Theory of competitive counterion adsorption on flexible polyelectrolytes: divalent salts

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

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

Ion Correlation-Driven Hysteretic Adhesion and Repulsion between Opposing Polyelectrolyte Brushes

Polyelectrolyte (PE) brushes are widely used in biomaterials and nanotechnology to regulate the surface properties and interactions. Here, we apply the electrostatic correlation augmented self-consistent field theory to investigate the interactions between opposing PE brushes in a mixture of 1:1 and 3:1 salt solutions. Our theory predicts a hysteretic feature of the normal stress induced by strong ion correlations. In the presence of trivalent ions, the force profile is discontinuous: repulsive in the compression branch and adhesive in the separation branch. The molecular origin of the hysteretic force is the coexistence of two collapsed modes: two separated condensed layers on each surface in the compression and a single bundled condensed layer in the separation. With the systematic inclusion of ion correlations, our theory captures well the hysteretic force, adhesive separation, "jump-in" and "jump-out" features, and the "specific ion effect", all in good agreement with the reported experimental results.

Beyond monopole electrostatics in regulating conformations of intrinsically disordered proteins

Conformations and dynamics of an intrinsically disordered protein (IDP) depend on its composition of charged and uncharged amino acids, and their specific placement in the protein sequence. In general, the charge (positive or negative) on an amino acid residue in the protein is not a fixed quantity. Each of the ionizable groups can exist in an equilibrated distribution of fully ionized state (monopole) and an ion-pair (dipole) state formed between the ionizing group and its counterion from the background electrolyte solution. The dipole formation (counterion condensation) depends on the protein conformation, which in turn depends on the distribution of charges and dipoles on the molecule. Consequently, effective charges of ionizable groups in the IDP backbone may differ from their chemical charges in isolation-a phenomenon termed charge-regulation. Accounting for the inevitable dipolar interactions, that have so far been ignored, and using a self-consistent procedure, we present a theory of charge-regulation as a function of sequence, temperature, and ionic strength. The theory quantitatively agrees with both charge reduction and salt-dependent conformation data of Prothymosin-alpha and makes several testable predictions. We predict charged groups are less ionized in sequences where opposite charges are well mixed compared to sequences where they are strongly segregated. Emergence of dipolar interactions from charge-regulation allows spontaneous coexistence of two phases having different conformations and charge states, sensitively depending on the charge patterning. These findings highlight sequence dependent charge-regulation and its potential exploitation by biological regulators such as phosphorylation and mutations in controlling protein conformation and function.

Structural dynamics of the intrinsically disordered linker region of cardiac troponin T

The cardiac troponin complex, composed of troponins I, T, and C, plays a central role in regulating the calcium-dependent interactions between myosin and the thin filament. Mutations in troponin can cause cardiomyopathies; however, it is still a major challenge for the field to connect how changes in sequence affect troponin's function. Recent high-resolution structures of the thin filament revealed critical insights into the structure-function relationship of the troponin complex, but there remain large, unresolved segments of troponin, including the troponin-T linker region that is a hotspot for several cardiomyopathy mutations. This unresolved yet functionally-significant linker region has been proposed to be intrinsically disordered, with behaviors that are not well described by traditional structural approaches; however, this proposal has not been experimentally verified. Here, we used a combination of single-molecule Förster resonance energy transfer (FRET), molecular dynamics simulations, and functional reconstitution assays to investigate the troponin-T linker region. We experimentally and computationally show that in the context of both isolated troponin and the fully regulated troponin complex, the linker behaves as a dynamic, intrinsically disordered region. This region undergoes polyampholyte expansion in the presence of high salt and distinct conformational changes during the assembly of the troponin complex. We also examine the ΔE160 hypertrophic cardiomyopathy mutation in the linker, and we demonstrate that this mutation does not affect the conformational dynamics of the linker, rather it allosterically affects interactions with other subunits of the troponin complex, leading to increased molecular contractility. Taken together, our data clearly demonstrate the importance of disorder within the troponin-T linker and provide new insights into the molecular mechanisms controlling the pathogenesis of cardiomyopathies.

eDNA Adsorption onto Microplastics: Impacts of Water Chemistry and Polymer Physiochemical Properties

Adsorption of biomacromolecules onto polymer surfaces, including microplastics (MPs), occurs in multiple environmental compartments, forming an ecocorona. Environmental DNA (eDNA), genetic material shed from organisms, can adsorb onto MPs which can potentially either (1) promote long-range transport of antibiotic resistant genes or (2) serve to gain insights into the transport pathways and origins of MPs by analyzing DNA sequences on MPs. However, little is known about the capacity of MPs to adsorb eDNA or the factors that influence sorption, such as polymer and water chemistries. Here we investigated the adsorption of extracellular linear DNA onto a variety of model MP fragments composed of three of the most environmentally prevalent polymers (polyethylene, polyethylene terephthalate, and polystyrene) in their pristine and photochemically weathered states. Batch adsorption experiments in a variety of water chemistries were complemented with nonlinear modeling to quantify the rate and extent of eDNA sorption. Ionic strength was shown to strongly impact DNA adsorption by reducing or inhibiting electrostatic repulsion. Polyethylene terephthalate exhibited the highest adsorption capacity when normalizing for MP specific surface area, likely due to the presence of ester groups. Kinetics experiments showed fast adsorption (majority adsorbed under 30 min) before eventually reaching equilibrium after 1-2 h. Overall, we demonstrated that DNA quickly binds to MPs, with pseudo-first- and -second-order models describing adsorption kinetics and the Freundlich model describing adsorption isotherms most accurately. These insights into DNA sorption onto MPs show that there is potential for MPs to act as vectors for genetic material of interest, especially considering that particle-bound DNA typically persists longer in the environment than dissolved DNA.

