Rapid Brownian Motion Primes Ultrafast Reconstruction of Intrinsically Disordered Phe-Gly Repeats Inside the Nuclear Pore Complex

Conformational behavior of intrinsically disordered proteins, such as Phe-Gly repeat domains, alters drastically when they are confined in, and tethered to, nan channels. This has challenged our understanding of how they serve to selectively facilitate translocation of nuclear transport receptor (NTR)-bearing macromolecules. Heterogeneous FG-repeats, tethered to the NPC interior, nonuniformly fill the channel in a diameter-dependent manner and adopt a rapid Brownian motion, thereby forming a porous and highly dynamic polymeric meshwork that percolates in radial and axial directions and features two distinguishable zones: a dense hydrophobic rod-like zone located in the center, and a peripheral low-density shell-like zone. The FG-meshwork is locally disrupted upon interacting with NTR-bearing macromolecules, but immediately reconstructs itself between 0.44 μs and 7.0 μs, depending on cargo size and shape. This confers a perpetually-sealed state to the NPC, and is solely due to rapid Brownian motion of FG-repeats, not FG-repeat hydrophobic bonds. Elongated-shaped macromolecules, both in the presence and absence of NTRs, penetrate more readily into the FG-meshwork compared to their globular counterparts of identical volume and surface chemistry, highlighting the importance of the shape effects in nucleocytoplasmic transport. These results can help our understanding of geometrical effects in, and the design of, intelligent and responsive biopolymer-based materials in nanofiltration and artificial nanopores.

The single spoke with disordered FG-Nups being in their initial, cramped configurations. FG-Nups are modeled as bead-spring. Upper and lower spokes are colored in dark and light green, with the tethering point of each FG-Nup colored in pink. Each bead in the FG-Nup represents a monomer with the following coloring scheme: brown is hydrophobic (HB), yellow is hydrophobic-negative (HN), cyan is hydrophobic-positive (HP), red is purely negative (PN), blue is purely positive (PP), and white is hydrophilic with zero net charge (HL).

