Efficiency, Selectivity, and Robustness
of Nucleocytoplasmic Transport
Anton Zilman1¤, Stefano Di Talia1, Brian T. Chait2, Michael P. Rout3*, Marcelo O. Magnasco1*
1 Laboratory of Mathematical Physics, The Rockefeller University, New York, New York, United States of America, 2 Laboratory of Mass Spectrometry and Gaseous Ion
Chemistry, The Rockefeller University, New York, New York, United States of America, 3 Laboratory of Cellular and Structural Biology, The Rockefeller University, New York,
New York, United States of America
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.
Citation: Zilman A, Di Talia S, Chait BT, Rout MP, Magnasco MO (2007) Efficiency, selectivity, and robustness of nucleocytoplasmic transport. PLoS Comput Biol 3(7): e125.
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 macro-
molecules (,40 kDa) , and small neutral particles (diameter
,5 nm) can diffuse unaided across the NPC , while larger
macromolecules (and even many small macromolecules) will
generally only be transported efficiently if they display a
particular transport signal sequence, such as a nuclear
localization signal (NLS) or nuclear export signal (NES).
Macromolecular cargoes carrying these signal sequences bind
cognate soluble transport factors that facilitate the passage of
the resulting transport factor–cargo complexes through the
NPC. The-best studied transport factors belong to a family of
structurally related proteins, collectively termed b-karyo-
pherins, although other transport factors can also mediate
nuclear transport, particularly the export of mRNAs (re-
viewed in [1,3–6]). NPCs can pass cargoes up to 30 nm
diameter (such as mRNA particles), at rates as high as several
hundred macromolecules per second—each transport factor–
cargo complex dwelling in the NPC for a time on the order of
10 ms [7,8].
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 karyo-
pherins 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 karyopher-
ins, 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 [1,3–6,9–15].
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 (Figure 1). The first and last stages have
been the subject of numerous studies, and are relatively well
Editor: Susan Wente, Vanderbilt University, United States of America
Received August 16, 2006; Accepted May 17, 2007; Published July 13, 2007
Copyright: ? 2007 Zilman et al. This is an open-access article distributed under the
terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author
and source are credited.
Abbreviations: FG repeats, phenylalanine–glycine repeats; FG nups, a class of
NPC-associated proteins containing FG repeats; NPC, nuclear pore complex
* To whom correspondence should be addressed. E-mail: email@example.com
(MPR); firstname.lastname@example.org (MOM)
¤ Current address: Theoretical Biology and Biophysics and Center for Nonlinear
Studies, Los Alamos National Laboratory, Los Alamos, New Mexico, United States of
PLoS Computational Biology | www.ploscompbiol.orgJuly 2007 | Volume 3 | Issue 7 | e1251281
understood, being soluble-phase reactions amenable to
biochemical characterization (reviewed in [1,3–6,15]). The
intermediate stage of transport is much less understood.
Nevertheless, it is clear that the ability of karyopherins (and
other transport factors) to bind a particular class of NPC-
associated proteins containing phenylalanine–glycine (FG)
repeats, known collectively as FG nups, is a key feature of the
transport process, and allows them to selectively and
efficiently pass with their cargoes through the NPC. In
particular, experiments in which the FG nup–binding sites on
the karyopherins were mutated show that disrupting the
binding of karyopherins to FG nups impairs transport
[6,12,13,15,16]. Current estimates of the binding affinity of
karyopherins to most FG nups are in the range 1–1,000 nM
(or 10–30 kBT per binding site), depending on the FG nup
and karyopherin type [18–20]. Each FG nup usually carries a
small region that anchors it to the body of the NPC, and a
larger region characterized by multiple FG repeats. These FG
repeat regions are natively disordered flexible chains or
filaments that contain binding sites for transport factors
(including karyopherins) and also appear to set up a barrier
at the entrance of the NPC for macromolecules that cannot
bind them [1,3,4,9,13,14,21–23]. The detailed physicochemical
nature of this barrier is still under active study, although FG
nups have been shown in vitro to form flexible polymer
brushes when grafted to a surface  or gels in bulk solution
. Importantly, it has been repeatedly demonstrated that
individual FG repeat regions can have a long reach, on the
order of many tens of nm, within the NPC [3,22,23,25]. What
is still needed is a quantitative theoretical explanation that
can account for the observed characteristics of facilitated
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) [7,8,12,13,26]?
