Kinetic cooperativity resolves bidirectional clogging within the nuclear pore complex

Abstract

As the main gatekeeper of the nucleocytoplasmic transport in eukaryotic cells, the nuclear pore complex (NPC) faces the daunting task of facilitating the bidirectional transport of a high volume of macromolecular cargoes while ensuring the selectivity, speed, and efficiency of this process. The competition between opposing nuclear import and export fluxes passing through the same channel is expected to pose a major challenge to transport efficiency. It has been suggested that phase separation-like radial segregation of import and export fluxes within the assembly of intrinsically disordered proteins that line the NPC pore could be a mechanism for ensuring efficient bidirectional transport. We examine the impact of radial segregation on the efficiency of bidirectional transport through the NPC using a coarse-grained computational model of the NPC. We find little evidence that radial segregation improves transport efficiency. By contrast, surprisingly, we find that NTR crowding may enhance rather than impair the efficiency of bidirectional transport although it decreases the available space in the pore. We identify mechanisms of this novel crowding-induced transport cooperativity through the self-regulation of cargo density and flux in the pore. These findings explain how the functional architecture of the NPC resolves the problem of efficient bidirectional transport, and provide inspiration for the alleviation of clogging in artificial selective nanopores.

Significance

The nuclear pore complex (NPC) facilitates a large volume of bidirectional macromolecular traffic between the nucleus and the cytoplasm. How this bidirectional traffic—that proceeds through the channel of the NPC—is regulated to avoid “traffic jams” and clogging is poorly understood. In this work, we investigate the potential mechanisms for improving the efficiency of bidirectional transport. We find no evidence that a commonly hypothesized mechanism—radial segregation of the import and export fluxes—improves transport efficiency. Instead, surprisingly, we find that nonlinear interactions between import and export fluxes naturally lead to cooperative crowding-alleviating mechanisms which render bidirectional transport largely robust against clogging.

Introduction

Compartmentalization of genetic material within the nucleus in eukaryotic cells necessitates the selective exchange of a large number of biomolecules involved in protein synthesis, gene regulation, and other cellular processes between the nucleus and cytoplasm (1). The nuclear pore complex (NPC) is a large macromolecular transporter embedded across the nuclear envelope, which facilitates this biomolecular exchange. It is therefore faced with the challenge of transporting a multitude of different cargoes in both directions, while ensuring that this transport is selective, efficient, and rapid (1). Disruptions in NPC function and nucleocytoplasmic transport have been linked to multiple cancers and neurodegenerative diseases (2,3,4,5,6,7).

The NPC is able to recognize and transport many different cargoes of various sizes through a unique transport mechanism. Structural nucleoporin proteins stabilize a roughly hourglass-shaped passage of a diameter between 30 and 50 nm (1,8), which connects the nucleus and the cytoplasm. This central channel is filled with a “nanomaterial” formed by an assembly of multiple copies of intrinsically disordered proteins known as phenylalanine and glycine (FG) nucleoporins (or FG nups) due to their enrichment of FG repeats (8,9). In terms of their material properties, the intrinsically disordered FG nups behave in many aspects as flexible polymer chains under the action of thermal fluctuations constrained by crowding and the cohesive intra- and interchain transient cross-links arising from FG-FG and other interactions (1,10,11,12,13,14,15,16). The “nanomaterial” formed by the FG assembly acts as a barrier that restricts the diffusion of nonspecific macromolecules between the nucleus and cytoplasm in a size-dependent way (17,18,19,20). FG nup material also provides a template for binding of a class of proteins—the nuclear transport receptors (NTRs)—predominantly through the transient binding of FG repeats to hydrophobic grooves on the NTR surfaces (1,9,11,21). NTRs also recognize nuclear localization or nuclear export sequences present on the proteins that are designated for nuclear import or export, respectively. Upon binding to a cargo molecule, the resulting NTR-cargo complex is able to partition into the FG nup material and travel via a diffusion-like mechanism between the nuclear and cytoplasmic sides facilitated by multivalent weak binding interactions with FG nups (1,10,22,23,24,25). The ability of the NTR-mediated transport cycle to directionally transport cargoes against their concentration gradients is maintained through an energy consuming out-of-equilibrium nucleocytoplasmic gradient of the GTPase Ran—the RanGTP form is primarily found in the nucleus as the cytoplasm contains factors that hydrolyze RanGTP into RanGDP. In the import cycle, importins (NTRs responsible for transporting cargoes from the cytoplasm to the nucleus) release their cargoes upon binding to RanGTP in the nucleus. Cargoes released from the NTRs are unable to travel through the FG nup assembly, and are therefore sequestered in their destination compartment. The resulting NTR-RanGTP complexes can translocate back to the cytoplasm where RanGTP is hydrolyzed, causing the complex to break apart. The resulting RanGDP is transported back into the nucleus by the specialized NTR, NTF2, where it is converted back to the RanGTP form by the nuclear factor RanGEF (1,10,26,27). A similar cycle maintains the export flux of cargoes from the nucleus to the cytoplasm, with an important difference that exportins (NTRs responsible for transporting cargoes from the nucleus to the cytoplasm) bind RanGTP in together with their export cargo.

This transport mechanism has multiple advantages: it allows a single transporter to selectively transport a multitude of different cargoes of different sizes and biophysical properties, and transport many cargoes simultaneously in both directions. However, it also raises several questions. With many cargoes simultaneously being transported in both directions through the same channel within limited space, how does the NPC regulate the bidirectional traffic flow to avoid “traffic jams”? Identifying and understanding the mechanisms through which the NPC regulates traffic is a key step toward decoding the functional design of this nanoscale “machine,” which could advance therapeutic treatments of NPC-related diseases, and improve the efficiency of NPC-inspired artificial nanopores and nanomaterials used in molecular filtering and transport (1,28). Our previous work (29) examined the question of how clogging is prevented in unidirectional transport through the NPC channel such as nuclear import. Here, we examine the full and more restrictive problem of the mechanisms of clogging alleviation in simultaneous bidirectional transport.

