Effects of multiple occupancy and interparticle interactions on selective transport through narrow channels: theory versus experiment

Evolutionary trajectory for nuclear functions of ciliary transport complex proteins

SUMMARYCilia and the nucleus were two defining features of the last eukaryotic common ancestor. In early eukaryotic evolution, these structures evolved through the diversification of a common membrane-coating ancestor, the protocoatomer. While in cilia, the descendants of this protein complex evolved into parts of the intraflagellar transport complexes and BBSome, the nucleus gained its selectivity by recruiting protocoatomer-like proteins to the nuclear envelope to form the selective nuclear pore complexes. Recent studies show a growing number of proteins shared between the proteomes of the respective organelles, and it is currently unknown how ciliary transport proteins could acquire nuclear functions and vice versa. The nuclear functions of ciliary proteins are still observable today and remain relevant for the understanding of the disease mechanisms behind ciliopathies. In this work, we review the evolutionary history of cilia and nucleus and their respective defining proteins and integrate current knowledge into theories for early eukaryotic evolution. We postulate a scenario where both compartments co-evolved and that fits current models of eukaryotic evolution, explaining how ciliary proteins and nucleoporins acquired their dual functions.

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.

Self-regulation of the nuclear pore complex enables clogging-free crowded transport

Nuclear pore complexes (NPCs) are the main conduits for macromolecular transport into and out of the nucleus of eukaryotic cells. The central component of the NPC transport mechanism is an assembly of intrinsically disordered proteins (IDPs) that fills the NPC channel. The channel interior is further crowded by large numbers of simultaneously translocating cargo-carrying and free transport proteins. How the NPC can efficiently, rapidly, and selectively transport varied cargoes in such crowded conditions remains ill understood. Past experimental results suggest that the NPC is surprisingly resistant to clogging and that transport may even become faster and more efficient as the concentration of transport protein increases. To understand the mechanisms behind these puzzling observations, we construct a computational model of the NPC comprising only a minimal set of commonly accepted consensus features. This model qualitatively reproduces the previous experimental results and identifies self-regulating mechanisms that relieve crowding. We show that some of the crowding-alleviating mechanisms-such as preventing saturation of the bulk flux-are "robust" and rely on very general properties of crowded dynamics in confined channels, pertaining to a broad class of selective transport nanopores. By contrast, the counterintuitive ability of the NPC to leverage crowding to achieve more efficient single-molecule translocation is "fine-tuned" and relies on the particular spatial architecture of the IDP assembly in the NPC channel.

Percolation transition prescribes protein size-specific barrier to passive transport through the nuclear pore complex

Nuclear pore complexes (NPCs) control biomolecular transport in and out of the nucleus. Disordered nucleoporins in the complex's pore form a permeation barrier, preventing unassisted transport of large biomolecules. Here, we combine coarse-grained simulations of experimentally derived NPC structures with a theoretical model to determine the microscopic mechanism of passive transport. Brute-force simulations of protein transport reveal telegraph-like behavior, where prolonged diffusion on one side of the NPC is interrupted by rapid crossings to the other. We rationalize this behavior using a theoretical model that reproduces the energetics and kinetics of permeation solely from statistics of transient voids within the disordered mesh. As the protein size increases, the mesh transforms from a soft to a hard barrier, enabling orders-of-magnitude reduction in permeation rate for proteins beyond the percolation size threshold. Our model enables exploration of alternative NPC architectures and sets the stage for uncovering molecular mechanisms of facilitated nuclear transport.

Observing capture with a colloidal model membrane channel

We use video microscopy to study the full capture process for colloidal particles transported through microfluidic channels by a pressure-driven flow. In particular, we obtain trajectories for particles as they move from the bulk into confinement, using these to map in detail the spatial velocity and concentration fields for a range of different flow velocities. Importantly, by changing the height profiles of our microfluidic devices, we consider systems for which flow profiles in the channel are the same, but flow fields in the reservoir differ with respect to the quasi-2D monolayer of particles. We find that velocity fields and profiles show qualitative agreement with numerical computations of pressure-driven fluid flow through the systems in the absence of particles, implying that in the regimes studied here particle-particle interactions do not strongly perturb the flow. Analysis of the particle flux through the channel indicates that changing the reservoir geometry leads to a change between long-range attraction of the particles to the pore and diffusion-to-capture-like behaviour, with concentration fields that show qualitative changes based on device geometry. Our results not only provide insight into design considerations for microfluidic devices, but also a foundation for experimental elucidation of the concept of a capture radius. This long standing problem plays a key role in transport models for biological channels and nanopore sensors.

