Progress curves of reactions catalyzed by unstable enzymes. A theoretical approach

When an enzyme is incubated with its substrate, the rate of catalysis will decline with time due to the combined effects of substrate utilization and product accumulation. These effects will be superimposed upon a progressive loss of catalytic activity if the enzyme is unstable, either spontaneously or as a result of an added reagent. In this report, the effect of enzyme inactivation on the progress curve for an enzyme-catalyzed reaction is considered. It is shown that under most circumstances catalysis will stop before the substrate is totally exhausted and that the amount of substrate remaining is related to the inactivation rate constants for various intermediates on the catalytic pathway. A graphical method for estimating these inactivation rate constants is suggested for several situations, including one which encompasses the effect of a suicide substrate. Expressions for the half time of the reaction are also given for some special cases.     

Rabbit muscle phosphofructokinase. 2. Inactivation by the affinity label 5'-[p-(fluorosulfonyl)benzoyl]-1,N6-ethenoadenosine

The reaction of the fluorescent affinity label 5'-[p-(fluorosulfonyl)benzoyl]-1,N6-ethenoadenosine with rabbit skeletal muscle phosphofructokinase results in an inactivation of the enzyme and in the covalent incorporation of up to one label/monomer. The substrates, MgATP and fructose 6-phosphate, each protect against inactivation of the enzyme, but neither diminishes the extent of covalent incorporation of the label, indicating that the inactivation is not the result of covalent incorporation of the label. Dithiothreitol reactivates the inactivated enzyme but does not reduce the extent of incorporation of the label. A determination of the number of free sulfhydryl groups on the enzyme as a function of the extent of inactivation by the reagent suggests that the inactivation is associated with the loss of two free sulfhydryl groups per phosphofructokinase monomer. The inactivation reaction appears to involve the reversible formation of an enzyme-reagent complex (Kd = 1.11 mM) prior to the conversion of the complex to inactive enzyme (k1 = 0.98 min-1). In view of the protection afforded by either substrate and the evidence suggesting the formation of an enzyme-reagent complex prior to inactivation, it would appear that the inactivation results from a reagent-mediated formation of a disulfide bond between two cysteinyl residues in close proximity, possibly in or near the catalytic site of the enzyme. The site of covalent attachment of the label appears to be the binding site specific for the activating adenine nucleotides cAMP, AMP, and ADP. The extent of covalent incorporation of the label at this site is diminished in the presence of cAMP, and phosphofructokinase modified at this site by this affinity label is no longer subject to activation by cAMP.     

Detergentless microemulsions as media for enzymatic reactions. Cholesterol oxidation catalyzed by cholesterol oxidase

Catalytic activity and stability of cholesterol oxidase dissolved in ternary systems composed of n-hexane, isopropanol, and water were studied. The dependence of catalytic activity on the composition of the system revealed two maxima, in contrast to the behaviour of previously studied enzymes where a single maximum has been observed. The stability profile of cholesterol oxidase showed a single sharp maximum coinciding with the microemulsion region of the phase diagram. Both catalytic activity and the first-order inactivation rate constant of cholesterol oxidase dissolved in n-hexane/isopropanol/water ternary systems were found to decrease with decreasing temperature. This decrease was more rapid for the inactivation rate constant than for catalytic activity, the activation energies being 200 and 60 kJ.mol-1, respectively. Preparative conversion of cholesterol to cholestenone catalyzed by cholesterol oxidase in n-hexane/isopropanol/water ternary systems was carried out with 100% yield. Decreased temperature and the presence of catalase were required to achieve high degrees of cholesterol conversion. A simple procedure suitable for rapid separation of the reaction product and recovery of the enzyme was developed.     

Experimental modelling of thermal inactivation of urease

Thermal inactivation of jack bean urease (EC 3.5.1.5) was investigated in a 0.1 M phosphate buffer with pH 7. An injection flow calorimetry method was adapted for the measurement of the enzyme activity. The inactivation curves were measured in the temperature range of 55 to 87.5 degrees C. The curves exhibited a biphasic pattern in the whole temperature range and they were well fitted with a biexponential model. A simultaneous fit of all inactivation data was based on kinetic models that were derived from different inactivation mechanisms and comprised the material balances of several enzyme forms and the enthalpy balance characterizing the initial heating period of enzyme solution. The multitemperature evaluation revealed that an adequate model had to incorporate at least three reaction steps. It was concluded that the key reaction steps at urease thermal inactivation were the reversible dissociation/denaturation of native form into an inactive denatured form, and irreversible association reactions of both the denatured and native forms.     

