Pore diffusion model for a two-substrate enzymatic reaction: Application to galactose oxidase immobilized on porous glass particles

An analysis of the pore diffusion model involving a two-substrate enzymatic reaction is presented. The resulting equations have been applied to the case of galactose oxidase catalyzed oxidation of galactose when the enzyme is immobilized on porous glass particles. The physical constants of the system were obtained by theoretical predictions and the enzyme concentration in the porous medium was derived from the experimental results. The calculations were performed with the assumption that the kinetic parameters of the enzyme remain unchanged upon immobilization. The theoretically calculated effectiveness factors were compared with the experimental effectiveness factors determined from the batch kinetic experiments and were found to be in agreement. The results are presented as effectiveness factor plots graphed as functions of bulk galactose and oxygen concentrations. The model was extended in order to study the effect of external mass transfer coefficients and pore enzyme concentrations on the effectiveness factors.     

Diffusion and chemical reaction rates with nonuniform enzyme distribution: an experimental approach

The study of the effects of nonuniform distributions of immobilized beta-galactosidase on the overall reaction rate of the hydrolysis of lactose are presented. Diffusion inside the particles has been characterized by measuring the diffusion rates of two beta-galactosidase substrates: lactose and ONPG in a commercial silica-alumina support. Effective diffusivities have been determined by the chromatographic method under inert conditions. The results obtained for tortuosity can be explained assuming that the transport only takes place in the macropores. The distribution of the immobilized enzyme has been measured by means of confocal microscopy technique. The enzyme has been tagged with FITC and immobilized in particles of different diameters, the internal local concentrations of the enzyme have been determined with the aid of an image computer program. As expected, a more nonuniform internal profile of the enzyme was found when the particle diameter was bigger. Experiments under reaction conditions were carried out in batch reactors using lactose and ONPG as substrates and particles of the immobilized beta-galactosidase of different diameter (1 x 10(-4) to 5 x 10(-3) m) as catalyst, employing a temperature of 40 degrees C for lactose and 25 and 40 degrees C for ONPG, respectively. The mass balance inside the particle for the substrates has been solved for the internal profiles of the immobilized enzyme inside particles of different size and the enzymatic reactions considered. The calculated and the experimental effectiveness factor values were similar when particles under 2.75 x 10(-3) m in diameter were employed. For the same Thiele modulus, a particle with nonuniform distribution of enzyme showed a higher effectiveness as a catalyst than particles with a more uniform distribution.     

Effect of mass transfer in a recirculation batch reactor system for immobilized penicillin amidase

The effect of external mass transfer resistance on the overall reaction rate of the immobilized whole cell penicillin amidase of E. coli in a recirculation batch reactor was investigated. The internal diffusional resistance was found negligible as indicated by the value of effectiveness factor, 0.95. The local environmental change in a column due to the pH drop was successfully overcome by employing buffer solution. The reaction rate was measured by pH-stat method and was found to follow the simple Michaelis-Menten law at the initial stage of the reaction. The values of the net reaction rate experimentally determined were used to calculate the substrate concentration at the external surface of the catalyst pellet and then to calculate the mass transfer coefficient, k(L), at various flow rates and substrate concentrations. The correlation proposed by Chilton and Colburn represented adequately the experimental data. The linear change of log j(D) at low log N(Re) with negative slope was ascribed to the fact that the external mass transfer approached the state of pure diffusion in the limit of zero superficial velocity.     

Effect of pore diffusion limitation on dextrin hydrolysis by immobilized glucoamylase

Data reported here and previously indicate that when dextrin is hydrolyzed in the presence of immobilized glucoamylase, use of a larger average molecular weight substrate leads to lower overall rates of hydrolysis, while the maltose concentration during the bulk of the reaction and the maximum glucose concentration are lower than when the soluble form of the enzyme is employed under the same conditions. Computer simulation of the system demonstrated that all three observations were caused by pore diffusion limitation: the first by slow diffusion of substrate, the second by slow diffusion of intermediates, and the third by slow diffusion of glucose. Follow-up experiments with glucoamylase immobilized to particles of different sizes confirmed this finding, as results with the smallest beads were identical to those with soluble glucoamylase.     

Application of immobilized chymotrypsin in a multistage fluidized-bed reactor

The stereospecific hydrolysis of D ,L -phenylalanine methylester with immobilized α-chymotrypsin was carried out as a model reaction for the racemate resolution of aromatic amino acids in a five staged fluidized-bed reactor (FBR). Owing to ester hydrolysis, a pH shift occurred along the reactor. Because of the pH-dependent enzyme activity a particular longitudinal pH profile had to be enforced by a proper entrance pH in order to gain an optimum conversion. In the FBR with optimum pH profile, higher conversions were achieved than in a continuous stirred tank reactor (CSTR) at the pH optimum and at the same contact time. By the application of a proton balance and the results of kinetic measurements a model was developed for the prediction of the optimum longitudinal pH profile with regard to the maximum conversion.     

