Charge regulation in biomolecular solution

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.

Direct prediction of intermolecular interactions driven by disordered regions

Intrinsically disordered regions (IDRs) are critical for a wide variety of cellular functions, many of which involve interactions with partner proteins. Molecular recognition is typically considered through the lens of sequence-specific binding events. However, a growing body of work has shown that IDRs often interact with partners in a manner that does not depend on the precise order of the amino acid order, instead driven by complementary chemical interactions leading to disordered bound-state complexes. Despite this emerging paradigm, we lack tools to describe, quantify, predict, and interpret these types of structurally heterogeneous interactions from the underlying amino acid sequences. Here, we repurpose the chemical physics developed originally for molecular simulations to develop an approach for predicting intermolecular interactions between IDRs and partner proteins. Our approach enables the direct prediction of phase diagrams, the identification of chemically-specific interaction hotspots on IDRs, and a route to develop and test mechanistic hypotheses regarding IDR function in the context of molecular recognition. We use our approach to examine a range of systems and questions to highlight its versatility and applicability.

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.

Modulation of the electrostatic potential around α-lactalbumin using oligoelectrolyte chains, pH and salt concentration

In this study, we conducted a comprehensive computational investigation of the interaction between α-lactalbumin, a small globular protein, and strong anionic oligoelectrolyte chains with a polymerization degree from 2 to 9. Both the protein and oligoelectrolyte chains are represented using coarse-grained models, and their properties were calculated by the Monte Carlo method under constant pH conditions. We were able to estimate the effects of this interaction on the electrostatic potential around the protein. At acidic pH, the protein had a net positive charge; therefore, the electrostatic potential around it was also positive. To neutralize or reverse this electrostatic potential, oligoelectrolyte chains with a minimum size of six monomers were necessary. Simultaneously, low salt concentrations were required as elevated salt levels led to a significant attenuation of the electrostatic interactions and the corresponding electrostatic potential.

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.

Interaction of Polyanionic and Polycationic Brushes with Globular Proteins and Protein-like Nanocolloids

A large number of experimental studies have demonstrated that globular proteins can be absorbed from the solution by both polycationic and polyanionic brushes when the net charge of protein globules is of the same or of the opposite sign with respect to that of brush-forming polyelectrolyte chains. Here, we overview the results of experimental studies on interactions between globular proteins and polycationic or polyanionic brushes, and present a self-consistent field theoretical model that allows us to account for the asymmetry of interactions of protein-like nanocolloid particles comprising weak (pH-sensitive) cationic and anionic groups with a positively or negatively charged polyelectrolyte brush. The position-dependent insertion free energy and the net charge of the particle are calculated. The theoretical model predicts that if the numbers of cationic and anionic ionizable groups of the protein are approximately equal, then the interaction patterns for both cationic and anionic brushes at equal offset on the "wrong side" from the isoelectric point (IEP), i.e., when the particle and the brush charge are of the same sign, are similar. An essential asymmetry in interactions of particles with polycationic and polyanionic brushes is predicted when fractions of cationic and anionic groups differ significantly. That is, at a pH above IEP, the anionic brush better absorbs negatively charged particles with a larger fraction of ionizable cationic groups and vice versa.

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.

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.

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.

Unveiling protein-protein interaction potential through Monte Carlo simulation combined with small-angle X-ray scattering

Protein interactions are investigated under different conditions of lysozyme concentration, temperature and ionic strength by means of in-solution small angle X-Ray scattering (SAXS) experiments and Monte Carlo (MC) simulations. Initially, experimental data were analysed through a Hard-Sphere Double Yukawa (HSDY) model combined with Random Phase Approximation (RPA), a closure relationship commonly used in the literature for monodisperse systems. We realized by means of MC that the HSDY/RPA modelling fails to describe the protein-protein pair potential for moderated and dense systems at low ionic strength, mainly due to inherent distortions of the RPA approximation. Our SAXS/MC results thus show that lysozyme concentrations between 2 (diluted) and 20 mg/mL (not crowded) present similar protein-protein pair potential preserving the values of surface net charge around 7 e, protein diameter of 28 Å, decay range of attractive well potential of 3 Å and a depth of the well potential varying from 1 to 5 kBT depending on temperature and salt addition. Noteworthy, we here propose a novel method to analyse the SAXS data from interacting proteins through MC simulations, which overcomes the deficiencies presented by the use of a closure relationship. Furthermore, this new methodology of combining SAXS with MC simulations gives a step forward to investigate more complex systems as those composed of a mixture of proteins of distinct species presenting different molecular weights (and hence sizes) and surface net charges at low, moderate and very dense systems.