Adsorption Isotherm and Mechanism of Ca2+ Binding to Polyelectrolyte

Polyelectrolytes, such as poly(acrylic acid) (PAA), can effectively mitigate CaCO3 scale formation. Despite their success as antiscalants, the underlying mechanism of binding of Ca2+ to polyelectrolyte chains remains unresolved. Through all-atom molecular dynamics simulations, we constructed an adsorption isotherm of Ca2+ binding to sodium polyacrylate (NaPAA) and investigated the associated binding mechanism. We find that the number of calcium ions adsorbed [Ca2+]ads to the polymer saturates at moderately high concentrations of free calcium ions [Ca2+]aq in the solution. This saturation value is intricately connected with the binding modes accessible to Ca2+ ions when they bind to the polyelectrolyte chain. We identify two dominant binding modes: the first involves binding to at most two carboxylate oxygens on a polyacrylate chain, and the second, termed the high binding mode, involves binding to four or more carboxylate oxygens. As the concentration of free calcium ions [Ca2+]aq increases from low to moderate levels, the polyelectrolyte chain undergoes a conformational transition from an extended coil to a hairpin-like structure, enhancing the accessibility to the high binding mode. At moderate concentrations of [Ca2+]aq, the high binding mode accounts for at least one-third of all binding events. The chain's conformational change and its consequent access to the high binding mode are found to increase the overall Ca2+ ion binding capacity of the polyelectrolyte chain.

Dilute polyelectrolyte solutions: recent progress and open questions

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

Multivalent Salt-Induced Self-Assembly of Amphiphilic Polyelectrolytes of Different Charge Fractions: A Coarse-Grained Molecular Dynamics Simulation Study

Amphiphilic polymers with both hydrophobic and hydrophilic blocks are of great interest for their potential applications in drug delivery. Their self-assembly behavior in response to environmental factors like ion charge and multivalent salt concentration has been the subject of recent investigation. Our study utilizes coarse-grained molecular dynamics simulations to investigate the aggregation behavior of amphiphilic copolymers upon introducing tetravalent salt at varying charge fractions. We identify a critical concentration, Cs*, where the aggregation number reaches its maximum for each charge fraction, followed by a subsequent decrease at the excessive salt regime. This study reveals distinct morphological transitions in response to increasing salt concentration and decreasing charged fractions, namely, (i) stable dispersed micelles, (ii) a singular micelle comprising all copolymer chains, and (iii) redispersed micelles, particularly evident at lower charged fractions. Our study highlights the significant influence of tetravalent salt and charge fractions of polyelectrolyte chains on the self-assembly behavior of polyelectrolyte copolymers.

Effect of counterion size on polyelectrolyte conformations and thermodynamics

We present a theoretical model to study the effect of counterion size on the effective charge, size, and thermodynamic behavior of a single, isolated, and flexible polyelectrolyte (PE) chain. We analyze how altering counterion size modifies the energy and entropy contributions to the system, including the ion-pair free energy, excluded volume interactions, entropy of free and condensed ions, and dipolar attraction among monomer-counterion pairs, which result in competing effects challenging intuitive predictions. The PE self-energy is calculated using the Edwards-Muthukumar Hamiltonian, considering a Gaussian monomer distribution for the PE. The condensed ions are assumed to be confined within a cylindrical volume around the PE backbone. The dipolar and excluded volume interactions are described by the second and third virial coefficients. The assumption of freely rotating dipoles results in a first-order coil-globule transition of the PE chain. A more realistic, weaker dipolar attraction, parameterized in our theory, shifts it to a second-order continuous transition. We calculate the size scaling-exponent of the PE and find exponents according to the relative dominance of the electrostatic, excluded volume, or dipolar effects. We further identify the entropy- and energy-driven regimes of the effective charge and conformation of the PE, highlighting the interplay of free ion entropy and ion-pair energy with varying electrostatic strengths. The crossover strength, dependent on the counterion size, indicates that diminishing sizes favor counterion condensation at the expense of free ion entropy. The predictions of the model are consistent with trends in simulations and generalize the findings of the point-like counterion theories.

Multivalent binding proteins can drive collapse and reswelling of chromatin in confinement

Collapsed conformations of chromatin have been long suspected of being mediated by interactions with multivalent binding proteins, which can bring together distant sections of the chromatin fiber. In this study, we use Langevin dynamics simulation of a coarse grained chromatin polymer to show that the role of binding proteins can be more nuanced than previously suspected. In particular, for chromatin polymer in confinement, entropic forces can drive reswelling of collapsed chromatin with increasing binder concentrations, and this reswelling transition happens at physiologically relevant binder concentrations. Both the extent of collapse, and also of reswelling depends on the strength of confinement. We also study the kinetics of collapse and reswelling and show that both processes occur in similar timescales. We characterise this reswelling of chromatin in biologically relevant regimes and discuss the non-trivial role of multivalent binding proteins in mediating the spatial organisation of the genome.

