Attraction of like-charged macroions in the strong-coupling limit

Ion correlation-driven like-charge attraction in multivalent salt solutions

The electrostatic double layer force is key to determining the stability and self-assembly of charged colloids and many other soft matter systems. Fully understanding the attractive force between two like-charged surfaces remains a great challenge. Here, we apply the modified Gaussian renormalized fluctuation theory to study ion correlation-driven like-charge attraction in multivalent salt solutions. The effects of spatially varying ion correlations on the structure of overlapping double layers and their free energy are self-consistently accounted for. In the presence of multivalent salts, increasing surface charge or counterion valency leads to a short-range attraction. We demonstrate that although both overcharging and like-charge attraction are outcomes of ion correlation, there is no causal relationship between them. Our theory also captures the non-monotonic dependence of like-charge attraction on multivalent salt concentration. The reduction of attraction at high salt concentrations could be a contributing factor toward the reentrant stability of charged colloidal suspensions. Our theoretical predictions are consistent with the observations reported in experiments and simulations.

Interactions between Rigid Polyelectrolytes Mediated by Ordering and Orientation of Multivalent Nonspherical Ions in Salt Solutions

Multivalent ions in solutions with polyelectrolytes (PEs) induce electrostatic correlations that can drastically change ion distributions around the PEs and their mutual interactions. Using coarse-grained molecular dynamics simulations, we show how in addition to valency, ion shape and concentration can be harnessed as tools to control rigid like-charged PE-PE interactions. We demonstrate a correlation between the orientational ordering of aspherical ions and how they mediate the effective PE-PE attraction induced by multivalency. The interaction type, strength, and range can thus be externally controlled in ionic solutions. Our results can be used as generic guidelines to tune the self-assembly of like-charged polyelectrolytes by variation of the characteristics of the ions.

Biocompatible Macroion/Growth Factor Assemblies for Medical Applications

Growth factors are a class of proteins that play a role in the proliferation (the increase in the number of cells resulting from cell division) and differentiation (when a cell undergoes changes in gene expression becoming a more specific type of cell) of cells. They can have both positive (accelerating the normal healing process) and negative effects (causing cancer) on disease progression and have potential applications in gene therapy and wound healing. However, their short half-life, low stability, and susceptibility to degradation by enzymes at body temperature make them easily degradable in vivo. To improve their effectiveness and stability, growth factors require carriers for delivery that protect them from heat, pH changes, and proteolysis. These carriers should also be able to deliver the growth factors to their intended destination. This review focuses on the current scientific literature concerning the physicochemical properties (such as biocompatibility, high affinity for binding growth factors, improved bioactivity and stability of the growth factors, protection from heat, pH changes or appropriate electric charge for growth factor attachment via electrostatic interactions) of macroions, growth factors, and macroion-growth factor assemblies, as well as their potential uses in medicine (e.g., diabetic wound healing, tissue regeneration, and cancer therapy). Specific attention is given to three types of growth factors: vascular endothelial growth factors, human fibroblast growth factors, and neurotrophins, as well as selected biocompatible synthetic macroions (obtained through standard polymerization techniques) and polysaccharides (natural macroions composed of repeating monomeric units of monosaccharides). Understanding the mechanisms by which growth factors bind to potential carriers could lead to more effective delivery methods for these proteins, which are of significant interest in the diagnosis and treatment of neurodegenerative and civilization diseases, as well as in the healing of chronic wounds.

Bridges of Calcium Bicarbonate Tightly Couple Dipolar Lipid Membranes

The interaction between lipid membranes and ions is associated with a range of key physiological processes. Most earlier studies have focused on the interaction of lipids with cations, while the specific effects of the anions have been largely overlooked. Owing to dissolved atmospheric carbon dioxide, bicarbonate is an important ubiquitous anion in aqueous media. In this paper, we report on the effect of bicarbonate anions on the interactions between dipolar lipid membranes in the presence of previously adsorbed calcium cations. Using a combination of solution X-ray scattering, osmotic stress, and molecular dynamics simulations, we followed the interactions between 1,2-didodecanoyl-sn-glycero-3-phosphocholine (DLPC) lipid membranes that were dialyzed against CaCl2 solutions in the presence and absence of bicarbonate anions. Calcium cations adsorbed onto DLPC membranes, charge them, and lead to their swelling. In the presence of bicarbonate anions, however, the calcium cations can tightly couple one dipolar DLPC membrane to the other and form a highly condensed and dehydrated lamellar phase with a repeat distance of 3.45 ± 0.02 nm. Similar tight condensation and dehydration has only been observed between charged membranes in the presence of multivalent counterions. Bridging between bilayers by calcium bicarbonate complexes induced this arrangement. Furthermore, in this condensed phase, lipid molecules and adsorbed ions were arranged in a two-dimensional oblique lattice.

