Monte Carlo study of titration of linear polyelectrolytes

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.

On the Validity of Constant pH Simulations

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

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

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

Canonical titration simulations

We present a Monte Carlo approach for performing titration simulations in the canonical ensemble. The standard constant pH (cpH) simulation methods are intrinsically grand canonical, allowing us to study the protonation state of molecules only as a function of pH in the reservoir. Due to the Donnan potential between a system and an (implicit) reservoir of a semi-grand canonical simulation, the pH of the reservoir can be significantly different from that of an isolated system, for an identical protonation state. The new titration method avoids this difficulty by using the canonical reactive Monte Carlo algorithm to calculate the protonation state of macromolecules as a function of the total number of protons present inside the simulation cell. The pH of an equilibrated system is then calculated using a new surface insertion Widom algorithm, which bypasses the difficulties associated with the bulk Widom particle insertion for intermediate and high pH values. To properly treat the long range Coulomb force, we use the Ewald summation method, showing the importance of the Bethe potential for calculating the pH of canonical systems.

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.

Accelerated simulation method for charge regulation effects

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

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

HYPOTHESIS

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

EXPERIMENTS

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

FINDINGS

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

Adsorption of lysozyme into a charged confining pore

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

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

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

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

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

On the distribution of hydrophilic polyelectrolytes and their counterions around zwitterionic micelles: the possible impact on the charge density in solution

Despite their charge neutrality, micelles composed of surfactants with zwitterionic headgroups selectively accumulate anions at their hydrophobic core/solution interphase due to electrostatic interactions if headgroup positive moieties are the innermost. This tendency may be markedly enhanced if polyions substitute simple ions. To investigate this possibility, solutions composed of zwitterionic micelles and hydrophilic polyanions have been investigated with Monte Carlo simulations representing the studied systems via primitive electrolyte models. Structural and energetic properties are obtained to highlight the impact of connecting simple ions into polyions on the interactions between electrolytes and micelles. Despite the latter, polyanions conserve their conformational properties. A marked increase in the concentration of charged species inside the micellar corona is, instead, found when polyions are present independently of their charge sign or the headgroup structure. Thus, polyelectrolytes act as "shuttle" for all charged species, with the potential of increasing reactions rates involving the latter due to mass effects. Besides, results for the polyions/micelles mixing free energy and Helmholtz energy profiles indicate that the critical micelle concentration is impacted minimally by hydrophilic polyelectrolytes, an outcome agreeing with experiments. This finding is entirely due to weak enthalpic effects while mixing hydrophilic polyions and micelles. A strong reduction in the screening of the micelle negative charge, acquired following the adsorption of anions in the corona and due to counterions layering just outside it (the so called "chameleon effect"), is forecasted when polyanions substitute monovalent anions.

Interface Counterion Localization Induces a Switch between Tight and Loose Configurations of Knotted Weak Polyacid Rings despite Intermonomer Coulomb Repulsions

Stochastic simulations have been used to investigate the conformational behavior of knotted weak polyacid rings as a function of pH. Different from the commonly expected ionization-repulsion-expansion scheme upon increasing pH, theoretical results suggest a nonmonotonic behavior of the gyration radius R_g². Polyelectrolyte recontraction at high ionization is induced by the weakening of Coulomb repulsion due to counterions (CIs) localizing at the interphase between the polymer and solvent, and the more marked it appears, the more complex is the knot topology. Compared with strong polyelectrolytic species of identical ionization, weak polyacids present tighter knots due to their ability to localize neutral monomers inside the tangled part. Increasing the solvent Bjerrum length enhances CIs localization, lowering the pH at which polyacids start decreasing their average size. A similar effect is also obtained by increasing the amount of "localizable" cations by adding salts.

Theoretical and Computational Insight into Solvent and Specific Ion Effects for Polyelectrolytes: The Importance of Local Molecular Interactions

Polyelectrolytes in solution show a broad plethora of interesting effects. In this short review article, we focus on recent theoretical and computational findings regarding specific ion and solvent effects and their impact on the polyelectrolyte behavior. In contrast to standard mean field descriptions, the properties of polyelectrolytes are significantly influenced by crucial interactions with the solvent, co-solvent and ion species. The corresponding experimental and simulation results reveal a significant deviation from theoretical predictions, which also highlights the importance of charge transfer, dispersion and polarization interactions in combination with solvation mechanisms. We discuss recent theoretical and computational findings in addition to novel approaches which help broaden the applicability of simple mean field theories.

Monte Carlo study of the effects of macroion charge distribution on the ionization and adsorption of weak polyelectrolytes and concurrent counterion release

HYPOTHESIS

Adsorption of weak polyelectrolytes onto charged nanoparticles, and concurrent effects such as spatial partitioning of ions may be influenced by details of the polyelectrolyte structure (linear or star-like) and size, by the mobility of the nanoparticle surface charge, or the valence of the nanoparticle counterions.

