article: 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.
  The nuclear pore complex (NPC; see Figure 1) is the exclusive gateway of material transport across the nuclear envelope (NE) [1], [2], [3]. This selective gateway plays a critical role in regulating transcription and protecting the genetic material of eukaryotic cells; consequently, its structure is highly conserved from yeast to vertebrates. However, the mechanism of transport across the NPC remains elusive and proposed models thus far remain incomplete and sometimes contradictory even for normal (wild-type) NPC [4], [5], [6], [7], [8], [9], [10], [11], [12]. 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 [13], [14], [15], [16]. Nups also play an important role in viral infections by providing docking sites for viral capsids, and also by blocking host cell mRNA export or inhibiting import of antiviral signals [17]. For a most recent comprehensive review about NPC-related disease see Jamali et al. , 2011 [18]. Along the same lines, Buehler et al. , 2010, have recently reviewed the role of protein mechanics in disease conditions [19]. The total mass of the NPC is species-dependent and is about 44 MDa and 60 MDa in yeast and vertebrates, respectively [20]. The dimensions of the NPC also depend on the species. For example, yeast NPCs are 15% smaller than Xenopus NPCs [21]. The subunits of the NPC structure are cytoplasmic filaments, a cytoplasmic ring, a central channel that includes the spoke domains, a nuclear ring, and a nuclear basket that is composed of nuclear rods and a distal ring (see Figure 2 for the detailed dimensions of Xenopus oocyte NPC). The NPC acts as a freeway for passive diffusion of molecules and ions smaller than ∼5 nm in diameter [22], [23], while actively controlling and facilitating the transport of larger cargos up to about 39–40 nm [7], [24] by discriminating between inert and karyopherin-bound cargos. Karyopherins (kaps) are a family of soluble proteins by which different cargos are shuttled between the cytoplasm and nucleoplasm via the NPC, in a process known as nucleocytoplasmic transport (NCT). Large cargo being imported/exported have to be bound to an appropriate kap in the cytoplasm/nucleoplasm via a signaling process to form the cargo-complex. A single NPC can accommodate a remarkable rate of transport on the order of ∼1000 translocations/sec, corresponding to a mass flow of ∼100 MDa/sec [7]. Importantly, almost 30% of nups include natively unfolded domains of phenylalanine-glycine (FG) repeats, collectively termed FG-repeat domains [25], [26]. It is believed that the key feature in nucleocytoplasmic transport is the interaction of kap with these natively unfolded repeat domains, and thus, FG-repeats are the central players in all proposed models. Indeed, distinctions between the different models originate from different speculations about the properties and spatial arrangement of FG-repeats within the NPC and the way they interact with kaps during the transport. While interacting with FG-repeats, kap escorts the cargo along the pore until it reaches the destination, where RanGTP dissociates the kap and the cargo is released [4]. 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 in vivo [7], [27], [28]. Current imaging techniques fail to capture both the time and spatial resolution necessary to understand NCT. The refined resolution of single molecule imaging techniques become costly when we recognize their poor time resolution [29] and invasiveness, rendering them inappropriate for in-vivo measurement. Furthermore, even if they come close to these resolutions, they fail to capture transient interactions happening between FG-repeats and the cargo-complex on the order of nanoseconds. These uncertainties about the molecular events happening during nucleocytoplasmic transport (NCT) have been the source of contrasting speculations about the transport process across the pore. 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 [30], [31], [32]. When it comes to the whole structure of the NPC, however, the best alternative to an all-atom MD is a coarse-grained model in which the atomic details are lost in order to obtain computational feasibility. Coarse-graining is known for its ability to study biological systems for time scales nearer to those in physiological conditions, though atomistic details are neglected [33]. Recently we showed that this methodology can be effectively employed to mimic the NPC functionality [34]. Here we present the details of the coarse-grained model of the unbiased functional state of the NPC along with new insights into nucleocytoplasmic transport. The model is based on an intensive literature survey on simulating biological phenomena via a coarse-grained approach and applying these methods to develop a coarse-grained model of the NPC. We look into the NCT mechanism at a fine resolution both in time and space to mimic the NPC from a biophysical perspective by implementing experimentally known data about the NPC and utilizing polymer physics principles. Once the model is established, we can use it to examine different hypotheses about transport as well as examining different factors that alter transport, i. e. changing molecular size, shape, hydrophobicity and so on. Former studies have been limited to a single sheet of FG-repeat being ruptured by a bead [35].
  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 (Figure 6-a), placed on top of each other [6], [46]: In our coarse-grained model, we have considered all these groups with their corresponding functions (Figure 6-b). In the following, we discuss the rationales behind the assumptions of our model.