Additive Modulation of DNA-DNA Interactions by Interstitial Ions

Quantitative understanding of biomolecular electrostatics, particularly involving multivalent ions and highly charged surfaces, remains lacking. Ion-modulated interactions between nucleic acids provide a model system in which electrostatics plays a dominant role. Using ordered DNA arrays neutralized by spherical cobalt³⁺ hexammine and Mg²⁺ ions, we investigate how the interstitial ions modulate DNA-DNA interactions. Using methods of ion counting, osmotic stress, and x-ray diffraction, we systematically determine thermodynamic quantities, including ion chemical potentials, ion partition, DNA osmotic pressure and force, and DNA-DNA spacing. Analyses of the multidimensional data provide quantitative insights into their interdependencies. The key finding of this study is that DNA-DNA forces are observed to linearly depend on the partition of interstitial ions, suggesting the dominant role of ion-DNA coupling. Further implications are discussed in light of physical theories of electrostatic interactions and like-charge attraction.

Self-assembly of charged colloidal cubes

In this work, we show how and why the interactions between charged cubic colloids range from radially isotropic to strongly directionally anisotropic, depending on tuneable factors. Using molecular dynamics simulations, we illustrate the effects of typical solvents to complement experimental investigations of cube assembly. We find that in low-salinity water solutions, where cube self-assembly is observed, the colloidal shape anisotropy leads to the strongest attraction along the corner-to-corner line, followed by edge-to-edge, with a face-to-face configuration of the cubes only becoming energetically favorable after the colloids have collapsed into the van der Waals attraction minimum. Analysing the potential of mean force between colloids with varied cubicity, we identify the origin of the asymmetric microstructures seen in experiment.

Attraction between Like-Charged Macroions Mediated by Specific Counterion Configurations

Attraction between like-charged macroions is fundamental to many processes in biology, chemistry, and physics. It also plays an important role in industrial applications such as ion-extraction processes or catalysis. In this work, we report a novel mechanism by which attraction can be realized between spherical macroions at high ionic strength. It consists of specific configurations of two, three, and more counterions that appear between macroions with high statistical probability. The attraction is manifested in a minimum in the potential of mean force between the macroions at short distances. Its depth increases with increasing charge of the macroion, demonstrating that the attraction is electrostatic in nature. It is shown that the implicit solvent model with a distance-dependent dielectric constant can capture both the geometry and thermodynamics of charge-stabilized macroion dimers on the qualitative level. The results obtained for a model colloid with a smooth surface are extrapolated to more realistic systems. Evidence is found that the reported mechanism can be observed in small chemical compounds with encapsulated ions such as fullerenes.

Structure and Interactions between Charged Lipid Membranes in the Presence of Multivalent Ions

When aqueous salt solutions contain multivalent ions (like Ca²⁺ or Mg²⁺), strong correlation effects may lead to ion-bridging, net attraction, and tight-coupling between like-charged interfaces. To examine the effects of surface charge density, temperature, salt type, and salt concentration on the structures of tightly coupled charged interfaces, we have used mixed lipid membranes, containing either saturated or unsaturated tails in the presence of multivalent ions. We discovered that tightly coupled membrane lamellar phases, dominated by attractive interactions, coexisted with weakly coupled lamellar phases, dominated by repulsive interactions. To control the membrane charge density, we mixed lipids with negatively charged headgroups, DLPS and DOPS, with their zwitterionic analogue having the same tails, DLPC and DOPC, respectively. Using solution X-ray scattering we measured the lamellar repeat distance, D, at different ion concentrations, temperatures, and membrane charge densities. The multivalent ions tightly coupled the mixed lipid bilayers whose charged lipid molar fraction was between 0.1 and 1. The repeat distance of the tightly coupled phase was about 4 nm for the DLPS/DLPC mixtures and about 5 nm for the DOPS/DOPC mixtures. In this phase, the repeat distance slightly increased with increasing temperature and decreased with increasing charge density. When the molar fraction of charged lipid was 0.1 or 0.25, a less tightly coupled phase coexisted with the tightly coupled phase. The weakly coupled lamellar phase had significantly larger D values, although they were consistently shorter than the D values in monovalent salt solutions with similar screening lengths.

