Effects of jamming on nonequilibrium transport times in nanochannels

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.

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.

Different time scales in dynamic systems with multiple outcomes

Stochastic biochemical and transport processes have various final outcomes, and they can be viewed as dynamic systems with multiple exits. Many current theoretical studies, however, typically consider only a single time scale for each specific outcome, effectively corresponding to a single-exit process and assuming the independence of each exit process. However, the presence of other exits influences the statistical properties and dynamics measured at any specific exit. Here, we present theoretical arguments to explicitly show the existence of different time scales, such as mean exit times and inverse exit fluxes, for dynamic processes with multiple exits. This implies that the statistics of any specific exit dynamics cannot be considered without taking into account the presence of other exits. Several illustrative examples are described in detail using analytical calculations, mean-field estimates, and kinetic Monte Carlo computer simulations. The underlying microscopic mechanisms for the existence of different time scales are discussed. The results are relevant for understanding the mechanisms of various biological, chemical, and industrial processes, including transport through channels and pores.

Impact of Interparticle Interaction on Thermodynamics of Nano-Channel Transport of Two Species

Understanding the function and control of channel transport is of paramount importance for cell physiology and nanotechnology. In particular, if several species are involved, the mechanisms of selectivity, competition, cooperation, pumping, and its modulation need to be understood. What lacks is a rigorous mathematical approach within the framework of stochastic thermodynamics, which explains the impact of interparticle in-channel interactions on the transport properties of the respective species. To achieve this, stochastic channel transport of two species is considered in a model, which different from mean field approaches, explicitly conserves the spatial correlation of the species within the channel by analysis of the stochastic dynamics within a state space, the elements of which are the channel's spatial occupation states. The interparticle interactions determine the stochastic transitions between these states. Local flow and entropy production in this state space reveal the respective particle flows through the channel and the intensity of the Brownian ratchet like rectifying forces, which these species exert mutually on each other, together with its thermodynamic effectiveness and costs. Perfect coupling of transport of the two species is realized by an attractive empty channel and strong repulsive forces between particles of the same species. This confines the state space to a subspace with circular topology, in which the concentration gradients as thermodynamic driving forces act in series, and channel flow of both species becomes equivalent. For opposing concentration gradients, this makes the species with the stronger gradient the driving, positive entropy producing one; the other is driven and produces negative entropy. Gradients equal in magnitude make all flows vanish, and thermodynamic equilibrium occurs. A differential interparticle interaction with less repulsive forces within particles of one species but maintenance of this interaction for the other species adds a bypass path to this circular subspace. On this path, which is not involved in coupling of the two species, a leak flow of the species with less repulsive interparticle interaction emerges, which is directed parallel to its concentration gradient and, hence, produces positive entropy here. Different from the situation with perfect coupling, appropriate strong opposing concentration gradients may simultaneously parallelize the flow of their respective species, which makes each species produce positive entropy. The rectifying potential of the species with the bypass option is diminished. This implies the existence of a gradient of the other species, above which its flow and gradient are parallel for any gradient of the less coupled species. The opposite holds for the less coupled species. Its flow may always be rectified and turned anti-parallel to its gradient by a sufficiently strong opposing gradient of the other one.

One-dimensional discrete aggregation-fragmentation model

We study here one-dimensional model of aggregation and fragmentation of clusters of particles obeying the stochastic discrete-time kinetics of the generalized totally asymmetric simple exclusion process (gTASEP) on open chains. The gTASEP is essentially the ordinary TASEP with backward-ordered sequential update (BSU), however, equipped with two hopping probabilities: p and p_{m}. The second modified probability p_{m} models a special kinematic interaction between the particles of a cluster in addition to the simple hard-core exclusion interaction, existing in the ordinary TASEP. We focus on the nonequilibrium stationary properties of the gTASEP in the generic case of attraction between the particles of a cluster. In this case the particles of a cluster have higher chance to stay together than to split, thus producing higher throughput in the system. We explain how the topology of the phase diagram in the case of irreversible aggregation, occurring when the modified probability equals unity, changes sharply to the one, corresponding to the ordinary TASEP with BSU, as soon as the modified probability becomes less than unity and aggregation-fragmentation of clusters appears. We estimate various physical quantities in the system and determine the parameter-dependent injection and ejection critical values by extensive computer simulations. With the aid of random walk theory, supported by the Monte Carlo simulations, the properties of the phase transitions between the three stationary phases are assessed.

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.

Hydrogenation Facilitates Proton Transfer through Two-Dimensional Honeycomb Crystals

Recent experiments have triggered a debate about the ability of protons to transfer easily through individual layers of graphene and hexagonal boron nitride (h-BN). However, state-of-the-art computer calculations have shown that the barriers to proton penetration can, at >3 eV, be excessively high. Despite considerable interest the origin of this apparent anomaly between experiment and simulation remains unclear. We offer a new perspective on this debate and show on the basis of first-principles calculations that the barrier for proton penetration is significantly reduced, to <1 eV, upon hydrogenation, even in the absence of pinholes in the lattice. Although hydrogenation has not been offered as an explanation before, analysis reveals that the barrier is reduced because hydrogenation destabilizes the initial state (a deep-lying chemisorption state) and expands the honeycomb lattice through which the protons penetrate. This study offers a rationalization of the fast proton transfer observed in experiments and highlights the ability of proton transport through single-layer materials in hydrogen-rich solutions.

