Transient model of thermal deactivation of enzymes

The kinetics of enzyme deactivation provide useful insights on processes that determine the level of biological function of any enzyme. Photinus pyralis (firefly) luciferase is a convenient enzyme system for studying mechanisms and kinetics of enzyme deactivation, refolding, and denaturation caused by various external factors, physical or chemical by nature. In this report we present a study of luciferase deactivation caused by increased temperature (i.e., thermal deactivation). We found that deactivation occurs through a reversible intermediate state and can be described by a Transient model that includes active and reversibly inactive states. The model can be used as a general framework for analysis of complex, multiexponential transient kinetics that can be observed for some enzymes by reaction progression assays. In this study the Transient model has been used to develop an analytical model for studying a time course of luciferase deactivation. The model might be applicable toward enzymes in general and can be used to determine if the enzyme exposed to external factors, physical or chemical by nature, undergoes structural transformation consistent with thermal mechanisms of deactivation.     

Mechanistic analysis of complex enzyme deactivations: influence of various parameters on series-type inactivations

A series-type enzyme deactivation model is used to model and to quantitate some more complex enzyme deacti-vations. The influence of temperature, pH, immobilization, chemical modifier (inhibitor or protector), substrate, and metal ion on the inactivation kinetics and on the parameter values is examined. In some cases the influence of two parameters on enzyme inactivations is presented. This provides further physical insights into enzyme inactivation and stabilization processes.     

Enhancing the feasibility of many biotechnological processes through enzyme deactivation studies

Enzymes are deactivated by different ways to an inactive state, which is a major constraint in the development of biotechnological processes. Understanding the complex process of enzyme deactivation will be helpful in enhancing the feasibility of many biological processes. This paper mainly deals with the different ways by which enzymes are inactivated, which includes the role of thermodynamics in deactivation. Different models namely, unified deactivation theory, single exponential model, rapid equilibrium model, isozyme model and bacterial contamination model used to describe the complex deactivation processes are also discussed in this communication. The complete understanding of deactivation process is very essential in commercialization because enzyme activity and stability of the enzyme play a critical role in economics of biotechnological processes. Activity and stability of the enzyme are conflicting properties and trade-off between these factors must be considered in the selection and design of enzymes.     

Graphical determination of mean activation energy and standard deviation in a microheterogeneity model of enzyme deactivation

Traditionally, enzyme populations have been treated as if they were either homogenous, or heterogeneous with distinct and separable subpopulations. The microheterogeneity model, however, assumes that there is a continuous distribution of properties in the population. In the area of enzyme deactivation kinetics, this model describes the heterogeneous population as having a continuous distribution of activation energy of deactivation. This distribution is characterized by mean activation energy, and a standard deviation of activation energy. The microheterogeneity model contains two parameters, (0) and sigma. Parameter (0) is the mean value of for a heterogeneous enzyme population; is the activation energy divided by absolute temperature and the ideal gas constant. Parameter sigma is the standard deviation of the Gaussian distribution of values in the population. If the population is homogeneous, then = (0) for all enzyme molecules and sigma = 0. There are certain ratios which are independent of (0) and dependent upon sigma. Two important ratios are t(1/4)/t(1/2) and t(1/2)/t(1/2) ('), where t(1/2) (') represents t(1/2) for a homogeneous enzyme population with the same mean ((0)), as the heterogeneous population. If there is experimental deactivation data for the heterogeneous population which is well behaved, the first ratio, t(1/4)/t(1/2), can be determined by estimating the time in minutes at which the enzyme has lost 25% of its activity (t(1/4)), and the time in minutes at which the enzyme has lost 50% of its activity (t(1/2)), and then taking the ratio t(1/4)/t(1/2). The corresponding value of sigma can be estimated from a graph. The ratio t(1/2)/t(1/2) (') can be found directly as a function of t(1/4)/t(1/2), and can be estimated from another graph. The value of (0) can then be calculated from the formulasgiven in the article.     

