Simulations of nuclear pore transport yield mechanistic insights and quantitative predictions

Abstract

To study transport through the nuclear pore complex, we developed a computational simulation that is based on known structural elements rather than a particular transport model. Results agree with a variety of experimental data including size cutoff for cargo transport with (30-nm diameter) and without (< 10 nm) nuclear localization signals (NLS), macroscopic transport rates (hundreds per second), and single cargo transit times (milliseconds). The recently observed bimodal cargo distribution is predicted, as is the relative invariance of single cargo transit times out to large size (even as macroscopic transport rate decreases). Additional predictions concern the effects of the number of NLS tags, the RanGTP gradient, and phenylalanine-glycine nucleopore protein (FG-Nup) structure, flexibility, and cross-linking. Results are consistent with and elucidate the molecular mechanisms of some existing hypotheses (selective phase, virtual gate, and selective gate models). A model emerges that is a hybrid of a number of preexisting models as well as a Brownian ratchet model, in which a cargo-karyopherin complex remains bound to the same FG-Nups for its entire trajectory through the nuclear pore complex until RanGTP severs the cargo-Nup bonds to effect release into the nucleus.

Significant advances in our understanding of the nuclear pore complex (NPC), which mediates all transport between nucleus and cytoplasm, include a cataloging of the structural components, characterization of the transport factors, assays for rates of transport, including measurements of single molecule transit, some preliminary reconstitutions of nuclear transport, structural studies both at the cryo-EM and the X-ray crystallographic level, and molecular dynamics simulations between select components (1).

Qualitative models to explain the selectivity of NPC transport for specifically tagged [nuclear localization signal (NLS)] cargo focus on the roles of the soluble factors and structural components of the pore. Two main soluble factors are Ran and the karyopherins (“kaps,” also known as exportins or importins). The kaps are transport receptors that bind with high affinity to NLS cargo, whereas Ran is a small GTPase that exists in a gradient of its GTP:GDP form from the nucleus to cytoplasm and is involved in cargo release. The structural components are flexible filamentous phenylalanine-glycine nucleopore proteins (FG-Nups) that fill the central core of the pore. They are considered relatively “unstructured”—in vitro they lack secondary structure—and they have a series of repeats of the amino acid motif FG, varying from 6 to 43 per filament (2) and of various forms such as FxFG, GLFG, PSFG, or xxFG. All of the FG-Nups are arranged in eightfold symmetry, with some as a single set and some as two or four rings. Although the FG-Nups are essential for selective transport through the nuclear pore, many are dispensable. In yeast, up to 50% of the FG-Nup mass can be deleted while still maintaining cell viability (3).

Despite the progress made in characterizing the nuclear pore complex, there is considerable disagreement on the mechanism for transport, and a number of different hypotheses have been offered. The selective phase model (4, 5) postulates that interactions between FG repeats on different FG-Nups result in the formation of a cross-linked gel. Cargo with an NLS, and in complex with a karyopherin, binds to FG motifs, competing for the FG-FG interactions, thereby allowing the cargo to melt into the gel, enabling transport through repeated steps of binding and melting. The virtual gate model (6) dispenses with the FG-FG interactions, maintaining that the very presence of unstructured FG-Nups prevents passage of cargo lacking a NLS by entropic exclusion. NLS cargo can bind FG-Nups, and this binding energy overcomes the entropic barrier for entering the pore. The competition model (7) maintains that the Nups can exclude cargo lacking an NLS only when cargo with an NLS is present. The reduction-of-dimensionality model (8) maintains that binding of NLS cargo to FG repeats lining the NPC effectively reduces their movement to a two-dimensional random walk, which would be significantly more efficient than the three-dimensional walk experienced by non-NLS cargo. The selective gating/collapse model (9) assumes the virtual gate entropic barrier but maintains that NLS-cargo passage is facilitated by a conformational change of FG-Nups that occurs when binding karyopherins. This binding causes collapse of the bound FG-Nups that reels in the NLS cargo toward the center of the NPC in what is termed “fly casting.” It has also been proposed that conformational changes of the entire pore itself result in changes in its effective diameter, helping to facilitate passage of cargo with NLS (10).

It has been previously proposed that nuclear transport may be the consequence of a Brownian ratchet: NLS cargo moving by thermal fluctuations with a chemical potential gradient biasing the net movement (11). However, this model did not specify the molecular mechanism by which the Brownian ratchet may function. Additionally, due to a lack of knowledge of the biophysics and physiology of transport through the pore, the model was not quantified to see if it recapitulated physiologically relevant events including transport rates and transit times.

Although no consensus exists on the mechanism of transport, the accepted experimental detail has reached a level to make the field ripe for simulations to bridge the gap between qualitative ideas and quantitative experiment. Models of nuclear transport take a few forms. Molecular dynamics provide insight into interactions between NPC components (12). However, their time scale (10^-9–10^-6 s) is out of the range of the millisecond transport events. Lower-resolution models of one (13) to three (14) dimensions have replicated specific hypotheses of transport. Here we present a low-resolution yet fully three-dimensional model of NPC transport that can address different transport hypotheses.

