THE NUCLEAR PORE COMPLEX

TOP ABSTRACT THE NUCLEAR PORE COMPLEX THE PLATE MODEL: EQUILIBRIUM... KINETICS DIFFUSION BY FORCE FLUCTUATIONS CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

Introduction

The nuclear pore complex (NPC) is among the largest of the molecular machines that are being probed by the methods of single-molecule biophysics (Elbaum, 2001; Salman et al., 2001). It plays the crucial role of regulating the import traffic of proteins from the cytoplasm across the nuclear envelope into the nucleus and the export traffic of gene transcription RNA strands from the nucleus into the cytoplasm (Panté and Aebi, 1996; Allen et al., 2000; Wente, 2000). The NPC has remarkable transport capabilities with two distinct modes: passive and facilitated. Passive transport is nonspecific and takes place by ordinary diffusion. Colloidal gold particles with radii up to 4 nm, and generic proteins up to 50 kDa in size, pass efficiently through the NPC in this way (Paine et al., 1975). Facilitated transport is highly specific and can proceed against concentration gradients (Dingwall et al., 1982). Macromolecules with sizes exceeding 1 MDa can be transported this way (Panté and Kann, 2002), while the mass-flow rate can reach the amazing level of 100 MDa/s with as many as 10³ macromolecules passing an individual NPC each second (Ribbeck and Görlich, 2001; Smith et al., 2002).

Because facilitated NPC transport requires a free energy source, it would seem that understanding the surprising transport properties of the NPC demands a full analysis of the intricate molecular machinery. As discussed below, the NPC is an "affinity switch" that relies on the concentration difference of another molecular species to act as the free energy source for essentially diffusive transport. The singular physical properties of a gel-like plug in the central core of the NPCthe transporteris the key determining factor for the specificity of facilitated transport, with the additional biochemical machinery responsible for the tagging of cargo molecules and the maintenance of the concentration differences. The present article will discuss the fact that the properties of the transporter appear to be in conflict with what is known about the rheology of gels and propose a solution, illustrated by a simple, analytical soluble model. We will start with a brief review of the basic structure of the NPC.

The NPC is a self-assembled, eightfold symmetric ring-like structure consisting of 30-50 different proteins with a total mass of ~125 MDa (Rout et al., 2000; Ryan and Wente, 2000), connecting the inner and outer nuclear membranes. On the inner (nuclear) side eight fibrils are connected to two rings, forming a basket structure with a size of ~100 nm. On the outer (cytoplasmic) side, another eight 100-nm fibrils also are connected to a ring. The coaxial rings themselves are attached on opposite sides of a spoke-complex with, at the center, a porous core called the transporter forming a selective permeability barrier. The free energy source that permits facilitated transport against a concentration gradient is the hydrolysis of guanosine triphosphate (GTP) molecules. First, a short peptide tag ("NLS") identifies cargo macromolecules for nuclear import. Tagged cargo macromolecules then form a complex with  importin, a transporter protein (Allen et al., 2000). Following import, the cargo/importin complex breaks apart when a GTP-associating protein, known as Ran (Azuma and Dasso, 2000), forms a complex with the importin. The Ran/GTP/importin complex is then exported through the NPC. Following GTP hydrolysis the importin is released, after which it is ready for another import cycle. The actual transport of the cargo/transportin complex across the NPC does not require GTP hydrolysis: no mechanical forces are exerted by the NPC during macromolecular traffic (Schwoebel et al., 1998; Nachury and Weis, 1999). The concentration difference of Ran/GTP and Ran/GDP across the nuclear envelope is the free energy source that compensates for unfavorable cargo molecule concentration differences. GTP hydrolysis is necessary only to maintain the high concentration of Ran/GTP inside the nucleus.

Recent experiments (Ribbeck and Görlich, 2001; Quimby et al., 2001; Smith et al., 2002) on the translocation of transporter proteins across the NPC in fact demonstrate that the traffic can be described as conventional permeation. However, compared to passive transport, the NPC permeability for transporter proteins is extremely high, about a quarter of the permeability of an equivalently sized water-filled cylindrical channel. In other words, the permeability barrier is selectively transparent for transporter proteins. The NPC transporter can switch between different levels of permeability, depending on whether or not it "recognizes" the macromolecule.