The biophysics of disordered proteins from the point of view of single-molecule fluorescence spectroscopy

Intrinsically disordered proteins (IDPs) and regions (IDRs) have emerged as key players across many biological functions and diseases. Differently from structured proteins, disordered proteins lack stable structure and are particularly sensitive to changes in the surrounding environment. Investigation of disordered ensembles requires new approaches and concepts for quantifying conformations, dynamics, and interactions. Here, we provide a short description of the fundamental biophysical properties of disordered proteins as understood through the lens of single-molecule fluorescence observations. Single-molecule Förster resonance energy transfer (FRET) and fluorescence correlation spectroscopy (FCS) provides an extensive and versatile toolbox for quantifying the characteristics of conformational distributions and the dynamics of disordered proteins across many different solution conditions, both in vitro and in living cells.

Ion-specific nanoscale compaction of cysteine-modified poly(acrylic acid) brushes revealed by 3D scanning force microscopy with frequency modulation detection

Stimuli-responsive polyelectrolyte brushes adapt their physico-chemical properties according to pH and ion concentrations of the solution in contact. We synthesized a poly(acrylic acid) bearing cysteine residues at side chains and a lipid head group at the terminal, and incorporated them into a phospholipid monolayer deposited on a hydrophobic silane monolayer. The ion-specific, nanoscale response of polyelectrolyte brushes was detected by using three-dimensional scanning force microscopy (3D-SFM) combined with frequency modulation detection. The obtained topographic and mechanical landscapes indicated that the brushes were uniformly stretched, undergoing a gradual transition from the brush to the bulk electrolyte in the absence of divalent cations. When 1 mM calcium ions were added, the brushes were uniformly compacted, exhibiting a sharper brush-to-bulk transition. Remarkably, the addition of 1 mM cadmium ions made the brush surface significantly rough and the mechanical landscape highly heterogeneous. Currently, cadmium-specific nanoscale compaction of the brushes is attributed to the coordination of thiol and carboxyl side chains with cadmium ions, as suggested for naturally occurring, heavy metal binding proteins.

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.

Kinetics of charged polymer collapse in poor solvents

Extensive molecular dynamics simulations, using simple charged polymer models, have been employed to probe the collapse kinetics of a single flexible polyelectrolyte (PE) chain under implicit poor solvent conditions. We investigate the role of the charged nature of PE chain (A), valency of counterions (Z) on the kinetics of such PE collapse. Our study shows that the collapse kinetics of charged polymers are significantly different from those of the neutral polymer and that the finite-size scaling behavior of PE collapse times does not follow the Rouse scaling as observed in the case of neutral polymers. The critical exponent for charged PE chains is found to be less than that of neutral polymers and also exhibits dependence on counterion valency. The coarsening of clusters along the PE chain suggests a multi-stage collapse and exhibits opposite behavior of exponents compared to neutral polymers: faster in the early stages and slower in the later stages of collapse.

Effects of temperature on chaotropic anion-induced shape transitions of star molecular bottlebrushes with heterografted poly(ethylene oxide) and poly(N,N-dialkylaminoethyl methacrylate) side chains in acidic water

This article reports a study of the effects of temperature on chaotropic anion (CA)-induced star-globule shape transitions in acidic water of three-arm star bottlebrushes composed of heterografted poly(ethylene oxide) (PEO) and either poly(2-(N,N-dimethylamino)ethyl methacrylate) (PDMAEMA) or poly(2-(N,N-diethylamino)ethyl methacrylate) (PDEAEMA) (the brushes denoted as SMB-11 and -22, respectively). The brush polymers were synthesized by grafting alkyne-end-functionalized PEO and PDMAEMA or PDEAEMA onto an azide-bearing three-arm star backbone polymer using the copper(i)-catalyzed alkyne-azide cycloaddition reaction. Six anions were studied for their effects on the conformations of SMB-11 and -22 in acidic water: super CAs [Fe(CN)6]3- and [Fe(CN)6]4-, moderate CAs PF6- and ClO4-, weak CA I-, and for comparison, kosmotropic anion SO42-. At 25 °C, the addition of super and moderate CAs induced shape transitions of SMB-11 and -22 in pH 4.50 water from a starlike to a collapsed globular state stabilized by PEO side chains, which was driven by the ion pairing of protonated tertiary amine groups with CAs and the chaotropic effect. The shape changes occurred at much lower salt concentrations for super CAs than moderate CAs. Upon heating from near room temperature to 70 °C, the super CA-collapsed brushes remained in the globular state, whereas the moderate CA-collapsed brushes underwent reversible globule-to-star shape transitions. The transition temperature increased with increasing salt concentration and was found to be higher for SMB-22 at the same salt concentration, presumably caused by the chaotropic effect. In contrast, I- and SO42- had small effects on the conformations of SMB-11 and -22 at 25 °C in the studied salt concentration range, and only small and gradual size variations were observed upon heating to 70 °C. The results reported here may have potential uses in the design of stimuli-responsive systems for substance encapsulation and release.