Supporting Information
Brownian dynamics (1) used to simulate the 3D model of the yeast NPC central channel with eight spoke domains including all of the central FG-Nups. The 3D map of the NPC central channel (2), the upper and the lower half-spoke domains along with the tethering points of FG-Nups therein (3), and the sequence of the disordered domain of each yeast FG-Nup along with their anchor domains (4) are incorporated in our model.
The central channel is composed of eight identical spoke domains, composed of upper and lower halfspokes, and five FG-Nups per half-spoke (3). In this work, we first made these two half-spokes with all of known FG-Nups tethered to them (Fig. S1), and then the entire spoke domain was rotationally repeated eight times around the central axis of the pore to form the central channel. This procedure is consistent with the proposed assembly of the NPC in vivo (3,5). The disordered domains of FG-Nups (4,6) were extracted from the original sequences available in UniProt (see Table1). The anchor domains were taken from the work of Yamada et al. (4), while positions of tethering points on the spoke domain were calculated from the work of Alber et al. (3). The initial configurations of disordered domains were set straight, but cramped so that they could be fit within the diameter of the channel, consistent with our previous works (Fig. S1) (7,8). Nonetheless, by running multiple independent initial conditions, we found that final conformations of FG-repeats are not sensitive to the initial configurations.
Eight spoke domains are modeled as rigid elements, while intrinsically disordered domains of FG-Nups are modeled as bead-spring elements (Fig. S1) using discrete wormlike chains (dWLC) with the persistence length of 4.3 p l = Å (9). Each bead represents a monomer consisting of five successive amino acids (AAs), featuring the net charge of constituting AAs at physiological pH, hydrophobicity, and the total vdW volume (Fig. S2).
There are six different types of monomers considered in the current model: i) hydrophobic (HB), which have one or two Phe residues with the rest of residues being polar; ii) hydrophobic-negative (HN), which have both Phe residue and at least one negatively charged residue (Asp, Glu), iii) hydrophobic-positive Figure S2. The 2D representation of an octagonal arm of the yeast NPC with a cytoplasmic filament, a spoke domain, and a nuclear basket rod. Each bead represents a monomer, which includes the positive and negative charges as well as the hydrophobicity of constituting residues. Positive and negative charges of a monomer are represented by adequate number of blue and red dots inside the oval. Hydrophobic monomers are tanned.
(HP), having Phe and at least one positively charged residue (Arg, Lys), iv) purely negative (PN), have at least one negatively charged residue without any hydrophobic residue, v) purely positive (PP) with at least one positively charge residue without any hydrophobic residue, and vi) hydrophilic (HL), which have only polar residues with the net zero charge.
The force-extension constitutive equation for dWLC is (7,8), where c dL is the segmental contour length for a single monomer and equals the sum of lengths of the N-to C-termini of five successive AAs (10).
The bending potential for dWLC is taken as with the pairwise cutoff radii being the sum of vdW radii of two approaching monomers (11). The same potential is applied between monomers and the channel rigid wall to prevent penetration into the wall and represent the confinement.
The electrostatic interaction between two monomers with radii i R and j R is modeled using the linearized Poisson-Boltzmann equation (12), i.e. Deybe-Hückel approximation, with the central net charges ( , ) i j q q and the additive pairwise potential (13,14): Here 0 ε is the vacuum permittivity, r ε the relative permittivity of the cytoplasmic environment that amounts to 45.0 (15), and 1 κ λ − = is the inverse Debye screening length, which is 1.0 nm for the cytoplasmic ionic strength (16).
However, if the surface-to-surface distance of two monomers is less than λ , then the unscreened Coulombic potential is applied (17). The FG-repeat domains are mainly positively charged as opposed to NTRs that possess net negative charges (18).
Hydrophobic effects are relatively more subtle to model since they are mainly entropic in nature. In the current study, we tested a wide range of hydrophobic force fields (19)(20)(21)(22)(23)(24). To determine the optimized force filed for FG-Nups five criteria were implemented: i) availability of the force field parameters for FG-Nups, ii) ability to represent the overall known physiological properties of FG-Nups, iii) minimal number of parameters involved, iv) numerical efficiency, and v) simplicity of implementation in the simulation. Remarkably, we found the double-parameter exponential potential energy was the most optimized force field for FG-Nups. This potential is proposed for purely hydrophobic interaction energy between two hydrophobic surfaces in a polar solution like water (25): representing the hydrophobic affinity per unit area and 1.0 nm µ = being the characteristic length of interaction.
In this work, we take into account the hydrophobic interaction between all Phe residues-the most abundant hydrophobic residue within the disorder domains of FG-Nups. Thus, the area of interaction is chosen as the solvent-accessible surface area (SASA) of the nonpolar side chain of Phe in its extended conformation, i.e. 2 175 Å (11).
The main parameters used in potential energies as described above are the same as those used in previous 2D model. For example, the persistence length of discretized wormlike chains, the repulsion potential energy, and the hydrophobic interaction potential energy could be borrowed from our 2D model (7,8).
The entire systems is placed in an aqueous bath with cytoplasmic viscosity and ionic strength at physiological pH = 7.3 and the body temperature of 37 C. The integrated form of the coupled Langevin equations of motion are then solved explicitly in time for all monomers: being the cytoplasmic viscosity and hyd i R the hydrodynamic radius of the monomer (26). In addition, i Λ is the random displacement vector that represents the random motion of the monomer in implicit solvent and follows a Gaussian distribution with a zero mean and the variance of 2 .
Cargos are also modeled as rigid objects with desired dimensions and shapes, globular or elongated. The repulsion, electrostatic, and hydrophobic potentials are applied between the cargo surface and monomers/wall in the same manner outlined above for monomer-monomer and monomer-wall interactions. To represent NTRs bound to the cargo adequate number of hydrophobic binding spots are placed on the cargo surface (see Table 1). The hydrophobic binding spots are restricted to only one side of the cargo surface, consistent with the localization of the hydrophobic binding patches on the convex surface of the kap β (27). To model the NTR-FG hydrophobic interactions, the same attractive interaction potential like the one used to model hydrophobic Phe-Phe interaction, , is used. However, the NTR-Phe hydrophobic affinity is set stronger than Phe-Phe affinity (6) to yield the nanomolar apparent dissociation constant between NTRs and FG-Nups (27) through This yields a hydrophobic affinity density between a NTR binding spot and a FG-motif of The characteristic length of hydrophobic interaction between NTR binding spot and FG, however, remains the same as that for Phe-Phe hydrophobic interaction and is Furthermore, the area of interaction is chosen as the solvent-accessible surface area (SASA) of the nonpolar side chain of Phe in its extended conformation, i.e. . This formulation has been successfully applied to elongated macromolecules in the context of Brownian dynamics (29). The hydrophobic side of the cargo is faced toward the FG-repeats upon entering the central channel.
To get statistically reliable results, independent simulations are launched with different initial conditions, meaning, the initial random seeds for each simulation is different.
Throughout the simulations, all physical parameters are dimensionless with respect to energy, time, and length with the unit values of 1.0 B k T , 0.1ns , and 1.0 nm respectively. After an equilibration of 10 µs, the simulation started and continued for 20 ms, long enough to span the entire nucleocytoplasmic transport process, and presumably allowing for the disordered domains to assume all possible conformations that can be adopted in the absence of any cargo.

