Conceived and designed the experiments: RMB MRKM. Performed the experiments: RMB YJ. Analyzed the data: RMB YJ RK MRKM. Contributed reagents/materials/analysis tools: RMB YJ RK MRKM. Wrote the paper: RMB YJ RK MRKM.
The authors have declared that no competing interests exist.
The nuclear pore complex (NPC) regulates molecular traffic across the nuclear envelope (NE). Selective transport happens on the order of milliseconds and the length scale of tens of nanometers; however, the transport mechanism remains elusive. Central to the transport process is the hydrophobic interactions between karyopherins (kaps) and Phe-Gly (FG) repeat domains. Taking into account the polymeric nature of FG-repeats grafted on the elastic structure of the NPC, and the kap-FG hydrophobic affinity, we have established a coarse-grained model of the NPC structure that mimics nucleocytoplasmic transport. To establish a foundation for future works, the methodology and biophysical rationale behind the model is explained in details. The model predicts that the first-passage time of a 15 nm cargo-complex is about 2.6±0.13 ms with an inverse Gaussian distribution for statistically adequate number of independent Brownian dynamics simulations. Moreover, the cargo-complex is primarily attached to the channel wall where it interacts with the FG-layer as it passes through the central channel. The kap-FG hydrophobic interaction is highly dynamic and fast, which ensures an efficient translocation through the NPC. Further, almost all eight hydrophobic binding spots on kap-β are occupied simultaneously during transport. Finally, as opposed to intact NPCs, cytoplasmic filaments-deficient NPCs show a high degree of permeability to inert cargos, implying the defining role of cytoplasmic filaments in the selectivity barrier.
Perforating and spanning the nuclear envelope (NE), the nuclear pore complex (NPC) is a supramolecular assembly that regulates all traffic between the nucleus and cytoplasm. As the unique gateway to the nucleus, NPC selectively facilitates the transport of large cargo while offering a relatively unobstructed pathway for small molecules and ions. Despite the high throughput of about 1000 translocations per NPC per second, the NPC strictly controls the passage of individual cargos. However, the dynamic mechanism of nucleocytoplasmic transport is poorly understood. It is too difficult to experiment on the transport mechanism within the confined geometry of this tiny pore
The nuclear pore complex (NPC; see
For more excellent descriptive figures of the NPC along with different biochemical agents see the recent comprehensive review by Jamali
The NPC is composed of about 30 distinct proteins collectively called nucleoporins (nups). Individual nups are directly related to several human diseases including influenza, cancers such as leukemia and inflammatory myofibroblastic tumors, and less frequent diseases like triple-A syndrome and primary biliary cirrhosis
The total mass of the NPC is species-dependent and is about 44 MDa and 60 MDa in yeast and vertebrates, respectively
Dimensions taken from Akey 1989 and Stoffler
The NPC acts as a freeway for passive diffusion of molecules and ions smaller than ∼5 nm in diameter
Importantly, almost 30% of nups include natively unfolded domains of phenylalanine-glycine (FG) repeats, collectively termed FG-repeat domains
The small size of the NPC and its compactness make it very hard to track individual cargo-complexes; hence, most studies are limited to bulk transport across many channels and cargos. However, a detailed understanding of the transport mechanism cannot be reached unless we examine individual cargos moving across the pore and inspect their interaction with surrounding nups, which can be genetically modified to model disease cases. While extensive efforts have been devoted to revealing the biochemical aspects of the transport mechanism, far less is known about its biophysical details.
