Molecular understanding of calorimetric protein unfolding experiments

Abstract

Testing and predicting protein stability gained importance because proteins, including antibodies, became pharmacologically relevant in viral and cancer therapies. Isothermal scanning calorimetry is the principle method to study protein stability. Here, we use the excellent experimental heat capacity C_p(T) data from the literature for a critical inspection of protein unfolding as well as for the test of a new cooperative model. In the relevant literature, experimental temperature profiles of enthalpy, H_cal(T), entropy, S_cal(T), and free energy, G_cal(T) are missing. First, we therefore calculate the experimental H_cal(T), S_cal(T), and G_cal(T) from published C_p(T) thermograms. Considering only the unfolding transition proper, the heat capacity and all thermodynamic functions are zero in the region of the native protein. In particular, the free energy of the folded proteins is also zero and G_cal(T) displays a trapezoidal temperature profile when cold denaturation is included. Second, we simulate the DSC-measured thermodynamic properties with a new molecular model based on statistical-mechanical thermodynamics. The model quantifies the protein cooperativity and predicts the aggregate thermodynamic variables of the system with molecular parameters only. The new model provides a perfect simulation of all thermodynamic properties, including the observed trapezoidal G_cal(T) temperature profile. Importantly, the new cooperative model can be applied to a broad range of protein sizes, including antibodies. It predicts not only heat and cold denaturation but also provides estimates of the unfolding kinetics and allows a comparison with molecular dynamics calculations and quasielastic neutron scattering experiments.

Figure 1

Differential scanning calorimetry of 50 μM lysozyme in 20% glycine buffer, pH 2.5. Black data points: DSC thermograms obtained with a heating rate of 1°C min⁻¹ and a step size of 0.17°C. Red lines: simulations with the Zimm-Bragg theory with h₀ = 1.0 kcal/mol, c_v = 6 cal/molK, σ = 1.0 × 10⁻⁶, ν = 129. (A) Heat capacity C_p(T). (B) Unfolding enthalpy H(T). (C) Unfolding entropy S(T). (D) Free energy of unfolding G(T). Blue line: ΔG(T) (Eq. 13) predicted by the two-state model calculated with a conformational enthalpy ΔH₀ = 107 kcal/mol and a heat capacity increase $\Delta {\text{C}}_{\text{p}}^0$ = 2.269 kcal/mol. Data taken from reference (10).

Figure 2

gpW62 DSC unfolding (7). (▲) Original C_p(T) data (taken from Fig. 4 A in reference (15)). (▪) Baseline-corrected data. Data points in (B–D) were calculated from data points in (A) with Eqs. 10, 11, and 12. Red lines: simulations of unfolding transition proper with the Zimm-Bragg theory (h₀ = 1.26 kcal/mol, c_v = 5 cal/molK, σ = 5.0 × 10⁻⁴, ν = 62 residues). Blue lines: two-state model (ΔH₀ = 49 kcal/mol, $\Delta {\text{C}}_{\text{p}}^0$= 0.9 kcal/molK). The red dotted lines in (A) are the differences between the experimental DSC data and the simulations. For better visibility, the blue dotted line is shifted by −1 kcal/molK. Green lines: sum of Zimm-Bragg theory and the contribution of the basic heat capacity of the native protein C_p = 2.7 kcal/molK (cf. Eqs. 7, 8, and 9).

Figure 3

Thermal unfolding of monoclonal antibody mAb at pH 6.2. (▪) DSC experiment. Solid lines are simulations with the Zimm-Bragg theory: green, pretransition; violet, main transition; red, sum of pre- and main transition. (A) Molar heat capacity. (B) Unfolding enthalpy. (C) Unfolding entropy. (D) Free energy. Simulation parameters: h₀ = 1.1 kcal/mol, c_v = 7.0 cal/molK. Pretransition: T₀ = 73°C, ν_pre = 263, σ = 5 × 10⁻⁵. Main transition: T₀ = 85.4°C, ν_main = 880, σ = 2 × 10⁻⁵. Data taken from reference (13).

Figure 4

Unfolding of metmyoglobin at acidic pH. (▪) Experimental data taken from reference (16). Red lines: simulations with the multistate cooperative Zimm-Bragg theory (fit parameters listed in Table 1). (red ▪) Differences between DSC data and the Zimm-Bragg simulation (shifted in B by −1 kcal/molK for better visibility). Blue lines: two-state model. (A) C_p(T) at pH 4.1 (Fig. 13 in reference (16)). (B) C_p(T) at pH 3.83 (Fig. 12 in reference (16)). (C and D) Free energies calculated using the heat capacity data shown in (A) and (B).

Figure 5

Unfolding entropy. (▪) Experimental data. (magenta ▪) Dynameonics Entropy Dictionary (18). (A) gpW62 (15). (B) Ubiquitin (17) (Fig. 1 in reference (17)). (C) Lysozyme (10) (D) Metmyoglobin (16) (Fig. 3 in reference (16), pH 10). Red lines: Zimm-Bragg total entropy. Brown lines: conformational entropy proper. Green lines: contribution of the heat capacity c_v.