A hard sphere model for single-file water transport across biological membranes.

We use Gürsey's statistical mechanics of a one-dimensional fluid to find a formula for the P f / P d ratio in the transport of hard spheres across a membrane through a narrow channel that can accommodate molecular movement only in single file. P f is the membrane permeability for osmotic flow and P d the permeability for exchange across the membrane in the absence of osmotic flow. The deviation of the ratio from unity indicates the degree of cooperative transport relative to ordinary diffusion of independent molecules. In contrast to an early idea that P f / P d must be equal to the number of molecules in the channel, regardless of the physical nature of the interactions among the molecules, we find a functional dependence on the fractional occupancy of the length of the channel by the hard spheres. We also attempt a random walk calculation for P d individually, which gives a result for P f as well when combined with the ratio.

The physical basis of osmosis.

Osmosis is an important force in all living organisms, yet the molecular basis of osmosis is widely misunderstood as arising from diffusion of water across a membrane separating solutions of differing osmolarities, and hence different water concentrations. In 1923, Peter Debye proposed a physical model for a semipermeable membrane emphasizing the repulsive forces between solute molecules and membrane that prevent the solute from entering the membrane. His work was hardly noticed at the time and slipped out of view. We show that Debye's analysis of van 't Hoff's law for osmotic equilibrium also provides a consistent and plausible mechanism for osmotic flow. A difference in osmolyte concentrations in solutions separated by a semipermeable membrane leads to different pressures at the two water-membrane interfaces because the total repulsive force between solute molecules and the membrane is different at the two interfaces. Water is therefore driven through the membrane for exactly the same reason that pure water flows in response to an imposed hydrostatic pressure difference. In this paper, we present the Debye model in both equilibrium and flow conditions. We point out its applicability regardless of the nature of the membrane with examples ranging from the predominantly convective flow of water through synthetic membranes and capillary walls to the purely diffusive flow of independent water molecules through a lipid bilayer and the flow of a single-file column of water molecules in narrow protein channels.

Editorial on Special Issue "New Era in the Volume Phase Transition of Gels".

The Special Issue of gels titled "Advancements in Gel Science" has been published from MDPI in 2019 [...].

On the thermodynamic stability of bubbles, immiscible droplets, and cavities.

Nanobubbles filled with air or a variety of pure gases are observed to persist in bulk water for weeks and months. Nanoemulsions consisting of oil droplets in water are also remarkably stable against coagulation, with lifetimes up to weeks even if not coated with surfactants. The inverse system of nanodroplets of water in oil is also accessible for study and application. Voids on the nanoscale are formed in simulations of water under strong tension and are stable during the time of the simulation. The stability of these nano-entities is ultimately determined by the molecular-level structure of their interfaces. However, a thermodynamic theory might also be capable of providing some insight. We therefore consider spherical gas nanobubbles, immiscible liquid nanodroplets, and nanocavities formed in water under negative pressure on the same footing, and give a unified thermodynamic analysis of these systems. In all cases, mechanical equilibrium (local free energy maximum or minimum) is expressed by the Laplace equation, and thermodynamic stability (local free energy minimum) follows from the radius dependence of surface tension. All of them would be unstable if their surface tensions were constant. Data from the literature allow construction of numerical examples for cavities and gas nanobubbles. Spectroscopic data are cited in support of an interfacial water structure in gas nanobubbles and water droplets in oil that differ from their flat surface counterparts. The observed longevity of nanobubbles in particular has been thought to violate fundamental principles of diffusion and solubility. A close look at the Laplace equation and its derivation shows why this widespread belief is incorrect.

Construction of a Universal Gel Model with Volume Phase Transition.