Using Molecular Simulations to Understand the Effect of Dodecyl Sulfate on the Calcium-Binding Ability of Polystyrene Sulfonate

We demonstrate the potential to tune the binding of calcium ions with polystyrene sulfonate (PSS) in the presence of dodecyl sulfate (DS). This can aid the design of surfactant-responsive water-softening agents for applications in detergency. We use molecular dynamics simulations to study the effect of the concentration of DS ions and the degree of sulfonation on the propensity of calcium ions toward PSS. We observe that the presence of DS ions increases the propensity of calcium ions toward 100% sulfonated PSS. The above phenomenon is due to the hydrophobic attraction between PSS and DS at low DS concentrations and the formation of calcium ion bridges between sulfonate and sulfate groups at moderate to high DS concentrations. We also observe the formation of calcium ion bridges between the sulfonate groups at high DS concentrations. The presence of DS ions also increases the propensity of calcium ions toward 20% sulfonated PSS. This is mainly due to the hydrophobic attraction between PSS and DS ions. The calcium ion bridges between sulfonate and sulfate groups are less prevalent than those of 100% sulfonated PSS. We do not observe calcium ion bridges between sulfonate groups of 20% sulfonated PSS. We use the above-mentioned insights to suggest potential strategies for designing an anionic polyelectrolyte having a suitable calcium-binding ability at a given concentration of the anionic surfactant. Finally, strong PSS-DS affinity is detrimental to the activity of surfactants because less surfactant ions are available for detergency. Our results also indicate the possibility of altering the PSS-DS affinity by changing the degree of sulfonation.

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.

Effects of Amino Acid Side-Chain Length and Chemical Structure on Anionic Polyglutamic and Polyaspartic Acid Cellulose-Based Polyelectrolyte Brushes

We used atomistic molecular dynamics (MD) simulations to study polyelectrolyte brushes based on anionic α,L-glutamic acid and α,L-aspartic acid grafted on cellulose in the presence of divalent CaCl₂ salt at different concentrations. The motivation is to search for ways to control properties such as sorption capacity and the structural response of the brush to multivalent salts. For this detailed understanding of the role of side-chain length, the chemical structure and their interplay are required. It was found that in the case of glutamic acid oligomers, the longer side chains facilitate attractive interactions with the cellulose surface, which forces the grafted chains to lie down on the surface. The additional methylene group in the side chain enables side-chain rotation, enhancing this effect. On the other hand, the shorter and more restricted side chains of aspartic acid oligomers prevent attractive interactions to a large degree and push the grafted chains away from the surface. The difference in side-chain length also leads to differences in other properties of the brush in divalent salt solutions. At a low grafting density, the longer side chains of glutamic acid allow the adsorbed cations to be spatially distributed inside the brush resulting in a charge inversion. With an increase in grafting density, the difference in the total charge of the aspartic and glutamine brushes disappears, but new structural features appear. The longer sides allow for ion bridging between the grafted chains and the cellulose surface without a significant change in main-chain conformation. This leads to the brush structure being less sensitive to changes in salt concentration.

Strong Reduction of the Chain Rigidity of Hyaluronan by Selective Binding of Ca2+ Ions

The biological functions of natural polyelectrolytes are strongly influenced by the presence of ions, which bind to the polymer chains and thereby modify their properties. Although the biological impact of such modifications is well recognized, a detailed molecular picture of the binding process and of the mechanisms that drive the subsequent structural changes in the polymer is lacking. Here, we study the molecular mechanism of the condensation of calcium, a divalent cation, on hyaluronan, a ubiquitous polymer in human tissues. By combining two-dimensional infrared spectroscopy experiments with molecular dynamics simulations, we find that calcium specifically binds to hyaluronan at millimolar concentrations. Because of its large size and charge, the calcium cation can bind simultaneously to the negatively charged carboxylate group and the amide group of adjacent saccharide units. Molecular dynamics simulations and single-chain force spectroscopy measurements provide evidence that the binding of the calcium ions weakens the intramolecular hydrogen-bond network of hyaluronan, increasing the flexibility of the polymer chain. We also observe that the binding of calcium to hyaluronan saturates at a maximum binding fraction of ∼10-15 mol %. This saturation indicates that the binding of Ca2+ strongly reduces the probability of subsequent binding of Ca2+ at neighboring binding sites, possibly as a result of enhanced conformational fluctuations and/or electrostatic repulsion effects. Our findings provide a detailed molecular picture of ion condensation and reveal the severe effect of a few, selective and localized electrostatic interactions on the rigidity of a polyelectrolyte chain.