Batch reactor performance for the enzymatic synthesis of cephalexin: influence of catalyst enzyme loading and particle size

A mathematical model is presented for the kinetically controlled synthesis of cephalexin that describes the heterogeneous reaction-diffusion process involved in a batch reactor with glyoxyl-agarose immobilized penicillin acylase. The model is based on equations considering reaction and diffusion components. Reaction kinetics was considered according to the mechanism proposed by Schro?n, while diffusion of the reacting species was described according to Fick's law. Intrinsic kinetic and diffusion parameters were experimentally determined in independent experiments. It was found that from the four kinetic constants, the one corresponding to the acyl-enzyme complex hydrolysis step had the greatest value, as previously reported by other authors. The effective diffusion coefficients of all substances were about 5×10(-10)m(2)/s, being 10% lower than free diffusion coefficients and therefore agreed with the highly porous structure of glyoxyl-agarose particles. Simulations made from the reaction-diffusion model equations were used to evaluate and analyze the impact of internal diffusional restrictions in function of catalyst enzyme loading and particle size. Increasing internal diffusional restrictions decreases the Cex synthesis/hydrolysis ratio, the conversion yield and the specific productivity. A nonlinear relationship between catalyst enzyme loading and specific productivity of Cex was obtained with the implication that an increase in catalyst enzyme loading will not increase the volumetric productivity by the same magnitude as it occurs with the free enzyme. Optimization of catalyst and reactor design should be done considering catalyst enzyme loading and particle size as the most important variables. The approach presented can be extended to other processes catalyzed by immobilized enzymes.     

Determination of total and active immobilized enzyme distribution in porous solid supports

The total and active immobilized enzyme (IME) distributions in porous supports are studied both theoretically and experimentally. In order to determine experimentally the enzyme distribution profiles within a single particle, we construct a diffusion cell containing controlled-pore glass particles such that the cell would mimic a large pellet support. Our purpose is to study the interplay between the diffusion process within the interparticle void space and immobilization process in the controlled-pore glass particles onto the evolution of the (total and active) enzyme distributions. A mathematical model is developed to describe the interaction of various processes within the diffusion cell. The immobilized enzymes are determined for a system of trypsin and controlled-pore glass particles. The total amount of enzymes are determined by the amino acid analysis, and the active fraction is obtained by an active-site titration. The experimentally measured total IME profiles compare very well with that predicted by the model. The determined active enzyme profile is found to be nonuniform one, and it represents about 40% of the total enzyme immobilized in the support particles.     

The effect of methylacrylate on the activity of glucomylase immobilized on granular polyacrylonitrile

Glucoamylase was immobilized on granular polyacrylonitrile (PAN) and the optimum condition in its immobilization reaction was determined. The effect of the ratio of the imidoester and methylester to the total cyanogen on the activity of the immobilized enzyme was studied. The activity of the immobilized enzyme increased in proportion to the molar number of imidoester and decreased with that of methylester. The K(m) and V(m) values of immobilized glucoamylase which were prepared at various conditions of immobilization were determined. There were opposite trends in K(m)S between glucoamylase immobilized on imidoester-rich support and immobilized on methylester in the support, evidenced as functions of temperature. This suggests that opposite charges in the support, produced by heat deformation of PAN by hydrolysis of methylester, were an influence on the apparent K(m) of immobilized glucoamylase, besides the diffusional limitation.     

Immobilized enzymes: electrokinetic effects on reaction rates in a porous medium

The effect of the internal diffusion and electrical surface charge on the overall rate of a reaction catalyzed by an enzyme immobilized on a porous medium are examined. Effectiveness factors have been calculated which compare the global reaction rate to that existing in the absence of the internal diffusion and/or the electrical field. The surface charge, assumed to arise from the dissociation equilibria of the acidic and basic surface groups of the enzyme, generates an electrical double layer at the pore surface. The double-layer potential is governed by the Poisson-Boltzmann equation. It is shown that the diffusion potential can be characterized by a modulus which depends upon the surface reaction rate, the charges and diffusivities of the substrate and products, the ionic strength, and the pore dimensions. The flux of a charged species in the pore occurs under the influences of the concentration gradient and the electrical potential gradient. The governing equations are solved by an iterative numerical method. The effects of pH, enzyme concentration, and substrate concentration on the rates of two different hydrolysis reactions catalyzed by immobilized papain are examined. The release of H(+) in one of the reactions causes the lowering of internal pH, and also a constancy of the internal pH when the external pH in creases beyond a certain value. The latter reaction also shows a maximum in the reaction rate with respect to enzyme concentration. The reaction not involving H(+) as a product shows a maximum in the reaction rate with respect to external pH, but a monotonic increase in the reaction rate as the enzyme concentration increases.     

