Comparison of the Poisson-Boltzmann Model and the Donnan Equilibrium of a Polyelectrolyte in Salt Solution

Counterion Release from Macroion Assemblies of Planar Geometry

Macroions such as nanoparticles, polyelectrolytes, ionic gels, and amphiphiles can form condensed, often self-assembled, phases that are embedded in a solvent region. The condensed phase contains not only the partially or fully immobile charges of their macroions but also corresponding counterions that are mobile and thus free to migrate out of their confinement into the solvent region where they benefit from high translational entropy. Based on the nonlinear Poisson-Boltzmann model for monovalent ions, we quantify the corresponding fraction of released counterions for a planar slab geometry of the macroion phase. Slab thickness, extension of the solvent phase, fixed background charge density provided by the macroions, and dielectric constants inside slab and solvent combine into three dimensionless parameters that the fraction of released counterions depends on. We calculate that fraction and analyze the limits of a thin macroion phase, a large solvent phase, and linearized theory, where simple analytic results become available. Of particular interest is the presence of a single-planar interface that separates a bulk macroion phase from an extended solvent region. We calculate the apparent surface charge density that emerges due to the released counterions. Our model yields a comprehensive description of counterion partitioning between a planar macroion phase and a solvent region on the level of mean-field electrostatics in the absence of added salt ions.

Electrostatic Theory of the Acidity of the Solution in the Lumina of Viruses and Virus-Like Particles

Recently, Maassen et al. measured an appreciable pH difference between the bulk solution and the solution in the lumen of virus-like particles, self-assembled in an aqueous buffer solution containing the coat proteins of a simple plant virus and polyanions (Maassen, S. J.; et al. Small 2018, 14, 1802081). They attribute this to the Donnan effect, caused by an imbalance between the number of negative charges on the encapsulated polyelectrolyte molecules and the number of positive charges on the RNA binding domains of the coat proteins that make up the virus shell or capsid. By applying Poisson-Boltzmann theory, we confirm this conclusion and show that simple Donnan theory is accurate even for the smallest of viruses and virus-like particles. This, in part, is due to the additional screening caused by the presence of a large number of immobile charges in the cavity of the shell. The presence of a net charge on the outer surface of the capsid we find in practice to not have a large effect on the pH shift. Hence, Donnan theory can indeed be applied to connect the local pH and the amount of encapsulated material. The large shifts up to a full pH unit that we predict must have consequences for applications of virus capsids as nanocontainers in bionanotechnology and artificial cell organelles.

Direct Osmotic Pressure Measurements in Articular Cartilage Demonstrate Nonideal and Concentration-Dependent Phenomena

The osmotic pressure in articular cartilage serves an important mechanical function in healthy tissue. Its magnitude is thought to play a role in advancing osteoarthritis. The aims of this study were to: (1) isolate and quantify the magnitude of cartilage swelling pressure in situ; and (2) identify the effect of salt concentration on material parameters. Confined compression stress-relaxation testing was performed on 18 immature bovine and six mature human cartilage samples in solutions of varying osmolarities. Direct measurements of osmotic pressure revealed nonideal and concentration-dependent osmotic behavior, with magnitudes approximately 1/3 those predicted by ideal Donnan law. A modified Donnan constitutive behavior was able to capture the aggregate behavior of all samples with a single adjustable parameter. Results of curve-fitting transient stress-relaxation data with triphasic theory in febio demonstrated concentration-dependent material properties. The aggregate modulus HA increased threefold as the external concentration decreased from hypertonic 2 M to hypotonic 0.001 M NaCl (bovine: HA=0.420±0.109 MPa to 1.266±0.438 MPa; human: HA=0.499±0.208 MPa to 1.597±0.455 MPa), within a triphasic theory inclusive of osmotic effects. This study provides a novel and simple analytical model for cartilage osmotic pressure which may be used in computational simulations, validated with direct in situ measurements. A key finding is the simultaneous existence of Donnan osmotic and Poisson-Boltzmann electrostatic interactions within cartilage.

