AZ, SDT, BTC, MPR, and MOM conceived and designed the experiments and formulated the model. AZ and SDT performed the calculations and simulations. AZ, BTC, MPR, and MOM wrote the paper.
¤ Current address: Theoretical Biology and Biophysics and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico, United States of America
The authors have declared that no competing interests exist.
All materials enter or exit the cell nucleus through nuclear pore complexes (NPCs), efficient transport devices that combine high selectivity and throughput. NPC-associated proteins containing phenylalanine–glycine repeats (FG nups) have large, flexible, unstructured proteinaceous regions, and line the NPC. A central feature of NPC-mediated transport is the binding of cargo-carrying soluble transport factors to the unstructured regions of FG nups. Here, we model the dynamics of nucleocytoplasmic transport as diffusion in an effective potential resulting from the interaction of the transport factors with the flexible FG nups, using a minimal number of assumptions consistent with the most well-established structural and functional properties of NPC transport. We discuss how specific binding of transport factors to the FG nups facilitates transport, and how this binding and competition between transport factors and other macromolecules for binding sites and space inside the NPC accounts for the high selectivity of transport. We also account for why transport is relatively insensitive to changes in the number and distribution of FG nups in the NPC, providing an explanation for recent experiments where up to half the total mass of the FG nups has been deleted without abolishing transport. Our results suggest strategies for the creation of artificial nanomolecular sorting devices.
The DNA at the heart of our cells is contained in the nucleus. This nucleus is surrounded by a barrier in which are buried gatekeepers, termed nuclear pore complexes (NPCs), which allow the quick and efficient passage of certain materials while excluding all others. It has long been known that materials must bind to the NPC to be transported across it, but how this binding translates into selective passage through the NPC has remained a mystery. Here we describe a theory to explain how the NPC works. Our theory accounts for the observed characteristics of NPC–mediated transport, and even suggests strategies for the creation of artificial nanomolecular sorting devices.
The contents of the eukaryotic nucleus are separated from the cytoplasm by the nuclear envelope. Nuclear pore complexes (NPCs) are large protein assemblies embedded in the nuclear envelope and are the sole means by which materials exchange across it. Water, ions, small macromolecules (<40 kDa) [
Here we focus on karyopherin-mediated import, although our conclusions pertain to other types of nucleocytoplasmic transport as well, including mRNA export. During import, karyopherins bind cargoes in the cytoplasm via their nuclear localization signals. The karyopherin–cargo complexes then translocate through NPCs to the nucleoplasm, where the cargo is released from the karyopherin by RanGTP, which is maintained in its GTP-bound form by a nuclear factor, RanGEF. The high affinity of RanGTP binding for karyopherins allows it to displace cargoes from the karyopherins in the nucleus. Subsequently, karyopherins with bound RanGTP travel back through the NPC to the cytoplasm, where conversion of RanGTP to RanGDP is stimulated by the cytoplasmic factor RanGAP. The energy released by GTP hydrolysis is used to dissociate RanGDP from the karyopherins, which are then ready for the next cycle of transport. Importantly, this GTP hydrolysis is the only step in the process of nuclear import that requires an input of metabolic energy. Overall, the energy obtained from RanGTP hydrolysis is used to create a concentration gradient of karyopherin–cargo complexes between the cytoplasm and the nucleus, so that the process of actual translocation across the NPC occurs purely by diffusion [
Conceptually, nuclear import can be divided into three stages: first, the loading of cargo onto karyopherins in the cytoplasm, second, the translocation of karyopherin–cargo complexes through the NPC, and, third, the release of cargo inside the nucleus (
(A) Schematic of the nuclear import process. The karyopherins bind the cargo in the cytoplasm and transport it to the nucleus, where the cargo is released by RanGTP.
(B) Schematic of the NPC. The nucleus and the cytoplasm are connected by a channel, which is filled with flexible, mobile filamentous proteins termed FG nups. The karyopherins carrying a cargo enter from the cytoplasm and hop between the binding sites on the FG nups until they either reach the nuclear side of the NPC or return to the cytoplasm.
