Crowding effects in non-equilibrium transport through nano-channels

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.

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.

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.

Importin-β modulates the permeability of the nuclear pore complex in a Ran-dependent manner

Soluble karyopherins of the importin-β (impβ) family use RanGTP to transport cargos directionally through the nuclear pore complex (NPC). Whether impβ or RanGTP regulate the permeability of the NPC itself has been unknown. In this study, we identify a stable pool of impβ at the NPC. A subpopulation of this pool is rapidly turned-over by RanGTP, likely at Nup153. Impβ, but not transportin-1 (TRN1), alters the pore's permeability in a Ran-dependent manner, suggesting that impβ is a functional component of the NPC. Upon reduction of Nup153 levels, inert cargos more readily equilibrate across the NPC yet active transport is impaired. When purified impβ or TRN1 are mixed with Nup153 in vitro, higher-order, multivalent complexes form. RanGTP dissolves the impβ•Nup153 complexes but not those of TRN1•Nup153. We propose that impβ and Nup153 interact at the NPC's nuclear face to form a Ran-regulated mesh that modulates NPC permeability.

DOI:http://dx.doi.org/10.7554/eLife.04052.001

In our cells, genetic material is contained within the nucleus, which is separated from the rest of the cell by a double-layered membrane called the nuclear envelope. Within this membrane there are pores that allow proteins and other molecules to enter and exit the nucleus.

Small molecules can pass through these pores unaided, which is known as ‘passive’ transport. However, larger cargos need help from transport receptor proteins in a process called ‘active’ transport. Large cargos bind to transport receptors, such as importin-β, in the cytoplasm and are then guided through the pore. Once the cargo and importin-β are inside the nucleus, a protein called RanGTP binds to importin-β to release the cargo.

It is thought that importin-β and RanGTP are only important for the active transport of cargo. Here, Lowe et al. studied how importin-β interacts with the pore. The experiments show that in the absence of RanGTP, importin-β accumulates inside the pore and binds to a protein called Nup153, which is part of the complex of proteins that makes up the pore. However, when RanGTP is present, some of the importin-β is displaced from Nup153 and leaves the pore, which makes it easier for cargo to pass through.

Further experiments show that when Nup153 and importin-β are mixed, they associate into a gel-like material that can be ‘melted’ by RanGTP. Lowe et al. propose a model for how RanGTP may control the flow of cargo through the nuclear pore by affecting the binding of importin-β to Nup153. Lowe et al.'s findings suggest that passive and active transport of cargo across the nuclear pore are fundamentally connected and suggest that RanGTP provides the cell with an additional layer of control over nucleocytoplasmic transport.

DOI:http://dx.doi.org/10.7554/eLife.04052.002

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.

Cooperative transport in nanochannels

Channel transport of different species of particles is viewed usually only in terms of competition and selectivity. In this paper we show that transport of one species may be promoted by the presence of another and that both may even mutually cooperate. We investigate a discretized Markovian model of nanochannel transport via in-channel sites, allowing for the simultaneous transport of several different species of particles; interaction between transported particles is included via the condition of single occupancy on a channel site. By numerically solving the model exactly, particularly an analysis of situations of crowding in the channel is possible and we observe three situations: mutual cooperation, promotion of one species at the cost of the other, and mutual competition. The physical situation has a strong nonequilibrium character as Onsager's relations on coupled flows do not hold.