The diffusion of ions across biological membranes

Summary The derivation of a diffusion equation is given for the transport of ions across biological membranes. It is suggested that the diffusion proceeds by the jumps of ions from one site to another and the number of such sites in the membrane is restricted.     

Fluctuations of barrier structure in ionic channels

In rate-theory analysis of ion transport in channels, the energy of binding sites and the height of activation barriers are usually considered to be time-independent and not influenced by the movement of the ion. The assumption of a fixed barrier structure seems questionable, however, in view of the fact that proteins may exist in a large number of conformational states and may rapidly move from one state to the other. In this study, some of the effects of multiple conformational states of a channel on ion transport are analyzed. In the first part of the paper, the ion permeability of a channel with n binding sites is treated on the assumption that interconversion of channel states is much faster than ion transfer between binding sites. Under this condition, the form of the flux equation remains the same as for a channel with fixed barriers, provided that the rate constants for ion jumps are replaced by weighted averages over the rate constants for the individual conformational states. In the second part, a channel with two (main) barriers and a single (main) binding site is considered, with the rates of conformational transitions being arbitrary. This case, in particular, includes the situation where a jump of the ion is followed by a slow transition to a more polarized state of the binding site. Under this condition, the conductance of the channel exhibits a nonlinear dependence on ion concentration which is different from a simple saturation behavior. Under non-stationary conditions damped oscillations may occur.     

Some Cation Interactions in Muscle

It has been possible to treat potassium, rubidium, and cesium ion entry into frog sartorius muscle by the use of a model which assumes a limited number of sites at the cell surface. The ion concentration in an outer surface layer is regarded as the main factor determining the rate of inward movement. It is supposed that the concentration of ions in the external solution is effective in promoting inward movement only to an extent determined by the fraction of sites occupied. Equations are derived from the model which fit the inward flux versus applied concentration curves experimentally determined for the three ions. The ions were found to compete for the postulated sites in various bi-ionic mixtures, the competition being satisfactorily described by equations derived from the model. The constants assigned to each ion remain invariant and independent of gradients in electrochemical potential. The order of decreasing exchange rate found is K > Rb > Cs. The order of decreasing site affinity found is Rb > K > Cs which is the same order as that observed for the ion selectivity deduced from analytical measurements of cation preference after equilibration in various equimolal mixtures (Lubin and Schneider (21)). The manner in which such a model might affect the application of a theory which assumes electrical driving forces as well is discussed.     

Kinetics of target site localization of a protein on DNA: a stochastic approach

It is widely recognized that the cleaving rate of a restriction enzyme on target DNA sequences is several orders-of-magnitude faster than the maximal one calculated from the diffusion-limited theory. It was therefore commonly assumed that the target site interaction of a restriction enzyme with DNA has to occur via two steps: one-dimensional diffusion along a DNA segment, and long-range jumps coming from association-dissociation events. We propose here a stochastic model for this reaction which comprises a series of one-dimensional diffusions of a restriction enzyme on nonspecific DNA sequences interrupted by three-dimensional excursions in the solution until the target sequence is reached. This model provides an optimal finding strategy which explains the fast association rate. Modeling the excursions by uncorrelated random jumps, we recover the expression of the mean time required for target site association to occur given by Berg et al. in 1981, and we explicitly give several physical quantities describing the stochastic pathway of the enzyme. For competitive target sites we calculate two quantities: processivity and preference. By comparing these theoretical expressions to recent experimental data obtained for EcoRV-DNA interaction, we quantify: 1), the mean residence time per binding event of EcoRV on DNA for a representative one-dimensional diffusion coefficient; 2), the average lengths of DNA scanned during the one-dimensional diffusion (during one binding event and during the overall process); and 3), the mean time and the mean number of visits needed to go from one target site to the other. Further, we evaluate the dynamics of DNA cleavage with regard to the probability for the restriction enzyme to perform another one-dimensional diffusion on the same DNA substrate following a three-dimensional excursion.     