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.

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.

Towards an optimal monoclonal antibody with higher binding affinity to the receptor-binding domain of SARS-CoV-2 spike proteins from different variants

A highly efficient and robust multiple scales in silico protocol, consisting of atomistic Molecular Dynamics (MD), coarse-grain (CG) MD, and constant-pH CG Monte Carlo (MC), has been developed and used to study the binding affinities of selected antigen-binding fragments of the monoclonal antibody (mAbs) CR3022 and several of its here optimized versions against 11 SARS-CoV-2 variants including the wild type. Totally 235,000 mAbs structures were initially generated using the RosettaAntibodyDesign software, resulting in top 10 scored CR3022-like-RBD complexes with critical mutations and compared to the native one, all having the potential to block virus-host cell interaction. Of these 10 finalists, two candidates were further identified in the CG simulations to be the best against all SARS-CoV-2 variants. Surprisingly, all 10 candidates and the native CR3022 exhibited a higher affinity for the Omicron variant despite its highest number of mutations. The multiscale protocol gives us a powerful rational tool to design efficient mAbs. The electrostatic interactions play a crucial role and appear to be controlling the affinity and complex building. Studied mAbs carrying a more negative total net charge show a higher affinity. Structural determinants could be identified in atomistic simulations and their roles are discussed in detail to further hint at a strategy for designing the best RBD binder. Although the SARS-CoV-2 was specifically targeted in this work, our approach is generally suitable for many diseases and viral and bacterial pathogens, leukemia, cancer, multiple sclerosis, rheumatoid, arthritis, lupus, and more.

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

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

Biomolecular Complexation on the "Wrong Side": A Case Study of the Influence of Salts and Sugars on the Interactions between Bovine Serum Albumin and Sodium Polystyrene Sulfonate

In the protein purification, drug delivery, food industry, and biotechnological applications involving protein-polyelectrolyte complexation, proper selection of co-solutes and solution conditions plays a crucial role. The onset of (bio)macromolecular complexation occurs even on the so-called "wrong side" of the protein isoionic point where both the protein and the polyelectrolyte are net like-charged. To gain mechanistic insights into the modulatory role of salts (NaCl, NaBr, and NaI) and sugars (sucrose and sucralose) in protein-polyelectrolyte complexation under such conditions, interaction between bovine serum albumin (BSA) and sodium polystyrene sulfonate (NaPSS) at pH = 8.0 was studied by a combination of isothermal titration calorimetry, fluorescence spectroscopy, circular dichroism, and thermodynamic modeling. The BSA-NaPSS complexation proceeds by two binding processes (first, formation of intrapolymer complexes and then formation of interpolymer complexes), both driven by favorable electrostatic interactions between the negatively charged sulfonic groups (-SO₃^-) of NaPSS and positively charged patches on the BSA surface. Two such positive patches were identified, each responsible for one of the two binding processes. The presence of salts screened both short-range attractive and long-range repulsive electrostatic interactions between both macromolecules, resulting in a nonmonotonic dependence of the binding affinity on the total ionic strength for both binding processes. In addition, distinct anion-specific effects were observed (NaCl < NaBr < NaI). The effect of sugars was less pronounced: sucrose had no effect on the complexation, but its chlorinated analogue, sucralose, promoted it slightly due to the screening of long-range repulsive electrostatic interactions between BSA and NaPSS. Although short-range non-electrostatic interactions are frequently mentioned in the literature in relation to BSA or NaPSS, we found that the main driving force of complexation on the "wrong side" are electrostatic interactions.