High-Performance Biomimetic Water Channel: The Constructive Interplay of Interaction Parameters and Hydrophilic Doping Levels

In this study, we introduce a superfast biomimetic water channel mimicking the hydrophobicity scales of the Aquaporin (AQP) pore lining. Molecular dynamics simulation is used to scrutinize the impact of hydrophilic doping level in the nanotube and the water-wall interaction strength on water permeability. In the designed biomimetic channel, the constructive interplay of Lennard-Jones (LJ) ε parameters and hydrophilic doping levels increased the possibility of ultrafast water transport. Moreover, a unique set of LJ parameters is discovered for each biomimetic channel with different hydrophilic doping levels, enhancing water permeation. Inside high-performance biomimetic channels, water distribution surprisingly implies a varying pore geometry that narrows down in the middle, mimicking the pattern obtained from GplF pore analysis, evoking the narrow pore induced by the aromatic/arginine selectivity filter. This exciting accordance occurred as a result of tailoring specific hydrophilic arrays within the hydrophobic channel backbone by mimicking the AQP pore interior. The main takeaway of hydrophilic doping arrays implanted within the hydrophobic nanotube is to break the large barrier in the water-wall vdW energy profile into multiple reduced ones to increase water conduction. Consequently, the "water jumping" phenomenon in the middle of the biomimetic channel occurs under specific circumstances. The biomimetic channel with the highest value of water permeability of about 13.67 ± 0.66 × 10^-13 cm³·s^-1 exhibits the best mechanism for artificial water channels (AWCs), serving superfast water transport considering the low entrance barrier and weak water-wall interaction.

Physics of the Nuclear Pore Complex: Theory, Modeling and Experiment

The hallmark of eukaryotic cells is the nucleus that contains the genome, enclosed by a physical barrier known as the nuclear envelope (NE). On the one hand, this compartmentalization endows the eukaryotic cells with high regulatory complexity and flexibility. On the other hand, it poses a tremendous logistic and energetic problem of transporting millions of molecules per second across the nuclear envelope, to facilitate their biological function in all compartments of the cell. Therefore, eukaryotes have evolved a molecular "nanomachine" known as the Nuclear Pore Complex (NPC). Embedded in the nuclear envelope, NPCs control and regulate all the bi-directional transport between the cell nucleus and the cytoplasm. NPCs combine high molecular specificity of transport with high throughput and speed, and are highly robust with respect to molecular noise and structural perturbations. Remarkably, the functional mechanisms of NPC transport are highly conserved among eukaryotes, from yeast to humans, despite significant differences in the molecular components among various species. The NPC is the largest macromolecular complex in the cell. Yet, despite its significant complexity, it has become clear that its principles of operation can be largely understood based on fundamental physical concepts, as have emerged from a combination of experimental methods of molecular cell biology, biophysics, nanoscience and theoretical and computational modeling. Indeed, many aspects of NPC function can be recapitulated in artificial mimics with a drastically reduced complexity compared to biological pores. We review the current physical understanding of the NPC architecture and function, with the focus on the critical analysis of experimental studies in cells and artificial NPC mimics through the lens of theoretical and computational models. We also discuss the connections between the emerging concepts of NPC operation and other areas of biophysics and bionanotechnology.

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.

Molecular determinants of large cargo transport into the nucleus

Nucleocytoplasmic transport is tightly regulated by the nuclear pore complex (NPC). Among the thousands of molecules that cross the NPC, even very large (>15 nm) cargoes such as pathogens, mRNAs and pre-ribosomes can pass the NPC intact. For these cargoes, there is little quantitative understanding of the requirements for their nuclear import, especially the role of multivalent binding to transport receptors via nuclear localisation sequences (NLSs) and the effect of size on import efficiency. Here, we assayed nuclear import kinetics of 30 large cargo models based on four capsid-like particles in the size range of 17-36 nm, with tuneable numbers of up to 240 NLSs. We show that the requirements for nuclear transport can be recapitulated by a simple two-parameter biophysical model that correlates the import flux with the energetics of large cargo transport through the NPC. Together, our results reveal key molecular determinants of large cargo import in cells.

Direct detection of molecular intermediates from first-passage times

All natural phenomena are governed by energy landscapes. However, the direct measurement of this fundamental quantity remains challenging, particularly in complex systems involving intermediate states. Here, we uncover key details of the energy landscapes that underpin a range of experimental systems through quantitative analysis of first-passage time distributions. By combined study of colloidal dynamics in confinement, transport through a biological pore, and the folding kinetics of DNA hairpins, we demonstrate conclusively how a short-time, power-law regime of the first-passage time distribution reflects the number of intermediate states associated with each of these processes, despite their differing length scales, time scales, and interactions. We thereby establish a powerful method for investigating the underlying mechanisms of complex molecular processes.