In order to capture transport events that occur on the millisecond time scale, we forego the high temporal and spatial resolution of the atomistic approach that is limited to the nanosecond to microsecond time scale and instead opt for a lower-resolution model that nonetheless aims to capture the essential physics and biology of NPC transport. Using simulation to explore relevant parameter spaces, we can then determine under what conditions, if any, the hypothesized mechanisms for transport emerge, whether these conditions are in fact physiologic, and if calculated results agree at least semiquantitatively with experiment. For parameters that have been determined experimentally, such values are employed. For those that are not yet determined, educated guesses are made. Even where parameters have been determined, we aim to vary them by orders of magnitude to explore the sensitivity and robustness of our results to these values. The results recapitulate many of the experimental observations on nuclear transport and demonstrate behaviors consistent with some, but not all, of the proposed models and provide molecular-level detail of how these models may operate.

The Model

Our approach is to create a fully three-dimensional physical space in which to simulate the dynamics of the FG-Nups and cargo with no a priori allegiance to any particular model of transport. We have constructed a model space that includes a single NPC, modeled as a cylinder containing rings of FG-Nup filaments. Although the NPC dimensions may be varied, they are generally 30 nm in length and 50 nm wide, in agreement with the structure of the yeast pore. Additional user-defined space on either side of the NPC completes the model space. A single FG-Nup is modeled as a flexible filament, using the Pairwise Agent Interaction with Rational Superposition (PAIRS) model (15). Filament flexibility is varied primarily through a single parameter (Ctheta); a lone filament anchored at one end to the inside shell of the model NPC cylinder will have different mean end-to-end distances as Ctheta is varied. The amino acid length as well as the filament radius and structure (helix versus extended chain) are varied to yield various FG-Nup structures. FG motifs are modeled as binding sites along the filament length (each structural filament segment is associated with a FG binding site). When FG motifs on the same or different Nups collide, they may bind. Subsequent bond dissociation is governed by a defined FG-FG off rate. Similarly, FG motifs can interact with kap binding sites on the cargo surface when they collide. Individual FG-Nups are anchored in rings of 8-fold symmetry along the NPC length, allowing for variation of the number of such rings as well as the spacing between them.

Cargo is modeled as a spherical molecule of defined diameter. It may have any number of NLS tags, corresponding to an association with a virtual karyopherin represented by FG-binding sites on the cargo surface. Generally, both NLS and non-NLS cargo are introduced in stochastic fashion via a defined rate, thus allowing for competition between the two and determination of selectivity of NLS over non-NLS cargo. RanGTP is introduced into the model in the form of a concentration gradient (maximum at the nucleus) whose shape and magnitude can be defined. Proportional to this concentration (and of actual interest in the model) is the rate at which RanGTP binds the NLS-cargo-kap complex, as this binding displaces the cargo-kap bond (the dissociation of which is exceedingly slow in the absence of RanGTP), with the consequence that the cargo can no longer bind (or rebind) FG-Nups. This displacement occurs whether or not the cargo-kap complex has already bound to FG-Nups (in fact, RanGTP induces a conformational change when binding to the karyopherin, which also affects dissociation, otherwise exceedingly slow, of any kap-FG bonds). See SI Methods for greater detail on the model construction. The outcomes generated by the simulations include the macroscopic rates of cargo transport, single cargo transit times, selectivity for NLS versus non-NLS cargo, spatial-temporal distributions of FG-Nups and cargo, and actual analysis of individual trajectories. These are studied as functions of variations in cargo size, number of NLS tags, individual FG-Nup structure (thickness and amino acid length) and dynamics (flexibility), number of FG-Nup filaments, FG-FG off rates, and the RanGTP gradient (Table 1). Details of all of the methodology for the modeling is given in SI Text.

Table 1.

Parameters varied in the simulations

Parameters varied	Range of values tested in our simulations
Number of rings	1 to 10 rings (8 to 80 FG-Nups; 104 to 2,080 FG repeats)
Spacing between the rings	1.78 to 3.33 nm
FG-Nup filament radius	0.3, 0.6, 1.2, and 3 nm
Ctheta	0.001 to 0.5
Filament amino acid length	150 to 1,800
Number of FG-Nup binding sites per karyopherin	10 (clustered)
Number of FG repeats per filament	13, 26
Off-rate for FG-FG interactions	102 to 10∞/s
Cargo diameter	6 to 48 nm
Ran gradient (distance from center of NPC at which RanGTP falls to 10% of nuclear concentration)	0 to 12.5 nm
Maximum Kap-RanGTP on rate	103 to 105/s
Width of the NPC	30 to 50 nm
Length of the space for the simulation beyond the 30-nm length of the nuclear pore	70 nm, 120 nm, 170 nm (total length with nuclear pore = 100 nm, 150 nm, 200 nm)

Results

The FG-Nup as Flexible Filament.

We first studied the dynamics of a single FG-Nup filament anchored at one of its ends to the inner rim of the cylindrical model NPC. A filament of length 55 nm and radius 3 nm (corresponding to approximately 1,800 amino acids), was seen to assume many different conformations of varying end-to-end distances (Fig. 1A and Movie S1). Similarly, its free end was seen to transit from one side of the NPC to the other. End-to-end distance was studied as a function of Ctheta (Fig. 1B). Decreasing Ctheta increases filament flexibility and decreases mean end-to-end distance. The relationship between Ctheta and mean EED was essentially unchanged when varying filament thickness as well as the number of FG domains.