Here, we develop a diffusion-based theory to explain the mechanism of the intermediate stage of nucleocytoplasmic transport—i.e., translocation through the NPC. A useful theory of NPC-mediated transport should provide insight into several major unresolved questions, including: (i) How does the NPC achieve high transport efficiency of cargoes of variable sizes and in both directions, through only diffusion of the transport factor–cargo complexes? (ii) How does binding of transport factors to FG nups facilitate transport efficiency while maintaining a high throughput (up to hundreds of molecules per second per NPC) [
Several theoretical models have been proposed for the mechanism of transport through the NPC. These include the Brownian Affinity Gate model [
The aim of the present paper is to establish a general quantitative framework for NPC transport that is consistent with well-established structural and functional properties of the NPC and its components. We explain how the binding of karyopherins to the FG nups' flexible filaments inside the NPC can give rise to efficient transport. We demonstrate that competition for the limited space and binding sites within the NPC leads to a novel, highly selective filtering process. Finally, we explain how the flexibility of the FG nups could account for the high robustness of NPC-mediated transport with respect to structural changes [
The NPC contains a central channel (approximately 35 nm in diameter) that connects the nucleoplasm with the cytoplasm. The internal volume of this channel, as well as large fractions of the nuclear and cytoplasmic surfaces of the NPC, is occupied by the flexible FG-repeat regions of the FG nups (i.e., that portion in each FG nup containing multiple FG repeats). Since these FG-repeat regions also protrude into the nucleus and the cytoplasm, the effective length of the NPC is estimated to be 70 nm [
We represent transport through the NPC as a combination of two independent processes contributing to the movement of the karyopherin–cargo complexes through the central channel of the NPC: (i) the binding and unbinding of the karyopherins to the FG-repeat regions, and (ii) the spatial diffusion of the complexes, either in the unbound state or while still bound to a flexible FG-repeat region. The complexes entering the NPC from the cytoplasm thus stochastically hop back and forth inside the channel until they either reach the nuclear side, where the cargo is released by RanGTP, or return to the cytoplasm. Detachment from the FG-repeat regions and exit from the NPC can be either thermally activated, or catalyzed by RanGTP directly at the nuclear exit of the NPC [
It is important to distinguish between two different properties of the transport process, namely, (i) the speed with which individual complexes traverse the NPC, and (ii) the probability that complexes, entering from the cytoplasm, arrive at the nuclear side [
For simplicity, we assume that the unbinding and rebinding occur faster than the lateral diffusion of karyopherin–cargo complexes along the channel (although our conclusions were verified by computer simulations for any ratio of binding–unbinding rate to diffusion rate, unpublished data). In this limit, movement through the NPC can be approximated by diffusion in an effective potential as explained below. The strength of the effective potential depends on the relative strength of two effects. The first effect is the entropic repulsion between karyopherin–cargo complexes and FG-repeat regions and between the FG-repeat regions themselves, as the karyopherin–cargo complexes have to compress and displace the FG-repeat region filaments to enter the channel. The second effect is an attraction due to the binding of karyopherin–cargo complexes to the FG-repeat regions, as illustrated in
The NPC channel is represented by a potential well U(x), shown in black. The complexes enter the NPC at x = R at an average rate J. A fraction of the entrance flux, JM, goes through to the nucleus. The rest return to the cytoplasm at an average rate, J0. The exit of the complexes from the channel into the nucleus occurs either due to thermal activation, with the rate JL, or by activated release by RanGTP, with the rate Je. Steady state particle density inside the channel, ρ(x), is shown in blue. It differs from what would be expected from equilibrium statistical mechanics as the complexes do not accumulate at the minimum of the potential but rather their density decreases toward the exit.
We represent the transport of karyopherin–cargo complexes through the NPC as diffusion in a one-dimensional potential, U(x) (expressed in units of kBT), in the interval 0 < x < L (
Thus, the entrance current, J, splits into J0 and JM, corresponding to the flux of complexes returning to the cytoplasm and going through to the nucleus, respectively. Active release of the karyopherin–cargo complexes from the NPC by the nuclear RanGTP is modeled by imposing an additional exit flux, Je (proportional to the nuclear concentration of RanGTP), at a position x = L − R. Therefore, the transmitted flux, JM, splits into Je and JL, which correspond respectively to the flux of karyopherin–cargo complexes released from the FG-repeat regions by RanGTP and to thermally activated release, as shown in
The efficiency of the transport through the NPC is determined by the fraction of the complexes that reach the nucleus, JM/J. We emphasize that we did not study the equilibrium thermodynamic properties of the channel, but rather the steady state, out-of-equilibrium behavior.
We neglected possible differences in the diffusion coefficient of the complexes inside and outside the NPC to focus on the role of karyopherin binding in the import process. We also assumed that no current enters the NPC from the nucleus as the cargoes are released from the karyopherins in the nucleus by RanGTP. Finally, we neglected variations of the potential in the direction perpendicular to the channel axis. The effects of these factors do not change our conclusions, and will be studied in detail elsewhere.