Field-driven tracer diffusion through curved bottlenecks: fine structure of first passage events

Using scaling arguments and extensive numerical simulations, we study the dynamics of a tracer particle in a corrugated channel represented by a periodic sequence of broad chambers and narrow funnel-like bottlenecks enclosed by a hard-wall boundary.

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.

Mapping Intrachannel Diffusive Dynamics of Interacting Molecules onto a Two-Site Model: Crossover in Flux Concentration Dependence

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.

Tracer diffusion in crowded narrow channels

We summarise different results on the diffusion of a tracer particle in lattice gases of hard-core particles with stochastic dynamics, which are confined to narrow channels-single-files, comb-like structures and quasi-one-dimensional channels with the width equal to several particle diameters. We show that in such geometries a surprisingly rich, sometimes even counter-intuitive, behaviour emerges, which is absent in unbounded systems. This is well-documented for the anomalous diffusion in single-files. Less known is the anomalous dynamics of a tracer particle in crowded branching single-files-comb-like structures, where several kinds of anomalous regimes take place. In narrow channels, which are broader than single-files, one encounters a wealth of anomalous behaviours in the case where the tracer particle is subject to a regular external bias: here, one observes an anomaly in the temporal evolution of the tracer particle velocity, super-diffusive at transient stages, and ultimately a giant diffusive broadening of fluctuations in the position of the tracer particle, as well as spectacular multi-tracer effects of self-clogging of narrow channels. Interactions between a biased tracer particle and a confined crowded environment also produce peculiar patterns in the out-of-equilibrium distribution of the environment particles, very different from the ones appearing in unbounded systems. For moderately dense systems, a surprising effect of a negative differential mobility takes place, such that the velocity of a biased tracer particle can be a non-monotonic function of the force. In some parameter ranges, both the velocity and the diffusion coefficient of a biased tracer particle can be non-monotonic functions of the density. We also survey different results obtained for a tracer particle diffusion in unbounded systems, which will permit a reader to have an exhaustively broad picture of the tracer diffusion in crowded environments.

Model of osmosis in a single-file pore

Single-file transport of water and other small molecules through narrow pores in osmosis has drawn considerable attention in recent years due to its extensive application in biology and industry. In this work, we propose a discrete model to describe nonideal osmosis through single-file pores. Every site is assumed to be occupied by a molecule according to experiments and simulations. Hence, a dense chain can always be found, and collective hopping is the only movement method enabling the molecular chain to move. The roles of solute in osmosis are clarified in this model. Those molecules reflected at the pore entrance produce osmotic pressure, and those inside the pore contribute to the flow resistance of the molecular chain. The solute molecules that can enter the pore but cannot penetrate it may significantly reduce the osmotic flux, although they are all rejected by the pore. This conclusion can help to clarify the emerging debate about whether the reflection coefficient of the fully rejected solute can be less than 1. The design of highly efficient membrane pores may also benefit from this study.

Membranes: A Variety of Energy Landscapes for Many Transfer Opportunities

A membrane can be represented by an energy landscape that solutes or colloids must cross. A model accounting for the momentum and the mass balances in the membrane energy landscape establishes a new way of writing for the Darcy law. The counter-pressure in the Darcy law is no longer written as the result of an osmotic pressure difference but rather as a function of colloid-membrane interactions. The ability of the model to describe the physics of the filtration is discussed in detail. This model is solved in a simplified energy landscape to derive analytical relationships that describe the selectivity and the counter-pressure from ab initio operating conditions. The model shows that the stiffness of the energy landscape has an impact on the process efficiency: a gradual increase in interactions (such as with hourglass pore shape) can reduce the separation energetic cost. It allows the introduction of a new paradigm to increase membrane efficiency: the accumulation that is inherent to the separation must be distributed across the membrane. Asymmetric interactions thus lead to direction-dependent transfer properties and the membrane exhibits diode behavior. These new transfer opportunities are discussed.

Particle transport across a channel via an oscillating potential

Membrane protein transporters alternate their substrate-binding sites between the extracellular and cytosolic side of the membrane according to the alternating access mechanism. Inspired by this intriguing mechanism devised by nature, we study particle transport through a channel coupled with an energy well that oscillates its position between the two entrances of the channel. We optimize particle transport across the channel by adjusting the oscillation frequency. At the optimal oscillation frequency, the translocation rate through the channel is a hundred times higher with respect to free diffusion across the channel. Our findings reveal the effect of time-dependent potentials on particle transport across a channel and will be relevant for membrane transport and microfluidics application.

