Relationship between preexponent and distribution over activation barrier energies for enzymatic reactions

A relationship between the preexponent of the rate constant and the distribution over activation barrier energies for enzymatic/protein reactions is revealed. We consider an enzyme solution is an ensemble of individual molecules with different values of the activation barrier energy described by the distribution. From solvent viscosity effect on the preexponent we derive the integral equation for the distribution and find its approximate solution. Our approach enables us to attain a twofold goal. Firstly it yields a simple interpretation of solvent viscosity dependence for enzymatic/protein reactions that requires neither a modification of the Kramers’ theory nor that of the Stokes law. Secondly our approach enables us to deduce the form of the distribution over activation barrier energies. The obtained function has a familiar bell-shaped form and is in qualitative agreement with results of single enzyme kinetics measurements. General formalism is exemplified by the analysis of literature experimental data.