(iii) NPCs largely exclude nonspecific macromolecules in
favor of transport factor–bound cargoes (reviewed in e.g., ).
How is this high degree of selectivity achieved? (iv) Neither
deletion of up to half the mass of the FG nups’ filamentous
unfolded regions, nor deletion of asymmetrically disposed FG
nups’ filamentous regions that potentially set up an affinity
gradient, abolish transport . Directionality of transport
across the NPC can even be reversed by reversing the
concentration gradient of RanGTP . How can we account
for such a high degree of robustness?
Several theoretical models have been proposed for the
mechanism of transport through the NPC. These include the
Brownian Affinity Gate model [4,14], Selective Phase models
[12,13,29,30], the Oily Spaghetti model , Affinity Gradient
models [10,11,15,20,31], the Dimensionality Reduction model
, and most recently a Two-Gate model . All these
models can be thought of as viewing the NPC as a ‘‘Virtual
Gate’’ [4,14], where the FG nups set up a barrier for entrance
into the NPC and transport through the NPC involves
facilitated diffusion controlled by association and disassoci-
ation of transport receptors with FG nups. They differ only in
specific assumptions, such as the conformation and spatial
deployment of the FG nups, their physicochemical state, or
the distribution of affinities of binding sites (reviewed in ).
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,
Figure 1. Main Features of the NPC and of Nuclear Import
(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.
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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
Physical Model of Transport through the NPC
we explain how the flexibility of the FG nups could account
for the high robustness of NPC-mediated transport with
respect to structural changes . We conclude by discussing
verifiable experimental predictions of the model.
Setting Up a Physical Model of NPC Transport
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 [1,3,4]. The details of the
distribution of the FG-repeat regions inside the central
channel and the external surfaces of the NPC, as well as the
exact number of binding sites on the karyopherins and the
number of the FG-repeats on the FG-repeat regions that are
accessible for binding, have not yet been well-established
(although the number of the FG repeats is in the range of 5–
50 per FG nup [6,27]). We made no specific assumptions
about the distribution of FG nups, interactions between
them, and their density, degree of flexibility, or conformation
within the NPC. As we will discuss, the general features of
transport through the NPC appear relatively insensitive to
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 [1,4]. A schematic illustration of
transport through the NPC is shown in Figure 1.
Enhanced Transport Efficiency Arises from the
Karyopherins’ Ability to Bind to FG nups
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 [1,4,9,33–35]. As we discuss below,
binding of karyopherins to the FG nups increases the
probability of the karyopherins traversing the NPC, i.e., their
transport efficiency; in the absence of such binding, the
probability of traversing the NPC is low.
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 them-
selves, 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 Figure 2.
We represent the transport of karyopherin–cargo com-
plexes through the NPC as diffusion in a one-dimensional
potential, U(x) (expressed in units of kBT), in the interval 0 ,
x , L (Figure 2). The shape of the potential in the NPC is
determined by the distribution of the FG nups along the
channel (an issue we address later). The actual length of the
NPC corresponds to the interval from x¼R to x¼L?R, and
the regions of length R (on the order of the width of the
channel [33,36]) at both ends of the interval correspond to the
distance outside the NPC over which the particles diffuse into
either the nucleoplasm or the cytoplasm. We did not directly
model the diffusion of complexes outside of the NPC. Instead,
we assumed that karyopherin–cargo complexes stochastically
entered the NPC from the cytoplasm, with an average rate J at
x ¼ R, where J is proportional to the concentration of the
karyopherin–cargo complexes in the cytoplasm [26,36].
Because of the random nature of movement inside the
channel, a certain fraction of the complexes impinging on the
channel entrance will not reach the nucleus and eventually
return to the cytoplasm. We modeled this event by imposing
absorbing boundary conditions at x ¼ 0 and x ¼ L that
correspond to a karyopherin–cargo complex returning back
to the cytoplasm, or going through to the nucleus, respec-
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
Figure 2. Transport through the NPC Is Modeled as Diffusion in an
The NPC channel is represented by a potential well U(x), shown in black.
The complexes enter the NPC at x¼R 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, q(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.
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Physical Model of Transport through the NPC
additional exit flux, Je(proportional to the nuclear concen-
tration of RanGTP), at a position x ¼ L ? R. Therefore, the
transmitted flux, JM,splits into Jeand 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 Figure 2.