Dependence of the Enzymatic Velocity on the Substrate Dissociation Rate

Enzymes are biological catalysts that play a fundamental role in all living systems by supporting the selectivity and the speed for almost all cellular processes. While the general principles of enzyme functioning are known, the specific details of how they work at the microscopic level are not always available. Simple Michaelis-Menten kinetics assumes that the enzyme-substrate complex has only one conformation that decays as a single exponential. As a consequence, the enzymatic velocity decreases as the dissociation (off) rate constant of the complex increases. Recently, Reuveni et al. [ Proc. Natl. Acad. Sci. USA 2014 , 111 , 4391 - 4396 ] showed that it is possible for the enzymatic velocity to increase when the off rate becomes higher, if the enzyme-substrate complex has many conformations which dissociate with the same off rate constant. This was done using formal mathematical arguments, without specifying the nature of the dynamics of the enzyme-substrate complex. In order to provide a physical basis for this unexpected result, we derive an analytical expression for the enzymatic velocity assuming that the enzyme-substrate complex has multiple states and its conformational dynamics is described by rate equations with arbitrary rate constants. By applying our formalism to a complex with two conformations, we show that the unexpected off rate dependence of the velocity can be readily understood: If one of the conformations is unproductive, the system can escape from this "trap" by dissociating, thereby giving the enzyme another chance to form the productive enzyme-substrate complex. We also demonstrate that the nonmonotonic off rate dependence of the enzymatic velocity is possible not only when all off rate constants are identical, but even when they are different. We show that for typical experimentally determined rate constants, the nonmonotonic off rate dependence can occur for micromolar substrate concentrations. Finally, we discuss the relation of this work to the problem of optimizing the flux through singly occupied membrane channels and transporters.

Importins promote high-frequency NF-κB oscillations increasing information channel capacity

Background

Importins and exportins influence gene expression by enabling nucleocytoplasmic shuttling of transcription factors. A key transcription factor of innate immunity, NF-κB, is sequestered in the cytoplasm by its inhibitor, IκBα, which masks nuclear localization sequence of NF-κB. In response to TNFα or LPS, IκBα is degraded, which allows importins to bind NF-κB and shepherd it across nuclear pores. NF-κB nuclear activity is terminated when newly synthesized IκBα enters the nucleus, binds NF-κB and exportin which directs the complex to the cytoplasm. Although importins/exportins are known to regulate spatiotemporal kinetics of NF-κB and other transcription factors governing innate immunity, the mechanistic details of these interactions have not been elucidated and mathematically modelled.

Results

Based on our quantitative experimental data, we pursue NF-κB system modelling by explicitly including NF-κB–importin and IκBα–exportin binding to show that the competition between importins and IκBα enables NF-κB nuclear translocation despite high levels of IκBα. These interactions reduce the effective relaxation time and allow the NF-κB regulatory pathway to respond to recurrent TNFα pulses of 45-min period, which is about twice shorter than the characteristic period of NF-κB oscillations. By stochastic simulations of model dynamics we demonstrate that randomly appearing, short TNFα pulses can be converted to essentially digital pulses of NF-κB activity, provided that intervals between input pulses are not shorter than 1 h.

Conclusions

By including interactions involving importin-α and exportin we bring the modelling of spatiotemporal kinetics of transcription factors to a more mechanistic level. Basing on the analysis of the pursued model we estimated the information transmission rate of the NF-κB pathway as 1 bit per hour.

Reviewers

This article was reviewed by Marek Kimmel, James Faeder and William Hlavacek.

Electronic supplementary material

The online version of this article (doi:10.1186/s13062-016-0164-z) contains supplementary material.

Single-File Escape of Colloidal Particles from Microfluidic Channels

Single-file diffusion is a ubiquitous physical process exploited by living and synthetic systems to exchange molecules with their environment. It is paramount to quantify the escape time needed for single files of particles to exit from constraining synthetic channels and biological pores. This quantity depends on complex cooperative effects, whose predominance can only be established through a strict comparison between theory and experiments. By using colloidal particles, optical manipulation, microfluidics, digital microscopy, and theoretical analysis we uncover the self-similar character of the escape process and provide closed-formula evaluations of the escape time. We find that the escape time scales inversely with the diffusion coefficient of the last particle to leave the channel. Importantly, we find that at the investigated microscale, bias forces as tiny as 10^{-15}  N determine the magnitude of the escape time by drastically reducing interparticle collisions. Our findings provide crucial guidelines to optimize the design of micro- and nanodevices for a variety of applications including drug delivery, particle filtering, and transport in geometrical constrictions.

