Schottky-like anomalies and phase transitions in ideal heteropolymer proteinogenic chains in terms of size and single-residue helical propensity

Proteins belong to a class of biological macromolecules capable to undergo thermodynamic phase transitions from random one-dimensional to three-dimensional non-periodic structures. For a large class of proteins, such configurational phase transitions can be directly associated with a passage from poor or non-biologically active states to highly biologically-active conformations. In this work we present a simple model suitable to study analytical and numerically thermodynamic properties for an ideal heteropolymer chain X1−X2⋯−XN composed by N aminoacid residues. This model includes as relevant parameters the single-residue helical propensity ωj associated to residue Xj and characteristic energies, ϵj and Uj corresponding to native and non-native contact energies, respectively. We derive analytical expressions for the partition function Z when two types of energy spectra E are considered. One corresponding to all-or-none transitions typically observed in protein folding and a second one which considers partial contributions to the folded state. Specific heat Cv, configurational entropy S, average energy 〈E〉 and average occupancy number N are discussed in order to analyze the onset of thermodynamic phase transitions and Schottky-like anomalies in polypeptide chains of different size.