These mechanisms are underlaid by the material properties of the FG nup assembly within the NPC pore. The FG nup material comprises multiple copies of approximately 10–15 types of different FG nups, intercalated by 100–300 copies of free and cargo-loaded NTRs translocating in both directions (1). Inter- and intrachain interactions as well as NTR-FG nup interaction can produce spatial inhomogeneities of the FG nup assembly in both the radial and axial directions within the FG assembly, possibly through a phase separation-like mechanism (14,30,31,32). Therefore, it has been suggested that different NTRs and NTR-cargo complexes may utilize mostly distinct regions within the channel through nanoscale phase separation and kinetic segregation mechanisms, leading to potentially decoupled import and export pathways (33,34,35). This offers a potential solution to the problem of bidirectional transport by reducing the problem to two largely decoupled unidirectional transport processes, which have been shown to be very robust to NTR crowding within the NPC despite the resulting significant reduction of the available space in the pore (29).

However, the effects of this potential nanoscale spatial segregation on transport dynamics are not yet well understood. Interactions between NTRs and FG nups modify the spatial density distribution of the FG assembly, which in turn affects the localization of different NTRs-cargo complexes. This network of collective effects within the FG nup material can give rise to nontrivial nonlinear effects in how the presence of one species within the pore impact the translocation dynamics of other species (36). While there has been increasing attention focused on ascertaining the spatial distribution of FG nups and NTRs in the pore (34,35,37), few efforts have been directed at determining the effects of segregation of import and export actually on the bidirectional transport efficiency.

The NPC transports many different cargoes carried by many different NTRs resulting in a potentially very complex multispecies kinetic environment within the pore (1). To understand the basic principles of species segregation and pathway separation in the NPC channel, we use a minimal coarse-grained model, reported and validated in (29), with a minimal number of variables and parameters that capture the essential features of the multispecies bidirectional flux through the pore. We find that, while NTRs of different sizes or different binding affinities to FG nups naturally segregate within the spatially inhomogeneous FG material, there is no clear correlation between the extent of radial segregation and the transport efficiency.

As we found little evidence to suggest that radial segregation offered a solution to the problem of bidirectional transport in the NPC, we then investigated more broadly the risk of clogging for the bidirectional NPC transport. Surprisingly, we found that, despite high level of pore occupancy by NTRs that decrease available space, significant attenuation of bidirectional transport occurred only at very high levels of bidirectional NTR crowding. Furthermore, at lower—and biologically relevant—NTR concentrations, we instead found a surprising cooperative regime where the transport in both directions could be enhanced by NTR crowding.

The paper is structured as follows: in the methods we introduce the model, and use it to examine the radial segregation of import and export fluxes and investigate how this radial segregation affects bidirectional transport efficiency in the pore in the effect of radial segregation on the bidirectional transport efficiency. Crowding-alleviating mechanisms in bidirectional transport examines the extent of clogging in bidirectional transport and analyzes the factors responsible for the surprising cooperativity that enhances flux at biologically relevant NTR concentrations. We summarize the findings and discuss their implication for NPC transport and bioinspired nanomaterial design in the discussion.

Methods

Based on previous work by us and others demonstrating that a small number of basic principles and variables are sufficient to capture the salient properties of the FG nups and their assemblies in the NPC and NPC-like in vitro mimics (14,18,25,33,38,39,40,41,42), we use a coarse-grained computational model of the NPC with a minimal number of features. The model comprises a pore containing grafted polymer-like chains, representing the disordered regions of FG nups, and spherical particles of appropriate size representing the NTRs (18,29,42,43,44). This minimal model subsumes the discrete FG repeats and the discrete binding sites on the NTRs into approximate average interaction parameters distributed over the particle surface, which has been shown to agree well with more detailed patchy models in their ability to describe the thermodynamic and kinetic behavior of individual NTR-FG nup interactions (25). This model of the NPC was previously successfully used by us in (29) to explain several experimental puzzles of crowded unidirectional NPC transport, and is described in detail therein.

In brief, each FG nup is represented by a chain of 200 monomers of a diameter of 1 nm each representing approximately four amino acids, corresponding to the typical size of an FG repeat. NTRs are modeled as spheres, with importin- and exportin-like NTRs having diameters of 6 nm and the corresponding volume $v=113$ nm³, and NTF2-like NTRs having diameters of 4 nm and volume $v=33.5$ nm³. While the material within the NPC is substantially more complex, here we include only these three main classes of NTRs as a minimal description of the NPC channel, with the aim to understand the principles of multispecies transport dynamics competition (see also (1)). In the minimal model used here the NTR-like particles model both cargo-bound and the unbound forms of the NTRs in the NPC.

As in (29), the interactions between all particles are modeled by a shifted Lennard-Jones-type potential (45):

$$U\left(r\right)=\{\begin{array}{c}{ϵ}_{\text{rep}}\left[{\left(\frac{d_{\text{ave}}}{r}\right)}^{12}-2{\left(\frac{d_{\text{ave}}}{r}\right)}^6\right]+{ϵ}_{\text{rep}}-{ϵ}_{\text{attr}},\\ r<d_{\text{ave}}\\ {ϵ}_{\text{attr}}\left[{\left(\frac{b}{r-r_0}\right)}^{12}-2{\left(\frac{b}{r-r_0}\right)}^6\right],r>d_{\text{ave}}\end{array}$$ (Equation 1)

where b is the monomer diameter, $d_{\text{ave}}$ is the average diameter of the two interacting particles, $r_0=\frac{d-b}{2}$ (where d is the NTR diameter), and r is the distance between the particles. The shift by $r_0$ accounts for the difference in particle sizes, and keeps the range of interaction independent of ${ϵ}_{\text{attr}}$. The strengths of the repulsive and attractive interactions are given, respectively, by ${ϵ}_{\text{rep}}$ and ${ϵ}_{\text{attr}}$, ${ϵ}_{\text{attr}}>0$ for cohesive interactions between monomers and monomers (${ϵ}_{\text{coh}}$), and the binding interactions between monomers and NTRs (${ϵ}_{\text{NTR}}$), while ${ϵ}_{\text{attr}}=0$ for the purely repulsive NTR-NTR interactions.

