Energy of an ion crossing a low dielectric membrane: the role of dispersion self-free energy

The Born charging equation predicts that the permeability of a cell membrane to ions by the solubility-diffusion mechanism depends on the ionic radius and on the dielectric constant of the membrane. However, experiments, for example, on red blood cells and on lysosome membranes, show that the permeability depends strongly on the choice of salt anion in a way that cannot be accommodated by differences in ionic size. We demonstrate that one step towards understanding this ion specificity is to take account of the previously ignored dispersion self-free energy of the ion. This is the quantum electrodynamic analogue of the (electrostatic) Born self-energy of an ion. We show that the dispersion self-free energy contribution can be and often is of the same order of magnitude as the Born contribution. To understand the observed specificity, it is essential to take into account of both ionic size and ionic polarizability. In parallel and to reinforce these observations, we also give simple estimates for how self-free energy changes that occur when an ion moves into the air-water interface region (which has a density profile for water molecules) can influence the surface tension of salt solutions. Consistency can be found between the Hofmeister sequences observed in ion permeation and in surface tension of electrolytes when these previously ignored self-free energies are included properly.     

ON THE CAUSE OF THE INFLUENCE OF IONS ON THE RATE OF DIFFUSION OF WATER THROUGH COLLODION MEMBRANES. II

1. It had been shown in previous publications that when pure water is separated from a solution of an electrolyte by a collodion membrane the ion with the same sign of charge as the membrane increases and the ion with the opposite sign of charge as the membrane diminishes the rate of diffusion of water into the solution; but that the relative influence of the oppositely charged ions upon the rate of diffusion of water through the membrane is not the same for different concentrations. Beginning with the lowest concentrations of electrolytes the attractive influence of that ion which has the same sign of charge as the collodion membrane upon the oppositely charged water increases more rapidly with increasing concentration of the electrolyte than the repelling effect of the ion possessing the opposite sign of charge as the membrane. When the concentration exceeds a certain critical value the repelling influence of the latter ion upon the water increases more rapidly with a further increase in the concentration of the electrolyte than the attractive influence of the ion having the same sign of charge as the membrane. 2. It is shown in this paper that the influence of the concentration of electrolytes on the rate of transport of water through collodion membranes in electrical endosmose is similar to that in the case of free osmosis. 3. On the basis of the Helmholtz theory of electrical double layers this seems to indicate that the influence of an electrolyte on the rate of diffusion of water through a collodion membrane in the case of free osmosis is due to the fact that the ion possessing the same sign of charge as the membrane increases the density of charge of the latter while the ion with the opposite sign diminishes the density of charge of the membrane. The relative influence of the oppositely charged ions on the density of charge of the membrane is not the same in all concentrations. The influence of the ion with the same sign of charge increases in the lowest concentrations more rapidly with increasing concentration than the influence of the ion with the opposite sign of charge, while for somewhat higher concentrations the reverse is true.     

Theoretical studies of enzymic reactions: Dielectric,electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme

A general method for detailed study of enzymic reactions is presented. The method considers the complete enzyme-substrate complex together with the surrounding solvent and evaluates all the different quantum mechanical and classical energy factors that can affect the reaction pathway. These factors include the quantum mechanical energies associated with bond cleavage and charge redistribution of the substrate and the classical energies of steric and electrostatic interactions between the substrate and the enzyme. The electrostatic polarization of the enzyme atoms and the orientation of the dipoles of the surrounding water molecules is simulated by a microscopic dielectric model. The solvation energy resulting from this polarization is considerable and must be included in any realistic calculation of chemical reactions involving anything more than an isolated molecule in vacuo. Without it, acidic groups can never become ionized and the charge distribution on the substrate will not be reasonable. The same dielectric model can also be used to study the reaction of the substrate in solution. In this way the reaction in solution can be compared with the enzymic reaction.In this paper we study the stability of the carbonium ion intermediate formed in the cleavage of a glycosidic bond by lysozyme. It is found that electrostatic stabilization is an important factor in increasing the rate of the reaction step that leads to the formation of the carbonium ion intermediate. Steric factors, such as the strain of the substrate on binding to lysozyme, do not seem to contribute significantly.     

