Three-dimensional numerical approach to investigate the substrate transport and conversion in an immobilized enzyme reactor

This numerical study evaluates the momentum and mass transfer in an immobilized enzyme reactor. The simulation is based on the solution of the three-dimensional Navier-Stokes equation and a scalar transport equation with a sink term for the transport and the conversion of substrate to product. The reactor consists of a container filled with 20 spherical enzyme carriers. Each of these carriers is covered with an active enzyme layer where the conversion takes place. To account for the biochemical activity, the sink term in the scalar transport equation is represented by a standard Michaelis-Menten approach. The simulation gives detailed information of the local substrate and product concentrations with respect to external and internal transport limitations. A major focus is set on the influence of the substrate transport velocity on the catalytic process. For reactor performance analysis the overall and the local transport processes are described by a complete set of dimensionless variables. The interaction between substrate concentration, velocity, and efficiency of the process can be studied with the help of these variables. The effect of different substrate inflow concentrations on the process can be seen in relation to velocity variations. The flow field characterization of the system makes it possible to understand fluid mechanical properties and its importance to transport processes. The distribution of fluid motion through the void volume has different properties in different parts of the reactor. This phenomenon has strong effects on the arrangement of significantly different mass transport areas as well as on process effectiveness. With the given data it is also possible to detect zones of high, low, and latent enzymatic activity and to determine whether the conversion is limited due to mass transfer or reaction resistances.     

Consecutive immobilized enzymatic reactions with and without enzyme denaturation

Consecutive biochemical reactions in an immobilized enzyme particle under the effects of internal and external diffusional resistances are analyzed. A rigorous nonlinear reaction kinetics is employed and the steady state effectiveness factor with negligible enzyme denaturation compared with the previous prediction by the first-order kinetics. It is found that the difference between them is rather substantial under most circumstances. The cases with significant enzyme denaturation are also investigated by using an unsteady state model. The substrate concentration responses to variation of the physical and kinetic parameters reveal many interesting characteristics of the reaction system.     

Effectiveness factor calculations for immobilized enzyme catalysts

The steady state, nonlinear diffusion equations which describe reactions in constrained enzyme solutions are of great interest in many biological and engineering applications. As in other types of nonlinear differential equations, exact analytical solutions do not exist except in some simplified cases. In this paper, a general procedure is presented for solving numerically for the substrate concentration profile and effectiveness factor utilizing the transformation method suggested by Na and Na. Design correlations for enzyme solutions constrained within spherical membranes are included. The use of a unique definition of the Thiele Modulus in these charts permits the clear illustration of the effects of substrate concentration and external mass transfer resistances on the overall effectiveness factor for the catalyst particle.     

External and internal diffusion in heterogeneous enzymes systems

The intrusion of diffusion in heterogeneous enzyme reactions, which follow. Michaelis-Menten kinetics, is quantitatively characterized by dimensionless parameters that are independent of the substrate concentration. The effects of these parameters on the overall rate of reaction is illustrated on plots commonly employed in enzyme kinetics. The departure from Michaelis-Menten kinetics due to diffusion limitations can be best assessed by using Hofstee plots which are also suitable to distinguish between internal and external transport effects. A graphical method is described for the evaluation of the reaction rate as a function of the surface concentration of the substrate from measured data.     

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.     

Diffusional falsification of kinetic constants on Lineweaver-Burk plots

The effect of mass transfer resistances on the Lineweaver-Burk plots in immobilized enzyme systems has been investigated numerically and with analytical approximate solutions. While Hamilton, Gardner & Colton (1974) studied the effect of internal diffusion resistances in planar geometry, our study was extended to the combined effect of internal and external diffusion in cylindrical and spherical geometries as well. The variation of Lineweaver-Burk plots with respect to the geometries was minimized by modifying the Thiele modulus and the Biot number with the shape factor. Especially for a small Biot number all the three Lineweaver-Burk plots fell on a single line. As was discussed by Hamilton, et al (1974), the curvature of the line for large external diffusion resistances was small enough to be assumed linear, which was confirmed from the two approximate solutions for large and small substrate concentrations. Two methods for obtaining intrinsic kinetic constants were proposed: First, we obtained both maximum reaction rate and Michaelis constant by fitting experimental data to a straight line where external diffusion resistance was relatively large, and second, we obtained Michaelis constant from apparent Michaelis constant from the figure in case we knew maximum reaction rate a priori.     