A finitely extensible nonlinear elastic potential, $U=-\frac{k}{2}{r}_{\max }^2\ln \left[1-{\left(\frac{r}{r_{\max }^2}\right)}^2\right]$, is applied to the neighboring monomers on each FG nup chain to represent the bonds forming the backbone of each polymer. We take the standard values in the field for $r_{\max }$, the maximum extension of the chain as $1.5b$ and k, the stiffness of the bond as 30 $\frac{k_{B}T}{{\text{nm}}^2}$, as described in (29,43,46,47).

As in (29), there are two classes of FG nups within our pore model: the six central FG nups of each spoke were modeled with a stronger cohesiveness (${ϵ}_{\text{coh}}=0.5k_{B}T$) than the two sets of four FG nups located closer to the channel exits (${ϵ}_{\text{coh}}=0.3k_{B}T$). This takes into account experimental observations, which indicate that the types of FG nups grafted near the center of the NPC tend to adopt collapsed-coil configurations in solution (corresponding to higher cohesion), while the types of FG nups located near the openings of the pore tend to adopt relaxed or extended-coil conformations (corresponding to lower cohesion) (9,12,13,30,48,49,50). NTRs interact attractively with FG monomers and, as detailed at the end of this section and in Fig. 2, the values of ${ϵ}_{\text{NTR}}$ are used in this work to tune the degree of segregation within the pore.

Figure 2.

Illustration of segregation simulation setups. (A) Setup for a respresentative set of simulations of bidirectional competition between same-sized importin-like and exportin-like NTRs. The red “probe” import NTR has a high affinity to the FG nups, while the affinity of the yellow “tuning” export NTR increases from low to high (low: ${ϵ}_{\text{NTR}}=0.85$$k_{B}T$; medium: ${ϵ}_{\text{NTR}}=0.95$$k_{B}T$; and high: ${ϵ}_{\text{NTR}}=1.0$$k_{B}T$). Additional simulation sets with other NTR pairs are reported in Fig. S1 (B) Competition of the importin-like (probe) NTR, with the interaction strength fixed at ${ϵ}_{\text{NTR}}=0.95$$k_{B}T$ (red), and a small (tuning) NTF2-like NTR (blue). The interaction strength of the tuning NTR varied from ${ϵ}_{\text{NTR}}=0.835$$k_{B}T$ to ${ϵ}_{\text{NTR}}=1.2$$k_{B}T$. To see this figure in color, go online.

As in (29), we use semi-periodic boundary conditions at the nuclear and the cytoplasmic ends of the simulation box to generate steady-state concentration gradient fluxes of NTRs through the NPC that mimic the steady-state nucleocytoplasmic fluxes without attempting to explicitly model the active Ran cycle. In this setup, NTRs corresponding to nuclear import are absorbed at the nuclear end of the simulation box and “recycled” into the cytoplasmic box, while the NTR corresponding to nuclear export is absorbed at the cytoplasmic end of the simulation box and “recycled” into the nuclear box (Fig. 1 C) This setup maintains constant bidirectional steady-state fluxes of NTRs between the cytoplasm and nucleus. The NTR flux through the box is calculated from the simulations as $J=1/⟨\Delta t⟩=M/T$, where $⟨\Delta t⟩$ is the average particle interarrival time, and M is the number of particles passing through the box boundary during the measurement time T. We stop the data collection when the difference between the fluxes estimated at T and $T/2$ is less than 1/ms or less than 1 SEM of the measured flux.

Figure 1.

Simulation setup for investigating the efficiency of bidirectional transport. (A) Sample snapshot of a simulation. The rigid structures of the pore are modeled by stationary particles (white), FG nups are modeled as polymers with lower cohesion in the pore peripheries (purple), and higher cohesion in the center (cyan). NTRs are modeled as spheres (red and yellow), which interact attractively with FG nups. (B) In the coarse-grained simulation model, the properties of each NTR are determined by two parameters: its size and its interaction strength with the monomers representing FG nups. (C) Illustration of the simulation box (not to scale) including the cytoplasmic and the nuclear compartments and the boundary conditions for simulating steady-state bidirectional fluxes. The import species (red) is absorbed at the nuclear end of the simulation box and “recycled” back into the cytoplasmic compartment, while the export species (yellow) is absorbed at the cytoplasmic end of the simulation box and “recycled” back into the nuclear compartment. The nuclear envelope extends from $z=-L$ to $z=L$, the FG assembly is contained within the region $z\in \left[-A,A\right]$, and R is the radius of the NTR undergoing translocation. As a measure of transport efficiency, we use the translocation probability (for import) as the proportion of trajectories that leaves the region containing the FG nup assembly on the nuclear side (red arrow) out of all the trajectories that enter fully into the NPC from the cytoplasmic side (red and gray arrows). For export, the same definitions hold, with directionalities reversed. (D) Schematic illustration of radial segregation of import and export pathways that has been hypothesize to be a mechanism for alleviating clogging in of bidirectional transport. To see this figure in color, go online.

