Lysozyme heat unfolding: Thermodynamic parameters obtained model-independently by DSC
Lysozyme (14.3 kDa) is a globular 129-residue protein with ~ 25% α-helix, ~ 40% β-structure and ~ 35% random coil in solution at room temperature.29 Upon unfolding, the α-helix is almost completely lost and the random coil content increases to ~ 60%. The DSC thermogram of lysozyme unfolding is shown in Fig. 1. The baseline-corrected heat capacity ΔCp(T) of the native protein is zero (for detail see ref.12), goes through a maximum at the midpoint temperature Tm = 62°C and levels off again. The heat capacity increases upon unfolding by \(\Delta C_{p}^{0}\) = 2.27 kcal/molK. (Literature: 1.54–2.27 kcal/molK5, 11, 29, 36–39). The enthalpy ΔH(T)DSC and entropy ΔS(T)DSC are evaluated model-independently with equations 2 and 3 and have sigmoidal shapes (Figs. 1B, 1C). The free energy ΔG(T)DSC (Eq. 4) of the native protein is zero, is slightly negative in the initial phase of unfolding, and decreases rapidly beyond the midpoint temperature Tm = 62°C (Fig. 1D).
The total unfolding enthalpy is \(\Delta {H_{DSC}}=\) 138 kcal/mol. It is composed of the conformational enthalpy proper, \(\Delta {H_0}\) and a contribution \(\Delta {H_{\Delta C_{p}^{0}}}\)caused by the heat capacity term \(\Delta C_{p}^{0}\).
$$\Delta {H_{DSC}}=\Delta {H_0}+\Delta {H_{\Delta C_{p}^{0}}}$$
25
The two enthalpies can be separated by applying the models described above. In the model-independent analysis, the following approximation (Eq. 26) is confirmed by a comparison with the predictions of the ΘU(T)-weighted chemical equilibrium model or the statistical-mechanical models.
$$\Delta {H_{\Delta C_{p}^{0}}} \approx \left( {\Delta C_{p}^{0}/3} \right)\left( {{T_{end}} - {T_{ini}}} \right)$$
26
For lysozyme with \(\Delta C_{p}^{0}\)= 2.269 kcal/molK, Tini = 318K, and Tend = 346 K this results in \(\Delta {H_{\Delta C_{p}^{0}}}\)= 21.2 kcal/mol (simulations yield 20–24 kcal/mol). The experimental data for lysozyme are summarized in Table 1A.
Table 1A - C
Of note, "unfolded" proteins are not completely unfolded, but contain residual structure.40, 41 Complete unfolding is difficult to achieve as many different physical and chemical factors contribute to protein stability.41 In the present evaluation the extent of unfolding is always ΘU > 0.9 as judged by applying the unfolding models.
β-Lactoglobulin cold denaturation - Thermodynamic parameters obtained model-independently by DSC
DSC data for cold denaturation are scarce. One of the best examples is the unfolding of β-lactoglobulin in urea solution.42 Bovine β-Lactoglobulin is a 18.4 kDa protein comprising 162 amino acids that fold up into an 8-stranded, antiparallel β-barrel with a 3-turn α-helix on the outer surface. A DSC cold-denaturation experiment of β-lactoglobulin is shown in Fig. 2 (data taken from reference42). The experiment starts with the native protein at ~ 35°C and the temperature is lowered gradually to -14°C. The heat capacity of the native protein is zero and all thermodynamic functions are necessarily also zero at ambient temperature.
Cold denaturation is an exothermic reaction. At the end of the DSC experiment at -14°C the released heat as evaluated with Eq. 1 is ΔHDSC =-69.5 kcal/mol (-291 kJ/mol, in agreement with Table 2 in42). The corresponding entropy change is ΔSDSC =-0.248 kcal/mol. The ratio ΔHDSC/ΔSDSC = 280K = 7°C is close to the experimental Cp(T) minimum at 277 K.
β-Lactoglobulin. Heat-induced folding and unfolding. Thermodynamic parameters obtained model-independently by DSC
The DSC experiment shown in Fig. 3 is unusual as it involves a disorder \(\to\)order transition at low temperature and the reverse order \(\to\)disorder transition at high-temperature.
Figure 3A reports the DSC experiment. The heat capacity Cp(T) is then used to calculate the thermodynamic properties in Figs. 3B − 3D. At the beginning of the DSC experiment at -9°C, the protein is cold-denaturated and disordered. Upon heating, the protein goes through a disorder\(\to\)order transition with a heat uptake of ΔHDSC = 78.3 kcal/mol. At 25°C the protein is in an ordered, native-like structure. Further heating induces new disorder with an enthalpy uptake of ΔHDSC = 104.1 kcal/mol. Cp(T) shows maxima at 4°C and 57°C. The entropies increase by ΔSDSC = 0.3283 kcal/molK and ΔSDSC = 0.313 kcal/molK, respectively. The ratio ΔHDSC/ΔSDSC is 277K = 4°C for the disorder\(\to\)order transition and 333K = 60°C for heat denaturation, in agreement with the heat capacity maxima.
