Solute translocation probability, lifetime, and "rectification" in membrane channels with localized constriction.

We study the translocation probability and lifetime of a solute molecule in a cylindrical membrane channel that contains a localized constriction at an arbitrary location. Using a one-dimensional continuous diffusion description of solute dynamics in the channel, we explore two models. The first one describes a molecule's interaction with the constriction in terms of a narrow rectangular barrier in the potential of mean force. The second novel model proposed here represents this interaction by introducing an infinitely thin permeable partition. It is shown that when the parameters of the two models are chosen to warrant the same translocation probability, both models predict the same mean lifetime of the molecule in the channel. While the translocation probability is independent of the constriction location, the mean lifetime is a function of the location. The benefit of the thin partition model is that it allows one to lump together the height and length of the potential barrier into a single parameter, which is the partition's permeability. It is shown that in the case of an asymmetric location of the localized constriction and strong repulsion between the solutes, the solute flux through the channel is a function of the direction in which it goes, analogous to the phenomenon known in ion channel electrophysiology as rectification.

Dimeric Tubulin Modifies Mechanical Properties of Lipid Bilayer, as Probed Using Gramicidin A Channel.

Using the gramicidin A channel as a molecular probe, we show that tubulin binding to planar lipid membranes changes the channel kinetics-seen as an increase in the lifetime of the channel dimer-and thus points towards modification of the membrane's mechanical properties. The effect is more pronounced in the presence of non-lamellar lipids in the lipid mixture used for membrane formation. To interpret these findings, we propose that tubulin binding redistributes the lateral pressure of lipid packing along the membrane depth, making it closer to the profile expected for lamellar lipids. This redistribution happens because tubulin perturbs the lipid headgroup spacing to reach the membrane's hydrophobic core via its amphiphilic α-helical domain. Specifically, it increases the forces of repulsion between the lipid headgroups and reduces such forces in the hydrophobic region. We suggest that the effect is reciprocal, meaning that alterations in lipid bilayer mechanics caused by membrane remodeling during cell proliferation in disease and development may also modulate tubulin membrane binding, thus exerting regulatory functions. One of those functions includes the regulation of protein-protein interactions at the membrane surface, as exemplified by VDAC complexation with tubulin.

Trapping of single diffusing particles by a circular disk on a reflecting flat surface. Absorbing hemisphere approximation.

Recent progress in biophysics (for example, in studies of chemical sensing and spatiotemporal cell-signaling) poses new challenges to statistical theory of trapping of single diffusing particles. Here we deal with one of them, namely, trapping kinetics of single particles diffusing in a half-space bounded by a reflecting flat surface containing an absorbing circular disk. This trapping problem is essentially two-dimensional and the question of the angular dependence of the kinetics on the particle starting point is highly nontrivial. We propose an approximate approach to the problem that replaces the absorbing disk by an absorbing hemisphere of a properly chosen radius. This replacement makes the problem angular-independent and essentially one-dimensional. After the replacement one can find an exact solution for the particle propagator (Green's function) that allows one to completely characterize the kinetics. Extensive testing of the theoretical predictions based on the absorbing hemisphere approximation against three-dimensional Brownian dynamics simulations shows excellent agreement between the analytical and simulation results when the particle starts sufficiently far away from the disk. Our approach fails and the angular dependence of the kinetics is important when the distance of the particle starting point from the disk center is comparable with the disk radius. However, even when the initial distance is only two disk radii, the maximum relative error of the theoretical predictions is about 10%. The relative error rapidly decreases as the initial distance increases.

Permeability and diffusion resistance of porous membranes: Analytical theory and its numerical test.