The model was implemented using the molecular dynamics package LAMMPS (51) using Brownian dynamics and an implicit solvent. As in (29), the FG nups were allowed to relax to equilibrium in the pore before NTRs were added, and the dynamic data collection began once the average number of NTRs inside the pore stabilized to steady state. Timescales in the simulation were determined by assigning the viscosity μ in the Stokes-Einstein equation, $D=\frac{k_{B}T}{6\pi \mu R}$ for determining the diffusion coefficients of the monomers and NTRs in the implicit solvent, as the typical viscosity of the cellular cytoplasm, taken to be 5cP (52). With this choice, one timestep of our simulation corresponds to $1.37\times {10}^{-7}$ ms.

Results

The effect of radial segregation on the bidirectional transport efficiency

Radial inhomogeneity of the FG assembly leads naturally to NTR segregation

To tune the extent of radial segregation between the two NTR species within the pore, we vary their relative interaction strengths with FG nups (${ϵ}_{\text{NTR}}$).

To this end, we have designed our simulations so that, of the two types of NTRs undergoing bidirectional transport, one type of NTR is the “probe” NTR, while the other type is the “tuning” NTR. Transport efficiency measurements are taken of the probe NTR, whose interaction strength with FG nups is held constant across each set of simulations, while the interaction strength of the tuning NTR is varied across simulations to vary the degree of radial segregation between the two types of NTRs. To isolate the effects of the pathway segregation on transport independent of the effects of NTR crowding that are known to affect transport efficiency directly (29,53), the numbers of both probe and tuning NTRs within the pore are also maintained constant across all simulations. Therefore, from the perspective of the probe NTR, each pore in a set of simulations is identical except for the extent of spatial segregation with the tuning NTR.

The number of NTRs within the pore is maintained to be 150 for importin- and exportin-sized NTRs, and 200 for NTF2-sized NTRs, informed by the experimental measurements (37). Thus, between 0.42 and 0.54 of the total volume inside the pore is occupied by NTRs and FG nups, consistent with high levels of crowding within the NPC in biological settings (for comparison, the random close packing volume fraction is around 0.6).

Maintaining the number of NTRs within the pore constant is achieved by varying the concentrations of NTRs. Therefore, in investigating the effects of pathway segregation, instead of quantifying transport efficiencies of different species by comparing transport fluxes (which depend on the NTR concentration outside the pore), we opted for the single-molecule translocation probability as a measure of the transport efficiency because it is independent of the NTR concentration outside the pore, and sensitive to jamming inside the NPC channel.

We first investigate spatial segregation in the case of same-sized importin-like and exportin-like NTRs that have differing NTR-FG nup interaction strengths as shown in Fig. 2 A. To sample various competition scenarios, the interaction strengths between NTRs and FG nups is varied from low to high, covering a range of interaction strengths. As our importin- and exportin-like NTRs are otherwise identical, this effectively results in three sets of simulations where, in each set, the interaction strength between the probe NTR and FG nups is kept constant, while the interaction strength between the tuning NTR and FG nups is varied (see also Fig. S1).

To investigate the effect of the NTR size on the spatial segregation, we explored the competition between the importin- and NTF2-like NTRs of different sizes (although in reality both importin and NTF2 travel in both directions through the NPC) (shown in Fig. 2 B). In each set of simulations, the interaction strength of the probe NTR is held constant, while the interaction strength of the tuning NTR is varied.

Fig. 3 demonstrates that radial segregation can be induced by the differences in the FG nup binding affinity between the probe and the tuning NTRs (see also Fig. S2). In the absence of NTRs, the FG nup material in our model has an inhomogeneous radial density profile of the torodial shape (Fig. S15) with a density dip along the pore axis consistent with previous observations (30,31,54). Collective effects induced by the NTRs shift this density distribution toward one with the peak at the center of the pore (see Fig. S5). Consequently, segregation arises as the NTR species with the higher effective affinity toward FG nups (due to differences in interaction strength and size) outcompete the species for the regions of high FG nup density within the pore. However, we note that it is possible that, under different conditions of high density FG nups, all NTRs might prefer to reside in the low density regions of the NPC, resulting in different segregation effects, which will be studied elsewhere.

Figure 3.

Spatial segregation in multispecies competition. (A) Normalized radial density profiles measured from the set of simulations with importin-like high-affinity “probe” NTRs (red) and exportin-like “tuning” NTRs (yellow) of variable FG nup affinity (see Fig. 2A). The radial densities of NTRs are calculated by averaging over all z-slices of the pore, and normalized such that the integral of the density over the cross-sectional area is 1.0. Radial smoothing has been applied to the density maps for better visualization of segregation to the center/peripheries (see also Fig. S3). The numbers of probe and tuning NTRs are kept constant at $150\pm 1$ particles in the pore for all parameter values to approximate biological conditions (see text). (B) Normalized radial density profiles measured from the simulation of differently sized NTRs between the probe importin-like NTR (red) and the tuning NTF2-like NTR (blue). $1-\eta$ defines the extent of radial segregation in each of the cases shown (see text). The radial densities of NTRs are calculated by averaging over all z-slices of the pore, and normalized such that the integral of the density over the cross-sectional area is 1.0. Fig. S2 shows the results for the other combinations of importin-like NTRs with exportin-like NTRs, and importin-like NTRs with NTF2-like NTRs. To see this figure in color, go online.

We quantify the extent of radial segregation between the two NTRs sharing the pore via the integral of the overlap of their probability density distributions (shown in Fig. 3), denoted as η; the extent of segregation is defined as $1-\eta$.

Transport efficiency is not correlated with the extent of radial segregation

To isolate the effect of segregation on the transport efficiency, we fix the number of NTRs in the pore to a biologically informed number (see radial inhomogeneity of the FG assembly leads naturally to NTR segregation (27,37)), and focus on the single-molecule quantity of translocation probability as a measure for transport efficiency as described in (29).

