Conceived and designed the experiments: TRL AS IB. Performed the experiments: TRL. Analyzed the data: TRL AS IB. Wrote the paper: TRL AS IB.
The authors have declared that no competing interests exist.
The nuclear pore complex (NPC) is the gate to the nucleus. Recent determination of the configuration of proteins in the yeast NPC at ∼5 nm resolution permits us to study the NPC global dynamics using coarse-grained structural models. We investigate these large-scale motions by using an extended elastic network model (ENM) formalism applied to several coarse-grained representations of the NPC. Two types of collective motions (global modes) are predicted by the ENMs to be intrinsically favored by the NPC architecture: global bending and extension/contraction from circular to elliptical shapes. These motions are shown to be robust against tested variations in the representation of the NPC, and are largely captured by a simple model of a toroid with axially varying mass density. We demonstrate that spoke multiplicity significantly affects the accessible number of symmetric low-energy modes of motion; the NPC-like toroidal structures composed of 8 spokes have access to highly cooperative symmetric motions that are inaccessible to toroids composed of 7 or 9 spokes. The analysis reveals modes of motion that may facilitate macromolecular transport through the NPC, consistent with previous experimental observations.
The nuclear pore complex (NPC) serves as the sole gateway to the cell nucleus, and its proper functioning is therefore crucial for gene expression and many vital signaling pathways. Although it is typically circular, the overall structure of the NPC has been observed to change in response to the presence of cargo. Recently, the molecular architecture of the yeast NPC, including the shapes and relative positions of its constituent proteins, has been resolved. These new structural data provide us with a first opportunity to construct an accurate dynamical model of a macromolecular machine containing hundreds of proteins. By modeling the NPC as a network of masses connected by springs, we investigate its probable large-scale dynamics. We start from a very coarse model and gradually refine it, observing how the structural details influence the calculated dynamics. We find that the NPC dynamics are quite similar to those of a flexible toroid with an uneven mass distribution, and that the 8-fold symmetry that is universally observed in NPCs enables them to undergo certain collective motions that are inaccessible to structures of other symmetries.
Any macromolecule entering or exiting the nucleus must pass through a nuclear pore complex (NPC). NPCs are formed by hundreds of proteins organized into cylindrically symmetric pores that act as gateways to the cell nucleus. The NPC has a mass of ∼50 MDa in yeast and up to 125 MDa in vertebrates, comparable to a small organelle. The yeast NPC is a ring of about 100 nm diameter, creating a central pore of ∼30 nm diameter. It contains approximately 450 proteins, termed nucleoporins, arranged into eight similar “spokes” extending from the NPC's central channel to its outer perimeter (
The yeast NPC consists of 456 proteins (nucleoporins), shown here as spheres, that form eight identical spokes. The cytoplasmic face of the NPC (facing the viewer) is colored pink, and the nuclear face is colored light blue. A single spoke is highlighted in red/blue. The lumenal ring, consisting of sixteen POM152 nucleoporins is colored orange. Half of the NE fragment is shown in silver, and the border of the NE fragment is colored black.
The NPC is somewhat plastic
Although conventional models of nuclear transport focus primarily on the local interactions of FG nucleoporins with karyopherins
A dynamical model that has enjoyed considerable success in elucidating the machinery of biomolecular assemblies near physiological conditions is the elastic network model (ENM)
Here, we examine the dynamics of the NPC at several levels of coarse-graining with a newly developed version of a classical ENM. We begin by modeling the NPC on the coarsest scale possible, i.e., representing the full complex as a flexible toroid. We then increment the level of detail to include non-uniform mass distributions, spokes, and finally the full NPC architecture. We find that the global modes of the NPC are largely captured by the modes of a simple toroid, suggesting that the global dynamics of the NPC does not depend on the details of the NPC architecture. We note that the eight-fold symmetry permits ease of certain cooperative motions that are inaccessible to structures with other rotational symmetries. Two highly robust modes of collective deformation emerge from the analysis, providing insight into the details of NPC motions that are observed with low-resolution tomography
We began by modeling the NPC as a uniform toroid consisting of about 3800 identical point masses spaced approximately 3 nm apart and enclosed in a torus with characteristic dimensions comparable to those of the NPC (major radius of 30 nm and minor radius of 13 nm). Each of the point masses was connected to its neighbors within 12 nm via elastic springs of uniform spring constant, and the normal modes of motion were calculated as described in
A system in thermal equilibrium is expected to fluctuate along all of its normal modes, with arbitrary phase, and amplitudes that are determined by the normal mode frequencies. The modes with lowest frequency are the most easily accessible from energetic point of view, and therefore dominate the global dynamics. The first mode of the homogeneous toroid is a 2-fold degenerate
(A) The first mode is a two-fold degenerate bending of the toroid. The centroids are colored blue to red to indicate increasing mobility. (B) The second mode is a two-fold degenerate stretching of the toroid. (C) The third mode is a non-degenerate rolling mode. The centroids here are colored from blue to red as a function of initial distance from the center of the toroid. The mass centroids on the top (visible) surface move along the toroid surface toward the central channel, and the centroids on the opposite side move away from the central channel toward the toroid perimeter. (D) Mode 6 is a non-degenerate dilation.