Role of electrostatic interactions in the adsorption of dye molecules by Ti3C2-MXenes

MXenes, a new class of 2D materials, have recently attracted increasing attention as promising adsorbents for environmental remediation. It has been previously demonstrated that MXenes can successfully capture selected organic dyes from aqueous media; however, to date, the adsorption performance of MXenes for a wide variety of dyes in simulated real-life aquatic environments other than clean laboratory deionized (DI) water has not been systematically investigated. In this study, we systematically investigated the adsorption performance of delaminated Ti3C2-MXenes for six different organic dyes in aquatic media at different pH levels and ionic strengths. Our results strongly suggest the importance of the electrostatic interactions between the ionizable functional groups of MXenes and dyes for removal efficiency. The electrostatic repulsions between negatively charged MXenes and certain anionic dyes reduced the removal efficiencies of MXenes for these dyes in DI water; however, the presence of divalent cations significantly improved the removal efficiencies, possibly owing to the charge screening effects and like-charge attractions mediated by cation binding to the functionalities of dyes and MXenes. These results provide a rational strategy for optimizing the conditions for efficient removal of different types of organic dyes using MXenes.

Ion-mediated interactions between like-charged polyelectrolytes with bending flexibility

Ion-mediated interactions between polyelectrolytes (PEs) are crucial to the properties of flexible biopolymers such as nucleic acids and proteins but the effect of PE flexibility on such interactions has not been explicitly addressed until now. In this work, the potentials of mean force (PMFs) between like-charged PEs with different bending flexibility have been investigated by Monte Carlo simulations and a cylindrical confinement around each PE was involved to model two PEs in an array. We found that in the absence of trivalent salt, the PMFs between like-charged PEs in an array are apparently repulsive while the bending flexibility can visibly decrease the repulsive PMFs. With the addition of high trivalent salt, the PMFs become significantly attractive whereas the attractive PMFs can be apparently weakened by the bending flexibility. Our analyses reveal that the effect of bending flexibility is attributed to the increased PE conformational space, which allows the PEs to fluctuate away to decrease the monovalent ion-mediated repulsion or to weaken the trivalent ion-mediated attraction through disrupting trivalent ion-bridging configuration. Additionally, our further calculations show that the effect of bending flexibility on the ion-mediated interactions is less apparent for PEs without cylindrical confinement.

Ion Pairing and the Structure of Gel Coacervates

A scaling model for the structure of coacervates is presented for mixtures of oppositely-charged polyelectrolytes of both symmetric and asymmetric charge-densities for different degrees of electrostatic strength and levels of added salt. At low electrostatic strengths, weak coacervates, with the energy of electrostatic interactions between charges less than the thermal energy, k B T, are liquid. At higher electrostatic strengths, strong coacervates are gels with crosslinks formed by ion pairs of opposite charges bound to each other with energy higher than k B T. Charge-symmetric coacervates are formed for mixtures of oppositely-charged polyelectrolytes with equal and opposite charge-densities. While charge-symmetric weak coacervates form a semidilute polymer solution with a correlation length equal to the electrostatic blob size, charge-symmetric strong coacervates form reversible gels with a correlation length on the order of the distance between bound ion pairs. Charge-asymmetric coacervates are formed from mixtures of oppositely-charged polyelectrolytes with different charge-densities. While charge-asymmetric weak coacervates form double solutions with two correlation lengths and qualitatively different chain conformations of polycations and polyanions, charge-asymmetric strong coacervates form bottlebrush and star-like gels. Unlike liquid coacervates, for which an increase in the concentration of added salt screens electrostatic interactions, causing structural rearrangement and eventually leads to their dissolution, the salt does not affect the structure of strong coacervates until ion pairs dissociate and the gel disperses.

Effect of Buffer Anions on Pearl-Necklace Morphology of Tertiary Amine-Containing Binary Heterografted Linear Molecular Bottlebrushes in Acidic Aqueous Buffers

Molecular bottlebrushes can exhibit a multitude of distinct conformations under different conditions, and precise control of their morphology can facilitate better use of such materials in potential applications. Herein, we report a study on the effect of buffer anions on the pearl-necklace morphology of linear binary heterografted molecular brushes consisting of pH-responsive poly(2-N,N-diethylamino)ethyl methacrylate) (PDEAEMA) with a pK_a of 7.40 and thermoresponsive poly(ethoxydi(ethylene glycol) acrylate) (PDEGEA) with a lower critical solution temperature of 9 °C as side chains in various acidic aqueous buffers at 0 °C. The molecular brushes, denotated as BMB, were prepared by a grafting-to approach using copper(I)-catalyzed azide-alkyne cycloaddition reaction. Dynamic light scattering studies showed that the apparent hydrodynamic size of BMB in aqueous buffers with a pH of 6.50 at 1 °C decreased with increasing valency of buffer anions, from acetate anions with a charge of 1-, to phosphate anions carrying charges of 2- and 1- and citrate anions bearing charges of 3- and 2- at pH = 6.50. Atomic force microscopy revealed that BMB exhibited a pearl-necklace morphology from all three aqueous buffers with a pH of 6.50 when spin-cast at 0 °C. Analysis of AFM images showed that the average length of BMB and the number of beads per brush molecule decreased with increasing valency of buffer anions while the size and height of the beads increased. The pearl-necklace morphology of BMB was believed to be the result of microphase separation of the neutral PDEGEA and the charged PDEAEMA side chains along the brush backbone. Multivalent kosmotropic buffer anions formed bridging linkages between protonated tertiary amine moieties and thus "crosslinked" the charged PDEAEMA side chains, resulting in the shrinkage of BMB and enhanced microphase separation of two side chain polymers.