We define the translocation probability as the fraction of successful translocation attempts out of all (successful and abortive) translocation attempts. As shown in Fig. 1 C, we consider a translocation attempt to begin once an NTR has entered completely into the pore on the cis side (i.e., the cytoplasmic side for import, and the nuclear side for export), i.e., once the center of the NTR reached a position of one NTR radius within the nuclear envelope. The translocation attempt ends once the NTR reaches the edge of the region containing the FG assembly (defined as the region containing 99% of FG nup monomers) at $z=A$ or $z=-A$. If the translocation attempt ended on the cis side, the translocation attempt is counted as abortive, while if the translocation attempt ended on the trans side, the translocation attempt is counted as successful.

The results are presented in Fig. 4, showing the lack of correlation of the transport efficiency and the degree of spatial segregation. Both in the case of competition of importin-like with exportin-like NTRs, and the competition of importin-like with NTF2-like NTRs, increased segregation of the two NTRs is not accompanied by increased translocation probabilities. Therefore, in our minimal model, radial segregation alone is not sufficient to serve as a mechanism for increasing bidirectional transport efficiency in crowded conditions.

Figure 4.

Translocation probabilities as a function of the extent of segregation across all simulations. The three sets of simulations of importin-like with exportin-like NTRs are labeled as “High affinity,” “Med affinity,” and “Low affinity” according to the binding affinity of the probe NTR in those simulations (see Fig. 2). The two sets of importin-like with NTF2-like NTR simulations are labeled by the “probe” NTR, followed by the “tuning” NTR in parentheses. The extent of segregation is defined according to the extent of nonoverlap of the radial density distributions (see text). Translocation probabilities appear uncorrelated with the extent of segregation of the two types of NTRs in the pore. To see this figure in color, go online.

Crowding-alleviating mechanisms in bidirectional transport

As our results suggested that radial segregation alone is unlikely to be a mechanism for alleviating potential clogging resulting from bidirectional competition, we next asked what effects might be responsible for the presence or absence of clogging.

We first quantified the impact of bidirectional fluxes of different NTRs on each other under different NTR concentrations outside the pore. To focus on the effects of bidirectional interference relative to the effects of NTR crowding per se, we compared the NTR fluxes in bidirectional conditions to the unidirectional fluxes through the NPC in the absence of competition (Eq. 2). To this end, we define the bidirectional flux attenuation as

$$\text{bidirectional}\text{flux}\text{attenuation}=1-\frac{J_{\text{bi},1}\left(c_1\right)+J_{\text{bi},2}\left(c_2\right)}{J_{\text{uni},1}\left(c_1\right)+J_{\text{uni},2}\left(c_2\right)}$$ (Equation 2)

where $c_{\text{bi}}$ and $J_{\text{bi}}$ are the NTR concentration outside the pore and flux through the pore, respectively, in the bidirectional conditions, and $J_{\text{uni}}\left(c_i\right)$ is the benchmark unidirectional flux through for the same NTR concentration outside the pore (see Fig. S7). Subscripts 1 and 2 refer to the two NTRs corresponding to import and export. The NTR concentration outside the pore is defined to be the average concentration of NTRs measured at steady state in the cytoplasmic compartment (for import NTRs) or the nuclear compartment (for export NTRs) in the volume beyond the extent of the FG assembly (defined as the area containing 99% of the FG monomers) (see Fig. 1 C). The number of NTRs in the whole simulation box is varied between different simulations to produce different steady-state concentrations.

In our minimal setup, we ran simulations in bidirectional transport conditions using two types of importin/exportin-like NTRs—a low-affinity one with ${ϵ}_{\text{NTR}}=0.85$ $k_{B}T$ (which we refer to as Kap-weak, where Kap is short for the karyopherin NTR family and refers to the larger-sized importin/exportin-like NTRs, inspired by yeast terminology), and a high-affinity one with ${ϵ}_{\text{NTR}}=0.95$ $k_{B}T$ (which we refer to as Kap-strong)—and one type of NTF2-like NTR, with ${ϵ}_{\text{NTR}}=1.0$ $k_{B}T$.

We ran four families of bidirectional transport simulations: two where the import and export NTRs are identical (Kap-weak with Kap-weak and Kap-strong with Kap-strong), and two where the imported and exported NTRs are different (Kap-weak with Kap-strong and Kap-strong with NTF2).

Fig. 5 shows the bidirectional flux attenuation for each of these cases represented as heatmaps. For technical reasons, as our simulations use a fixed number of particles in the whole simulation box, concentrations of NTRs in the cytoplasmic/nuclear compartments cannot be independently and separately fixed but depend on the interaction parameters. Thus, instead of a regular grid-based heatmap, we have used Voronoi tessellation between simulation data points that sample the concentration space. We also note that, as the bidirectional flux attenuation is a symmetric quantity, in the cases where the import and export NTRs are identical, the data points on the upper and lower diagonal come from the same simulation, and are repeated for ease of visualization.

Figure 5.

Bidirectional flux attenuation. (A) Competition of Kap-weak with Kap-weak (the two directions are differentiated as “A” and “B”) as a function of their concentrations outside the pore. (B) Competition of Kap-weak with Kap-strong as a function of their concentrations outside the pore. (C) Competition of Kap-strong with Kap-strong as a function of their concentrations outside the pore. (D) Competition of Kap-strong with NTF2 as a function of their concentrations outside the pore. In all panels the results are from Eq. 2. We note that, while the values for flux attenuation may drop below $-1$, the color map is cut off such that positive values are shown in red and negative values in blue. To see this figure in color, go online.