Simple biophysics underpins collective conformations of the intrinsically disordered proteins of the Nuclear Pore Complex

Nuclear Pore Complexes (NPCs) are key cellular transporter that control nucleocytoplasmic transport in eukaryotic cells, but its transport mechanism is still not understood. The centerpiece of NPC transport is the assembly of intrinsically disordered polypeptides, known as FG nucleoporins, lining its passageway. Their conformations and collective dynamics during transport are difficult to assess in vivo. In vitro investigations provide partially conflicting results, lending support to different models of transport, which invoke various conformational transitions of the FG nucleoporins induced by the cargo-carrying transport proteins. We show that the spatial organization of FG nucleoporin assemblies with the transport proteins can be understood within a first principles biophysical model with a minimal number of key physical variables, such as the average protein interaction strengths and spatial densities. These results address some of the outstanding controversies and suggest how molecularly divergent NPCs in different species can perform essentially the same function.

DOI:http://dx.doi.org/10.7554/eLife.10785.001

Animal, plant and fungal cells contain a structure called the nucleus, inside which the genetic material of the cell is stored. For the cell to work properly, certain proteins and other molecules need to be able to enter and exit the nucleus. This transport is carried out by pore-like molecular “devices” known as Nuclear Pore Complexes, whose architecture and mode of operation are unique among cellular transporters.

Nuclear Pore Complexes are charged with a daunting task of deciding which of the hundreds of molecules it conducts per second should go through and which should not. Small molecules can pass freely through Nuclear Pore Complexes. However, larger molecules can only pass through the pore efficiently if they are bound to specialized transport proteins that interact with the proteins – called FG nucleoporins – that line the pore. A unique feature of the FG nucleoporins is that, unlike typical proteins, they do not have a defined three-dimensional structure. Instead, they form a soft and pliable lining inside the Nuclear Pore Complex passageway.

Exactly how interacting with transport proteins affects the structure and spatial arrangements of the FG nucleoporins in a way that allows them to control transport is not well understood. This is in part because existing experimental techniques are unable to study the structures of the FG nucleoporins in enough detail to track how they change during transport. The complexity and the diversity of the FG nucleoporins also make them difficult to model in detail.

Vovk, Gu et al. have developed a theoretical model that is based on just three basic physical properties of the FG nucleoporins – their flexibility, their ability to interact with each other, and their binding with the transport proteins. Future work can refine the model by incorporating further molecular details about the interactions between FG nucleoporins and transport proteins.

The predictions made by this simple model agree well with experimental results in a wide range of situations – from single molecules to complex spatial assemblies. They also explain why some of the experimental results appear to contradict each other and suggest how several outstanding controversies in the field can be reconciled. Because the model invokes only fundamental physical principles of FG nucleoporin assemblies, it shows that some of their general properties do not depend on the exact conditions. In particular, this might shed light on why Nuclear Pore Complexes in different organisms perform essentially the same function, although the details of their molecular structure may differ. This also suggests how the FG nucleoporins can be manipulated to build artificial devices based on the same principles.

DOI:http://dx.doi.org/10.7554/eLife.10785.002

Deciphering the Structure and Function of Nuclear Pores Using Single-Molecule Fluorescence Approaches

Due to its central role in macromolecular trafficking and nucleocytoplasmic information transfer, the nuclear pore complex (NPC) has been studied in great detail using a wide spectrum of methods. Consequently, many aspects of its architecture, general function, and role in the life cycle of a cell are well understood. Over the last decade, fluorescence microscopy methods have enabled the real-time visualization of single molecules interacting with and transiting through the NPC, allowing novel questions to be examined with nanometer precision. While initial single-molecule studies focused primarily on import pathways using permeabilized cells, it has recently proven feasible to investigate the export of mRNAs in living cells. Single-molecule assays can address questions that are difficult or impossible to answer by other means, yet the complexity of nucleocytoplasmic transport requires that interpretation be based on a firm genetic, biochemical, and structural foundation. Moreover, conceptually simple single-molecule experiments remain technically challenging, particularly with regard to signal intensity, signal-to-noise ratio, and the analysis of noise, stochasticity, and precision. We discuss nuclear transport issues recently addressed by single-molecule microscopy, evaluate the limits of existing assays and data, and identify open questions for future studies. We expect that single-molecule fluorescence approaches will continue to be applied to outstanding nucleocytoplasmic transport questions, and that the approaches developed for NPC studies are extendable to additional complex systems and pathways within cells.