Predicting Experimental B22 Values and the Effects of Histidine Charge States for Monoclonal Antibodies Using Coarse-Grained Molecular Simulations

Static light scattering (SLS) was used to characterize five monoclonal antibodies (MAbs) as a function of total ionic strength (TIS) at pH values between 5.5 and 7.0. Second osmotic virial coefficient (B₂₂) values were determined experimentally for each MAb as a function of TIS using low protein concentration SLS data. Coarse-grained molecular simulations were performed to predict the B₂₂ values for each MAb at a given pH and TIS. To include the effect of charge fluctuations of titratable residues in the B₂₂ calculations, a statistical approach was introduced in the Monte Carlo algorithm based on the protonation probability based on a given pH value and the Henderson-Hasselbalch equation. The charged residues were allowed to fluctuate individually, based on the sampled microstates and the influence of electrostatic interactions on net protein-protein interactions during the simulations. Compared to static charge simulations, the new approach provided improved results compared to experimental B₂₂ values at pH conditions near the pK_a of titratable residues.

Adsorption of flexible proteins in the 'wrong side' of the isoelectric point: Casein macropeptide as a model system

We analyze the conditions of the adsorption of a flexible peptide onto a charged substrate in the 'wrong side' of the isoelectric point (WSIP), i.e. when surface and peptide charges have the same sign. As a model system, we focus on the casein macropeptide (CMP), both in the aglycosylated (aCMP) and fully glycosydated (gCMP) forms. We model the substrate as a uniformly charged plane while CMP is treated as a bead-and-spring model including electrostatic interactions, excluded volume effects and acid/base equilibria. Adsorption coverage, aminoacid charges and concentration profiles are computed by means of Monte Carlo simulations at fixed pH and salt concentration. We conclude that for different reasons the CMP can be adsorbed to both positively and negatively charged surfaces in the WSIP. For negatively charged surfaces, WSIP adsorption is due to the patchy distribution of charges: the peptide is attached to the surface by the positively charged end of the chain, while the repulsion of the surface for the negatively charged tail is screened by the small ions of the added salt. This effect increases with salt concentration. Conversely, a positively charged substrate induces strong charge regulation of the peptide: the acidic groups are deprotonated, and the peptide becomes negatively charged. This effect is stronger at low salt concentrations and it is more intense for gCMP than for aCMP, due to the presence of the additional sialic groups in gCMP.

A method for quantifying how the activity of an enzyme is affected by the net charge of its nearest crowded neighbor

The electrostatic effects of protein crowding have not been systematically explored. Rather, protein crowding is generally studied with co‐solvents or crowders that are electrostatically neutral, with no methods to measure how the net charge (Z) of a crowder affects protein function. For example, can the activity of an enzyme be affected electrostatically by the net charge of its neighbor in crowded milieu? This paper reports a method for crowding proteins of different net charge to an enzyme via semi‐random chemical crosslinking. As a proof of concept, RNase A was crowded (at distances ≤ the Debye length) via crosslinking to different heme proteins with Z = +8.50 ± 0.04, Z = +6.39 ± 0.12, or Z = −10.30 ± 1.32. Crosslinking did not disrupt the structure of proteins, according to amide H/D exchange, and did not inhibit RNase A activity. For RNase A, we found that the electrostatic environment of each crowded neighbor had significant effects on rates of RNA hydrolysis. Crowding with cationic cytochrome c led to increases in activity, while crowding with anionic “supercharged” cytochrome c or myoglobin diminished activity. Surprisingly, electrostatic crowding effects were amplified at high ionic strength (I = 0.201 M) and attenuated at low ionic strength (I = 0.011 M). This salt dependence might be caused by a unique set of electric double layers at the dimer interspace (maximum distance of 8 Å, which cannot accommodate four layers). This new method of crowding via crosslinking can be used to search for electrostatic effects in protein crowding.