Ion movement through gramicidin A channels. Studies on the diffusion-controlled association step

The permeability characteristics of gramicidin A channels are generally considered to reflect accurately the intrinsic properties of the channels themselves; i.e., the aqueous convergence regions are assumed to be negligible barriers for ion movement through the channels. The validity of this assumption has been examined by an analysis of gramicidin A single-channel current-voltage characteristics up to very high potentials (500 mV). At low permeant ion concentrations the currents approach a voltage-independent limiting value, whose magnitude is proportional to the permeant ion concentration. The magnitude of this current is decreased by experimental maneuvers that decrease the aqueous diffusion coefficient of the ions. It is concluded that the magnitude of this limiting current is determined by the diffusive ion movement through the aqueous convergence regions up to the channel entrance. It is further shown that the small-signal (ohmic) permeability properties also reflect the existence of the aqueous diffusion limitation. These results have considerable consequences for the construction of kinetic models for ion movement through gramicidin A channels. It is shown that the simple two-site-three-barrier model commonly used to interpret gramicidin A permeability data may lead to erroneous conclusions, as biionic potentials will be concentration dependent even when the channel is occupied by at most one ion. The aqueous diffusion limitation must be considered explicitly in the analysis of gramicidin A permeability characteristics. Some implications for understanding the properties of ion-conducting channels in biological membranes will be considered.     

The structural basis of secondary active transport mechanisms

Secondary active transporters couple the free energy of the electrochemical potential of one solute to the transmembrane movement of another. As a basic mechanistic explanation for their transport function the model of alternating access was put forward more than 40 years ago, and has been supported by numerous kinetic, biochemical and biophysical studies. According to this model, the transporter exposes its substrate binding site(s) to one side of the membrane or the other during transport catalysis, requiring a substantial conformational change of the carrier protein. In the light of recent structural data for a number of secondary transport proteins, we analyze the model of alternating access in more detail, and correlate it with specific structural and chemical properties of the transporters, such as their assignment to different functional states in the catalytic cycle of the respective transporter, the definition of substrate binding sites, the type of movement of the central part of the carrier harboring the substrate binding site, as well as the impact of symmetry on fold-specific conformational changes. Besides mediating the transmembrane movement of solutes, the mechanism of secondary carriers inherently involves a mechanistic coupling of substrate flux to the electrochemical potential of co-substrate ions or solutes. Mainly because of limitations in resolution of available transporter structures, this important aspect of secondary transport cannot yet be substantiated by structural data to the same extent as the conformational change aspect. We summarize the concepts of coupling in secondary transport and discuss them in the context of the available evidence for ion binding to specific sites and the impact of the ions on the conformational state of the carrier protein, which together lead to mechanistic models for coupling.     

Rate theory models for ion transport through rigid pores. I. Time-dependent analysis in the case of vanishing interactions

In this first of a series of papers concerning the theoretical analysis of rate theory models for ion transport through rigid pores, the case of vanishing interactions is investigated. "Rigidity" means that ions crossing membranes through pores see a fixed structure of the pores, not changing in time. A single pore is considered to be a sequence of (n + 1) activation barriers separated by n energy minima. The explicit analytical treatment is restricted to pores with regular internal barrier structure, including the nonequilibrium situation of an applied electric field. In this case the connection with continuum diffusion models is demonstrated by performing in the limit n leads to infinity (n = number of binding sites within the pores) the transition to continuum. Thus, from diffusion equations describing a discrete number of jumps, the corresponding diffusion-like partial differential equations and boundary conditions are generated. For regular pores, from the time dependent solutions of the discrete equations, the corresponding solutions of the continuum equations are explicitly generated. The time-dependent relaxation behaviour of the discrete model is in good agreement with the continuum model if one assumes more than two binding sites in the pores.     

Tuning the voltage dependence of tetraethylammonium block with permeant ions in an inward-rectifier K+ channel.

To understand the role of permeating ions in determining blocking ion-induced rectification, we examined block of the ROMK1 inward-rectifier K+ channel by intracellular tetraethylammonium in the presence of various alkali metal ions in both the extra- and intracellular solutions. We found that the channel exhibits different degrees of rectification when different alkali metal ions (all at 100 mM) are present in the extra- and intracellular solution. A quantitative analysis shows that an external ion site in the ROMK1 pore binds various alkali metal ions (Na+, K+, Rb+, and Cs+) with different affinities, which can in turn be altered by the binding of different permeating ions at an internal site through a nonelectrostatic mechanism. Consequently, the external site is saturated to a different level under the various ionic conditions. Since rectification is determined by the movement of all energetically coupled ions in the transmembrane electrical field along the pore, different degrees of rectification are observed in various combinations of extra- and intracellular permeant ions. Furthermore, the external and internal ion-binding sites in the ROMK1 pore appear to have different ion selectivity: the external site selects strongly against the smaller Na+, but only modestly among the three larger ions, whereas the internal site interacts quite differently with the larger K+ and Rb+ ions.     