Theoretical analysis of selectivity mechanisms in molecular transport through channels and nanopores

Selectivity is one of the most fundamental concepts in natural sciences, and it is also critically important in various technological, industrial, and medical applications. Although there are many experimental methods that allow to separate molecules, frequently they are expensive and not efficient. Recently, a new method of separation of chemical mixtures based on utilization of channels and nanopores has been proposed and successfully tested in several systems. However, mechanisms of selectivity in the molecular transport during the translocation are still not well understood. Here, we develop a simple theoretical approach to explain the origin of selectivity in molecular fluxes through channels. Our method utilizes discrete-state stochastic models that take into account all relevant chemical transitions and can be solved analytically. More specifically, we analyze channels with one and two binding sites employed for separating mixtures of two types of molecules. The effects of the symmetry and the strength of the molecular-pore interactions are examined. It is found that for one-site binding channels, the differences in the strength of interactions for two species drive the separation. At the same time, in more realistic two-site systems, the symmetry of interaction potential becomes also important. The most efficient separation is predicted when the specific binding site is located near the entrance to the nanopore. In addition, the selectivity is higher for large entrance rates into the channel. It is also found that the molecular transport is more selective for repulsive interactions than for attractive interactions. The physical-chemical origin of the observed phenomena is discussed.

Determinants of cation transport selectivity: Equilibrium binding and transport kinetics

The crystal structures of channels and transporters reveal the chemical nature of ion-binding sites and, thereby, constrain mechanistic models for their transport processes. However, these structures, in and of themselves, do not reveal equilibrium selectivity or transport preferences, which can be discerned only from various functional assays. In this Review, I explore the relationship between cation transport protein structures, equilibrium binding measurements, and ion transport selectivity. The primary focus is on K(+)-selective channels and nonselective cation channels because they have been extensively studied both functionally and structurally, but the principles discussed are relevant to other transport proteins and molecules.

Channel-facilitated diffusion boosted by particle binding at the channel entrance

We investigate single-file diffusion of Brownian particles in arrays of closely confining microchannels permeated by a variety of attractive optical potentials and connecting two baths with equal particle concentration. We simultaneously test free diffusion in the channel, diffusion in optical traps coupled in the center of the channel, and diffusion in traps extending into the baths. We found that both classes of attractive optical potentials enhance the translocation rate through the channel with respect to free diffusion. Surprisingly, for the latter class of potentials we measure a 40-fold enhancement in the translocation rate with respect to free diffusion and find a sublinear power law dependence of the translocation rate on the average number of particles in the channel. Our results reveal the function of particle binding at the channel entrances for diffusive transport and open the way to a better understanding of membrane transport and design of synthetic membranes with enhanced diffusion rate.

Anisotropic diffusion of spherical particles in closely confining microchannels

We present here the measurement of the diffusivity of spherical particles closely confined by narrow microchannels. Our experiments yield a two-dimensional map of the position-dependent diffusion coefficients parallel and perpendicular to the channel axis with a resolution down to 129 nm. The diffusivity was measured simultaneously in the channel interior, the bulk reservoirs, as well as the channel entrance region. In the channel interior we found strongly anisotropic diffusion. While the perpendicular diffusion coefficient close to the confining walls decreased down to approximately 25% of the value on the channel axis, the parallel diffusion coefficient remained constant throughout the entire channel width. In addition to the experiment, we performed finite element simulations for the diffusivity in the channel interior and found good agreement with the measurements. Our results reveal the distinctive influence of strong confinement on Brownian motion, which is of significance to microfluidics as well as quantitative models of facilitated membrane transport.

Nucleocytoplasmic transport: a role for nonspecific competition in karyopherin-nucleoporin interactions

Nucleocytoplasmic transport occurs through the nuclear pore complex (NPC), which in yeast is a ~50 MDa complex consisting of ~30 different proteins. Small molecules can freely exchange through the NPC, but macromolecules larger than ~40 kDa must be aided across by transport factors, most of which belong to a related family of proteins termed karyopherins (Kaps). These transport factors bind to the disordered phenylalanine-glycine (FG) repeat domains in a family of NPC proteins termed FG nups, and this specific binding allows the transport factors to cross the NPC. However, we still know little in terms of the molecular and kinetic details regarding how this binding translates to selective passage of transport factors across the NPC. Here we show that the specific interactions between Kaps and FG nups are strongly modulated by the presence of a cellular milieu whose proteins appear to act as very weak competitors that nevertheless collectively can reduce Kap/FG nup affinities by several orders of magnitude. Without such modulation, the avidities between Kaps and FG nups measured in vitro are too tight to be compatible with the rapid transport kinetics observed in vivo. We modeled the multivalent interactions between the disordered repeat binding sites in the FG nups and multiple cognate binding sites on Kap, showing that they should indeed be sensitive to even weakly binding competitors; the introduction of such competition reduces the availability of these binding sites, dramatically lowering the avidity of their specific interactions and allowing rapid nuclear transport.