EXPERIMENTS

Ionization and complexation of weak polyelectrolytes on spherical macroions with monovalent and divalent countrions has been studied with constant-pH Monte Carlo titrations and primitive electrolyte models for linear and star-like polymers capable, also, of forming charged hydrogen bonds. Nanoparticles surface charge has been represented either as a single colloid-centered total charge (CCTC) or as surface-tethered mobile monovalent spherical charges (SMMSC).

FINDINGS

Differences in the average number of adsorbed polyelectrolyte arms and their average charge, and in the relative amount of macroion counterions (m-CI's) released upon polymer adsorption are found between CCTC and SMMSC nanoparticles. The amount of the counterions released also depends on the polymer structure. As CCTC adsorbs a lower number of star-like species arms, the degree of condensation of polymer counterions (p-CI's) onto the polyelectrolyte is also substantially higher for the CCTC colloid, with a concurrent decrease of the osmotic coefficient values.

Simulations of ionization equilibria in weak polyelectrolyte solutions and gels

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

Direct Demonstration of DNA Compaction Mediated by Divalent Counterions

We unambiguously demonstrated DNA attraction and its regulation mediated by divalent cations Mg²⁺ and Ca²⁺ by tethering a DNA single chain at various pH solutions. It is found that DNA is compacted when the pH of the solution containing these divalent counterions is decreased below 5. When the pH of the medium is ∼4, DNA is in an unstable transition state, being able to switch between compact and extensible states. We can also regulate the DNA attraction through a cyclic process of DNA compaction and unraveling by alternating the pH of the solution between 3 and 8. The corresponding change of morphology of DNA modulated by pH is also confirmed by atomic force microscopy (AFM). In the theoretical aspect, the present experimental finding is consistent with the coarse-grained simulation of Langevin dynamics on the effect of pH on DNA in a solution of divalent counterions.

pH-Dependent Polyelectrolyte Bridging of Charged Nanoparticles

Systems comprised of polyelectrolytes and charged nanoparticles are of great technological interest, being common components in formulations among other uses. The colloidal stability of formulations is an important issue, and thus a lot of effort has been made to study the interactions of individual components in these systems. Here, the complexation and adsorption of an annealed (pH-dependent) polyelectrolyte to two spherical nanoparticles has been studied using coarse-grained Monte Carlo simulations. This has been done mainly by varying the solution pH and separation distance (concentration) between the nanoparticles. The polyelectrolyte charge distribution is seen to vary with nanoparticle separation distance, and its ability to bridge both nanoparticles changes with pH. The flexible polyelectrolyte creates compact, multilink bridges at short nanoparticle separation distances and evolves to a stretched single-link bridge at longer distances, where a larger fraction of the polyelectrolyte wraps around the nanoparticles. The annealed polyelectrolyte is also compared with a quenched polyelectrolyte of similar fixed fractional charge. Here, a difference is found in the adsorption ability at low pH/ionization due to the ability of the annealed polyelectrolytes to concentrate charges in the vicinity of the nanoparticle. At intermediate polyelectrolyte charge fractions and with increasing nanoparticle separation distances, the annealed system is able to link nanoparticles at larger distances as compared to the quenched, in good agreement with experimental observations. The results in this work contribute to the understanding of the effect of annealed polyelectrolytes and pH variations in the phase behavior of polyelectrolyte-nanoparticle systems, potentially aiding in the design and optimization of pH-responsive systems.

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

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

Coupling of Charge Regulation and Conformational Equilibria in Linear Weak Polyelectrolytes: Treatment of Long-Range Interactions via Effective Short-Ranged and pH-Dependent Interaction Parameters

The classical Rotational Isomeric State (RIS) model, originally proposed by Flory, has been used to rationalize a wide range of physicochemical properties of neutral polymers. However, many weak polyelectrolytes of interest are able to regulate their charge depending on the conformational state of the bonds. Recently, it has been shown that the RIS model can be coupled with the Site Binding (SB) model, for which the ionizable sites can adopt two states: protonated or deprotonated. The resulting combined scheme, the SBRIS model, allows for analyzing ionization and conformational equilibria on the same foot. In the present work, this approach is extended to include pH-dependent electrostatic Long-Range (LR) interactions, ubiquitous in weak polyelectrolytes at moderate and low ionic strengths. With this aim, the original LR interactions are taken into account by defining effective Short-Range (SR) and pH-dependent parameters, such as effective microscopic protonation constants and rotational bond energies. The new parameters are systematically calculated using variational methods. The machinery of statistical mechanics for SR interactions, including the powerful and fast transfer matrix methods, can then be applied. The resulting technique, which we will refer to as the Local Effective Interaction Parameters (LEIP) method, is illustrated with a minimal model of a flexible linear polyelectrolyte containing only one type of rotating bond. LEIP reproduces very well the pH dependence of the degree of protonation and bond probabilities obtained by semi-grand canonical Monte Carlo simulations, where LR interactions are explicitly taken into account. The reduction in the computational time in several orders of magnitude suggests that the LEIP technique could be useful in a range of areas involving linear weak polyelectrolytes, allowing direct fitting of the relevant physical parameters to the experimental quantities.