Influence of Calcium Binding on Conformations and Motions of Anionic Polyamino Acids. Effect of Side Chain Length

Investigation of the effect of CaCl2 salt on conformations of two anionic poly(amino acids) with different side chain lengths, poly-(α-l glutamic acid) (PGA) and poly-(α-l aspartic acid) (PASA), was performed by atomistic molecular dynamics (MD) simulations. The simulations were performed using both unbiased MD and the Hamiltonian replica exchange (HRE) method. The results show that at low CaCl2 concentration adsorption of Ca2+ ions lead to a significant chain size reduction for both PGA and PASA. With the increase in concentration, the chains sizes partially recover due to electrostatic repulsion between the adsorbed Ca2+ ions. Here, the side chain length becomes important. Due to the longer side chain and its ability to distance the charged groups with adsorbed ions from both each other and the backbone, PGA remains longer in the collapsed state as the CaCl2 concentration is increased. The analysis of the distribution of the mineral ions suggests that both poly(amino acids) should induce the formation of mineral with the same structure of the crystal cell.

Ionic effects on synthetic polymers: from solutions to brushes and gels

The ionic effects on synthetic polymers have attracted extensive attention due to the crucial role of ions in the determination of the properties of synthetic polymers. This review places the focus on specific ion effects, multivalent ion effects, and ionic hydrophilicity/hydrophobicity effects in synthetic polymer systems from solutions to brushes and gels. The specific ion effects on neutral polymers are determined by both the direct and indirect specific ion-polymer interactions, whereas the ion specificities of charged polymers are mainly dominated by the specific ion-pairing interactions. The ionic cross-linking effect exerted by the multivalent ions is widely used to tune the properties of polyelectrolytes, while the reentrant behavior of polyelectrolytes in the presence of multivalent ions still remains poorly understood. The ionic hydrophilicity/hydrophobicity effects not only can be applied to make strong polyelectrolytes thermosensitive, but also can be used to prepare polymeric nano-objects and to control the wettability of polyelectrolyte brush-modified surfaces. The not well-studied ionic hydrogen bond effects are also discussed in the last section of this review.

3D structure stability of the HIV-1 TAR RNA in ion solutions: A coarse-grained model study

As an extremely common structural motif, RNA hairpins with bulge loops [e.g., the human immunodeficiency virus type 1 (HIV-1) transactivation response (TAR) RNA] can play essential roles in normal cellular processes by binding to proteins and small ligands, which could be very dependent on their three-dimensional (3D) structures and stability. Although the structures and conformational dynamics of the HIV-1 TAR RNA have been extensively studied, there are few investigations on the thermodynamic stability of the TAR RNA, especially in ion solutions, and the existing studies also have some divergence on the unfolding process of the RNA. Here, we employed our previously developed coarse-grained model with implicit salt to predict the 3D structure, stability, and unfolding pathway for the HIV-1 TAR RNA over a wide range of ion concentrations. As compared with the extensive experimental/theoretical results, the present model can give reliable predictions on the 3D structure stability of the TAR RNA from the sequence. Based on the predictions, our further comprehensive analyses on the stability of the TAR RNA as well as its variants revealed that the unfolding pathway of an RNA hairpin with a bulge loop is mainly determined by the relative stability between different states (folded state, intermediate state, and unfolded state) and the strength of the coaxial stacking between two stems in folded structures, both of which can be apparently modulated by the ion concentrations as well as the sequences.

Apparent repulsion between equally and oppositely charged spherical polyelectrolytes in symmetrical salt solutions

Ion-mediated interactions are very important for the properties of colloids and biomacromolecules such as nucleic acids and proteins. In this work, the ion-mediated interactions between equally and oppositely charged spherical polyelectrolytes (SPEs) in symmetrical divalent electrolytes have been investigated by Monte Carlo simulations, and an unexpected apparent repulsion was observed at high divalent salt concentration. Our investigations also show that the effective repulsion becomes more pronounced for SPEs with higher charge densities and for counterions with larger sizes and was found to be tightly accompanied with the over-neutralization to SPEs by condensed counterions and their release upon the approach of SPEs. Such attractive interaction can be reproduced by our proposed modified Poisson-Boltzmann model and is mainly attributed to the increase in the electrostatic repulsion between on charged SPE and the over-neutralized counterions around the other oppositely SPE with the approach of the two SPEs.

Small ion effects on self-coacervation phenomena in block polyampholytes

Self-coacervation is a phenomenon in which a solution of polyampholytes spontaneously phase separates into a dense liquid coacervate phase, rich in the polyampholyte, coexisting with a dilute supernatant phase. Such coacervation results in the formation of membraneless organelles in vivo and has further been applied industrially as synthetic encapsulants and coatings. It has been suggested that coacervation is primarily driven by the entropy gain from releasing counter-ions upon complexation. Using fully fluctuating field-theoretic simulations employing complex Langevin sampling and complementary molecular dynamics simulations, we have determined that the small ions contribute only weakly to the self-coacervation behavior of charge-symmetric block polyampholytes in solution. Salt partitioning between the supernatant and coacervate is also found to be negligible in the weak-binding regime at low electrostatic strengths. Asymmetries in charge distribution along the polyampholytes can cause net-charges that lead to "tadpole" configurations in dilute solution and the suppression of phase separation at low salt content. The field and particle-based simulation results are compared with analytical predictions from the random phase approximation (RPA) and postulated scaling relationships. The qualitative trends are mostly captured by the RPA, but the approximation fails at low concentration.