This study is devoted to the transport of neutral solutes through porous flat membranes, driven by the solute concentration difference in the reservoirs separated by the membrane. Transport occurs through membrane channels, which are assumed to be non-overlapping, identical, straight cylindrical pores connecting the reservoirs. The key quantities characterizing transport are membrane permeability and its diffusion resistance. Such transport problems arising in very different contexts, ranging from plant physiology and cell biology to chemical engineering, have been studied for more than a century. Nevertheless, an expression giving the permeability for a membrane of arbitrary thickness at arbitrary surface densities of the channel openings is still unknown. Here, we fill in the gap and derive such an expression. Since this expression is approximate, we compare its predictions with the permeability obtained from Brownian dynamics simulations and find good agreement between the two.

Restricting α-synuclein transport into mitochondria by inhibition of α-synuclein-VDAC complexation as a potential therapeutic target for Parkinson's disease treatment.

Involvement of alpha-synuclein (αSyn) in Parkinson's disease (PD) is complicated and difficult to trace on cellular and molecular levels. Recently, we established that αSyn can regulate mitochondrial function by voltage-activated complexation with the voltage-dependent anion channel (VDAC) on the mitochondrial outer membrane. When complexed with αSyn, the VDAC pore is partially blocked, reducing the transport of ATP/ADP and other metabolites. Further, αSyn can translocate into the mitochondria through VDAC, where it interferes with mitochondrial respiration. Recruitment of αSyn to the VDAC-containing lipid membrane appears to be a crucial prerequisite for both the blockage and translocation processes. Here we report an inhibitory effect of HK2p, a small membrane-binding peptide from the mitochondria-targeting N-terminus of hexokinase 2, on αSyn membrane binding, and hence on αSyn complex formation with VDAC and translocation through it. In electrophysiology experiments, the addition of HK2p at micromolar concentrations to the same side of the membrane as αSyn results in a dramatic reduction of the frequency of blockage events in a concentration-dependent manner, reporting on complexation inhibition. Using two complementary methods of measuring protein-membrane binding, bilayer overtone analysis and fluorescence correlation spectroscopy, we found that HK2p induces detachment of αSyn from lipid membranes. Experiments with HeLa cells using proximity ligation assay confirmed that HK2p impedes αSyn entry into mitochondria. Our results demonstrate that it is possible to regulate αSyn-VDAC complexation by a rationally designed peptide, thus suggesting new avenues in the search for peptide therapeutics to alleviate αSyn mitochondrial toxicity in PD and other synucleinopathies.

Beta-Barrel Channel Response to High Electric Fields: Functional Gating or Reversible Denaturation?

Ion channels exhibit gating behavior, fluctuating between open and closed states, with the transmembrane voltage serving as one of the essential regulators of this process. Voltage gating is a fundamental functional aspect underlying the regulation of ion-selective, mostly α-helical, channels primarily found in excitable cell membranes. In contrast, there exists another group of larger, and less selective, ß-barrel channels of a different origin, which are not directly associated with cell excitability. Remarkably, these channels can also undergo closing, or "gating", induced by sufficiently strong electric fields. Once the field is removed, the channels reopen, preserving a memory of the gating process. In this study, we explored the hypothesis that the voltage-induced closure of the ß-barrel channels can be seen as a form of reversible protein denaturation by the high electric fields applied in model membranes experiments-typically exceeding twenty million volts per meter-rather than a manifestation of functional gating. Here, we focused on the bacterial outer membrane channel OmpF reconstituted into planar lipid bilayers and analyzed various characteristics of the closing-opening process that support this idea. Specifically, we considered the nearly symmetric response to voltages of both polarities, the presence of multiple closed states, the stabilization of the open conformation in channel clusters, the long-term gating memory, and the Hofmeister effects in closing kinetics. Furthermore, we contemplate the evolutionary aspect of the phenomenon, proposing that the field-induced denaturation of membrane proteins might have served as a starting point for their development into amazing molecular machines such as voltage-gated channels of nerve and muscle cells.

Counter-Intuitive Features of Particle Dynamics in Nanopores.