The permeability of liposomes to nonelectrolytes. I. Activation energies for permeation.

The effect of temperature on the permeability of nonelectrolytes across liposomal membranes above and below their transition temperature has been studied by using an osmotic method. Below their transition temperature, liposomes are osmotically insensitive structures but, on addition of gramicidin A, the water permeability so increased that the permeability of solutes could be studied. The measured activation energies for permeation of a variety of nonelectrolytes has been found to increase when a) there is an increase in the capability of the solutes to form hydrogen bonds in water, b) the cholesterol concentration in the membranes increases and c) the membranes pass from a liquid-crystalline to a solid-crystalline state. The change in the activation energy for permeation per hydrogen bond is about 1.8 kcal/mole for all the different liposome systems investigated; the only solute tested that deviated from this correlation was urea, whose activation energy for permeation across a gramicidin-containing system was much lower than expected from its hydrogen-bonding capacity. This finding suggests that urea is permeating across the gramicidin pore. Although the literature contains only incomplete data relating the activation energies for permeation of nonelectrolytes across biological membranes to their hydrogen-bonding capacity, the available evidence suggests that there is a similar correlation to that found in liposomes. Thus, the average increase in the activation energy per hydrogen bond for permeation across ox red cell membranes (Jacobs, Glassman & Parpart, J. Cell. Comp. Physiol. 7:197, 1935) is 2.2 plus or minus 0.4 kcal/mole, a value that is similar to that obtained in liposomes. However, the activation energies for water and urea are - in such a system - very much lower than expected, suggesting that they, too, are permeating by some parallel route such as an aqueous pore.     

The Double Fixed Charge Membrane: Solution-Membrane Ion Partition Effects and Membrane Potentials

An analysis is made of the effect of solution-membrane partition of ions on the electrostatic potential and ion concentration profiles in fixed charge membranes. It is shown that the inclusion of partition effects gives rise to large solution-membrane “Donnan” potentials even when the concentration of fixed charges is of the same order as the concentration of the external solution. This effect renders the system and the simplified analysis of the double fixed charge membrane (FCM) previously given more applicable to biological membranes. An analysis is also given of the voltage dependence of the fluxes of individual ion species in the double FCM when it separates different ionic solutions and an expression is deduced for the membrane resting potential. Although the latter is similar in form to the Goldman-Hodgkin-Katz (GHK) equation the corresponding value of the permeability ratio P_C1/P_K is under certain specified conditions both concentration and potential dependent.     

The State of Water in the Isolated Toad Bladder in the Presence and Absence of Vasopressin

An attempt has been made to assess the validity of applying the frictional and viscous coefficients of bulk water to the movement of water and solutes through the urinary bladder of the toad. The temperature dependence of diffusion of THO, C¹⁴-urea, C¹⁴-thiourea, and net water transfer across the bladder was determined in the presence and absence of vasopressin. The activation energy for diffusion of THO was 9.8 kcal per mole in the absence of vasopressin and 4.1 kcal per mole with the hormone present. Activation energies simultaneously determined following vasopressin for diffusion and net transfers of water were similar, and in the same range as known activation energies for diffusion and viscous flow in water. Urea had activation energies for diffusion of 4.1 and 3.9 kcal per mole in the absence and presence of vasopressin, respectively. Thiourea had a high activation energy for diffusion of 6.3 kcal per mole, which was unchanged, 6.6 kcal per mole, following hormone. These findings suggest that in its rate-limiting permeability barrier, water is present in a structured state, offering a high resistance to penetration by water. Vasopressin enlarges the aqueous channels so that the core of water they contain possesses the physical properties of ordinary bulk water. Urea penetrates the tissue via these aqueous channels while thiourea is limited by some other permeability barrier.     

Charge density-dependent strength of hydration and biological structure.