The Ribbeck-Görlich (RG) Model

Ribbeck and Görlich (2001) proposed the model for the permeability of NPC switching shown in Fig. 1. The permeable core of an NPC has a length of ~40 nm and a comparable diameter. It contains a 12-MDa low-density peptide "plug" consisting of long diblock copolymers rich in hydrophobic phenylalanine-glycine repeat units separated by hydrophilic spacers (Bayliss et al., 1999; Rout et al., 2000). These hydrophobic repeat units attract each other with a low binding energy (of order k_BT). The NPC loses its selectivity when this plug is removed, in which case the transport kinetics of the NPC reduces to that of a water-filled pore. In the RG model, the plug is treated as a homogeneous polymer network with a mesh size in the range of 3 nm, as deduced from the fact that there are ~10³ hydrophobic units per NPC. Generic proteins (or gold particles) smaller than this mesh size can pass easily while generic proteins with larger size are increasingly blocked. Transporter proteins have a selective affinity for the hydrophobic repeat units, through surface residues such as tryptophane, which causes them to be incorporated into the bonding network. The recognition of the transporter protein by the gel allows it to enter the permeability barrier. The opening/closing kinetics of the weak bonds then produces efficient diffusive transport of the transporter protein across the barrier.

View larger version (15K): [in this window] [in a new window]   FIGURE 1   Selective phase model of Ribbeck and Görlich for the transporter. The transporter is filled with long diblock copolymers, rich in hydrophobic phenylalanine-glycine repeat units. These units can form weak bonds, which leads to formation of a reversible gel. Particles smaller than the mesh size of the network can diffuse freely through the pore. Generic macromolecules larger than the mesh size cannot pass the transporter, but macromolecules with chemical affinity for the linker units are incorporated into the network. Adapted from Ribbeck and Görlich, 2001.

From the viewpoint of conventional polymer rheology, the RG model cannot work. The proposed structure of the NPC core corresponds to what is known as a reversible gel in the polymer physics literature (Tanaka and Edwards, 1992; Semenov et al., 1995; Semenov and Rubinstein, 1998). The diffusion constant D₀ of a sphere with no recognition groups, moving through a reversible gel having a mesh size small compared to the radius r, equals k_BT/(6r) with  = G the gel viscosity, G the plateau modulus, and  the stress relaxation time (de Gennes, 1979). If reversible recognition groups are placed on the macromolecule surface, connecting it to the gel, then the diffusion constant should be reduced by a factor P = D/D₀ equal to the probability of the sphere to be "free" (Leibler et al., 1991). A macromolecule that is recognized by a reversible gel should have a lower, not a higher, mobility. The interaction does indeed allow a sphere with linker groups to enter the reversible gel more easily, but in Appendix A we show that the net permeability for a sphere to pass a cylindrical pore containing a reversible gel remains highest when the sphere does not carry any recognition groups. This is in direct contradiction with the in vitro experiments on NPCs, so either the RG model for facilitated NPC transport is incorrect or the permeability barrier is not a conventional reversible gel, the option pursued in this paper.

Reversible gels in poor solvent and force fluctuations

We would like to propose that the RG model still may work provided the permeability plug is not a conventional gel, but a confined reversible gel in poor solvent (de Gennes, 1979). Consider a reversible gel in aqueous solvent with the polymers constituting the gel consisting of hydrophobic groups separated by hydrophilic spacers, as in the RG model. Let a plug of the gel be confined to the interior of a hollow cylinder with the chain ends grafted to the cylinder surface. It is well known that under bulk conditions, a reversible gel undergoes a collapse transition upon reduction of the "solvent quality" (the average solubility per monomer) (Rubinstein and Dobrynin, 1999). In a confined geometry with the polymers pinned to the container walls, a collapse transition is not possible. Instead, the polymer chains are placed under tension at this point: by stretching selective chains, a larger number of paired hydrophobic groups can be achieved. It should be emphasized at this point that the stability of such stretched structure requires the chain extremities to be held at a fixed position. Indeed, releasing this constraint would lead to a collapse of the gel to its desired density into a cylindrical ring.