Stability and catalytic activity of alpha-amylase from barley malt at different pressure-temperature conditions

The impact of high hydrostatic pressure and temperature on the stability and catalytic activity of alpha-amylase from barley malt has been investigated. Inactivation experiments with alpha-amylase in the presence and absence of calcium ions have been carried out under combined pressure-temperature treatments in the range of 0.1-800 MPa and 30-75 degrees C. A stabilizing effect of Ca(2+) ions on the enzyme was found at all pressure-temperature combinations investigated. Kinetic analysis showed deviations of simple first-order reactions which were attributed to the presence of isoenzyme fractions. Polynomial models were used to describe the pressure-temperature dependence of the inactivation rate constants. Derived from that, pressure-temperature isokinetic diagrams were constructed, indicating synergistic and antagonistic effects of pressure and temperature on the inactivation of alpha-amylase. Pressure up to 200 MPa significantly stabilized the enzyme against temperature-induced inactivation. On the other hand, pressure also hampers the catalytic activity of alpha-amylase and a progressive deceleration of the conversion rate was detected at all temperatures investigated. However, for the overall reaction of blocked p-nitrophenyl maltoheptaoside cleavage and simultaneous occurring enzyme inactivation in ACES buffer (0.1 M, pH 5.6, 3.8 mM CaCl(2)), a maximum of substrate cleavage was identified at 152 MPa and 64 degrees C, yielding approximately 25% higher substrate conversion after 30 min, as compared to the maximum at ambient pressure and 59 degrees C.     

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.     

Affinity labeling of the catalytic and AMP allosteric sites of 3-hydroxy-3-methylglutaryl-coenzyme A reductase kinase by 5'-p-fluorosulfonylbenzoyladenosine

The nucleotide analogue 5'-p-fluorosulfonylbenzoyladenosine (FSBA) reacts irreversibly with rat liver cytosolic 3-hydroxy-3-methylglutaryl-CoA (HMG-CoA) reductase kinase, causing a rapid loss of the AMP activation capacity and a slower inactivation of the catalytic activity. The rate constant for loss of AMP activation is about 10 times higher (kappa 1 = 0.112 min-1) than the rate constant of inactivation (kappa 2 = 0.0106 min-1). There is a good correspondence between the time-dependent inactivation of reductase kinase and the time-dependent incorporation of 5'-p-sulfonylbenzoyl[14C]adenosine ([14C]SBA). An average of 1.65 mol of reagent/mol of enzyme subunit is bound when reductase kinase is completely inactivated. The time-dependent incorporation is consistent with the postulate that covalent reaction of 1 mol of SBA/mol of subunit causes complete loss of AMP activation, whereas reaction of another mole of SBA/mol of subunit would lead to total inactivation. Protection against inactivation by the reagent is provided by the addition of Mg2+, AMP, Mg-ATP, or Mg-AMP to the incubation mixtures. In contrast, addition of ATP, 2'-AMP, or 3'-AMP has no effect on the rate constants. Mg-ATP protects preferentially the catalytic site against inactivation, whereas Mg-AMP at low concentration protects preferentially the allosteric site. Mg-ADP affords less protection than Mg-AMP to the allosteric site when both nucleotides are present at a concentration of 50 microM with 7.5 mM Mg2+. Experiments done with [14C]FSBA in the presence of some protectants have shown that a close correlation exists between the pattern of protection observed and the binding of [14C]SBA. The postulate is that there exists a catalytic site and an allosteric site in the reductase kinase subunit and that Mg-AMP is the main allosteric activator of the enzyme.     