The physical principle underlying the familiar condensation transition from vapor to liquid is the competition between the energetic tendency to condense owing to attractive forces among molecules of the fluid and the entropic tendency to disperse toward the maximum volume available as limited only by the walls of the container. Van der Waals incorporated this principle into his equation of state and was thus able to explain the discontinuous nature of condensation as the result of instability of intermediate states. The volume phase transition of gels, also discontinuous in its sharpest manifestation, can be understood similarly, as a competition between net free energy attraction of polymer segments and purely entropic dissolution into a maximum allowed volume. Viewed in this way, the gel phase transition would require nothing more to describe it than van der Waals' original equation of state (with osmotic pressure Π replacing pressure P). But the polymer segments in a gel are networked by cross-links, and a consequent restoring force prevents complete dissolution. Like a solid material, and unlike a van der Waals fluid, a fully swollen gel possesses an intrinsic volume of its own. Although all thermodynamic descriptions of gel behavior contain an elastic component, frequently in the form of Flory-style rubber theory, the resulting isotherms usually have the same general appearance as van der Waals isotherms for fluids, so it is not clear whether the solid-like aspect of gels, that is, their intrinsic volume and shape, adds any fundamental physics to the volume phase transition of gels beyond what van der Waals already knew. To address this question, we have constructed a universal chemical potential for gels that captures the volume transition while containing no quantities specific to any particular gel. In this sense, it is analogous to the van der Waals theory of fluids in its universal form, but although it incorporates the van der Waals universal equation of state, it also contains a network elasticity component, not based on Flory theory but instead on a nonlinear Langevin model, that restricts the radius of a fully swollen spherical gel to a solid-like finite universal value of unity, transitioning to a value less than unity when the gel collapses. A new family of isotherms arises, not present in a preponderately van der Waals analysis, namely, profiles of gel density as a function of location in the gel. There is an abrupt onset of large amplitude density fluctuations in the gel at a critical temperature. Then, at a second critical temperature, the entire swollen gel collapses to a high-density phase.

A bead-spring chain as a one-dimensional polyelectrolyte gel.

The physical principles underlying expansion of a single-chain polyelectrolyte coil caused by Coulomb repulsions among its ionized groups, and the expansion of a cross-linked polyelectrolyte gel, are probably the same. In this paper, we analyze a "one-dimensional" version of a gel, namely, a linear chain of charged beads connected by Hooke's law springs. In the Debye-Hückel range of relatively weak Coulomb strength, where counterion condensation does not occur, the springs are realistically stretched on a nanolength scale by the repulsive interactions among the beads, if we use a spring constant normalized by the inverse square of the solvent Bjerrum length. The persistence length and radius of gyration counter-intuitively decrease when Coulomb strength is increased, if analyzed in the framework of an OSF-type theory; however, a buckling theory generates the increase that is consistent with bead-spring simulations.

Condensation of Counterions Gives Rise to Contraction Transitions in a One-Dimensional Polyelectrolyte Gel.

The equilibrium volume of a polyelectrolyte gel results from a balance between the tendency to swell caused by outbound polymer/counterion diffusion along with Coulomb interactions on the one hand; and, on the other, the elastic resilience of the cross-linked polymer network. Direct Coulomb forces contribute both to non-ideality of the equilibrated Donnan osmotic pressure, but also to stretching of the network. To isolate the effect of polyelectrolyte expansion, we have analyzed a "one-dimensional" version of a gel, a linear chain of charged beads connected by Hooke's law springs. As in the range of weak Coulomb strengths previously studied, the springs are significantly stretched by the repulsive interactions among the beads even when the Coulomb strength is strong enough to cause condensation of counterions. There is a quasi-abrupt transition from a stretched state to a partially collapsed state in a transition range between weak and strong Coulomb strengths. Fluctuations between stretched and contracted conformations occur within the transition range. As the solvent quality decreases past the transition range, a progressive collapse can result if the condensed counterions strengthen the spring constant.

Universal McMillan-Mayer van der Waals Langevin Gel.

We present a theory for a universal gel based on a McMillan-Mayer treatment of a solute-solvent fluid as a generalization of the universal van der Waals equation of state for a pure liquid/vapor system. The elastic resilience of the networked gel is modeled by a universal Langevin function. This combination of van der Waals interactions and nonlinear Langevin elasticity produces an abrupt onset of large-amplitude density fluctuations deep in the interior of the gel at a critical temperature. Then, at a second, lower, critical temperature, the entire swollen gel collapses to a high-density phase. The universal gel has an "upper" critical temperature behavior, meaning that the gel transition to high density occurs on decreasing the temperature. At the cost of loss of universality, the theory is generalized to predict lower critical temperature dependence, whereby an aqueous hydrophobic gel exhibits phase coexistence when the temperature is raised. The theory is consistent with the Gibbs phase rule, suitably generalized to coexisting phases that are not at the same pressure in equilibrium conditions.

Predicting Salt Permeability Coefficients in Highly Swollen, Highly Charged Ion Exchange Membranes.

