Inspired by the protein folding and DNA packing, we have systematically studied the thermodynamic and kinetic behaviors of single semiflexible homopolymers by Langevin dynamics simulations. In line with experiments, a rich variety of folding products, such as rod-like bundles, hairpins, toroids, and a mixture of them, are observed in the complete diagram of states. Moreover, knotted structures with a significant population are found in a certain range of bending stiffness in thermal equilibrium. As the solvent quality becomes poorer, the population of the intermediate occurring in the folding process increases, which leads to a severe chevron rollover for the folding arm. However, the population of the intermediates in the unfolding process is very low, insufficient to induce unfolding arm rollover. The total types of folding pathways from the coil state to the toroidal state for a semiflexible polymer chain remain unchanged by varying the solvent quality or temperature, whereas the kinetic partitioning into different folding events can be tuned significantly. In the process of knotting, three types of mechanisms, namely, plugging, slipknotting, and sliding, are discovered. Along the folding evolution, a semiflexible homopolymer chain can knot at any stage of folding upon leaving the extended coil state, and the probability to find a knot increases with chain compactness. In addition, we find rich types of knotted topologies during the folding of a semiflexible homopolymer chain. This study should be helpful in gaining insight into the general principles of biopolymer folding.

In vitro, kinetically significant non-native interactions have been identified experimentally during the folding of proteins Im7, Im9, and A39V/N53P/V55L Fyn SH3 domain. To understand the role of non-native interactions on the folding of some frustrated proteins in chaperone, we employed native-centric models with and without additional transferable, sequence-dependent non-native hydrophobic interactions to comparatively study the folding behaviors of the three proteins confined in spherical cages. Under purely repulsive confinement, as a decrease of cavity size, the non-native interactions increase, especially in the unfolded state, enhancing the roughness of the folding energy landscape. As a result, the increase in native stability for the three proteins by the model incorporated non-native interactions (db + MJ hϕ model) is much smaller than that by the purely native-centric model (desolvation-barrier (db) model); the acceleration of folding simulated by the db + MJ hϕ model is much slower than that via the db model; in particular, the folding rate of Im7 decreases when reducing the cavity size under zero-denaturant condition. The repulsive confinement can also promote formation of specific non-native contacts in the transition state and favor more folding pathways passing through the misfolded state, leading to a higher population of the misfolded intermediate. In an attractive cage, the attractive interactions could inhibit the formation of intrachain non-native contacts and provide alternate folding pathways to the native state so that the population of the misfolded intermediate decreases when increasing the strength of attractive interaction between the substrate protein and cavity wall. This study should be helpful in general to understand how the chaperonins reshape the folding energy landscape of some frustrated proteins.