Fig. 1.

(A) End-to-end distance (EED) dynamics for a single FG-Nup anchored to the inner rim of the model NPC (filament radius 3 nm, length 55 nm, Ctheta = 0.002) and (B) as a function of Ctheta (mean EED ± SD).

Populating the NPC with Many FG-Nups.

FG-Nup dynamics, cross-linking, and distribution were studied as a function of FG-FG binding by varying the FG-FG off rate for systems of 1 to 10 rings of 8 FG-Nups each for a total of 8 to 80 Nups or a total of 104 to 2,080 FG domains. In the absence of FG-FG interactions (FG-FG off rate = 10^∞/s), the filaments were observed to move dynamically from one side of the pore to the other for a 1-ring (Fig. 2A, Top, Movie S2), a 3-ring (Movie S3), or 10-ring system (80 FG-Nups, Movie S4). The speed and extent of movement was restricted as the FG-FG off rate was slowed (Fig. 2A, Bottom) from 10^∞/s (purple) to 10⁶/s (green) to 10⁵/s (red) to 10⁴/s (blue) for the 1-ring (Fig. 2B Left, Movie S5), 3-ring (Movie S6), and 10-ring systems (Fig. 2B, Right, Movie S7). The observations were similar for off rates of 10⁵/s or faster. Analysis of the bonds between FG-Nups (Fig. 2C) revealed extensive linkage for off rates slower than 10⁵/s and few bonds for faster off rates.

Fig. 2.

FG-Nup interactions and dynamics as a function of FG-FG off rate. (A) Snapshots of FG-Nup dynamics looking at a side view of the NPC (from within the plane of the nuclear membrane) or into the NPC (from the cytosol or nucleus), for FG-FG off rate 10⁴/s and for no FG-FG interactions. Scale bar is 30 nm. (B) FG repeat domain density histograms for a variety of FG-FG off rates. The beige box marks the thickness of a NPC. (C) The fraction of FG repeat domains that are bound to other FG repeat domains and absolute number of FG-FG bonds (Inset) as a function of FG-FG off rate for different FG-Nup landscapes.

The FG-FG off rate modulated the FG-Nup dynamics. At one end of this continuum (< 10⁴/s), dynamics emerged that were reminiscent of the selective phase model, a cross-linked gel. At the other end of the spectrum (> 10⁵/s), the dynamics resembled the virtual gate model. The 10⁵/s off rate is a transition in which there was a gel-like cross-linking, although looser and of greater spatial extent than denser, more cohesive gels found at lower off rates.

Evidence for the Brownian Ratchet: The RanGTP/RanGDP Gradient Is Necessary for Transport

For a nuclear pore devoid of FG-Nups or RanGTP, 10-nm diameter cargo transported at approximately 280 particles/s (Fig. 3B, yellow line). In the presence of a single ring of noninteracting FG-Nups (i.e., FG-FG interactions turned off), cargo without an NLS transited the pore at 80 particles/s (Fig. 3B, black line); for three or more rings of FG-Nups, 10-nm diameter cargo did not transit at all (the non-NLS size cutoff was smaller, i.e., 6 nm, as in Figs. S1 and S2). Cargo with and without an NLS were stochastically introduced to a pore with FG-Nups present but lacking RanGTP. Transport of either cargo was zero (Fig. 3B, purple line). This was a consequence of the karyopherin/FG-Nup off rate (< 1/s; see SI Methods). The pore was jammed with NLS cargo, preventing other cargo from transiting. There was no net transport in the absence of a gradient of RanGTP:RanGDP.

Fig. 3.

Effects of varying the RanGTP gradient. (A) Gradients of different width for maximum kap-RanGTP on rate of 10⁴/s. Maximum [RanGTP], and hence maximum kap-RanGTP on rate, occurs at the nucleoplasmic entrance to the NPC, at position +15 nm along the nuclear axis. Wider gradients penetrate farther in that [RanGTP], and hence the kap-RanGTP on rate, falls to 10% of the maximum closer to the center of the NPC (position 0). For example, the red curve corresponds to a width of 15 nm because the on rate falls to 10% at position 0. (B) Effects on transport rates for one NLS versus zero cargo (10-nm diameter) simultaneously introduced into the pore (compare solid and dashed lines of the same color) for the various RanGTP gradients in the 1-ring system, no FG-FG interactions, and Ctheta = 0.02. Each point corresponds to a different RanGTP gradient: The maximum kap-RanGTP on rate is indicated by the color, as defined in the legend, whereas the gradient width is indicated by the x-axis value, as in A; i.e., the position along the nuclear axis at which [RanGTP] falls to 10% of its maximum. Also graphed are the rate of zero NLS cargo alone (black line), the rate in the absence of RanGTP (purple line) (illustrated for zero NLS cargo and identical to the rate for the competing one NLS cargo), and the rate in the absence of FG-Nups (yellow line).