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 coef-
ficient 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 . Impor-
tantly, the model can be solved for a potential of an arbitrary
shape, allowing us to model different distributions of binding
sites within the NPC.
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, q(x)
where the local flux of the complexes within the NPC, J(x), is
JðxÞ ¼ ?D@qðxÞ
The first term in Equation 2 describes the random thermal
motion of the complexes, and the second term stands for the
variations in the flux due to local variations of the potential
U(x); D is the diffusion coefficient of the complexes inside the
The steady state density of the complexes in the channel,
obtained by solving Equations 1 and 2, with entrance flux J
and satisfying the boundary conditions q(0) ¼ q(L) ¼ 0, is
jJ0jR ? JM
for R,x,L ? R
for L ? R,x,L
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
jJ0jþJM¼J; similarly, JM?JL¼Je. 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
potential outside the channel is zero (U(x)¼0) for 0 , x , R
and L?R , x , L, and using the continuity of q(x) at x¼L?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):
Ptr¼ JM=J ¼
2 ? K=ð1 þ KÞ þ1
where K ¼ JranR2/D exp(?U(L?R)).
Equation 4 is the main result of this section and has several
important consequences. The probability of traversing the
NPC, Ptr, defines the transport efficiency. This efficiency is
seen to increase with the potential depth E, (defined as E ¼
?minxU(x), Figure 2), proportional to the binding strength of
the karyopherin–cargo complexes to the FG-repeat regions.
In the absence of binding, Ptr is small (;R/L), so that a
complex will, on average, return to the cytoplasm soon after
entering the NPC. Notably, an attractive potential inside the
NPC increases the time the complex spends inside the NPC
and thus increases the probability that it reaches the nuclear
side, rather than returns to the cytoplasm.
When RanGTP only releases cargo from its karyopherin,
but not from the FG-repeat regions (i.e., Je¼ 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
The second important consequence of Equation 4 is that
Ptrdepends only weakly on the shape of the potential, U(x).
This can account for why the transport properties of the NPC
are relatively insensitive to the details of the distribution of
FG-repeat regions inside the NPC, and to the distribution of
the binding sites on the FG-repeat regions.
A Mechanism for Selectivity is Provided by the Limited
Space and Binding Capacity within the NPC
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 (Text S1), and leads to an accumulation
of karyopherin–cargo complexes inside the channel. How-
ever, the space and the number of available binding sites
inside the channel are limited. As the number of these
complexes in the channel increases, they start to interfere
with the passage of each other due to molecular crowding.
This molecular crowding results from two different sources.
One is the repulsion that the entering macromolecules feel
from the FG nups that set up the permeability barrier. The
second is competition for the limited space inside the
channel between the karyopherin–cargo complexes them-
selves, and which we now demonstrate can determine the
selectivity. In effect, these two factors represent the entropic
exclusion that we have discussed previously [1,3,4,9,13,14,21–
To quantitatively investigate how mutual interference
between translocating karyopherin–cargo complexes and
molecular crowding affect transport efficiency, we performed
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Physical Model of Transport through the NPC
dynamic Monte Carlo simulations of the diffusion of
complexes inside the NPC, in the potential U(x), using a
variant of the Gillespie algorithm [37–39]. The simulations
are a discrete version of the continuum formulation of the
previous section. The interval [0,L] is represented by N
discrete positions, which is a standard way to approximate
the continuous diffusion; it is important to appreciate that
these sites do not represent the actual binding sites, but
correspond to the length of a diffusion step. We allowed only
a limited number, nmax, of complexes at each position at any
moment of time, which models the competition between
complexes for the limited space and the accessible binding
sites inside the channel. In line with our analytical model
above, karyopherin–cargo complexes were deposited at the
position iR if it was unoccupied by a complex, with a
probability of JL2/(DN2) per simulation step. When a complex
reached position i ¼ 0 (cytoplasm) or i ¼ N (nucleus), it was
removed from the channel. In addition, the complexes
present at the position i ¼ N ? iRcould be removed directly,
with the probability JranL2/(DN2), which models the effect of
the release of the complexes from FG-repeat regions by
nuclear RanGTP. Once inside the channel, a complex present
at site i could hop to an adjacent unoccupied site, i 6 1, with
the following probability:
Pði ! i61Þ ¼
J þ JranxN?iRþ
where xiis the site occupancy: xi¼0 if the site is unoccupied,
xi¼ n if n complexes are present at the position i, up to the
nmaxcomplexes. The transition rates from a site i to a site i 6
1 were ri;i61¼
zero, if xi61¼ nmax[37–39].
