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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.

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.

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, β-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 β-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.

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.

The geometry of ion and metabolite channels in membranes of biological cells and organelles is usually far from that of a regular right cylinder. Rather, the channels have complex shapes that are characterized by the so-called vestibules and constriction zones which play roles of molecular filters determining the channel selectivity. In the present paper we discuss several channel structures with varying radius that approximate most of the cases found in nature, specifically, channels of smoothly varying radius and channels composed of multiple cylindrical sections of different lengths and radii including channels containing very thin circular constrictions. We consider diffusive transport of electrically neutral molecules driven by the concentration gradient and derive analytical expressions for the diffusion resistance - the integral parameter that describes steady-state transport properties of membrane channels.

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.

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.

Being motivated by recent progress in nanopore sensing, we develop a theory of the effect of large analytes, or blockers, trapped within the nanopore confines, on diffusion flow of small solutes. The focus is on the nanopore diffusion resistance which is the ratio of the solute concentration difference in the reservoirs connected by the nanopore to the solute flux driven by this difference. Analytical expressions for the diffusion resistance are derived for a cylindrically symmetric blocker whose axis coincides with the axis of a cylindrical nanopore in two limiting cases where the blocker radius changes either smoothly or abruptly. Comparison of our theoretical predictions with the results obtained from Brownian dynamics simulations shows good agreement between the two.

The voltage-dependent anion channel (VDAC) is a β-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 β-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 β-barrel. Residue K12 is bistable; its motions between two widely separated positions along the pore axis enhance the fluctuations of the β-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 β-barrel fluctuations, governed particularly by residue K12, drive VDAC gating transitions.

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., (© 2022. This is a U.S. Government work and not under copyright protection in the US; foreign copyright protection may apply.)

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 β-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., (© 2021 Wiley-VCH GmbH.)

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.

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.

Regulation of VDAC by α-synuclein (αSyn) is a rich and instructive example of protein-protein interactions catalyzed by a lipid membrane surface. αSyn, a peripheral membrane protein involved in Parkinson's disease pathology, is known to bind to membranes in a transient manner. αSyn's negatively charged C-terminal domain is then available to be electromechanically trapped by the VDAC β-barrel, a process that is observed in vitro as the reversible reduction of ion flow through a single voltage-biased VDAC nanopore. Binding of αSyn to the lipid bilayer is a prerequisite of the channel-protein interaction; surprisingly, however, we find that the strength of αSyn binding to the membrane does not correlate in any simple way with its efficiency of blocking VDAC, suggesting that the lipid-dependent conformations of the membrane-bound αSyn control the interaction. Quantitative models of the free energy landscape governing the capture and release processes allow us to discriminate between several αSyn (sub-) conformations on the membrane surface. These results, combined with known structural features of αSyn on anionic lipid membranes, point to a model in which the lipid composition determines the fraction of αSyn molecules for which the charged C terminal domain is constrained to be close, but not tightly bound, to the membrane surface and thus readily captured by the VDAC nanopore. We speculate that changes in the mitochondrial membrane lipid composition may be key regulators of the αSyn-VDAC interaction and consequently of VDAC-facilitated transport of ions and metabolites in and out of mitochondria and, i.e. mitochondrial metabolism., (Copyright © 2021 Elsevier B.V. All rights reserved.)

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.

We study diffusion of a Brownian particle in a two-dimensional periodic channel of abruptly alternating width. Our main result is a simple approximate analytical expression for the particle effective diffusivity, which shows how the diffusivity depends on the geometric parameters of the channel: lengths and widths of its wide and narrow segments. The result is obtained in two steps: first, we introduce an approximate one-dimensional description of particle diffusion in the channel, and second, we use this description to derive the expression for the effective diffusivity. While the reduction to the effective one-dimensional description is standard for systems of smoothly varying geometry, such a reduction in the case of abruptly changing geometry requires a new methodology used here, which is based on the boundary homogenization approach to the trapping problem. To test the accuracy of our analytical expression and thus establish the range of its applicability, we compare analytical predictions with the results obtained from Brownian dynamics simulations. The comparison shows excellent agreement between the two, on condition that the length of the wide segment of the channel is equal to or larger than its width.