Control of protein activity by photoinduced spin polarized charge reorganization

Considerable electric fields are present within living cells, and the role of bioelectricity has been well established at the organismal level. Yet much remains to be learned about electric-field effects on protein function. Here, we use phototriggered charge injection from a site-specifically attached ruthenium photosensitizer to directly demonstrate the effect of dynamic charge redistribution within a protein. We find that binding of an antibody to phosphoglycerate kinase (PGK) is increased twofold under illumination. Remarkably, illumination is found to suppress the enzymatic activity of PGK by a factor as large as three. These responses are sensitive to the photosensitizer position on the protein. Surprisingly, left (but not right) circularly polarized light elicits these responses, indicating that the electrons involved in the observed dynamics are spin polarized, due to spin filtration by protein chiral structures. Our results directly establish the contribution of electrical polarization as an allosteric signal within proteins. Future experiments with phototriggered charge injection will allow delineation of charge rearrangement pathways within proteins and will further depict their effects on protein function.

Predicting the Orientation of Adsorbed Proteins Steered with Electric Fields Using a Simple Electrostatic Model

Under the most common experimental conditions, the adsorption of proteins to solid surfaces is a spontaneous process that leads to a rather compact layer of randomly oriented molecules. However, controlling such orientation is critically important for the development of catalytic surfaces. In this regard, the use of electric fields is one of the most promising alternatives. Our work is motivated by experimental observations that show important differences in catalytic activity of a trypsin-covered surface, which depended on the applied potential during the adsorption. Even though adsorption results from the combination of several processes, we were able to determine that (under the selected conditions) mean-field electrostatics play a dominant role, determining the orientation and yielding a difference in catalytic activity. We simulated the electrostatic potential numerically, using an implicit-solvent model based on the linearized Poisson-Boltzmann equation. This was implemented in an extension of the code PyGBe that included an external electric field, and rendered the electrostatic component of the solvation free energy. Our model (extensions available at the Github repository) allowed estimating the overall affinity of the protein with the surface, and their most likely orientation as a function of the potential applied. Our results show that the active sites of trypsin are, on average, more exposed when the electric field is negative, which agrees with the experimental results of catalytic activity, and confirm the premise that electrostatic interactions can be used to control the orientation of adsorbed proteins.

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.

Improved Random Batch Ewald Method in Molecular Dynamics Simulations

The random batch Ewald (RBE) is an efficient and accurate method for molecular dynamics (MD) simulations of physical systems at the nano/microscale. The method shows great potential to solve the computational bottleneck of long-range interactions, motivating a necessity to accelerate short-range components of the nonbonded interactions for a further speedup of MD simulations. In this work, we present an improved RBE method for the nonbonding interactions by introducing the random batch idea to constructing neighbor lists for the treatment of both the short-range part of the Ewald splitting and the Lennard-Jones potential. The efficiency that the novel neighbor list algorithm owes to the stochastic minibatch strategy can significantly reduce the total number of neighbors. We obtain the error estimate and convergence by theoretical analysis and implement the improved RBE method in the LAMMPS package. Benchmark simulations are performed to demonstrate the accuracy and stability of the algorithm. Numerical tests on computer performance by conducting large-scaled MD simulations for systems including up to 0.1 billion water molecules are run on massive clusters with up to 50000 CPU cores, demonstrating the attractive features such as the high parallel scalability and memory-saving of the method in comparison to the existing methods.

Impact of Charge Regulation on Self-Assembly of Zwitterionic Nanoparticles

Zwitterionic modification of colloids with weak acids and bases represents a promising strategy in creating functional materials with tunable properties and modeling the self-organization of charged proteins. However, accurate incorporation of the dynamic dissociation or association of ionization groups known as charge regulation (CR) is often intractable in theoretical and computational investigations since charge redistribution and configuration need to be evolved self-consistently. Using hybrid Monte Carlo and molecular dynamics simulations, we demonstrate that a dilute suspension of overall charge-neutral zwitterionic Janus nanoparticles shows a conformational transition from an open assembly of string or bundle to compact cluster along with the variation in pH. The behavior under CR is qualitatively different from the commonly employed constant charge condition where the transition is absent. The CR-induced clustering is due to the inhomogeneous and fluctuating charges localized near the equatorial boundary of the Janus particle. These features are enhanced particularly at low salt concentration and high electrostatic coupling strength. Our results indicate the critical role of charge regulation in the spatial self-organization of zwitterionic nanoparticles.

