Refinement of noncalorimetric determination of the change in heat capacity, DeltaC(p), of protein unfolding and validation across a wide temperature range

Abstract

The change in heat capacity, DeltaC(p), on protein unfolding has been usually determined by calorimetry. A noncalorimetric method which employs the Gibbs-Helmholtz relationship to determine DeltaC(p) has seen some use. Generally, in this method the free energy change on unfolding of the protein is determined at a variety of temperatures and the temperature at which DeltaG is zero, T(m), and change in enthalpy at T(m) are determined by thermal denaturation and DeltaC(p) is then calculated using the Gibbs-Helmholtz equation. We show here that an abbreviated method with stability determinations at just two temperatures gives values of DeltaC(p) consistent with values from free energy change on unfolding determination at a much wider range of temperatures. Further, even the free energy change on unfolding from a single solvent denaturation at the proper temperature, when coupled with the melting temperature, T(m), and the van't Hoff enthalpy, DeltaH(vH), from a thermal denaturation, gives a reasonable estimate of DeltaC(p), albeit with greater uncertainty than solvent denaturations at two temperatures. We also find that nonlinear regression of the Gibbs-Helmholtz equation as a function of stability and temperature while simultaneously fitting DeltaC(p), T(m), and DeltaH(vH) gives values for the last two parameters that are in excellent agreement with experimental values.

Figure 1

Plot of ΔGH₂O against temperature for wild-type staphylococcal nuclease and five representative quadruple mutants. The mutant 23I/25I/66L/72L was chosen as it has a low ΔC_p and subjectively amongst the worst fits. The mutant 23L/25I/66I/72V has a markedly lower stability than wild-type, yet has a ΔC_p that is nearly the same and a subjectively good fit. The mutants 66I/72V/92V/99I and 66I/72V/92V/99L are poor and good fits respectively but both with high ΔC_p. The mutant 23L/25I/66I/72L has a low ΔC_p and a good fit. The lines are the best fit against the Gibbs-Helmholtz equation obtained by non-linear regression of predicted ΔG against ΔGH₂O for all measured temperatures while defining T_m and ΔH as the values obtained from thermal denaturation experiments. The points indicate experimental data obtained at various temperatures. The standard deviation in ΔGH₂O found in repeated determinations of wild-type stability at 20°C is ± 0.1 kcal/mol.

Figure 2

Comparison of ΔC_p values with and without fixed, experimental ΔH and T_m values. In all panels the x axis is the global ΔC_p. This is the value of ΔC_p obtained by fitting predicted ΔG against ΔGH₂O for all measured temperatures while defining T_m and ΔH as the values obtained from thermal denaturation experiments. In panel A, the global ΔC_p is plotted against the value of ΔC_p returned by the regression when ΔH is fixed as the experimental value obtained from thermal denaturation and T_m is fit as well. In panel B, the global ΔC_p is plotted against the value of ΔC_p found by the regression when T_m is fixed as the value obtained from thermal denaturation and ΔH is now fit. In panel C, the global ΔC_p is plotted against the value of ΔC_p returned when ΔC_p, ΔH, and T_m are all fit by non-linear regression at once. In all cases, the error bars are those given by the regression and the lines are the least square regression of ΔC_p versus ΔC_p.

Figure 3

Comparison of experimental T_m values to those derived from non-linear regression. In both panels the x axis is the experimental T_m. In panel A, the experimental T_m is plotted against the value of T_m returned by the regression when ΔH is fixed as the value obtained from thermal denaturation while ΔC_p is fit as well. In panel B, the experimental T_m is plotted against the value of T_m returned when ΔC_p, ΔH, and T_m are all fit by non-linear regression at once. The error bars in the y axis are those given by the regressions, the error in the x axis is the ± 0.4° standard deviation found in repeated thermal denaturations of wild-type staphylococcal nuclease. The lines are the least square regression of T_m versus T_m.

Figure 4

Comparison of experimental ΔH values to those derived from non-linear regression. In both panels the x axis is the experimental ΔH. In panel A, the experimental ΔH is plotted against the value of ΔH returned by the regression when T_m is fixed as the value obtained from thermal denaturation while ΔC_p is fit as well. In panel B, the experimental ΔH is plotted against the value of ΔH returned when ΔC_p, ΔH, and T_m are all fit simultaneously by non-linear regression. The error bars in the y axis are those given by the regressions, the error in the x axis is the ± 2.6 kcal/mol standard deviation found in repeated thermal denaturations of wild-type staphylococcal nuclease. The lines are the least square regression of ΔH versus ΔH.

Figure 5

Plot of ΔC_p versus temperature for all 22 mutants and wild-type. Each ΔC_p value is calculated from the experimental ΔH and T_m and the ΔGH₂O for that protein at each temperature.

Figure 6

Comparison of ΔC_p values calculated with ΔGH₂O values determined at different temperatures and the values of ΔH and T_m from a thermal denaturation. In all panels the x axis is the global ΔC_p. This is the value of ΔC_p obtained by fitting predicted ΔG against ΔGH₂O for all measured temperatures while defining T_m and ΔH as the values obtained from thermal denaturation experiments. In panel A, the global ΔC_p is plotted against the value of ΔC_p returned by the regression against just the stabilities determined at 15 and 20°C. In panel B, the global ΔC_p is plotted against the value of ΔC_p returned by the regression against just the stabilities determined at 20 and 30°C. In panel C, the global ΔC_p is plotted against the value of ΔC_p calculated from just the stability determined at 20°C. In all cases, the error bars are those given by the regression and the lines are the least square regression of ΔC_p versus ΔC_p.