We claim that this arrangement is characterized by strong thermal fluctuations in terms of the network connectivity, somewhat in the nature of a spin glass. There are in general many different arrangements, with comparable free energies, to pair-off the hydrophobic groups. If one link connecting two groups would fail, due to a thermal fluctuation, or if a new pair forms, then the network connectivity would locally change with a corresponding rearrangement of the tensions in the connecting spacer groups (see Fig. 2). A tense, reversible gel of this type thus would be characterized as well by force fluctuations.

View larger version (14K): [in this window] [in a new window]   FIGURE 2   Connectivity fluctuations. If a link connecting a pair of hydrophobic groups opens because of a thermal fluctuation, then the tensions of the polymers in the local neighborhood are rearranged.

Now assume that a macromolecule contains reversible linker groups allowing weak links with the gel. The force exerted by the fluctuating network on the macromolecule is the sum of the tension forces exerted by polymers attached to its surface. Rearrangements of the tension field due to the thermal configuration fluctuations would produce a time-dependent fluctuating force on the macromolecule. In the neighborhood of a given local minimum of the position of the macromolecule, there would be neighboring minima characterized by local changes in the gel connectivity. Unlike a generic macromolecule, a macromolecule with recognition groups could move from local minimum to local minimum.

The remainder of the paper is dedicated to the analysis of a simple "toy" model of a reversible gel, introduced in the next section, with just two minima for an embedded plate-like macromolecule. The aim of the model is to demonstrate that the form of transport outlined above, with hopping between different local minima, can produce enhanced diffusion constants. The model is certainly too simple to describe the complex properties of confined gels in poor solvent outlined above, but it has the virtue that it can be examined on an analytical basis. The model demonstrates that enhanced diffusion is possible, but only by a careful tuning of the system parameters, such as the linker binding energy. The model was inspired by a recent experimental and numerical study of the forces between two surfaces covered by an array of tethered ligands and receptors and in principle could be studied experimentally or numerically (Jeppesen et al., 2001).

The Kinetics section discusses the model in terms of the "chemical noise" generated by the opening/closing kinetics of the ligand-receptor pairs. In the subsequent section we apply the model to estimate the anomalous diffusion constant of a plate embedded in a tense reversible gel. We find that, as a function of the binding energy and the degree of stretching of the network polymers, there indeed is an intermediate range where the diffusion coefficient is significantly enhanced.

	THE PLATE MODEL: EQUILIBRIUM PROPERTIES

TOP ABSTRACT THE NUCLEAR PORE COMPLEX THE PLATE MODEL: EQUILIBRIUM... KINETICS DIFFUSION BY FORCE FLUCTUATIONS CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

We represent the macromolecule as a flat plate of area  confined between two parallel walls with spacing 2L (see Fig. 3). The position X of the plate is defined with respect to the middle of the gap, and is allowed to vary between L and +L. Each side of the plate has a surface concentration  = n/ of receptor groups. Ideal polymers with radius of gyration R are grafted on both walls with the same surface coverage. The free end of each grafted polymer contains a ligand group that can bind to a receptor group on the plate with a binding energy .

View larger version (11K): [in this window] [in a new window]   FIGURE 3   Plate model. The plate can move freely between two walls, each covered with n grafted polymers. The free end of each chain can adsorb onto the plate with binding energy . The equilibrium position of the plate is determined by the balance between compression of the two polymer layers and the stretching of bound chains.