Like-Charge Attraction between Metal Nanoparticles in a 1∶1 Electrolyte Solution

We calculate the force between two spherical metal nanoparticles of charge Q_{1} and Q_{2} in a dilute 1∶1 electrolyte solution. Numerically solving the nonlinear Poisson-Boltzmann equation, we find that metal nanoparticles with the same sign of charge can attract one another. This is fundamentally different from what is found for like-charged, nonpolarizable, colloidal particles, the two-body interaction potential for which is always repulsive inside a dilute 1∶1 electrolyte. Furthermore, the existence of like-charge attraction between spherical metal nanoparticles is even more surprising in view of the result that such attraction is impossible between parallel metal slabs, showing the fundamental importance of curvature. To overcome a slow convergence of the numerical solution of the full nonlinear Poisson-Boltzmann equation, we developed a modified Derjaguin approximation which allows us to accurately and rapidly calculate the interaction potential between two metal nanoparticles or between a metal nanoparticle and a phospholipid membrane.

Charged nanorods at heterogeneously charged surfaces

We study the spatial and orientational distribution of charged nanorods (rodlike counterions) as well as the effective interaction mediated by them between two plane-parallel surfaces that carry fixed (quenched) heterogeneous charge distributions. The nanorods are assumed to have an internal charge distribution, specified by a multivalent monopolar moment and a finite quadrupolar moment, and the quenched surface charge is assumed to be randomly distributed with equal mean and variance on the two surfaces. While equally charged surfaces are known to repel within the traditional mean-field theories, the presence of multivalent counterions has been shown to cause attractive interactions between uniformly charged surfaces due to the prevalence of strong electrostatic couplings that grow rapidly with the counterion valency. We show that the combined effects due to electrostatic correlations (caused by the coupling between the mean surface field and the multivalent, monopolar, charge valency of counterions) as well as the disorder-induced interactions (caused by the coupling between the surface disorder field and the quadrupolar moment of counterions) lead to much stronger attractive interactions between two randomly charged surfaces. The interaction profile turns out to be a nonmonotonic function of the intersurface separation, displaying an attractive minimum at relatively small separations, where the ensuing attraction can exceed the maximum strong-coupling attraction (produced by multivalent monopolar counterions between uniformly charged surfaces) by more than an order of magnitude.

Rodlike Counterions near Charged Cylinders: Counterion Condensation and Intercylinder Interaction

We study a system composed of like-charged cylinders and dumbbell-like counterions, with the focus laid on the role of the internal structure of counterions, using Monte Carlo simulations. The dumbbell ions are found to exhibit novel counterion condensation behavior governed by their length. Effective electrostatic interactions mediated between charged parallel cylinders also turn out significantly different from the case of pointlike ions, as a result of the complex interplay between the spatially separated charge distribution in the dumbbell counterions, their orientation, and the curvature of the charged cylinder. We show that at a weak-to-moderate electrostatic coupling strength, where effective like-charge interactions are usually found to be repulsive, the intercylinder interaction can become attractive and display a distinctive sensitivity to the cylinder curvature and dumbbell size, proving the significant effect of ion structure.

Efficient simulation method for nano-patterned charged surfaces in an electrolyte solution

We present a method to efficiently simulate nano-patterned charged surfaces inside an electrolyte solution. Simulations are performed in the grand canonical ensemble and are used to calculate the force between surfaces with various charge patterns. The electric field produced by the surfaces is calculated analytically and is used as an external potential. To treat the long range Coulomb interaction between the ions we use a modified 3d Ewald summation method. The force between the surfaces is found to depend strongly on the specific charge pattern, on the surface alignment and separation.

Thermodynamically consistent Langevin dynamics with spatially correlated noise predicting frictionless regime and transient attraction effect

While the origins of temporal correlations in Langevin dynamics have been thoroughly researched, the understanding of spatially correlated noise (SCN) is rather incomplete. In particular, very little is known about the relation between friction and SCN. In this article, starting from the microscopic, deterministic model, we derive the analytical formula for the spatial correlation function in the particle-bath interactions. This expression shows that SCN is the inherent component of binary mixtures, originating from the effective (entropic) interactions. Further, employing this spatial correlation function, we postulate the thermodynamically consistent Langevin equation driven by the Gaussian SCN and calculate the adequate fluctuation-dissipation relation. The thermodynamical consistency is achieved by introducing the spatially variant friction coefficient, which can be also derived analytically. This coefficient exhibits a number of intriguing properties, e.g., the singular behavior for certain types of interactions. Eventually, we apply this new theory to the system of two charged particles in the presence of counter-ions. Such particles interact via the screened-charge Yukawa potential and the inclusion of SCN leads to the emergence of the anomalous frictionless regime. In this regime the particles can experience active propulsion leading to the transient attraction effect. This effect suggests a nonequilibrium mechanism facilitating the molecular binding of the like-charged particles.