We observe that, in all cases, within biologically relevant concentration values ($<100$ $\mu \mathrm{M}$), flux attenuation appears to be minimal, although at higher concentrations it can become significant. We observe the highest attenuation within the biological concentration range for the competition of Kap-strong with NTF2, largely resulting from the attenuation of NTF2 by Kap-strong (see Fig. S10, E and F), which is consistent with previous observations of Karyopherin-β1 excluding NTF2 from FG nup assemblies (55). These results suggest the possibility that bidirectional NTR crowding may be less of a problem than previously thought, and the NPC may not require extra mechanisms to enhance bidirectional transport efficiency.

Combination of kinetic cooperativity and quasiequilibrium inside the pore at low NTR concentrations result in bidirectional flux enhancement

Rather surprisingly, we further observed instances of flux enhancement, where the bidirectional flux attenuation becomes negative (blue regions in Fig. 5, A and C). In these regimes, the total bidirectional flux $J_{\text{bi},1}\left(c_1\right)+J_{\text{bi},2}\left(c_2\right)$ is greater than the sum of the individual unidirectional particle fluxes if they were to occur separately through two separate NPCs without bidirectional competition, $J_{\text{uni},1}\left(c_1\right)+J_{\text{uni},2}\left(c_2\right)$. The highest bidirectional flux enhancement occurs during the competition of Kap-strong with Kap-strong (note that coincidentally in this case there is no radial segregation as import and export NTRs have the same size and the same FG nup binding affinity). Notably, this is in a direct contrast to the expectation that decoupling of import and export would result in the maximal efficiency of bidirectional transport.

These counterintuitive results can be traced to the highly surprising superlinear dependence of the unidirectional flux $J_{\text{uni}}\left(c\right)$ on the concentration c, as illustrated in Fig. 6 on the example of NTRs that interact strongly with the FG nups (Kap-strong). Fig. 6 A shows that, in the absence of bidirectional competition, unidirectional flux increases superlinearly with the NTR concentration outside the NPC, so that $J_{\text{uni}}\left(c_1+c_2\right)>J_{\text{uni}}\left(c_1\right)+J_{\text{uni}}\left(c_2\right)$, which we term kinetic cooperativity.

Figure 6.

Kinetic cooperativity explains the flux enhancement of high-affinity NTRs. (A) Unidirectional flux versus concentration for high-affinity Kap-strong (black) is superlinear, such that, for concentrations $c_1$ and $c_2$, $J_{\text{uni}}\left(c_1\right)+J_{\text{uni}}\left(c_2\right)<J_{\text{uni}}\left(c_1+c_2\right)$. The red lines illustrate this disparity for various choices of $c_1$ and $c_2$. (B) In the definition of bidirectional flux attenuation (Eq. 2), the linear sum $J_{\text{uni}}\left(c_1\right)+J_{\text{uni}}\left(c_2\right)$ was used to compare against the total bidirectional flux $J_{\text{bi},1}\left(c_1\right)+J_{\text{bi},2}\left(c_2\right)$, and flux enhancement was observed. Here, if we account for kinetic cooperativity by comparing the total bidirectional flux with $J_{\text{uni}}\left(c_1+c_2\right)$, we observe that the total bidirectional flux is always lower. To see this figure in color, go online.

Furthermore, the bidirectional flux through the pore possesses another unusual feature. As shown in Fig. 6 B, at low concentrations $J_{\text{bi}}\left(c_1\right)+J_{\text{bi}}\left(c_2\right)\approx J_{\text{uni}}\left(c_1+c_2\right)$ indicating that the overall degree of NTR crowding rather than the asymmetry of the fluxes affects the transport. We refer to this regime as quasiequilibrium; in this regime the NTR densities inside the pore are minimally affected by the asymmetry of concentrations outside. It is important to emphasize that quasiequilibrium does not imply a lack of interactions between the NTRs in the pore but occurs because at low concentrations the NTRs are primarily crowded at the center of the pore at, and the density distribution is minimally affected by flux directionality. In fact, NTR-NTR crowding is substantial in the quasiequilbirum regime as the volume fraction occupied within the pore is $\sim 40$% already at concentrations lower than 100 $\mu \mathrm{M}$ (see Fig. S11), and crowding-induced interactions can also be seen in the substantial change in the individual NTR transport times in the low-density regime, as shown in Fig. S12.

In this quasiequilibrium regime, kinetic cooperativity translates into the bidirectional flux enhancement of the total bidirectional flux: $J_{\text{bi}}\left(c_1\right)+J_{\text{uni}}\left(c_2\right)\approx J_{\text{uni}}\left(c_1+c_2\right)>J_{\text{uni}}\left(c_1\right)+J_{\text{uni}}\left(c_2\right)$ such that the bidirectional flux attenuation, $1-\frac{J_{\text{bi},1}\left(c_1\right)+J_{\text{bi},2}\left(c_2\right)}{J_{\text{uni},1}\left(c_1\right)+J_{\text{uni},2}\left(c_2\right)}$, is negative.

By contrast, at high concentrations, quasiequilibrium breaks down as NTRs progressively occupy not only the center of the pore but also its peripheries, so that $J_{\text{bi},1}\left(c_1\right)+J_{\text{bi},2}\left(c_2\right)<J_{\text{uni}}\left(c_1\right)+J_{\text{uni}}\left(c_2\right)<J_{\text{uni}}\left(c_1+c_2\right)$ and the outcome is always attenuation instead of enhancement. Therefore, at high NTR concentrations, bidirectional fluxes obstruct each other, and in this regime bidirectional transport is significantly less efficient than unidirectional transport. In bidirectional flux attenuation at high concentrations and is linked to the high asymmetry in nonequilibrium NTR density profiles we discuss the mechanistic underpinnings of these results.

