Rate-dependent force-extension models for single-molecule force spectroscopy experiments.

Single-molecule force spectroscopy techniques allow for the measurement of several static and dynamic features of macromolecules of biological origin. In particular, atomic force microscopy, used with a variable pulling rate, provides valuable information on the folding/unfolding dynamics of proteins. We propose here two different models able to describe the out-of-equilibrium statistical mechanics of a chain composed of bistable units. These latter represent the protein domains, which can be either folded or unfolded. Both models are based on the Langevin approach and their implementation allows for investigating the effect of the pulling rate and of the device intrinsic elasticity on the chain unfolding response. The theoretical results (both analytical and numerical) have been compared with experimental data concerning the unfolding of the titin and filamin proteins, eventually obtaining a good agreement over a large range of the pulling rates.

Thermodynamics of small systems with conformational transitions: The case of two-state freely jointed chains with extensible units.

Several experimental methods are usually applied for stretching single molecules and provide valuable insights about the static and dynamic responses induced by externally applied forces. This analysis is even more important for macromolecules exhibiting conformational transitions, thereby corresponding to folding/unfolding processes. With the aim of introducing the statistical mechanics of such phenomena, we apply here the spin variables approach based on a set of discrete quantities able to identify the folded or unfolded state of the chain units. First, we obtain the macroscopic thermodynamics of the chain from its microscopic description. For small systems, far from the thermodynamic limit, this result depends on the applied boundary condition (e.g., isometric or isotensional), which corresponds to the considered statistical ensemble. Then, we develop the theory for the two-state extensible freely jointed chain, where the elastic constant of the units, a property often neglected, plays a central role in defining the force-extension curve. For this system, the partition function of the isometric ensemble can be written in closed form in terms of the natural generalization of the Hermite polynomials, obtained by considering negative indices. These results are relevant for the interpretation of stretching experiments, operated from the entropic regime up to the unfolding processes.