The blue data points in Fig. 3A are integrated with equations 2–4 result in the black data points in panels 3B − 3D. The comparison with cold denaturation in Fig. 2 suggests a shift of the enthalpy by -78.3 kcal/mol, the enthalpy of cold denaturation. This scale shift (Fig. 3B, red data points) leads to a zero enthalpy for the native protein and makes Fig. 3 consistent with Fig. 2. Likewise, the entropy in Fig. 2C is shifted by -0.283 kcal/molK. The entropy of the native protein is now also zero. With these scale shifts the recalculated free energy is given by the red data points in Fig. 3D. The free energy shows a trapezoidal temperature profile.
Figure 3A is almost a quantitative mirror image of cold-denaturation in Fig. 2A. Not surprisingly, the red data points in Fig. 3 related to cold denaturation are consistent with the direct measurements in Fig. 2.
The experimental thermodynamic data for β-lactoglobulin are summarized in Table 1B.
Analysis Of Dsc Thermograms With Three Different Models
Lysozyme heat unfolding.
Figure 4 compares the experimental data of Fig. 1 with the ΘU(T)-weighted chemical equilibrium model (magenta lines), the statistical-mechanical two-state model (red lines), and the multistate cooperative model (green lines). The simulations cover a large temperature range, predicting both heat and cold denaturation. However, no experimental data are available for lysozyme cold denaturation. Figure 4A shows virtually identical simulations of the heat capacity by the three models. The conformational parameters of the two-state models are almost identical (ΔH0 = 107 kcal/mol, ΔE0 = 110 kcal/mol). The simulation parameters are listed in Table 1B, C for the two-state models and in Table 2 for the multistate cooperative model.
Table 2
The three models provide good fits of all experimental thermodynamic properties (Figs. 4B-4D), predicting sigmoidal temperature profiles for enthalpy and entropy and a trapezoidal shape for the free energy. The models show differences with respect to cold denaturation. The statistical-mechanical models predict cold denaturation 20°C − 50°C lower than the ΘU(T)-weighted chemical equilibrium two-state model (cf. Figures 4B-4C).
A further difference between the three models is shown in Fig. 5, displaying the free energy at enhanced resolution. The DSC experiment reports a zero free energy for the native lysozyme, which becomes immediately negative upon unfolding. Of note, the experimental free energy is always negative, never positive. Both statistical-mechanical models reproduce this result correctly. In contrast, the ΘU(T)-weighted chemical equilibrium model displays a small positive peak in the vicinity of Tm. Consequently, the free energies at the midpoint of unfolding are also different. The experimental free energy at Tm = 62°C is \(\Delta H{\left( {{T_m}} \right)_{DSC}}\)= -0.76 kcal/mol. The multistate cooperative model predicts correctly \(\Delta F\left( {{T_m}} \right)\)= 0.73 kcal/mol and the statistical-mechanical two-state model \(\Delta F({T_m})= - R{T_m}\ln 2\)= -0.46 kcal/mol. In contrast, the ΘU-weighted chemical equilibrium two-state model yields exactly ΔGΘ(Tm) = 0 kcal/mol. At Tm all three models predict the extent of unfolding as ΘU(Tm) = 1/2. The protein is partially denatured at Tm and its free energy is necessarily negative.
The parabolic profile of the Gibbs free energy, which is predicted by the standard chemical equilibrium two-state model (Eq. 9), deviates even more from the DSC result and is hence not included in Figs. 4D and 5.
β-Lactoglobulin. Cold denaturation analysed with different models
Cold denaturation is analysed with three different models. All models provide good fits of the thermodynamic properties. However, the ΘU(T)-weighted chemical equilibrium two-state model predicts some positive free energy, which is not supported by the DSC experiment. The multistate cooperative model provides the best simulation.
β-Lactoglobulin. Heat-induced folding and unfolding analysed with different models
The simultaneous analysis of two heat-induced unfolding transitions is shown in Fig. 7A for the ΘU(T)-weighted chemical equilibrium model (Eq. 11) and in Fig. 7B for the statistical-mechanical models. All three models describe the temperature-profile of the heat capacity Cp(T) equally well.
A criterion for protein stability is the temperature difference between heat and cold denaturation. DSC yields a temperature difference of ΔT = 53°C between the heat capacity maxima. The prediction of the ΘU(T)-weighted chemical equilibrium model is \(\Delta T \approx 2{T_0}(1 - {\operatorname{e} ^{\frac{{ - \Delta {H_0}}}{{{T_0}\Delta C_{p}^{0}}}}})\)= 45°C, that of the statistical-mechanical two-state model ΔT = ΔE0/Cv = 46°C, and that of the multistate cooperative model ΔT\(\approx\)h0/cv = 48°C.
The simulations of the three models overlap almost completely for heat capacity Cp(T) and enthalpy ΔH(T)DSC (Fig. 7C). In contrast, the free energy prediction of the ΘU(T)-weighted chemical equilibrium model deviates from the experimental result in the vicinity of the phase transitions (Fig. 7D). The DSC-derived free energy is zero or negative, never positive. The small positive peaks of the ΘU(T)-weighted chemical equilibrium two-state model disagree with this experimental result.
The total enthalpy of heat unfolding at 57°C is ΔHDSC=104 kcal/mol, but the conformational enthalpy is only ΔH0 = 5 6 kcal/mol. The large difference is presumably caused by the binding of urea molecules and is \(\Delta {H_{\Delta C_{p}^{0}}}\)~ 50 kcal/mol.
The thermodynamic data and the fit parameters for β-lactoglobulin are summarized in Table 1B and 1C for the two-state models and in Table 2 for the multistate cooperative model.