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.

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.

Diffusion Resistance of Segmented Channels.

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.

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.

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.

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.

Blocker Effect on Diffusion Resistance of a Membrane Channel: Dependence on the Blocker Geometry.

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.

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.

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.

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.

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.

Effective diffusivity of a Brownian particle in a two-dimensional periodic channel of abruptly alternating width.

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.

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.

Exploring lipid-dependent conformations of membrane-bound α-synuclein with the VDAC nanopore.

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 ß-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.

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.

Trapping of particles diffusing in two dimensions by a hidden binding site.

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.

Evaluating diffusion resistance of a constriction in a membrane channel by the method of boundary homogenization.

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.

α-Synuclein emerges as a potent regulator of VDAC-facilitated calcium transport.

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 CaCl2 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 CaCl2 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.