Voltage-dependent anion channel (VDAC) is the most ubiquitous channel at the mitochondrial outer membrane, and is believed to be the pathway for calcium entering or leaving the mitochondria. Therefore, understanding the molecular mechanisms of how VDAC regulates calcium influx and efflux from the mitochondria is of particular interest for mitochondrial physiology. When the Parkinson's disease (PD) related neuronal protein, alpha-synuclein (αSyn), is added to the reconstituted VDAC, it reversibly and partially blocks VDAC conductance by its acidic C-terminal tail. Using single-molecule VDAC electrophysiology of reconstituted VDAC we now demonstrate that, at CaCl 2 concentrations below 150 mM, αSyn reverses the channel's selectivity from anionic to cationic. Importantly, we find that the decrease in channel conductance upon its blockage by αSyn is hugely overcompensated by a favorable change in the electrostatic environment for calcium, making the blocked state orders-of-magnitude more selective for calcium and thus increasing its net flux. -Our findings with higher calcium concentrations also demonstrate that the phenomenon of "charge inversion" is taking place at the level of a single polypeptide chain. Measurements of ion selectivity of three VDAC isoforms in CaCl 2 gradient show that VDAC3 exhibits the highest calcium permeability among them, followed by VDAC2 and VDAC1, thus pointing to isoform-dependent physiological function. Mutation of the E73 residue - VDAC1 purported calcium binding site - shows that there is no measurable effect of the mutation in either open or αSyn-blocked VDAC1 states. Our results confirm VDACs involvement in calcium signaling and reveal a new regulatory role of αSyn, with clear implications for both normal calcium signaling and PD-associated mitochondrial dysfunction., (Published by Elsevier Ltd.)

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.

Voltage-dependent anion channel (VDAC), the most abundant mitochondrial outer membrane protein, is important for a variety of mitochondrial functions including metabolite exchange, calcium transport, and apoptosis. While VDAC's role in shuttling metabolites between the cytosol and mitochondria is well established, there is a growing interest in understanding the mechanisms of its regulation of mitochondrial calcium transport. Here we review the current literature on VDAC's role in calcium signaling, its biophysical properties, physiological function, and pathology focusing on its importance in cardiac diseases. We discuss the specific biophysical properties of the three VDAC isoforms in mammalian cells-VDAC 1, 2, and 3-in relationship to calcium transport and their distinct roles in cell physiology and disease. Highlighting the emerging evidence that cytosolic proteins interact with VDAC and regulate its calcium permeability, we advocate for continued investigation into the VDAC interactome at the contact sites between mitochondria and organelles and its role in mitochondrial calcium transport., (Copyright © 2021 Elsevier Ltd. All rights reserved.)

We demonstrate that a naturally occurring nanopore, the voltage-dependent anion channel (VDAC) of the mitochondrion, can be used to electromechanically trap and interrogate proteins bound to a lipid surface at the single-molecule level. Electromechanically probing α-synuclein (αSyn), an intrinsically disordered neuronal protein intimately associated with Parkinson's pathology, reveals wide variation in the time required for individual proteins to unbind from the same membrane surface. The observed distributions of unbinding times span up to 3 orders of magnitude and depend strongly on the lipid composition of the membrane; surprisingly, lipid membranes to which αSyn binds weakly are most likely to contain subpopulations in which electromechanically driven unbinding is very slow. We conclude that unbinding of αSyn from the membrane surface depends not only on membrane binding affinity but also on the conformation adopted by an individual αSyn molecule on the membrane surface.

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.