Interaction modes between asymmetrically and oppositely charged rods

The interaction of oppositely and asymmetrically charged rods in salt-a simple model of (bio)macromolecular assembly-is observed via simulation to exhibit two free energy minima, separated by a repulsive barrier. In contrast to similar minima in the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, the governing mechanism includes electrostatic attraction at large separation, osmotic repulsion at close range, and depletion attraction near contact. A model accounting for ion condensation and excluded volume is shown to be superior to a mean-field treatment in predicting the effect of charge asymmetry on the free-energy profile.

Ion-mediated interactions between net-neutral slabs: Weak and strong disorder effects

We investigate the effective interaction between two randomly charged but otherwise net-neutral, planar dielectric slabs immersed in an asymmetric Coulomb fluid containing a mixture of mobile monovalent and multivalent ions. The presence of charge disorder on the apposed bounding surfaces of the slabs leads to substantial qualitative changes in the way they interact, as compared with the standard picture provided by the van der Waals and image-induced, ion-depletion interactions. While, the latter predict purely attractive interactions between strictly neutral slabs, we show that the combined effects from surface charge disorder, image depletion, Debye (or salt) screening, and also, in particular, their coupling with multivalent ions, give rise to a more diverse behavior for the effective interaction between net-neutral slabs at nano-scale separations. Disorder effects show large variation depending on the properly quantified strength of disorder, leading either to non-monotonic effective interaction with both repulsive and attractive branches when the surface charges are weakly disordered (small disorder variance) or to a dominating attractive interaction that is larger both in its range and magnitude than what is predicted from the van der Waals and image-induced, ion-depletion interactions, when the surfaces are strongly disordered (large disorder variance).

Strong coupling electrostatics for randomly charged surfaces: antifragility and effective interactions

We study the effective interaction mediated by strongly coupled Coulomb fluids between dielectric surfaces carrying quenched, random monopolar charges with equal mean and variance, both when the Coulomb fluid consists only of mobile multivalent counterions and when it consists of an asymmetric ionic mixture containing multivalent and monovalent (salt) ions in equilibrium with an aqueous bulk reservoir. We analyze the consequences that follow from the interplay between surface charge disorder, dielectric and salt image effects, and the strong electrostatic coupling that results from multivalent counterions on the distribution of these ions and the effective interaction pressure they mediate between the surfaces. In a dielectrically homogeneous system, we show that the multivalent counterions are attracted towards the surfaces with a singular, disorder-induced potential that diverges logarithmically on approach to the surfaces, creating a singular but integrable counterion density profile that exhibits an algebraic divergence at the surfaces with an exponent that depends on the surface charge (disorder) variance. This effect drives the system towards a state of lower thermal 'disorder', one that can be described by a renormalized temperature, exhibiting thus a remarkable antifragility. In the presence of an interfacial dielectric discontinuity, the singular behavior of counterion density at the surfaces is removed but multivalent counterions are still accumulated much more strongly close to randomly charged surfaces as compared with uniformly charged ones. The interaction pressure acting on the surfaces displays in general a highly non-monotonic behavior as a function of the inter-surface separation with a prominent regime of attraction at small to intermediate separations. This attraction is caused directly by the combined effects from charge disorder and strong coupling electrostatics of multivalent counterions, which dominate the surface-surface repulsion due to the (equal) mean charges on the two surfaces and the osmotic pressure of monovalent ions residing between them. These effects can be quite significant even with a small degree of surface charge disorder relative to the mean surface charge. The strong coupling, disorder-induced attraction is typically much stronger than the van der Waals interaction between the surfaces, especially within a range of several nanometers for the inter-surface separation, where such effects are predicted to be most pronounced.

