A Molecular Thermodynamic Model of Complexation in Mixtures of Oppositely Charged Polyelectrolytes with Explicit Account of Charge Association/Dissociation

Charge Regulation Triggers Condensation of Short Oligopeptides to Polyelectrolytes

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

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

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

Design Rules for the Sequestration of Viruses into Polypeptide Complex Coacervates

Encapsulation is a strategy that has been used to facilitate the delivery and increase the stability of proteins and viruses. Here, we investigate the encapsulation of viruses via complex coacervation, which is a liquid-liquid phase separation resulting from the complexation of oppositely charged polymers. In particular, we utilized polypeptide-based coacervates and explored the effects of peptide chemistry, chain length, charge patterning, and hydrophobicity to better understand the effects of the coacervating polypeptides on virus incorporation. Our study utilized two nonenveloped viruses, porcine parvovirus (PPV) and human rhinovirus (HRV). PPV has a higher charge density than HRV, and they both appear to be relatively hydrophobic. These viruses were compared to characterize how the charge, hydrophobicity, and patterning of chemistry on the surface of the virus capsid affects encapsulation. Consistent with the electrostatic nature of complex coacervation, our results suggest that electrostatic effects associated with the net charge of both the virus and polypeptide dominated the potential for incorporating the virus into a coacervate, with clustering of charges also playing a significant role. Additionally, the hydrophobicity of a virus appears to determine the degree to which increasing the hydrophobicity of the coacervating peptides can enhance virus uptake. Nonintuitive trends in uptake were observed with regard to both charge patterning and polypeptide chain length, with these parameters having a significant effect on the range of coacervate compositions over which virus incorporation was observed. These results provide insights into biophysical mechanisms, where sequence effects can control the uptake of proteins or viruses into biological condensates and provide insights for use in formulation strategies.

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.

Counterion-controlled phase equilibria in a charge-regulated polymer solution

We study phase equilibria in a minimal model of charge-regulated polymer solutions. Our model consists of a single polymer species whose charge state arises from protonation-deprotonation processes in the presence of a dissolved acid, whose anions serve as screening counterions. We explicitly account for variability in the polymers' charge states. Homogeneous equilibria in this model system are characterised by the total concentration of polymers, the concentration of counter-ions and the charge distributions of polymers which can be computed with the help of analytical approximations. We use these analytical results to characterise how parameter values and solution acidity influence equilibrium charge distributions and identify for which regimes uni-modal and multi-modal charge distributions arise. We then study the interplay between charge regulation, solution acidity and phase separation. We find that charge regulation has a significant impact on polymer solubility and allows for non-linear responses to the solution acidity: Re-entrant phase behaviour is possible in response to increasing solution acidity. Moreover, we show that phase separation can yield to the coexistence of local environments characterised by different charge distributions.

Theoretical treatment of complex coacervate core micelles: structure and pH-induced disassembly

Complex coacervate core micelles (C3Ms) are supramolecular soft nanostructures formed by the assembly of a block copolymer and an oppositely charged homopolymer. The coacervation of the charged segments in both macromolecules drives the formation of the core of the C3M, while the neutral block of the copolymer forms the corona. This work introduces a molecular theory (MOLT) that predicts the internal structure and stimuli-responsive properties of C3Ms and explicitly considers the chemical architecture of the polyelectrolytes, their acid-based equilibria and electrostatic and non-electrostatic interactions. In order to accurately predict complex coacervation, the correlations between charged species are incorporated into MOLT as ion-pairing processes, which are modeled using a coupled chemical equilibrium formalism. Very good agreement was observed between the experimental results in the literature and MOLT predictions for the scaling relationships that relate the dimensions of the micelle (aggregation number and sizes of the micelle and the core) to the lengths of the different blocks. MOLT was used to study the disassembly of the micelles when the solution pH is driven away from the value that guarantees the charge stoichiometry of the core. This study reveals that very sharp disassembly transitions can be obtained by tuning the length or architecture of the copolymer component, thereby suggesting potential routes to design C3Ms capable of releasing their components at very precise pH values.