Discretization scheme
To ensure our results are not dependent on spatial resolution of the bead-spring model (the level of discretization), a sensitivity analysis was performed. Specifically, a coarser discretization level, i.e. 15 AAs/monomer, was performed and the time evolution of the averaged radii of gyration of FG-repeats, g R , was examined over 400 s µ . It appears that the time evolution as well as the average values of g R exhibit similar patterns in both of the discretization levels (Fig. S3), and thus the model is not sensitive to the discretization level, suggesting that the spatial resolution does not bias the conformational behavior of FG-repeats. It must be noted that the discretization cannot go too fine as the model should be coarse enough to assure that the assumption of eliminating atomistic hydrogen bonds between solventsolvent, solvent-reside, and residue-residue is valid. We thus adhere to a five-AAs/monomer coarsegraining level throughout the presented simulations, which is coarse enough to safely ignore the explicit hydrogen bonds, treating them with the stochastic Langevin treatment, and is fine enough to show the dynamics and distribution of hydrophobic and charged residues.

Pre-exposed hydrophobic and charged monomers of cramped FG-repeats in the confined geometry of the central channel
The lengths of FG-repeats localized to the NPC central channel are in the range of 85 to 272 nm (Table 1) with an average length of 180 nm. Thus, the initial configurations of FG-repeats in our simulations are cramped so that they can be fit in the 40-nm wide central channel, leading to hydrophobic as well as the charged monomers being pre-exposed.
We thus asked whether the formation of the observed dense FG-meshwork inside the central channel was potentially an artifact caused by the pre-exposed hydrophobic as well as charged residues from the same or different FG-Nups due to their initially cramped configurations within the channel. To address this question, we removed the initial pre-exposing by artificially enlarging the channel diameter, and thus, relaxing the crampedness while keeping the positions of tethering points unaltered. The artificial channel had the same height as the central channel (40 nm) but its diameter is 550 nm (Fig. S4-A), a bit more than twice the longest disordered FG-Nup, i.e. Nup 100 (4). The initial configurations of disordered FG-Nups were set to their fully extended lengths, consistent with previous molecular dynamics studies performed in an open (unconfined) geometry (30,31), ensuring that hydrophobic as well as charged residues along the same or different FG-Nups were not initially pre-exposed ( Fig. S4-A).
FG-Nups within the artificial channel were first equilibrated for 5 µs (Fig. S4-B), during which they rapidly shrunk and deposited on the channel wall. After that, the channel diameter was shrunk in a stepwise manner to reach its native dimension, i.e. 40 nm. In each step, FG-Nups inside the new artificial channel were equilibrated for another 5 µs (Fig. S4-C to G). Finally, after the diameter was downsized to 40 nm the FG-meshwork formed and occupied the channel with similar average density and conformations as those in the native channel. This proved that formation of the FG-meshwork inside the native central channel in our model is not an artifact of the pre-exposed hydrophobic or charged residues. Figure S4. Starting from an artificially large channel, the channel diameter is decreased stepwise, and FG-Nups are equilibrated in each step. A) The initial configuration of the artificial central channel with unstructured domains of FG-Nups being in their fully extended length configuration. The diameter of the artificial channel is 550 nm, which guarantees monomers do not meet nor the unstructured domains cross each other. The height of the artificial channel is kept the same as that for the original yNPC central channel. B) The artificial channel after equilibration. C-G) The diameter of the artificial channel is shrunk stepwise to the final value of 40 nm. In each step, the system is equilibrated for 5 µs before shrinking the diameter to the next step. In the final step, the channel-filling meshwork is reached, as in the original NPC channel.