Hysteresis behavior of complex I in delta mu H+-dependent reduction of NAD+ succinate

It was shown that the membrane-bound complex I is fully inactive in the absence of NADH during the reverse electron transfer from succinate to NAD+. The enzyme activation is attained by preincubation of submitochondrial particles with low concentrations of NADH; the activating effect persists after a complete oxidation of the latter during long-term (several hours) aerobic incubation. The experimental results suggest that complex I contains a redox component, whose reduction by NADH and aerobic oxidation are not involved in the overall catalytic reaction. An experimental scheme is proposed, according to which the key role of such a component is ascribed to the tightly bound ubiquinone; the activation and inactivation of the enzyme are due to a slow reversible redox conversion (ubiquinone in equilibrium ubisemiquinone), whereas the catalytic act involves a rapid reversible conversion (ubisemiquinone in equilibrium ubiquinol). It was demonstrated that the "redox" mechanism of the inactivation-activation reaction determines the strong dependence of activity of the reverse electron transfer on the mode of preparation of submitochondrial particles. The coupling properties of the submitochondrial particulate membrane and the activities of enzymes involved in the reverse electron transfer are stable at room temperature for over 14 hours.     

The sulphatase of ox liver. XXI: kinetic studies of the substrate-induced inactivation of sulphatase A.

The theoretical basis is given for methods of determining the apparent velocity constant, k*, for the substrate-induced inactivation of sulphatase A (aryl-sulphate sulphohydrolase, EC 3.1.6.1) and the initial velocity, vo, of the catalytic reaction. The expression is of the same form as the empirical relationships previously used but the significance of the various terms is clearly established. The method has been applied to the characterisation of the inactivation occurring during the hydrolysis of a number of substrates and it has been shown that k* varies with so in a hyperbolic relationship described by k, a velocity constant at infinite substrate concentrations and by K, a constant analogous to the Michaelis constant. Although K varies considerably for different substrates, and is consistently less than the corresponding Km, k is almost constant at 0.23 min-1. It is therefore suggested that the inactivation of the enzyme does not proceed through an enzyme . substrate complex but through the enzyme . SO2-4 complex produced during the catalytic reaction. The effects of several variables on these parameters are described.     

Chemical modification of rabbit skeletal muscle phosphorylase kinase with phenylglyoxal

Nonactivated phosphorylase kinase from rabbit skeletal muscle is inactivated by treatment with phenylglyoxal. Under mild reaction conditions, a derivative that retains 10-15% of the pH 8.2 catalytic activity is obtained. The kinetics of inactivation profile, differential effects of modification on pH 6.8 and 8.2 catalytic activities, and the insensitiveness of the modified enzyme to activation by ADP reveal that the 10-15% of catalytic activity remaining is very likely due to intrinsic catalytic activity of the derivative rather than to the presence of unmodified enzyme molecules. The kinetic results also suggest that the inactivation is correlatable with the reaction of one molecule of the reagent with the enzyme without any prior binding of phenylglyoxal. The phenylglyoxal modification reduces the autophosphorylation rate of the kinase. Autophosphorylated phosphorylase kinase is inactivated by phenylglyoxal at a much slower rate than the inactivation of nonactivated kinase. Thus, phenylglyoxal modification influences the phosphorylation and vice versa. The modified enzyme can be reactivated by treatment with trypsin or by dissociation using chatropic salts. The activity of the phenylglyoxal-modified enzyme after trypsin digestion or dissociation with LiBr reaches the same level as that of the native enzyme digested with trypsin or treated with LiBr under identical conditions. The results suggest that the effect of modification is overcome by dissociation of the subunits of phosphorylase kinase and that the catalytic site is not modified under conditions when 85% of the pH 8.2 catalytic activity is lost. Among various nucleotides and metal ions tested, only ADP, with or without Mg2+, afforded effective protection against inactivation with phenylglyoxal. At pH 6.8, 1 mM ADP afforded complete protection against inactivation. Experiments with 14C-labeled phenylglyoxal revealed that ADP seemingly protects one residue from modification. This result is in agreement with the kinetic result that the inactivation seemingly is due to reaction of one molecule of the reagent with the enzyme. The results confirm the existence of a high-affinity ADP binding site on nonactivated phosphorylase kinase and suggest the involvement of a functional arginyl residue at or near the ADP binding site in the regulation of of pH 8.2 catalytic activity of the enzyme.     