Role of geometrical shape in like-charge attraction of DNA

While the phenomenon of like-charge attraction of DNA is clearly observed experimentally and in simulations, mean-field theories fail to predict it. Kornyshev et al. argued that like-charge attraction is due to DNA's helical geometry and hydration forces. Strong-coupling (SC) theory shows that attraction of like-charged rods is possible through ion correlations alone at large coupling parameters, usually by multivalent counterions. However for SC theory to be applicable, counterion-counterion correlations perpendicular to the DNA strands need to be sufficiently small, which is not a priori the case for DNA even with trivalent counterions. We study a system containing infinitely long DNA strands and trivalent counterions by computer simulations employing varying degrees of coarse-graining. Our results show that there is always attraction between the strands, but its magnitude is indeed highly dependent on the specific shape of the strand. While discreteness of the charge distribution has little influence on the attractive forces, the role of the helical charge distribution is considerable: charged rods maintain a finite distance in equilibrium, while helices collapse to close contact with a phase shift of π, in full agreement with SC predictions. The SC limit is applicable because counterions strongly bind to the charged sites of the helices, so that helix-counterion interactions dominate over counterion-counterion interactions. Thus DNA's helical geometry is not crucial for like-charge DNA attraction, but strongly enhances it, and electrostatic interactions in the strong-coupling limit are sufficient to explain this attraction.

DNA under Force: Mechanics, Electrostatics, and Hydration

Quantifying the basic intra- and inter-molecular forces of DNA has helped us to better understand and further predict the behavior of DNA. Single molecule technique elucidates the mechanics of DNA under applied external forces, sometimes under extreme forces. On the other hand, ensemble studies of DNA molecular force allow us to extend our understanding of DNA molecules under other forces such as electrostatic and hydration forces. Using a variety of techniques, we can have a comprehensive understanding of DNA molecular forces, which is crucial in unraveling the complex DNA functions in living cells as well as in designing a system that utilizes the unique properties of DNA in nanotechnology.

Water-induced correlation between single ions imaged at the solid-liquid interface

When immersed into water, most solids develop a surface charge, which is neutralized by an accumulation of dissolved counterions at the interface. Although the density distribution of counterions perpendicular to the interface obeys well-established theories, little is known about counterions' lateral organization at the surface of the solid. Here we show, by using atomic force microscopy and computer simulations, that single hydrated metal ions can spontaneously form ordered structures at the surface of homogeneous solids in aqueous solutions. The structures are laterally stabilized only by water molecules with no need for specific interactions between the surface and the ions. The mechanism, studied here for several systems, is controlled by the hydration landscape of both the surface and the adsorbed ions. The existence of discrete ion domains could play an important role in interfacial phenomena such as charge transfer, crystal growth, nanoscale self-assembly and colloidal stability.

Ion competition in condensed DNA arrays in the attractive regime

Physical origin of DNA condensation by multivalent cations remains unsettled. Here, we report quantitative studies of how one DNA-condensing ion (Cobalt(3+) Hexammine, or Co(3+)Hex) and one nonDNA-condensing ion (Mg(2+)) compete within the interstitial space in spontaneously condensed DNA arrays. As the ion concentrations in the bath solution are systematically varied, the ion contents and DNA-DNA spacings of the DNA arrays are determined by atomic emission spectroscopy and x-ray diffraction, respectively. To gain quantitative insights, we first compare the experimentally determined ion contents with predictions from exact numerical calculations based on nonlinear Poisson-Boltzmann equations. Such calculations are shown to significantly underestimate the number of Co(3+)Hex ions, consistent with the deficiencies of nonlinear Poisson-Boltzmann approaches in describing multivalent cations. Upon increasing the concentration of Mg(2+), the Co(3+)Hex-condensed DNA array expands and eventually redissolves as a result of ion competition weakening DNA-DNA attraction. Although the DNA-DNA spacing depends on both Mg(2+) and Co(3+)Hex concentrations in the bath solution, it is observed that the spacing is largely determined by a single parameter of the DNA array, the fraction of DNA charges neutralized by Co(3+)Hex. It is also observed that only ∼20% DNA charge neutralization by Co(3+)Hex is necessary for spontaneous DNA condensation. We then show that the bath ion conditions can be reduced to one variable with a simplistic ion binding model, which is able to describe the variations of both ion contents and DNA-DNA spacings reasonably well. Finally, we discuss the implications on the nature of interstitial ions and cation-mediated DNA-DNA interactions.