Owing to the symmetry of the toroid, the majority of the slowest modes are two-fold degenerate; these motions are not invariant under rotations about the toroid central axis, and the two-fold degeneracy is required to span the plane perpendicular to the cylindrical axis. Non-degenerate modes in the uniform model, on the other hand, correspond to motions that are invariant under rotations about the central axis. There are three non-degenerate modes among the slowest 20 modes of the uniform toroid model. The first of these is the rolling mode 3 (
Next, we investigated the effects of the mass distribution on the toroid modes. We used three models, each of which preserves the particle density and network topology of the uniform-density model, but has a different distribution of mass within the toroid. The mass was made to be a linear function of position such that the minimum and maximum values of centroid masses were the same in all three models.
In the first case, the mass density is a linear function of distance from the toroid surface, such that the toroid's mass is concentrated about its interior and decreases toward the toroid surface. This distribution preserves much of the symmetry of the uniform-density toroid and therefore also reproduces its dynamics. The 10 slowest modes are qualitatively identical to those of the uniform-density model described above, and the mobilities induced by the first three modes have correlations of 1.000, 1.000 and 0.903 with those induced by their counterparts in the uniform-density model. Thus, biasing the mass toward the toroid's interior does not affect the slow modes.
In the second case, the toroid's mass density is made a linear function of radial coordinate, such that the mass is concentrated on the surface facing the central axis. Here, too, the slow modes are qualitatively identical to those of the uniform-density model, and the mobilities induced by the slowest three modes exhibit correlations of 1.000, 0.998 and 0.989 with their uniform-density counterparts. The earliest deviation from the uniform-density model is the order of modes 10 and 12: Although the shapes of these modes are identical in both models, their positions within the eigenvalue spectrum are swapped.
A more discernible deviation from the uniform-density model is observed when the toroid mass density varies linearly with the axial coordinate, such that the upper half of the toroid has twice the mass of the lower half. The mobilities induced by the three slowest modes in this case have correlations of 0.931, 0.442 and 0.923 with their uniform-density counterparts. The difference in mobility can be detected starting with the first mode, which preferentially mobilizes the more massive face of the toroid (
(A) Three views are shown for the second mode of a toroid with axially varying mass density. In each view, the z-axis is shown (by the dot or arrow). The uneven mass distribution causes increased mobility on the lighter side of the toroid (compare to
We further considered toroids composed of 7, 8 or 9 loosely connected spokes of uniform mass density. These models were constructed using the same characteristic toroid dimensions as those used for the continuum model, and with an inter-node spacing of 3 nm. To emphasize the physical division of the toroid into smaller components, distinctive gaps of 5 nm were left between adjacent spokes and the inter-spoke spring constant was taken to be 1% of the intra-spoke spring constant. We found that the correlation between the modes of the spoke-based models and those of the uniform-density toroid increases with the number of spokes.
The three panels show the correlation cosines between the slowest modes of toroids composed of 7 (panel A), 8 (panel B) and 9 (panel C) uniform-density spokes with those of a uniform toroid. Perfect agreement between the modes of two models would be indicated by the unit matrix. For clarity, the blocks corresponding to different modes are separated by thin lines. The 2×2 blocks refer to 2-fold degenerate modes, and the smaller squares within each of these capture the mode overlap as well as an arbitrary phase separation. Here it can be seen that increasing the number of spokes reduces the number of off-diagonal elements in the matrix, indicating a higher overlap between the two sets.