Modified activity of Aeromonas aminopeptidase: metal ion substitutions and role of substrates

Aeromonas aminopeptidase contains two nonidentical metal binding sites that have been shown by both spectroscopy and kinetics to be capable of interacting with one another [Prescott, J.M., Wagner, F.W., Holmquist, B., & Vallee, B.L. (1985) Biochemistry 24, 5350-5356]. The effects of metal ion substitutions on the susceptibility of the p-nitroanilides of L-alanine, L-valine, and L-leucine--substrates that are hydrolyzed at widely differing rates by native Aeromonas aminopeptidase--were studied by determining values of kcat and Km for the 16 metalloenzymes that result from all possible combinations of Zn2+, Co2+, Ni2+, and Cu2+ in each of the two sites. The different combinations of metal ions and substrates yield a broad range in kinetic values; kcat varies by more than 1800-fold, Km by 3000-fold, and kcat/Km ratios by more than 10,000. L-Leucine-p-nitroanilide is by far the most susceptible of the three substrates, and the hyperactivation previously observed with aminopeptidase containing either Ni2+ or Cu2+ in the first binding site and Zn2+ in the second site occurs only with the two poorer substrates, L-alanine-p-nitroanilide and L-valine-p-nitroanilide. Although the enzyme with Zn2+ in both sites hydrolyzes the substrates with N-terminal alanine and valine poorly, it is extremely effective toward L-leucine-p-nitroanilide. Neither metal binding site can be identified as controlling either Km or kcat; both parameters are influenced by the identity of the metal ions, by the site each occupies, and, most strongly, by the substrate.(ABSTRACT TRUNCATED AT 250 WORDS)     

Dynamic ion-ion and water-ion interactions in ion channels.

The dynamic interactions among ions and water molecules in ion channels are treated based on an assumption that ions at binding sites can be knocked off by both transient entering ions and local water molecules. The theory, when applied to a single-site model K+ channel, provides solutions for super- and subsaturations, flux-ratio exponent (n') greater than 1, osmotic streaming current, activity-dependent reversal potentials, and anomalous mole-fraction behavior. The analysis predicts that: (a) the saturation may but, in general, does not follow the Michaelis-Menten relation; (b) streaming current results from imbalanced water-ion knock-off interactions; (c) n' greater than 1 even for single-site channels, but it is unlikely to exceed 1.4 unless the pore is occupied by one or more ion(s); (d) in the calculation involving two permeant ion species with similar radii, the heavier ions show higher affinity; the ion-ion knock-off dissociation from the site is more effective when two interacting ions are identical. Therefore, the "multi-ion behaviors" found in most ion channels are the consequences of dynamic ion-ion and water-ion interactions. The presence of these interactions does not require two or more binding sites in channels.     

Effect of 3-phenylindole on lipophilic ion and carrier-mediated ion transport across bilayer lipid membranes

The physical effects of 3-phenylindole, an antimicrobial compound which interacts with phospholipids, on ion transport across phosphatidylcholine-cholesterol bilayers have been investigated using three lipophilic ions and one ion-carrier complex. It was found that 3-phenylindole increased membrane electrical conductance of positively charged membrane probes and decreased electrical conductance of negatively charged probes. The enhancement of conductance detected by nonactin-K+ complex and tetraphenylarsonium+ was several orders of magnitude, whereas the suppression of conductance due to tetraphenylborate- and dipicrylamine- was less than a factor of ten. Presence of 3-phenylindole in aqueous phase slightly decreased adsorption of tetraphenylborate- and dipicrylamine- at the membrane surface. From the voltage dependence of the steady-state conductance it was shown that 3-phenylindole induced kinetic limitation of membrane transport of potassium mediated by nonactin. No such limitation was found in the case of tetraphenylarsonium+ transport. These results are shown to be consistent with the present concept of ion diffusion in membranes and the assumption that 3-phenylindole decreases the electric potential in the membrane interior. The asymmetry of the effect of 3-phenylindole on the magnitude of conductance changes for positively and negatively charged membrane permeable ions is also discussed as a reflection of the discreteness of both the absorbed 3-phenylindole and lipid dipoles.     