The results of our simulations for the experimentally
relevant range of interaction strength E and incoming flux J
are shown in Figure 3 (see Discussion for an explanation of
our choice for the parameter values). Figure 3 shows the
results for nmax¼ 1; the results did not change substantially
for higher allowed local occupancy (Text S1 and Figure S5).
For low interaction strength, E, the translocation probability
curves for all entrance fluxes, J, collapse onto a single line
(which is predicted by the analytical solution from the
previous section), because in this regime there are few
complexes simultaneously present in the channel, and
molecular crowding is negligible. For stronger binding,
karyopherin–cargo complexes accumulate inside the channel,
blocking the inflow of additional complexes, and the channel
becomes jammed as reflected in the decrease of the
probability to traverse the NPC.
The main conclusion of the simulations, as shown in Figure
3A, is that transport through the NPC is maximal for an
optimal value of the interaction strength Ec. Therefore, our
theory rigorously confirms the intuitive notion of the
existence of an optimal binding that balances increased
transport probability with increased time spent within the
NPC. In our simulations, the optimal binding strength
depends on the entrance flux, and thus on the abundance
of complexes of a particular type in the cytoplasm. In
particular, the optimal interaction strength is higher for low
entrance fluxes, as illustrated in Figure 3B.
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.
ðL=NÞ2expððUi? Ui61Þ=2Þxi, if xi61, nmax, and
Competition between Non-Specifically Binding
Macromolecules and Karyopherins Enhances the
Selectivity of the NPC
As discussed in the previous section, the binding of
karyopherins to FG-repeat regions provides a mechanism of
selectivity. However, the maximum depicted in Figure 3A is
Figure 3. Transport Efficiency Is Determined by the Interaction Strength
(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)¼1.5. 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 Equation 4. For any
entrance rate, the transport efficiency is maximal at a specific value of
the interaction strength, which provides a mechanism of selectivity.
(B) Optimal interaction strength of (A) as a function of the incoming rate
J (in units of 10?416D/R2), for JranL2/(N2D) ¼ 1.5; parabolic potential
shape. See Discussion for the corresponding actual values of the flux
through the pore. Black dots are simulation results.
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Physical Model of Transport through the NPC
broad; the translocation probability is significant even for
binding strengths considerably lower than the optimal one.
For instance, if the optimal interaction strength is E ¼ 15kT
(Kd ’ 300 nM), macromolecules whose interaction strength is
7kT (Kd ’ 1 mM)  have a probability of reaching the
nucleus that is more than half the optimal one. On one hand,
this broad maximum allows NPC-mediated import to
function efficiently across a broad range of transport factor
binding strengths. On the other hand, this might also permit
passage of macromolecules that bind nonspecifically to FG-
repeat regions (e.g., due to electrostatic interactions). How-
ever, proper functioning of living cells requires a high
selectivity of the NPC—how might this be achieved? So far,
Figure 3 only takes into account the competition between
complexes of identical binding strength for space inside the
channel. However, in a situation where optimally binding
karyopherins compete for space and binding sites inside the
channel with other, weakly binding macromolecules, the
passage of the latter is sharply reduced, which significantly
increases the selectivity of the NPC. Qualitatively, because the
strongly binding karyopherins and karyopherin–cargo com-
plexes spend more time in the NPC, a weakly binding
macromolecule entering the channel will—with high proba-
bility—find it occupied by karyopherin–cargo complexes.
Therefore, because the residence time of a low affinity
macromolecule is relatively short, there is a high probability
that it will return to the cytoplasm before the channel clears.
On the other hand, if a karyopherin–cargo complex enters a
channel that is already occupied by other strongly binding
complexes, there is still a high probability that, due to its
relatively high residence time, it will reside inside the channel
long enough for the complexes that are already inside the
NPC to get through. As free karyopherins exchange back and
forth across the NPC constantly, there will always be
karyopherins (or other transport factors) binding in the
NPC and so excluding nonspecific macromolecules, making
the NPC a remarkably efficient filter.
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 Text S2
for the actual code).