Nucleocytoplasmic transport (NCT) happens in a channel about 50 nm wide and in a millisecond time scale
Simulation models can probe into the narrow NPC channel to examine the sequential events leading to the NCT cycle. Molecular dynamics simulations are not applicable since both the NPC size and the transport time are beyond available computational recourses. However, molecular dynamic (MD) simulations have been used to examine the behavior of limited arrays of FG-repeats
Recently we showed that this methodology can be effectively employed to mimic the NPC functionality
Brownian dynamics simulations are carried out with 1
In addition to the axial extension, the bending rigidity is taken into account by the following cosine-based potential energy between two consecutive segments (
For FG-repeats, axial extension is modeled by discrete wormlike chains (WLC) that are governed by the following force-law (
The inter-FG as well as kap-FG hydrophobic affinity is modeled by the following long-range potential energy
In addition, a short-range pairwise repulsive potential is applied between beads to avoid collision
In the absence of inertial effects (i.e, diffusive regime or Stokes flow), the Langevin equations of motion are solved explicitly forward in time for every bead
To develop an accurate coarse-gained model of the NPC, we need to consider its structural features. The NPC structure is composed of three different groups of nups (
Pore membrane proteins (Poms) that anchor the NPC scaffold to the NE and surround the central scaffold.
Structural proteins that give the NPC shape and strength and maintain the main scaffold.
FG-nups that provide the binding sites for NTRs and are central to selectivity.
In our coarse-grained model, we have considered all these groups with their corresponding functions (
Poms are modeled as a set of parallel Hookean springs that on one end are fixed to the NE and on the other end are connected to the central channel, and therefore, do not allow the NPC to detach (
The NPC main scaffold, consisting of the cytoplasmic filaments, central channel, and nuclear basket, is also assumed elastic
In addition to the elastic extension in the backbone, we also consider the bending rigidity of the NPC with the bending force constant
As the third group of nups, FG-repeat domains are the innermost layer, and are believed to be the key players in the facilitated transport. They are natively unfolded domains
FG-motifs are scattered across the NPC structure and electron microscopy has localized FG-nups to the central channel as well as the cytoplasmic and nuclear faces
In a recent AFM study, Lim
Each bead in the discrete WLC represents an FG-motif. The total number of FG-motifs in our model is 720. Sine our model has two spokes, this is in agreement with the density of FG-motifs in the wild NPC, i.e., ∼2700–3000/NPC
The hydrophobic potential energy is long-range, and its parameters for FG-FG interactions are chosen such that the weak, transient interaction between FG motifs
While the FG-beads serve as FG-motifs and interact with each other, with the solvent, and with cargo, the WLC springs represent the entropic effects of the FG-repeat domains. In fact, the WLC springs play the role of the hydrophilic linkers between FG-motifs. Hydrophilic linkers are a sequence of 5–30 charged amino acids between FG-motifs
Additionally, the persistence length (
Since we are using the discrete WLC, individual FG-repeats are discretized into several WLC segments, each with a corresponding contour length,
The finite-extensibility of the WLC imposes a singularity on its force-extension law, such that as the chain approaches the contour length, the axial force goes to infinity. This finite-extensibility is a well-known behavior of (bio)polymers. It has an important role in their rheology and is one of the key differences between the linear and non-linear springs
For the discrete WLC, it is also necessary to account for the bending potential energy between successive segments
In Eq. (2), the choice of the cosine-based potential energy for the bending rigidity of FG-repeats has some advantages. In addition to being computationally efficient
The chain curvature in the WLC is based on thermal fluctuations
Overall, the WLC is a powerful model to understand the properties of a wide range of synthetic and biological polymers
To replicate the excluded volume effects of polymer aggregations in a bead-spring model, several potentials, namely Lennard-Jones (LJ), Morse, and other exponential forms are used
In addition to the structural features of the NPC, the cargo-complex and its interaction with the FG-repeats is another important factor that should be thought of in developing an accurate coarse-grained model. We consider the complex of NLS-cargo and kap-β as a solid sphere interacting with the solvent as well as the FG-repeat domains.
