In the first part of the project we have mostly focused on methodology development. These first result have laid the foundations for the future investigation of knotted proteins, tuning well-suited techniques to address this and other related biophysical problems. We have first investigated the folding mechanism of Granulocyte-macrophage colony-stimulating factor, a glycoprotein that handles diverse functions in the human body. This protein folds in a rather common self-entangled conformation named Complex-Lasso. Understanding how a polypeptide encodes into its sequence the capability of tying itself into such kind of self-entangled structures represents a major advancement in the comprehension of protein folding. We have studied this mechanism using both a well-known minimalistic model of the protein, and an alternative model, specifically designed by us to highlight the preferential pathways of entangled folding. Our calculations have shown how the protein can avoid the kinetic traps related to self-entanglement, managing to fold in a reproducible and efficient way. In this work we have employed a genetic optimization strategy which was developed to tune the parameters of the coarse-grained protein model, favoring the most efficient and reproducible pathways towards the native state. At the same time, we have developed two suitable descriptors to classify the topological state of the protein backbone. Such topological variables have allowed to monitor the evolution of the protein entanglement topology along the folding trajectories. Nonetheless, the definition of such kind of descriptors is crucial for the future application of enhanced sampling techniques, as these are based on the definition of a bias potential that acts on the relevant degrees of freedom of the folding dynamics. The topological variables represent the ideal degrees of freedom in the framework of self-entangled proteins. This work outlined an effective strategy for the study of further systems, which have been carried out in the following part of the project. In the following, we have addressed the coarse-grained molecular modeling of Leptin, a self-entangled hormone which is known to regulate energetic processes in human cells. Leptin presents a self-entangled native structure as well. Thanks to the developed methodologies we could study the folding of this protein, obtaining precious information on its dynamics and preferential folding pathways. These fruitful techniques were also applied to another self-entangled molecule, namely Human Interleukin 3, a signaling protein that regulates the production, differentiation and function of granulocyte and macrophages in the human cells. Also in these case we could enlighten a detailed map of the kinetics of entangled folding. In all these cases, the obtained results could be exploited to direct molecular dynamics studied using more detailed atomistic models of the molecules. In the final work period we have also directed our work towards the study of Human Ubiquitin C-terminal Hydrolase, an enzyme whose expression is highly specific to neurons and to cells of the diffuse neuroendocrine system and their tumors. We have applied our well-tuned modeling techniques, starting the development of a coarse-grained representation capable to fold in an efficient way towards its complex native topology. The outcomes of the action were disseminated through a methodological paper, a review paper, and a number of seminars in conferences, workshops and research groups. In connection to the project and its results, a mini-course on enhanced sampling methods has also been delivered to M.Sc. and PhD students in the University of Trento.