Using the framework of a continuous diffusion model based on the Smoluchowski equation, we analyze particle dynamics in the confinement of a transmembrane nanopore. We briefly review existing analytical results to highlight consequences of interactions between the channel nanopore and the translocating particles. These interactions are described within a minimalistic approach by lumping together multiple physical forces acting on the particle in the pore into a one-dimensional potential of mean force. Such radical simplification allows us to obtain transparent analytical results, often in a simple algebraic form. While most of our findings are quite intuitive, some of them may seem unexpected and even surprising at first glance. The focus is on five examples: (i) attractive interactions between the particles and the nanopore create a potential well and thus cause the particles to spend more time in the pore but, nevertheless, increase their net flux; (ii) if the potential well-describing particle-pore interaction occupies only a part of the pore length, the mean translocation time is a non-monotonic function of the well length, first increasing and then decreasing with the length; (iii) when a rectangular potential well occupies the entire nanopore, the mean particle residence time in the pore is independent of the particle diffusivity inside the pore and depends only on its diffusivity in the bulk; (iv) although in the presence of a potential bias applied to the nanopore the "downhill" particle flux is higher than the "uphill" one, the mean translocation times and their distributions are identical, i.e., independent of the translocation direction; and (v) fast spontaneous gating affects nanopore selectivity when its characteristic time is comparable to that of the particle transport through the pore.

Voltage-activated complexation of α-synuclein with three diverse ß-barrel channels: VDAC, MspA, and α-hemolysin.

Voltage-activated complexation is the process by which a transmembrane potential drives complex formation between a membrane-embedded channel and a soluble or membrane-peripheral target protein. Metabolite and calcium flux across the mitochondrial outer membrane was shown to be regulated by voltage-activated complexation of the voltage-dependent anion channel (VDAC) and either dimeric tubulin or α-synuclein (αSyn). However, the roles played by VDAC's characteristic attributes-its anion selectivity and voltage gating behavior-have remained unclear. Here, we compare in vitro measurements of voltage-activated complexation of αSyn with three well-characterized ß-barrel channels-VDAC, MspA, and α-hemolysin-that differ widely in their organism of origin, structure, geometry, charge density distribution, and voltage gating behavior. The voltage dependences of the complexation dynamics for the different channels are observed to differ quantitatively but have similar qualitative features. In each case, energy landscape modeling describes the complexation dynamics in a manner consistent with the known properties of the individual channels, while voltage gating does not appear to play a role. The reaction free energy landscapes thus calculated reveal a non-trivial dependence of the αSyn/channel complex stability on the surface density of αSyn.

The Single Residue K12 Governs the Exceptional Voltage Sensitivity of Mitochondrial Voltage-Dependent Anion Channel Gating.

The voltage-dependent anion channel (VDAC) is a ß-barrel channel of the mitochondrial outer membrane (MOM) that passively transports ions, metabolites, polypeptides, and single-stranded DNA. VDAC responds to a transmembrane potential by "gating," i.e. transitioning to one of a variety of low-conducting states of unknown structure. The gated state results in nearly complete suppression of multivalent mitochondrial metabolite (such as ATP and ADP) transport, while enhancing calcium transport. Voltage gating is a universal property of ß-barrel channels, but VDAC gating is anomalously sensitive to transmembrane potential. Here, we show that a single residue in the pore interior, K12, is responsible for most of VDAC's voltage sensitivity. Using the analysis of over 40 µs of atomistic molecular dynamics (MD) simulations, we explore correlations between motions of charged residues inside the VDAC pore and geometric deformations of the ß-barrel. Residue K12 is bistable; its motions between two widely separated positions along the pore axis enhance the fluctuations of the ß-barrel and augment the likelihood of gating. Single channel electrophysiology of various K12 mutants reveals a dramatic reduction of the voltage-induced gating transitions. The crystal structure of the K12E mutant at a resolution of 2.6 Å indicates a similar architecture of the K12E mutant to the wild type; however, 60 µs of atomistic MD simulations using the K12E mutant show restricted motion of residue 12, due to enhanced connectivity with neighboring residues, and diminished amplitude of barrel motions. We conclude that ß-barrel fluctuations, governed particularly by residue K12, drive VDAC gating transitions.