Driving forces of the complex formation between highly charged disordered proteins

Highly disordered complexes between oppositely charged intrinsically disordered proteins present a new paradigm of biomolecular interactions. Here, we investigate the driving forces of such interactions for the example of the highly positively charged linker histone H1 and its highly negatively charged chaperone, prothymosin α (ProTα). Temperature-dependent single-molecule Förster resonance energy transfer (FRET) experiments and isothermal titration calorimetry reveal ProTα-H1 binding to be enthalpically unfavorable, and salt-dependent affinity measurements suggest counterion release entropy to be an important thermodynamic driving force. Using single-molecule FRET, we also identify ternary complexes between ProTα and H1 in addition to the heterodimer at equilibrium and show how they contribute to the thermodynamics observed in ensemble experiments. Finally, we explain the observed thermodynamics quantitatively with a mean-field polyelectrolyte theory that treats counterion release explicitly. ProTα-H1 complex formation resembles the interactions between synthetic polyelectrolytes, and the underlying principles are likely to be of broad relevance for interactions between charged biomolecules in general.

Phase Behavior and Electrochemical Properties of Highly Asymmetric Redox Coacervates

This work reports the phase behavior and electrochemical properties of liquid coacervates made of ferricyanide and poly(ethylenimine). In contrast to the typical polyanion/polycation pairs used in liquid coacervates, the ferricyanide/poly(ethylenimine) system is highly asymmetric because poly(ethylenimine) has approximately 170 charges per molecule, while ferricyanide has only 3. Two types of phase diagrams were measured and fitted with a theoretical model. In the first type of diagram, the stability of the coacervate was studied in the plane given by the concentration of poly(ethylenimine) versus the concentration of ferricyanide for a fixed concentration of added monovalent salt (NaCl). The second type of diagram involved the plane given by the concentration of poly(ethylenimine) vs the concentration of the added monovalent salt for a fixed poly(ethyleneimine)/ferricyanide ratio. Interestingly, these phase diagrams displayed qualitative similarities to those of symmetric polyanion/polycation systems, suggesting that coacervates formed by a polyelectrolyte and a small multivalent ion can be treated as a specific case of polyelectrolyte coacervate. The characterization of the electrochemical properties of the coacervate revealed that the addition of monovalent salt greatly enhances charge transport, presumably by breaking ion pairs between ferricyanide and poly(ethylenimine). This finding highlights the significant influence of added salt on the transport properties of coacervates. This study provides the first comprehensive characterization of the phase behavior and transport properties of asymmetric coacervates and places these results within the broader context of the better-known symmetric polyelectrolyte coacervates.

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.

The effect of polymer length in liquid-liquid phase separation

Understanding the thermodynamics that drive liquid-liquid phase separation (LLPS) is quite important given the number of diverse biomolecular systems undergoing this phenomenon. Many studies have focused on condensates of long polymers, but very few systems of short-polymer condensates have been observed and studied. Here, we study a short-polymer system of various lengths of poly-adenine RNA and peptides formed by the RGRGG sequence repeats to understand the underlying thermodynamics of LLPS. Using the recently developed COCOMO coarse-grained (CG) model, we predicted condensates for lengths as short as 5-10 residues, which was then confirmed by experiment, making this one of the smallest LLPS systems yet observed. A free-energy model reveals that the length dependence of condensation is driven primarily by entropy of confinement. The simplicity of this system will provide the basis for understanding more biologically realistic systems.