Effect of Plasma Membrane Semipermeability in Making the Membrane Electric Double Layer Capacitances Significant

Electric double layers (or EDLs) formed at the membrane-electrolyte interface (MEI) and membrane-cytosol interface (MCI) of a charged lipid bilayer plasma membrane develop finitely large capacitances. However, these EDL capacitances are often much larger than the intrinsic capacitance of the membrane, and all of these capacitances are in series. Consequently, the effect of these EDL capacitances in dictating the overall membrane-EDL effective capacitance C_eff becomes negligible. In this paper, we challenge this conventional notion pertaining to the membrane-EDL capacitances. We demonstrate that, on the basis of the system parameters, the EDL capacitance for both the permeable and semipermeable membranes can be small enough to influence C_eff. For the semipermeable membranes, however, this lowering of the EDL capacitance can be much larger, ensuring a reduction of C_eff by more than 20-25%. Furthermore, for the semipermeable membranes, the reduction in C_eff is witnessed over a much larger range of system parameters. We attribute such an occurrence to the highly nonintuitive electrostatic potential distribution associated with the recently discovered phenomena of charge-inversion-like electrostatics and the attainment of a positive zeta potential at the MCI for charged semipermeable membranes. We anticipate that our findings will impact the quantification and the identification of a large number of biophysical phenomena that are probed by measuring the plasma membrane capacitance.

Electric double layer electrostatics of lipid-bilayer-encapsulated nanoparticles: Toward a better understanding of protocell electrostatics

Lipid-bilayer-encapsulated nanoparticles (LBLENPs) or NP-supported LBL systems, such as protocells (which are lipid bilayer encapsulated mesoporous silica nanoparticles or MSNPs) have received extensive attention for applications like targeted drug and gene deliveries, multimodal diagnostics, characterization of membrane-geometry sensitive molecules, etc. Very often electrostatic-mediated interactions have been hypothesized to play key roles in the functioning of these LBLENPs. Despite that, very little has been done to theoretically quantify the fundamental electric double layer (EDL) electrostatics of such LBLENPs. In this study, we develop an EDL theory to describe the electrostatics of such LBLENPs. We show that the electrostatics is a manifestation of the charged/dielectric nature of the NP, LBL structure and charging, and the ionic environment in which the LBLENPs are present. We also establish that for certain conditions of charging of the NP one witnesses a most remarkable charge inversion like electrostatics within the LBL membrane or the NP itself. We anticipate that our findings will provide an extremely useful platform for better understanding the fabrication and functioning of such LBLENPs and discuss examples where our theory can be useful.

Lamellar cationic lipid-DNA complexes from lipids with a strong preference for planar geometry: A Minimal Electrostatic Model

We formulate and analyze a minimal model, based on condensation theory, of the lamellar cationic lipid (CL)-DNA complex of alternately charged lipid bilayers and DNA monolayers in a salt solution. Each lipid bilayer, composed by a random mixture of cationic and neutral lipids, is assumed to be a rigid uniformly charged plane. Each DNA monolayer, located between two lipid bilayers, is formed by the same number of parallel DNAs with a uniform separation distance. For the electrostatic calculation, the model lipoplex is collapsed to a single plane with charge density equal to the net lipid and DNA charge. The free energy difference between the lamellar lipoplex and a reference state of the same number of free lipid bilayers and free DNAs, is calculated as a function of the fraction of CLs, of the ratio of the number of CL charges to the number of negative charges of the DNA phosphates, and of the total number of planes. At the isoelectric point the free energy difference is minimal. The complex formation, already favoured by the decrease of the electrostatic charging free energy, is driven further by the free energy gain due to the release of counterions from the DNAs and from the lipid bilayers, if strongly charged. This minimal model compares well with experiment for lipids having a strong preference for planar geometry and with major features of more detailed models of the lipoplex.