Glucose isomerase immobolized on porous glass

Partially purified glucose isomerase from a Streptomyces species was immobilized on porous glass particles and studied for various characteristics concerning its use as an industrial catalyst. The activities were investigated in relation to the reaction parameters and the enzyme deactivation was studied systematically under various reaction conditions. The half-life of the immobilized enzyme was found to exceed 200 days at 50°C. The rate equation of the reversible glucose ? fructose reaction was derived and the kinetic constants were determined. The rate equation was found to be in good agreement with experimental data for both forward and reverse reactions. The degree of diffusional effects was experimentally measured and theoretically analyzed.     

Analysis of the transient response of a CSTR containing immobilized enzyme particles: Part II. Minimum existence criterion and determination of substrate effective diffusivity and main reaction rate constant

A minimum existence criterion in the transient response of the bulk substrate concentration in a CSTR containing immobilized enzyme (IMEs) in porous solid supports has been obtained from simulation results using several kinetic expressions for the main reaction and the enzyme deactivation reaction. A simple method for the determination of the substrate effective diffusivity and the reaction rate constant is also presented, and applied to the decomposition of hydrogen peroxide, that reacts in a CSTR that contains silica–alumina porous catalyst particles, in which horseradish peroxidase enzyme had been previously immobilized.     

The kinetics of penicillin-V deacylation on an immobilized enzyme

An immobilized Penicillin-V-acylase (commercial name, Novozym 217) with high specificity for the phenoxyacetyl-(V)- side chain was investigated in a recycle reactor and in a batch reactor to find the enzymatic reaction rate as a function of conversion, x, substrate concentration, c(A) (0) and pH. The reaction rate depends strongly on pH, and both products, phenoxy-acetic acid and 6-APA, inhibit the reaction. Nonspecific side reactions amount to only a few per cent when c(A) (0) <150mM and pH& gt; 6.5. The effectiveness factor for commercial-size particles is found to be about 0.65, and a value of 1.3mM is obtained for the equilibrium constant, K(eq), of the deacylation reaction. A kinetic model for the deacylation process which includes the effect of pH and of the reverse (acylation) reaction is proposed. Rate data for particles of different size are fitted to the nonlinear model. Five kinetic parameters and an effective diffusivity for the immobilized enzyme particles are determined.     

Effect of internal diffusion in heterogeneous enzyme systems: Evaluation of true kinetic parameters and substrate diffusivity

The effect of internal diffusion on the overall reaction rate in spherical particles and membranes containing immobilized enzymes has been investigated theoretically. Since they represent open systems, the MichaelisMenten kinetics is obeyed in the absence of diffusional effects at steady state even at high enzyme concentrations. When internal diffusion perturbs the reaction, the system can not be described any more by K_M and V_max? alone, but is conveniently characterized by the modulus. Assuming that only internal diffusion interferes with the enzyme reaction, the effect of the modulus on the overall rate of reaction is illustrated by the results of computer calculations. Plots of the overall reaction rate against the substrate concentration are hyperbolas at various moduli for both membranes and spherical particles and no sigmoidal curves are obtained with immobilized enzyme systems. Since the conventional plots of enzyme kinetics do not yield straight lines under such conditions, a graphical method is proposed to determine K_M and V_max? as well as the substrate diffusivity in the enzymic medium.     

Effectiveness factors for substrate and product inhibition

The transformation technique of Na and Na (Math. Biosci., 6 , 25, 1970) is extended to convert boundary-value problems involving the steady-state diffusion equation for spherical immobilized enzyme particles exhibiting substrate and product inhibition to initial-value problems. This allows a study of the influence of external mass transfer resistances on the effectiveness factors. It also considerably reduces the number of calculations required to investigate the effect of changes in the kinetic parameters on the overall rate of reaction. The existence of multiple steady states for substrate inhibition kinetics in spherical catalyst particles is illustrated and a criterion for uniqueness of steady states is developed. Effectiveness factors for competitive and noncompetitive product inhibition increase with increasing value of the Sherwood number for the substrate and increasing value of the ratio of substrate to product effective diffusivities within the particle.     

Estimation of intrinsic kinetic constants for pore diffusion-limited immobilized enzyme reactions

A simple method is presented that establishes intrinsic rate parameters when slow pore diffusion of substrate limits immobilized enzyme reactions that obey Michaelis-Menten kinetics. The Aris-Bischoff modulus is employed. Data at high substrate concentrations, where the enzyme would be saturated in the absence of diffusion limitation, and at low substrate concentrations, where effectiveness factors are inversely proportional to reaction modulus, are used to determine maximum rate and Michaelis constant, respectively. Because Michaelis-Menten and Langmuir-Hinshelwood kinetics are formally identical, this method may be used to estimate intrinsic rate parameters of many heterogeneous catalysts. The technique is demonstrated using experimental data from the hydrolysis of maize dextrin with diffusion-limited immobilized glucoamylase. This system yields a Michaelis constant of 0.14%, compared to 0.11% for soluble glucoamylase and 0.24% for immobilized glucoamylase free of diffusional effects.     