Polyelectrolyte-multivalent molecule complexes: physicochemical properties and applications

The complexation of polyelectrolytes with other oppositely charged structures gives rise to a great variety of functional materials with potential applications in a wide spectrum of technological fields. Depending on the assembly conditions, polyelectrolyte complexes can acquire different macroscopic configurations such as dense precipitates, nanosized colloids and liquid coacervates. In the past 50 years, much progress has been achieved to understand the principles behind the phase separation induced by the interaction of two oppositely charged polyelectrolytes in aqueous solutions, especially for symmetric systems (systems in which both polyions have similar molecular weight and concentration). However, in recent years, the complexation of polyelectrolytes with alternative building blocks such as small charged molecules (multivalent inorganic species, oligopeptides, and oligoamines, among others) has gained attention in different areas. In this review, we discuss the physicochemical characteristics of the complexes formed by polyelectrolytes and multivalent small molecules, putting a special emphasis on their similarities with the well-known polycation-polyanion complexes. In addition, we analyze the potential of these complexes to act as versatile functional platforms in various technological fields, such as biomedicine and advanced materials engineering.

Polyelectrolyte complexation of two oppositely charged symmetric polymers: A minimal theory

Interplay of Coulomb interaction energy, free ion entropy, and conformational elasticity is a fascinating aspect in polyelectrolytes (PEs). We develop a theory for complexation of two oppositely charged PEs, a process known to be the precursor to the formation of complex coacervates in PE solutions, to explore the underlying thermodynamics of complex formation, at low salts. The theory considers general degrees of solvent polarity and dielectricity within an implicit solvent model, incorporating a varying Coulomb strength. Explicit calculation of the free energy of complexation and its components indicates that the entropy of free counterions and salt ions and the Coulomb enthalpy of bound ion-pairs dictate the equilibrium of PE complexation. This helps decouple the self-consistent dependency of charge and size of the uncomplexed parts of the polyions, derive an analytical expression for charge, and evaluate the free energy components as functions of chain overlap. Complexation is observed to be driven by enthalpy gain at low Coulomb strengths, driven by entropy gain of released counterions but opposed by enthalpy loss due to reduction of ion-pairs at moderate Coulomb strengths, and progressively less favorable due to enthalpy loss at even higher Coulomb strengths. The total free energy of the system is found to decrease linearly with an overlap of chains. Thermodynamic predictions from our model are in good quantitative agreement with simulations in literature.

Modeling Concentration-dependent Phase Separation Processes Involving Peptides and RNA via Residue-Based Coarse-Graining

Biomolecular condensation, especially liquid-liquid phase separation, is an important physical process with relevance for a number of different aspects of biological functions. Key questions of what drives such condensation, especially in terms of molecular composition, can be addressed via computer simulations, but the development of computationally efficient yet physically realistic models has been challenging. Here, the coarse-grained model COCOMO is introduced that balances the polymer behavior of peptides and RNA chains with their propensity to phase separate as a function of composition and concentration. COCOMO is a residue-based model that combines bonded terms with short- and long-range terms, including a Debye-Hückel solvation term. The model is highly predictive of experimental data on phase-separating model systems. It is also computationally efficient and can reach the spatial and temporal scales on which biomolecular condensation is observed with moderate computational resources.

Driving force and pathway in polyelectrolyte complex coacervation

There is notable discrepancy between experiments and coarse-grained model studies regarding the thermodynamic driving force in polyelectrolyte complex coacervation: experiments find the free energy change to be dominated by entropy, while simulations using coarse-grained models with implicit solvent usually report a large, even dominant energetic contribution in systems with weak to intermediate electrostatic strength. Here, using coarse-grained, implicit-solvent molecular dynamics simulation combined with thermodynamic analysis, we study the potential of mean force (PMF) in the two key stages on the coacervation pathway for symmetric polyelectrolyte mixtures: polycation-polyanion complexation and polyion pair-pair condensation. We show that the temperature dependence in the dielectric constant of water gives rise to a substantial entropic contribution in the electrostatic interaction. By accounting for this electrostatic entropy, which is due to solvent reorganization, we find that under common conditions (monovalent ions, room temperature) for aqueous systems, both stages are strongly entropy-driven with negligible or even unfavorable energetic contributions, consistent with experimental results. Furthermore, for weak to intermediate electrostatic strengths, this electrostatic entropy, rather than the counterion-release entropy, is the primary entropy contribution. From the calculated PMF, we find that the supernatant phase consists predominantly of polyion pairs with vanishingly small concentration of bare polyelectrolytes, and we provide an estimate of the spinodal of the supernatant phase. Finally, we show that prior to contact, two neutral polyion pairs weakly attract each other by mutually induced polarization, providing the initial driving force for the fusion of the pairs.