Kap-β has a boat-like shape
The localization of binding spots on the kap convex surface has resulted in the idea of a “coherent FG-binding stripe” instead of discrete binding spots
Kap-β has a boat-like shape
Different energy and force-laws along with the parameters that have been used in the model are shown in
Parameter | Symbol/Formulation | Value | Reference(s) |
Young's modulus of the NPC main scaffold |
|
17 kPa | Estimated based on the |
Bending force constant of the NPC main scaffold |
|
100 |
Estimated |
Bending force constant of the discrete WLC |
|
0.04 |
Estimated based on the |
Contour length of the WLC: Peripheral FG-repeats |
|
200 nm | Estimated based on the |
Contour length of the WLC: Central FG-repeats |
|
50 nm | Estimated based on the |
Persistence length of the WLC |
|
0.43 nm |
|
Kap-FG hydrophobic affinity strength |
|
2 |
Estimated based on the |
Kap-FG hydrophobic characteristic length |
|
1.5 nm | Estimated based on the |
FG-FG hydrophobic affinity strength |
|
1.5 |
Estimated based on the |
FG-FG hydrophobic characteristic length |
|
1 nm | Estimated based on the |
Density of protein |
|
1.35 |
|
Cellular viscosity |
|
5 cP |
|
Hydrodynamic radius: FG-beads |
|
0.9 nm | Calculated based on the |
Hydrodynamic radius: main scaffold beads |
|
1.2–4.7 nm | Calculated based on the |
Drag coefficient: FG-beads |
|
|
Calculated based on the |
Drag coefficient: main scaffold beads |
|
|
Calculated based on the |
Diffusion coefficient: FG-beads |
|
50 |
Calculated based on the |
Diffusion coefficient: main scaffold beads |
|
8–42 |
Calculated based on the |
Depth of bead-bead repulsive potential well |
|
100 |
Estimated based on the |
Characteristic length of the repulsive potential |
|
1.0 nm | Estimated based on the |
The computational cost of the model is about 15 CPU-hours per one millisecond of transport time. Moreover, approximately a constant memory resource of about 5 MB is required for each simulation. These data are calculated based on computing nodes at the San Diego Supercomputer Center (SDSC).
As the first step to validate our model, we explored the transport of a single, common-size cargo-complex 15 nm in diameter. The model's prediction of the mean transport time in the absence of molecular traffic and competing factors is in agreement with the experimental observations.
To be statistically reliable, we ran 150 independent simulations, over which the first-passage time is averaged with the standard error of the mean (SEM) about 5%. Each run was continued until the cargo-complex was successfully imported into the nucleus. To save computational time, the cargo-complex is initially placed in the vicinity of the NPC entry, on the cytoplasmic side. The starting time of transport was defined as when the cargo-complex and FG-repeat domains in the cytoplasmic filaments interact for the first time. Accordingly, the end time was defined as when the cargo-complex passes the pore and is completely loaded into the nuclear basket for the first time. As soon as the cargo-complex is completely loaded into the basket, we assume that it is disassembled by interaction of RanGTP with kap, and therefore, the cargo is released and the simulation is ended. Under these conditions, our model predicts that the first-passage time of the cargo-complex is
As it can be seen, the transport time is scattered over a wide range from 0.5 ms to 8 ms. The transport time can be viewed as the first passage time
Next, we changed the cargo size and investigated the active transport of 9 and 20 nm cargo-complexes, independently. For each size, 50 independent simulations were carried out with the same conditions as aforementioned. We obtained the first-passage time of 9 nm and 20 nm cargo-complexes to be
Furthermore, we examined the capability of our model to conduct passive transport of small cargos (≤3 nm), which are known to diffuse freely across the NPC
|
9 | 15 | 20 |
|
10.0 | 6.1 | 4.5 |
|
|
|
|
|
7.90 | 7.89 | 7.90 |
|
|
|
|
|
|
|
|
|
36.0 | 7.3 | 10.7 |
Our simulation results show that the first-passage time distribution of the cargo-complex obeys an inverse Gaussian distribution (
Not coincidentally, this probability distribution function has been originally derived to predict the distribution of the first-passage time of a particle undergoing Brownian motion with a positive drift of
We analyzed the path of the cargo-complex in the central channel and averaged it over all independent simulations for each cargo size to find the radial probability distribution. The model suggests that inside the channel, the cargo-complex is most likely found near the wall, where it hydrophobically interacts with the FG-repeat layer on the wall. For a 15 nm cargo-complex, the radial probability distribution is maximized around
The cargo diameter is indicated inside the plot area. Each diagram is averaged over 50 independent simulations. Histograms are fit with Gaussian distributions (red dash line). The peak ± SD is recorded on top of histograms for each cargo size. Color bars on the right side show the radial probability distribution inside the central channel geometry. The more reddish, the higher the probability density. As it can be seen both in histograms and color bars, the large cargo is more likely to attach to the wall and less likely to disperse in the channel as opposed to the smaller cargo. This is partly due to its larger surface area and smaller diffusion coefficient, which make it less mobile compared the smaller cargo.