Functional role for transporter isoforms in optimizing membrane transport

Quantitative analysis of carrier parameters demonstrates that with decreasing substrate concentration the optimal strength of substrate-carrier interaction which maximizes the flux across the membrane increases and requires less fine-tuning than at higher concentrations of the substrate.

Single molecule studies of nucleocytoplasmic transport

Molecular traffic between the cytoplasm and the nucleoplasm of eukaryotic cells is mediated by nuclear pore complexes (NPCs). Hundreds, if not thousands, of molecules interact with and transit through each NPC every second. The pore is blocked by a permeability barrier, which consists of a network of intrinsically unfolded polypeptides containing thousands of phenylalanine-glycine (FG) repeat motifs. This FG-network rejects larger molecules and admits smaller molecules or cargos bound to nuclear transport receptors (NTRs). For a cargo transport complex, minimally consisting of a cargo molecule plus an NTR, access to the permeability barrier is provided by interactions between the NTR and the FG repeat motifs. Numerous models have been postulated to explain the controlled accessibility and the transport characteristics of the FG-network, but the amorphous, flexible nature of this structure has hindered characterization. A relatively recent development is the ability to monitor the real-time movement of single molecules through individual NPCs via single molecule fluorescence (SMF) microscopy. A major advantage of this approach is that it can be used to continuously monitor a series of specific molecular interactions in an active pore with millisecond time resolution, which therefore allows one to distinguish between kinetic and thermodynamic control. Novel insights and prospects for the future are outlined in this review. This article is part of a Special Issue entitled: Regulation of Signaling and Cellular Fate through Modulation of Nuclear Protein Import.

Thermodynamics of competitive molecular channel transport: application to artificial nuclear pores

In an analytical model channel transport is analyzed as a function of key parameters, determining efficiency and selectivity of particle transport in a competitive molecular environment. These key parameters are the concentration of particles, solvent-channel exchange dynamics, as well as particle-in-channel- and interparticle interaction. These parameters are explicitly related to translocation dynamics and channel occupation probability. Slowing down the exchange dynamics at the channel ends, or elevating the particle concentration reduces the in-channel binding strength necessary to maintain maximum transport. Optimized in-channel interaction may even shift from binding to repulsion. A simple equation gives the interrelation of access dynamics and concentration at this transition point. The model is readily transferred to competitive transport of different species, each of them having their individual in-channel affinity. Combinations of channel affinities are determined which differentially favor selectivity of certain species on the cost of others. Selectivity for a species increases if its in-channel binding enhances the species' translocation probablity when compared to that of the other species. Selectivity increases particularly for a wide binding site, long channels, and fast access dynamics. Recent experiments on competitive transport of in-channel binding and inert molecules through artificial nuclear pores serve as a paradigm for our model. It explains qualitatively and quantitatively how binding molecules are favored for transport at the cost of the transport of inert molecules.

Crowding effects in non-equilibrium transport through nano-channels

Transport through nano-channels plays an important role in many biological processes and industrial applications. Gaining insights into the functioning of biological transport processes and the design of man-made nano-devices requires an understanding of the basic physics of such transport. A simple exclusion process has proven to be very useful in explaining the properties of several artificial and biological nano-channels. It is particularly useful for modeling the influence of inter-particle interactions on transport characteristics. In this paper, we explore several models of the exclusion process using a mean field approach and computer simulations. We examine the effects of crowding inside the channel and in its immediate vicinity on the mean flux and the transport times of single molecules. Finally, we discuss the robustness of the theory's predictions with respect to the crucial characteristics of the hindered diffusion in nano-channels that need to be included in the model.

Fluxes of non-interacting and strongly repelling particles through a single conical channel: Analytical results and their numerical tests

Using a diffusion model of particle dynamics in the channel, we study entropic effects in channel-facilitated transport. We derive general expressions for the fluxes of non-interacting particles and particles that strongly repel each other through the channel of varying cross section area, assuming that the transport is driven by the difference in particle concentrations on the two sides of the membrane. For a special case of a right truncated cone expanding in the left-to-right direction, we show how the fluxes depend on the geometric parameters of the channel and on the particle concentrations. For non-interacting particles the flux is direction-independent in the sense that inversion of the concentration difference leads to the inversion of the direction of the flux without changing its magnitude. This symmetry is broken for repelling particles: The flux in the left-to-right direction exceeds its right-to-left counterpart. Our theoretical predictions are supported by three-dimensional Brownian dynamics simulations.