The ligand-receptor statistical mechanics will be described by treating the polymer population as a two-state system with polymers either paired (+) or unpaired (). Let w₊(y) and w(y) be the free energies of polymers belonging to the (+), respectively, () population with y equal to L  X on the right side of the plate and equal to L + X on the left side. In particular, w₊(y) is the configurational free energy of an ideal polymer whose endgroups are attached to two plates separated by a distance y, while w(y) is the configurational free energy with only one of the two ends fixed. In the small y limit (i.e., y R) w₊(y)  w(y)  k_BT(y/R)², while for y > R, w₊(y)  k_BT(y/R)² and w(y)  0 (de Gennes, 1979). The two-state free energy per polymer g(y) is then given by

(1)

with  = 1/(k_BT). A reasonable interpolation formula for g(y) valid for general y is then g(y)  u(y)  k_BT ln(1 + e^(v(y))), where u(y) = 1/2k_BT(y/R)² and v(y) = 1/2k_BT(y/R)². Adding the contributions coming from polymers on both sides of the plate, we find for the total free energy V(X) = n{g(L  X) + g(L + X)}

(2)

where E(y) =  + v(y) is shown in Fig. 4. We can consider V(X) as the "potential of mean force" for the plate. It is an even function of X, diverging at X = ±L, whose form depends sensitively on the binding energy .

View larger version (12K): [in this window] [in a new window]   FIGURE 4   Two-state model for the polymer chains. Unpaired, relaxed polymers (top) correspond to zero on our energy scale. If a polymer is stretched (bottom), then the free energy increases initially due to the entropic elasticity of the chain until the endpoint can make a bond with the plate where the polymer gains a binding energy . Depending on the degree of stretching  (ratio of the end-to-end length and the unstretched radius of gyration), binding may or may not lower the free energy of the polymer.

Single-minimum regime

In the regimes of /k_BT large or small compared to one, V(X) adopts the following limiting forms:

(3)

for /k_BT 1, and

(4)

for /k_BT 1. Because both u(y) and v(y) are convex functions of y, V(X) is a convex function as well in these two limits. Because it is even, V(X) has a single minimum at X = 0. For the case /k_BT 1, the free energy minimum is located at X = 0 because that maximizes the number of links, while for /k_BT 1, the minimum is located at X = 0 because that minimizes the configurational free energy of the two confined polymer brushes. It also can be shown from Eq. 2 that the even, convex form for V(X) is encountered for general values of /k_BT in the "low-tension" limit  = L/R < _c = , where the stretching energy of a paired polymer chain is small compared to the thermal energy. There are no multiple minima in the free energy landscape of the plate in these regimes and the equilibrium position of the plate is at X = 0 with no anomalous kinetic properties. In the following, we will assume that /k_BT is of order one and  > _c.

Multiple-minima regime

Expand the free energy V(X) to fourth-order in the dimensionless plate displacement x = X/L

(5)

where a₂[, ] and a₄[, ] are dimensionless functions of  and . For /k_BT either large or small compared to one, a₂ is positive. Because a₄ is always positive, it follows that V(X) again has a single minimum at X = 0 in these regimes, as noted earlier. Over an intermediate energy interval ₁ <  < ₂, the coefficient a₂ is negative: it follows that the function V(x) has two symmetrically placed minima X  ±, separated by a maximum at X = 0. Fig. 5 shows the position of these minima as a function of . The location of the two bifurcation points  = _1,2 is given by

(6)

The two bifurcations correspond to continuous transitions where the symmetry between +X and X is broken spontaneously. In this intermediate regime, the plate chooses to associate itself with either of the two boundaries, relieving the tension on one side of the plate at the cost of losing the pairing energy on the other side of the plate.

View larger version (8K): [in this window] [in a new window]   FIGURE 5   Plate positions minimizing the free energy as a function of the adsorption energy  (in units of kBT) for  = 1.75. Two degenerate minima appear in the range 1 <  < 2, where 1 and 2 are functions of the parameter  given in the text.

	KINETICS

TOP ABSTRACT THE NUCLEAR PORE COMPLEX THE PLATE MODEL: EQUILIBRIUM... KINETICS DIFFUSION BY FORCE FLUCTUATIONS CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

The plate is exposed to two types of noise: the usual thermal fluctuations of the surrounding solvent and the chemical noise of the on-off fluctuations of the linker groups. To describe these tension-sensitive fluctuations, we assume that the plate position X(t) can be treated as an adiabatic, collective degree of freedom (Risken, 1996) with a separation in time scales between the microscopic chain degrees of freedomtreated again as a dynamical two-level systemand the macroscopic plate position.