As Figure 4 shows, competition for the space inside the
channel between the translocating particles dramatically
narrows the selectivity curve as compared with Figure 3. This
effect is a novel mechanism for the enhancement of transport
selectivity beyond what is expected from the equilibrium
binding affinity differences alone. In contrast to other
mechanisms of specificity enhancement (e.g., kinetic proof-
reading ), no additional metabolic energy is required for
this enhanced discrimination. Instead, selectivity is achieved
by competition producing a differential NPC response to two
ranges of binding affinities. There remains a broad range of
higher binding affinities, occupied by transport factors,
wherein passage across the NPC is efficient; however, in the
low range of affinities, transmission is effectively prevented.
We emphasize that this is an essentially nonequilibrium
effect, and the selectivity enhancement goes far beyond the
difference in the equilibrium binding affinities. Importantly,
the enhancement of the selectivity persists even when high
local occupancies are allowed (Text S1 and Figure S4).
The Flexibility of FG-Repeat Regions May Account for the
Robustness of NPC-Mediated Transport
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 be-
tween 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
[3,22,23]. This flexibility allows the complexes to diffuse along
the channel while remaining bound to an FG-repeat region. A
complex can also unbind from an FG-repeat region and
rebind again to the same or a neighboring FG-repeat region,
moving while unbound by passive diffusion. In this section,
we elucidate how the number of FG nups inside the channel
Under these assumptions, the translocation of karyopher-
in–cargo complexes through the NPC can be described as
diffusion in an array of potentials, as illustrated in Figure 5,
where each potential well Ui(x) of a width pirepresents an
FG-repeat region. The shape of each well depends on the
Figure 4. Karyopherins Efficiently Exclude Nonspecifically Binding
Macromolecules from the NPC
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
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Physical Model of Transport through the NPC
number of the binding sites on each FG-repeat region, the
binding strength of the karyopherin–cargo complex, and the
rigidity and the length of an FG-repeat region, which
determine the cost of its entropic stretching in the process
of spatial fluctuations. This description allows for the
possibility of having several binding sites on an FG nup
which affects only the shape of the wells (Text S1 and Figure
S1). The blue line in Figure 5 corresponds to the unbound
state. Although for the purposes of illustration all the wells
are shown to have the same form, the subsequent results are
valid for an arbitrary distribution of potential shapes. We
shall denote the density of the karyopherin–cargo complexes
in the i-th well as qi(x) and the density of unbound complexes
as q0(x). The lateral diffusion of the complexes, combined
with the binding and unbinding to the FG-repeat regions is
then described by the following equations :
½ri0ðxÞqiðxÞ ? r0iðxÞq0ðxÞ?
@xeUiðxÞqiðxÞ þ r0iðxÞq0ðxÞ ? ri0ðxÞqiðxÞ
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, r0i(x)/ri0(x) ¼ e?Ui(x)þU0(x), which reflects the energy
difference between the bound and unbound states. If the
unbinding rates are fast compared with the diffusion time
across the wells, the relative densities of bound and unbound
complexes are at their local equilibrium Boltzmann ratio :
PLoS Computational Biology | www.ploscompbiol.org
the complexes at a position x. Adding up Equations 6, one
obtains an equation for the total density of the complexes at
position x, q(x):
of translocation through an array of flexible FG nups within
the NPC can be described as simple lateral diffusion in the
effective potential Ueff, which leads to Equations 1 and 2 in
the first section of this manuscript, which are essentially
identical to Equation 7, even though they are written in a
slightly different form.
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 Equations 6
and 7 that the shape of the effective potential depends only
weakly on the number and the shape of the overlapping
potential wells corresponding to the FG-repeat regions.
Even more strikingly, in our model the transport proper-
ties 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 . This
was in striking contrast to the case of the inflexible FG-repeat
regions when the binding sites are sparsely distributed
without a large degree of overlap (Text S1 and Figure S2).
We illustrate this point through a limiting case where the FG-
repeat regions barely touch, represented by the potential
shown by the blue line in Figure 6A. The flat central part of
each potential well corresponds to where the karyopherin–
cargo complex is diffusing in the channel while bound to an
FG-repeat region, andthe sharply rising regions at the borders
correspond to unbinding of the complex and its transfer to
the next filament. Narrow wells correspond to filaments with
limited reach, while wide wells correspond to filaments that
can stretch a long distance without significant entropic cost.