A gradient of RanGTP was introduced as a Gaussian function decreasing from the nuclear to cytoplasmic side (Fig. 3A). As described in SI Methods, a particular gradient is characterized by two parameters: the maximum kap-RanGTP on-rate (representing the RanGTP concentration at the nucleoplasmic side) and the width of the gradient, in particular the position along the nuclear axis (cytoplasm = -15 nm, NPC center = 0 nm, nucleus = +15) at which the RanGTP concentration (and hence kap-RanGTP on rate) drops to one-tenth of the maximum value found at the nucleoplasmic side. The cyan, green, and red curves in Fig. 3B correspond to gradients with maximum kap-RanGTP on rates of 10⁵/s, 10⁴/s, and 10³/s, respectively. Each point in each of these curves corresponds to a gradient of different width, derived from the point’s x-axis value, as described in the figure legend.

The presence of a RanGTP gradient accelerated the rate of transport of cargo with one NLS (the solid cyan, green, and red curves in Fig. 3B) to levels even faster than transport in the absence of FG-Nups (Fig. 3B, yellow line). Increasing the maximum kap-RanGTP on rate increased the rate of transport, although increasing the maximum on rate above 10⁴/s had little effect (compare cyan, green, and red solid curves in Fig. 3B). Allowing RanGTP to diffuse further into the pore affected the transport rate as well, with wider gradients (i.e., moving to the left on the x axis in Fig. 3B) generally yielding increased transport rates. However, when RanGTP penetrated too far into the pore, the simulation predicted a decrease in the transport rate (i.e., rate decreases when moving from 2.5 to 0 nm along the x axis for both the cyan and green solid curves). This decrease is a consequence of the RanGTP releasing some cargo-NLS-karyopherin complexes from the FG-Nups on the cytoplasmic side of the pore.

Cargo with and without a single NLS were introduced simultaneously and stochastically into the pore (compare solid and dashed lines of corresponding color) in order to allow for competition between them (see SI Methods). When RanGTP minimally diffused into the pore (i.e., positions 7.5 to 12.5 nm along the x axis), NLS cargo bound the FG-Nups essentially irreversibly and transport of NLS and non-NLS cargo was blocked (solid and dashed curves approach zero rate). When RanGTP entered farther into the pore (position 5 nm and below along the x axis), there was transport of NLS-containing cargo and competition with the non-NLS cargo was reduced (dashed lines approach black line, i.e., the rate of 0 NLS cargo alone). The on rates of RanGTP also affected competition. Specifically, as the on rate increased from 10³/s to 10⁵/s, the transport of cargo without an NLS increased (compare dashed cyan, green, and red curves in Fig. 3B).

These results support the rudimentary Brownian ratchet model. Namely, cargo moves with a filament in the pore by thermal fluctuations. The gradient of RanGTP concentration decreasing from the nucleoplasmic to cytoplasmic side of the NPC is necessary to release the cargo from the filament, ratcheting the cargo on the nuclear side, resulting in NLS-cargo transport. Simulation in the model NPC allows for even finer resolution, currently unavailable through experiment, which elucidates an actual mechanism for the Brownian ratchet.

A Molecular Mechanism for the Brownian Ratchet Model Emerges in Analysis of Single Cargo Trajectories.

The transit of individual cargo molecules was followed with 1 (Movie S8), 3 (Fig. 4, Fig. S3 B and C, and Movies S9, S10, and S11), and 10 rings (Movie S12) of FG-Nups (Fig. S3A). All simulations were run with an equal concentration of cargo with and without a NLS. Irrespective of variation in FG-FG interactions, number of filaments, filament flexibility (i.e., Ctheta), or dynamics of RanGTP interactions, the behavior of cargo with an NLS was similar under all conditions and could be described by five phases.

Fig. 4.

Analysis of cargo trajectories. (A) Snapshots from the trajectory of a 10-nm diameter cargo with one NLS (green sphere) in the 3-ring system (no FG-FG interactions, Ctheta = 0.02) (red sphere = other NLS cargo; pink filaments are bound to NLS cargo). (B) Quantitative analysis of the highlighted (green) trajectory in A: The black curve indicates the position over time, whereas each of the other curves represents the number of bonds between the cargo and a particular FG-Nup. (C) Histogram from combined transporting cargo trajectories, illustrating the cargo position density (i.e., probability of finding a single cargo at a given position).

Both NLS and non-NLS-cargo approach the NPC by thermal fluctuations. Phase 1 begins when NLS cargo first binds (is captured by) an FG-Nup. Once one binding site on a cargo-bound karyopherin binds an FG-Nup, subsequent sites rapidly (< 0.2 ms) bind other FG domains, most on the same filament (blue line in Fig. 4B and Fig. S3), but occasionally on other filaments (red and green lines). The cargo-NLS-karyopherin moves by thermal fluctuations. However, with each additional bond between the karyopherin and an FG domain, the cargo is pulled deeper into the NPC. Thus, in phase 1, cargo moves in a directed manner as if reeled in by the filament (black line in Fig. 4B and Fig. S3). This movement resembles fly casting and is accomplished by the FG-Nups rapidly binding sites on the karyopherin, winding around and effectively shortening in length, potentially accounting for the FG-Nup collapse observed by Lim et al. (9).