Small ions of high charge density (kosmotropes) bind water molecules strongly, whereas large monovalent ions of low charge density (chaotropes) bind water molecules weakly relative to the strength of water-water interactions in bulk solution. The standard heat of solution of a crystalline alkali halide is shown here to be negative (exothermic) only when one ion is a kosmotrope and the ion of opposite charge is a chaotrope; this standard heat of solution is known to become proportionally more positive as the difference between the absolute heats of hydration of the corresponding gaseous anion and cation decreases. This suggests that inner sphere ion pairs are preferentially formed between oppositely charged ions with matching absolute enthalpies of hydration, and that biological organization arises from the noncovalent association of moieties with matching absolute free energies of solution, except where free energy is expended to keep them apart. The major intracellular anions (phosphates and carboxylates) are kosmotropes, whereas the major intracellular monovalent cations (K+; arg, his, and lys side chains) are chaotropes; together they form highly soluble, solvent-separated ion pairs that keep the contents of the cell in solution.     

Hydration and conformational equilibria of simple hydrophobic and amphiphilic solutes.

We consider whether the continuum model of hydration optimized to reproduce vacuum-to-water transfer free energies simultaneously describes the hydration free energy contributions to conformational equilibria of the same solutes in water. To this end, transfer and conformational free energies of idealized hydrophobic and amphiphilic solutes in water are calculated from explicit water simulations and compared to continuum model predictions. As benchmark hydrophobic solutes, we examine the hydration of linear alkanes from methane through hexane. Amphiphilic solutes were created by adding a charge of +/-1e to a terminal methyl group of butane. We find that phenomenological continuum parameters fit to transfer free energies are significantly different from those fit to conformational free energies of our model solutes. This difference is attributed to continuum model parameters that depend on solute conformation in water, and leads to effective values for the free energy/surface area coefficient and Born radii that best describe conformational equilibrium. In light of these results, we believe that continuum models of hydration optimized to fit transfer free energies do not accurately capture the balance between hydrophobic and electrostatic contributions that determines the solute conformational state in aqueous solution.     

Permeability and Electrical Properties of Thin Lipid Membranes

We present and discuss the permeability and electrical properties of thin lipid membranes, and the changes induced in these properties by several agents added to the aqueous phases after the membranes have formed. The unmodified membrane is virtually impermeable to ions and small "hydrophilic" solutes, but relatively permeable to water and "lipophilic" molecules. These properties are consistent with those predicted for a thin film of hydrocarbon through which matter is transported by dissolving in the membrane phase and then diffusing through it. The effect of cholesterol in reducing the water and "lipophilic" solute permeability is attributed to an increase of the "viscosity" of the hydrocarbon region, thus reducing the diffusion coefficient of molecules within this phase. The selective permeability of the membrane to iodide (I^-) in the presence of iodine (I₂) is attributed to the formation of polyiodides (perhaps I₅ ^-), which are presumed to be relatively soluble in the membrane because of their large size, and hence lower surface charge density. Thus, I₂ acts as a carrier for I^-. The effects of "excitability-inducing material" and the depsipeptides (particularly valinomycin) on ion permeability are reviewed. The effects of the polyene antibiotics (nystatin and amphotericin B) on ion permeability, discussed in greater detail, are the following: (a) membrane conductance increases with the 10th power of nystatin concentration; (b) the membrane is anion-selective but does not discriminate completely between anions and cations; (c) the membrane discriminates among anions on the basis of size; (d) membrane conductance decreases extraordinarily with increasing temperatures. Valinomycin and nystatin form independent conductance pathways in the same membrane, and, in the presence of both, the membrane can be reversibly shifted between a cation and anion permeable state by changes in temperature. It is suggested that nystatin produces pores in the membrane and valinomycin acts as a carrier.     

Kinetic criteria of solid state mechanisms in biology

Kinetic criteria for solid state physical mechanisms of electron and ion transport in biological systems are summarized, and the mechanisms are discussed. A reaction which is rate-limited by electron or ion transport across a particle or membrane in accord with Ohm's law will show first order kinetics, with an hyperbolic relationship between rate constant and the sum of substrate plus product. Larger initial substrate concentrations produce smaller rate constants, thus giving the appearance of substrate inhibition. Examples are cytochrome oxidase and peroxidase, and pyruvate carboxylase. Ohmic transport mechanisms may be caused by electron conduction or superconduction through protein, by electron conduction through water, or by conduction of ions through membranes. A reaction which is rate-limited by charge transport across an activation energy barrier at an interface in accord with a logarithmic voltage-current law will show reaction kinetics conforming to the Elovich equation, and will have the appearance of a pair of simultaneous first order processes. Examples include decay of photogenerated free radicals in eye melanin particles and in photosynthetic particles of bacteria, and sodium and potassium ion transport across cell surfaces. The logarithmic voltage-current law may be regarded as an empirical relationship describing behavior of interfaces, justified by extensive experimental data on many types of interfaces, or it may be derived theoretically for individual cases from statistical mechanical and/or solid state physical considerations. Dedicated to Prof. N. Rashevsky and to his enlightened editorial policy, especially to his policy of publishing that which is new, even when he disagrees with it.     