Experimental and theoretical evidence for convective nutrient transport in an immobilized cell support.

Even though immobilized-cell reactors possess several engineering advantages over free-cell reactors, their full potential has not been realized because mass transfer often limits the rate of nutrient supply and product removal from immobilized cell supports. We studied the interaction between mass transfer and reaction kinetics in the anaerobic conversion of glucose to CO2 and ethanol by yeast immobilized in a porous rotating disk on the agitator shaft of a conventional CSTR. A Sherwood number correlation was used to show that external mass-transfer resistances were negligible under typical operating conditions. The modulus of Weisz based on observable reaction parameters was used to gauge the importance of pore diffusion limitations. Under conditions for which significant pore diffusion effects and hence low effectiveness factors (eta = ca. 0.1) would be predicted, the observed reaction rates were much higher than expected (eta = ca. 1), suggesting that pore diffusion limitations were at least partially relieved by convective transport of glucose into the support. Two possible mechanisms of convective transport are discussed. We hypothesize that gas evolution was responsible for the convective enhancement of glucose supply.     

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.     

Solid-state fermentation in rotating drum bioreactors: operating variables affect performance through their effects on transport phenomena

Aspergillus oryzae ACM 4996 was grown on an artificial gel-based substrate and on steamed wheat bran during solid-state fermentations in 18.7 L rotating drum bioreactors. For gel fermentations fungal growth decreased as rotational speed increased, presumably due to increased shear. For wheat bran fermentations fungal growth improved under agitated compared to static culture conditions, due to superior heat and mass transfer. We conclude that the effects of operational variables on the performance of SSF bioreactors are mediated by their effects on transport phenomena such as mixing, shear, heat transfer, and mass transfer within the substrate bed. In addition, the substrate characteristics affect the need for and the rates of these transport processes. Different transport phenomena may be rate limiting with different substrates. This work improves understanding of the effects of bioreactor operation on SSF performance.     

Influence of substrate and product diffusion on the heterogeneous kinetics of enzymic reversible reactions

When a reversible reaction is catalyzed by a surface bound enzyme, the diffusion of both substrate and product can considerably modify the kinetic properties of the reaction. According to this theoretical analysis, the enzyme activity is decreased due to the presence of substrate and product concentration gradients in the enzyme microenvironment, and the relative kinetic importance of the two diffusion steps mainly depend on the value of a dimensionless criterion inversely proportional to the equilibrium constant. Moreover diffusional effects increase with increasing bound enzyme activity, but decrease with increasing substrate and product concentration. Analytical expressions are established for the limiting values of substrate and product concentrations in the enzyme microenvironment, as well as for the increase in half-maximal-activity substrate concentration in the presence of substrate and product diffusional limitations.     

Mass transport in a microchannel enzyme reactor with a porous wall: Hydrodynamic modeling and applications

A two-dimensional flow model, incorporating mass transport, has been developed to simulate a microchannel enzyme reactor with a porous wall. A two-domain approach based on the finite volume method was implemented. Two parameters are defined to characterize the mass transports in the fluid and porous regions: the porous Damkohler number and the fluid Damkohler number. For reactions close to first-order type (enzyme reactor), the concentration results are found to be well correlated by the use of a reaction–convection distance parameter which incorporates the effects of axial distance, substrate consumption and convection. The reactor efficiency reduces with reaction–convection distance parameter because of reduced reaction (or flux) due to the lower concentration. Increased fluid convection improves the efficiency but it is limited by the diffusion in the fluid region. The correlated results can find applications for the design of enzyme reactors with a porous wall.     