Influence of charge sequence on the adsorption of polyelectrolytes to oppositely-charged polyelectrolyte brushes

When a solution of polyanionic chains is placed in contact with a polycationic brush, the polyanions adsorb into the brush. We investigate the influence of the charge sequences of the free and bound species on the thermodynamics of polyelectrolyte adsorption. As model systems, we consider free and brush polyelectrolytes with either block or alternating charge sequences, and study the adsorption process using coarse-grained Langevin dynamics with implicit solvent, explicit counterions, and excess salt. Free energy, internal energy, and entropy of adsorption are computed using umbrella sampling methods. When the number of polyanions exceed the number of polycations, the brush becomes overcharged. Free chains adsorb most strongly when both free and tethered chains have a block charge sequence, and most weakly when both species have an alternating sequence. Adsorption is stronger when the free polyanion has a block sequence and the tethered polycation is alternating than in the reverse case of an alternating free polymer and a tethered block copolymer. Sequence-dependent effects are shown to be largely energetic, rather than entropic, in origin.

Tunable Coacervation of Well-Defined Homologous Polyanions and Polycations by Local Polarity

The ionic complexation of polyelectrolytes is an important mechanism underlying many important biological processes and technical applications. The main driving force for complexation is electrostatic, which is known to be affected by the local polarity near charge centers, but the impact of which on the complexation of polyelectrolytes remains poorly explored. We developed a homologous series of well-defined polyelectrolytes with identical backbone structures, controlled molecular weights, and tunable local polarity to modulate the solvation environment near charged groups. A multitude of systematic, accurate phase diagrams were obtained by spectroscopic measurements of polymer concentrations via fluorescent labeling of polycations. These phase diagrams unambiguously revealed that the liquidlike coacervation is more stable against salt addition at reduced local polarity over a wide range of molecular weights. These trends were quantitatively captured by a theory of complexation that incorporates the effects of dispersion interactions, charge connectivity, and reversible ion-binding, providing the microscopic design rules for tuning molecular parameters and local polarity.

Sequence charge decoration dictates coil-globule transition in intrinsically disordered proteins

We present an analytical theory to compute conformations of heteropolymers-applicable to describe disordered proteins-as a function of temperature and charge sequence. The theory describes coil-globule transition for a given protein sequence when temperature is varied and has been benchmarked against the all-atom Monte Carlo simulation (using CAMPARI) of intrinsically disordered proteins (IDPs). In addition, the model quantitatively shows how subtle alterations of charge placement in the primary sequence-while maintaining the same charge composition-can lead to significant changes in conformation, even as drastic as a coil (swelled above a purely random coil) to globule (collapsed below a random coil) and vice versa. The theory provides insights on how to control (enhance or suppress) these changes by tuning the temperature (or solution condition) and charge decoration. As an application, we predict the distribution of conformations (at room temperature) of all naturally occurring IDPs in the DisProt database and notice significant size variation even among IDPs with a similar composition of positive and negative charges. Based on this, we provide a new diagram-of-states delineating the sequence-conformation relation for proteins in the DisProt database. Next, we study the effect of post-translational modification, e.g., phosphorylation, on IDP conformations. Modifications as little as two-site phosphorylation can significantly alter the size of an IDP with everything else being constant (temperature, salt concentration, etc.). However, not all possible modification sites have the same effect on protein conformations; there are certain "hot spots" that can cause maximal change in conformation. The location of these "hot spots" in the parent sequence can readily be identified by using a sequence charge decoration metric originally introduced by Sawle and Ghosh. The ability of our model to predict conformations (both expanded and collapsed states) of IDPs at a high-throughput level can provide valuable insights into the different mechanisms by which phosphorylation/charge mutation controls IDP function.

Molecular weight dependence of chain conformation of strong polyelectrolytes

Using sodium polystyrene sulfonate (NaPSS) and quarternized poly 4-vinylpyridine (QP4VP) as model systems, the chain conformation of polyelectrolytes under finite salt concentrations is investigated at a single molecular level. By fluorescence correlation spectroscopy (FCS), the hydrodynamic radius (R _h) of the samples with the molecular weight ranging more than one order of magnitude was measured. The variations of R _h as a function of molecular weight reveal the molecular weight dependence: under moderate salt concentrations (such as 10^-4 and 0.1M), the shorter chains of both NaPSS and QP4VP take the rod-like conformation, while the longer chains take the coiled conformation (random coil or swelled random coil conformation, respectively). At high enough salt levels, both the charged chains take the coiled conformations. Photon counting histogram (PCH) measurements of the local pH value at the vicinity of the NaPSS chain expose the higher extent of counterion adsorption for longer chains as well as higher salt concentrations, telling that the charge regularization process is the major governing factor.