In this paper we analyze diffusive transport of noninteracting electrically uncharged solute molecules through a cylindrical membrane channel with a constriction located in the middle of the channel. The constriction is modeled by an infinitely thin partition with a circular hole in its center. The focus is on how the presence of the partition slows down the transport governed by the difference in the solute concentrations in the two reservoirs separated by the membrane. It is assumed that the solutions in both reservoirs are well stirred. To quantify the effect of the constriction we use the notion of diffusion resistance defined as the ratio of the concentration difference to the steady-state flux. We show that when the channel length exceeds its radius, the diffusion resistance is the sum of the diffusion resistance of the cylindrical channel without a partition and an additional diffusion resistance due to the presence of the partition. We derive an expression for the additional diffusion resistance as a function of the tube radius and that of the hole in the partition. The derivation involves the replacement of the nonpermeable partition with the hole by an effective uniform semipermeable partition with a properly chosen permeability. Such a replacement makes it possible to reduce the initial three-dimensional diffusion problem to a one-dimensional one that can be easily solved. To determine the permeability of the effective partition, we take advantage of the results found earlier for trapping of diffusing particles by inhomogeneous surfaces, which were obtained with the method of boundary homogenization. Brownian dynamics simulations are used to corroborate our approximate analytical results and to establish the range of their applicability.

We study trapping of particles diffusing in a two-dimensional rectangular chamber by a binding site located at the end of a rectangular sleeve. To reach the site a particle first has to enter the sleeve. After that it has two options: to come back to the chamber or to diffuse to the site where it is trapped. The main result of the present work is a simple expression for the mean particle lifetime as a function of its starting position and geometric parameters of the system. This expression is obtained by an approximate reduction of the initial two-dimensional problem to the effective one-dimensional one which can be solved with relative ease. Our analytical predictions are tested against the results obtained from Brownian dynamics simulations. The test shows excellent agreement between the two for a wide range of the geometric parameters of the system.

The voltage-dependent anion channel (VDAC) is the most abundant protein in the mitochondrial outer membrane and an archetypical β-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 Q 10  = 1.22 ± 0.02, which exceeds that of the bathing electrolyte solution conductivity, Q 10  = 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 β-barrel channel gating, suggest either an entropy-driven gating mechanism or a role for enthalpy-entropy compensation., (Copyright © 2020 Biophysical Society. Published by Elsevier Inc. All rights reserved.)

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

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 β-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.

Post-translational modifications (PTMs) of proteins are recognized as crucial components of cell signaling pathways through modulating folding, altering stability, changing interactions with ligands, and, therefore, serving multiple regulatory functions. PTMs occur as covalent modifications of the protein's amino acid side chains or the length and composition of their termini. Here we study the functional consequences of PTMs for α-synuclein (αSyn) interactions with the nanopore of the voltage-dependent anion channel (VDAC) of the outer mitochondrial membrane. PTMs were mimicked by a divalent Alexa Fluor 488 sidechain attached separately at two positions on the αSyn C-terminus. Using single-channel reconstitution into planar lipid membranes, we find that such modifications change interactions drastically in both efficiency of VDAC inhibition by αSyn and its translocation through the VDAC nanopore. Analysis of the on/off kinetics in terms of an interaction "quasipotential" allows the positions of the C-terminal modifications to be determined with an accuracy of about three residues. Moreover, our results uncover a previously unobserved mechanism by which cytosolic proteins control β-barrel channels and thus a new regulatory function for PTMs.

There is accumulating evidence that endogenous steroids and non-polar drugs are involved in the regulation of mitochondrial physiology. Many of these hydrophobic compounds interact with the Voltage Dependent Anion Channel (VDAC). This major metabolite channel in the mitochondrial outer membrane (MOM) regulates the exchange of ions and water-soluble metabolites, such as ATP and ADP, across the MOM, thus governing mitochondrial respiration. Proteomics and biochemical approaches together with molecular dynamics simulations have identified an impressively large number of non-polar compounds, including endogenous, able to bind to VDAC. These findings have sparked speculation that both natural steroids and synthetic hydrophobic drugs regulate mitochondrial physiology by directly affecting VDAC ion channel properties and modulating its metabolite permeability. Here we evaluate recent studies investigating the effect of identified VDAC-binding natural steroids and non-polar drugs on VDAC channel functioning. We argue that while many compounds are found to bind to the VDAC protein, they do not necessarily affect its channel functions in vitro . However, they may modify other aspects of VDAC physiology such as interaction with its cytosolic partner proteins or complex formation with other mitochondrial membrane proteins, thus altering mitochondrial function., (Copyright © 2020 Rostovtseva, Queralt-Martín, Rosencrans and Bezrukov.)