Using our model, we looked into the kap-FG hydrophobic interactions, which are believed to be intermittent and weak
The model predicts that during transport, almost all binding spots on kap-β simultaneously participate in the kap-FG hydrophobic interaction. In effect, once the cargo is hydrophobically engaged, on average about 7.89 out of eight binding spots are interacting simultaneously with FG-repeats (
Further, based on the cargo-complex trajectory during transport (
The NPC structure is sketched on the right to illustrate the corresponding location of the cargo-complex within the NPC. It can be seen that the cargo-complex jumps back and forth tens of times before it released in the nuclear basket. The majority of its time is spent in the central channel.
To shed light on the biophysical basis of the selectivity mechanism and the potential role of cytoplasmic filaments there, we investigated the diffusion of inert cargos across the NPC with and without cytoplasmic filaments.
First, in the intact NPC, we ran 50 independent simulations for inert cargo having a diameter of 15 nm, all with the same conditions as before. Next, we removed the cytoplasmic filaments from the NPC and carried out another 50 independent simulations with the same inert cargo. The only difference in these two sets of simulations, therefore, was the presence and the absence of cytoplasmic filaments. In both sets, each simulation was allowed to be run up to 8 ms, which is the maximum time an active cargo with the same size transported across the NPC (see
In intact NPCs, only 7% of inert cargos could diffuse through the NPC and reach the nuclear basket with an average time of about
The problem of the first-passage time
It should be noted that the first-passage time reported here does not take into account the time for the NLS-cargo and the kap to search for each other. This phenomenon has been suggested to increase the time between import cycles up to ∼10 s
In the absence of molecular traffic and competitive factors, which for simplicity are ignored here, our model predicts that the mean first-passage time for a common-size cargo-complex of 15 nm is
Furthermore, single molecular imaging techniques
Interestingly, for a smaller cargo with the diameter of 9 nm the first-passage time is almost the same as 15 nm, whereas for a larger cargo of 20 nm a 27% increase is observed (
The model predicts that when the cargo-complex is passing through the central channel, it is mainly found near the wall where an FG-layer is formed. In effect, the long-range hydrophobic interaction between kap-β and FG-repeats is enough to attract the cargo-complex and keep it near the wall. Therefore, instead of randomly exploring inside the channel space, the cargo-complex ‘slides’ over the wall and diffuses back and forth until it reaches the nuclear basket. This scenario is an indication of the reduction-of-dimensionality phenomenon
Due to the hydrophobic affinity for FG-nups, the cargo-complex transiently binds to the FG-repeats and diffuses back and forth until it leaves the pore. Our model corroborates the highly transient nature of kap-FG interactions and predicts that a single hydrophobic bond between a binding spot on kap and an FG-motif persists, on average, only for about 1.5 ns. Indeed, during the millisecond event of NCT, millions of these fleeting interactions that are on the order of thermal noise happen between the kap and FG-regions. These super-fast interactions may explain the high transport rate (1000 translocations/NPC/s
Importantly, during translocation when the cargo-complex interacts hydrophobically with FG-repeats, almost all of its binding spots simultaneously participate in interaction (see
Additionally, the transitory kap-FG hydrophobic bonds bring about numerous back and forth movements of the cargo-complex inside the NPC. The shorter the length scale over which the cargo-complex is observed, the bigger the captured number of these jumps is. A meaningful length scale is the distance of the NPC entry to the middle of the channel. We found that less than half of the cargo-complexes undergo at least one back-and-forth in this region (
Our model suggests that cytoplasmic filaments play an important role in selectivity by repelling inert cargos and preventing them from entering the central channel. Upon removing these filaments in the model, the possibility of inert cargos to enter the central channel and passively diffuse across the NPC significantly increased (7% versus 46%).