The average configurational entropic cost sets the theoretical minimum number of hydrophobic binding spots: larger macromolecules need more NTR to overcome permeability barrier.
When a large active cargo enters the central channel, the disordered FG-repeats in its vicinity experience a reduction in their configurational entropy to locally accommodate the cargo. The decrease in configurational entropy of a single monomer, mon s Δ , can be estimated by calculating the reduction in the available space to the monomer (32) as: where c V is the volume of the active cargo, and 0 V the available space to the monomer before the cargo enters the channel, taking into account the excluded volume effects of other monomers: . Therefore, at the cell temperature the free energy gain required to overcome this entropic barrier must be greater than 12 B k T . Considering the NTR-FG-repeats hydrophobic interaction, at least six hydrophobic patches would be required on the surface of a 15-nm cargo for the energy gain of 12 B k T . However, in reality this number of hydrophobic patches is not sufficient for this cargo to overcome the entropic barrier and perforate the FG-meshwork, at least during the simulation times in the current study. Yet, this calculation can set a theoretical lower bound to estimate the minimal number of hydrophobic binding spots required for a cargo with a certain size to overcome the entropic barrier of the NPC. It should be remarked, however, that this is a rough estimation; for simplicity we used an average value of the density for the monomers inside the channel and also we ignored the end-tethering of the FG-repeats, which otherwise would not be straightforward to find the entropic cost. Moreover, the solvent translation entropy was ignored in our implicit solvent model. Nevertheless, this theoretical approach shows that larger cargos need more number of hydrophobic binding spots, and thus more NTRs, to overcome the permeability barrier, as was observed experimentally (33).   Figure S5. The dependence of the intra-FG-meshwork zone formation on the diameter of the channel. The channel diameter is increased from 42 nm in panel A to 64 nm in panel L. From each panel to the next, the increment in the channel diameter is 2 nm. Colorbars in all panels have the same limits (0 to 2.5 M), and x and y axes' limits are -30 to 30 nm.   Figure S8. The pair distribution function (PDF) is used to calculate the frequencies and distances of particles' pairwise separations inside a fluid with a given structure and density relative to a fluid with a completely random structure at the same density. Thus, it can provide profound insights into the microstructure of the FG-meshwork in the current study. The monomer-monomer PDF plot of the FG-meshwork for A) the entire meshwork (rectangles), B) high-density zone in the middle (circles), and C) low-density zone near the wall (triangles). Figure S9. The reconstruction patter of the non-FG-meshwork, where all of the Phe residues are replaced by polar residues with the same vdW dimension. The cargo in this case is a 20nm NTR-bearing globular macromolecule.

Movies' legends
Movie S1: Concentration fluctuations of unstructured domains in the central channel.