Extrapolation to water of kinetic and equilibrium data for the unfolding of barnase in urea solutions

Assumptions about the dependence of protein unfolding on theconcentration of urea have been examined by an extensive surveyof the equilibrium unfolding of barnase and many of its mutantsmeasured by urea denaturation and differential scanning calorimetry.The free energy of equilibrium unfolding and the activationenergy for the kinetics of unfolding of proteins are generallyassumed to change linearly with [urea]. A slight downward curvatureis detected, however, in plots of highly precise measurementsof logjtu versus [urea] (where k_u is the observed rate constantfor the unfolding of barnase). The data fit the equation logkk_u= logk_u^H₂O* + mk_u^*.[urea] – 0.014[urea]², where mk_u^*is a variable which depends on the mutation. The constant 0.014 was measured directly on four destabilized mutants and wildtype, and was also determined from a global analysis of data from>60 mutants of barnase. Any equivalent deviations from linearityin the equilibrium unfolding are small and in the same region,as determined from measurements on 166 mutants. The free energyof unfolding of barnase, G_U–F, appears significantly largerby 1.6 kcal mol^–1 when measured by calorimetry than whendetermined by urea denaturation. However, the changes in G_U–Fon mutation, G_U–F, determined by calorimetry and by ureadenaturation are identical. We show analytically how, hi general,the curvature in plots of activation or equilibrium energiesagainst [denaturant] should not affect the changes of thesevalues on mutation provided measurements are made over the sameconcentration ranges of denaturant and the curvature is independentof mutation.