This study presents a framework for predicting salt permeability coefficients in ion exchange membranes in contact with an aqueous salt solution. The model, based on the solution-diffusion mechanism, was tested using experimental salt permeability data for a series of commercial ion exchange membranes. Equilibrium salt partition coefficients were calculated using a thermodynamic framework (i.e., Donnan theory), incorporating Manning's counterion condensation theory to calculate ion activity coefficients in the membrane phase and the Pitzer model to calculate ion activity coefficients in the solution phase. The model predicted NaCl partition coefficients in a cation exchange membrane and two anion exchange membranes, as well as MgCl2 partition coefficients in a cation exchange membrane, remarkably well at higher external salt concentrations (>0.1 M) and reasonably well at lower external salt concentrations (<0.1 M) with no adjustable parameters. Membrane ion diffusion coefficients were calculated using a combination of the Mackie and Meares model, which assumes ion diffusion in water-swollen polymers is affected by a tortuosity factor, and a model developed by Manning to account for electrostatic effects. Agreement between experimental and predicted salt diffusion coefficients was good with no adjustable parameters. Calculated salt partition and diffusion coefficients were combined within the framework of the solution-diffusion model to predict salt permeability coefficients. Agreement between model and experimental data was remarkably good. Additionally, a simplified version of the model was used to elucidate connections between membrane structure (e.g., fixed charge group concentration) and salt transport properties.

An ionic-chemical-mechanical model for muscle contraction.

The dynamic process underlying muscle contraction is the parallel sliding of thin actin filaments along an immobile thick myosin fiber powered by oar-like movements of protruding myosin cross bridges (myosin heads). The free energy for functioning of the myosin nanomotor comes from the hydrolysis of ATP bound to the myosin heads. The unit step of translational movement is based on a mechanical-chemical cycle involving ATP binding to myosin, hydrolysis of the bound ATP with ultimate release of the hydrolysis products, stress-generating conformational changes in the myosin cross bridge, and relief of built-up stress in the myosin power stroke. The cycle is regulated by a transition between weak and strong actin-myosin binding affinities. The dissociation of the weakly bound complex by addition of salt indicates the electrostatic basis for the weak affinity, while structural studies demonstrate that electrostatic interactions among negatively charged amino acid residues of actin and positively charged residues of myosin are involved in the strong binding interface. We therefore conjecture that intermediate states of increasing actin-myosin engagement during the weak-to-strong binding transition also involve electrostatic interactions. Methods of polymer solution physics have shown that the thin actin filament can be regarded in some of its aspects as a net negatively charged polyelectrolyte. Here we employ polyelectrolyte theory to suggest how actin-myosin electrostatic interactions might be of significance in the intermediate stages of binding, ensuring an engaged power stroke of the myosin motor that transmits force to the actin filament, and preventing the motor from getting stuck in a metastable pre-power stroke state. We provide electrostatic force estimates that are in the pN range known to operate in the cycle.

Partitioning of mobile ions between ion exchange polymers and aqueous salt solutions: importance of counter-ion condensation.

Equilibrium partitioning of ions between a membrane and a contiguous external solution strongly influences transport properties of polymeric membranes used for water purification and energy generation applications. This study presents a theoretical framework to quantitatively predict ion sorption from aqueous electrolytes (e.g., NaCl, MgCl2) into charged (i.e., ion exchange) polymers. The model was compared with experimental NaCl, MgCl2, and CaCl2 sorption data in commercial cation and anion exchange membranes. Ion sorption in charged polymers was modeled using a thermodynamic approach based on Donnan theory coupled with Manning's counter-ion condensation theory to describe non-ideal behavior of ions in the membrane. Ion activity coefficients in solution were calculated using the Pitzer model. The resulting model, with no adjustable parameters, provides remarkably good agreement with experimental values of membrane mobile salt concentration. The generality of the model was further demonstrated using literature data for ion sorption of various electrolytes in charged polymers, including HCl sorption in Nafion.

The energy of naturally curved elastic rods with an application to the stretching and contraction of a free helical spring as a model for DNA.

We give a contemporary and direct derivation of a classical, but insufficiently familiar, result in the theory of linear elasticity-a representation for the energy of a stressed elastic rod with central axis that intrinsically takes the shape of a general space curve. We show that the geometric torsion of the space curve, while playing a crucial role in the bending energy, is physically unrelated to the elastic twist. We prove that the twist energy vanishes in the lowest-energy states of a rod subject to constraints that do not restrict the twist. The stretching and contraction energies of a free helical spring are computed. There are local high-energy minima. We show the possibility of using the spring to model the chirality of DNA. We then compare our results with an available atomic level energy simulation that was performed on DNA unconstrained in the same sense as the free spring. We find some possible reflections of springlike behavior in the mechanics of DNA, but, unsurprisingly, the base pairs lend a material substance to the core of DNA that a spring does not capture.