Kinetic cooperativity is a consequence of competition-induced release

In this section, we show that kinetic cooperativity can be traced to the crowding-alleviating effect of the competition-induced NTR release that we discovered in an earlier investigation of unidirectional transport through the NPC (29). In brief, competition-induced release refers to the speed up of the NTR release from the pore with an increase in NTR crowding. This speed up can be understood by viewing the translocation of NTRs through the NPC as diffusive transport in an effective potential (as described in (29) and supporting material, section 1), which subsumes all NTR-FG nup and NTR-NTR interactions and space constraints. Steric repulsion and competition for space between the NTRs reduces the depth of the effective potential, thus making the escape time—that to the first-order scales as $\sim \exp \left[U_{\min }\right]$ with the potential well depth $U_{\min }$ (29,56)—more rapid.

Furthermore, for sufficiently strong interactions, the flux through the pore can be heuristically approximated as $J\propto \frac{\mathcal{N}}{\tau }$, where $\mathcal{N}$ is the average number of NTRs within the pore, and τ is their average residence time (29). Because $\mathcal{N}$ saturates already at low concentrations ($\sim 10$ $\mu \mathrm{M}$), the unidirectional flux through the pore scales as $J\sim \exp \left[-U_{\min \left(c\right)}\right]$.

These heuristic arguments are verified numerically in Fig. S8, which shows the changes in the Boltzmann-Arrhenius factor of the maximal depth of the effective potential ($B\left(c\right)\equiv \exp \left[U_{\min \left(c\right)}\right]$) with the NTR concentration c for Kap-weak, Kap-strong, and NTF2-like NTRs. We observe that for Kap-weak and Kap-strong NTRs, the change in $B\left(c\right)$ with NTR concentration exhibits a nonlinearity that parallels the nonlinearity of the respective unidirectional flux-concentration curves (the kinetic cooperativity), as shown in Fig. S7 and discussed in the previous section (see also Fig. S9 A). Intuitively, this nonlinearity is present only for Kap-strong NTRs as the effective potential at low concentrations needs to be sufficiently deep for a crowding-sensitive transition to occur before the pore reaches occupancy and the effective potential flattens. Kinetic cooperativity is therefore closely linked to competition-induced release.

Bidirectional flux attenuation at high concentrations is linked to the high asymmetry in nonequilibrium NTR density profiles

Substantial bidirectional flux attenuation does not appear until very high concentrations outside the biologically relevant range ($<100$ $\mu \mathrm{M}$, Fig. 5) for all NTR combinations (except NTF2). As shown in Figs. 7 and S13, the extent of bidirectional flux attenuation correlates with the extent of axial asymmetry in the density distributions of the NTRs (see supporting material, section 2 for details).

Figure 7.

Total density asymmetries of NTRs undergoing bidirectional transport (see supporting material, section 2). (A) Density asymmetry for the bidirectional transport of Kap-weak with Kap-weak. (B) Density asymmetry for the bidirectional transport of Kap-strong with Kap-strong. We note the high degree of similarity with the corresponding bidirectional flux attenuation heatmaps shown in Fig. 5, A and C. See Fig. S13 for the remaining density asymmetries.

The transition from the quasiequilibrium regime at low NTR concentrations to the highly nonequilibrium asymmetric regime at high NTR concentrations can be explained by the biphasic and spatially inhomogeneous filling of the pore by the NTRs. At low concentrations, due to the attractive interactions between the NTRs and the FG nups with the FG assembly, most of the available NTRs are pulled into the center of the pore, resulting in minimal density asymmetry, even though their numbers there are sufficient for substantial NTR-NTR crowding and the resulting decrease in the depth of the effective potentials experienced by the NTRs. Much higher NTR concentrations are needed for a significant accumulation near the pore exits (see Fig. S14). The density asymmetry in the accumulation of NTRs on the trans side at the pore periphery results in a bias in the effective potential, favoring abortive translocations and resulting in the observed flux attenuation.

Discussion

With the importance of NPC transport emerging in drug design and the design of bioinspired nanotechnology pores and materials, building first-principles understanding of the biophysical underpinnings of the nucleocytoplasmic transport is critical. With multiple families of NTRs transporting a multitude of different cargoes between the nucleus and the cytoplasm in both directions, the interior of the NPC is highly crowded. This can potentially lead to the clogging of the NPC, slowing or disrupting nucleocytoplasmic transport, with serious implications for the entire cell. Therefore, prevention of NPC clogging during bidirectional transport is a key objective in the regulation of nucleocytoplasmic transport. Beyond its biological significance, understanding the mechanisms of crowded transport through complex multicomponent materials is central for clogging and fouling prevention in biomaterial design and development.

In this work, we used a minimal model of the NPC which encapsulates the main principles of NPC organization and function to investigate the effects of bidirectional competition on NPC transport efficiency. In the effect of radial segregation on the bidirectional transport efficiency, we investigate a long-held hypothesis within the field that decoupling the import and export fluxes through radial segregation inside the pore may provide a mechanism for alleviating clogging in bidirectional transport. While we observed that tunable radial segregation naturally arises through a nanoscale phase separation-like mechanism based on the differences in binding affinity and/or size between NTRs, we found little evidence to suggest that radial segregation was correlated with transport efficiency.

As radial segregation did not appear to offer a promising solution to the problem of bidirectional transport, in crowding-alleviating mechanisms in bidirectional transport, we further investigated under what conditions of bidirectional transport crowding would be expected to result in substantial clogging, and possible mechanisms of clogging suppression. We found that, while the attenuation of bidirectional flux could be significant at very high concentrations, in most cases it was not substantial at low concentrations ($<100$ $\mu \mathrm{M}$) encompassing the biologically relevant range. Furthermore, surprisingly, at low concentrations, in the case of competition of two NTRs that both strongly interact with the FG nups, we observed the opposite of attenuation: the flux was enhanced under bidirectional transport conditions.