Phase 2 initiates once the cargo-NLS-karyopherin binds all of its FG-binding sites. This phase is marked by two behaviors. First, the cargo fluctuates in the pore from thermal motion, but constrained by its bonds to the filaments (note limited Δposition from 16.6 to 17 ms for the cargo in Fig. 4B, from 3 to 8 ms for the cargo in Fig. S3A, from 4.5 to 10.5 ms for the cargo in Fig. S3B, and from 9.125 to 9.25 ms for the cargo in Fig. S3C). The second behavior is the surrounding unbound FG-Nups, through thermal fluctuations, slowly enveloping the cargo-NLS-karyopherin and reorganizing around it. The cargo thus spends time around 10–15 nm from the center on the cytoplasmic side.

In phase 3, the FG-Nups have reorganized and present less of a barrier to movement of the cargo to the nucleoplasmic side. The cargo can move by thermal fluctuations from its relatively stable point on the cytoplasmic side to an equivalent stable position on the nucleoplasmic side of the central core of the FG filaments. This FG-Nup reorganization is consistent with the virtual gate/entropic exclusion model and in fact provides the molecular mechanism by which it operates. Non-NLS cargo cannot bind FG-Nups; consequently, most will diffuse back out of the NPC before FG-Nups reorganize around them to enable passage through the NPC.

In phase 4, the cargo fluctuates around this stable point until its karyopherin binds RanGTP, enabling release and subsequent movement by thermal fluctuation away from the NPC (phase 5). In the absence of RanGTP, the particle fluctuates back and forth from its position in phase 2 to that of phase 4. Because waiting for FG-Nup reorganization and waiting for RanGTP binding are the slowest parts of transport, cargo is predicted to spend most of its time in either phase 2 or phase 4, confirmed by the spatial cargo distribution plotted in Fig. 4C. This bimodal distribution is in agreement with recent single molecule experiments, published while this paper was in review (16). Throughout, cargo is bound primarily to one or at most a few FG-Nups until its release by RanGTP. The robustness of this mechanism is demonstrated in the 10-ring system, where despite the large density of FG-Nups seen by the approaching cargo, it nonetheless is bound to the same three filaments throughout its translocation of the NPC (Fig. S3A); most of those bonds are in fact associated with one particular filament.

The mechanism elucidated here is consistent with the Brownian ratchet model: Namely, NLS cargo moves by thermal motion through the NPC, and transport can occur only in the presence of a RanGTP gradient. It furthermore offers unique insights as to how the ratchet works. The free ends of FG-Nups in fact transit from one side of the pore to the other through thermal motion. An NLS cargo is seen to bind one, or at most a few, FG-Nups on the cytoplasmic side of the NPC, to which it remains bound for its entire transit through the NPC. Neighboring FG-Nups must reorganize around the cargo-Nup complex, and this is in fact a rate-limiting step that, together with time spent waiting for RanGTP release, rationalizes the predicted bimodal spatial-temporal cargo distribution. In the absence of RanGTP, the cargo-Nup complex diffuses back and forth repeatedly. However, in the presence of a gradient of RanGTP, contact with RanGTP severs the cargo-Nup and cargo-kap bonds and allows for release to the nucleoplasmic side. Although RanGTP was proposed earlier to act in some fashion as a Brownian ratchet, the transport mechanism that emerges in these simulations demonstrates how this actually occurs and as such constitutes a detailed explanation for how it operates. In fact, these simulations also delineate molecular mechanisms for the other consistent transport models: FG-Nup reorganization explains the virtual gate/entropic exclusion model, and the hypothesized fly casting of the selective gate/collapse model is clearly seen.

Brownian Ratchet Predicts Cargo Size Cutoffs and Transport Rate Variation.

When the FG-Nups were omitted from the pore (Fig. 5A, purple boxes), there was a weak size dependence for cargo to successfully transit the pore. Macroscopic transport rates were on the order of 280/s for 6-nm diameter cargo and a little over 100/s for 34-nm diameter cargo. Introducing the eight FG-Nups into a single ring reduced the size of non-NLS cargo that transited the pore below 10-nm diameter (Fig. 5A, black lines; the non-NLS size cutoff decreases to 6 nm with increasing number of rings of FG-Nups, i.e., Figs. S1 and S2 and with decreasing pore diameter, i.e., Fig. S4). Cargo with and without a single NLS were introduced into pores with [RanGTP] decreasing to 10% at the NPC center. Cargo with a single NLS (red lines) transited the pore at rates exceeding 800/s, and cargo up to 26-nm diameter (the size of a ribosome) could now transit the nuclear pore complex. When cargo containing two NLS was introduced, even larger cargo (up to 30-nm diameter) could readily transit the pore (cyan lines). In each case the macroscopic transport rates and size selectivity of cargo without an NLS were barely affected by the presence of NLS cargo (compare black and green lines). These trends held with 1, 3, or 10 rings of FG-Nups, as well as for varying distances between these rings (Figs. S1 and S2).

Fig. 5.

