Interactions between a fluctuating polymer barrier and transport factors together with enzyme action are sufficient for selective and rapid transport through the nuclear pore complex

The nuclear pore complex, the only pathway for transport between the nucleus and cytoplasm, functions as a highly selective gate that blocks nonspecific macromolecules while allowing the rapid transport of tagged [transport factor (TF) bound] cargo up to an order of magnitude larger. The mechanism of this gate's operation is not yet fully understood and progress has been primarily hindered by the inherent complexity and multiscale nature of the problem. One needs to consider the hundreds of disordered proteins (phenylalanine glycine nucleoporins or FG nups) lining the pore, as well as their overall architecture and dynamics at the microsecond scale, while also accounting for transport at the millisecond scale across the entire pore. Here we formulate an approach that addresses transport properties over a large range of length and time scales. We do this by incorporating microscopic biophysical details, such as charge and specific TF-FG nup interactions, to compute the free energy landscape encountered by the cargo. We connect this to macroscopic transport by treating cargo translocation as a stochastic barrier crossing process and computing the current and the translocation time. We then identify distinct transport regimes (fast permeable, slow permeable, and impermeable) determined by the cargo size, TF affinity for FG nups, and the activity of the enzymes that cleave TFs from cargo. Our results, therefore provide an integrated picture of transport through the NPC, while highlighting how FG nup interactions with TFs and enzyme activity cooperate to produce selectivity and efficiency.

Figure 1

(a) Snapshots (top view) of FG nups from simulations for various hydrophobic interaction strengths $\varepsilon$. Different colors indicate physical properties of monomers: hydrophobic (dark gray), negatively charged (blue), positively charged (red), and electrically neutral (light gray). For weak interactions ($\varepsilon =1.0$), FG nups are isolated, resulting in an open NPC. For increasing interaction strengths, FG nups are partially entangled ($\varepsilon =1.7$), or form a central plug ($\varepsilon =2.4$). (b) Occurrence counts of the FG-nup configurations having connectivity $\theta$, defined in Eq. (2), showing how the typical configurations change from the open state to the closed state as $\varepsilon$ increases. For intermediate $\varepsilon$ ($\varepsilon =1.7$), the connectivity of FG nups has a bimodal distribution. (c) The average connectivity $\langle \theta \rangle$ and its fluctuation $\sqrt{\langle {\theta }^2\rangle -{\langle \theta \rangle }^2}$ vs $\varepsilon$. Near $\varepsilon =1.7$, $\langle \theta \rangle$ rapidly changes, signaling a conformational transition that is accompanied by pronounced fluctuations. (d) Configurations of FG nup at $\varepsilon =1.7$, represented by thick orange lines, closely resemble the conformational fluctuations observed in recent experiments [36].

Figure 2

(a) Schematic of a spherical cargo translocating through a model NPC with FG nups anchored at $z=0$. (b) Free energy difference, $\mathrm{\Delta }F\left(z\right)=F\left(z\right)-F\left(z_0\right)$, as a function of the cargo position $z$, for inert cargo (${\varepsilon }_{\mathrm{TF}}=0$) of radius ($R=5,8,11$). Here, $z_0=-40$. A free energy barrier is developed around the anchoring position of the FG nups due to steric interactions between the cargo and the FG nups. Consequently, the barrier is higher for a larger cargo. (c) Free energy profiles of interacting cargo of radius $R=8$ with different ${\varepsilon }_{\mathrm{TF}}$. For the interacting cargo, not only is the barrier at $z=0$ lowered, but also wells (the regions of $\mathrm{\Delta }F<0$) are created.

Figure 3

Current and translocation time of cargo through our model NPC (obtained numerically as described in Sec. 3c). We set $k=0$ and $p_0/\pi R_{\mathrm{pore}}^2=1\mu \mathrm{M}$. (a) Current $J$ as a function of the cargo radius $R$. At a given ${\varepsilon }_{\mathrm{TF}}$, the currents are reduced as $R$ increases. At a given radius, cargos more strongly interacting with FG nups yield larger currents. In the $R$ dependence of $J$, there exists a crossover from slow to fast decay that represents the size-selective permeability. The dashed line corresponds to the current amplitude at the crossover for an inert cargo. (b) Translocation time $\tau$ as a function of $R$. For an inert cargo, $\tau$ monotonically increases with $R$; its dependence on $R$ also displays crossover behavior (the dashed line is the translocation time at the crossover). In contrast to the inert cargos, the translocation times of interacting cargos have minima. The descending behavior of $\tau$ for small $R$ is a consequence of the well formation shown in Fig. 2.

Figure 4

Diagrams to characterize the cargo transport through NPC (a) in the absence of RanGTP ($k=0$) and (b) in the presence of RanGTP ($k={10}^3$) based on current and translocation time obtained numerically as described in Sec. 3c). Cargos with $R$ and ${\varepsilon }_{\mathrm{TF}}$ in the fast permeable region are efficiently transported through NPC, yielding high currents and short translocation time $\tau <{\tau }^{*}$. Boundaries between permeable and impermeable regions are indicated by setting $J=J^{*}$ or $J=10J^{*}$, respectively. Here the criteria $J^{*}$ and ${\tau }^{*}$ are marked in Fig. 3. (c) Schematic pictures of free energy profiles of translocation for inert cargo (gray dashed), and interacting cargo without RanGTP (black solid) and with RanGTP (red dashed). Here $\mathrm{\Delta }{F}_B$ is the central barrier height for an inert cargo. On the trans side, RanGTP dissociates a cargo from TF and replaces the free energy profile with that for an inert cargo. Upper panel: The rate-limiting process is the crossing of the central barrier; therefore, the presence of RanGTP hardly affects the translocation time. Lower panel: For a strongly trapped cargo inside the well, the rate-limiting process is the escape from the well. In this case, the presence of RanGTP accelerates the translocation.

Figure 5

(a) Hydrophobicity, charge, and FG-motif distributions on the monomers of a coarse-grained Nsp1. (b) Accumulated hydrophobicity $A_H\left(i\right)\equiv {\sum }_{j=1}^i{\eta }_j$, evaluated for the original amino acid sequence of Nsp1 (black line) and for the coarse-grained monomer sequence (red line), as function of the normalized bead index $i/N$. The numbers of beads are given as $N=617$ for real Nsp1 and $N=103$ for model Nsp1, respectively. The accumulated hydrophobicity is also normalized by the magnitude of its minimum value, $A_{H,\min }$. Bipartite nature of the hydrophobicity distribution is clearly shown; about half of beads toward the anchoring end are hydrophilic, while the other half of beads are hydrophobic. It also turns out that the distribution of hydrophobicity in the original sequence is preserved in coarse-grained FG nup. (c) Distribution of 12 binding sites for FG motifs on a cargo-TF complex. Binding sites are closely packed along two stripes beside a great circle on the spherical cargo. The distance between the stripes is set to $1.2a$.

Figure 6

Radii of gyration for various coarse-grained FG nups as a function of the monomer size $a$. We set the hydrophobic interaction strength to be $\varepsilon =1.7$. For all FG nups examined here, when $a$ is taken to be 1 nm, radii of gyration of various FG nups are compatible with experimentally measured Stokes radii. In simulations, the radius of the pore is set to be $R_{\mathrm{pore}}=25a$, yielding $R_{\mathrm{pore}}/R_g\left(\mathrm{Nsp}1\mathrm{m}\right)=3.55$, which is in reasonable accordance with experimental values, 3.83.

Figure 7

Probability distributions of work done by moving an inert cargo with radius $R=6a$ from $z_0=-40a$ to $z=0$ [see Fig. 2 and related explanation in main text]. The vertical dashed line is the free energy difference estimated through Eq. (C6). At the estimated free energy difference, the work distribution from the forward process $p_f\left(w\right)$ (black lines) intersects with the distribution of negative work from the corresponding backward process $p_b(-w)$ (red lines). This confirms the validity of the estimated $\mathrm{\Delta }F$.

Figure 8

The free energy landscapes for cargos of various sizes and interaction strengths.

Figure 9

The same diagrams with Fig. 4 of the main text, indicating distinct transport regimes for various values of $k$. Difference between the diagrams for $k=0$ and $k={10}^2$ is hardly appreciable. Also for the case of $k={10}^4$, each region in the diagram is almost the same as that for $k={10}^3$ case.