Knotting and unknotting proteins in the chaperonin cage: Effects of the excluded volume

Molecular dynamics simulations are used to explore the effects of chaperonin-like cages on knotted proteins with very low sequence similarity, different depths of a knot but with a similar fold, and the same type of topology. The investigated proteins are VirC2, DndE and MJ0366 with two depths of a knot. A comprehensive picture how encapsulation influences folding rates is provided based on the analysis of different cage sizes and temperature conditions. Neither of these two effects with regard to knotted proteins has been studied by means of molecular dynamics simulations with coarse-grained structure-based models before. We show that encapsulation in a chaperonin is sufficient to self-tie and untie small knotted proteins (VirC2, DndE), for which the equilibrium process is not accessible in the bulk solvent. Furthermore, we find that encapsulation reduces backtracking that arises from the destabilisation of nucleation sites, smoothing the free energy landscape. However, this effect can also be coupled with temperature rise. Encapsulation facilitates knotting at the early stage of folding and can enhance an alternative folding route. Comparison to unknotted proteins with the same fold shows directly how encapsulation influences the free energy landscape. In addition, we find that as the size of the cage decreases, folding times increase almost exponentially in a certain range of cage sizes, in accordance with confinement theory and experimental data for unknotted proteins.