  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 ms (mean SEM, and hereafter). Distribution of the first-passage time is scattered over a wide range from 0. 5 ms to 8 ms (see Figure 8), i. e. a 16-fold variation, which is an indication of the stochastic nature of nucleocytoplasmic transport. Based on our results, on average, 53% of transport time is spent in the central channel. 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 ms and ms, respectively. For an in-depth study of the size effects on nucleocytoplasmic transport see [34]. 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 [22]. For the small cargo, we removed the hydrophobic affinity from its surface and ran 50 independent simulations under the same conditions as mentioned above. The first-passage time for the passive diffusion of 3 nm cargo was obtained to be ms (Table 2). Our simulation results show that the first-passage time distribution of the cargo-complex obeys an inverse Gaussian distribution (Figure 8) with the following probability distribution function: (12) where ms is the mean and ms is the scale parameter. 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, where is the distance from an absorbing wall and is the mean first-passage time [80]. In our model nm is the length over which the cargo-complex diffuses until it reaches the nuclear compartment (absorbing wall). Therefore, according to this distribution, in the absence of any molecular traffic and competing factors, a positive drift of is expected for a single cargo-complex. 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 nm (SD) and it rarely visits the channel axis. Similar results were obtained for 9 and 20 nm cargo-complexes (see Figure 9). Using our model, we looked into the kap-FG hydrophobic interactions, which are believed to be intermittent and weak [7], [11], [57]. The model corroborates the highly transient nature of these bonds and suggests that the average lifetime of a hydrophobic bond between a single binding spot on the kap-β and an FG-motif during transport is approximately 1. 5 ns with a standard deviation of about 17 ns. The minimum interaction lifetime is 0. 01 ns (equal to the simulation time step), while its maximum is approximately ns (Table 2). 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 (Table 2). Further, based on the cargo-complex trajectory during transport (Figure 10), we quantified back-and-forth fluctuations of the cargo-complex before exiting the NPC. The number of fluctuations is dependent on the length scale we are looking at. We found that 36% of 9 nm cargos, 7. 3% of 15 nm cargos, and 10. 7% of 20 nm cargos shuttle back and forth, at least once, between the NPC entry and the middle of 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 Figure 8). In intact NPCs, only 7% of inert cargos could diffuse through the NPC and reach the nuclear basket with an average time of about ms, while the rest (93%) were effectively inhibited and could not enter the central channel to reach its middle part. In NPCs lacking cytoplasmic filaments, however, the percentage of successful inert cargos significantly increased. Indeed, 46% of inert cargos could enter the central channel and diffuse to reach into the nuclear basket with an average time of ms.
  The problem of the first-passage time [80] in our model becomes more clear if we imagine an absorbing wall in the nuclear compartment where RanGTP dissociates the kap from the cargo. 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 [11]. Moreover, our reported values do not include the time a cargo-complex needs to search for the NPC entry, because in our model the cargo-complex is initially placed near the cytoplasmic entry to save computational time. It also does not include the time for the cargo-complex disassembly in the basket as well as the time cargo needs to diffuse fully out of the basket and appear on the nuclear side. Therefore, it is important to note that there is a slight difference between the ‘first-passage time’ calculated herein and the ‘residence/dwell time’ reported experimentally in other works, which was found to be 1 to 9 ms depending on cargo and import agent concentrations as well as spatiotemporal resolution of their experimental setup [22], [27], [28], [81], [82], [83]. 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 ms. This is about 20% more than the time required for the same particle to freely diffuse the corresponding distance of 160 nm in the cytoplasmic viscosity: ms. The slight difference reveals that NCT is near the free diffusion regime. This would imply that the presence of FG-repeat domains might not anomalously affect the normal diffusion of the cargo-complex. Instead, it provides attractive binding sites, helping the cargo-complex pass through the NPC rather than repelling it away. Curiously, this also substantiates the common notion that the transport per se within the channel is energy-independent and does not need any motor protein [4], [46], [61], [62], [78], [79]. Furthermore, single molecular imaging techniques [27] have illustrated that the majority of the transport time is spent within the central channel. In our model, the same behavior is predicted and the cargo-complex spends approximately 53% of the total time in the channel, whereas the remaining time is shared between the cytoplasmic and nuclear compartments. 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 (Table 2). For a passive cargo of 3 nm in diameter, the mean first passage-time across the NPC is 0. 7 ms, which is surprisingly 70% more than the same particle freely diffusing the same distance (0. 4 ms). This is most likely due to the cramped structure of the NPC and the fact that passive cargo should pass through the confined channel. 