A transition path is a part of a one-dimensional trajectory of a diffusing particle, which starts from point a and is terminated as soon as it comes to point b for the first time. It is the trajectory's final segment that leaves point a and goes to point b without returning to point a . The duration of this segment is called transition path time or, alternatively, direct transit time. We study the mean transition path time in monotonically increasing entropy potentials of the narrowing cones in spaces of different dimensions. We find that this time, normalized to its value in the absence of the potential, monotonically increases with the barrier height for the entropy potential of a narrowing two-dimensional cone, is independent of the barrier height for a narrowing three-dimensional cone, and monotonically decreases with the barrier height for narrowing cones in spaces of higher dimensions. Moreover, we show that as the barrier height tends to infinity, the normalized mean transition path time approaches its universal limiting value n /3, where n = 2, 3, 4, ... is the space dimension. This is in sharp contrast to the asymptotic behavior of this quantity in the case of a linear potential of mean force, for which it approaches zero in this limit.

Voltage-dependent anion channel (VDAC) is the major pathway for the transport of ions and metabolites across the mitochondrial outer membrane. Among the three known mammalian VDAC isoforms, VDAC3 is the least characterized, but unique functional roles have been proposed in cellular and animal models. Yet, a high-sequence similarity between VDAC1 and VDAC3 is indicative of a similar pore-forming structure. Here, we conclusively show that VDAC3 forms stable, highly conductive voltage-gated channels that, much like VDAC1, are weakly anion selective and facilitate metabolite exchange, but exhibit unique properties when interacting with the cytosolic proteins α-synuclein and tubulin. These two proteins are known to be potent regulators of VDAC1 and induce similar characteristic blockages (on the millisecond time scale) of VDAC3, but with 10- to 100-fold reduced on-rates and altered α-synuclein blocking times, indicative of an isoform-specific function. Through cysteine scanning mutagenesis, we found that VDAC3's cysteine residues regulate its interaction with α-synuclein, demonstrating VDAC3-unique functional properties and further highlighting a general molecular mechanism for VDAC isoform-specific regulation of mitochondrial bioenergetics., (© 2019 Queralt-Martín et al.)

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.

For a particle diffusing in one dimension, the distribution of its first-passage time from point a to point b is determined by the durations of the particle trajectories that start from point a and are terminated as soon as they touch point b for the first time. Any such trajectory consists of looping and direct-transit segments. The latter is the final part of the trajectory that leaves point a and goes to point b without returning to point a. The rest of the trajectory is the looping segment that makes numerous loops which begin and end at the same point a without touching point b. In this article we discuss general relations between the first-passage time distribution and those for the durations of the two segments. These general relations allow us to find exact solutions for the Laplace transforms of the distributions of the first-passage, direct-transit, and looping times for transitions between two points separated by a symmetric cusp potential barrier or well of arbitrary height and depth, respectively. The obtained Laplace transforms are inverted numerically, leading to nontrivial dependences of the resulting distributions on the barrier height and the well depth.

It is well established that α-synuclein (α-syn) binding from solution to the surface of membranes composed of negatively charged and/or non-lamellar lipids can be characterized by equilibrium dissociation constants of tens of micromolar. Previously, we have found that a naturally occurring nanopore of the mitochondrial voltage-dependent anion channel (VDAC), reconstituted into planar bilayers of a plant-derived lipid, responds to α-syn at nanomolar solution concentrations. Here, using lipid mixtures that mimic the composition of mitochondrial outer membranes, we show that functionally important binding does indeed take place in the nanomolar range. We demonstrate that the voltage-dependent rate at which a membrane-embedded VDAC nanopore captures α-syn is a strong function of membrane composition. Comparison of the nanopore results with those obtained by the bilayer overtone analysis of membrane binding demonstrates a pronounced correlation between the two datasets. The stronger the binding, the larger the on-rate, but with some notable exceptions. This leads to a tentative model of α-syn-membrane interactions, which assigns different lipid-dependent roles to the N- and C-terminal domains of α-syn accounting for both electrostatic and hydrophobic effects. As a result, the rate of α-syn capture by the nanopore is not simply proportional to the α-syn concentration on the membrane surface but found to be sensitive to the specific interactions of each domain with the membrane and nanopore.