Intrinsic diffusion resistance of a membrane channel, mean first-passage times between its ends, and equilibrium unidirectional fluxes.

Diffusive flux of solute molecules through a membrane channel driven by the solute concentration difference on the two sides of the membrane is inversely proportional to the channel diffusion resistance. We show that the intrinsic, channel proper, part of this resistance is the ratio of the sum of the mean first-passage times of the molecule between the channel ends and the molecule partition function in the channel. This is derived without appealing to any specific model of the channel and, therefore, is applicable to transport in channels of arbitrary shape and tortuosity and at arbitrary interaction strength of solute molecules with the channel walls.

Capturing single molecules by nanopores: measured times and thermodynamics.

In numerous nanopore sensing applications transient interruptions in ion current through single nanopores induced by capturing solute molecules are a source of information on how solutes interact with the nanopores. We show that the distribution of time spent by a single captured solute molecule in a nanopore is bimodal with the majority of capture events being too fast to be experimentally resolved. As a result, the exact mean durations of the event and inter-event interval are orders of magnitude shorter than their measured values. Moreover, the exact and measured mean durations have qualitatively different dependences on the molecule diffusivity. This leads to a formal contradiction with the thermodynamics of molecule partitioning between the bulk and the nanopore. Here we resolve this controversy. We also demonstrate that, surprisingly, the probability of finding a molecule in the nanopore, obtained from the ratio of the measured mean durations of the capture event and interevent interval, is essentially identical to the exact equilibrium thermodynamic probability.

Surface-facilitated trapping by active sites: From catalysts to viruses.

Trapping by active sites on surfaces plays important roles in various chemical and biological processes, including catalysis, enzymatic reactions, and viral entry into host cells. However, the mechanisms of these processes remain not well understood, mostly because the existing theoretical descriptions are not fully accounting for the role of the surfaces. Here, we present a theoretical investigation on the dynamics of surface-assisted trapping by specific active sites. In our model, a diffusing particle can occasionally reversibly bind to the surface and diffuse on it before reaching the final target site. An approximate theoretical framework is developed, and its predictions are tested by Brownian dynamics computer simulations. It is found that the surface diffusion can be crucial in mediating trapping by active sites. Our theoretical predictions work reasonably well as long as the area of the active site is much smaller than the overall surface area. Potential applications of our approach are discussed.

Localized potential well vs binding site: Mapping solute dynamics in a membrane channel onto one-dimensional description.

In the one-dimensional description, the interaction of a solute molecule with the channel wall is characterized by the potential of mean force U(x), where the x-coordinate is measured along the channel axis. When the molecule can reversibly bind to certain amino acid(s) of the protein forming the channel, this results in a localized well in the potential U(x). Alternatively, this binding can be modeled by introducing a discrete localized site, in addition to the continuum of states along x. Although both models may predict identical equilibrium distributions of the coordinate x, there is a fundamental difference between the two: in the first model, the molecule passing through the channel unavoidably visits the potential well, while in the latter, it may traverse the channel without being trapped at the discrete site. Here, we show that when the two models are parameterized to have the same thermodynamic properties, they automatically yield identical translocation probabilities and mean translocation times, yet they predict qualitatively different shapes of the translocation time distribution. Specifically, the potential well model yields a narrower distribution than the model with a discrete site, a difference that can be quantified by the distribution's coefficient of variation. This coefficient turns out to be always smaller than unity in the potential well model, whereas it may exceed unity when a discrete trapping site is present. Analysis of the translocation time distribution beyond its mean thus offers a way to differentiate between distinct translocation mechanisms.

Correction to: Molecular mechanism of olesoxime-mediated neuroprotection through targeting α-synuclein interaction with mitochondrial VDAC.

In the published article, an error was noticed and this has been corrected with this erratum publication.