The response of DNA length and twist to changes in ionic strength.

We examine twist-stretch coupling of unconstrained DNA using polyelectrolyte theory as applied to a line-charge model along with published data on the ionic-strength dependence of the twist angle. We conclude that twist-stretch coupling is negative: environmental changes that stretch free DNA, unconstrained by externally applied pulling or twisting forces, are accompanied by unwinding of the double helix. We also analyze a helical model and conclude that the observed unwinding of the DNA helix when ionic strength is decreased is driven by radial swelling of the helix.

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.

Excess counterion condensation on polyelectrolyte kinks and branch points and the interaction of skewed charged lines.

We have developed explicit formulas for the excess number of counterions condensed on kinked and intersecting charged lines caused by the more intense electric field in the neighborhood of the kink or intersection. As expected, the number of additionally bound counterions is greater for more pronounced kinks, and also increases with the number of lines that intersect at a common point. We have also analyzed the electrostatic interaction potential as a function of distance between two charged lines in skewed orientation. Our finding in this case is that in a range of close distances the lines must cross a free energy barrier in order to separate.

Theoretical assessment of the oligolysine model for ionic interactions in protein-DNA complexes.

The observed salt dependence of charged ligand binding to polyelectrolytes, such as proteins to DNA or antithrombin to heparin, is often interpreted by means of the "oligolysine model." We review this model as derived entirely within the framework of the counterion condensation theory of polyelectrolytes with no introduction of outside assumptions. We update its comparison with experimental data on the structurally simple systems for which it was originally intended. We then compute the salt dependence of the binding free energy for a variety of protein-DNA complexes with nonlinear Poisson-Boltzmann (NLPB) simulation methods. The results of the NLPB calculations confirm the central prediction of the oligolysine model that the net charge density of DNA remains invariant to protein binding. Specifically, when a cationic protein residue penetrates the layer of counterions condensed on DNA, a counterion is released to bulk solution, and when an anionic protein residue penetrates the condensed counterion layer, an additional counterion is condensed from bulk solution. We also conclude, however, that the cumulative effect of charged protein residues distant from the binding interface makes a significant contribution to the salt dependence of binding, an observation not accommodated by the oligolysine model.

The interaction between a charged wall and its counterions: a condensation theory.

We use the theory of counterion condensation to calculate the potential of mean force between a charged planar wall and an oppositely charged ion of unsigned valence Z. We find two solutions to this problem, an outer potential when the Z-ion is relatively far from the wall and an inner one when it is relatively close. There is a discontinuous upward jump from one to the other branch, as the Z-ion approaches the wall. The jump must occur at the wall itself, and we attribute it to interaction with the condensed layer of counterions which resides on the wall. At an infinitesimal distance from the wall, the Z-ion encounters a repulsive force resisting penetration of the condensed layer. Then, when the Z-ion penetrates the condensed layer, Z univalent counterions are released from the condensed layer into the bulk solution, maintaining the effective charge on the wall at an invariant critical value. We discuss the possible connections of the theory with published data from molecular dynamics simulations on lipid bilayers.

On the apparent saturation of the dipole induced in a rodlike polyion at high electric fields.

It has been inferred from electric birefringence and electric dichroism data that the field-induced dipole moment on charged rodlike particles saturates at higher field strengths. We give theoretical justification for this interpretation of the data. We calculate the threshold field marking the onset of saturation and compare the result with measured values.

Counterion condensation on charged spheres, cylinders, and planes.

We use the framework of counterion condensation theory, in which deviations from linear electrostatics are ascribed to charge renormalization caused by collapse of counterions from the ion atmosphere, to explore the possibility of condensation on charged spheres, cylinders, and planes immersed in dilute solutions of simple salt. In the limit of zero concentration of salt, we obtain Zimm-Le Bret behavior: a sphere condenses none of its counterions regardless of surface charge density, a cylinder with charge density above a threshold value condenses a fraction of its counterions, and a plane of any charge density condenses all of its counterions. The response in dilute but nonzero salt concentrations is different. Spheres, cylinders, and planes all exhibit critical surface charge densities separating a regime of counterion condensation from states with no condensed counterions. The critical charge densities depend on salt concentration, except for the case of a thin cylinder, which exhibits the invariant criticality familiar from polyelectrolyte theory.