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

Abstract

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.

Isotensional and isometric force-extension response of chains with bistable units and Ising interactions

“…(5) and by considering its approximation for finite relatively high values of κ. The details of this calculation can be found in recent literature [82]. Typical values of κ for real macromolecules are compatible with such approximations [89].…”

“…where a ∈ R, b > 0, N ∈ N and the path Γ is given in Fig.3. An application of the complex variable method allows us to obtain the closed form expression for this integral, as follows [82]…”

“…From the historical point of view, the first model based on a discrete quantity, similar to a spin variable, has been performed to predict the response of skeletal muscles [77,78]. This method has been recently applied to different two-state systems and molecular chains as well [79][80][81][82]. Both Gibbs and Helmholtz ensembles can be considered by the spin variables approach, allowing to draw direct comparisons between isotensional and isometric conditions at thermodynamic equilibrium.…”

Full Statistics of Conjugated Thermodynamic Ensembles in Chains of Bistable Units

“…The mathematical details about this idea can be found in Refs. [74][75][76]. In the present analysis, this approach leads to closed form expressions for the probability densities defined above, and the final results can be interpreted by introducing a form of duality between the two ensembles, useful to better understand the specific features of the isotensional and isometric conditions.…”

“…Of course, when we adopt the approximation of the energy wells with two quadratic functions, we lose the information about the energy barrier between the wells and therefore we can not use this version of our model to deal with out-of-equilibrium regimes [55]. This approach has been recently used to investigate the properties of several two-state systems and macromolecular chains [74][75][76][77][78]. Both the Gibbs and the Helmholtz ensembles can be studied by the spin variables methodology, permitting to draw direct comparisons between isotensional and isometric conditions, provided that we work at thermodynamic equilibrium.…”