The binding of a second divalent metal ion is necessary for the activation of ATP hydrolysis and its inhibition by tightly bound ADP in the ATPase from Halobacterium saccharovorum.

A binding site for divalent metal ions on the ATPase from Halobacterium saccharovorum is proposed. This site is different from the catalytic site which binds ATP and a complexed divalent metal ion. Occupation of the second site greatly stimulates the rate of ATP hydrolysis and the affinity of the catalytic site for the metal ion-ATP complex. The time-dependent inhibition of the ATPase, which occurs during catalysis and which is known to be caused by the retention of ADP, is also dependent on the occupation of this metal ion binding site. The binding of the metal ion apparently induces extremely tight binding of ADP after the departure of Pi. Mg2+, Mn2+, Zn2+, Co2+, and Ca2+ were tested and showed both the activating and the inhibitory effects, although their binding constants for ATP and the second metal ion binding site were quite different. The characteristic shapes of the nonlinear ATP hydrolysis curves obtained with different metal ions, and different ratios of metal ion and ATP, could be explained with the established dissociation constants. On this basis, a model for the ATPase was developed with two catalytic cycles: one in which the second metal ion binding site is occupied, and another in which it is empty. These pathways are connected by metal ion-dependent equilibria.     

Metal ions binding to NAD-glycohydrolase from the venom of Agkistrodon acutus: regulation of multicatalytic activity

AA-NADase from Agkistrodon acutus venom is a unique multicatalytic enzyme with both NADase and AT(D)Pase activities. Among all identified NADases, only AA-NADase contains Cu(2+) ions that are essential for its multicatalytic activity. In this study, the interactions between divalent metal ions and AA-NADase and the effects of metal ions on its structure and activity have been investigated by equilibrium dialysis, isothermal titration calorimetry, fluorescence, circular dichroism, dynamic light scattering and HPLC. The results show that AA-NADase has two classes of Cu(2+) binding sites, one activator site with high affinity and approximately six inhibitor sites with low affinity. Cu(2+) ions function as a switch for its NADase activity. In addition, AA-NADase has one Mn(2+) binding site, one Zn(2+) binding site, one strong and two weak Co(2+) binding sites, and two strong and six weak Ni(2+) binding sites. Metal ion binding affinities follow the trend Cu(2+) > Ni(2+) > Mn(2+) > Co(2+) > Zn(2+), which accounts for the existence of one Cu(2+) in the purified AA-NADase. Both NADase and ADPase activities of AA-NADase do not have an absolute requirement for Cu(2+), and all tested metal ions activate its NADase and ADPase activities and the activation capacity follows the trend Zn(2+) > Mn(2+) > Cu(2+) ~Co(2+) > Ni(2+). Metal ions serve as regulators for its multicatalytic activity. Although all tested metal ions have no obvious effects on the global structure of AA-NADase, Cu(2+)- and Zn(2+)-induced conformational changes around some Trp residues have been observed. Interestingly, each tested metal ion has a very similar activation of both NADase and ADPase activities, suggesting that the two different activities probably occur at the same site.     

The Origin of Bioelectrical Potentials in Plant and Animal Cells

The electrical potential difference across a plant or animalcell membrane can be caused by at least three different mechanisms,acting alone or in concert. First, a Donnan equilibrium canaccount for a sizable membrane potential without the participationof any active transport process. In a Donnan equilibrium themembrane potential is generated by the diffusion of permeatingions down their concentration gradients. The asymmetric distributionof permeating ions is caused by the presence of charged, nondiffusibleions, e.g., proteins inside the cell. The second mechanism isan electrically neutral ion pump, e.g., the coupled sodium-potassiumpump found in many types of cells. An electrically neutral pumpcan generate a large membrane potential if the membrane hasa high passive permeability to one of the actively transportedions, usually potassium. The third mechanism is an electrogenicion pump, which makes a substantial contribution to the membranepotential in several types of plant and animal cells. An electrogenicpump directly causes a net movement of charge across the cellmembrane. The membrane voltage generated by the pump then causesa passive flow of diffusible ions which partially short circuitsthe potential difference generated by the pump.     