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

In this study, we investigated the effect of pH on single weak acidic polyelectrolyte microgels under salt-free conditions with (i) varying microgel concentration, (ii) varying content of acidic groups and (iii) different crosslinking densities using Monte Carlo simulations under explicit consideration of the protonation/deprotonation reaction. We assessed both global properties, such as the degree of ionization, the degree of swelling and the counterion distribution, and local properties such as the radial network ionization profile and the ionization along the polymer chains as a function of pH. We found a pronounced suppression of the pH-dependent ionization of the microgels, as compared to the ideal titration behavior and a shift of the titration curve to a higher pH originating in the proximity of acidic groups in the microgel. In contrast to macroscopic gels, counterions can leave the microgel, resulting in an effective charge of the network, which hinders the ionization. A decreasing microgel concentration leads to an increased effective charge of the microgel and a more pronounced shift of the titration curve. The number of acidic groups showed only a weak effect on the ionization behavior of the microgels. For two different microgels with different crosslinking densities, similar scaling of the gel size was observed. A distinct transition from an uncharged and unswollen to a highly charged and expanded polymer network was observed for all investigated microgels. The degree of swelling mainly depends on the degree of ionization. An inhomogeneous distribution of the degree of ionization along the radial profile of the microgel was found.

Computer Simulations of Static and Dynamical Properties of Weak Polyelectrolyte Nanogels in Salty Solutions

We investigate the chemical equilibria of weak polyelectrolyte nanogels with reaction ensemble Monte Carlo simulations. With this method, the chemical identity of the nanogel monomers can change between neutral or charged following the acid-base equilibrium reaction HA ⇌ A- + H⁺. We investigate the effect of changing the chemical equilibria by modifying the dissociation constant K a . These simulations allow for the extraction of static properties like swelling equilibria and the way in which charge-both monomer and ionic-is distributed inside the nanogel. Our findings reveal that, depending on the value of K a , added salt can either increase or decrease the gel size. Using the calculated mean-charge configurations of the nanogel from the reaction ensemble simulation as a quenched input to coupled lattice-Boltzmann molecular dynamics simulations, we investigate dynamical nanogel properties such as the electrophoretic mobility μ and the diffusion coefficient D.

An accurate coarse-grained model for chitosan polysaccharides in aqueous solution

Computational models can provide detailed information about molecular conformations and interactions in solution, which is currently inaccessible by other means in many cases. Here we describe an efficient and precise coarse-grained model for long polysaccharides in aqueous solution at different physico-chemical conditions such as pH and ionic strength. The Model is carefully constructed based on all-atom simulations of small saccharides and metadynamics sampling of the dihedral angles in the glycosidic links, which represent the most flexible degrees of freedom of the polysaccharides. The model is validated against experimental data for Chitosan molecules in solution with various degree of deacetylation, and is shown to closely reproduce the available experimental data. For long polymers, subtle differences of the free energy maps of the glycosidic links are found to significantly affect the measurable polymer properties. Therefore, for titratable monomers the free energy maps of the corresponding links are updated according to the current charge of the monomers. We then characterize the microscopic and mesoscopic structural properties of large chitosan polysaccharides in solution for a wide range of solvent pH and ionic strength, and investigate the effect of polymer length and degree and pattern of deacetylation on the polymer properties.

Wang-Landau Reaction Ensemble Method: Simulation of Weak Polyelectrolytes and General Acid-Base Reactions

We present a novel method for the study of weak polyelectrolytes and general acid-base reactions in molecular dynamics and Monte Carlo simulations. The approach combines the advantages of the reaction ensemble and the Wang-Landau sampling method. Deprotonation and protonation reactions are simulated explicitly with the help of the reaction ensemble method, while the accurate sampling of the corresponding phase space is achieved by the Wang-Landau approach. The combination of both techniques provides a sufficient statistical accuracy such that meaningful estimates for the density of states and the partition sum can be obtained. With regard to these estimates, several thermodynamic observables like the heat capacity or reaction free energies can be calculated. We demonstrate that the computation times for the calculation of titration curves with a high statistical accuracy can be significantly decreased when compared to the original reaction ensemble method. The applicability of our approach is validated by the study of weak polyelectrolytes and their thermodynamic properties.

