A modified Ising model for the thermodynamic properties of local and global protein folding–unfolding observed by circular dichroism and small-angle X-ray scattering

Abstract

Based on the mean‐field approximation, we have applied a modified Ising model to describe general protein unfolding behavior at thermodynamic equilibrium with the free energy contributed by the subgroup units (amino acids or peptide bonds) of the protein. With the thermodynamic properties of the protein, this model can associate the stepwise change of an unfolding fraction ratio profile with the local and global conformation unfolding. Taking cytochrome c (cyt c) as a model protein, we have observed, using small‐angle X‐ray scattering and circular dichroism (CD), the global and local structure changes for the protein in three kinds of denaturant environments: acid, urea and guanidine hydrochloride. The small‐angle X‐ray scattering and CD results are mapped to the unfolding fractions as a function of the pH value or denaturant concentration, from which we have extracted local and global unfolding free energies of cyt c in different denaturant environments using a modified Ising model. Based on the characteristics of the thermodynamic properties deduced from the local and global protein folding–unfolding, we discuss the thermodynamic stabilities of the protein in the three denaturant environments, and the possible correlation between the global conformation change of the protein and the local unfolding activities of the S—Fe bond in the Met80‐heme and the α‐helices.

Global and Local Structural Changes of Cytochrome c and Lysozyme Characterized by a Multigroup Unfolding Process

“…Analysis of SAXS data for the global structural information follows that reported previously (22,28). SAXS intensity distribution for monodisperse protein solutions is modeled as IðQÞ ¼ I oP ðQÞSðQÞ; with the scattering vector Q ¼ 4psinu/l defined by scattering angle 2u and x-ray wavelength l, the normalized form factorPðQÞ; and the structure factor S(Q).…”

“…The peak of Soret band of the heme group located at 409 nm in native cytochrome c was attributed to the in-plane pÀp* transition of porphyrin. Due to the interactions of the protein with the denaturants, the Soret peak of the final unfolded protein showed a blue shift of 4 nm in 10 M urea and 14 nm at pH 1.8 (22,30).…”

“…When all units of a protein are in the folded (or unfolded) state, the protein behaves as fully folded (or unfolded); for a partially folded protein, 0 , f u , 1. The evolution of protein morphology, in terms of the unfolded fraction f u , during the unfolding process is then governed by the free energy change of the protein units (20)(21)(22)(23).…”

“…where E i stands for the total energy of the ith unit (20)(21)(22)(23). Under mean-field approximation and the assumption that the canonical ensemble lies not far from thermodynamic equilibrium, the expectation state (nonbinary in value) of the 'th group can be expressed as…”

“…The model could provide a topology of protein foldingunfolding and the related calorimetric change of enthalpy; because of a rather complicated form of the partition function, it is, however, not straightforward to associate the protein unfolding free energy and denaturant concentration with the unfolding fraction deduced from experiment. On the other hand, the Ising model with a mean-field approximation developed by Lin's group can describe kinetics and thermodynamics of protein folding-unfolding influenced by denaturant effects (20)(21)(22)(23) or calorimetric change of enthalpy, as well as the unfolding in atomic force measurements (24,25), via a microscopic point of view. In this study, we find that the mean-field Ising model can provide a simple yet coherent description on the multigroup unfolding behaviors of the model proteins we have observed at not only global but also local scales as experimentally accessed via SAXS and spectroscopic measurements.…”