Molecular mechanism of olesoxime-mediated neuroprotection through targeting α-synuclein interaction with mitochondrial VDAC.

An intrinsically disordered neuronal protein α-synuclein (αSyn) is known to cause mitochondrial dysfunction, contributing to loss of dopaminergic neurons in Parkinson's disease. Through yet poorly defined mechanisms, αSyn crosses mitochondrial outer membrane and targets respiratory complexes leading to bioenergetics defects. Here, using neuronally differentiated human cells overexpressing wild-type αSyn, we show that the major metabolite channel of the outer membrane, the voltage-dependent anion channel (VDAC), is a pathway for αSyn translocation into the mitochondria. Importantly, the neuroprotective cholesterol-like synthetic compound olesoxime inhibits this translocation. By applying complementary electrophysiological and biophysical approaches, we provide mechanistic insights into the interplay between αSyn, VDAC, and olesoxime. Our data suggest that olesoxime interacts with VDAC ß-barrel at the lipid-protein interface thus hindering αSyn translocation through the VDAC pore and affecting VDAC voltage gating. We propose that targeting αSyn translocation through VDAC could represent a key mechanism for the development of new neuroprotective strategies.

Regulation of Mitochondrial Respiration by VDAC Is Enhanced by Membrane-Bound Inhibitors with Disordered Polyanionic C-Terminal Domains.

The voltage-dependent anion channel (VDAC) is the primary regulating pathway of water-soluble metabolites and ions across the mitochondrial outer membrane. When reconstituted into lipid membranes, VDAC responds to sufficiently large transmembrane potentials by transitioning to gated states in which ATP/ADP flux is reduced and calcium flux is increased. Two otherwise unrelated cytosolic proteins, tubulin, and α-synuclein (αSyn), dock with VDAC by a novel mechanism in which the transmembrane potential draws their disordered, polyanionic C-terminal domains into and through the VDAC channel, thus physically blocking the pore. For both tubulin and αSyn, the blocked state is observed at much lower transmembrane potentials than VDAC gated states, such that in the presence of these cytosolic docking proteins, VDAC's sensitivity to transmembrane potential is dramatically increased. Remarkably, the features of the VDAC gated states relevant for bioenergetics-reduced metabolite flux and increased calcium flux-are preserved in the blocked state induced by either docking protein. The ability of tubulin and αSyn to modulate mitochondrial potential and ATP production in vivo is now supported by many studies. The common physical origin of the interactions of both tubulin and αSyn with VDAC leads to a general model of a VDAC inhibitor, facilitates predictions of the effect of post-translational modifications of known inhibitors, and points the way toward the development of novel therapeutics targeting VDAC.

VDAC Gating Thermodynamics, but Not Gating Kinetics, Are Virtually Temperature Independent.

The voltage-dependent anion channel (VDAC) is the most abundant protein in the mitochondrial outer membrane and an archetypical ß-barrel channel. Here, we study the effects of temperature on VDAC channels reconstituted in planar lipid membranes at the single- and multichannel levels within the 20°C to 40°C range. The temperature dependence of conductance measured on a single channel in 1 M KCl shows an increase characterized by a 10°C temperature coefficient Q10 = 1.22 ± 0.02, which exceeds that of the bathing electrolyte solution conductivity, Q10 = 1.17 ± 0.01. The rates of voltage-induced channel transition between the open and closed states measured on multichannel membranes also show statistically significant increases, with temperatures that are consistent with activation energy barriers of ∼10 ± 3 kcal/mol. At the same time, the gating thermodynamics, as characterized by the gating charge and voltage of equipartitioning, does not display any measurable temperature dependence. The two parameters stay within 3.2 ± 0.2 elementary charges and 30 ± 2 mV, respectively. Thus, whereas the channel kinetics, specifically its conductance and rates of gating response to voltage steps, demonstrates a clear increase with temperature, the conformational voltage-dependent equilibria are virtually insensitive to temperature. These results, which may be a general feature of ß-barrel channel gating, suggest either an entropy-driven gating mechanism or a role for enthalpy-entropy compensation.