When the inert cargo is in the cytoplasmic periphery it is repelled by cytoplasmic filaments and their FG-repeats. This implies the defining role of cytoplasmic filaments in the selectivity barrier. In effect, FG-repeats localized to cytoplasmic filaments tend to dangle out into the cytoplasmic area and sample it with their thermal motions. In this regard, the role of cytoplasmic filaments is significant because by their vacillating motions, they help cytoplasmic FG-repeats cover a wider area around the NPC entry. Therefore, through their thermally vibrated motions, these filaments, along with the attached FG-repeat domains, can effectively reject inert cargos.
Indeed, at the subcellular spatiotemporal scale, viscous forces are dominant compared to inertial forces
In addition to repelling inert cargos, cytoplasmic filaments provide initial docking sites for cargo-complexes, as proposed elsewhere
From an active transport perspective, there are two common viewpoints about the functional role of cytoplasmic filaments. The first is that these filaments along with their pliable structure provide the initial docking sites for import complexes and thereby increase the transport efficiency
Our model is intended to be a platform to examine transport hypotheses and to suggest a plausible transport mechanism by scrutinizing these hypotheses
Detailed understanding of the NCT mechanism may allow for development of a new set of drugs that work by intervening in nucleocytoplasmic transport. Many current drugs work by manipulating transport channels that are located on the cell membrane. We can envision a next generation of drugs based on manipulating transport across the nuclear membrane. To design such drugs, as a barrier against viral infection and to modulate gene expression, a thorough understanding of NCT mechanism is necessary. Recent development in virtual patient models have been used in the pharmaceutical industry to accelerate drug testing
Moreover, some open questions can be investigated using the coarse-grained approach that otherwise are not reachable by biochemical experiments. An important question in this realm is about the unique geometry of the NPC structure and its plausible role in transport. What is the biological explanation behind the hourglass shape of the central channel? What advantages does this specific shape give to the transport of a cargo-complex? While there is symmetry around the planes perpendicular to the NE, why is there an asymmetry relative to the NE, i.e. why are the cytoplasmic and nuclear peripheries different? Does this asymmetry have a role in transport? Indeed, the answers to these questions await further investigation.
In this regard, a more interesting question is: how does the complicated functionality of a large-scale proteinous complex (here, the NPC) emerges from simple elements (here, individual nups)? This kind of biological paradigm is receiving attention in literature
On the other hand, of main limitations to a coarse-graining approach to the NPC, is poor information about the structural properties of the NPC and its building block, i.e. nups. This limitation can be addressed by experimental studies measuring material properties of the NPC under the same physiological conditions inside the cell.
On the cargo transport through the NPC: Analysis of mean square displacement and the biphasic effect of the kap-FG avidity.
(DOCX)
Nucleocytoplasmic transport of a 15-nm cargo-complex through the NPC.
(SWF)
Fruitful discussions with M. Azimi, J. Golji, C. Zhao and other members of the Molecular Cell Biomechanics Laboratory are gratefully acknowledged.