Blockage coefficient of cylindrical blocker and diffusion resistance of membrane channels.

This study deals with potential flow of ideal fluid in an infinite cylindrical tube in the presence of a blocking object. The blockage effect of the object on the flow can be characterized by a lump parameter, blockage coefficient, which accounts for the object shape and size. For a cylindrical blocking object, analytical results for the blockage coefficient are known only in three limiting cases: for a long thin cylinder and for small and large blocking disks. We propose a simple analytical expression for the blockage coefficient of a cylindrical blocker of arbitrary length and radius that reduces to the known asymptotic results in the corresponding limits.

Brownian motion in confined geometries.

In a great number of technologically and biologically relevant cases, transport of micro- or nanosized objects is governed by both omnipresent thermal fluctuations and confining walls or constrictions limiting the available phase space. The present Topical Issue covers the most recent applications and theoretical findings devoted to studies of Brownian motion under confinement of channel-like geometries.

Diffusion in a tube of varying cross section: numerical study of reduction to effective one-dimensional description.

Brownian dynamics simulations of the particle diffusing in a long conical tube (the length of the tube is much greater than its smallest radius) are used to study reduction of the three-dimensional diffusion in tubes of varying cross section to an effective one-dimensional description. The authors find that the one-dimensional description in the form of the Fick-Jacobs equation with a position-dependent diffusion coefficient, D(x), suggested by Zwanzig [J. Phys. Chem. 96, 3926 (1992)], with D(x) given by the Reguera-Rubi formula [Phys. Rev. E 64, 061106 (2001)], D(x)=D/sq rt1+R'(x)2, where D is the particle diffusion coefficient in the absence of constraints, and R(x) is the tube radius at x, is valid when |R'(x)|1, higher spatial derivatives of the one-dimensional concentration in the effective diffusion equation cannot be neglected anymore as was indicated by Kalinay and Percus [J. Chem. Phys. 122, 204701 (2005)]. Thus the reduction to the effective one-dimensional description is a useful tool only when |R'(x)|

Analytical theory of hysteresis in ion channels: two-state model.

Channel-forming proteins in a lipid bilayer of a biological membrane usually respond to variation of external voltage by changing their conformations. Periodic voltages with frequency comparable with the inverse relaxation time of the protein produce hysteresis in the occupancies of the protein conformations. If the channel conductance changes when the protein jumps between these conformations, hysteresis in occupancies is observed as hysteresis in ion current through the channel. We develop an analytical theory of this phenomenon assuming that the channel conformational dynamics can be described in terms of a two-state model. The theory describes transient behavior of the channel after the periodic voltage is switched on as well as the shape and area of the hysteretic loop as functions of the frequency and amplitude of the applied voltage. The area vanishes as the voltage period T tends to zero and infinity. Asymptotic behaviors of the loop area A in the high- and low-frequency regimes, respectively, are A approximately T and A approximately T(-1).

VDAC channels differentiate between natural metabolites and synthetic molecules.

VDAC provides the major permeability pathway through the mitochondrial outer membrane by forming voltage-gated channels with pore radius of 1.2-1.5 nm. We find that VDAC can select among comparably-charged molecules with a much smaller effective radius, 0.4-0.5 nm. The molecules studied were the nucleotides, ATP, UTP, NADH and synthetic anions, tetraglutamate (T-Glu) and 1-hydroxypyrene-3,6,8-trisulfonate (HPTS). VDAC channels were reconstituted into planar phospholipid membranes bathed in 1.0 M NaCl (buffered to pH 8.0). The nucleotides decreased the conductance of VDAC for NaCl demonstrating that they could permeate into the channel. In contrast, T-Glu and HPTS did not change the single-channel conductance, indicating exclusion from the channel. Reversal potential measurements report near ideal selectivity of Na + over T-Glu. The nucleotides increased single-channel noise as they penetrated into the channel, while T-Glu had no effect. HPTS increased noise, but unlike NADH, this was not voltage-dependent when HPTS was added asymmetrically, indicating no penetration into the channel. The differences in effective size and charge cannot explain the difference in permeation characteristics. Thus VDAC must select among these based on shape and charge distribution. We propose that the electrostatic environment within the channel has been evolutionarily selected to favor the passage of adenine nucleotides.