Design of immobilized enzyme reactors for the continuous production of fructose syrup from whey permeate

Biocatalyst inactivation is inherent to continuous operation of immobilized enzyme reactors, meaning that a strategy must exist to ensure a production of uniform quality and constant throughput. Flow rate can be profiled to compensate for enzyme inactivation maintaining substrate conversion constant. Throughput can be maintained within specified margins of variation by using several reactors operating in parallel but displaced in time. Enzyme inactivation has been usually modeled under non-reactive conditions, leaving aside the effect of substrate and products on enzyme stability. Results are presented for the design of enzyme reactors under the above operational strategy, considering first-order biocatalyst inactivation kinetics modulated by substrate and products. The continuous production of hydrolyzed-isomerized whey permeate with immobilized lactase and glucose isomerase in sequential packed-bed reactors is used as a case study. Kinetic and inactivation parameters for immobilized lactase have been determined by the authors; those for glucose isomerase were taken from the literature. Except for lactose, all other substrates and products were positive modulators of enzyme stability. Reactor design was done by iteration since it depends on enzyme inactivation kinetics. Reactor performance was determined based on a preliminary design considering non-modulated first-order inactivation kinetics and confronted to such pattern. The new pattern of inactivation was then used to redesign the reactor and the process repeated until reactor performance (considering modulation) matched the assumed pattern of inactivation. Convergence was very fast and only two iterations were needed.     

Catalytic activity of beta-amylase from barley in different pressure/temperature domains

The depolymerization of starch by beta-amylase during exposure to hydrostatic pressure up to 700 MPa and within a temperature range from 20 to 70 degrees C has been investigated. Inactivation of the enzyme as well as alterations in conversion speed in response to combined pressure-temperature treatments were assessed by analyzing the kinetic rate constants. At 200 MPa a significant stabilization of the enzyme against heat inactivation was observed. However, high pressure also impedes the catalytic reaction and a progressive reduction of the conversion rate constants with increasing pressure was found at all temperatures investigated. For the overall reaction of maltose liberation from soluble starch in ACES buffer at pH 5.6 an optimum was identified at 106 MPa and at 63 degrees C, which is approximately 7 degrees C above the local maximum at ambient pressure (0.1 MPa). Gelatinization of nonsoluble starch granules in response to pressure-temperature (p-T) treatment has been inspected by phase-contrast microscopy and yielded circular curves of identical effect in the p-T plane.     

Inactivation of glutamine synthetases by an NAD:arginine ADP-ribosyltransferase

Glutamine synthetase from ovine brain has a critical arginine residue at the catalytic site (Powers, S. G., and Riordan, J.F. (1975) Proc. Natl. Acad. Sci. U.S. A. 72, 2616-2620). This enzyme is now shown to be a substrate for a purified NAD:arginine ADP-ribosyltransferase from turkey erythrocyte cytosol that catalyzes the transfer of ADP-ribose from NAD to arginine and purified proteins. The transferase catalyzed the inactivation of the synthetase in an NAD-dependent reaction; ADP-ribose and nicotinamide did not substitute for NAD. Agmatine, an alternate ADP-ribose acceptor in the transferase-catalyzed reaction, prevented inactivation of glutamine synthetase. MgATP, a substrate for the synthetase which was previously shown to protect that enzyme from chemical inactivation, also decreased the rate of inactivation in the presence of NAD and ADP-ribosyltransferase. Using [32P]NAD, it was observed that approximately 90% inactivation occurred following the transfer of 0.89 mol of [32P]ADP-ribose/mol of synthetase. The erythrocyte transferase also catalyzed the NAD-dependent inactivation of glutamine synthetase purified from chicken heart; 0.60 mol of ADP-ribose was transferred per mol of enzyme, resulting in a 95% inactivation. As noted with the ovine brain enzyme, agmatine and MgATP protected the chicken synthetase from inactivation and decreased the extent of [32P]ADP-ribosylation of the synthetase. These observations are consistent with the conclusion that the NAD:arginine ADP-ribosyltransferase modifies specifically an arginine residue involved in the catalytic site of glutamine synthetase. Although the transferase can use numerous proteins as ADP-ribose acceptors, some characteristics of this particular arginine, perhaps the same characteristics that are involved in its function in the catalytic site, make it a favored ADP-ribose acceptor site for the transferase.     

Thermostability of Bacillus subtilis neutral protease.