We showed in kinetic cooperativity is a consequence of competition-induced release that this flux enhancement can be traced to the superlinear dependence of the unidirectional flux on NTR concentration for the strongly binding NTRs (Fig. 6 A), an effect that we termed kinetic cooperativity. This kinetic cooperativity of Kap-strong is closely related to competition-induced NTR release (first discussed in (29)) whereby the effective potentials experienced by NTRs within the NPC become shallower due to crowding, thus speeding up the NTR escape.

Based on these results, the FG assembly emerges as a surprisingly effective regulator of the effects of NTR crowding on nucleocytoplasmic transport. Due to the attractive pull the FG assembly exerts on the NTRs, and the higher density of the FG nups at the center of the pore, the NPC fills in stages. At low to moderate levels of crowding (corresponding to biological NTR concentrations within our simulations), the accumulation of NTRs occurs mostly in the center of the channel, which increases competition between the NTRs in this region, reducing the depth of the effective potential experienced by NTRs (29).

The resulting competition-induced release results in faster transport times and, for sufficiently strongly binding NTRs, kinetic cooperativity that leads to bidirectional flux enhancement. In this regime, as NTR accumulation mostly occurs in the center of the pore, the asymmetry of density distributions of NTRs remains low, resulting in minimal attenuation of bidirectional flux. Only at much higher NTR concentrations that result in crowding of the pore peripheries and exterior, we observe negative effect of the bidirectional competition on the transport efficiency. These insights are summarized in Fig. 8.

Figure 8.

A summary of the impacts of NTR crowding on the bidirectional NPC transport. The interior of the pore fills before the peripheries and exit adjacent regions of the pore, resulting in two distinct regimes of bidirectional transport. (Left) At low to moderate NTR concentrations (corresponding to the biological range in our simulations), weakly binding NTRs experience increasingly rapid translocation times as a result of competition-induced release, see combination of kinetic cooperativity and quasiequilibrium inside the pore at low NTR concentrations result in bidirectional flux enhancement. As most of the NTR accumulation occurs in the center of the pore, the NTR density distribution remains mostly symmetric in this quasiequilibrium regime, and the bidirectional flux attenuation remains low (Fig. 5). In contrast, at very high NTR concentrations, the peripheries and exit adjacent regions of the pore also begin to fill, shifting it to the fully nonequilibrium regime with high nucleocytoplasmic asymmetry of the NTR density distributions. This results in import and export fluxes obstructing each other, leading to a strong attenuation of the bidirectional flux. (Right) For strongly binding NTRs, in the low to moderate concentration regime, competition-induced release further leads to kinetic cooperativity, resulting in flux enhancement (Fig. 6). To see this figure in color, go online.

These theoretical results provide verifiable experimental predictions regarding the concentration dependence of fluxes of NTR mixtures that can be measured both in intact NPCs and nanopore mimics. However, we note that experimental observation of a superlinear dependence of flux on concentration for NTRs, may require very low measurement uncertainty (for instance, larger errors bars on the data points in Fig. 6 B would make a linear fit appear equally appropriate) or can be confounded by the presence of endogenous NTRs in the pore (57). One also should bear in mind that both experimentally and computationally the NTR flux is measured within a certain time window, which is determined by the specific constraints of a particular system. Therefore, at very low NTR concentrations, it is possible that the measured flux is effectively carried by the NTRs in the “fast” population (29).

While our computational model makes several minimalist simplifications such as modeling NTRs as uniform spheres which interact with FG nups over the entire surface instead of through discrete binding sites, previous work has indicated that such “smoothing” of the FG nup-NTR interaction adequately captures the physical behaviors both for individual FG nup-NTR interactions (25) and their NPC-like assemblies (14,29). The key property is the ability of FG nups to form mulitvalent interactions with NTRs, which is common to both uniform and patchy models, therefore we expect to see similar results from more refined models. We have also assumed only two different types of FG nups. It is possible that different types NTRs interact with different affinities with different FG nup types or with different regions of the FG nups based on different types of FG repeats or other local physical properties of the chains. This could result in additional spatial patterning of the FG nup-NTR assembly. Further models beyond the minimalist ones studied here will investigate the potential effects of the discrete “patchiness” of the NTR surface, the sequence heterogeneity of the FG nups and the multiple FG nup types on the dynamics of crowded NPC transport.

Furthermore, synthetic nanopores and materials inspired by the NPC are finding increasing application in molecular filtering and transport (1,28). The theoretical possibilities uncovered in this work may be of use in extending nanopore designs for complex scenarios of multispecies transport and providing new modes of concentration-dependent control of pore permeability (see also supporting material, section 3).

In this work, we see further potential for the seemingly simple design principles of the NPC to combine to produce novel regulation mechanisms where NTR crowding is key to achieving efficient transport, rather than an obstacle. To fully understand the implications of this work for nucleocytoplasmic transport and bioinspired nanopores and materials, further experimental studies—in particular single-molecule studies—will be required to ascertain the regime in which the NPC is operating in and to constrain and fine-tune the model. On the theoretical side, future work will examine more complex NTR and cargo combinations and the explicit modeling of the nonequilibrium nature of the NPC transport cycle.

Data and code availability

All simulation data and analysis code are available at https://doi.org/10.5683/SP3/WWNJBH. Simulation data files can be found at https://github.com/tz545/NPC-bidirectional-simulation.

Author contributions

A.Z. and T.Z. devised the research. T.Z. performed the calculations and the simulations and carried out data analysis. T.Z. and A.Z. wrote the paper.

Editor: Ben O'Shaughnessy.