We study trapping of particles diffusing in a spherical cavity with an absorbing wall containing small static spherical absorbers localized in a spherical region in the center of the cavity. The focus is on the competition between the absorbers and the cavity wall for diffusing particles. Assuming that the absorbers and, initially, the particles are uniformly distributed in the central region, we derive an expression for the particle trapping probability by the cavity wall. The expression gives this probability as a function of two dimensionless parameters: the transparency parameter, characterizing the efficiency of the particle trapping by the absorbers, and the ratio of the absorber-containing region radius to that of the cavity. This work is a generalization of a recent study by Krapivsky and Redner [J. Chem. Phys. 147, 214903 (2017)] who considered the case where the absorber-containing region occupies the entire cavity. The expression for the particle trapping probability is derived in the framework of a steady-state approach which, in our opinion, is much simpler than the time-dependent approach used in the above-mentioned study.

The voltage-dependent anion channel (VDAC) is the most abundant protein of the mitochondrial outer membrane (MOM) where it regulates transport of ions and metabolites in and out of the organelle. VDAC function is extensively studied in a lipid bilayer system that allows conductance monitoring of reconstituted channels under applied voltage. The process of switching from a high-conductance state, open to metabolites, to a variety of low-conducting states, which excludes metabolite transport, is termed voltage gating and the mechanism remains poorly understood. Recent studies have implicated the involvement of the membrane-solvated residue E73 in the gating process through β-barrel destabilization. However, there has been no direct experimental evidence of E73 involvement in VDAC1 voltage gating. Here, using electrophysiology measurements, we exclude the involvement of E73 in murine VDAC1 (mVDAC1) voltage gating process. With an established protocol of assessing voltage gating of VDACs reconstituted into planar lipid membranes, we definitively show that mVDAC1 gating properties do not change when E73 is replaced by either a glutamine or an alanine. We further demonstrate that cholesterol has no effect on mVDAC1 gating characteristics, though it was shown that E73 is coordinating residue in the cholesterol binding site. In contrast, we found a pronounced gating effect based on the charge of the phospholipid headgroup, where the positive charge stimulates and negative charge suppresses gating. These findings call for critical evaluation of the existing models of VDAC gating and contribute to our understanding of VDAC's role in control of MOM permeability and regulation of mitochondrial respiration and metabolism., (Copyright © 2018. Published by Elsevier B.V.)

This study focuses on how interactions of solute molecules affect the concentration dependence of their flux through narrow membrane channels. It is assumed that the molecules cannot bypass each other because of their hard-core repulsion. In addition, other short- and long-range solute-solute interactions are included into consideration. These interactions make it impossible to develop an analytical theory for the flux in the framework of a diffusion model of solute dynamics in the channel. To overcome this difficulty, we course-grain the diffusion model by mapping it onto a two-site one, where the rate constants describing the solute dynamics are expressed in terms of the parameters of the initial diffusion model. This allows us (i) to find an analytical solution for the flux as a function of the solute concentration and (ii) to characterize the solute-solute interactions by two dimensionless parameters. Such a characterization proves to be very informative as it results in a clear classification of the effects of the solute-solute interactions on the concentration dependence of the flux. Unexpectedly, this dependence can be nonmonotonic, exhibiting a sharp maximum in a certain parameter range. We hypothesize that such a behavior may constitute an element of a regulatory mechanism, wherein maximal flux reports on the optimal solute concentration in the bulk near the channel entrance.

The microtubule protein tubulin is a heterodimer comprising α/β subunits, in which each subunit features multiple isotypes in vertebrates. For example, seven α-tubulin and eight β-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/β-tubulin constructs with α-CTTs, β-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 β-CTTs block the VDAC pore and that the efficiency of blockage by individual CTTs spans 2 orders of magnitude, depending on the CTT isotype. β-Tubulin constructs, notably β3, 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.