On the nature of screening in Voorn–Overbeek type theories

By using a recently formulated Legendre transform approach to the thermodynamics of charged systems, we explore the general form of the screening length in the Voorn–Overbeek-type theories, which remains valid also in the cases where the entropy of the charged component(s) is not given by the ideal gas form as in the Debye–Hückel theory. The screening length consistent with the non-electrostatic terms in the free energy ansatz for the Flory–Huggins and Voorn–Overbeek type theories, derived from the local curvature properties of the Legendre transform, has distinctly different behavior than the often invoked standard Debye screening length, though it reduces to it in some special cases.

Field theory description of ion association in re-entrant phase separation of polyampholytes

Phase separation of several different overall neutral polyampholyte species (with zero net charge) is studied in solution with two oppositely charged ion species that can form ion-pairs through an association reaction. A field theory description of the system, that treats polyampholyte charge sequence dependent electrostatic interactions as well as excluded volume effects, is hereby given. Interestingly, analysis of the model using random phase approximation and field theoretic simulation consistently show evidence of a re-entrant polyampholyte phase separation at high ion concentrations when there is an overall decrease of volume upon ion-association. As an illustration of the ramifications of our theoretical framework, several polyampholyte concentration vs ion concentration phase diagrams under constant temperature conditions are presented to elucidate the dependence of phase separation behavior on polyampholyte sequence charge pattern as well as ion-pair dissociation constant, volumetric effects on ion association, solvent quality, and temperature.

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.

A simple simulation model for complex coacervates

When oppositely charged polyelectrolytes mix in an aqueous solution, associative phase separation gives rise to coacervates. Experiments reveal the phase diagram for such coacervates, and determine the impact of charge density, chain length and added salt. Simulations often use hybrid MC-MD methods to produce such phase diagrams, in support of experimental observations. We propose an idealized model and a simple simulation technique to investigate coacervate phase behavior. We model coacervate systems by charged bead-spring chains and counterions with short-range repulsions, of size equal to the Bjerrum length. We determine phase behavior by equilibrating a slab of concentrated coacervate with respect to swelling into a dilute phase of counterions. At salt concentrations below the critical point, the counterion concentration in the coacervate and dilute phases are nearly the same. At high salt concentrations, we find a one-phase region. Along the phase boundary, the total concentration of beads in the coacervate phase is nearly constant, corresponding to a "Bjerrum liquid''. This result can be extended to experimental phase diagrams by assigning appropriate volumes to monomers and salts.

The desalting/salting pathway: a route to form metastable aggregates with tuneable morphologies and lifetimes

We investigate the formation/re-dissociation mechanisms of hybrid complexes made from negatively charged PAA_2k coated γ-Fe₂O₃ nanoparticles (NP) and positively charged polycations (PDADMAC) in aqueous solution in the regime of very high ionic strength (I). When the building blocks are mixed at large ionic strength (1 M NH₄Cl), the electrostatic interaction is screened and complexation does not occur. If the ionic strength is then lowered down to a targeted ionic strength I_target, there is a critical threshold I_c = 0.62 M at which complexation occurs, that is independent of the charge ratio Z and the pathway used to reduce salinity (drop-by-drop mixing or fast mixing). If salt is added back up to 1 M, the transition is not reversible and persistent out-of-equilibrium aggregates are formed. The lifetimes of such aggregates depends on I_target: the closer I_target to I_c is, the more difficult it is to dissolve the aggregates. Such peculiar behavior is driven by the inner structure of the complexes that are formed after desalting. When I_target is far below I_c, strong electrostatic interactions induce the formation of dense, compact and frozen aggregates. Such aggregates can only poorly reorganize further on with time, which makes their dissolution upon resalting almost reversible. Conversely, when I_target is close to I_c more open aggregates are formed due to weaker electrostatic interactions upon desalting. The system can thus rearrange with time to lower its free energy and reach more stable out-of-equilibrium states which are very difficult to dissociate back upon resalting, even at very high ionic strength.