Atypical kinetics of immobilized firefly luciferase

The kinetic properties of collagen-bound firefly luciferase have been investigated. Under definite hydrodynamic conditions with low agitation in the reaction medium, the observed behavior is modified compared to the enzyme free in solution: reducing the stirring rate decreases the observed enzymatic activity. But diffusional resistances alone cannot account for these atypical kinetics though mass transfer may certainly play an important role during the transient state of the bioluminescent reaction. After immobilization, the time necessary to reach the steady state increased from 300 ms to 3 min and the two substrates, luciferin and ATP, behave differently with respect to the enzyme: The nature of the saturating substrate first in contact with the bound enzyme is not indifferent suggesting that immobilization can reveal behaviors or mechanisms which are not visualized with the free enzyme.     

Diffusional influences on the parametric sensitivity of immobilized enzyme catalysts

Immobilization of an enzyme within an insoluble material permeable to substrate can change the apparent behavior of the enzyme. In particular, external mass transfer and intraparticle diffusion effects can significantly influence the dependence of observed reaction rate on operating parameters such as temperature and pH. This analysis shows that, under very general conditions, the influence of diffusion alone is to diminish the sensitivity of the observed rate to any parameter that is uniform throughout the catalyst particle. However, the optimal value of the parameter is not changed because of intrapellet diffusional effects.     

Modeling intrinsic kinetics of enzymatic cellulose hydrolysis

A multistep approach was taken to investigate the intrinsic kinetics of the cellulase enzyme complex as observed with hydrolysis of noncrystalline cellulose (NCC). In the first stage, published initial rate mechanistic models were built and critically evaluated for their performance in predicting time-course kinetics, using the data obtained from enzymatic hydrolysis experiments performed on two substrates: NCC and alpha-cellulose. In the second stage, assessment of the effect of reaction intermediates and products on intrinsic kinetics of enzymatic hydrolysis was performed using NCC hydrolysis experiments, isolating external factors such as mass transfer effects, physical properties of substrate, etc. In the final stage, a comprehensive intrinsic kinetics mechanism was proposed. From batch experiments using NCC, the time-course data on cellulose, cello-oligosaccharides (COS), cellobiose, and glucose were taken and used to estimate the parameters in the kinetic model. The model predictions of NCC, COS, cellobiose, and glucose profiles show a good agreement with experimental data generated from hydrolysis of different initial compositions of substrate (NCC supplemented with COS, cellobiose, and glucose). Finally, sensitivity analysis was performed on each model parameter; this analysis provides some insights into the yield of glucose in the enzymatic hydrolysis. The proposed intrinsic kinetic model parametrized for dilute cellulose systems forms a basis for modeling the complex enzymatic kinetics of cellulose hydrolysis in the presence of limiting factors offered by substrate and enzyme characteristics.     

The kinetics of the beta-galactoside-proton symport of Escherichia coli.

beta-Galactoside transport by Escherichia coli occurs with the concomitant uptake of a proton. The kinetics of beta-galactoside uptake at various values of external pH are interpreted in terms of a model in which both the galactoside and the proton are substrates of the transport reaction. The values of some of the kinetic constants for this two-substrate reaction were determined. The observed effects of the protonmotive force on the apparent Michaelis constant for galactoside can be explained in terms of the proton being a substrate of the transport reaction.     

Enzyme reactions at the surface of living cells. I. Electric repulsion of charged ligands and recognition of signals from the external milieu

The dynamic behaviour of a polyelectrolyte-bound enzyme is studied when diffusion of substrate or diffusion of product is coupled to electric repulsion and to Michaelis-Menten enzyme reaction. The definition of the classical concepts of electric partition coefficients and Donnan potential of a polyelectrolyte membrane has been extended under global non-equilibrium conditions. This extension is permissible when a strong repulsion exists of substrate and product by the fixed negative charges of the membrane. Coupling between product diffusion, electric repulsion and enzyme reaction at constant advancement may result in a hysteresis loop of the partition coefficient as the product concentration is increased in the reservoir. This hysteresis loop vanishes as the rate of product diffusion increases. No hysteresis loop may occur when electric repulsion effects are coupled to substrate diffusion and reaction. The existence of multiple values of the partition coefficient for a fixed concentration of product implies that the membrane may store short-term memory of the former product concentration present in the external milieu. The occurrence of hysteresis generated by coupling enzyme reaction, product diffusion, electric partition effects at constant advancement of the reaction may be viewed as a sensing device of product concentration in the external milieu. Surprisingly, non-linearities required to generate this sensing device come from electrostatic effects and not from enzyme kinetics.     