Kinetics of the modulation of chloroplastic fructose-1,6-bisphosphatase activity by thioredoxin fb

The inactivation of reduced chloroplast fructose-bisphosphatase by oxidized thioredoxin fb has been studied during the enzyme reaction along the principle of Tian and Tsou [Biochemistry (1982) 21, 1028-1032]. A minimum model for this process is presented and its kinetic and equilibrium parameters have been determined. Thioredoxin fb binding to the enzyme is fast relative to catalysis and product desorption. Under quasi-equilibrium conditions oxidized thioredoxin is a non-competitive inhibitor of the enzyme reaction and must bind to a regulatory 'thioredoxin site'. The slow deactivation is thermodynamically favoured, and as expected from binding data, slowed down by the presence of substrate, fructose bisphosphate. The desorption of thioredoxin fb from the enzyme is extremely slow and this small protein may be regarded as a 'regulatory' subunit of fructose-bisphosphatase.     

Effects of chemical modification on enzymatic activities and stabilities

The influence of chemical modification on the initial specific activity, residual activity, and deactivation kinetics of various enzymes is analyzed using a series mechanism. This straightforward multistate sequential model presented is consistent with the enzyme deactivation data obtained from different fields. The enzymes are placed in five different categories depending on the effect of chemical modification on initial specific activity and residual activity or stability. Wherever possible, structure-function relationships are described for the enzymes in the different categories. The categorization provides one avenue that leads to further physical insights into enzyme deactivation processes and into the enzyme structure itself.     

Thermal denaturation: is solid-state fermentation really a good technology for the production of enzymes?

The potential for thermal denaturation to cause enzyme losses during solid-state fermentation processes for the production of enzymes was examined, using the protease of Penicillium fellutanum as a model system. The frequency factor and activation energies for the first-order denaturation of this enzyme were determined as 3.447 x 10(59) h(-1) and 364,070 Jmol(-1), respectively. These values were incorporated into a mathematical model of enzyme deactivation, which was used to investigate the consequences of subjecting this protease to temporal temperature profiles reported in the literature for mid-height in a 34 cm high packed-bed bioreactor of 150 mm diameter. In this literature source, temperature profiles were measured for 5, 15 and 25 liters per minute of air and enzyme activities were measured as a function of time. The enzyme activity profiles predicted by the model were distributed similarly, one relative to the other, as had been found in the experimental study, with substantial amounts of denaturation being predicted when the substrate temperature exceeded 40 degrees C, which occurred at the lower two airflow rates. A mathematical model of a well-mixed bioreactor was used to explore the difficulties that would be faced at large scale. It suggests that even with airflows as high as one volume per volume per minute, up to 85% of the enzyme produced by the microorganism can be denatured by the end of the fermentation. This work highlights the extra care that must be taken in scaling up solid-state fermentation processes for the production of thermolabile products.     

Effect of pH on the production of lipolyzed butter oil by a calf pregastric esterase immobilized in a hollow-fiber reactor: I. uniresponse kinetics

A calf pregastric esterase immobilized in a hollow-fiber reactor was employed to hydrolyze milkfat, thereby producing a lipolyzed butteroil. The reaction kinetics can be modeled by a two-parameter model of the general Michaelis-Menten form based on a ping-pong bi-bi mechanism; the rate of enzyme deactivation can be modeled as a first-order reaction. The initial concentration of accessible glyceride bonds, [G](O), was estimated by complete saponification of the substrate butteroil as 2400 mM. An extra sum of squares test indicated that not only the parameters of the kinetic generalized Michaelis-Menten model, but also the deactivation-rate constant varied significantly with pH. The optimum pH, for lypolysis is near 6.0 at a temperature of 40 degrees C because at this pH the rate of deactivation of the esterase is minimized.     

pH-Dependent enzyme deactivation models

A pH-dependent "series-type" enzyme deactivation model using rapid protonation and deprotonation equilibria and the relatively slower inactivation rates is presented. From the enzyme activity-time trajectories at different pH the models presented permit the evaluation of some of the protonation and inactivation rate constants as well as the specific activities of the different enzyme forms. pH dependence of enzyme deactivations may also exhibit deactivation disguised kinetics. Three different examples of pH-dependent enzyme deactivations available in the literature are appropriately modeled to indicate the general applicability of the model. The model presented is consistent with the data and provides mechanistic insights into the pH-dependent deactivation of different enzymes.     