Explicit characterization of counterion dynamics around a flexible polyelectrolyte

The article presents a comprehensive study of counterion dynamics around a generic linear polyelectrolyte chain with the help of coarse-grained computer simulations. The ion-chain coupling is discussed in the form of binding time, mean-square displacement (MSD) relative to the chain, local ion transport coefficient, and spatiotemporal correlations in the effective charge. We have shown that a counterion exhibits subdiffusive behavior 〈δR^{2}〉∼t^{δ}, δ≈0.9 w.r.t. chain's center of mass. The MSD of ions perpendicularly outward from the chain segment exhibits a smaller subdiffusive exponent compared to the one along the chain backbone. The effective diffusivity of ion is the lowest in chain's close proximity, extending up to the length-scale of radius of gyration R_{g}. Beyond R_{g} at larger distances, they attain diffusivity of free ion with a smooth cross-over from the adsorbed regime to the free ion regime. We have shown that the effective diffusivity drastically decreases for the multivalent ions, while the crossover length scale remains the same. Conversely, with increasing salt concentration the coupling-length scale reduces, while the diffusivity remains unaltered. The effective diffusivity of adsorbed-ion reveals an exponential reduction with electrostatic interaction strength. We further corroborate this from the binding time of ions on the chain, which also grows exponentially with the coupling strength of the ion-polymer duo. Moreover, the binding time of ions exhibits a weak dependence with salt concentration for the monovalent salt, while for multivalent salts the binding time decreases dramatically with concentration. Our work also elucidates fluctuations in the effective charge per site, where it exhibits strong negative correlations at short length-scales.

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.

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.

Reactive Monte Carlo simulations for charge regulation of colloidal particles

We use a reactive Monte Carlo simulation method and the primitive model of electrolyte to study acid-base equilibrium that controls charge regulation in colloidal systems. The simulations are performed in a semi-grand canonical ensemble in which colloidal suspension is in contact with a reservoir of salt and strong acid. The interior of colloidal particles is modeled as a low dielectric medium, different from the surrounding water. The effective colloidal charge is calculated for different numbers of surface acidic groups, pH, salt concentrations, and types of electrolyte. In the case of potassium chloride, the titration curves are compared with the experimental measurements obtained using potentiometric titration. A good agreement is found between simulations and experiments. In the case of lithium chloride, the specific ionic adsorption is taken into account through the partial dehydration of lithium ion.

Quantifying charge state heterogeneity for proteins with multiple ionizable residues

Ionizable residues can release and take up protons and this has an influence on protein structure and function. The extent of protonation is linked to the overall pH of the solution and the local environments of ionizable residues. Binding or unbinding of a single proton generates a distinct charge microstate defined by a specific pattern of charges. Accordingly, the overall partition function is a sum over all charge microstates and Boltzmann weights of all conformations associated with each of the charge microstates. This ensemble-of-ensembles description recast as a q-canonical ensemble allows us to analyze and interpret potentiometric titrations that provide information regarding net charge as a function of pH. In the q-canonical ensemble, charge microstates are grouped into mesostates where each mesostate is a collection of microstates of the same net charge. Here, we show that leveraging the structure of the q-canonical ensemble allows us to decouple contributions of net proton binding and release from proton arrangement and conformational considerations. Through application of the q-canonical formalism to analyze potentiometric measurements of net charge in proteins with repetitive patterns of Lys and Glu residues, we determine the underlying mesostate pKa values and, more importantly, we estimate relative mesostate populations as a function of pH. This is a strength of using the q-canonical approach that cannot be replicated using purely site-specific analyses. Overall, our work shows how measurements of charge equilibria, decoupled from measurements of conformational equilibria, and analyzed using the framework of the q-canonical ensemble, provide protein-specific quantitative descriptions of pH-dependent populations of mesostates. This method is of direct relevance for measuring and understanding how different charge states contribute to conformational, binding, and phase equilibria of proteins.