Chemical noise

Let p(y, t) be the fraction of chains in the paired state and 1  p(y, t) the unpaired fraction. If k(y) is the "off-rate" for the break-up of a paired state and r(y) the "on-rate" for the formation of the paired state, then the stochastic equation of motion for p(y, t) is

(7)

Here, µ_y(t) is a zero-average Gaussian noise source

(8)

which describes the chemical noise whose intensity (y) remains to be determined. The on- and off-rates depend on the collective coordinate X(t) through the variable y(t), which equals L + X(t) or L  X(t), depending which side of the plate the chain is located on. The on-rate is

(9)

where  is the attempt frequency that will be identified with the inverse of the Rouse relaxation time _R of the polymer chains (proportional to N² with N the number of Kuhn lengths). The off-rate then follows from the condition that the steady-state solution of Eq. 7, p_y = r(y)/(k(y) + r(y)), must be consistent with the pairing probability as predicted by the Boltzmann distribution

(10)

From this condition it follows that the relaxation time (y) = {r(y) + k(y)}¹ equals

(11)

We can determine the autocorrelation function for fluctuations in the population of bound and free chains by integrating the equation of motion Eq. 7

(12)

where p(t) = p(t)  p_y. Finally, the noise intensity (y) in Eq. 7 is given by the fluctuation-dissipation theorem

(13)

Force fluctuations

We now turn to the macroscopic equation of motion for the plate and the force fluctuations exerted by the binding and unbinding of the polymers. We will assume that solvent and chemical noise operate in an additive way on the plate. The equation of motion of the plate then takes the form

(14)

On the left side, (X) is the effective friction coefficient of the plate in the presence of the chemical noise. In the first term on the right-hand side, the potential U(X) equals n{u(L  X) + u(L + X)} and describes the compression of the polymer layers. The second term F(X, t) is the stochastic tension force exerted by the chains on the plate

(15)

Here, f(y) = dv/dy is the restoring force of a stretched polymer chain. The last term of Eq. 14 represents the conventional hydrodynamic noise

(16)

where the hydrodynamic friction coefficient ₀ of the plate moving through chain-free solvent is assumed to be known.

Separate the tension force F(X, t) = F(X) + F_X(t) into an average and a random force of zero mean, then the Langevin equation simplifies to

(17)

with V(X) the equilibrium potential energy determined earlier. The autocorrelation function of the random force F_X(t) follows from the autocorrelation function (Eq. 12) for the ligand-receptor fluctuations

(18)

The RMS of the chemical noise force exerted on the plate is thus of order f(L)n^1/2, as is reasonable on intuitive grounds. By assumption, the kinetics of the plate is slow compared to that of the chains, which means that we can approximate the force autocorrelation function by a delta function

(19)

To compare the chemical and hydrodynamic noise levels in Eq. 17, let r be the size of the macromolecule. The solvent contribution to the friction is then of order ₀ ~ r₀, with ₀ the solvent viscosity. The intensity of hydrodynamic force fluctuations is then of order k_BTr₀ (see Eq. 16). However, the intensity of the on-off force fluctuations is, according to Eq. 19, of order nf². The relaxation time  is estimated as the Rouse time _R = ₀N²a³ of the polymers of the gel, where a is the Kuhn segment length of the polymer. The typical tension level in the polymer chains is f ~ L/(Na²). Assuming, moreover, that the area density of binding sites on the surface of the macromolecule is matched to the gel mesh size (i.e., that n is of order (r/L)²), the ratio of chemical over hydrodynamic noise intensities is of order ~r/a. For the experiments on NPCs described in the Introduction, this is of order of 10¹-10², so we expect the chemical fluctuations to dominate.

	DIFFUSION BY FORCE FLUCTUATIONS

TOP ABSTRACT THE NUCLEAR PORE COMPLEX THE PLATE MODEL: EQUILIBRIUM... KINETICS DIFFUSION BY FORCE FLUCTUATIONS CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

Friction coefficient and transition rate

The stochastic differential equation (Eq. 17) for X(t) combined with the noise correlation function (Eq. 19) represents a Brownian walk in a potential V(X) with a position-dependent Gaussian noise-source (van Kampen, 1992; Risken, 1996). Using well-established methods (see Appendix B), it can now be shown that the solution of the Fokker-Planck equation for the probability distribution of the stochastic variable X(t) only leads to the Boltzmann distribution in steady state, provided the friction coefficient has the form

(20)

The friction coefficient is a non-monotonic function of X, as shown in Fig. 6. Using this expression for the friction coefficient, we now can treat the kinetics of the plate.