The potential wells can have different widths, pi,so that their
combined width is equal to the total length of the channel,
potential at a point x ¼P
in Figure 6A do not depend on the number of wells. Both the
translocation probability and the residence time are equiv-
alent for the multiwell potential shown in blue, and the
single-well potential shown in red, obtained by rescaling an
individual blue well to the whole length, L ? 2R, of the
channel. Indeed, it follows from Equation 4 that the trans-
location probability for the multiwell potential with n wells, is
where Ueff¼ ?lnðe?U0ðxÞþ
e?UiðxÞÞ . Thus, the process
i¼1pi¼ L ? 2R. All the potential wells have the same shape,
U0, re-scaled to the width of an individual well, so that the
j,ipjþ Dx is: U(x) ¼U0(Dx/pi).
Crucially, the transport properties of the potential shown
2 ? K=ð1 þ KÞ þ1
2 ? K=ð1 þ KÞ þ1
which is independent of the number and the width of the
exp(?U(L ? R)). We prove in Text S1 that the residence time
is similarly independent of the number of wells. Since both
ipi ¼ L ? 2R; as before, K ¼ JranR2/D
Figure 5. Discrete Overlapping FG-Repeat Regions Can Be Approximated
by a Smooth Effective Potential
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.
July 2007 | Volume 3 | Issue 7 | e1251287
Physical Model of Transport through the NPC
translocation probability and residence time are independ-
ent of the number of wells, the transport properties do not
depend on the number of wells, even for high entrance rates
or binding strengths, when jamming becomes important, as
verified by computer simulation (Figure 6B).
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 [1,3–6,9–15,29–32,48]. Most
propose that this transport relies on diffusion of the
transport factor–cargo complexes in the environment of
flexible FG-repeat regions of the FG nups, controlled by
transient binding to the FG nups these flexible regions [1,3–
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 distribu-
tion 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 perturba-
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. Anal-
ogous ideas have been suggested to account for the transport
properties of ion channels and porins [14,33–35,42]. In our
model, the probability of a karyopherin–cargo complex
reaching the nucleus increases with the binding strength of
the transport factors to FG-repeat regions, but at the expense
of increased residence time inside the NPC; eventually,
complexes spend so much time in the NPC that they impede
the passage of other complexes through the channel. There-
fore, there is an optimal value of the binding strength of
karyopherins to FG-repeat regions that maximizes their
transport efficiency through the NPC. For karyopherin–cargo
complexes with lower entrance fluxes, the optimal binding
strength is higher because at low fluxes the accumulating
complexes can reside longer in the channel without blocking
it (Figure 3). This correlation of optimal binding strength and
entrance flux could explain why there are different karyo-
pherin types; the binding strength of each karyopherin type
might be related to its cargoes’ relative flux and hence the
abundance of its cargoes.
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 . The diffusion
coefficient D of the complexes inside the channel can be
estimated to lie in the range 1–10 l2/s, typical for protein
diffusion in the crowded environment of the cytoplasm
[43,44]. In vivo, the total cargo flux through an NPC ranges
from several molecules per second up to several hundred
molecules per second [12,26,45–47]. Accordingly, we per-
formed simulations for values of the incoming flux J in the
range 0.3–10 in units of (10?416D/R2), which corresponds to a
flux through the NPC in the range of 10–1,000 molecules per
second. This results in predicted residence times in the NPC
(given by LR/Dexp(E/kT)) of approximately 0.01–1 s, con-
sistent with experimentally determined residence times [7,8].
For incoming flux values in this range, our model predicts
optimal interaction strengths in the range of 5–15kBT
(Figure 3). Molecular dynamics calculations predict higher
binding energies , but the effective interaction strength
in our model is reduced by entropic effects from the flexible
Our model can also explain the high specificity of
facilitated transport through the NPC, wherein each NPC
Figure 6. The Number of FG nups Does Not Significantly Affect the
Transport Properties of NPCs
(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
PLoS Computational Biology | www.ploscompbiol.orgJuly 2007 | Volume 3 | Issue 7 | e1251288
Physical Model of Transport through the 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 macro-
molecules; they compete for space and binding sites in the
channel. As a consequence of their stronger binding, trans-
port 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 there-
fore effectively excludes nonspecifically binding macromole-
cules 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 , the resulting selectivity is much higher
than might be expected from just the different binding
affinities of transport factors and nonspecific macromole-
cules (Figure 4).
