Statistical Study of the Unfolding of Multimodular Proteins and their Energy Landscape by Atomic Force Microscopy

The aim of the present thesis is to investigate several aspects of: the proteins mechanics, interprotein interactions and to study also new techniques, theoretical and technical, to obtain and analyze the force spectroscopy experiments. The first section is dedicated to the statistical properties of the unfolding forces in a chain of homomeric multimodular proteins. The basic idea of this kind of statistic is to divide the peaks observed in a force extension curve in separate groups and then analyze these groups considering their position in the force curves. In fact in a multimodular homomeric protein the unfolding force is related to the number of not yet unfolded modules (we call it "N"). Such effect yields to a linear dependence of the most probable unfolding force of a peak on ln(N). We demonstrate how such dependence can be used to extract the kinetic parameters and how, ignoring it, could lead to significant errors. Following this topic we continue with non kinetic methods that, using the resampling from the rupture forces of any peak, could reconstruct the rupture forces for all the other peaks in a chain. Then a discussion about the Monte Carlo simulation for protein pulling is present. In fact a theoretical framework for such methodology has to be introduced to understand the various simulations done. In this chapter we also introduce a methodology to study the ligand receptor interactions when we directly functionalize the AFM tip and the substrate. In fact, in many of our experiments, we see a "cloud of points" in the force vs loading rate graph. We have modeled a system composed by "N" parallel springs, and studying the distribution of forces obtained in the force vs loading rate graph we have establish a procedure to restore the kinetic parameters used. Such procedure has then been used to discuss real experiments similar to biotin-avidin interaction. In the following chapter we discuss a first order approximation of the Bell-Evans model where a more explicit form of the potential is considered. In particular the dependence of the curvature of the potential on the applied force at the minimum and at the metastable state is considered. In the well known Bell-Evans model the prefactors of the transition rate are fixed at any force, however this is not what happen in nature, where the prefactors (that are the second local derivative of the interacting energy with respect to the reaction coordinate in its minimum and maximum) depend on the force applied. The results obtained with the force spectroscopy of the Laminin-binding-protein are discussed, in particular this protein showed a phase transition when the pH was changed. The behavior of this protein changes, from a normal WLC behavior to a plateau behavior. The analysis of the force spectroscopy curves shows a distribution of length where the maximum of the first prominent peak correspond to the full length of the protein. However, length that could be associated with dimers and trym