This paper deals with the effect of stochastic gating on the flux of solute molecules through a membrane channel. According to the conventional approach, this flux is given by the product of the flux through the open channel and the probability of finding the channel in the open state. Recently we derived an expression for the flux through a stochastically gated channel (Berezhkovskii and Bezrukov 2017 J. Chem. Phys. 147 084109) that showed that the conventional approach may underestimate the flux at fast gating by orders of magnitude. The present work proposes a novel approach to the problem: while our initial derivation of the expression for the flux focuses on the molecule propagator in the channel, here we treat the problem by considering the steady-state flux through an ensemble of identical stochastically gated channels. We show now that the effect of gating on the flux is independent of the gate position, i.e. whether the gate is located at the channel entrance or exit.

We study transitions of diffusing particles between the left and right ends of expanding and narrowing conical tubes. In an expanding tube, such transitions occur faster than in the narrowing tube of the same length and radius variation rate. This happens because the entropy potential pushes the particle towards the wide tube end, thus accelerating the transitions in the expanding tube and slowing them down in the narrowing tube. To gain deeper insight into how the transitions occur, we divide each trajectory into the direct-transit and looping segments. The former is the final part of the trajectory, where the particle starting from the left tube end goes to the right end without returning to the left one. The rest of the trajectory is the looping segment, where the particle, starting from the left tube end, returns to this end again and again until the direct transition happens. Our focus is on the durations of the two segments and their sum, which is the duration of the particle first passage between the left and right ends of the tube. We approach the problem using the one-dimensional description of the particle diffusion along the tube axis in terms of the modified Fick-Jacobs equation. This allows us to derive analytical expressions for the Laplace transforms of the probability densities of the first-passage, direct-transit, and looping times, which we use to find the mean values of these random variables. Our results show that the direct transits are independent of the entropy potential and occur as in free diffusion. However, this "free diffusion" occurs with the effective diffusivity entering the modified Fick-Jacobs equation, which is smaller than the particle diffusivity in a cylindrical tube. This is the only way how the varying tube geometry manifests itself in the direct transits. Since direct-transit times are direction-independent, the difference in the first-passage times in the tubes of the two types is due to the difference in the durations of the looping segments in the expanding and narrowing tubes. Obtained analytical results are supported by three-dimensional Brownian dynamics simulations.

Recent experiments with single biological nanopores, as well as single-molecule fluorescence spectroscopy and pulling studies of protein and nucleic acid folding raised a number of questions that stimulated theoretical and computational investigations of barrier crossing dynamics. The present paper addresses a closely related problem focusing on trajectories of Brownian particles that escape from a cylindrical trap in the presence of a force F parallel to the cylinder axis. To gain new insights into the escape dynamics, we analyze the "fine structure" of these trajectories. Specifically, we divide trajectories into two segments: a looping segment, when a particle unsuccessfully tries to escape returning to the trap bottom again and again, and a direct-transit segment, when it finally escapes moving without touching the bottom. Analytical expressions are derived for the Laplace transforms of the probability densities of the durations of the two segments. These expressions are used to find the mean looping and direct-transit times as functions of the biasing force F. It turns out that the force-dependences of the two mean times are qualitatively different. The mean looping time monotonically increases as F decreases, approaching exponential F-dependence at large negative forces pushing the particle towards the trap bottom. In contrast to this intuitively appealing behavior, the mean direct-transit time shows rather counterintuitive behavior: it decreases as the force magnitude, |F|, increases independently of whether the force pushes the particles to the trap bottom or to the exit from the trap, having a maximum at F = 0.

Signaling proteins and neurotransmitter receptors often associate with saturated chain and cholesterol-rich domains of cell membranes, also known as lipid rafts. The saturated chains and high cholesterol environment in lipid rafts can modulate protein function, but evidence for such modulation of ion channel function in lipid rafts is lacking. Here, using raft-forming model membrane systems containing cholesterol, we show that lipid lateral phase separation at the nanoscale level directly affects the dissociation kinetics of the gramicidin dimer, a model ion channel.