Concepts and protocols for electrostatic free energies

Electrostatic free energies play an essential role in numerous biomolecular processes occurring in solution. Difficulties arise when the long-range Coulomb interaction is computed for idealised infinite simulation models with periodic boundary conditions. To maintain a neutral simulation box and a finite per-box energy, a neutralising charge density or ‘gellium’ is commonly used, leading to a mean box potential that is constrained to be rigorously equal to zero at all times. Thus, in considering quantities such as ion solvation free energy, the potential drop to move from solvent into the usual, gas phase reference state is missing. In fact, for an infinite molecular system, the electrostatic potential itself is not uniquely defined, but takes the form of an infinite series that is only conditionally convergent. This leads to several possible computational conventions that give different values for the potential and field, all mathematically valid. For experimentally measurable quantities, however, unique results are obtained when sufficiently large simulation boxes are utilised. These concepts are detailed, as well as a fundamental, linear response theoretical framework that provides qualitative understanding of the physical processes involved, especially dielectric relaxation of the environment in response to a new solute charge. Illustrative applications to ligand binding and biomolecular electron transfer are described.     

A semi-microscopic Monte Carlo study of permeation energetics in a gramicidin-like channel: the origin of cation selectivity.

The influence of a gramicidin-like channel former on ion free energy barriers is studied using Monte Carlo simulation. The model explicitly describes the ion, the water dipoles, and the peptide carbonyls; the remaining degrees of freedom, bulk electrolyte, non-polar lipid and peptide regions, and electronic (high frequency) permittivity, are treated in continuum terms. Contributions of the channel waters and peptide COs are studied both separately and collectively. We found that if constrained to their original orientations, the COs substantially increase the cationic permeation free energy; with or without water present, CO reorientation is crucial for ion-CO interaction to lower cation free energy barriers; the translocation free energy profiles for potassium-, rubidium-, and cesium-like cations exhibit no broad barriers; the lipid-bound peptide interacts more effectively with anions than cations; anionic translocation free energy profiles exhibit well defined maxima. Using experimental data to estimate transfer free energies of ions and water from bulk electrolyte to a non-polar dielectric (continuum lipid), we found reasonable ion permeation profiles; cations bind and permeate, whereas anions cannot enter the channel. Cation selectivity arises because, for ions of the same size and charge, anions bind hydration water more strongly.     

The basis of the hydrophobic effect

The property of a molecule that most reliably determines the magnitude of the hydrophobic effect that it will experience is the number of hydrogen–carbon bonds it contains not the accessible surface area of its nonpolar portions. This conclusion follows from an examination of the standard free energies of transfer of alkanes, alkenes, alkadienes, and arenes from water to hexadecane. When the standard free energies of transfer for hydrocarbons in these different classes are plotted as a function of the number of hydrogen–carbon bonds they contain, all of the data fall upon the same line. These standard free energies of transfer are also directly proportional to the number of hydrogen–carbon bonds the hydrocarbons contain. When the same standard free energies of transfer are plotted as a function of the accessible surface areas of the hydrocarbons, the data do not fall upon the same line nor are the standard free energies of transfer directly proportional to the accessible surface areas. An examination of the standard free energies of transfer for the different classes of hydrocarbons from the gas phase to water and from the gas phase to hexadecane reinforces the conclusion that the number of hydrogen–carbon bonds in a molecule rather than its accessible surface area is the basis of the hydrophobic effect. Consequently, estimates of the contribution of different functional groups to the hydrophobic effect providing the free energy of folding of a molecule of protein or providing the free energy of dissociation for the association of two proteins or the association of a ligand with a protein should be made by counting the number of hydrogen–carbon bonds excluded from water rather than computing the accessible surface areas excluded from water.     