Polypeptide-Nanoparticle Interactions and Corona Formation Investigated by Monte Carlo Simulations

Biomacromolecule activity is usually related to its ability to keep a specific structure. However, in solution, many parameters (pH, ionic strength) and external compounds (polyelectrolytes, nanoparticles) can modify biomacromolecule structure as well as acid/base properties, thus resulting in a loss of activity and denaturation. In this paper, the impact of neutral and charged nanoparticles (NPs) is investigated by Monte Carlo simulations on polypeptide (PP) chains with primary structure based on bovine serum albumin. The influence of pH, salt valency, and NP surface charge density is systematically studied. It is found that the PP is extended at extreme pH, when no complex formation is observed, and folded at physiological pH. PP adsorption around oppositely-charged NPs strongly limits chain structural changes and modifies its acid/base properties. At physiological pH, the complex formation occurs only with positively-charged NPs. The presence of salts, in particular those with trivalent cations, introduces additional electrostatic interactions, resulting in a mitigation of the impact of negative NPs. Thus, the corona structure is less dense with locally-desorbed segments. On the contrary, very limited impact of salt cation valency is observed when NPs are positive, due to the absence of competitive effects between multivalent cations and NP.

Electrostatic complexation of polyelectrolyte and magnetic nanoparticles: from wild clustering to controllable magnetic wires

We present the electrostatic complexation between polyelectrolytes and charged nanoparticles. The nanoparticles in solution are γ-Fe2O3 (maghemite) spheres with 8.3 nm diameter and anionic surface charges. The complexation was monitored using three different formulation pathways such as direct mixing, dilution, and dialysis. In the first process, the hybrids were obtained by mixing stock solutions of polymers and nanoparticles. A 'destabilization state' with sharp and intense maximum aggregation was found at charges stoichiometry (isoelectric point). While on the two sides of the isoelectric point, 'long-lived stable clusters state' (arrested states) were observed. Dilution and dialysis processes were based on controlled desalting kinetics according to methods developed in molecular biology. Under an external magnetic field (B = 0.3 T), from dialysis at isoelectric point and at arrested states, cationic polyelectrolytes can 'paste' these magnetic nanoparticles (NPs) together to yield irregular aggregates (size of 100 μm) and regular rod-like aggregates, respectively. These straight magnetic wires were fabricated with diameters around 200 nm and lengths comprised between 1 μm and 0.5 mm. The wires can have either positive or negative charges on their surface. After analyzing their orientational behavior under an external rotating field, we also showed that the wires made from different polyelectrolytes have the same magnetic property. The recipe used a wide range of polyelectrolytes thereby enhancing the versatility and applied potentialities of the method. This simple and general approach presents significant perspective for the fabrication of hybrid functional materials.

Understanding the protonation behavior of linear polyethylenimine in solutions through Monte Carlo simulations

The success of polyethyleneimine (PEI) as a nonviral-based gene delivery vector has been attributed to its proton buffering capacity. Despite the great interest in PEI for its use in nonviral-based gene delivery, the protonation behavior of PEI in solution is not well understood. Earlier experimental studies have reported inconsistent values of the protonation state of PEI. In this work, we report our investigation of the protonation behavior of a realistic linear PEI (lPEI) with computational approaches. Reported experimental pK(a) values of several diamine compounds are first examined. A screened Coulombic interaction with a distance dependence dielectric is shown to reproduce the shifted pK(a) values of the model diamine compounds. Then atomistic molecular dynamic simulations of lPEI chain with 20 repeating units are performed and the results are used to provide parameters for a coarse-grained polyamine model. The screened Coulombic interaction is then incorporated in the coarse-grained lPEI chain and computational titrations are performed. The obtained computational titration curves of lPEI in solutions were found to be in best agreement with experimental results by Smits et al., but the computational titration curves have too strong of a dependence on salt concentration compared to the experimental results by Smits et al. Disregarding the discrepancy in the salt dependence, our computational titrations reveal that approximately 55% of the lPEI amine groups are protonated under physiological conditions in solution with a nearly alternating arrangement of protonated and nonprotonated amines. Titrations of lPEI in the presence of a polyanion are also performed to determine how the charge state of lPEI could be affected by complexation with DNA in gene therapy preparations. While the presence of the polyanion increases the degree of protonation of the PEI, many of PEI amines remain unprotonated under physiological conditions, providing evidence that PEI complexed with DNA could still have proton buffering capacity. Potential sources of error that have resulted in the inconsistency of previously reported protonation states of PEI were also discussed.