More than just a vulnerable pipeline: xylem physiology in the light of ion-mediated regulation of plant water transport

Major restrictions to the hydraulic conductance of xylem (K(XYL)) in vascular plants have traditionally been attributed to anatomical constraints. More recently, changes in the cationic concentration of xylem sap have been suggested to be responsible for short-term changes in K(XYL) based on data for 35 dicot species, and very few gymnosperms and ferns, indicating that xylem water transport may no longer be considered as an entirely passive process. Recent studies have revealed that this so-called ionic effect: (i) varies from little or no increase to >30%, (ii) is species specific, (iii) changes on a seasonal basis, (iv) depends on the cationic concentration, (v) is enhanced in embolized stems, and (vi) is positively correlated with vessel grouping. Furthermore, the ionic effect has been suggested to play functional roles in planta with respect to: (i) phloem-mediated control of xylem hydraulic properties, (ii) compensation of cavitation-induced loss of hydraulic conductance, with the result of optimizing light and water utilization, and (iii) differential regulation of water delivery to branches exposed to different levels of light. Pits are likely to play a key role in the ionic effect, which has largely been explained as a consequence of the poly-electrolytic nature and hydrogel properties of the pectic matrix of interconduit pit membranes, despite little evidence that pit membrane pectins remain present after cell hydrolysis. More research is needed to address the ionic effect in more species, physico-chemical properties of pit membranes, and how the ionic effect may increase xylem hydraulic conductance 'on demand'.

Reduced content of homogalacturonan does not alter the ion-mediated increase in xylem hydraulic conductivity in tobacco

Xylem hydraulic conductivity (K(s)) in stems of tobacco (Nicotiana tabacum) wild-type SR1 was compared to that of PG7 and PG16, two transgenic lines with increased levels of expression of the gene encoding the Aspergillus niger endopolygalacturonase (AnPGII). Activity of AnPGII removes in planta blocks of homogalacturonan (HG) with deesterified carboxyls, thus increasing the degree of neutrality of pectins. The effect of K+ was tested in increasing stem K(s) using model plants with more neutral polysaccharides in primary walls and, hence, in intervessel pit membranes. K(s) measured with deionized water was compared to that with KCl solutions at increasing concentrations (DeltaK(s), %). Plants transformed for HG degree of neutrality showed a dwarfed phenotype, but DeltaK(s) did not differ among the three experimental groups. The ion-mediated hydraulic effect saturated at a KCl concentration of 25 mm in SR1 plants. All the three tobacco lines showed DeltaK(s) of around +12.5% and +17.0% when perfused with 10 and 25 mm KCl, respectively. Because modification of HG content did not influence ion-mediated hydraulic enhancement, we suggest that pectin components other than HG, like rhamnogalacturonan-I and/or rhamnogalacturonan-II, might play important roles in the hydrogel behavior of pit membranes.

Ion-mediated increase in the hydraulic conductivity of Laurel stems: role of pits and consequences for the impact of cavitation on water transport

Changes in hydraulic conductivity (K(h)) were measured in stems of Laurus nobilis L. during perfusion with KCl, NaCl or sucrose solutions. Ionic solutes induced marked increase of K(h) with respect to deionized water but sucrose had no effect. The kinetics of KCl-induced K(h) increase was measured together with changes in [K(+)] of the perfused solution. K(h) increases were paralleled by increases in the [K(+)](out)/[K(+)](in) ratio. Samples of different lengths or with increasing percentage loss of conductivity (PLC) due to xylem cavitation were tested, with the aim of increasing radial flow through intervessel pits. KCl solutions enhanced the K(h) of 12-cm-long samples with a concentration-dependent effect up to 100 mm KCl. DeltaK(h) increased from 3 to 30% in 1.5- and 12-cm-long samples, respectively and remained constant for longer samples. Increasing PLC induced an exponential increase in DeltaK(h). PLC measured with KCl solutions was significantly less than that measured with deionized water, suggesting that measurements of PLC can be affected by the composition of the perfused solution. Experiments support the hypothesis that the 'ionic effect' is mediated by physico-chemical changes of pectins of the pit membranes and raise the possibility that plants might alter the ionic composition of the xylem sap to alleviate the hydraulic impact of cavitation.