Red Chlorophylls in the Exciton Model of Photosystem I

Structural arrangement of pigment molecules of Photosystem I of photosynthetic cyanobacterium Synechococcus elongatus is used for theoretical modeling of the excitation energy spectrum. It is demonstrated that a straightforward application of the exciton theory with the assumption of the same molecular transition energy does not describe the red side of the absorption spectrum. Since the inhomogeneity in the molecular transition energies caused by a dispersive interaction with the molecular surrounding cannot be identified directly from the structural model, the evolutionary search procedure is used for fitting the low temperature absorption and circular dichroism spectra. As a result, one dimer, three trimers and one tetramer of chlorophyll molecules responsible for the red side of the absorption spectrum with their assignment to the spectroscopically established three bands at 708, 714 and 719 nm are determined. All of them are found to be situated not in the very close vicinity of the reaction center but are encircling it almost at the same distance. In order to explain the unusual broadening on the red side of the spectrum the exciton state mixing with the charge transfer (CT) states is considered. It is shown that two effects can be distinguished as caused by mixing of those states: (i) the oscillator strength borrowing by the CT state from the exciton transition and (ii) the borrowing of the high density of the CT state by the exciton state. The intermolecular vibrations between two counter-charged molecules determine the high density in the CT state. From the broad red absorption wing it is concluded that the CT state should be the lowest state in the complexes under consideration. Such mixing effect enables resolving the diversity in the molecular transition energies as determined by different theoretical approaches.     

Many-body energies during proton transfer in an aqueous system

The energetics of the mechanism of proton transfer from a hydronium ion to one of the water molecules in its first solvation shell are studied using density functional theory and the Møller–Plesset perturbation (MP2) method. The potential energy surface of the proton transfer mechanism is obtained at the B3LYP and MP2 levels with the 6-311++G** basis set. Many-body analysis is applied to the proton transfer mechanism to obtain the change in relaxation energy, two-body, three-body and four-body energies when proton transfer occurs from the hydronium ion to one of the water molecules in its first solvation shell. It is observed that the binding energy (BE) of the complex decreases during the proton transfer process at both levels of theory. During the proton transfer process, the % contribution of the total two-body energy to the binding energy of the complex increases from 62.9 to 68.09% (39.9 to 45.95%), and that of the total three-body increases from 25.9 to 27.09% (24.16 to 26.17%) at the B3LYP/6-311++G** (MP2/ 6-311++G**) level. There is almost no change in the water–water–water three-body interaction energy during the proton transfer process at both levels of theory. The contribution of the relaxation energy and the total four-body energy to the binding energy of the complex is greater at the MP2 level than at the B3LYP level. Significant differences are found between the relaxation energies, the hydronium–water interaction energies and the four-body interaction energies at the B3LYP and MP2 levels.     

Analysis of structure-permeability correlation of nitrophenol analogues in newborn rat abdominal skin using semiempirical molecular orbital calculation

Theoretical analysis of the permeation process in percutaneous absorption is important for the molecular design of bioavailable transdermal drugs. In the present study, we examined the difference in permeability of nitrophenols across newborn rat abdominal skin and analyzed the structure-permeability correlation by molecular orbital calculation. The permeable rate of o-, n-, p-nitrophenol and 2,4-, 2,5-, 2,6-dinitrophenols was 0.291, 0.212, 0.085 and 0.042, 0.109, 0.027 micromol/cm2/h, respectively. The permeability of the nitrophenols correlated with their pKa values, indicating the ionizing process related to the permeation through the skin. The pKa values better correlated with the ionization potential (IP) energies than lowest unoccupied molecular orbital (LUMO) energies. Solvation free energies (dGW) of molecular form nitrophenols correlated better with permeabilities (pA) than partition coefficients (log P). In analyzing the dGW values with the permeability at pH 7.4, o-nitrophenol outlay from the theoretical line by ortho-effect. We conclude solvation free energy is a practical parameter and very useful for the molecular design of transdermal drugs.     

