Compact tube geometries in crowded environments

Abstract

We study the effects of crowding on a hard semi-flexible tube. We use the tube, to model polymers such as proteins, in the regime where its width is comparable to its length. In this regime the polymer does not form a coiled ball. We use the depletion volume interaction between the tube and a solution of small hard spheres to model the effects of crowding. The tube bends into a compact configuration in order to maximize the entropy of the spheres. We analyze these compact geometries for various size crowding spheres. We find that at some tube lengths a tight helix reminiscent of α-helices in proteins can be formed. We then elaborate on the crowding effect by constraining the system in a tight cylinder. The tight boundaries increases the drive toward helix formation and there is now an interplay between the two relevant length scales: the sphere sizes and the cylinder width. We apply the model to tight tunnels seen in the cell, such as the ribosomal exit tunnel. The tunnel has a width comparable to the α-helices which form as the nascent protein traverses the tunnel. In our simplified model the tight boundaries in the tunnel show a large free energy gain with helix formation. We compare the entropic drive towards helix formation with an electrostatic repulsion between the tube and cylinder walls. This allows us to compare two of the dominant forces in the tunnel. We do this in a simplified model where the helical tube is approximated s a straight cylinder that is concentric with the tunnel. We also smooth out the charges to give a homogeneous charge distribution along the cylinder walls. Using numerical solutions of the Poisson-Boltzmann equation we see that in the tight tunnel the counter-ions screen most of the charge so that in our model the charge would not be enough to overcome the entropic drive towards helix formation. The screening in the tight confines of the tunnel causes the electrostatic potential in the tunnel to depend logarithmically on the charges on the tunnel walls