Cargo dynamics: macroscopic transport rates (A, B, and D) and mean transit times (C and D) as a function of cargo size and number of NLS tags for different FG-FG off rates (A and C) and FG-Nup structure (r = FG-Nup filament radius) (B) and flexibility (Ctheta) (D). Results for A were produced in the 1-ring system, whereas those for B and D are from the 3-ring system (C features results from both). Unless otherwise specified, Ctheta = 0.02. FG-FG interactions are varied in A and C and turned off in B and D.

Selective Phase Dynamics (FG-FG Cross-Linking) Decreases Transport Rate.

To test for effects of FG-FG interactions on transport, the FG-FG bond off rate was varied (Fig. 5). Cargo up to 18-nm diameter (one NLS) and 26-nm diameter (two NLS) successfully transported through the pore at least as fast in the presence of FG-Nups as it did through an empty pore devoid of FG-Nup filaments. The fastest rate of transport occurred in the absence of interactions between the FG groups on the Nups (k_off for FG-FG interactions = 10^∞). Slowing the k_off between the filaments slowed the rate of transport of NLS-containing cargo (Fig. 5A, compare FG-FG off rate of 10⁵/s, dashed lines, and 10^∞/s, solid lines). This decreased transport was observed with two NLS (cyan line, Fig. 5A) or one (red line, Fig. 5A). For cargo without an NLS, there was no detectable effect of varying the FG-FG off rate on the rate of transport (compare green lines). These trends held for FG-Nups of varying flexibility (compare Fig. 5A and Fig. S5). Increasing filament flexibility (decreasing Ctheta and EED) generally decreased transport rate, though the relationship is more complex (see below).

Brownian Ratchet Predicts That FG-Nup Structure Effects Transport Rate.

In addition to results for 3-nm radius FG-Nup filaments (17, 18) shown throughout, we also studied 1.2-nm radius filaments, 0.6-nm radius (an alpha helix), and 0.3-nm radius (an extended peptide chain) (Fig. 5B and Fig. S2), all of the same static length. As described in SI Methods, this corresponds to filaments ranging between 150 and 1,800 amino acids. Results for different thickness filaments were qualitatively similar. Quantitatively, thinner diameter FG-Nups allowed a faster transport rate for larger NLS-containing cargo and a slightly slower rate for smaller cargo (Fig. 5B). The Brownian ratchet model not only predicts this variation but also explains it: Owing to their smaller cross-sectional area, thinner filaments are less efficient at capturing incoming NLS cargo but are more efficient at reorganizing around it and each other. As such, thinner filaments yield a slightly diminished transport rate at lower cargo size, where the discrepancy in capturing cargo dominates. Larger cargo, owing to its larger cross-sectional area, will be captured by filaments of essentially any thickness. Consequently, thinner filaments yield increasing transport rates owing to their greater efficiency at reorganization. Trends delineated above for FG-FG interactions as well as increasing number of NLS tags held for the thinner filaments, as demonstrated in Fig. S2 and Table S1.

Brownian Ratchet Predicts Single Molecule Transit Times.

Histograms were generated for the transit times for individual cargo molecules (Fig. S6; 10-nm diameter cargo, NLS = 1, Ctheta = 0.02) and the mean transit time plotted as a function of the cargo size and the number of NLS (Fig. 5C). The mean transit time is relatively unchanged over a large range of cargo sizes even as macroscopic transport rate decreases, a prediction recently verified in a paper published while this manuscript was in review (19). For cargo > 18-nm diameter, there was a substantial increase in the transit time. The presence of two NLS decreased the mean transit time, especially for larger cargo (diameter ≥26 nm) for which the transit time with one NLS was long enough that experimentally it might not be observed to transit the pore. These simulations of transit times are on the millisecond time scale, in agreement with recent single molecule measurements (20, 21). Stabilizing the FG-FG cross-linking by decreasing the FG-FG off rate (10⁵/s) slowed transport rate (see Fig. 5A and Fig. S5) and increased mean transit time. For the 10-ring system (Table S2), stabilizing FG-FG interactions slowed transport rate by > 50% (325 to 114/s) and increased the mean transit time by approximately 33% (27 to 35 ms). Note that individual non-NLS-cargo molecules transited faster than NLS cargo, a prediction that has also been verified while this manuscript was in review (19), representing the fact that free diffusion around FG-Nups is faster than transport mediated by binding to the Nups. NLS cargo, after all, is slowed down by the actual events of binding and RanGTP-mediated unbinding. Thus, exclusion of individual non-NLS cargo is a consequence of the low probability of such transport occurring, not the time scale of such transit when it does in fact occur.

As mentioned above, the low probability of non-NLS-cargo transport is explained by the Brownian ratchet molecular mechanism: A non-NLS cargo, not being bound to FG-Nups, is likely to diffuse back out to the cytoplasm in the time it takes for neighboring FG-Nups to reorganize around it (phase 2 of the Brownian ratchet mechanism delineated above). Similarly, cargo with two NLS transits faster and with increased macroscopic rate compared to cargo with one tag. More bonds between two NLS cargo and bound FG-Nups stabilizes its position relative to one NLS cargo, which compared with two NLS cargo is more of a moving target. Reorganization of neighboring FG-Nups around the more stationary two NLS cargo is therefore more efficient.