View larger version (10K): [in this window] [in a new window]   FIGURE 6   The dimensionless contribution to the friction coefficient due to reversible binding of the polymers to the plate, shown as a function of plate position, for chain extension  = 1.75 and binding energies  = 0.5, 1, and 2 kBT (from bottom to top). The plots are shown in dimensionless units,  = L2/(nR), with R the Rouse relaxation time of the chain.

Recall that if a₂ < 0, V(X) has two minima X  ±. The chemical noise fluctuations will cause a transitions between the two minima. The crossing rate  can be computed by solving the Fokker-Planck equation using the Kramers method (see Appendix B)

(21)

where V = V(0)  V(X_min). The reason that only the friction coefficient at X = 0 appears in Eq. 21 is that, during a successful transition between the two wells, the system spends more time at the "transition state" X = 0 separating the two wells than elsewhere.

Diffusion coefficient

In the first section we proposed that the diffusion constant of a macromolecule embedded in a confined, reversible gel under poor solvent conditions could be anomalously high and that it took place by random "hopping" between adjacent minima of an effective, three-dimensional potential. We will now apply the plate model, in the bifurcation regime, to estimate the diffusion coefficient as D  X², with X the typical spacing between the two adjacent minima. Using Eqs. 5 and 21, this leads to

(22)

with the friction coefficient given by Eq. 20.

According to Eq. 22, the diffusion coefficient has a non-monotonic dependence on the number of linker units. The diffusion coefficient vanishes exponentially in the large n limit, because the energy barrier separating the two minima is proportional to n. However, unlike conventional rheology, D increases with n for small n. In the regime where friction is dominated by the chemical noise, i.e., for  in the range ₁ <  < ₂, maximizing Eq. 22 with respect to n gives an optimal choice for the number of linkers n = 4a₄/|a₂|², independent of the size of the macromolecule. This diffusion "magnification" disappears at the boundaries of the bifurcation regime. As shown in Fig. 7, the specific diffusion coefficient is highly sensitive to small variations of the binding energy  and the extension rate .

View larger version (8K): [in this window] [in a new window]   FIGURE 7   Specific diffusion coefficient as a function of  (in units of kBT) for dimensionless chain extensions  of 1.75, 1.77, and 1.79 (from bottom to top). We have fixed n = 5 and the ratio RD0/L2 = 102, where D0 is the diffusion coefficient in pure solvent.

	CONCLUSIONS

TOP ABSTRACT THE NUCLEAR PORE COMPLEX THE PLATE MODEL: EQUILIBRIUM... KINETICS DIFFUSION BY FORCE FLUCTUATIONS CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

The results obtained from the plate model indicate that enhanced diffusion coefficients for embedded macromolecules are, at least in principle, possible. It must be clear though that our results on the plate model do not constitute a proof of the proposed mechanism. We have not demonstrated that the "energy landscape" of the plate model is realistic for real reversible gels in poor solvents. The description for such gels offered in the first section bears similarity with the physics of spin glasses, which are characterized by complex energy landscapes (Kumar and Douglas, 2001). Whether transport is determined by a "typical" energy barrieras assumed implicitly in the present paperis far from obvious. A second important point concerns the fact that in the plate model, unbinding events of different linkers are correlated only in a mean-field sense through the position X of the plane. Actually, the unbinding of a ligand group should affect its neighbors over a distance of the order of a certain correlation length , that may diverge at a stress percolation transition. Because the plate model is tractable, however, it would be very interesting if it could be realized experimentally, for instance by trapping a colloid between two plates that have grafted layers of polymers with recognition groups at their extremities. The mobility of the colloid, both in the plane of the plates and in the perpendicular direction, could be measured as a function of the plate spacing to control the tension of the polymers.