Model for solvent viscosity effect on enzymatic reactions

Why reaction rate constants for enzymatic reactions are typically inversely proportional to fractional power exponents of solvent viscosity remains to be already a thirty years old puzzle. Available interpretations of the phenomenon have not led to consensus among researches about its origin. They invoke to either a modification of 1. the conventional Kramers’ theory or that of 2. the Stokes law. We show that there is an alternative interpretation of the phenomenon at which neither of these modifications is in fact indispensable. Basing on an analogy from the theory of adsorption on heterogeneous surfaces we reconcile 1. and 2. with the experimentally observable dependence. We assume that an enzyme solution in solvent with or without cosolvent molecules is an ensemble of samples with different values of the viscosity for the movement of the system along the reaction coordinate. We assume that this viscosity consists of the contribution with the weight q from cosolvent molecules and that with the weight 1−q from protein matrix and solvent molecules. We introduce heterogeneity in our system with the help of a distribution over the weight q. The function of the distribution is a unique characteristic of the solution of enzymes (type of the enzyme, cosolvent molecular weight, pH, temperature, etc.). We verify the obtained solution of the integral equation for the unknown function of the distribution by direct substitution. We conclude that even at linear relationship between the solvent viscosity and that for the movement of the system along the reaction coordinate our approach enables us to obtain the required dependence. All parameters of the model are related to experimentally observable values. The meaning of fractional exponents appears to be the characteristic for the behavior of the distribution with the variation of the weight q. Our approach yields the existence of the limit value for the fractional power exponent with the decrease of cosolvent molecular weight that is in agreement experimental data known from the literature. This limit value is determined by the properties of the protein structure and thus is a unique characteristic of the type of the enzyme only rather than that of the solution. General formalism is exemplified by the analysis of literature experimental data for oxygen escape from hemerythin.