Effects of Varying the Flexibility of the FG-Nup Filaments Are Explained by the Brownian Ratchet Mechanism.

Ctheta is a parameter that tunes the flexibility of individual filaments. Lowering Ctheta results in individual filaments with smaller end-to-end distance. Ctheta was varied from 0.001 to 0.2, corresponding to end-to-end distances from 22 ± 7 nm to 46 ± 2 nm (Fig. 1B). The mean transit time decreases with decreased flexibility (increased Ctheta) (Fig. 5D, purple line). The dependence of transport rate on Ctheta is more complicated (Fig. 5D, red line), with a maximum occurring at intermediate flexibility. The reasons for these relationships become clear from observing the trajectories of individual cargo (compare Movies S9, S10, and S11 and corresponding Fig. 4B and Fig. S3 B and C) that illustrate the Brownian ratchet molecular mechanism. More rigid filaments extend further, contact an individual cargo earlier in its trajectory, and reel in the cargo more quickly (Fig. S3C, black line). This accounts for the decrease in mean transit time of a single cargo molecule, i.e., phase 1 of the Brownian ratchet mechanism is faster. However, owing to its stiffness, a rigid filament (with a high Ctheta) takes longer to “reset,” i.e., to return to a position at which it can bind a new cargo. Put another way, the translocation of FG-Nup free ends through the pore by Brownian motion is slower for the stiffer filaments. This results in less efficiency and an overall decrease in macroscopic transport rate. Optimization of these opposing trends gives the maximum transport rate at intermediate Ctheta, as seen in Fig. S3C. Comparing Movie S10 and Fig. S3C also explains why, for Ctheta = 0.2, the cargo seems to linger (though for a relatively short time) on the nucleoplasmic side once released by RanGTP binding (Fig. S3C). Namely, diffusion of the cargo away from its release point is momentarily hindered by the many surrounding stiff filaments that have yet to reset to the cytoplasmic side of the NPC.

Discussion

Numerous models have been proposed to explain selective transport across the nuclear pore. There has been little progress in resolving between these models for two reasons. First, the models are not formulated in a manner that allows them to make quantitative predictions that can be falsified. This quantification is essential for resolving between a model that might be “possible” but insufficient to account for physiological observed rates. Second, many of them have not been formulated in sufficient biophysical detail to make it possible to evaluate molecular mechanism. Recent significant advances in the ability to make biophysical measurements of the transit times and rates of single cargo provide information that can be used to test these models, if they are properly formulated. Our computational model of the NPC, created without a priori fidelity to a particular hypothesized mechanism, provides a quantitative tool for testing these models and yields results that are in agreement with different experimental data and provide insight into the molecular mechanisms underlying transport.

Based on these results, it is possible to draw some conclusions regarding the utility of competing models. Macroscopic transport rate was seen to increase as one moved from FG-FG off rates of ≤ 10⁵/s [resembling the selective phase (4, 5)] to faster off rates [resembling the virtual gate model (6)]. Without knowing the true FG-FG off rate inside of the NPC, it is not possible to say which model better approximates dynamics of the FG-Nups in the NPC. Recent applications of fluorescence anisotropy to the NPC offer the potential for resolving the dynamics of the FG-Nups (22, 23). However, even in the absence of such information, we can conclude that the higher-level cross-links of the selective phase mechanism are not required to explain NPC transport. That is, doing away entirely with FG-FG interactions not only maintains selectivity in transport but in fact increases overall rates of transport of NLS cargo. Thus, it is not necessary to invoke FG-FG interactions to account for the rates or selectivity of transport. The Brownian ratchet model, which emerges in the analysis of single cargo trajectories, provides an explanation for this result. FG-FG cross-linking prolongs phase 2 of transport, the FG-Nup reorganization around transiting cargo, and as such slows transport.

The “competition model” (7) is tested throughout by simultaneously introducing both NLS and non-NLS cargo into the pore. Over the wide range of conditions tested, the presence of FG-Nups alone was sufficient to limit non-NLS transport. Thus it is entropic exclusion by FG-Nup filaments (the virtual gate) that is sufficient to exclude non-NLS cargo and ensure selectivity of the NPC. Generally in our simulations, NLS cargo was seen to limit transport of non-NLS cargo only under nonphysiologic conditions, specifically only in the presence of nonphysiologic RanGTP gradients that also severely limited NLS cargo.

