Decreases in molecularity promote conversion when reactions are catalyzed by enzymes immobilized in slab-shaped beads

Application of Fick's ®rst law to substrate undergoing chemical reaction catalyzed by an enzyme immobilized in a porous slab-shaped bead leads to substrate concentration pro®les that are ̄atter when the ratio of stoichiometric coef®cients of product to reactant (m) is lower. Since the actual effectiveness factor decreases when m increases (at approximately the same rate irrespective of the value taken by the Michaelis-Menten parameter), then considerable overprediction of the conversion levels obtained within a given reaction timeframe will occur if the simplistic (and more easily modelled) situation of equimolar counterdiffusion is taken for modelling purposes when m > 1. List of symbols A cross-sectional area of the catalyst slab ‰m2Š Ctot total molar density of the mixture ‰mol mÿ3Š DSP binary diffusivity of S and P ‰m2 sÿ1Š E enzyme kcat ®rst order kinetic constant associated with formation of P ‰sÿ1Š Km Michaelis-Menten constant associated with dissociation of enzyme/substrate complex ‰mol mÿ3Š K m normalized value of Km ‰ÿŠ L half-thickness of the slab ‰mŠ N total number of iterations ‰ÿŠ NP molar ̄ux of P ‰mol mÿ2 sÿ1Š NS molar ̄ux of S ‰mol mÿ2 sÿ1Š NS;z uni-dimensional component of vector NS ‰mol mÿ2 sÿ1Š P product S substrate Th Thiele modulus ‰ÿŠ vmax maximum rate of reaction ‰mol mÿ3 sÿ1Š vP molar volume of P ‰m3 molÿ1Š vS molar volume of S ‰m3 molÿ1Š xS molar fraction of S ‰ÿŠ yS auxiliary variable ‰ÿŠ y …i† S value of yS at the i-th iteration ‰ÿŠ z unidimensional linear coordinate [m] z normalized value of z ‰ÿŠ Greek letters g effectiveness factor of catalyst slab ‰ÿŠ m stoichiometric coef®cient of P ‰ÿŠ 1 Introduction Although enzymes are synthesized and operate in vivo at concentration levels that are high when compared with the actual concentration levels of their substrates, industrial practice has it that the concentration of substrate is to be raised to the highest possible value, so as to approach the maximum possible rate ever, and the concentration of catalyst is to be maintained as low as possible, so as to approach the minimum operating costs ever. The reason for these apparently opposing behaviors hinges on the constraints posed on the aforementioned processes, which are metabolic in the former case and economic in the latter. Since enzymes are molecules possessing a paramount number of degrees of freedom in virtue of their size and tridimensional nature as peptide bonds, partial or total deactivation is easy and occurs at high rates unless the rigidity of their amino acid residue backbone is constrained via an externally engineered microenvironment; the easiest way to accomplish this deed is via immobilization of the enzyme onto a solid support. Since the shape of the immobilization bead is not of crucial importance provided that the characteristic length scale for intraparticle diffusion and the speci®c volumetric activity of enzyme are similar (Smith, 1981), a slab-shaped bead seems appropriate for modelling purposes because of the mathematical simplicity derived from its unidimensional nature and constant cross section along the direction of ̄ow, and has accordingly been considered by several authors (e.g. Malcata, 1991; Moreira and Malcata, 1996). The case of 1:1 stoichiometry has dominated essentially all theoretical analyses; in situations where a different stoichiometry exists, description of molecular transport has traditionally been effected via Fick's law of equimolar counterdiffusion, but the results obtained in terms of effectiveness factors may be excessively conservative when the molecularity of the chemical reaction increases considerably from reactants to products. It is the purpose of this communication to mathematically assess the effect of stoichiometry upon the substrate concentration gradient within a slab-shaped bead where an enzyme-catalyzed reaction following Michaelis-Menten kinetics takes place, and use such reasoning to predict the variation of the effectiveness factor of the enzyme bead R.M. Barros, F.X. Malcata Escola Superior de Biotecnologia, Rua Dr. AntoÂnio Bernardino de Almeida, P-4200 Porto, Portugal Correspondence to: F. Xavier Malcata with stoichiometric coef®cients and physicochemical and processing parameters. 2 Theory Consider a chemical reaction that takes place in a binary solution of substrate S and product P and is brought about by an enzyme E according to the following mechanism: E‡ S ! Km ES ! kcat E‡ mP ; …1† where Km is the equilibrium constant associated with dissociation of the enzyme-substrate complex (ES), kcat is the intrinsic kinetic constant associated with formation of product, and m is the stoichiometric coef®cient of product P. Consider, in addition, that said enzyme is uniformly immobilized in a microporous bead; in this situation, transport of molecular species S and P occurs by diffusion only, according to Fick's ®rst law: NS ˆ xS…NS ‡ NP† ÿ Ctot DSP r xS ; …2† where NS and NP denote the diffusion ̄ux of S and P, respectively, relative to stationary coordinates, xS the molar fraction of S;Ctot the total molar density of the mixture of S and P, and DSP the diffusivity in the binary system formed by S and P. The stoichiometry relationship between S and P as apparent in Eq. (1) allows one to write: