Percolation transition prescribes protein size-specific barrier to passive transport through the nuclear pore complex

Abstract

Nuclear pore complexes (NPCs) control biomolecular transport in and out of the nucleus. Disordered nucleoporins in the complex's pore form a permeation barrier, preventing unassisted transport of large biomolecules. Here, we combine coarse-grained simulations of experimentally derived NPC structures with a theoretical model to determine the microscopic mechanism of passive transport. Brute-force simulations of protein transport reveal telegraph-like behavior, where prolonged diffusion on one side of the NPC is interrupted by rapid crossings to the other. We rationalize this behavior using a theoretical model that reproduces the energetics and kinetics of permeation solely from statistics of transient voids within the disordered mesh. As the protein size increases, the mesh transforms from a soft to a hard barrier, enabling orders-of-magnitude reduction in permeation rate for proteins beyond the percolation size threshold. Our model enables exploration of alternative NPC architectures and sets the stage for uncovering molecular mechanisms of facilitated nuclear transport.

Fig. 1. Computational model of a composite NPC structure.

a Equilibration simulation. The computational model consists of a nuclear envelope (green), structured protein scaffold (cyan), and disordered FG-nups mesh consisting of 32 copies each of Nup145N (orange), Nsp1 (yellow), Nup49 (magenta), Nic96 (dark blue), and Nup57 (pink). Labels for the coordinate system (on the left) are in nm. Inset shows a magnified view of Nup57; each bead represents one amino acid. b Cross sections of the average FG-nup amino-acid density. The 3D density map was generated by averaging instantaneous configurations of the computational model every 0.1 μs over the final 6 ms fragments of two independent 7.5 ms simulations. The side and top view cross sections were additionally averaged along the y and z coordinate, respectively, within the [−7.5, +7.5] nm range. c Average density maps of individual FG-nup species generated using the same protocols as the density map shown in panel b. Black brackets define the "inner ring'' and "outer rings'' regions of the NPC scaffold. d Representative configuration of the NPC mesh, where individual FG-nups making at least one contact with another FG-nup are shown in blue and those without such contacts are shown in orange. Scale bar, 20 nm. e The fraction of nups forming at least one interchain contact, by species. Data presented as mean ± SD, based on N = 65,000 frames. The dashed line shows the fraction averaged over all species. An interchain contact was defined as having two residues from different chains within 0.8 nm of one another.

Fig. 2. CG simulations of passive diffusion across the NPC.

a To-scale structural representation of all proteins used for CG simulations of passive diffusion. b Side (top) and overhead (bottom) view of the simulation systems. Black and red lines indicate the approximate location of a cylindrical confinement potential of 25 and 50 nm radius, respectively. Blue crosses mark the two initial locations of the protein (at z = ±50 nm). c Rigid-body model of a maltose-binding protein (MBP) where each bead represents one amino acid. Pc1, pc2, and pc3 denote the principal axes of the protein. Black scale bar, 1 nm. d Center-of-mass z coordinate of each protein (colors defined in panel a) versus simulation time. The simulation traces in the two columns differ by the initial placement of the protein. The simulations were performed in the presence of a 50 nm-radius confinement potential. e Normalized distribution of the CoM z coordinate of thioredoxin. The black dotted and solid lines show the distribution extracted directly from the simulations and the symmetrized distribution, respectively. The blue line shows a symmetrized distribution for the simulation carried out in the absence of the FG-nup mesh. f Potential of mean force (PMF) for thioredoxin transport across the NPC. A PMF barrier is defined as the average value within ∣z∣ < 5 nm. g PMF barrier versus protein molecular mass determined from CG simulations of protein diffusion through our composite Lin2016 NPC model (black squares) and the model devoid of all FG-nups (blue circles). Data presented as mean values ± the average point-by-point difference of the unsymmetrized PMF values from − 50 < z < 0 nm and 0 < z < 50 nm intervals. PMF curves derived from N = 140,000 frames. N = 10 points along those curves defined the mean barrier, and N = 100 points defined the average point-by-point difference. Line shows a power-law fit to the data. h Mean first-passage time (MFPT) versus protein molecular mass, both axes logarithmic. Power-law fits are shown as dashed lines in panels g and h. In panel h only, the power-law fit to the "nups present'' data included proteins up to 62 kDa (hemoglobin).

Fig. 3. Timescale of protein crossings.

a Successful translocation of a maltose-binding protein (MBP, red) through the NPC. Three instantaneous configurations of the NPC are shown using white for the FG-nups, cyan for the protein scaffold and green for the lipid bilayer. The circle indicates the location of the MBP protein at each configuration. Red and pink beads illustrates the configurations explored by the protein between the instantaneous configurations (sampled every 25 ns). Scale bar, 20 nm. b CoM z coordinate of MBP simulated under a 25 nm-radius confinement potential. Two crossing events form this trace are shown in detail in panels c and d. c Zoomed-in on the crossing event trace. The same event is illustrated by snapshots (i, ii, iii) in a. d Example of another crossing event. The gray rectangles in panels c and d illustrate the time interval defined as a crossing time for the analysis shown in e. e Distribution of crossing times for six protein species. N specifies the total number of crossing events used to construct each histogram. Each normalized histogram was constructed using 15 evenly-spaced bins, from 0 to 30 μs, and the crossing time data from both confinement potential simulations. The average of each histogram is shown as a vertical line. The dashed lines show the distributions of first-passage time for a freely diffusing particle of the same diffusion constant as that of the corresponding protein.

Fig. 4. Void model of the translocation barrier.

a Void analysis map of an instantaneous NPC configuration computed using a spherical probe of 22.4 Å radius. The volume available to accommodate the probe (void) is shown in red, the volume excluded in blue, FG-nups in white, the scaffold in cyan and the lipid bilayer in green. The image shows a 2D section of a 3D map. b The fraction of the NPC volume that can accommodate the probe without clashes as a function of the pore axis coordinate. The fraction was computed by splitting the void analysis map into cylindrical segments of 50 nm radius and 0.6 nm height, co-axial with the pore. The data shown were computed for the instantaneous NPC configuration displayed in panel a. c Trajectory-averaged probability of accommodating the probe as a function of the pore axis coordinate, $\overline{P}\left(z\right)$, computed by averaging instantaneous void analysis maps over the last 6 ms of the NPC equilibration trajectory, sampled every 1.0 μs. d PMF of the spherical probe derived by void analysis. e Cut-away view of the NPC system containing no nups, only the scaffold and nuclear envelope potentials (gray). f Symmetrized PMF of three protein species (aprotinin, magenta; thioredoxin, orange; and hemoglobin, light blue) derived from brute-force CG simulations (left) and of the three spherical probes of approximately the same radius (R_p = 12.75, 15.71, and 30.04 Å) derived from void analysis (right). g PMF barrier versus protein mass. Interpolation was used to find void analysis PMF barriers for the proteins simulated using the CG method (Supplementary Fig. 10). Lines are guides to the eye. Both axes use logarithmic scale. h–j Same as in e–g but for the NPC model that includes nups.

Fig. 5. Fokker-Planck model of passive diffusion and the percolation transition.

a, b Comparison of the mean first-passage times (MFPT) calculated from CG simulations to those from our Fokker-Planck void analysis model for the Lin2016 structure without FG-nups (a) and with FG-nups present (b). The black line indicates perfect agreement. Data in a, b presented as mean values ± SEM, based on traces involving N = 140,000 frames. c MFPT from CG simulations (blue) and using our Fokker-Planck approach (red) as a function of protein molecular mass. Note the logarithmic scale of the axes. Power law fits, and their slopes, specified in the figure. d Same as in panel c but for the Lin2016 structure with FG-nups present. The black line shows a single fit to all of Fokker-Planck data, see text for the functional form. The three regions (i, ii, iii) correspond to power law, transition and exponential scaling behavior. e Connectivity map of an instantaneous NPC configuration computed for three different protein probe radii (specified under each map). Each map generated for ∣z∣ < 40 nm. f Probability of finding a complete, open path through the NPC versus protein radius and molecular mass (red, top) obtained from the analysis of NPC equilibration trajectories. Note the logarithmic scale of the horizontal axes. The three regions (i, ii, iii) correspond to those in panel d. g Location of each FG-nup species in one sixteenth of the CG model. Scale bar, 10 nm. h MFPT versus molecular mass for an NPC model devoid of one FG-nup species (colors) and with all FG-nups present (dashed black line). i MFPT for the deletion mutants normalized by the MFPT of the complete NPC model. j PMF of a 64 kDa (r = 3.0 nm) protein for all species present NPC (dashed black line) and the Nup145N deletion mutant (orange). The gray shaded region (∣z∣ < 20 nm) indicates the region used to define the pore length of the central channel. All data in this figure were obtained under a 50 nm-radius confinement potential. Interpolation was used to express the results of the Fokker-Planck void analysis model in terms of molecular mass (Supplementary Fig. 12).

Fig. 6. Passive diffusion of proteins through yeast NPC.

a Representative configuration of the CG model of the integrative Kim2018 structure of yeast NPC. The FG-nups are shown in white, protein scaffold in cyan and lipid bilayer in green. b Cross-section of the average FG-nup amino-acid density. The 3D density map was generated by averaging instantaneous configurations of the Kim2018 computational model every 0.1 μs over the final 6 ms fragment of the 7.5 ms simulation. The cross-section was averaged along the y coordinate within the (−7.5, +7.5) nm range. c Potentials of mean force along the pore axis of the Kim2018 model, calculated by void analysis for probe radii of 12.75, 15.71, and 30.04 Å (magenta, orange, and light blue, respectively). d PMF barrier as a function of protein mass, calculated by void analysis for the Kim2018 (red) and Lin2016 (gray) models. e Mean first-passage time versus protein mass for the Kim2018 (red) and Lin2016 (gray) models. The black line is a single fit to all of the Kim2018 data, the functional form is provided in the text. The fitting yields τ₀ = 3.8 μs/Å, R₀ = 19.0 Å, and α = 2.05. f Probability of finding a complete, open path through the Kim2018 model of the NPC versus protein radius. The three regions (i, ii, iii) correspond to power law, transition and exponential scaling behavior. To enable direct quantitative comparison of the Kim2018 and Lin2016 models, we defined the central channel to span the ∣z∣ < 20 nm region in both models. All data reported in this figure were obtained under a 50 nm-radius confinement potential.