Biomolecular electrostatics and solvation: a computational perspective

An understanding of molecular interactions is essential for insight into biological systems at the molecular scale. Among the various components of molecular interactions, electrostatics are of special importance because of their long-range nature and their influence on polar or charged molecules, including water, aqueous ions, proteins, nucleic acids, carbohydrates, and membrane lipids. In particular, robust models of electrostatic interactions are essential for understanding the solvation properties of biomolecules and the effects of solvation upon biomolecular folding, binding, enzyme catalysis, and dynamics. Electrostatics, therefore, are of central importance to understanding biomolecular structure and modeling interactions within and among biological molecules. This review discusses the solvation of biomolecules with a computational biophysics view toward describing the phenomenon. While our main focus lies on the computational aspect of the models, we provide an overview of the basic elements of biomolecular solvation (e.g. solvent structure, polarization, ion binding, and non-polar behavior) in order to provide a background to understand the different types of solvation models.

Effect of charge inhomogeneity and mobility on colloid aggregation

The aggregation of inhomogeneously charged colloids with the same average charge is analyzed using Monte Carlo simulations. We find aggregation of colloids for sizes in the range 10-200 nm, which is similar to the range in which aggregation is observed in several experiments. The attraction arises from the strongly correlated electrostatic interactions associated with the increase in the counterion density in the region between the particles; this effect is enhanced by the discreteness and mobility of the surface charges. Larger colloids attract more strongly when their surface charges are discrete. We study the aggregation as functions of the surface charge density, counterion valence, and volume fraction.

Macroion solutions in the cell model studied by field theory and Monte Carlo simulations

Aqueous solutions of charged spherical macroions with variable dielectric permittivity and their associated counterions are examined within the cell model using a field theory and Monte Carlo simulations. The field theory is based on separation of fields into short- and long-wavelength terms, which are subjected to different statistical-mechanical treatments. The simulations were performed by using a new, accurate, and fast algorithm for numerical evaluation of the electrostatic polarization interaction. The field theory provides counterion distributions outside a macroion in good agreement with the simulation results over the full range from weak to strong electrostatic coupling. A low-dielectric macroion leads to a displacement of the counterions away from the macroion.

Interactions between charged surfaces with ionizable sites

A key factor controlling the interactions between surfaces in aqueous solutions is the surface charge density. Surfaces typically become charged though a titration process where surface groups can become ionized based on their dissociation constant and the pH of the solution. In this work, we use a Monte Carlo method to treat this process in a system with two planar surfaces with explicitly described ionizable sites in a salt solution. We focus on a system with a surface density of ionizable sites set to 4.8 nm(-2), corresponding to silica. We find that the surface charge density changes as the surfaces come close to contact due to interactions between the ionizable groups on each surface. In addition, we observe an attraction between the surfaces above a threshold surface charge, in good agreement with previous theoretical predictions based on uniformly charged surfaces. However, close to contact we find the force is significantly different than for the uniformly charged case.

Divalent counterion-induced condensation of triple-strand DNA

Understanding and manipulation of the forces assembling DNA/RNA helices have broad implications for biology, medicine, and physics. One subject of significance is the attractive force between dsDNA mediated by polycations of valence ≥ 3. Despite extensive studies, the physical origin of the "like-charge attraction" remains unsettled among competing theories. Here we show that triple-strand DNA (tsDNA), a more highly charged helix than dsDNA, is precipitated by alkaline-earth divalent cations that are unable to condense dsDNA. We further show that our observation is general by examining several cations (Mg(2+), Ba(2+), and Ca(2+)) and two distinct tsDNA constructs. Cation-condensed tsDNA forms ordered hexagonal arrays that redissolve upon adding monovalent salts. Forces between tsDNA helices, measured by osmotic stress, follow the form of hydration forces observed with condensed dsDNA. Probing a well-defined system of point-like cations and tsDNAs with more evenly spaced helical charges, the counterintuitive observation that the more highly charged tsDNA (vs. dsDNA) is condensed by cations of lower valence provides new insights into theories of polyelectrolytes and the biological and pathological roles of tsDNA. Cations and tsDNAs also hold promise as a model system for future studies of DNA-DNA interactions and electrostatic interactions in general.

Charge renormalization of helical macromolecules

Some time ago a theory of electrostatic interaction between helical macromolecules was proposed (Kornyshev and Leikin 1997 J. Chem. Phys. 107 3656): the Kornyshev-Leikin (KL) theory. We place this theory on a more rigorous statistical mechanical grounding, starting from the free energy that can be derived from a grand partition function. We see that the long range behaviour of the force is indeed given by the KL theory, no matter whether the distributions of 'condensed' ionic charge are at the surface of the macromolecule or extend away from it. Thus, for the limiting behaviour, we need only self-consistently calculate the distribution of the condensed fraction of ions for a single macro-ion. This distribution can be related back to interaction parameters: KL parameters. Furthermore, we are able to see within the formalism where corrections due to the hard core radius of the ion enter. For the adjustment of the 'condensed' ions, we show an expression for the leading order contribution, as well as relevant decay lengths. As a demonstration of the theoretical 'machinery', as well as a study of qualitative effects, we calculate the KL parameters in one instance. We use a DNA-like surface charge distribution, where a fraction of the ions are assumed to be bound in the grooves at the surface of a DNA molecule, whereas the rest of the charge distribution is calculated self-consistently. Also, the electrostatic contribution to the counter-ion binding potentials that ions experience within the grooves can be calculated.

Dressed counterions: strong electrostatic coupling in the presence of salt

We reformulate the theory of strong electrostatic coupling in order to describe an asymmetric electrolyte solution of monovalent salt ions and polyvalent counterions using field-theoretical techniques and Monte Carlo simulations. The theory is based on an asymmetric treatment of the different components of the electrolyte solution. The weak coupling Debye-Hückel approach is used in order to describe the monovalent salt ions while a strong coupling approach is used to tackle the polyvalent counterions. This combined weak-strong coupling approach effectively leads to dressed interactions between polyvalent counterions and thus directly affects the correlation attraction mediated by polyvalent counterions between like-charged objects. The general theory is specifically applied to a system composed of two uniformly charged plane-parallel surfaces in the presence of salt and polyvalent counterions. In the strong coupling limit for polyvalent counterions, the comparison with Monte Carlo simulations shows good agreement for large enough values of the electrostatic coupling parameter. We delineate two limiting laws that in fact encompass all the Monte Carlo data.

Colloidal gas-liquid condensation of polystyrene latex particles with intermediate kappa a values (5 to 160, a >> kappa(-1))

Polystyrene latex particles showed gas-liquid condensation under the conditions of large particle radius (a >> kappa(-1)) and intermediate kappa a, where kappa is the Debye-Hückel parameter and a is the particle radius. The particles were dissolved in deionized water containing ethanol from 0 to 77 vol %, settled to the bottom of the glass plate within 1 h, and then laterally moved toward the center of a cell over a 20 h period in reaching a state of equilibrium condensation. All of the suspensions that were 1 and 3 microm in diameter and 0.01-0.20 vol % in concentration realized similar gas-liquid condensation with clear gas-liquid boundaries. In 50 vol % ethanol solvent, additional ethanol was added to enhance the sedimentation force so as to restrict the particles in a monoparticle layer thickness. The coexistence of gas-liquid-solid (crystalline solid) was microscopically recognized from the periphery to the center of the condensates. A phase diagram of the gas-liquid condensation was created as a function of KCl concentration at a particle diameter of 3 microm, 0.10 vol % concentration, and 50:50 water/ethanol solvent at room temperature. The miscibility gap was observed in the concentration range from 1 to 250 microM. There was an upper limit of salt concentration where the phase separation disappeared, showing nearly critical behavior of macroscopic density fluctuation from 250 microM to 1 mM. These results add new experimental evidence to the existence of colloidal gas-liquid condensation and specify conditions of like-charge attraction between particles.

Polyelectrolyte-induced aggregation of liposomes: a new cluster phase with interesting applications

Different charged colloidal particles have been shown to be able to self-assemble, when mixed in an aqueous solvent with oppositely charged linear polyelectrolytes, forming long-lived finite-size mesoscopic aggregates. On increasing the polyelectrolyte content, with the progressive reduction of the net charge of the primary polyelectrolyte-decorated particles, larger and larger clusters are observed. Close to the isoelectric point, where the charge of the adsorbed polyelectrolytes neutralizes the original charge of the particles' surface, the aggregates reach their maximum size, while beyond this point any further increase of the polyelectrolyte-particle charge ratio causes the formation of aggregates whose size is progressively reduced. This re-entrant condensation behavior is accompanied by a significant overcharging. Overcharging, or charge inversion, occurs when more polyelectrolyte chains adsorb on a particle than are needed to neutralize its original charge so that, eventually, the sign of the net charge of the polymer-decorated particle is inverted. The stability of the finite-size long-lived clusters that this aggregation process yields results from a fine balance between long-range repulsive and short-range attractive interactions, both of electrostatic nature. For the latter, besides the ubiquitous dispersion forces, whose supply becomes relevant only at high ionic strength, the main contribution appears due to the non-uniform correlated distribution of the charge on the surface of the polyelectrolyte-decorated particles ('charge-patch' attraction). The interesting phenomenology shown by these system has a high potential for biotechnological applications, particularly when the primary colloidal particles are bio-compatible lipid vesicles. Possible applications of these systems as multi-compartment vectors for the simultaneous intra-cellular delivery of different pharmacologically active substances will be briefly discussed.