A Fortran Program for Evaluation of the Effectiveness Factor of Biocatalysts in the Presence of External and Internal Diffusional Limitations

A Fortran program called SPEFF for evaluation of the effectiveness factor of immobilized enzyme preparations of spherical form in the presence of external and internal mass transfer resistances is described, and a listing of the program is given. Enzyme distribution in the bioparticle may be uniform or nonuniform. In the latter case the enzyme distribution is approximated by fifth-order polynomial. In the program differential equations are replaced by the system of non-linear algebraic equations, and the latter are solved by Newton iteration technique. The program is developed for Michaelis-Menten kinetics with allowance for competitive product inhibition and substrate inhibition. After slight modifications the program can be used for computation of the effectiveness factor of a membrane with an immobilized enzyme, or in the case when the enzyme kinetics are more complex. A typical run on a PDP-11/45 computer took 10-20 seconds. A typical computation time in the case of IBM-compatible TURBO PC was 15-30 seconds.     

Consecutive immobilized enzymatic reactions in continuous stirred tank reactor systems

The performance characteristics of two-enzyme reaction in a continuous stirred tank reactor (CSTR) are analytical investigated in this work. A model is formulated to describe the substrate concentration variations by taking into account the external and internal diffusion resistances. It is found that the reaction system exhibits the characteristics of reaction control or diffusion control depending on the operating conditions. The single CSTR model is also extended to describe the multiple CSTR system. The latter model enables the prediction of the number of CSTRs in series required to achieve a prescribed substrate conversion.     

Diffusional effects on the heterogeneous kinetics of two-substrate enzymic reactions.

When a two-substrate reaction is catalyzed by a surface bound enzyme, the diffusion of both substrates can considerably modify the kinetic properties of the reaction. According to this theoretical analysis, limitations in substrate diffusion yield very different effects depending whether the two substrates have similar or different affinities for the enzyme. With two substrates of comparable affinities the diffusion of the two substrates can be limiting, and similar activity dependences on the two substrate concentrations are obtained. Under these conditions, diffusional limitations may only slightly influence half-maximal-activity substrate concentrations. With two substrates of widely different affinities, on the contrary, the rate of the enzymic reaction can only be limited by the diffusion of the high-affinity substrate, used at the lower concentration. Under these conditions, in the presence of diffusional limitations the activity dependence on the two substrate concentrations are highly different, and the half-maximal-activity concentration is increased and decreased for the high- and low-affinity substrates, respectively. The theoretical results are verified by experimental data previously obtained with collagen-bound aspartate aminotransferase and sorbitol dehydrogenase.     

Enzyme stabilization by linear chain polymers in ultrafiltration membrane reactors

The experimental results discussed in this article concern p-nitrophenylphosphate hydrolysis by acid phosphatase in an Ultrafiltration membrane reactor. The basic conclusions drawn are: (1) Linking the enzyme to a soluble support does not give rise to an increase in its stability while the chemical manipulations involved result in marked reductions in enzymic activity. (2) Enzyme entrapment within a proteic gel produces a considerable increase in its thermal stability as compared to the diluted native enzyme; this presumably stems from drastic reductions in enzyme mobility. (3) Correspondingly, considerable reductions occur in enzyme activity that depend on substrate mass transfer resistances within the gel layer. (4) Small amounts of linear chain water-soluble synthetic polymers (polyacrylamides) give rise to high macromolecular concentration levels in the reactor region where the enzyme is dynamically immobilized and produce the same enzyme stabilization as gel entrapment. (5) Only minor substrate mass transfer limitations take place in this region and hence enzyme activity is virtually unaffected. (6) Both effects (stabilization and slight activity reduction) seem not to depend strongly on the characteristics of the soluble polymer (molecular weight and ionic character).