Optimal distribution of immobilized enzyme in a pellet for a substrate‐inhibited reaction

Abstract

The possibility of using nonuniform distributions of immobilized enzyme in porous supports has recently been examined by various investigators (ref. l), as an attempt to improve enzyme effectiveness. For reactions with Michaelis-Menten kinetics operating in the masstransfer controlled regime, Horvath and Engasser2y3 have shown that enzyme effectiveness could be significantly improved by locating the enzyme in an outer shell of the inert support. An enzyme distribution of this type reduces the substrate diffusion path, which enhances the overall reaction rate, by maintaining high concentration of the substrate where the enzyme is located.More recently, Park and co-workers4 have pointed out that this same nonuniform enzyme distribution is actually the best for all reactions of positive order with respect to the substrate. These authors also investigated the case of substrate-inhibited reactions, using the kinetic model proposed by Haldane.s Many reactions of practical import fall in the case,(' such as hydrolysis of sucrose by in~e r t a s e ,~ hydrolysis of benzil penicillin by amidase,8 and phenol d e g r a d a t i~n .~ A general kinetic expression for substrate-inhibited reactions can be written as followslO( 1) where k ( x ) is the reaction rate constant which depends on enzyme concentration, and thus in the case of nonuniformly distributed matrix, on position within the pellet. Parameter K is the Michaelis-Menten constant, K' is a substrate-inhibition constant, and K " is a constant which basically accounts for the residual reaction rate which some systems exhibit at very large substrate concentration. 10 It may be noted that this expression includes as particular cases several widely used kinetic models, such as the first-order ( K " = 0; K' = 0; K >> S ) , the MichaelisMenten or Monod ( K " = 0; K' = O), and the Haldane ( K " = 0) kinetics. GENERAL BEHAVIOR OF SUBSTRATE-INHIBITED REACTIONSAs indicated by eq. ( l ) , substrate-inhibited reactions exhibit the abnormal feature of a maximum in the rate as the substrate concentration is varied. Thus, for relatively low concentrations of the substrate, the reaction rate increases as the substrate concentration increases. However, as the substrate concentration increases further, the rate first reaches a maximum and then continuously decreases. This feature leads to multiple operating steady states under certain isothermal conditions, and this specific aspect of the problem has received considerable attention in the This peculiar behavior of the reaction rate function also implies that an improved pellet effectiveness can be obtained if the enzyme is located in a zone within the interior of the support, rather than on its outer shell. Such an enzyme distribution would take advantage of reaction rate enhancement that would occur purely as a result of decreased concentration of the substrate caused by diffusional resistance. This intuitive result has been confirmed by Park and c o -~o r k e r s ,~ by numerically computing the effectiveness of enzyme immobilized in ...