Correlation effects, image charge effects and finite size in the macro-ion-electrolyte system: a field-theoretic approach

We consider a model of a macro-ion surrounded by small ions of an electrolyte solution. The finite size of ionic charge distributions of ions, and image charge effects are considered. From such a model it is possible to construct a statistical field theory with a single fluctuating field and derive physical interpretations for both the mean field and two-point correlation function. For point-like charges, at the level of a Gaussian (or saddle point) approximation, we recover the standard Poisson-Boltzmann equation. However, to include ionic correlation effects, as well as image charge effects of individual ions, we must go beyond this. From the field theory considered, it is possible to construct self-consistent approximations. We consider the simplest of these, namely the Hartree approximation. The Hartree equations take the form of two coupled equations. One is a modified Poisson-Boltzmann equation; the other describes both image charge effects on the individual ions, as well as correlations. Such equations are difficult to solve numerically, so we develop an (a WKB-like) approximation for obtaining approximate solutions. This, we apply to a uniformly charged rod in univalent electrolyte solution, for point like ions, as well as for extended spherically symmetric distributions of ionic charge on electrolyte ions. The solutions show how correlation effects and image charge effects modify the Poisson-Boltzmann result. Finite-size charge distributions of the ions reduce both the effects of correlations and image charge effects. For point charges, we test the WKB approximation by calculating a leading-order correction from the exact Hartree result, showing that the WKB-like approximation works reasonably well in describing the full solution to the Hartree equations. From these solutions, we also calculate an effective charge compensation parameter in an analytical formula for the interaction of two charged cylinders.

Interactions of like-charged rods at low temperatures: analytical theory vs. simulations

We investigate a system consisting of two like-charged infinitely long rods and neutralizing counterions at low temperatures, using both analytic theory and simulations. With some reasonable approximations we can analytically solve for several ground-state structures of the model, starting with states where all counterions are lined up in the gap between the rods, over planar configurations, where the counterions are divided up into a fraction which resides between the rods, and counterions which are located on the outer surfaces, up to configurations which cover the full rod surfaces. Using parallel tempering simulations, we are able to study the system over a wide range of temperatures. At low temperatures we find good agreement with our T = 0 results. At higher temperatures, the strong coupling (SC) theory delivers qualitatively better results. We furthermore demonstrate that for the SC theory and our ground-state approximations to yield quantitative agreement, three parameters are required to be large, the strong-coupling parameter, the Rouzina-Bloomfield parameter, and the ratio of the average distance of the counterions to the radius of the rods. In the case of the latter ratio being small, our T = 0 results show better agreement with the simulation data at very low temperatures.

Strong repulsive interactions in polyelectrolyte-liposome clusters close to the isoelectric point: a sign of an arrested state

Charged colloidal particles whose interacting potential is governed by a short-range attraction and a long-range screened electrostatic repulsion contributions form aggregates whose shape, size, and overall charge are sensitively dependent on the balance between attraction and repulsion. In some cases, this class of colloidal systems shows an equilibrium cluster phase, where particles associate and dissociate reversibly into clusters. When the aggregation of the charged particles is induced by adding an oppositely charged polyion, very close to the isoelectric condition, the interaggregate interactions become very strong and a dynamical arrested state seems to occur. We provide some experimental evidences of this structural arrest in a colloid system composed by vesicles built up by a cationic lipid stuck together by an oppositely charged linear polyion, by means of the combined use of static and dynamic light scattering technique complemented by laser Doppler electrophoretic measurements. Our results show that the second virial coefficient, which is related to the potential of mean force between two adjacent aggregates, markedly increases in the vicinity of the isoelectric point. We interpret this increase as a print of strong interparticle interactions, yielding to a dynamical arrested state via cluster growth.