Alamethicin channel conductance modified by lipid charge.

The membrane surface charge modifies the conductance of ion channels by changing the electric potential and redistributing the ionic composition in their vicinity. We have studied the effects of lipid charge on the conductance of a multi-state channel formed in planar lipid bilayers by the peptide antibiotic alamethicin. The channel conductance was measured in two lipids: in a neutral dioleoylphosphatidylethanolamine (DOPE) and a negatively charged dioleoylphosphatidylserine (DOPS). The charge state of DOPS was manipulated by the pH of the membrane-bathing solution. We find that at high salt concentrations (e.g., 2 M NaCl) the effect of the lipid charge is below the accuracy of our measurements. However, when the salt concentration in the membrane-bathing solution is decreased, the surface charge manifests itself as an increase in the conductance of the first two channel levels that correspond to the smallest conductive alamethicin aggregates. Our analysis shows that both the salt and pH dependence of the surface charge effect can be rationalized within the nonlinear Poisson-Boltzmann approach. Given channel conductance in neutral lipids, we use different procedures to account for the surface charge (e.g., introduce averaging over the channel aperture and take into account Na+ adsorption to DOPS heads), but only one adjustable parameter: an effective distance from the nearest lipid charge to the channel mouth center. We show that this distance varies by 0.3-0.4 nm upon channel transition from the minimal conducting aggregate (level L0) to the next larger one (level L1). This conclusion is in accord with a simple geometrical model of alamethicin aggregation.

Kinetics of opening and closure of syringomycin E channels formed in lipid bilayers.

A cyclic lipodepsipeptide, syringomycin E (SME), incorporated into planar lipid membranes forms two types of channels ("small" and "large") different in their conductance by approximately a factor of six (Biophys. J. 74:2918-2925 (1998)). We analysed the dynamics of the SME-induced transmembrane current under voltage-clamp conditions to clarify the mechanisms of formation of these channels. The voltage-dependent opening/closure of SME channels in lipid bilayers are interpreted in terms of transitions between three types of clusters including 6-7 SME molecules and some lipid molecules. The initial cluster, the precursor of the other two, was in equilibrium with SME monomer molecules at the membrane surface. The other two types of clusters (State 1 and State 2) were formed from the precursor and also during their interconversions (the consecutive-parallel mechanism of transitions). State 1 was a non-conducting state in equilibrium with small channels, which partially determined the ionic conductance of lipid bilayers modified by SME. State 2 corresponded to large SME channels, major contributors to the conductance of a bilayer. The results of the theoretical analysis based on the chemical kinetics concepts were consistent with experimental observations. Such properties of the SME-induced channels as cluster organisation, voltage dependence and the existence of a non-conducting state are all features shared by many ion channels in biological membranes. This makes it possible to use SME channels as a model to study naturally occurring ion channels.

Controlling membrane permeability with bacterial porins: application to encapsulated enzymes.

Recent achievements of membrane protein science allow easy protein modification by genetic engineering and, for some proteins, their production in large quantities. We regard these features as the basic requirements for applications of membrane proteins in materials science. Here, we demonstrate a possible application of membrane proteins, inserting porins from the outer cell wall of Escherichia coli into the walls of liposomes. Encapsulation of enzymes into liposomes or polymer nanocapsules protects them against proteases and denaturation. Functional reconstitution of porins into the capsule shell allows to control the rate and selectivity of substrate permeation, and thus to control the enzyme reaction kinetics. We suggest that this technique can prove to be useful in the area of biosensors, providing enzymatic stability while keeping the functionality or even enhancing the sensitivity by substrate preselection. Another application of this kind of stabilisation is in the field of single enzyme activity recording.

A voltage-dependent channel involved in nutrient uptake by red blood cells infected with the malaria parasite.