How do calcium channels transport calcium ions?

Calcium channel activity is crucial for many fundamental physiological processes ranging from the heart beat to synaptic transmission. The channel-forming protein, of about 2000 amino acids, comprises four domains internally homologous to each other. Voltage-dependent Ca2+ channels are the most selective ion channels known. Under physiological conditions, they prefer Ca2+ over Na+ by a ratio of about 1000:1. To explain at the same time the exquisite ion selectivity and the large Ca2+ ion turnover rate of Ca2+ channels (approximately 3 x 10(6) ions/s), two kind models have been proposed. In one, the conduction pathway possesses two high-affinity binding sites. When two Ca2+ ions are bound to each site, the mutual repulsion between them speeds the exit rate for the ions, causing greater ion permeation through the pore. The second model hypothesizes the existence of a single site having a charged structure able to attract multiple, interacting ions, simultaneously. Recent studies that combine mutagenesis and electrophysiology show that the high-affinity binding site is formed by a ring of glutamate residues located in the pore forming region of the Ca2+ channel. As proposed in the second class of models, the results suggest that four glutamate residues, one glutamate donated by each repeat, combine to form a single high-affinity site. In this review the different conduction models for Ca2+ channels are discussed and confronted with structural data.     

Calcium ion binding to pancreatic phospholipase A2 and its zymogen: a 43Ca NMR study

Calcium ion binding to phospholipase A2 and its zymogen has been studied by 43Ca NMR. The temperature dependence of the band shape of the calcium-43 NMR signal has been used to calculate the calcium ion exchange rate. The on-rate was calculated to be 5 X 10(6) M-1 s-1, which is 2 orders of magnitude less than the diffusion limit of the hydrated Ca2+ ion in water. The 43Ca quadrupole coupling constant for calcium ions bound to phospholipase, chi = 1.4 MHz, is significantly larger than those found for EF-hand proteins, indicating a less symmetric site. For prophospholipase A2, we found chi = 0.8 MHz, indicating a calcium binding site, which is somewhat more symmetric than the EF-hand sites. The dependence of the 43Ca NMR band shape on the calcium ion concentration showed that there are two cation binding sites on the phospholipase A2 molecule: K1 = 4 X 10(3) M-1 and K2 = 20 M-1. The strong site was found to be affected by a pKa = 6.5 and the weak site by pKa = 4.5.     

NMR evaluation of ammonium ion movement within a unimolecular G-quadruplex in solution

d[G4(T4G4)3] has been folded into a unimolecular G-quadruplex in the presence of 15NH4+ ions. NMR spectroscopy confirmed that its topology is the same as the solution state structure determined earlier by Wang and Patel (J. Mol. Biol., 1995; 251: 76-94) in the presence of Na+ ions. The d[G4(T4G4)3] G-quadruplex exhibits four G-quartets with three 15NH4+-ion-binding sites (O1, I and O2). Quantitative analysis utilizing 15NH4+ ions as a NMR probe clearly demonstrates that there is no unidirectional 15NH4+ ion movement through the central cavity of the G-quadruplex. 15NH4+ ions move back and forth between the binding sites within the G-quadruplex and exchange with ions in bulk solution. 15NH4+ ion movement is controlled by the thermodynamic preferences of individual binding sites, steric restraints of the G-quartets for 15NH4+ ion passage and diagonal versus edge-type arrangement of the T4 loops. The movement of 15NH4+ ions from the interior of the G-quadruplex to bulk solution is faster than exchange within the G-quadruplex. The structural details of the G-quadruplex define stiffness of individual G-quartets that intimately affects 15NH4+ ion movement. The stiffness of G-quartets and steric hindrance imposed by thymine residues in the loops contribute to the 5-fold difference in the exchange rate constants through the outer G-quartets.     