Mechanisms of cooperation and competition of two-species transport in narrow nanochannels

Flow of particles of two different species through a narrow channel with solely two discrete spatial positions is analyzed with respect to the species' capability to cooperate or compete for transport. The origin of the latter arises from particle-channel and interparticle interactions within the channel, i.e., blocking the position of a particle, and its interaction with its neighbors in the channel. The variety of occupation options within the channel defines the state space. The transition dynamics within is considered as a continuous Markov process, which, in contrast to mean-field approaches, conserves explicitly spatial correlations. A strong repulsive interaction between particles of the same kind and a very attractive empty channel imply a strong entanglement of transport of both species. In the limiting case of perfect coupling, flows in state space are restricted to a cyclic subspace, where they become all equivalent in the steady state. In particular, this implies equal particle flows of the two species. Entanglement of transport implies that the species mutually exert entropic forces on each other. For parallel directed concentration gradients this implies that the species' ability to cooperate increases with the degree of entanglement. Thus, the gradient of one species reciprocally induces a higher flow of the other species when compared to that in its absence. The opposite holds for antiparallel gradients where species mutually hamper their transport. For a sufficient strong coupling, the species under the influence of the stronger concentration gradient drives the other against its gradient, i.e., the positive mixing entropy production of the driving species becomes the motor for the negative mixing entropy production of the driven one. The degree of effectiveness by which negative entropy production emerges at the cost of positive entropy production increases with the coupling strength. This becomes evident from location and connectivity of the sources of entropy production in state space.

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.

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.

Role of the potential landscape on the single-file diffusion through channels

Transport of colloid particles through narrow channels is ubiquitous in cell biology as well as becoming increasingly important for microfluidic applications or targeted drug delivery. Membrane channels in cells are useful models for artificial designs because of their high efficiency, selectivity, and robustness to external fluctuations. Here, we model the passive channels that let cargo simply diffuse through them, affected by a potential profile along the way. Passive transporters achieve high levels of efficiency and specificity from binding interactions with the cargo inside the channel. This however leads to a paradox: why should channels which are so narrow that they are blocked by their cargo evolve to have binding regions for their cargo if that will effectively block them? Using Brownian dynamics simulations, we show that different potentials, notably symmetric, increase the flux through narrow passive channels - and investigate how shape and depth of potentials influence the flux. We find that there exist optimal depths for certain potential shapes and that it is most efficient to apply a small force over an extended region of the channel. On the other hand, having several spatially discrete binding pockets will not alter the flux significantly. We also explore the role of many-particle effects arising from pairwise particle interactions with their neighbours and demonstrate that the relative changes in flux can be accounted for by the kinetics of the absorption reaction at the end of the channel.

Everlasting effect of initial conditions on single-file diffusion

We study the dynamics of a tagged particle in an environment of point Brownian particles with hard-core interactions in an infinite one-dimensional channel (a single-file model). In particular, we examine the influence of initial conditions on the dynamics of the tagged particle. We compare two initial conditions: equal distances between particles and uniform density distribution. The effect is shown by the differences of mean-square-displacement and correlation function for the two ensembles of initial conditions. We discuss the violation of Einstein relation, and its dependence on the initial condition, and the difference between time and ensemble averaging. More specifically, using the Jepsen line, we will discuss how transport coefficients, like diffusivity, depend on the initial state. Our work shows that initial conditions determine the long time limit of the dynamics, and in this sense the system never forgets its initial state in complete contrast with thermal systems (i.e., a closed system that attains equilibrium independent of the initial state).

"Targeting" nanoparticles: the constraints of physical laws and physical barriers

In comparison to the complexities of the body, its organs, its normal and aberrant cells, many nanoparticles will appear to be relatively simple objects. This view is deceptive because the physicochemical properties of nanosystems, although quite well understood on the basis of material science, surface science and colloid theory, are far from simple in practice. While their properties are largely controllable in vitro, often purportedly "designed", their administration by any route changing environments conspires to produce additional layers of complexity. Some of the key physical laws and physicochemical parameters governing the fate of nanoparticles on their journey from point of intravenous administration to desired destinations such as tumors are discussed. Much of the science relevant to nanocarrier based targeting has been elaborated in studying purely physical phenomena, but there can be found therein many analogies with biological systems. These include factors that impede quantitative targeting: diffusion in complex media, aggregation and flocculation, hindered behavior of particles in confined spaces, jamming and dispersion in flow. All of these have the ability to influence fate and destination. Most of the critical processes are particle size dependent but not always linearly so. Virtually all processes in vivo involve an element of probability. Particle size and properties can be controlled to a large extent, but stochastic processes cannot by definition. Progress has been made, but the quantitative delivery of a nanocarrier to defined sites in tumors is neither inevitable nor yet predictable.

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.

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.

Diffusion of tagged particle in an exclusion process

We study the diffusion of tagged hard-core interacting Brownian point particles under the influence of an external force field in one dimension. Using the Jepsen line we map this many-particle problem onto a single particle one. We obtain general equations for the distribution and the mean-square displacement <(xT)2> of the tagged center particle valid for rather general external force fields and initial conditions. The case of symmetric distribution of initial conditions around the initial position of the tagged particle on x=0 and symmetric potential fields V(x)=V(-x) yields zero drift <xT>=0 and is investigated in detail. We find <(xT)2>=R(1-R)/2Nr2 where 2N is the (large) number of particles in the system. R is a single particle reflection coefficient, i.e., the probability that a particle free of collisions starts on x0>0 and remains in x>0 while r is the probability density of noninteracting particles on the origin. We show that this equation is related to the mathematical theory of order statistics and it can be used to find <(xT)2> even when the motion between collision events is not Brownian (e.g., it might be ballistic or anomalous diffusion). As an example we derive the Percus relation for non-Gaussian diffusion. A wide range of physical behaviors emerge which are very different than the classical single file subdiffusion <(xT)2> approximately t1/2 found for uniformly distributed particles in an infinite space and in the absence of force fields.