Nuclear pore complexes (NPCs) are very selective filters that monitor the transport between the cytoplasm and the nucleoplasm. Two models have been suggested for the plug of the NPC. They are (i) it is a reversible hydrogel or (ii) it is a polymer brush. We propose a mesoscopic model for the transport of a protein through the plug, that is general enough to cover both. The protein stretches the plug and creates a local deformation. The bubble so created (prtoein+deformation) executes random walk in the plug. We find that for faster relaxation of the gel, the diffusion of the bubble is greater. Further, on using parameters appropriate for the brush, we find that the diffusion coefficient is much lower. Hence the gel model seems to be more likely explanation for the workings of the plug.
The nuclear envelope in all eucaryotes is perforated with nuclear pores [1,2,3,4,5,6]. Each pore has a selective filter, referred to as the nuclear pore complex (NPC). The NPC is a self-assembled, eightfold symmetric ringlike structure consisting of eight copies each of 30-50 different proteins, connecting the inner and outer nuclear membranes. It regulates the import and export traffic of proteins and has two distinct modes of transport: 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 mass, pass efficiently through the NPC in this way [7]. In contrast, facilitated translocation allows the passage of objects as large as several megadaltons. Proteins having a short amino acid sequence known as nuclear localization signal (NLS) form a complex with transportin [5] (a transporter protein rich in hydrophobic units) and are transported in this mode.
The transport requires specific interactions between the translocating species and constituents of the NPC and consequently is highly selective. Gold particles of up to 32-36 nm in diameter are able to pass through some NPC if they are coated with nucleoplasmin-importin complexes [3]. This suggests that the interaction of the protein-transportin complex with NPC is essential for transport. Passage of proteins through the NPC has attracted considerable experimental and theoretical attention [8,9,10,11]. According to Ribbeck and Görlich (RG) [8] the central plug of the nuclear pore is made of long diblock copolymers rich in hydrophobic phenylalanine-glycine (FG) units forming a meshwork. Only the macromolecules which can form hydrophobic contacts with the FG units are incorporated into this network and get transported. In an interesting paper, Bickel and Bruinsma (BB) [10] point out that such a model would lead to a lower rate of diffusion.
BB suggest that the central plug is a reversible polymer gel in a poor solvent and a protein in it experiences an extra noise (they call it “chemical noise”) arising from the fluctuations of the FG contacts and this extra noise enhances the diffusion of the protein within the NPC. Single molecule fluorescence microscopy by Yang, Gelles and Musser [9] shows that the protein executes random walk inside the central core of the NPC. An alternate model for the plug suggests that it is not a gel but a polymer brush [12]. Surprisingly, there have been experimental support for both gel and brush models. In interesting experiments, Frey et al. [13,14,15] have shown that the nucleoporins form a hydrogel in vitro, offering support to the model of RG and BB. On the other hand a beautiful study by Lim et al. [16] has found that the proetins, when grafted to a surface behave like an un-cross-linked brush. Also, it is known that the interaction between the transportin and the FG residues is not just hydrophobic, but involves hydrogen bonding, electrostatic and van der Waals interactions [17,18] and that there are extremely hydrophilic portions in between the FG units.
In the following we study a minimalistic model for the transport in the NPC. Our model is quite general, and would be applicable whether the plug is a gel or a brush. The actual values of the parameters in the model would depend on whether it is a gel or a brush. We find that the it is possible for the protein to diffuse rapidly within a reversible gel, while the diffusion would be much slower within the brush.
We take the pore complex to be infinitely long (end-effects neglected) and shall adopt a continuum description for the plug. We use x to denote position along the direction of the axis of the NPC. To make the problem one dimensional, we imagine the cross section of the pore to be a square, with width L Y in the Y and height L Z (= L Y ) in the Z directions. We shall assume that the particle has a length 2R 0 in the X direction, causes a distortion of height ∼ α and fills the pore fully in the Y direction. Further, to simplify the analysis, we assume that periodic boundary conditions are imposed in this direction, with a period L Y . With these, the problem is reduced to two dimensions (X and Z). The size of the distortion needed to create a cavity to accommodate the particle will be our important variable. Let φ(x) denote the height of the cavity in the Z-direction at the position x. The simplest possible expression for the energy of distortion 2 of the particle at R with the plug can compensate for the distortion energy. Q(y) with y = x -R is a function that determines the interaction between the particle and the plug.
We assume Q(y) to be a symmetric function of y, having maximum value Q(0) = 1. Further, we assume Q(±∞) = 0. When the particle enters the plug it would have to break hydrophobic contacts that may be there, and the energy expenditure for that may be met by formation of new contacts of the nucleoporins with the particle. All t
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