Ionic selectivity, saturation, and block in sodium channels. A four- barrier model

Ionic fluxes in Na channels of myelinated axons show ionic competition, block, and deviations from simple flux independence. These phenomena are particularly evident when external Na+ ions are replaced by other permeant or impermeant ions. The observed currents require new flux equations not based on the concepts of free diffusion. A specific permeability model for the Na channel is developed from Eyring rate theory applied to a chain of saturable binding sites. There are four energy barriers in the pore and only one ion is allowed inside at a time. Deviations from independence arise from saturation. The model shows that ionic permeability ratios measured from zero-current potentials can differ from those measured from relative current amplitudes or conductances. The model can be fitted to experiments with various external sodium substitutes by varying only two parameters: For each ion the height of the major energy barrier (the selectivity filter) determines the biionic zero-current potential and the depth of the energy well (binding site) just external to that barrier then determines the current amplitudes. Voltage clamp measurements with myelinated nerve fibers are given showing numerous examples of deviations from independence in ionic fluxes. Strong blocks of ionic currents by guanidinium compounds and Tl+ ions are fitted by binding within the channel with apparent dissociation constants in the range 50-122 mM. A small block with high Na+ concentrations can be fitted by Na+ ion binding with a dissociation constant of 368 mM. The barrier model is given a molecular interpretation that includes stepwise dehydration of the permeating ion as it interacts with an ionized carboxylic acid.     

3D Holey‐Graphene Architecture Expedites Ion Transport Kinetics to Push the OER Performance

The kinetics process of heterogeneous catalysis involves several steps including adsorption, diffusion, and surface chemical reactions. Current studies generally aim at increasing active site amount and improving intrinsic activity. However, the ion diffusion kinetics at the electrode/electrolyte interface as a bottleneck has been rarely directly addressed. Here, a 3D holey‐graphene framework is demonstrated as a catalyst‐loading platform, with nanoscale holes that can be elaborately tuned via facile aqueous‐phase chemical etching. This enables the ions to be efficiently transported to deeply buried active sites to mitigate their insufficient supply. With systematical electrochemical investigations tuned by varied pore structures, a series of models from a simplified equivalent circuit to complicate realistic one are proposed to figure out the modulation rules of weakened electrochemical diffusion domination and identify the ion transport resistance as well. Moreover, given the inevitable negative effect on the conductivity of graphene skeleton by introducing nanoscale holes, the balance between the outside ion transport and the inside charge transport of electrode is highlighted. Such a protocol represents a synergistic modulation of catalytic performance from both the supply side (reactive ion transport) and the consuming side (active site), and provides striking information for the precise design of catalyst electrodes toward further pushing the oxygen evolution reaction performance limit.     

Preference of Cd(II) and Zn(II) for the two metal sites in Bacillus cereus beta-lactamase II: A perturbed angular correlation of gamma-rays spectroscopic study

Cd-substituted forms of the Bacillus cereus metallo-beta-lactamases (BCII) were studied by perturbed angular correlation of gamma-rays (PAC) spectroscopy. At very low [Cd]:[apo-beta-lactamase] ratios, two nuclear quadrupole interactions (NQI) were detected. For [Cd]:[apo-beta-lactamase] ratios between 0.8 and 3.0, two new NQIs appear, and the spectra show that up to 2 cadmium ions can be bound per molecule of apoenzyme. These results show the existence of two interacting Cd-binding sites in BCII. The relative populations of the two NQIs found at low [Cd]:[apo-beta-lactamase] ratios yielded a 1:3 ratio for the microscopic dissociation constants of the two different metal sites (when only one cadmium ion is bound). X-ray diffraction data at pH 7.5 demonstrate that also for Zn(II) two binding sites exist, which may be bridged by a solvent molecule. The measured NQIs could be assigned to the site with three histidines as metal ligands (three-His site) and to the site with histidine, cysteine, and aspartic acid as metal ligands (Cys site), respectively, by PAC measurements on the Cys168Ala mutant enzyme. This assignment shows that cadmium ions preferentially bind to the Cys site. This is in contrast to the preference of Zn(II) in the hybrid Zn(II)Cd(II) enzyme, where an analysis of the corresponding PAC spectrum showed that Cd(II) occupied the Cys site, whereby Zn(II) occupied the site with three histidines. The difference between Zn(II) and Cd(II) in affinity for the two sites is combined with the kinetics of hydrolysis of nitrocefin for different metal ion substitutions (Zn(2)E, ZnE, Cd(2)E, CdE, and ZnCdE) to study the function of the two metal ion binding sites.