Effects of Ca2+ Ion Condensation on the Molecular Structure of Polystyrene Sulfonate at Air-Water Interfaces

The structure of poly(sodium 4-styrenesulfonate) (NaPSS) polyelectrolytes at air-water interfaces was investigated with tensiometry, ellipsometry, and vibrational sum-frequency generation (SFG) in the presence of low and high CaCl₂ concentrations. In addition, we have studied the foaming behavior of 20 mM NaPSS solutions to relate the PSS molecular structure at air-water interfaces to foam properties. PSS polyelectrolytes without additional salt exhibited significant surface activity, which can be tuned further by additions of CaCl₂. The hydrophobicity of the backbone due to incomplete sulfonation during synthesis is one origin, whereas the effective charge of the polyelectrolyte chain is shown to play another major role. At low salt concentrations, we propose that the polyelectrolyte is forming a layered structure. The hydrophobic parts are likely to be located directly at the interface in loops, whereas the hydrophilic parts are at low concentrations stretched out into near-interface regions in tails. Increasing the Ca²⁺ concentration leads to ion condensation, a collapse of the tails, and likely to Ca²⁺ intra- and intermolecular bridges between polyelectrolytes at the interface. The increase in both surface excess and foam stability originates from changes in the polyelectrolyte's hydrophobicity due to Ca²⁺ condensation onto the PSS polyanions. Consequently, charge screening at the interface is enhanced and repulsive electrostatic interactions are reduced. Furthermore, SFG spectra of O-H stretching bands reveal a decrease in intensity of the low-frequency branch when c(Ca²⁺) is increased whereas the high-frequency branch of O-H stretching modes persists even for 1 M CaCl₂. This originates from the remaining net charge of the PSS polyanions at the air-water interface that is not fully compensated by condensation of Ca²⁺ ions and leads to electric-field-induced contributions to the SFG spectra of interfacial H₂O. A charge reversal of the PSS net charge at the air-water interface is not observed and is consistent with bulk electrophoretic mobility measurements.

Modulating charge patterning and ionic strength as a strategy to induce conformational changes in intrinsically disordered proteins

We present an analytical theory to describe conformational changes as a function of salt for polymers with a given sequence of charges. We apply this model to describe Intrinsically Disordered Proteins (IDPs) by explicitly accounting for charged residues and their exact placement in the primary sequence while approximating the effect of non-electrostatic interactions at a mean-field level by effective short-range (two body and three-body) interaction parameters. The effect of ions is introduced by treating electrostatic interactions within Debye-Huckle approximation. Using typical values of the short-range mean-field parameters derived from all-atom Monte Carlo simulations (at zero salt), we predict the conformational changes as a function of salt concentration. We notice that conformational transitions in response to changes in ionic strength strongly depend on sequence specific charge patterning. For example, globule to coil transition can be observed upon increasing salt concentration, in stark contrast to uniformly charged polyelectrolyte theories based on net charge only. In addition, it is possible to observe non-monotonic behavior with salt as well. Drastic differences in salt-induced conformational transitions is also evident between two doubly phosphorylated sequences-derived from the same wild type sequence-that only differ in the site of phosphorylation. Similar effects are also predicted between two sequences derived from the same parent sequence differing by a single site mutation where a negative charge is replaced by a positive charge. These effects are purely a result of charge decoration and can only be understood in terms of metrics based on specific placement of charges, and cannot be explained by models based on charge composition alone. Identifying sequences and hot spots within sequences-for post translational modification or charge mutation-using our high-throughput theory will yield fundamental insights into design and biological regulation mediated by phosphorylation and/or local changes in salt concentration.

Ability of the Poisson-Boltzmann equation to capture molecular dynamics predicted ion distribution around polyelectrolytes

Here, we examine polyelectrolyte (PE) and ion chemistry specificity in ion condensation via all-atom molecular dynamics (MD) simulations and assess the ability of the Poisson-Boltzmann (PB) equation to describe the ion distribution predicted by the MD simulations. The PB model enables the extraction of parameters characterizing ion condensation. We find that the modified PB equation which contains the effective PE radius and the energy of the ion-specific interaction as empirical fitting parameters describes ion distribution accurately at large distances but close to the PE, especially when strongly localized charge or specific ion binding sites are present, the mean field description of PB fails. However, the PB model captures the MD predicted ion condensation in terms of the Manning radius and fraction of condensed counterions for all the examined PEs and ion species. We show that the condensed ion layer thickness in our MD simulations collapses on a single master curve for all the examined simple, monovalent ions (Na⁺, Br⁺, Cs⁺, Cl^-, and Br^-) and PEs when plotted against the Manning parameter (and consequently the PE line charge density). The significance of this finding is that, contrary to the Manning radius extracted from the mean field PB model, the condensed layer thickness in the all atom detail MD modelling does not depend on the PE chemistry or counterion type. Furthermore, the fraction of condensed counterions in the MD simulations exceeds the PB theory prediction. The findings contribute toward understanding and modelling ion distribution around PEs and other charged macromolecules in aqueous solutions, such as DNA, functionalized nanotubes, and viruses.