Enhancement of transport selectivity through nano-channels by non-specific competition

The functioning of living cells requires efficient and selective transport of materials into and out of the cell, and between different cellular compartments. Much of this transport occurs through nano-scale channels that do not require large scale molecular re-arrangements (such as transition from a 'closed' to an 'open' state) and do not require a direct input of metabolic energy during transport. Nevertheless, these 'always open' channels are highly selective and pass only their cognate molecules, while efficiently excluding all others; indeed, these channels can efficiently transport specific molecules even in the presence of a vast excess of non-specific molecules. Such biological transporters have inspired the creation of artificial nano-channels. These channels can be used as nano-molecular sorters, and can also serve as testbeds for examining modes of biological transport. In this paper, we propose a simple kinetic mechanism that explains how the selectivity of such 'always open' channels can be based on the exclusion of non-specific molecules by specific ones, due to the competition for limited space inside the channel. The predictions of the theory account for the behavior of the nuclear pore complex and of artificial nanopores that mimic its function. This theory provides the basis for future work aimed at understanding the selectivity of various biological transport phenomena.

Nonlinear hopping transport in ring systems and open channels

We study the nonlinear hopping transport in one-dimensional rings and open channels. Analytical results are derived for the stationary current response to a constant bias without assuming any specific coupling of the rates to the external fields. It is shown that anomalous large effective jump lengths, as observed in recent experiments by taking the ratio of the third-order nonlinear and the linear conductivity, can occur already in ordered systems. Rectification effects due to site energy disorder in ring systems are expected to become irrelevant for large system sizes. In open channels, in contrast, rectification effects occur already for disorder in the jump barriers and do not vanish in the thermodynamic limit. Numerical solutions for a sinusoidal bias show that the ring system provides a good description for the transport behavior in the open channel for intermediate and high frequencies. For low frequencies temporal variations in the mean particle number have to be taken into account in the open channel, which cannot be captured in the more simple ring model.

Functionalization of a nanopore: the nuclear pore complex paradigm

Biological cells maintain a myriad of nanopores which, although relying on the same basic small-hole principle, serve a large variety of functions. Here we consider how the nuclear pore complex (NPC), a large nanopore mediating the traffic between genetic material and protein synthesizing apparatus, is functionalized to carry out a set of transport functions. A major parameter of NPC functionalization is a lining of it external and internal surfaces with so-called phenylalanine glycine (FG) proteins. FG proteins integrate a multitude of transport factor binding sites into intrinsically disordered domains. This surprising finding has given rise to a number of transport models assigning direct gating functions to FG proteins. However, recent data suggest that the properties of FG proteins cannot be properly assessed by considering only the purified, transport-factor-stripped NPC. At physiological conditions transport factors may shape FG proteins in a way allotting an essential role to surface diffusion, reconciling tight binding with efficient transport. Thus, NPC studies are revealing both general traits and novel aspects of nanopore functionalization. In addition, they inspire artificial molecule sorters for proteomic and pharmaceutical applications.

Effects of jamming on nonequilibrium transport times in nanochannels

Many biological channels perform highly selective transport without direct input of metabolic energy and without transitions from a "closed" to an "open" state during transport. Mechanisms of selectivity of such channels serve as an inspiration for creation of artificial nanomolecular sorting devices and biosensors. To elucidate the transport mechanisms, it is important to understand the transport on the single molecule level in the experimentally relevant regime when multiple particles are crowded in the channel. In this Letter we analyze the effects of interparticle crowding on the nonequilibrium transport times through a finite-length channel by means of analytical theory and computer simulations.

Cargo surface hydrophobicity is sufficient to overcome the nuclear pore complex selectivity barrier

To fulfil their function, nuclear pore complexes (NPCs) must discriminate between inert proteins and nuclear transport receptors (NTRs), admitting only the latter. This specific permeation is thought to depend on interactions between hydrophobic patches on NTRs and phenylalanine-glycine (FG) or related repeats that line the NPC. Here, we tested this premise directly by conjugating different hydrophobic amino-acid analogues to the surface of an inert protein and examining its ability to cross NPCs unassisted by NTRs. Conjugation of as few as four hydrophobic moieties was sufficient to enable passage of the protein through NPCs. Transport of the modified protein proceeded with rates comparable to those measured for the innate protein when bound to an NTR and was relatively insensitive both to the nature and density of the amino acids used to confer hydrophobicity. The latter observation suggests a non-specific, small, and plant interaction network between cargo and FG repeats.

Entropic effects in channel-facilitated transport: interparticle interactions break the flux symmetry

We analyze transport through conical channels that is driven by the difference in particle concentrations on the two sides of the membrane. Because of the detailed balance, fluxes of noninteracting particles through the same channel, inserted into the membrane in two opposite orientations, are equal. We show that this flux symmetry is broken by particle-particle interactions so that one of the orientations can be much more efficient for transport under the same external conditions. The results are obtained analytically using a one-dimensional diffusion model and confirmed by three-dimensional Brownian dynamics simulations.