Kinetics of milk coagulation: I. The kinetics of kappa casein hydrolysis in the presence of enzyme deactivation

The kinetics of the primary phase of the enzymatic coagulation of milk, i.e., kappa-casein hydrolysis, was investigated in the presence and in the absence of concurrent enzyme deactivation processes. For conditions under which the enzyme is stable, the rate of hydrolysis can be described by Michaelis-Menten kinetics, as has been reported by previous investigators. A mathematical model, experimental data, and parameter estimates are provided for kappa-casein hydrolysis in the presence of concurrent deactivation of enzyme. The model accurately describes the experimental results when porcine pepsin was used as the renneting enzyme. The model and the experimental results indicate that samples of milk treated under conditions where deactivation of enzyme is significant can have fractional conversions of kappa-casein ranging from zero to unity and yet contain no active enzyme at the termination of the treatment.     

Integrating multiple signals into cell decisions by networks of protein modification cycles

Posttranslational protein modifications play a key role in regulating cellular processes. We present a general model of reversible protein modification networks and demonstrate that a single protein modified by several enzymes is capable of integrating multiple signals into robust digital decisions by switching between multiple forms that can activate distinct cellular processes. First we consider two competing protein modification cycles and show that in the saturated regime, the protein is concentrated into a single form determined by the enzyme activities. We generalize this to protein modification networks with tree structure controlled by multiple enzymes that can be characterized by their phase diagram, which is a partition of the space of enzyme activities into regions corresponding to different dominant forms. We show that the phase diagram can be obtained analytically from the wiring diagram of the modification network by recursively solving a set of balance equations for the steady-state distributions and then applying a positivity condition to determine the regions corresponding to different responses. We also implement this method in a computer algebra system that automatically generates the phase diagram as a set of inequalities. Based on this theoretical framework, we determine some general properties of protein modification systems.     

Compartmental analysis in photophysics: kinetics and identifiability of models for quenching of fluorescent probes in micelles

The parameters describing the kinetics of excited-state processes can possibly be recovered by analysis of the fluorescence decay surface measured as a function of the experimental variables. The identifiability analysis of a photophysical model assuming errorless time-resolved fluorescence data can verify whether the model parameters can be determined. In this work, we have used the methods of similarity transformation and Taylor series to investigate the identifiability of two models utilized to describe the time-resolved fluorescence quenching of stationary probes in micelles. The first model assumes that exchange of the quencher between micelles is much slower than the fluorescence decay of the unquenched probe (the 'immobile' quencher model). The second model assumes that quenchers exchange between the aqueous and micellar phases (the 'mobile' quencher model). For the 'immobile' quencher model, the rate constants for deactivation (k(0)) and quenching (k(q)) of the excited probe are uniquely identified together with the average number of quencher molecules per micelle. For the 'mobile' quencher model, the rate constants k(0) and k(q) are uniquely identified, as are the rate constants for entry (k(+)) and exit (k(-)) of one quencher molecule into and from a micelle, and the micellar aggregation number. The concomitant rate equations describing the time-resolved fluorescence are solved using z-transforms.     

Categorization of enzyme deactivations using a series-type mechanism

A series-type enzyme deactivation model involving an active enzyme precursor and a final enzyme state with possible non-zero activity is proposed to categorize enzyme deactivation curves. The enzyme activity is a weighted function of the active enzyme states. The deactivation curves may be broadly classified into two major categories wherein the activity is either always less than or it may be more than the initial activity for some time period. Data taken from the literature may be classified into 14 cases. Complex enzyme deactivation curves exhibiting enzyme stabilization and a flex are some of the features that are classified.     

Analysis of heterogeneity in nonspecific PEGylation reactions of biomolecules

The compositional heterogeneity associated with polymer conjugation reactions of biomolecules is analyzed for the particular case of nonspecific PEGylation reactions. It is shown that the distribution of the number of PEG moieties grafted to biomolecules such as proteins is a binomial‐type function of two parameters—the reaction efficiency as well as the number of binding sites per biomolecule. The nature of this distribution implies that uniform compositions are favored for increasing number of coupling sites per biomolecule as well as for increasing efficiency of the modification process. Therefore, the binomial distribution provides a rationale for the pronounced heterogeneity that is observed for PEGylated small enzyme systems even at high coupling efficiencies. For the particular case of PEGylated trypsin it is shown that the heterogeneity results in a broad distribution of deactivation times that is captured by a stretched exponential decay model. The presented analysis is expected to apply to general modification processes of compounds in which partial functionalization of a fixed number of reactive sites is achieved by means of a nonspecific coupling reaction. © 2012 Wiley Periodicals, Inc. Biopolymers 99: 427–435, 2013.     