The differences in the models become more prevalent starting with the sixth mode (
After exploring the intrinsic dynamics of simple toroids and systems of spokes, we investigated the slow modes of the yeast NPC using an ENM. This model highlights how the relative positioning and mass distribution of the nucleoporins impact the inherent dynamics of the NPC. As detailed in
The slow dynamics is dominated by the first two modes (
Three views of the two slowest modes of the NPC without (A and D) and with (B and E) the NE fragment are shown. The z-axis is shown in panel A, and orientations are the same in panels B, D and E. Mode 1 is a “stretching/elongation” mode that mobilizes the nuclear face (z<0) more than the cytoplasmic face, whether or not the NE fragment is included in the model. Mode 2 is a “bending” mode that elongates the NPC in the absence of the NE (D), but not when the NE is included (E). For a clearer visualization of the motion, the lower half of the structure is shown in translucid colors in panel E. Scaled mobilities are plotted against axial position in panels C and F for the NPC alone (blue) and the NPC/NE (red). The addition of the NE decreases mobility of the POM rings in the first mode, as shown by the absence of a peak near z = 0 in panel C. In the second mode, the NE creates a more even distribution of motion in the cytoplasmic and nuclear faces, as shown by the peaks near z = +/−12 for the red curve in panel F.
Further comparative analysis reveals that each of the 10 slowest NPC modes can be matched directly with one of the slow modes of a toroid with axially dependent mass density, but this similarity vanishes in the higher frequency modes. In the NPC model, the higher modes tend to preferentially mobilize the least constrained proteins, particularly the 8 copies of Nup85 on the nuclear face and the ring of Pom152 that wraps around the NPC equator (colored orange in
In our final model, we surrounded the yeast NPC with a fragment of the NE. The NE fragment included in our model is only a stand-in for the full NE, which we assumed is placid far away from the NPC. Our NE model consists of 2070 mass centroids, each of which represents an area of about 0.6 nm2 on the NE surface. The NE mass centroids are connected to each other and to proximal nucleoporins via elastic springs, and the NE fragment is given periodic boundary conditions as described in
The dynamics are in this case also dominated by the two slowest modes, both of which are two-fold degenerate (
One of the primary effects of the NE on the NPC dynamics is the stabilization of the POM rings. Each of the first two modes of the NPC/NE has 0.87 overlap with its counterpart in the NPC modeled alone, and much of the difference is owed to decreased motion of the POM rings in the presence of the NE. Although the POM nucleoporins are mobile in the first mode of the NPC/NE, in the second mode they move axially but not radially. The reduced motion of the POM nucleoporins is exhibited throughout the slow modes and results in dynamics for the NPC/NE that is not present in the NPC-only model. The tradeoff that comes with increased stability of the POM rings is increased mobility in other outlying nucleoporins, such as Nup82, located on the cytoplasmic face, and nuclear copies of Nup85.
We next used our model to investigate which nucleoporins have the largest influence on NPC motions. We constructed 30 new models, each lacking all copies of one protein from the NPC, and investigated the resulting dynamics in terms of the dynamics of the unperturbed NPC.
The dominant modes of the 30 structures with deleted nucleoporins have overlap values between 0.534 and 0.920 with their counterparts in the unperturbed NPC. The most significant changes in dynamics coincide with deletions of proteins that are found in the NPC scaffold. The proteins Nup157 and Nup192 each significantly affect the motions and are both centrally located in the NPC and form the “inner rings” of the scaffolding
Another set of scaffold proteins that affect the NPC dynamics are those that form the Nup84 complex forming the outer rings of the NPC scaffold
The plasticity and large-scale conformational changes of the NPC are topics of considerable recent interest
In the present study, we have examined the global dynamics of the NPC using the recently determined yeast NPC architecture to construct an elastic network model, which in turn has been tuned to retrieve mechanisms of motions that are least sensitive to structural details. Our investigations reveal a set of equilibrium motions that are intrinsically accessible to toroidal structures modeled as elastic materials. As illustrated in
Our study is explorative in that it is based on a theory of normal modes and a low-resolution structural model. Although the structure was determined by satisfying restraints derived from experimental observations about the NPC, its low resolution imposes limits on its accuracy and thus its utility. Nonetheless, many of the gross features of our predictions are reproduced in even coarser models: the mass distribution and 8-fold symmetry of the NPC are the key elements in determining its most likely equilibrium fluctuations. We also had to assume the harmonic potential that we have used in all of our calculations and which provides all of the resultant motions. Similar potentials have time and again proved their worth in coarse-grained dynamical calculations of macromolecules. Our key results are robust to variations in the model and themselves provide testable predictions as to the nature of large-scale motions of the NPC.