Structural features and lipid binding domain of tubulin on biomimetic mitochondrial membranes.

Dimeric tubulin, an abundant water-soluble cytosolic protein known primarily for its role in the cytoskeleton, is routinely found to be associated with mitochondrial outer membranes, although the structure and physiological role of mitochondria-bound tubulin are still unknown. There is also no consensus on whether tubulin is a peripheral membrane protein or is integrated into the outer mitochondrial membrane. Here the results of five independent techniques-surface plasmon resonance, electrochemical impedance spectroscopy, bilayer overtone analysis, neutron reflectometry, and molecular dynamics simulations-suggest that α-tubulin's amphipathic helix H10 is responsible for peripheral binding of dimeric tubulin to biomimetic "mitochondrial" membranes in a manner that differentiates between the two primary lipid headgroups found in mitochondrial membranes, phosphatidylethanolamine and phosphatidylcholine. The identification of the tubulin dimer orientation and membrane-binding domain represents an essential step toward our understanding of the complex mechanisms by which tubulin interacts with integral proteins of the mitochondrial outer membrane and is important for the structure-inspired design of tubulin-targeting agents.

Sequence diversity of tubulin isotypes in regulation of the mitochondrial voltage-dependent anion channel.

The microtubule protein tubulin is a heterodimer comprising α/ß subunits, in which each subunit features multiple isotypes in vertebrates. For example, seven α-tubulin and eight ß-tubulin isotypes in the human tubulin gene family vary mostly in the length and primary sequence of the disordered anionic carboxyl-terminal tails (CTTs). The biological reason for such sequence diversity remains a topic of vigorous enquiry. Here, we demonstrate that it may be a key feature of tubulin's role in regulation of the permeability of the mitochondrial outer membrane voltage-dependent anion channel (VDAC). Using recombinant yeast α/ß-tubulin constructs with α-CTTs, ß-CTTs, or both from various human tubulin isotypes, we probed their interactions with VDAC reconstituted into planar lipid bilayers. A comparative study of the blockage kinetics revealed that either α-CTTs or ß-CTTs block the VDAC pore and that the efficiency of blockage by individual CTTs spans 2 orders of magnitude, depending on the CTT isotype. ß-Tubulin constructs, notably ß3, blocked VDAC most effectively. We quantitatively described these experimental results using a physical model that accounted only for the number and distribution of charges in the CTT, and not for the interactions between specific residues on the CTT and VDAC pore. Based on these results, we speculate that the effectiveness of VDAC regulation by tubulin depends on the predominant tubulin isotype in a cell. Consequently, the fluxes of ATP/ADP through the channel could vary significantly, depending on the isotype, thus suggesting an intriguing link between VDAC regulation and the diversity of tubulin isotypes present in vertebrates.

Blocker escape kinetics from a membrane channel analyzed by mapping blocker diffusive dynamics onto a two-site model.

When a large solute molecule enters a membrane channel from the membrane-bathing electrolyte solution, it blocks the small-ion current flowing through the channel. If the molecule spends in the channel sufficiently long time, individual blockades can be resolved in single-channel experiments. In this paper, we develop an analytical theory of the blocker escape kinetics from the channel, assuming that a charged blocking molecule cannot pass through a constriction region (bottleneck). We focus on the effect of the external voltage bias on the blocker survival probability in the channel. The bias creates a potential well for the charged blocker in the channel with the minimum located near the bottleneck. When the bias is strong, the well is deep, and escape from the channel is a slow process that allows for time-resolved observation of individual blocking events. Our analysis is performed in the framework of a two-site model of the blocker dynamics in the channel. Importantly, the rate constants, fully determining this model, are derived from a more realistic continuum diffusion model. This is done by mapping the latter onto its two-site counterpart which, while being much simpler, captures the main features of the blocker escape kinetics at high biases.