Adsorption of lysozyme into a charged confining pore

Several applications arise from the confinement of proteins on surfaces because their stability and biological activity are enhanced. It is also known that the way in which a protein adsorbs on the surface is important for its biological function since its active sites should not be obstructed. In this study, the adsorption properties of hen egg-white lysozyme, HEWL, into a negatively charged silica pore is examined by employing a coarse-grained model and constant-pH Monte Carlo simulations. The role of electrostatic interactions is taken into account via including the Debye-Hückel potentials into the Cα structure-based model. We evaluate the effects of pH, salt concentration, and pore radius on the protein preferential orientation and spatial distribution of its residues regarding the pore surface. By mapping the residues that stay closer to the pore surface, we find that the increase of pH leads to orientational changes of the adsorbed protein when the solution pH gets closer to the HEWL isoelectric point. Under these conditions, the pK_a shift of these important residues caused by the adsorption into the charged confining surface results in a HEWL charge distribution that stabilizes the adsorption in the observed protein orientation. We compare our observations to the results of the pK_a shift for HEWL available in the literature and to some experimental data.

Site Correlations, Capacitance, and Polarizability From Protein Protonation Fluctuations

We generalize the Kirkwood-Shumaker theory of protonation fluctuation for an anisotropic distribution of dissociable charges on a globular protein. The fluctuations of the total charge and the total dipole moment, in contrast to their average values, depend on the same proton occupancy correlator, thus exhibiting a similar dependence also on the solution pH. This has important consequences for the Kirkwood-Shumaker interaction and its dependence on the bathing solution conditions.

Protein Transport upon Advection at the Air/Water Interface: When Charge Matters

The formation of dense protein interfacial layers at a free air-water interface is known to result from both diffusion and advection. Furthermore, protein interactions in concentrated phases are strongly dependent on their overall positive or negative net charge, which is controlled by the solution pH. As a consequence, an interesting question is whether the presence of an advection flow of water toward the interface during protein adsorption produces different kinetics and interfacial structure of the adsorbed layer, depending on the net charge of the involved proteins and, possibly, on the sign of this charge. Here we test a combination of the following parameters using ovalbumin and lysozyme as model proteins: positive or negative net charge and the presence or absence of advection flow. The formation and the organization of the interfacial layers are studied by neutron reflectivity and null-ellipsometry measurements. We show that the combined effect of a positive charge of lysozyme and ovalbumin and the presence of advection flow does induce the formation of interfacial multilayers. Conversely, negatively charged ovalbumin forms monolayers, whether advection flow is present or not. We show that an advection/diffusion model cannot correctly describe the adsorption kinetics of multilayers, even in the hypothesis of a concentration-dependent diffusion coefficient as in colloidal filtration, for instance. Still, it is clear that advection is a necessary condition for making multilayers through a mechanism that remains to be determined, which paves the way for future research.

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.

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.

Measuring how two proteins affect each other's net charge in a crowded environment

Theory predicts that the net charge (Z) of a protein can be altered by the net charge of a neighboring protein as the two approach one another below the Debye length. This type of charge regulation suggests that a protein's charge and perhaps function might be affected by neighboring proteins without direct binding. Charge regulation during protein crowding has never been directly measured due to analytical challenges. Here, we show that lysine specific protein crosslinkers (NHS ester-Staudinger pairs) can be used to mimic crowding by linking two non-interacting proteins at a maximal distance of ~7.9 Å. The net charge of the regioisomeric dimers and preceding monomers can then be determined with lysine-acyl "protein charge ladders" and capillary electrophoresis. As a proof of concept, we covalently linked myoglobin (Z_monomer  = -0.43 ± 0.01) and α-lactalbumin (Z_monomer  = -4.63 ± 0.05). Amide hydrogen/deuterium exchange and circular dichroism spectroscopy demonstrated that crosslinking did not significantly alter the structure of either protein or result in direct binding (thus mimicking crowding). Ultimately, capillary electrophoretic analysis of the dimeric charge ladder detected a change in charge of ΔZ = -0.04 ± 0.09 upon crowding by this pair (Z_dimer  = -5.10 ± 0.07). These small values of ΔZ are not necessarily general to protein crowding (qualitatively or quantitatively) but will vary per protein size, charge, and solvent conditions.

