point of charge neutrality. The width of the U-shaped swelling transition, however, is found to depend on the chosen relative acid and base dissociation constants through which the extent of the favorable electrostatic intramolecular interaction of the ionized acidic and basic groups is altered. The pH-dependent swelling transition of the microgels is found to become broader, the stronger the intramolecular electrostatic interaction of the oppositely charged ionized species is. In addition, the intramolecular charge compensation of the acidic and basic groups of the microgels allows their counterions to abandon the microgel and the associated gain in translational entropy further amplifies the broadening of the pH-dependent swelling transition. The analysis of the radial ionization profiles of the acidic and basic groups of the differently composed microgels reveals a variety of radial ionization patterns with a dependence on the overall charge of the microgels.

In silico assessment of dehalogenase from Bacillus thuringiensis H2 in relation to its salinity-stability and pollutants degradation

Increased scientific interest has led to the rise in biotechnological uses of halophilic and halotolerant microbes for hypersaline wastewater bioremediation. Hence, this study performed molecular docking, molecular dynamic (MD) simulations, and validation by Molecular Mechanic Poisson-Boltzmann Surface Area (MM-PBSA) calculations on the DehH2 from Bacillus thuringiensis H2. We aimed to identify the interactions of DehH2 with substrates haloacids, haloacetates, and chlorpyrifos under extreme salinity (35% NaCl). MD simulations revealed that DehH2 preferentially degraded haloacids and haloacetates (-6.3 to -4.7 kcal/mol) by forming three or four hydrogen bonds to the catalytic triad, Asp125, Arg201, and Lys202. Conversely, chlorpyrifos was the least preferred substrate in both MD simulations and MM-PBSA calculations. MD simulation results ranked the DehH2-L-2CP complex (RMSD □0.125-0.23 nm) as the most stable while the least was the DehH2-chlorpyrifos complex (RMSD 0.32 nm; RMSF 0.0 - 0.29). The order of stability was as follows: DehH2-L-2CP > DehH2-MCA > DehH2-D-2CP > DehH2-3CP > DehH2-2,2-DCP > DehH2-2,3-DCP > DehH2-TCA > DehH2-chlorpyrifos. The MM-PBSA calculations further affirmed the DehH2-L-2CP complex's highest stability with the lowest binding energy of -45.14 kcal/mol, followed closely by DehH2-MCA (-41.21 kcal/mol), DehH2-D-2CP (-31.59 kcal/mol), DehH2-3CP (-30.75 kcal/mol), DehH2-2,2- DCP (-29.72 kcal/mol), DehH2-2,3-DCP (-22.20 kcal/mol) and DehH2-TCA (-18.46 kcal/mol). The positive binding energy of the DehH2-chlorpyrifos complex (+180.57 kcal/mol) proved the enzyme's non-preference for the substrate. The results ultimately illustrated the unique specificity of the DehH2 to degrade the above-said pollutants under a hypersaline condition.Communicated by Ramaswamy H. Sarma.

Charge Regulation Effects in Nanoparticle Self-Assembly

Nanoparticles in solution acquire charge through the dissociation or association of surface groups. Thus, a proper description of their electrostatic interactions requires the use of charge-regulating boundary conditions rather than the commonly employed constant-charge approximation. We implement a hybrid Monte Carlo/molecular dynamics scheme that dynamically adjusts the charges of individual surface groups of objects while evolving their trajectories. Charge regulation effects are shown to qualitatively change self-assembled structures due to global charge redistribution, stabilizing asymmetric constructs. We delineate under which conditions the conventional constant-charge approximation may be employed and clarify the interplay between charge regulation and dielectric polarization.

Understanding the Interaction of Polyelectrolyte Architectures with Proteins and Biosystems

The counterions neutralizing the charges on polyelectrolytes such as DNA or heparin may dissociate in water and greatly influence the interaction of such polyelectrolytes with biomolecules, particularly proteins. In this Review we give an overview of studies on the interaction of proteins with polyelectrolytes and how this knowledge can be used for medical applications. Counterion release was identified as the main driving force for the binding of proteins to polyelectrolytes: Patches of positive charge become multivalent counterions of the polyelectrolyte and lead to the release of counterions from the polyelectrolyte and a concomitant increase in entropy. This is shown from investigations on the interaction of proteins with natural and synthetic polyelectrolytes. Special emphasis is paid to sulfated dendritic polyglycerols (dPGS). The Review demonstrates that we are moving to a better understanding of charge-charge interactions in systems of biological relevance. Research along these lines will aid and promote the design of synthetic polyelectrolytes for medical applications.

Influence of macromolecular crowding on the charge regulation of intrinsically disordered proteins

In this work we study the coupling between ionization and conformational properties of two IDPs, histatin-5 and β-amyloid 42, in the presence of neutral and charged crowders. The latter is modeled to resemble bovine serum albumin (BSA). With this aim, semi-grand canonical Monte Carlo simulations are performed, so that the IDP charge is a dynamic property, undergoing protonation/deprotonation processes. Both ionization properties (global and specific amino acid charge and binding capacitance) and radius of gyration are analyzed in a large range of pH values and salt concentrations. Without crowder agents, the titration curve of histatin-5, a polycation, is salt-dependent while that of β-amyloid 42, a polyampholyte, is almost unaffected. The salt concentration is found to be particularly relevant at pH values where the protein binding capacitance (directly linked with charge fluctuation) is larger. Upon addition of neutral crowders, charge regulation is observed in histatin-5, while for β-amyloid 42 this effect is very small. The main mechanism for charge regulation is found to be the effective increase in the ionic strength due to the excluded volume. In the presence of charged crowders, a significant increase in the charge of both IDPs is observed in almost all the pH range. In this case, the IDP charge is altered not only by the increase in the effective ionic strength but also by its direct electrostatic interaction with the charged crowders.