The computer simulations are most consistent with a hybrid of some of the existing models: the virtual gate (entropic exclusion), selective gate (collapse/fly casting), selective phase, and Brownian ratchet models. In fact, analysis of single cargo trajectories generated in these simulations actually reveals molecular mechanisms for these models. The Brownian ratchet model, demonstrated here in molecular detail, initially proposed that NLS cargo diffuses through the NPC until it is released by RanGTP (the Brownian ratchet) on the nucleoplasmic side (11). Thermal ratchets were initially proposed as a “thought experiment” that would lead to a perpetual motion machine (review in ref. 24). The term Brownian ratchet was coined to describe how a molecular motor, working in a realm whose motion is dominated by Brownian motion, can take advantage of thermal motion and a chemical potential gradient to do work (11). The Brownian ratchet was applied earlier to the movement of molecules across membranes, in particular the endoplasmic reticulum (11) although it was suggested to apply to transport of macromolecules across other membranes including transport through the nuclear pore (11). Recently, a Brownian ratchet was invoked to explain nuclear export (25). This formulation was not sufficiently detailed to allow it to be quantitatively tested. The detailed mechanism of transport that emerges in our simulations finally enables understanding of a mechanism for a Brownian ratchet model. Namely, RanGTP severs the bonds between NLS cargo and the FG-Nup (or few FG-Nups) to which it remains bound and transits with together throughout its motion in the NPC. Without this release, the cargo-Nup complex would diffuse back and forth repeatedly.

A Brownian ratchet model provides a solution to previously puzzling aspects of nuclear transport. Many of the proposed models, (i.e., selective phase and reduction of dimensionality) require multiple rounds of karyopherin-FG-Nup binding/unbinding during each transport event. However, the observed nanomolar dissociation constant for the kap-FG bond (corresponding to an off rate of 1/s or less) with multiple rounds of binding and unbinding would take many orders of magnitude longer than the observed millisecond transport rates. The Brownian ratchet model posits that there are no unbinding events during transport, thus avoiding the problem of the apparently slow dissociation rate of karyopherin from the FG-Nup. Whichever FG-Nup filament the cargo binds to, it remains bound to until it reaches the nucleoplasmic side of the nuclear pore and is released by RanGTP (see Fig. S7).

Other aspects of NPC transport are similarly naturally explained by the molecular mechanism underlying the Brownian ratchet model. The flexibility of FG-Nups is thought to be central to their function. In the Brownian ratchet model, this flexibility is obviously necessary for FG-Nup/cargo translocation across the pore, enabling FG-Nups to bind NLS cargo and transit with it until being released by RanGTP. Flexibility is furthermore necessary for the reorganization of neighboring FG-Nups around transiting NLS-cargo-Nup complexes.

Recent experimental data that are not readily explained by other models are predicted and explained by the Brownian ratchet mechanism discovered here. The bimodal spatial-temporal distribution of NLS cargo is predicted by the cargo-Nup complex waiting for neighboring FG-Nups to reorganize (waiting at the cytoplasmic side) and then waiting for RanGTP to bind (waiting at the nucleoplasmic side) (phases 2 and 4 of the Brownian ratchet molecular mechanism). This polymer reorganization around a FG-Nup bound cargo molecule is a physical embodiment of the virtual gate model. This result of pausing on one side, not obvious in other models, is quantitatively predicted in our simulations and was confirmed in experiments published while this paper was in review (16). The FG-Nup reorganization that is central to the Brownian ratchet mechanism explains a variety of phenomena (described above), including why FG-FG cross-linking central to the selective phase model actually slows down transport and why two NLS cargo is faster than one NLS cargo. It also explains the result predicted in our simulations and confirmed while our paper was in review (19) that out to relatively large cargo size, the mean transit time of individual NLS cargo is essentially constant (Fig. 5C). The slow step of reorganization is a characteristic of the FG-Nups themselves and not of the transiting cargo (and therefore not of the cargo’s size).

Interestingly, the Brownian ratchet mechanism allows for some rapprochement between the virtual gate and selective phase models. In these simulations the microenvironment in the pore is dominated by the density of FG-Nups, whether or not they are in the gel (cross-linked) state posited by the selective phase model. Cargo molecules that do not bind to a filament will not freely diffuse through, as posited by the “entropic exclusion” model. Once the cargo is bound to a filament, the other filaments slowly reform, or reorganize, around it—the transiting NLS cargo can be viewed as a melting into and through this FG-Nup polymer. Thus, the lowering of energy for entropic exclusion is the result of the polymers reforming around it, allowing the cargo to melt through the FG-Nup polymers. In this way FG-Nup reorganization is reminiscent of the melting that occurs in the selective phase model while not requiring that model’s posited cycles of binding and unbinding of NLS cargo to FG repeats to melt the existing FG-FG bonds. Not only does transport not require a gel-like state (where a gel implies cross-linking), but our results in fact demonstrate that cross-linking between the polymers, far from facilitating NLS-cargo transport, actually hinders it.

Our simulations provide a molecular mechanism for the Brownian ratchet model and elucidate the molecular mechanisms of the virtual gate model and selective gate model (fly casting) while providing predictions to guide further study. Discussed above, these include the extent of RanGTP diffusion into the NPC, quantitative comparison of NLS cargo with different numbers of NLS tags, experimental determination of the effects of FG-FG cross-linking on transport (predicted here to slow it down), and effects of varying FG-Nup structure and flexibility. The latter may be particularly important in light of the recent proposal that FG-Nups of distinct structure types are distributed in a nonrandom way within the pore (26). The central tenet of the Brownian ratchet model molecular mechanism elucidated in our simulations, that a cargo molecule remains bound to the same FG-Nups throughout its trajectory through the NPC, has yet to be verified experimentally.