Extracting free energies of counterion binding to polyelectrolytes by molecular dynamics simulations

We use all-atom molecular dynamics simulations to extract ΔG^eff, the free energy of binding of potassium ions K⁺ to the partially charged polyelectrolyte poly(acrylic acid), or PAA, in dilute regimes. Upon increasing the charge fraction of PAA, the chains adopt more extended conformations, and simultaneously, potassium ions bind more strongly (i.e., with more negative ΔG^eff) to the highly charged chains to relieve electrostatic repulsions between charged monomers along the chains. We compare the simulation results with the predictions of a model that describes potassium binding to PAA chains as a reversible reaction whose binding free energy (ΔG^eff) is adjusted from its intrinsic value (ΔG) by electrostatic correlations, captured by a random phase approximation. The bare or intrinsic binding free energy ΔG, which is an input in the model, depends on the binding species and is obtained from the radial distribution function of K⁺ around the charged monomer of a singly charged, short PAA chain in dilute solutions. We find that the model yields semi-quantitative predictions for ΔG^eff and the degree of potassium binding to PAA chains, α, as a function of PAA charge fraction without using fitting parameters.

Aqueous Formulation of Concentrated Semiconductive Fluid Using Polyelectrolyte Coacervation

Conjugated polyelectrolytes (CPEs), which combine π-conjugated backbones with ionic side chains, are intrinsically soluble in polar solvents and have demonstrated tunability with respect to solution processability and optoelectronic performance. However, this class of polymers often suffers from limited solubility in water. Here, we demonstrate how polyelectrolyte coacervation can be utilized for aqueous processing of conjugated polymers at extremely high polymer loading. Sampling various mixing conditions, we identify compositions that enable the formation of complex coacervates of an alkoxysulfonate-substituted PEDOT (PEDOT-S) with poly(3-methyl-1-propylimidazolylacrylamide) (PA-MPI). The resulting coacervate is a viscous fluid containing 50% w/v polymer and can be readily blade-coated into films of 4 ± 0.5 μm thick. Subsequent acid doping of the film increased the electrical conductivity of the coacervate to twice that of a doped film of neat PEDOT-S. This higher conductivity of the doped coacervate film suggests an enhancement in charge carrier transport along PEDOT-S backbone, in agreement with spectroscopic data, which shows an enhancement in the conjugation length of PEDOT-S upon coacervation. This study illustrates the utilization of electrostatic interactions in aqueous processing of conjugated polymers, which will be useful in large-scale industrial processing of semiconductive materials using limited solvent and with added enhancements to optoelectronic properties.

Multiphase Coacervates Driven by Electrostatic Correlations

The liquid-liquid phase separation of a polyelectrolyte solution containing one type of negatively and two types of positively charged polymers with different charge densities is studied theoretically by random phase approximation (RPA). It is predicted that multicoacervate phases could coexist, driven purely by electrostatic correlations. The asymmetry of the linear charge density could induce an effective immiscibility between two positively charged polyelectrolytes, leading to the multiphase separation. Adding salt will induce the disappearance of the dilute phase, forming two coexisting complex phases, instead of fusion between coacervates. Raising temperature could either induce a two coexisting complex phase, or a dilute phase coexisting with a coacervate phase, depending on the bulk concentration. Our predictions are in good agreement with experiments and provide insights in the further designing of the multiphase coacervation system.