Under the above assumptions, the model can be solved using standard theory of stochastic processes [
The transport of the karyopherin–cargo complexes through the NPC was then described by the diffusion equation for the density of complexes inside the channel, ρ(x)
The steady state density of the complexes in the channel, obtained by solving
The sum of the flux of karyopherin–cargo complexes going through the NPC, and of that returning to the cytoplasm, is equal to the total flux of complexes entering the NPC; hence |J0| + JM = J; similarly, JM − JL = Je. The flux, Je, is proportional to the number of complexes present at the nuclear exit, and to the frequency, Jran, with which RanGTP molecules hit the nuclear exit of the NPC: Je = Jranρ(x = L −R)R. Recalling that the potential outside the channel is zero (U(x) = 0) for 0 < x < R and L − R < x < L, and using the continuity of ρ(x) at x = L − R, one obtains for Ptr, the probability of a given karyopherin–cargo complex reaching the nucleus (i.e., the fraction of complexes reaching the nucleus):
When RanGTP only releases cargo from its karyopherin, but not from the FG-repeat regions (i.e., Je = 0); the maximal translocation probability, Ptr, is 0.5. However, in the case when RanGTP also releases karyopherin–cargo complexes from FG-repeat regions, the translocation probability, Ptr, can reach unity. Importantly, the latter effect is more pronounced for a large K, that is, for strong binding at the exit. We shall discuss the practical implications of this result later.
The second important consequence of
The previous section does not take into account the interference between karyopherin–cargo complexes inside the channel. Although a large interaction strength, E, increases transport efficiency, this increase is at the expense of an increased transport time T(E), which grows roughly exponentially with E (
To quantitatively investigate how mutual interference between translocating karyopherin–cargo complexes and molecular crowding affect transport efficiency, we performed dynamic Monte Carlo simulations of the diffusion of complexes inside the NPC, in the potential U(x), using a variant of the Gillespie algorithm [
The results of our simulations for the experimentally relevant range of interaction strength E and incoming flux J are shown in
(A) Transport efficiency, as given by the probability to reach the nucleus, is shown as a function of the interaction strength. RanGTP activity in the nucleus is represented by using JranL2/(N2D) = 1.5. The curves correspond to four different values of the entrance rate, J (measured in units of 10−416D/R2); the red line is the low-rate limit of
(B) Optimal interaction strength of (A) as a function of the incoming rate J (in units of 10−416D/R2), for JranL2/(N2D) = 1.5; parabolic potential shape. See
The main conclusion of the simulations, as shown in
Existence of an optimal interaction strength provides a mechanism for the selectivity of NPC-mediated transport. Karyopherin–cargo complexes tuned for a particular strength of interaction with FG-repeat regions have a high translocation probability, while macromolecules that do not interact with FG nups are less likely to cross. We elaborate on this finding in the next section.
As discussed in the previous section, the binding of karyopherins to FG-repeat regions provides a mechanism of selectivity. However, the maximum depicted in
These heuristic arguments were verified via computer simulations, using the algorithm of the previous section, adapted to account for two species of particles of different binding strengths. Two species of particles of different binding affinities (representing a karyopherin–cargo complex and another macromolecule that can bind nonspecifically (and weakly) to the FG-repeat regions), are deposited stochastically at the NPC entrance with the same average rate J. As in the previous section, the particles diffuse inside the channel until they either reach the nucleus, or return to the cytoplasm. Due to limited space, each position can be occupied by only a limited number of particles (see
As
Shown is the transport efficiency of particles across the NPC as a function of interaction strength with the FG-repeat regions, either in the presence or absence of competing particles. Gray line: transport efficiency of particles as a function of interaction strength in the absence of competition. Red line: transport efficiency of a weakly binding species in an equal mixture of weakly and strongly binding species, as a function of the interaction strength of the weakly binding species; the interaction strength for the strongly binding species is 12kBT. Translocation of the weakly binding species is sharply reduced in the presence of the strongly binding species, until its binding strength approaches that of the strongly binding species. No RanGTP activity was included in these simulations, hence lowering the transport efficiency compared with
In the previous sections, we used a continuous potential to model transport through the NPC. However, in reality the translocating karyopherin–cargo complexes likely hop between discrete binding sites that are located on the separate (or on the same) flexible FG-repeat regions, which fluctuate in space around their anchor points due to thermal motion [
Under these assumptions, the translocation of karyopherin–cargo complexes through the NPC can be described as diffusion in an array of potentials, as illustrated in
Transport through the NPC can be represented as diffusion in an array of potential wells (solid black lines) that represent flexible FG-repeat regions whose fluctuation regions overlap. The red dotted arrows correspond to the complexes unbinding from and rebinding to the FG-repeat regions. The solid blue line represents the unbound state. The solid red line shows the equivalent potential in the case when the unbinding of the complexes from the FG-repeat regions is much faster than the lateral diffusion across an individual well.
