Cooperative Interactions between Different Classes of Disordered Proteins Play a Functional Role in the Nuclear Pore Complex of Baker’s Yeast

Nucleocytoplasmic transport is highly selective, efficient, and is regulated by a poorly understood mechanism involving hundreds of disordered FG nucleoporin proteins (FG nups) lining the inside wall of the nuclear pore complex (NPC). Previous research has concluded that FG nups in Baker’s yeast (S. cerevisiae) are present in a bimodal distribution, with the “Forest Model” classifying FG nups as either di-block polymer like “trees” or single-block polymer like “shrubs”. Using a combination of coarse-grained modeling and polymer brush modeling, the function of the di-block FG nups has previously been hypothesized in the Di-block Copolymer Brush Gate (DCBG) model to form a higher-order polymer brush architecture which can open and close to regulate transport across the NPC. In this manuscript we work to extend the original DCBG model by first performing coarse grained simulations of the single-block FG nups which confirm that they have a single block polymer structure rather than the di-block structure of tree nups. Our molecular simulations also demonstrate that these single-block FG nups are likely cohesive, compact, collapsed coil polymers, implying that these FG nups are generally localized to their grafting location within the NPC. We find that adding a layer of single-block FG nups to the DCBG model increases the range of cargo sizes which are able to translocate the pore through a cooperative effect involving single-block and di-block FG nups. This effect can explain the puzzling connection between single-block FG nup deletion mutants in S. cerevisiae and the resulting failure of certain large cargo transport through the NPC. Facilitation of large cargo transport via single-block and di-block FG nup cooperativity in the nuclear pore could provide a model mechanism for designing future biomimetic pores of greater applicability.


Free energy of stalk chain stretching
The per chain stretching free energy is (1): With n b the number of blobs per nup and where the stretching free energy per nup is equal to Boltzmann's constant times the temperature times the number of blobs in the stalk region of a nup. The number of blobs in the stalk brush if we have a flat brush is: With L the end to end distance of stalk chains and ξ the radius of the stalk brush blobs. The number of monomers per extended chain is: With a equal to the length of monomers in the chains.
⇒ N = (L/ξ)(ξ/a) 5/3 (5) ⇒ ξ = (N/L) 3/2 a 5/2 (6) Now we can consider the cylindrical brush case. First we consider blob size changes along the stalk brush as a function of the contour length s, which is a function of monomer number m. i.e. m is the mth monomer starting from the direction of the center of the cylinder with m = 0 the free end of the stalk chain: In direct analogue to the flat brush case, Eq. 6: We can now calculate the change in the number of blobs as a function of s, analogous to Eq. 3, as: Which after substitution from Eq. 18 becomes (2): We can now solve for the stretching free energy in Eq. 2 Which after the mean field approximation of: With H the height of the cylindrical stalk brush. Eq. 12 reduces to: Where we can now conclude that the stretching free energy per chain is:

Free energy of excluded volume interactions
For a flat brush the free energy per chain of excluded volume interactions is (1): With ρ blob the volume fraction of blobs in the brush. This can be generalized to a per chain free energy valid for cylindrical brushes: With n b the number of blobs per nup. The local volume faction can be defined as: Which implies that: Which simplifies to: Which after substitution for dn b ds by Eq. 8 equals: Which after substitution for ξ from Eq. 18: Which equals: Which after the mean field approximation of (2): and becomes: Which after substitution for the number of chains in the brush N c : leads to a per chain excluded volume free energy of:

Free energy of tip-tip cohesive interactions, f coh
We define f coh to be the absolute value of the energy density of sticky tip interactions, representing part of the F total coh term. Similar to the flat brush case, for the sticky tip to sticky tip interactions we can define an energy density of blob interactions to have an energy of k b T : We have c = N c N b /V the concentration of blobs (for N b the total number of blobs per chain). For each chain or FG nup there exists only one sticky tip, therefore N b =1 in this case. The volume the sticky tips can take on is approximated as an extended cylindrical region atop the stalk brush region whose volume is V = 2π(R − H)Lδ, with L equal to the height of the brush region axially along the pore.
N c is the number of chains determined by the grafting distance d, with The concentration c is therefore: Which results in a free energy density of blob interactions of: Which can be simplified to: The total cohesive energy per chain is then equal to the volume of a sticky tip blob times the cohesive free energy density of all blob interactions divided by the number of chains.
4 Free energy of tip-shrub cohesive interactions, f shrub We define f shrub to be the absolute value of the energy density of sticky tip to shrub interactions, representing part of the F total coh term. Similar to the tip-tip free energy we can define an energy density of shrub blob to tip blob interactions given that blob-blob interactions have an energy of s k b T : Where V o is the overlap volume of the different types of blobs.
For δ equal to the blob size of sticky tips, we have c t from Equ. 31: We have c s = N c N b /V the concentration of shrub blobs (for N b the total number of blobs per chain). For each chain or FG nup there exists only one shrub, therefore N b =1 in this case. The volume the shrubs can take on is approximated as a hallowed extended cylindrical region extending from the wall of the pore to radius R s whose volume is V = 2πR s Lδ s , with L equal to the height of the brush region axially along the pore.
N c is the number of chains determined by the grafting distance d, with The concentration c s is therefore: Which results in a free energy density of blob interactions of: Which can be simplified to: The overlap volume is approximately for the Heaviside step function Θ(x) which is 1 if x is positive and 0 if x is negative. The free energy density is therefore: The total cohesive energy per chain is then equal to the volume of a sticky tip blob times the cohesive free energy density of all blob interactions divided by the number of chains.

Free energy for the total brush
We can now solve for the total free energy of the cylindrical brush per chain: Simulation Equilibration  We use the first microsecond of our CG simulations as an equilibration period where data is not taken for characterization of FG nup properties. Only the last 4 microseconds of a CG simulation are considered the production run and have the molecular data analyzed. Shown is the radius of gyration for Nup42, starting as a fully extended polymer at time zero, which rapidly converges to a collapsed polymer structure after around 0.5 microseconds. Other FG nups which were simulated are of comparable size and converge on the same 0.5 microsecond timescale, which is well within the 1 microsecond initial equilibration time period.