Transfer Matrix Model of pH Effects in Polymeric Complex Coacervation

Oppositely charged polyelectrolytes can undergo an associative phase separation, in a process known as polymeric complex coacervation. This phenomenon is driven by the electrostatic attraction between polyanion and polycation species, leading to the formation of a polymer-dense coacervate phase and a coexisting polymer-dilute supernatant phase. There has been significant recent interest in the physical origin and features of coacervation; yet notably, experiments often use weak polyelectrolytes the charge state of which depends on solution pH, while theoretical or computational efforts typically assume strong polyelectrolytes that remain fully charged. There have been only a few efforts to address this limitation, and thus there has been little exploration of how pH can affect complex coacervation. In this paper, we modify a transfer matrix theory of coacervation to account for acid-base equilibria, taking advantage of its ability to directly account for some local ion correlations that will affect monomer charging. We show that coacervation can stabilize the charged state of a weak polyelectrolyte via the proximity of oppositely charged monomers, and can lead to asymmetric phase diagrams where the positively and negatively charged polyelectrolytes exhibit different behaviors near the pK_a of either chain. Specifically, there is a partitioning of one of the salt species to a coacervate to maintain electroneutrality when one of the polyelectrolytes is only partially charged. This results in the depletion of the same salt species in the supernatant, and overall can suppress phase separation. We also demonstrate that, when one of the species is only partially charged, mixtures that are off-stoichiometric in volume fraction but stoichiometric in charge exhibit the greatest propensity to form coacervate phases.

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.

Looping-in complexation and ion partitioning in nonstoichiometric polyelectrolyte mixtures

A wide variety of intracellular membraneless compartments are formed via liquid-liquid phase separation of charged proteins and nucleic acids. Understanding the stability of these compartments, while accounting for the compositional heterogeneity intrinsic to cellular environments, poses a daunting challenge. We combined experimental and theoretical efforts to study the effects of nonstoichiometric mixing on coacervation behavior and accurately measured the concentrations of polyelectrolytes and small ions in the coacervate and supernatant phases. For synthetic polyacrylamides and polypeptides/DNA, with unequal mixing stoichiometry, we report a general "looping-in" phenomenon found around physiological salt concentrations, where the polymer concentrations in the coacervate initially increase with salt addition before subsequently decreasing. This looping-in behavior is captured by a molecular model that considers reversible ion binding and electrostatic interactions. Further analysis in the low-salt regime shows that the looping-in phenomenon originates from the translational entropy of counterions that are needed to neutralize nonstoichiometric coacervates.

Molecular Mass Dependence of Interfacial Tension in Complex Coacervation

The interfacial tension of coacervates, the liquidlike phase composed of oppositely charged polymers that coexists at equilibrium with a supernatant, forms the basis for multiple technologies. Here we present a comprehensive set of experiments and molecular dynamics simulations to probe the effect of molecular mass on interfacial tension γ, far from the critical point, and derive γ=γ_{∞}(1-h/N), where N is the degree of polymerization, γ_{∞} is the infinite molecular mass limit, and h is a constant that physically corresponds to the number of monomers of one chain within the coacervate correlation volume.

Coacervate formation studied by explicit solvent coarse-grain molecular dynamics with the Martini model

Complex coacervates are liquid-liquid phase separated systems, typically containing oppositely charged polyelectrolytes. They are widely studied for their functional properties as well as their potential involvement in cellular compartmentalization as biomolecular condensates. Diffusion and partitioning of solutes into a coacervate phase are important to address because their highly dynamic nature is one of their most important functional characteristics in real-world systems, but are difficult to study experimentally or even theoretically without an explicit representation of every molecule in the system. Here, we present an explicit-solvent, molecular dynamics coarse-grain model of complex coacervates, based on the Martini 3.0 force field. We demonstrate the accuracy of the model by reproducing the salt dependent coacervation of poly-lysine and poly-glutamate systems, and show the potential of the model by simulating the partitioning of ions and small nucleotides between the condensate and surrounding solvent phase. Our model paves the way for simulating coacervates and biomolecular condensates in a wide range of conditions, with near-atomic resolution.

