Papain-Collodion Membranes. II. Analysis of the Kinetic Behavior of Enzymes Immobilized in Artificial Membranes

A theoretical treatment of the kinetic behavior of membranes with enzymic activity is presented. It is based on the assumption that a stationary state is attained within the membrane, and that the local concentrations of substrate and product are determined by the rate of the catalytic reaction and by the rates of diffusion of substrate and product. Two types of membrane are discussed. The first is a one-layer enzyme-membrane in which the enzyme is distributed homogeneously, and the second type is a two-layer enzyme-membrane in which one of the layers is devoid of enzyme. The mode of action of a one-layer enzyme-membrane separating two compartments of infinite volume (compartments 1 and 2) is analyzed for the following boundary conditions. Case Ia: The concentrations of substrate and product in compartment 1 exceed the corresponding concentrations in compartment 2. Case Ib: Substrate is present only in compartment 1, and both compartments are devoid of product. Case Ic: Both compartments contain the same concentrations of substrate and product. The mode of action of a two-layer enzyme-membrane separating two compartments of infinite volume is analyzed for a case (II) for which it is assumed that substrate is present only in compartment 1, and that both compartments are devoid of product. For all of the above cases expressions are derived for the sum of flows of substrate and product and the sum of substrate and product concentrations at any point in the membrane. With the assumption that local reaction rates obey first-order kinetics, explicit expressions are derived for the separate flows and concentration profiles of substrate and product in the membrane. The over-all rate of the membrane enzymic reaction is also evaluated. In case Ic an estimate is made of the over-all rate of enzymic reaction for a more general case in which the total enzymic reaction obeys Michaelis-Menten kinetics. A theoretical analysis is also made for an enzyme-membrane catalyzing a reaction in which acid is formed. The validity of some of the conclusions drawn was ascertained experimentally for papain-collodion membranes of different thickness acting on benzoyl-L-argininamide, benzoyl-L-arginine ethyl ester, and acetyl-L-glutamic acid diamide.