The first term in the first equation describes the diffusion in the unbound state, while the second and the third terms describe the unbinding and the binding to the wells. Similarly, the second equation describes the diffusion while still bound to the i-th well (FG-repeat region). The local unbinding and binding rates from the i-th well are ri0(x) and r0i(x), respectively. They are related by the detailed balance condition,
As we have proposed in previous sections, the transport properties of the NPC are relatively insensitive to the detailed shape of the effective potential. It follows from
Even more strikingly, in our model the transport properties of the NPC are not very sensitive to the number of the FG-repeat regions. This is robust with respect to the variations in the number of the FG-repeat regions [
(A) Effective potential for sparse flexible FG-repeat regions is shown as a blue line. Each well corresponds to an FG-repeat region. The transport properties in this multiwell potential are independent of the number of wells, and hence equivalent to those in the single-well potential, shown as a red line.
(B) Numerical simulations show essentially identical transport efficiencies in the multiwell potential (blue line) and in the single-well potential (red line).
Crucially, the transport properties of the potential shown in
This result highlights the robustness of our model of NPC transport; in multiwell potentials of this type, the NPC's transport properties do not depend on the specific number of FG-repeat regions, so long as they are flexible enough for their fluctuation regions to overlap, permitting complexes to freely transfer from one filament to the next, which might explain the puzzling degree of robustness of the NPC transport with respect to the deletion of FG repeat regions [
Several conceptual models have been proposed to describe transport through the NPC [
We have formulated and solved a rigorous mathematical model of transport through the NPC that depends on the physics of diffusion in a channel combined with binding to the flexible filamentous FG-repeat regions (without making detailed assumptions about the conformation and distribution of FG-repeat regions inside the channel). Our model applies to both export and import processes and explains the main features of NPC-mediated transport; namely, its high selectivity for cargoes bound to transport factors, its efficiency and directionality, and its robustness to perturbations.
We propose that the selectivity of the NPC arises from a balance between the probability (efficiency) and the speed of transport of individual karyopherin–cargo complexes. Analogous ideas have been suggested to account for the transport properties of ion channels and porins [
By considering known parameters of the nuclear transport machinery, we can test whether our simulations are consistent with the experimentally observed values of the flux through the NPC, and the residence time inside it. We take the effective length of the NPC as L ∼ 70 nm and its effective passive diffusion diameter as R ∼ 7 nm, within the range observed for different NPCs [
Our model can also explain the high specificity of facilitated transport through the NPC, wherein each NPC permits the passage of transport factor–cargo complexes but efficiently filters out macromolecules that do not bind specifically to the FG-repeat regions. The difference in binding energy between specifically and nonspecifically binding macromolecules can be as little as a few kBT, which may not seem enough for such efficient discrimination. However, we have uncovered an additional mechanism that we believe significantly enhances the specificity of NPC transport. This mechanism relies on the direct competition between transport factors and nonspecifically binding macromolecules; they compete for space and binding sites in the channel. As a consequence of their stronger binding, transport factors have a longer residence time within the channel as compared with nonspecifically binding macromolecules, which are therefore outcompeted for space and binding sites within the channel. The constant flux of cargo bound or free transport factors between the nucleus and cytoplasm therefore effectively excludes nonspecifically binding macromolecules from the channel. We emphasize that this selectivity enhancement is essentially a nonequilibrium kinetic effect. Hence, although no metabolic energy is expended in this filtering process [
In the case of karyopherin-mediated import, the transport efficiency is enhanced when RanGTP directly releases karyopherins from their binding sites on FG-repeat regions at the NPC exit [
Although the transport properties of the NPC depend strongly on the magnitude of the interaction strengths between transport factors and FG-repeat regions, we predict that transport depends only weakly on spatial variations of the binding strength along the channel. In particular, a gradient of binding affinity across the NPC should not, by itself, increase throughput compared with a uniform distribution of the same sites. This could explain how transport can be reversed across the NPC simply by reversing the gradient of RanGTP [
Although transport relies on the flexibility of the FG-repeat regions, it is relatively insensitive to the number of flexible FG-repeat regions inside the NPC—as long as their fluctuation regions can overlap (
Experimental tests for our model's predictions include varying the effective potential experienced by transport factor–cargo complexes inside the NPC by systematically introducing mutations into the binding sites [
The simulations were written in C language and run on a cluster of UNIX processors. The simulation algorithm is described in the text; see
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The authors are thankful to J. Aitchison, S. Bohn, T. Chou, J. Novatt, R. Peters, S. Shvartsman, G. Stolovitzky, and B. Timney for helpful comments. This work was supported by US National Institutes of Health grants RR00862 (BTC), GM062427 (MPR), GM071329 (MPR, BTC, AZ, MOM), and RR022220 (MPR, BTC).
phenylalanine–glycine repeats
a class of NPC-associated proteins containing FG repeats
nuclear pore complex
Reference [48] is cited out of order in the article because it was added while the article was in proof.