Complete suboptimal folding of RNA and the stability of secondary structures

An algorithm is presented for generating rigorously all suboptimal secondary structures between the minimum free energy and an arbitrary upper limit. The algorithm is particularly fast in the vicinity of the minimum free energy. This enables the efficient approximation of statistical quantities, such as the partition function or measures for structural diversity. The density of states at low energies and its associated structures are crucial in assessing from a thermodynamic point of view how well-defined the ground state is. We demonstrate this by exploring the role of base modification in tRNA secondary structures, both at the level of individual sequences from Escherichia coli and by comparing artificially generated ensembles of modified and unmodified sequences with the same tRNA structure. The two major conclusions are that (1) base modification considerably sharpens the definition of the ground state structure by constraining energetically adjacent structures to be similar to the ground state, and (2) sequences whose ground state structure is thermodynamically well defined show a significant tendency to buffer single point mutations. This can have evolutionary implications, since selection pressure to improve the definition of ground states with biological function may result in increased neutrality.     

Ab initio molecular orbital study of the interaction of Li, Na and K with the pore components of ion channels: Consideration of the size, structure and selectivity of the pore of the channels

Ab initio molecular orbital calculations were made for the various types of structures of the pore of the ion channel and the results were applied to the permeability model by Hille, an extension of the Eyring rate theory. In Hille's model, ion passage through the channel is regarded as a kinetic process. Accordingly, it is thought that the interaction energy between cation and ligand, the easier the passage is, due to the lower activation energy. The calculated interaction energy was in the order Li+ greater than Na+ greater than K+ for all models. The optimum size of the pore determined from the interaction energy depends on the structure of the filter. The size for the pentagon was largest, followed by the hexagon and tetragon. On the other hand, the size depends hardly at all on the kind of ligand molecules. In the case of the tetragon, the sizes for the Na and K channels were nearly the same as those estimated from the model building and inhibitor-blocking experiment. The interaction energy between the ionized carboxyl group and the cation was extremely large, clearly reflecting the experimental fact that the carboxyl group in the pore has an important role in making the passage of the cation through the channel easier by dehydrating the water molecules. By analysis of the interaction energy, it was revealed that the contribution of the electrostatic energy was predominant, although the contributions of the other effects might not be negligible. Among these effects, the value of the charge transfer energy is largest, and this is noteworthy in connection with the selective transmission of cations through the overlap of orbitals. It is concluded that the quantum-chemical indices such as interaction energy and the electronic charge calculated by the sophisticated ab initio method help to shed light on the nature of the pore of the ion channel.     

The Molecular Mechanism of Ion-Dependent Gating in Secondary Transporters

LeuT-like fold Na-dependent secondary active transporters form a large family of integral membrane proteins that transport various substrates against their concentration gradient across lipid membranes, using the free energy stored in the downhill concentration gradient of sodium ions. These transporters play an active role in synaptic transmission, the delivery of key nutrients, and the maintenance of osmotic pressure inside the cell. It is generally believed that binding of an ion and/or a substrate drives the conformational dynamics of the transporter. However, the exact mechanism for converting ion binding into useful work has yet to be established. Using a multi-dimensional path sampling (string-method) followed by all-atom free energy simulations, we established the principal thermodynamic and kinetic components governing the ion-dependent conformational dynamics of a LeuT-like fold transporter, the sodium/benzyl-hydantoin symporter Mhp1, for an entire conformational cycle. We found that inward-facing and outward-facing states of Mhp1 display nearly the same free energies with an ion absent from the Na2 site conserved across the LeuT-like fold transporters. The barrier separating an apo-state from inward-facing or outward-facing states of the transporter is very low, suggesting stochastic gating in the absence of ion/substrate bound. In contrast, the binding of a Na2 ion shifts the free energy stabilizing the outward-facing state and promoting substrate binding. Our results indicate that ion binding to the Na2 site may also play a key role in the intracellular thin gate dynamics modulation by altering its interactions with the transmembrane helix 5 (TM5). The Potential of Mean Force (PMF) computations for a substrate entrance displays two energy minima that correspond to the locations of the main binding site S1 and proposed allosteric S2 binding site. However, it was found that substrate''s binds to the site S1 ∼5 kcal/mol more favorable than that to the site S2 for all studied bound combinations of ions and a substrate.