Ligand- or voltage-driven stochastic gating-the structural rearrangements by which the channel switches between its open and closed states-is a fundamental property of biological membrane channels. Gating underlies the channel's ability to respond to different stimuli and, therefore, to be functionally regulated by the changing environment. The accepted understanding of the gating effect on the solute flux through the channel is that the mean flux is the product of the flux through the open channel and the probability of finding the channel in the open state. Here, using a diffusion model of channel-facilitated transport, we show that this is true only when the gating is much slower than the dynamics of solute translocation through the channel. If this condition breaks, the mean flux could differ from this simple estimate by orders of magnitude.

Surface transport, when the particle is allowed to leave the surface, travel in the bulk for some time, and then return to the surface, is referred to as bulk-mediated surface transport. Recently, we proposed a formalism that significantly simplifies analysis of bulk-mediated surface diffusion [A. M. Berezhkovskii, L. Dagdug, and S. M. Bezrukov, J. Chem. Phys. 143, 084103 (2015)]. Here this formalism is extended to bulk-mediated surface transport in the presence of bias, i.e., when the particle has arbitrary drift velocities on the surface and in the bulk. A key advantage of our approach is that the transport problem reduces to that of a two-state problem of the particle transitions between the surface and the bulk. The latter can be solved with relative ease. The formalism is used to find the Laplace transforms of the first two moments of the particle displacement over the surface in time t at arbitrary values of the particle drift velocities and diffusivities on the surface and in the bulk. This allows us to analyze in detail the time dependence of the effective drift velocity of the particle on the surface, which can be highly nontrivial.

Any trajectory of a diffusing particle making a transition between two end points of an interval can be divided into two segments, which we call direct-transit and looping parts. The former is the final segment of the trajectory, when the particle goes from one end point of the interval to the opposite end point, without retouching the starting point. The rest of the trajectory is the looping part. We study mean durations of the two parts in the presence of a symmetric linear cusp potential which, depending on the parameter values, forms either a barrier or a well between the end points. For the cusp barrier, we find that the mean direct-transit time decreases as the barrier height increases at a fixed interval length. This happens because the increase in the barrier height results in the increase of the magnitude of the force acting on the particle on both sides of the barrier. Interestingly, though the mean looping and direct-transit times are different in the case of the barrier and well potentials with equal height and depth, respectively, the mean first-passage times for the two cases are identical.

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., Competing Interests: The authors declare no conflict of interest.

Using two approaches, small-angle neutron scattering (SANS) from bulk solutions and nanopore conductance-fluctuation analysis, we studied structural and dynamic features of poly(ethylene glycol) (PEG) water/salt solutions in the dilute and semidilute regimes. SANS measurements on PEG 3400 at the zero-average contrast yielded the single chain radius of gyration ( R g ) over 1-30 wt %. We observed a small but statistically reliable decrease in R g with increasing PEG concentration: at 30 wt % the chain contracts by a factor of 0.94. Analyzing conductance fluctuations of the α -hemolysin nanopore in the mixtures of PEG 200 with PEG 3400, we demonstrated that polymer partitioning into the nanopore is mostly due to PEG 200. Specifically, for a 1:1 wt/wt mixture the smaller polymer dominates to the extent that only about 1/25 of the nanopore volume is taken by the larger polymer. These findings advance our conceptual and quantitative understanding of nanopore polymer partitioning; they also support the main assumptions of the recent "polymers-pushing-polymers" model., Competing Interests: Notes The authors declare no competing financial interest.

Specificity of small ions, the Hofmeister ranking, is long-known and has many applications including medicine. Yet it evades consistent theoretical description. Here we study the effect of Hofmeister anions on gramicidin A channels in lipid membranes. Counterintuitively, we find that conductance of this perfectly cation-selective channel increases about two-fold in the H 2 PO 4 - - ≈Br - ≈NO 3 - 4 - - series. Channel dissociation kinetics show even stronger dependence, with the dwell time increasing about 20-fold. While the conductance can be quantitatively explained by the changes in membrane surface potential due to exclusion of kosmotropes from (or accumulation of chaotropes at) the surface, the kinetics proved to be more difficult to treat. We estimate the effects of changes in the energetics at the bilayer surfaces on the channel dwell time, concluding that the change would have to be greater than typically observed for the Hofmeister effect outside the context of the lipid bilayer., (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.)