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 [1,4,20,31]—an enhancement that increases
with the binding strength at the nuclear exit. High affinity
binding sites at the nuclear exit of the NPC decrease the
probability of return, once a complex has reached the nuclear
side. This result may account for the observed high affinity
binding sites that are localized at the nuclear side on the NPC
in import pathways and at the cytoplasmic side in export
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 distri-
bution of the same sites. This could explain how transport
can be reversed across the NPC simply by reversing the
gradient of RanGTP . Only a high affinity trap at the exit
of the NPC in combination with the action of RanGTP in
releasing the karyopherins from this trap can improve the
throughput through the NPC.
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 fluctua-
tion regions can overlap (Figures 5 and 6). This could account
for recent experiments in which up to half the total mass of
the flexible FG-repeat regions in NPCs were deleted without
abrogating nucleocytoplasmic transport . In particular, it
follows from Equation 4 that the probability of traversing the
NPC is low if the binding sites are sparse and stationary,
unless they are so dense that they occupy almost all the
available length of the channel. However, in this case,
transport is sensitive to the number of sites. Thus, a theory
that neglects the flexibility of the FG nups is incapable of
explaining how the NPC can sustain a high throughput and be
relatively insensitive to the removal of up to half the binding
sites. Importantly, this result does not depend on the speed of
the diffusion of the karyopherin–cargo complexes between
the binding sites. By contrast, a model that relies on
stationary binding sites predicts that transport will not be
robust to deletion of half the binding sites (Figures S2 and
S3). Hence, we show that not every diffusion-based mecha-
nism can explain the robustness of transport with respect to
deletion of the FG-nups. Moreover, different karyopherins
can bind different specific FG-nups. Thus, they can follow
different pathways within the NPC channel, each reliant on a
small subset of specific FG-nups ; we thus predict that only
removal of this small subset would prevent that karyopherin
from transiting the NPC. One of the conclusions drawn by
 from their results is that the lethal deletions have
removed all the preferred FG-binding sites on an essential
pathway; our model is thus completely in line with their work.
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 , changing
the cargo size , or using cells with genetically modified
numbers of the FG-repeat regions . Finally, any device
built according to the principles outlined above would
possess the transport properties described by our model,
suggesting strategies for the creation of highly selective
artificial nanomolecular sieves.
Materials and Methods
The simulations were written in C language and run on a cluster of
UNIX processors. The simulation algorithm is described in the text;
see Text S2 for the actual code. Analytical calculations were in part
performed with pencil and paper, or in some cases using Mathema-
tica version 5.1.
Figure S1. Several Binding Sites on a Single FG-Repeat Region Result
in an Effective Potential
Found at doi:10.1371/journal.pcbi.0030125.sg001 (492 KB EPS).
Figure S2. Array of Discrete Bindings Sites
Found at doi:10.1371/journal.pcbi.0030125.sg002 (381 KB EPS).
Figure S3. Transport Efficiency in the Case of Stationary Binding
Found at doi:10.1371/journal.pcbi.0030125.sg003 (902 KB EPS).
Figure S4. Karyopherins Efficiently Exclude Nonspecifically Binding
Macromolecules from the NPC, Irrespective of the Permitted Local
Found at doi:10.1371/journal.pcbi.0030125.sg004 (590 KB EPS).
Figure S5. Translocation Probability for High Local Occupancy
Found at doi:10.1371/journal.pcbi.0030125.sg005 (1.4 MB EPS).
Text S1. Additional Information
Found at doi:10.1371/journal.pcbi.0030125.sd001 (422 KB DOC).
Text S2. Simulation Code for the Diffusion of Complexes through the
Found at doi:10.1371/journal.pcbi.0030125.sd002 (61 KB DOC).
The authors are thankful to J. Aitchison, S. Bohn, T. Chou, J. Novatt,
R. Peters, S. Shvartsman, G. Stolovitzky, and B. Timney for helpful
PLoS Computational Biology | www.ploscompbiol.org July 2007 | Volume 3 | Issue 7 | e1251289
Physical Model of Transport through the NPC
comments. This work was supported by US National Institutes of Download full-text
Health grants RR00862 (BTC), GM062427 (MPR), GM071329 (MPR,
BTC, AZ, MOM), and RR022220 (MPR, BTC).
Author contributions. 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.
Funding. The authors received no specific funding for this study.
Competing interests. The authors have declared that no competing
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Physical Model of Transport through the NPC