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 [84]. The idea is that reducing the dimensionality in which diffusion happens ensures a more rapid search toward the destination and thus enables a more efficient transport. It has been proposed elsewhere that nucleocytoplasmic transport is conducted by reduction of dimensionality [8], [46], which is also supported by our model. In line with this, remarkably, in a recent single-molecule imaging study the transport of cargo-free and cargo-bound kap-β was fluorescently tracked. Authors found that while inside the central channel, both kap-β and kap-β-cargo complex primarily locate near the channel wall and rarely occupy the axial channel [82]. Our results for different cargo sizes (Figure 9) corroborate their findings. While Figure 9 shows the radial probability distribution of the cargo-complex inside the central channel, in Figure 7 we plotted the mean trajectory of a 15 nm cargo-complex obtained from averaging over 150 independent transport simulations. 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 [7]) and the fact that the cargo-complex is not stuck inside the channel during translocation. This is in harmony with the perception that NCT is dependent on a sequence of relatively weak kap-FG interactions [4], [6], [7], [11], [57]. Interestingly, from this perspective the NPC can be imagined as a repository of billions of short-lived hydrophobic interactions per second. Nevertheless, rarely these weak bonds can last for up to several microseconds (see Table 2). Importantly, during translocation when the cargo-complex interacts hydrophobically with FG-repeats, almost all of its binding spots simultaneously participate in interaction (see Table 2), which is partly due to the dense layer of FG-repeats. Experimentally, however, the number of binding spots that simultaneously participate in transport is still unknown [54]. Interestingly, the number of binding spots is directly related to the kap-FG avidity and our ongoing simulations (data not shown) show that this avidity has a biphasic effect on transport time: when there is no binding spot (inert cargo with zero avidity), transport is prohibited and the cargo is rejected; when there are too many binding spots (high avidity) the cargo-complex becomes trapped within the pore (see Text S1). 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 (Table 2). This means that once a cargo reaches the center of the channel it is more likely for it to reach into the nuclear basket than being aborted to the cytoplasm. This can put an upper-limit for abortive transports. 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 [42], [85], and thus, in the absence of long-range hydrophobic affinity, an ‘impact’ by a cytoplasmic filament or their FG-repeats is enough to repel the inert cargo. It should be noted that the source of impacts at this scale is the steric repulsions between filaments and the cargo surface. This view is consistent with the virtual gate model, in which the peripheral FG-nups are suggested to act as “entropic bristles” that by their “push”, keep inert cargos away from the central channel [6]. In addition to repelling inert cargos, cytoplasmic filaments provide initial docking sites for cargo-complexes, as proposed elsewhere [86], [87], and possibly direct them toward the channel entrance (see Video S1). Importantly, this implies the possible role of these filaments in transport efficiency by making an ‘attractive trap’ for active cargos while constructing an ‘entropic barrier’ to inert cargos. This means that these filaments are not only essential for rejecting inert cargos, but are also likely to contribute to an efficient transport of active cargo. 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 [86], [87]. Along the same lines, it has been recently suggested that cytoplasmic filaments increase the “capture area” for active cargos in the cytoplasmic periphery [88]. The second hypothesis is that they are dispensable for kap-α/β-dependent import because the import of NLS-cargo shows a slight reduction in cytoplasmic filament-deficient NPCs [89]. From an inert cargo point of view, however, to our best knowledge, there has been no study so far on the potential role of these filaments in inhibition of inert cargo transport. Here, we propose that cytoplasmic filaments along with their localized FG-repeat domains are ‘mechanical components’ of the selectivity barrier. In other words, while the inner channel might be sufficient to create biochemical selectivity, our results show the necessity of cytoplasmic filaments in explaining mechanical selectivity. Our model is intended to be a platform to examine transport hypotheses and to suggest a plausible transport mechanism by scrutinizing these hypotheses in silico. At the first attempt, we have suggested elsewhere that the NPC works as a “lubricated gate” [34]. The model can also be used as a platform to study mechanics of nups in NCT and related disease, since the protein mechanics has been suggested to have a role in disease conditions [19]. 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 [90], and we believe that we need a virtual test tube to examine NCT. 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 [91]. 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.
summary: 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 in vivo. Currently, only computational techniques can elucidate the detailed events happening at this tiny pore with a refined spatiotemporal resolution to account for transient bonds. Based on experimental data regarding the NPC structure and nucleocytoplasmic transport, we have established a coarse-grained model of the functional state of the NPC. The model mimics nucleocytoplasmic transport and allows us to directly observe the processes happening within the pore from a biophysical perspective. The first-passage time of a single cargo-complex is found to be about 2. 6 ms. Furthermore, kap-FG hydrophobic bonds are highly dynamic and short-lived, ensuring efficient transport.
keywords: biology
  cell mechanics
  biophysics simulations
  biophysics
  biomechanics
title: Brownian Dynamics Simulation of Nucleocytoplasmic Transport: A Coarse-Grained Model for the Functional State of the Nuclear Pore Complex