The thermostability of the B. subtilis neutral protease was studied under various conditions. At elevated temperatures the enzyme was inactivated as a result of autolysis. The rate of inactivation did not depend on the enzyme concentration and the enzyme was most stable near its pH optimum. The rate of inactivation was unaffected by the presence of a second protease during the incubation at high temperatures. The results indicate that the rate of thermal inactivation of the neutral protease is determined by the kinetics of local unfolding processes that precede autolysis rather than by the catalytic rate of the autodigestion reaction or an irreversible unfolding step.     

The histidine residues of phospholipase C from Bacillus cereus.

The inactivation of phospholipase C from Bacillus cereus at pH6 by diethyl pyrocarbonate parallelled the N-ethoxyformylation of a single histidine residue in the enzyme. The inactivation arose from a decrease in the maximum velocity of the enzymic reaction with no effect on the Km value. The inactivation did not apparently alter the ability of the enzyme to bind to a substrate-based affinity gel. The native enzyme contained only one reactive histidine residue. Removal of the two zinc atoms from the enzyme increased the number of reactive histidine residues to five, whereas in the totally denatured enzyme nearly eight such residues were available for reaction with diethyl pyrocarbonate. The enzyme thus appears to contain one histidine residue that is essential for catalytic activity and four that may be involved in co-ordinating the zinc atoms in the structure.     

Catalytic power of enzymes decreases with temperature: New insights for understanding soil C cycling and microbial ecology under warming

Most current models of soil C dynamics predict that climate warming will accelerate soil C mineralization, resulting in a long‐term CO₂ release and positive feedback to global warming. However, ecosystem warming experiments show that CO₂ loss from warmed soils declines to control levels within a few years. Here, we explore the temperature dependence of enzymatic conversion of polymerized soil organic C (SOC) into assimilable compounds, which is presumed the rate‐limiting step of SOC mineralization. Combining literature review, modelling and enzyme assays, we studied the effect of temperature on activity of enzymes considering their thermal inactivation and catalytic activity. We defined the catalytic power of enzymes (E_power) as the cumulative amount of degraded substrate by one unit of enzyme until its complete inactivation. We show a universal pattern of enzyme's thermodynamic properties: activation energy of catalytic activity (EA_cat) < activation energy of thermal inactivation (EA_inact). By investing in stable enzymes (high EA_inact) having high catalytic activity (low EA_cat), microorganisms may maximize the E_power of their enzymes. The counterpart of such EAs’ hierarchical pattern is the higher relative temperature sensitivity of enzyme inactivation than catalysis, resulting in a reduction in E_power under warming. Our findings could explain the decrease with temperature in soil enzyme pools, microbial biomass (MB) and carbon use efficiency (CUE) reported in some warming experiments and studies monitoring the seasonal variation in soil enzymes. They also suggest that a decrease in soil enzyme pools due to their faster inactivation under warming contributes to the observed attenuation of warming effect on soil C mineralization. This testable theory predicts that the ultimate response of SOC degradation to warming can be positive or negative depending on the relative temperature response of E_power and microbial production of enzymes.     

Use of Swinbourne plots to study potential suicide substrates: effects of ATP and ADP on yeast mitochondrial F1-ATPase

A simple analytical procedure for comparing the rates of inactivation of an enzyme in the presence and absence of its substrate is proposed. The rapid inactivation of yeast F1-ATPase during the catalytic reaction was found to be due to certain anions rather than due to ATP or ADP. MgATP failed to protect the enzyme but substituting sulfate, acetate, bicarbonate, or N-tris(hydroxymethyl)methyl-2-aminoethane sulfonate anions and preincubation with ADP prevented the inactivation.     

Cysteine in the manganous-isocitrate binding site of pig heart TPN-specific isocitrate dehydrogenase: I. Kinetics of chemical modification and properties of thiocyano enzyme