A new radiochemical method to investigate ion binding with polyelectrolytes

A new method for investigating the binding of ions with polyelectrolytes has been developed. This method, based on Donnan equilibrium and an isotope exchange between the electrolyte and polyelectrolyte, can distinguish territorial from specific binding of ions and can determine fractions of ions bound with the polyion. This method can determine ion binding with polyelectrolytes in a wide range of polyelectrolyte concentrations in multicomponent solutions. The method was tested with radioactive tracers 22Na+, 36Cl- and heparin sodium salt. The influence of the ionic strength on the Na+ binding with heparin was investigated at 310 K. In the limit of zero ionic strength, all Na+ ions are bound to heparin, but only 45% of them are exchangeable. Thus Na+ ions can be bound both territorially and specifically. The fraction of bound ions decreases rapidly with increasing ionic strength. The fraction of the specifically bound ions becomes negligible when the ionic strength exceeds 0.01 M, whereas the fraction of territorially bound ions can be neglected at ionic strengths higher than 0.45 M.

Probing hydration of monovalent cations condensed around polymeric nucleic acids

We report the relative molar sound velocity increments, [U], partial molar volumes, V(o), and partial molar adiabatic compressibilities, K(S)(o), of the Li(+), Na(+), K(+), Rb(+), Cs(+), NH(4)(+), and N(CH(3))(4)(+) salts of poly(dAdT)poly(dAdT), poly(dGdC)poly(dGdC), poly(dIdC)poly(dIdC), poly(rA)poly(rU), poly(rG)poly(rC), poly(rI)poly(rC), and poly(rU) at 25 degrees C. When analyzing these data, we take into account the Donnan membrane equilibrium effect. Comparison between the values of [U], V(o), and K(S)(o) exhibited by the nucleic acid salts and respective chlorides (LiCl, NaCl, KCl, RbCl, CsCl, NH(4)Cl, and N(CH(3))(4)Cl) yields information about the state of counterion hydration in the vicinity of each nucleic acid structure studied here. Our analysis reveals that the poly(dGdC)poly(dGdC), poly(dIdC)poly(dIdC), and poly(rI)poly(rC) duplexes and single-stranded poly(rU) do not significantly influence the hydration properties of their condensed counterions. In the vicinity of these polymers, counterions retain their full hydration shells (within +/-15%). By contrast, counterions condensed around the poly(dAdT)poly(dAdT), poly(rA)poly(rU), and poly(rG)poly(rC) duplexes are significantly dehydrated and retain, respectively, only 65(+/-18)%, 34(+/-21)%, and 33(+/-9)% of their original hydration shells. Taken together, the volumetric data reported here provide important new information that ultimately may help us understand the central role that hydration and counterions play in modulating the conformational preferences of nucleic acids and the energetics of DNA recognition events.

Thermodynamics of the Laminar Donnan System

Thermodynamic quantities of a polyelectroyte immersed in salt solution are derived modeling the polyelectrolyte by a sequence of charged parallel flat plates. The starting point for the analysis is the derivation of the Gibbs free enthalpy in its canonic variables pressure (p) and temperature (T), i.e., as a thermodynamic potential. From this, further thermodynamic quantities such as Helmoltz free energy, entropy, internal energy, compressibility, isobar and isochor heat capacities, and expansive force are derived in analytical expressions by differentiation. All these formulas contain the parameter plate surface charge density (sigma) that provides a measure of the discontinuity of the polymer charge distribution that can be used to fit the theory to experimental data. Thermodynamic quantities are also known from the classical Donnan equilibrium that treats the polyelectroyte charge network as a charge continuum. A limiting process is used to perform the transition from the laminar Poisson- Boltzmann model to the continuous Donnan equilibrium. In general, the expressions of the Donnan system are recovered for plate charge density sigma-->0, number of plates Z-->infinity, and sigma Z=constant. Copyright 2000 Academic Press.