Growth of the malaria parasite in human red blood cells (RBCs) is accompanied by an increased uptake of many solutes including anions, sugars, purines, amino acids and organic cations. Although the pharmacological properties and selectivity of this uptake suggest that a chloride channel is involved, the precise mechanism has not been identified. Moreover, the location of this uptake in the infected RBC is unknown because tracer studies are complicated by possible uptake through fluid-phase pinocytosis or membranous ducts. Here we have studied the permeability of infected RBCs using the whole-cell voltage-clamp method. With this method, uninfected RBCs had ohmic whole-cell conductances of less than 100 pS, consistent with their low tracer permeabilities. In contrast, trophozoite-infected RBCs exhibited voltage-dependent, non-saturating currents that were 150-fold larger, predominantly carried by anions and abruptly abolished by channel blockers. Patch-clamp measurements and spectral analysis confirmed that a small (< 10 pS) ion channel on the infected RBC surface, present at about 1,000 copies per cell, is responsible for these currents. Because its pharmacological properties and substrate selectivities match those seen with tracer studies, this channel accounts for the increased uptake of small solutes in infected RBCs. The surface location of this new channel and its permeability to organic solutes needed for parasite growth indicate that it may have a primary role in a sequential diffusive pathway for parasite nutrient acquisition.

Examining noise sources at the single-molecule level: 1/f noise of an open maltoporin channel.

We have studied the phenomenological origin of 1/f noise in a solute-specific bacterial ion channel, maltoporin. We show that after excision of small, but resolvable stepwise changes in the recordings of the current through a single open channel, the 1/f noise component disappears and the channel exhibits noise that is "white" below 100 Hz. Combined with results of a recent noise study of several bacterial porins, our observations suggest that 1/f noise is caused by the equilibrium conductance fluctuations related to the conformational flexibility of the channel pore structural constituents.

Probing sugar translocation through maltoporin at the single channel level.

Sugar permeation through maltoporin of Escherichia coli, a trimer protein that facilitates maltodextrin translocation across outer bacterial membranes, was investigated at the single channel level. For large sugars, such as maltohexaose, elementary events of individual sugar molecule penetration into the channel were readily observed. At small sugar concentrations an elementary event consists of maltoporin channel closure by one third of its initial conductance in sugar-free solution. Statistical analysis of such closures at higher sugar concentrations shows that all three pores of the maltoporin channel transport sugars independently. Interestingly, while channel conductance is only slightly asymmetric showing about 10% higher values at -200 mV than at +200 mV (from the side of protein addition), asymmetry in dependence of the sugar binding constant on the voltage polarity is about 20 times higher. Combining our data with observations made with bacteriophage-lambda we conclude that the sugar residence time is much more sensitive to (and is decreased by) voltages that are negative from the intra-cell side of the bacterial membrane.

Positive cooperativity without domains or subunits in a monomeric membrane channel.

The monomeric VDAC channel shows an accelerated pH titration of its transport properties with a Hill coefficient of about 2. This manifests itself as a sharp peak in conductance noise as well as a fast change in channel selectivity with pH. On the basis of the known structure of this channel, we propose that this cooperativity arises from a mechanically linked mobile pair of ionizable groups. Concerted movement of these groups between two states changes the distance from nearby electrostatic charge to influence the pK of the groups. This model of pH-dependent motion produces positive cooperative behavior that fits the observations without need for subunits or identifiable domains within the protein. The mathematical formalism has never required such domains, but these are generally considered an essential part of cooperative behavior in proteins. The present proposal reduces the size of a cooperative unit to a minimum, extending the limits of what is perceived to be possible. Together with large-scale conformational transitions, these subtle cooperative structural changes may allow proteins to adapt, with high sensitivity, to changes in their environment. They might also be relatively easy to engineer into a protein.

Polymeric nonelectrolytes to probe pore geometry: application to the alpha-toxin transmembrane channel.