Pig heart TPN-dependent isocitrate dehydrogenase is inactivated by reaction with 5,5′-dithiobis (2-nitrobenzoic acid) (DTNB). The dependence of the rate constant for inactivation on the reagent concentration is nonlinear, and can be analyzed in terms of the existence of two mechanisms for reaction with the enzyme, one involving reversible binding prior to inactivation and the other a bimolecular reaction. Cyanide reacts with the inactive modified enzyme to yield thiocyano-isocitrate dehydrogenase without increasing the catalytic activity; this result suggests that inactivation by DTNB is not due to steric hindrance by the bulky thionitrobenzoate group bound to the enzyme. The inactive thiocyano enzyme binds manganous ion normally. In contrast to its effect on native enzyme, however, isocitrate does not strengthen the binding of Mn²⁺ to the thiocyano enzyme; the tightened binding of manganous-isocitrate may be critical for the catalytic activity of the enzyme. Protection against inactivation by DTNB is provided by isocitrate plus the activator, manganous ion, or the competitive inhibitor, calcium ion. The concerted inhibitors oxalacetate and glyoxylate, when present together with Mn²⁺ and TPN, also protect against loss of activity. A marked decrease in the inactivation rate constant to a finite limiting value is caused by saturating concentrations of TPNH and Mn²⁺, indicating that these ligands do not bind directly at the sites attacked by DTNB. The number of cysteine residues which react with DTNB concomitant with inactivation depends on the ligands present in the reaction mixture. In all cases, the equivalent of one -SH reacts without affecting activity. In the presence of Mn²⁺ and α-ketoglutarate, which do not appreciably affect the inactivation rate, loss of activity is proportional to reaction with two -SH groups. These results suggest that the integrity of a maximum of two cysteine residues is essential for the function of the pig heart isocitrate dehydrogenase, and that at least one cysteine residue may be located within the manganous-isocitrate binding site.     

4-Aminobutyrate aminotransferase reaction of sulfhydryl residues connected with catalytic activity

4-Aminobutyrate aminotransferase is inactivated by preincubation with N-(1-pyrene)maleimide (mixing molar ratio 10:1) at pH 7. The reaction with N-(1-pyrene)maleimide was monitored by fluorescence spectroscopy and the degree of labeling of the enzyme determined by absorption spectroscopy. The blocking of 2 cysteinyl residues/enzyme dimer is needed for inactivation of the aminotransferase. The time course of the reaction is significantly affected by the substrate alpha-ketoglutarate, which afforded complete protection against the loss of catalytic activity. Trypsin digestion of pyrene-labeled aminotransferase, followed by gel filtration and "fingerprint" analysis, revealed the presence of only one peptide tagged with the fluorescent probe. The reaction of approximately 1.9 SH residues/dimer with iodosobenzoate resulted in enzyme inactivation together with a formation of an oligomeric species of Mr = 100,000 detectable by sodium dodecyl sulfate-polyacrylamide gel electrophoresis. The cross-linked subunits are dissociated by addition of 2-mercaptoethanol which also restores full catalytic activity. Altogether, these observations are consistent with the concept that inactivation of 4-aminobutyrate aminotransferase by iodosobenzoate proceeds through disulfide bond formation between vicinal cysteinyl residues of the protein. It is postulated that the critical sulfhydryl groups of the enzyme are situated on opposite sides of the dimeric structure at the subunit interfaces.     

Evaluation of half-life of immobilized enzyme during continuous reaction in bioreactors: A theoretical study

A theoretical study has been carried out on the evaluation of the apparent half-life of immobilized enzyme activity during continuous reaction both in a plug-flow reactor (PFR) and in a continuous-flow stirred-tank reactor (CSTR). Two apparent half-lives have been defined: the elapsed time at which the feedrate becomes half of the initial one when the feedrate of the substrate solution is lowered to keep the conversion fixed (constant-conversion policy), and the elapsed time at which the conversion becomes half of the initial one when the feedrate (or space velocity) is kept constant (constant-feedrate policy or constant-space-velocity policy). Under no intraparticle diffusional limitation, the constant-conversion policy of operation in the PFR and CSTR gives the same half-life as that of the enzyme inactivation regardless of the formula of the reaction rate, and the constant-feedrate policy of operation in the PFR and CSTR offers the same half-life as that of the enzyme inactivation only when the reaction is zero-order. Under intra-particle diffusional limitation, apparent half-lives are always greater than that of enzyme denaturation, depending on many factors such as order of reaction, feeding policy (constant-conversion and constant-feedrate policies), initial conversion, and bioreactor configuration. It is suggested to perform the continuous operation with changing feedrate to keep the conversion (or outlet substrate concentration) fixed under the domain of zero-order kinetics so as to obtain an apparent half-life as close to the real one in industrial operation.