Quantifying viscosity and surface tension of multicomponent protein-nucleic acid condensates

Living cells organize their internal space into dynamic condensates through liquid-liquid phase separation of multivalent proteins in association with cellular nucleic acids. Here, we study how variations in nucleic acid (NA)-to-protein stoichiometry modulate the condensed phase organization and fluid dynamics in a model system of multicomponent heterotypic condensates. Employing a multiparametric approach comprised of video particle tracking microscopy and optical tweezer-induced droplet fusion, we show that the interfacial tension, but not viscosity, of protein-NA condensates is controlled by the NA/protein ratio across the two-phase regime. In parallel, we utilize fluorescence correlation spectroscopy to quantify protein and NA diffusion in the condensed phase. Fluorescence correlation spectroscopy measurements reveal that the diffusion of the component protein and NA within the condensate core is governed by the viscosity, and hence, also remains insensitive to the changes in NA-to-protein stoichiometry. Collectively, our results provide insights into the regulation of multicomponent heterotypic liquid condensates, reflecting how the bulk mixture composition affects their core versus surface organization and dynamical properties.

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.

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.

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.

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.

Interfacial water and ion distribution determine ζ potential and binding affinity of nanoparticles to biomolecules

The molecular features that dictate interactions between functionalized nanoparticles and biomolecules are not well understood. This is in part because for highly charged nanoparticles in solution, establishing a clear connection between the molecular features of surface ligands and common experimental observables such as ζ potential requires going beyond the classical models based on continuum and mean field models. Motivated by these considerations, molecular dynamics simulations are used to probe the electrostatic properties of functionalized gold nanoparticles and their interaction with a charged peptide in salt solutions. Counterions are observed to screen the bare ligand charge to a significant degree even at the moderate salt concentration of 50 mM. As a result, the apparent charge density and ζ potential are largely insensitive to the bare ligand charge densities, which fall in the range of ligand densities typically measured experimentally for gold nanoparticles. While this screening effect was predicted by classical models such as the Manning condensation theory, the magnitudes of the apparent surface charge from microscopic simulations and mean-field models are significantly different. Moreover, our simulations found that the chemical features of the surface ligand (e.g., primary vs. quaternary amines, heterogeneous ligand lengths) modulate the interfacial ion and water distributions and therefore the interfacial potential. The importance of interfacial water is further highlighted by the observation that introducing a fraction of hydrophobic ligands enhances the strength of electrostatic binding of the charged peptide. Finally, the simulations highlight that the electric double layer is perturbed upon binding interactions. As a result, it is the bare charge density rather than the apparent charge density or ζ potential that better correlates with binding affinity of the nanoparticle to a charged peptide. Overall, our study highlights the importance of molecular features of the nanoparticle/water interface and underscores a set of design rules for the modulation of electrostatic driven interactions at nano/bio interfaces.

How charge regulation and ion-surface affinity affect the differential capacitance of an electrical double layer

The differential capacitance of an electrical double layer is a topic of great importance to develop more efficient and environment-friendly energy storage devices: electric double layer supercapacitors. In addition to the bare electrostatic interactions, recent experimental and computational studies suggest that electrodes covered by ionizable groups do interact selectively with specific ion types, an effect that can increase the maximal conductivity and voltage of a supercapacitor. Inspired by this, in the present work we investigate how ion-specific non-electrostatic interactions modify the differential capacitance of a flat electrode whose surface is covered by ionizable groups subject to a charge regulation process. The incorporation of hydration interactions by means of ion-specific Yukawa potential into the Poisson-Boltzmann theory allows our model to describe different scenarios of ion-surface affinity and, hence, the selective depletion or accumulation of specific ion types close to a charged surface. We obtained larger capacitance values when considering electrodes that favor the accumulation of cations and the depletion of anions.