Investigation of the transition between the unfolded and folded states of the protein molecules by using statistical mechanical methods

A theoretical investigation is presented into the transition between the unfolded and folded states of protein molecules in aqueous environments using a statistical mechanical framework. The model constructs a canonical ensemble in which protein molecules are treated as electric dipoles interacting with an effective electrostatic field that simulates hydration effects. The canonical partition function is derived by integrating over dipole orientations and is analytically related to thermodynamic quantities such as free energy, internal energy, enthalpy, and heat capacity. To incorporate the influence of the chemical environment, the model extends to a grand canonical ensemble by introducing pH-dependent behavior through the concept of proton fugacity. The impact of pH on enthalpic stability is analyzed across multiple temperature conditions. All theoretical expressions are fitted to experimental calorimetric data reported for lysozyme within the pH range of 0–6. The results demonstrate that the folded state is thermodynamically favorable under acidic conditions, characterized by negative enthalpy changes and reduced heat capacity. In contrast, the unfolded state becomes dominant at higher pH levels. This approach successfully captures the enthalpic features of folding transitions and provides valuable insights into the cooperative effects of hydration on protein stability.