To provide a more accurate description of the detailed dynamics of the NPC, our model must be refined using a higher resolution structure, not yet available. Even better descriptions would include information about the elastic properties of the individual proteins, as well as details about the FG repeats in the FG nucleoporins, the nuclear basket, and cytoplasmic filaments. Such refinements inevitably require the accumulation of more data, and efforts are currently under way to produce a higher-resolution structure of NPC. More importantly, our work demonstrates that the dynamics of biological systems can be modeled at the nanometer scale, provided that structural data are available.
For the system's equilibrium conformation, we begin with the configuration of 456 nucleoporins in the yeast NPC
Because the NPC functions embedded in the NE, we also consider the local effects of the NE when modeling NPC dynamics. The NE consists of two parallel lipid bilayers separated by a ∼40 nm thick perinuclear cisterna. In the vicinity of the NPC, the cytoplasmic and nuclear faces of the NE merge, forming a curved surface that resembles the axial face of a torus with characteristic radii of 54 nm and 15 nm (see
We propose a variation of the anisotropic network model (ANM)
We define the mobility,
The network topology defined by connecting pairs of amino acids that are separated by a distance of roughly 15 Å or less has been shown in various applications to lead to structural dynamics that agree well with experimental data
Using the nucleoporins only, we vary the cutoff distance from 10 nm to 18 nm and calculate the resulting dynamics. We find that the slow dynamics are most robust against changes in the spring constant between 13 nm and 15 nm, and we select a cutoff distance of 14 nm for our model. We separately model the NPC with a fragment of the NE, varying the cutoff distance of the NPC and NE from 12 nm to 18 nm and from 8 nm to 12 nm, respectively. We find that the slow NPC dynamics are least sensitive to the details of the NE boundary conditions when the NE cutoff value is 8 nm, and we empirically select the values of 14 nm for nucleoporin centroids and 8 nm for NE centroids in the mixed NPC/NE models. If the cutoff distance for NE centroids is small compared to that for the nucleoporins, the NE has little perturbative effect on the NPC slow modes, but if the NE centroid cutoff distance is too large, the NPC slow modes are determined almost exclusively by the NE dynamics. Our selected cutoff values permit the NE to influence the NPC slow modes without dominating them. The nodes in the network resulting from our selected cutoff values are each connected to between 13 and 51 neighbors, with an average coordination number of 20. This method of assigning edges sufficiently constrains even the most remote nodes without over-constraining those in the dense regions of the structure. It is also robust against small variations in the selected cutoff distances: Both cutoff distances can be altered by a few nm without significantly affecting the shapes of the slowest modes.
Once the network topology is defined, the only free parameters in the model are the spring constants γij. In accord with conventional ENMs, we assume that γij depends exclusively on the types of the interacting molecules, proteins or lipids, that the particular spring (or network edge) associates. The adoption of different spring constants for different molecules is in accord with previous applications of ENMs to complexes formed by proteins and DNA/RNA, where different force constants have been adopted for amino acid pairs, nucleotide pairs, and interfacial amino acid-nucleotide pairs
The purpose of including a fragment of the surrounding NE in our model is to further account for the effects of environment on the NPC dynamics; the dynamics of the NE fragment itself are not of particular interest in this study. We therefore employ the following method, detailed elsewhere
Three models of the NE fragment. Top row: Cartoons of the NE fragment models, colored to indicate mobilities induced by the slowest degenerate mode (A) Free boundaries shown at far left, lead to large motions at the border of the NE fragment, indicated by red coloring. (B) Bath-coupled boundaries, center, couple the centroids on the NE fragment border (
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This file details the models of the NE that were used in this work, as well as the method of comparing motions for structures of different sizes.
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The authors would like to thank Frank Alber for providing data files and useful insights, and Mike Rout, Brian Chait and Ohad Medalia for stimulating discussions.