Single-step unimolecular non-first-order enzyme deactivation kinetics

A two-parameter deactivation model is proposed to describe the kinetics of activity stabilization for some enzymes. The single-step unimolecular mechanism exhibits non-first-order deactivation kinetics since the final enzyme state, E(1) is not completely inactivated. The usefulness of the model is demonstrated by applying it to the inactivation of different enzymes. The influence of the concentration of active ester, ionic strength, and pH on the model parameters is examined during the inactivation of electric eel acetylcholinesterase.(25) In general, inactivators would decrease the level of activity stabilization, alpha(1), and increase the first-order inactivation rate constant, k(1). The effect of protecting agents would be to increase alpha(1) and to decrease k(1).     

Polymerase chain reaction engineering

A mathematical model for polymerase chain reaction (PCR) is developed, taking into account the three steps in this process: melting of DNA; primer annealing; and DNA synthesis (polymerization). Activity and deactivation of the polymerase enzyme as a function of temperature is incorporated in the kinetic model to get a better understanding of the amplification of DNA. Computer simulation of the model is carried out to determine the effects of various parameters, such as the cycle number, initial DNA concentration (copynumber), initial enzyme concentration, extension time, temperature ramp, and enzyme deactivation on the DNA generation. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 55: 359-366, 1997.     

Suicide-peroxide inactivation of horseradish peroxidase in the presence of sodium n-dodecyl sulphate: a study of the enzyme deactivation kinetics

In the presence of the anionic surfactant sodium n-dodecyl sulphate (SDS), horseradish peroxidase (HRP) undergoes a deactivation process. Suicide inactivation of horseradish peroxidase by hydrogen peroxide(3 mM) was monitored by the absorbance change in product formation in the catalytic reaction cycle. The progress curve of the catalytic reaction cycle was obtained at 27degrees C and phosphate buffer 2.5 mM (pH = 7.0). The corresponding kinetic parameters i.e., intact enzyme activity (alpha i); the apparent rate constant of suicide inactivation by peroxide (ki); and the apparent rate constants of enzyme deactivation by surfactant (kd) were evaluated from the obtained kinetic equations. The experimental data are accounted for by the equations used in this investigation. Addition of SDS to the reaction mixture intensified the inactivation process. The deactivation ability of denaturant could be resolved from the observed inactivation effect of the suicide substrate by applying the proposed model. The results indicate that the deactivation and the inactivation processes are independent of each other.     

Enzymatic synthesis of perlauric acid using Novozym 435

For the biocatalytic synthesis of perlauric acid, Novozym 435 (Candida antarctica lipase immobilized on polyacrylic resin) was found to be the most active catalyst and toluene was the most suitable solvent. The effects of various parameters on conversion and rates of reaction were studied in the absence of mass transfer limitations. Reusability studies indicated that there was enzyme deactivation and from a preliminary study of the deactivation, it was observed that the deactivation obeys a pseudo-first-order model. Lineweaver–Burk plots indicated the formation of a ternary complex. A model based on ordered bi–bi mechanism was found to fit the initial rate data very well and using this model, some kinetic parameters were calculated by non-linear regression analysis.     

The effect of water concentration on the activity and stability of CLECs in supercritical CO2 in continuous operation

In this study the effect of the water concentration on a crystallized enzyme of Candida antarctica lipase B (ChiroCLEC™-CAB) in supercritical carbon dioxide (scCO₂) is studied. The model reaction used is the enantioselective esterification of racemic 1-phenyl ethanol with vinyl acetate; the reaction is performed in scCO₂ at 40 °C and 90 bar in batch and in continuous operation. Kinetic parameters have been derived from continuous experiments, leading to a catalytic turnover number of 0.95 s⁻¹. The optimum activity is reached at low water concentrations (0.05 g L⁻¹). At lower concentrations, CO₂ is stripping water from the enzyme leading to deactivation. However, adding a small amount of water to the substrates can reverse this deactivation and the enzyme activity is restored.     

General rate equations and rejection criteria for the rapid equilibrium carrier model of cotransport

It is demonstrated that under fixed activator conditions the general flux equation for the rapid equilibrium carrier model of cotransport can be written entirely in terms of five independent kinetic constants. Thus the kinetic parameters from any experiment carried out under the same activator conditions must necessarily be expressible in terms of these five constants. These predicted relationships between experimental kinetic parameters provide rejection criteria for the model, a number of which are derived here. Generalization of the treatment to the case where a competitive substrate is present on both sides of the membrane is also given.