Immobilized glucose oxidase–catalase and their deactivation in a differential-bed loop reactor

Glucose oxidase containing catalase was immobilized with a copolymer of phenylenediamine and glutaraldehyde on pumice and titania carrier to study the enzymatic oxidation of glucose in a differential-bed loop reactor. The reaction rate was found to be first order with respect to the concentration of limiting oxygen substrate, suggesting a strong external mass-transfer resistance for all the flow rates used. The partial pressure of oxygen was varied from 21.3 up to 202.6 kPa. The use of a differential-bed loop reactor for the determination of the active enzyme concentration in the catalyst with negligible internal pore diffusion resistance is shown. Catalyst deactivation was studied, especially with respect to the presence of catalase. It is believed that the hydrogen peroxide formed in the oxidation reaction deactivates catalase first; if an excess of catalase is present, the deactivation of glucose oxidase remains small. The mathematical model subsequently developed adequately describes the experimental results.     

Operational stability of immobilized d-glucose isomerase in a continuous feed stirred tank reactor

d-Glucose isomerization has been studied using immobilized cells of Streptomyces phaeochromogenes in a continuous feed stirred tank reactor (CSTR) where the external film diffusion resistance was negligible. Experiments conducted with various sizes of enzyme particles indicated that a strong internal diffusion resistance improved the apparent stability of these particles. The performance equations of the CSTR were constructed by associating the material balances for the inside porous support matrix with the bulk liquid phase, and enzyme deactivation was also taken into consideration. An iterative method together with the orthogonal collocation method is proposed for the evaluation of effectiveness factor and the substrate concentration profile within the enzyme particles. The numerical results offer an alternative analytical proof for the observation that under strong internal diffusion control the apparent operational stability of immobilized enzyme is improved.     

Operational effectiveness factors of immobilized enzyme systems

The steady-state and operational effectiveness factors for hydrolytic enzymes immobilized in spherical gel particles have been calculated by the collocation method for a wide range of microenvironmental conditions (given by the Thiele modulus) and macroenvironmental conditions (given by the Sherwood number and the relative substrate content). The operational effectiveness factor is a measure of the ratio of the times required to convert a defined amount of substrate with the same amount of free and immobilized enzyme, respectively. Calculations were made for reactors where the diffusion layers of the different enzyme-containing gel particles do not overlap. The theoretical values were compared with experimental values for stirred reactors with chymotrypsin and trypsin immobilized in spherical particles (Sepharose and Sephadex). Low molecular weight substrates were used. The theoretical and experimental values were found to agree within the experimental error. This demonstrates the predictive capacity of the collocation method in estimating steady-state and operational effectiveness factors for enzyme reactors. The microenvironment and macroenvironment were both found to influence the effectiveness over a wide range of substrate concentrations. However, the macroenvironmental influence is negligible when the Sherwood number of the reactor is larger than ～50. Then, the diffusion layer thickness is small compared with the dimensions of the enzyme-containing particles. The effectiveness factors calculated here can also be used to predict the performance of continuous stirred tank and plug-flow reactors.     

Non-porous magnetic micro-particles: comparison to porous enzyme carriers for a diffusion rate-controlled enzymatic conversion

In this study the kinetics of conversion of a low-soluble substrate by an immobilized enzyme was investigated with respect to the diffusion limitation within porous and non-porous carriers. Non-porous micro-magnetic beads in comparison to conventional porous supports like Eupergit and Sepharose were tested. Due to their small diameters and their magnetic properties, micro-magnetic beads are especially applicable in diffusion rate-controlled processes in biological suspensions. The enzymatic reaction studied was the conversion of emulsified dirhamnolipid by immobilized Naringinase from Penicillium decumbens to monorhamnolipid and L-rhamnose. Taking into account mass transfer phenomena, the variation of the reaction effectiveness factor with increasing enzyme loading was estimated and compared with experimental efficiencies utilizing different enzyme loaded immobilized preparations. For comparison, carrier activities were also determined with the model substrate p-nitro-phenyl-rhamnoside. Intrinsic enzyme activities were thereby evaluated for porous supports. Highest specific activities were obtained with the micro-magnetic beads. These non-porous micro-beads demonstrated to be the most suitable carrier for bioconversion of a low-soluble substrate like rhamnolipids, where mass diffusional resistances in the three-phase reaction process are completely overcome. However, the smaller particle surface available limited the specific activity obtained at high protein loadings.