Asymmetrical (one-sided) application of penetrating water-soluble polymers, polyethylene glycols (PEGs), to a well-defined channel formed by Staphylococcus aureus alpha-toxin is shown to probe channel pore geometry in more detail than their symmetrical (two-sided) application. Polymers added to the cis side of the planar lipid membrane (the side of protein addition) affect channel conductance differently than polymers added to the trans side. Because a satisfactory theory quantitatively describing PEG partitioning into a channel pore does not exist, we apply the simple empirical rules proposed previously (, J. Membr. Biol. 161:83-92) to gauge the size of pore openings as well as the size and position of constrictions along the pore axis. We estimate the radii of the two openings of the channel to be practically identical and equal to 1. 2-1.3 nm. Two apparent constrictions with radii of approximately 0. 9 nm and approximately 0.6-0.7 nm are inferred to be present in the channel lumen, the larger one being closer to the cis side. These structural findings agree well with crystallographic data on the channel structure (, Science. 274:1859-1866) and verify the practicality of polymer probing. The general features of PEG partitioning are examined using available theoretical considerations, assuming there is no attraction between PEG and the channel lumen. It is shown that the sharp dependence of the partition coefficient on polymer molecular weight found under both symmetrical and asymmetrical polymer application can be rationalized within a "hard sphere nonideal solution model." This finding is rather surprising because PEG forms highly flexible coils in water with a Kuhn length of only several Angstroms.

Membrane surface-charge titration probed by gramicidin A channel conductance.

We manipulate lipid bilayer surface charge and gauge its influence on gramicidin A channel conductance by two strategies: titration of the lipid charge through bulk solution pH and dilution of a charged lipid by neutral. Using diphytanoyl phosphatidylserine (PS) bilayers with CsCl aqueous solutions, we show that the effects of lipid charge titration on channel conductance are masked 1) by conductance saturation with Cs+ ions in the neutral pH range and 2) by increased proton concentration when the bathing solution pH is less than 3. A smeared charge model permits us to separate different contributions to the channel conductance and to introduce a new method for "bilayer pKa" determination. We use the Gouy-Chapman expression for the charged surface potential to obtain equilibria of protons and cations with lipid charges. To calculate cation concentration at the channel mouth, we compare different models for the ion distribution, exact and linearized forms of the planar Poisson-Boltzmann equation, as well as the construction of a "Gibbs dividing surface" between salt bath and charged membrane. All approximations yield the intrinsic pKain of PS lipid in 0.1 M CsCl to be in the range 2.5-3.0. By diluting PS surface charge at a fixed pH with admixed neutral diphytanoyl phosphatidylcholine (PC), we obtain a conductance decrease in magnitude greater than expected from the electrostatic model. This observation is in accord with the different conductance saturation values for PS and PC lipids reported earlier (, Biochim. Biophys. Acta. 552:369-378) and verified in the present work for solvent-free membranes. In addition to electrostatic effects of surface charge, gramicidin A channel conductance is also influenced by lipid-dependent structural factors.

Cluster organization of ion channels formed by the antibiotic syringomycin E in bilayer lipid membranes.

The cyclic lipodepsipeptide, syringomycin E, when incorporated into planar lipid bilayer membranes, forms two types of channels (small and large) that are different in conductance by a factor of sixfold. To discriminate between a cluster organization-type channel structure and other possible different structures for the two channel types, their ionic selectivity and pore size were determined. Pore size was assessed using water-soluble polymers. Ion selectivity was found to be essentially the same for both the small and large channels. Their reversal (zero current) potentials with the sign corresponding to anionic selectivity did not differ by more than 3 mV at a twofold electrolyte gradient across the bilayer. Reduction in the single-channel conductance induced by poly(ethylene glycol)s of different molecular weights demonstrated that the aqueous pore sizes of the small and large channels did not differ by more than 2% and were close to 1 nm. Based on their virtually identical selectivity and size, we conclude that large syringomycin E channels are clusters of small ones exhibiting synchronous opening and closing.

ATP transport through a single mitochondrial channel, VDAC, studied by current fluctuation analysis.

The "molecular Coulter counter" concept has been used to study transport of ATP molecules through the nanometer-scale aqueous pore of the voltage-dependent mitochondrial ion channel, VDAC. We examine the ATP-induced current fluctuations and the change in average current through a single fully open channel reconstituted into a planar lipid bilayer. At high salt concentration (1 M NaCl), the addition of ATP reduces both solution conductivity and channel conductance, but the effect on the channel is several times stronger and shows saturation behavior even at 50 mM ATP concentration. These results and simple steric considerations indicate pronounced attraction of ATP molecules to VDAC's aqueous pore and permit us to evaluate the effect of a single ATP molecule on channel conductance. ATP addition also generates an excess noise in the ionic current through the channel. Analysis of this excess noise shows that its spectrum is flat in the accessible frequency interval up to several kilohertz. ATP exchange between the pore and the bulk is fast enough not to display any dispersion at these frequencies. By relating the low-frequency spectral density of the noise to the equilibrium diffusion of ATP molecules in the aqueous pore, we calculate a diffusion coefficient D = (1.6-3.3)10(-11) m2/s. This is one order of magnitude smaller than the ATP diffusion coefficient in the bulk, but it agrees with recent results on ATP flux measurements in multichannel membranes using the luciferin/luciferase method.

Lipid packing stress and polypeptide aggregation: alamethicin channel probed by proton titration of lipid charge.

Lipid membranes are not passive, neutral scaffolds to hold membrane proteins. In order to examine the influence of lipid packing energetics on ion channel expression, we study the relative probabilities of alamethicin channel formation in dioleoylphosphatidylserine (DOPS) bilayers as a function of pH. The rationale for this strategy is our earlier finding that the higher-conductance states, corresponding to larger polypeptide aggregates, are more likely to occur in the presence of lipids prone to hexagonal HII-phase formation (specifically DOPE), than in the presence of lamellar L alpha-forming lipids (DOPC). In low ionic strength NaCl solutions at neutral pH, the open channel in DOPS membranes spends most of its time in states of lower conductance and resembles alamethicin channels in DOPC; at lower pH, where the lipid polar groups are neutralized, the channel probability distribution resembles that in DOPE. X-Ray diffraction studies on DOPS show a progressive decrease in the intrinsic curvature of the constituent monolayers as well as a decreased probability of HII-phase formation when the charged lipid fraction is increased. We explore how proton titration of DOPS affects lipid packing energetics, and how these energetics couple titration to channel formation.

Voltage-dependent ion channel formation by rigid rod-shaped polyols in planar lipid bilayers.

In this Letter, we describe the appearance of large, voltage-dependent currents in BLM induced by rigid rod-shaped polyols that function without charge and permanent dipole moment. The capacity of these symmetrical, nonpeptide models to form either short-living nanopores or small ion channels is shown to depend critically on the length of rigid-rod scaffold as well as the nature of the lateral side chains.

Signal transduction across alamethicin ion channels in the presence of noise.

We have studied voltage-dependent ion channels of alamethicin reconstituted into an artificial planar lipid bilayer membrane from the point of view of electric signal transduction. Signal transduction properties of these channels are highly sensitive to the external electric noise. Specifically, addition of bandwidth-restricted "white" noise of 10-20 mV (r.m.s.) to a small sine wave input signal increases the output signal by approximately 20-40 dB conserving, and even slightly increasing, the signal-to-noise ratio at the system output. We have developed a small-signal adiabatic theory of stochastic resonance for a threshold-free system of voltage-dependent ion channels. This theory describes our main experimental findings giving good qualitative understanding of the underlying mechanism. It predicts the right value of the output signal-to-noise ratio and provides a reliable estimate for the noise intensity corresponding to its maximum. Our results suggest that the alamethicin channel in a lipid bilayer is a good model system for studies of mechanisms of primary electrical signal processing in biology showing an important feature of signal transduction improvement by a fluctuating environment.