The effect of ion pairs on coacervate-driven self-assembly of block polyelectrolytes

The incorporation of oppositely charged polyelectrolytes into a block copolymer system can lead to formation of microphase separated nanostructures driven by the electrostatic complex between two oppositely charged blocks. It is a theoretical challenge to build an appropriate model to handle such coacervate-driven self-assembly, which should capture the strong electrostatic correlations for highly charged polymers. In this paper, we develop the self-consistent field theory considering the ion paring effect to predict the phase behavior of block polyelectrolytes. In our model, two types of ion pairs, the binding between two oppositely charged monomers and the binding between charged monomers and counterions, are included. Their strength of formation is controlled by two parameters K_aa and K_ac, respectively. We give a detailed analysis about how the binding strength K_ac and K_aa and salt concentration affect the self-assembled nanostructure of diblock polyelectrolyte systems. The results show that the binding between two oppositely charged blocks provides driven force for microphase separation, while the binding between charged monomers and counterions competes with the polyion pairing and thus suppresses the microphase separation. The addition of salt has a shielding effect on the charges of polymers, which is a disadvantage to microphase separation. The phase diagrams as a function of polymer concentration and salt concentration at different situations are constructed, and the influence of K_aa, K_ac, and charged block composition f_a is analyzed in depth. The obtained phase diagrams are in good agreement with currently existing experimental and theoretical results.

Dynamic Coupling in Unentangled Liquid Coacervates Formed by Oppositely Charged Polyelectrolytes

We develop a scaling theory that predicts the dynamics of symmetric and asymmetric unentangled liquid coacervates formed by solutions of oppositely-charged polyelectrolytes. Symmetric coacervates made from oppositely-charged polyelectrolytes consist of polycations and polyanions with equal and opposite charge densities along their backbones. These symmetric coacervates can be described as mixtures of polyelectrolytes in the quasi-neutral regime with a single correlation length. Asymmetric coacervates are made from polycations and polyanions with unequal charge densities. The difference in charge densities results in a double semidilute structure of asymmetric coacervates with two correlation lengths, one for the high-charge-density and the other for the low-charge-density polyelectrolytes. We predict that the double-semidilute structure in asymmetric coacervates results in a dynamic coupling which increases the friction of the high-charge-density polyelectrolyte. This dynamic coupling increases the contribution to the zero-shear viscosity of the high-charge-density polyelectrolyte. The diffusion coefficient of the high-charge-density polyelectrolyte is predicted to depend on the concentration and degree of polymerization of the low-charge-density polyelectrolyte in the coacervate if the size of the low-charge-density polymer is smaller than the correlation length of the high-charge-density polymer. We also predict a non-monotonic salt concentration dependence of the zero-shear viscosity of asymmetric coacervates.

Overcharging of polyelectrolyte complexes: an entropic phenomenon

Overcharging in complex coacervation, in which a polyelectrolyte complex coacervate (PEC) initially containing equal moles of the cationic and anionic monomers absorbs a large excess of one type of polyelectrolyte species, is predicted using a recently developed thermodynamic model describing complexation through a combination of reversible ion binding on the chains and long-range electrostatic correlations. We show that overcharging is favored roughly equally by the translational entropy of released counterions and the binding entropy of polyelectrolytes in the polyelectrolyte complex, thus helping resolve competing explanations for overcharging in the literature. We find that the extent of overcharging is non-monotonic in the concentration of added salt and increases with both strength of ion-pairing between polyions and chain hydrophobicity. The predicted extent of overcharging of the PEC is directly compared with that of multilayers made of poly(diallyldimethylammonium), PDADMA, and poly(styrene-sulfonate), PSS, overcompensated by the polycation in two different salts: KBr and NaCl. Accounting for the specificity of salt ion interactions with the polyelectrolytes, we find good qualitative agreement between theory and experiment.