Mathematical modeling of a single-cell enzyme assay

A quantitative assay of beta-galactosidase activity in single cells of Saccharomyces cerevisiae has been developed using a fluorogenic substrate and flow cytometry [reported in Wittrup & Bailey, Cytometry, 9,394 (1988)]. The beta-galactosidase activity is expressed in yeast from the Escherichia coli lacZ gene under the control of the yeast GAL10 promoter, and is used as a marker for multicopy plasmid content. A nonfluorescent fluorogenic substrate is enzymatically cleaved by intracellular beta-galactosidase to form a fluorescent product. The accumulation of fluorescent product in single cells was found to depend on bulk substrate concentration and single-cell enzyme activity in a fashion that could not be described by a Michaelis-Menten kinetic rate form. It has been demonstrated that diffusion limitation rather than enzyme activity can determine the level of single-cell fluorescence under certain assay conditions, and a mathematical model has; been formulated which accounts for substrate and product diffusion. Guided by the mathematical model, the assay conditions were modified to allow measurement of single-cell enzyme activity rather than diffusion rates.     

A kinetic model for enzyme interfacial activity and stability: pa-hydroxynitrile lyase at the diisopropyl ether/water interface

A kinetic framework is developed to describe enzyme activity and stability in two-phase liquid-liquid systems. In particular, the model is applied to the enzymatic production of benzaldehyde from mandelonitrile by Prunus amygdalus hydroxynitrile lyase (pa-Hnl) adsorbed at the diisopropyl ether (DIPE)/aqueous buffer interface (pH = 5.5). We quantitatively describe our previously obtained experimental kinetic results (Hickel et al., 1999; 2001), and we successfully account for the aqueous-phase enzyme concentration dependence of product formation rates and the observed reaction rates at early times. Multilayer growth explains the early time reversibility of enzyme adsorption at the DIPE/buffer interface observed by both enzyme-activity and dynamic-interfacial-tension washout experiments that replace the aqueous enzyme solution with a buffer solution. The postulated explanation for the unusual stability of pa-Hnl adsorbed at the DIPE/buffer interface is attributed to a two-layer adsorption mechanism. In the first layer, slow conformational change from the native state leads to irreversible attachment and partial loss of catalytic activity. In the second layer, pa-Hnl is reversibly adsorbed without loss in catalytic activity. The measured catalytic activity is the combined effect of the deactivation kinetics of the first layer and of the adsorption kinetics of each layer. For the specific case of pa-Hnl adsorbed at the DIPE/buffer interface, this combined effect is nearly constant for several hours resulting in no apparent loss of catalytic activity. Our proposed kinetic model can be extended to other interfacially active enzymes and other organic solvents. Finally, we indicate how interfacial-tension lag times provide a powerful tool for rational solvent selection and enzyme engineering.     

Microscopic diffusion-reaction coupling in steady-state enzyme kinetics

The theory of diffusion-controlled processes is applied to describe the steady state of a reversible enzymatic reaction with special emphasis on the effects of enzyme saturation. A standard macroscopic steady-state treatment requires only that the average diffusional influx of substrate equals the net reaction flux as well as the average diffusional efflux of product. In contrast, the microscopic diffusion-reaction coupling used here takes properly into account the conditional concentration distributions of substrate and product: Only when the enzyme is unoccupied will there be a diffusional association flux; when the enzyme is occupied, the concentration distributions will relax towards their homogeneous bulk values. In this way the relaxation effects of the non-steady state will be constantly reoccurring as the enzyme shifts between unoccupied and occupied states. Thus, one is forced to describe the steady state as the weighted sum of properly time-averaged non-stationary conditional distributions. The consequences of the theory for an appropriate assessment of the parameters obtained in Lineweaver-Burk plots are discussed. In general, our results serve to justify the simpler macroscopic coupling scheme. However, considerable deviations between the standard treatment and our analysis can occur for fast enzymes with an essentially irreversible product release.     

The electrode adsorption method for determination of enzyme activity: a study of substrate requirements

The electrode adsorption method for the determination of enzyme activity requires substrates that, besides having good kinetics constants for the enzyme, also show good adsorption/desorption kinetics to the electrode surface and adsorb in such a way that they change the double-layer capacitance of the electrode. A series of peptide substrates containing one to three aromatic groups has been synthesized. Our results show that the aromatic groups are of crucial importance for the capacitance change caused by the adsorbing/desorbing substrate. Thus, the tripeptide substrate, Bz-Phe(NO2)-Val-Arg-pNA, with three aromatic groups is superior to the other synthesized substrates containing only one or two aromatic groups. Our desorption experiments show that several factors determine the rate of capacitance increase observed when thrombin is added to a substrate solution in equilibrium with a substrate-covered electrode. The kinetic constants of the substrate determine how the substrate concentration in the solution decreases and, consequently, determine the spontaneous desorption measured as capacitance increase. Thrombin does not seem to split adsorbed substrate molecules but it adsorbs to the substrate-covered surface and in that way causes a capacitance decrease counteracting the change caused by desorption of substrate.     

Enzyme recovery in high-solids enzymatic hydrolysis of steam-pretreated willow: Requirements for the enzyme composition

In order to reduce the total enzyme consumption in high-solids static hydrolysis of nonwashed steam-exploded willowSalix caprea by mixed cellulase ofTrichoderma reesei + Aspergillus foetidus, two different approaches were proposed. In the first case, the enzyme activity adsorbed on residual solids after extended hydrolysis was used for hydrolysis of the newly added substrate. The initial mixing of fresh and hydrolyzed substrates was sufficient for the adsorbed enzyme redistribution and conversion of the new substrate portion, and constant mechanical stirring was not required. Feeding of two additional portions of the exploded hardwood adjusted to pH 4 with dry caustic into the reactor with simultaneous replacement of accumulated sugars with fresh buffer (pH 4.5) resulted, on average, in a 90% conversion of cellulose at the final enzyme loading of 8 IFPU per g ODM substrate, an average sugar concentration of 12%, and a glucose/xylose ratio of 5 : 1. In the second approach, weakly adsorbed cellulase fractions were used for static high-solids hydrolysis followed by their ultrafiltration recovery from the resultant sugar syrup. In contrast to the initial cellulase mixture whose residual activity in a syrup did not exceed 5–10% at the end of hydrolysis (48 h), up to 60% of weakly adsorbed enzyme fraction could be separated from sugar syrups by ultrafiltration and then reused. Weakly adsorbed enzymes displayed a hydrolysis efficiency of not less than 80% per IFPU enzyme consumed in extended hydrolysis of pretreated willow as compared to the original enzyme mixture. An electrophoretic study of the weakly adsorbed enzyme fraction identifiedT. reesei cellobiohydrolase II as the predominant component, whereas clear domination ofT. reesei cellobiohydrolase I was found by electrophoresis of proteins tightly bound to residual hydrolysis solids. Deceased     

A general solution for the steady-state kinetics of immobilized enzyme systems

A power series solution is presented which describes the steady-state concentration profiles for substrate and product molecules in immobilized enzyme systems. Diffusional effects and product inhibition are incorporated into this model. The kinetic consequences of diffusion limitation and product inhibition for immobilized enzymes are discussed and are compared to kinetic behavior characteristic of other types of effects, such as substrate inhibition and substrate activation.     

Enzyme-substrate interaction in lipid monolayers. II. Binding and activity of lipase in relation to enzyme and substrate concentration and to other factors.

With the limited stirring procedure used in the present work, substrate and enzyme together form a segregated and well-defined system on the surface. The lipase molecules responsible for the lipolysis are only those that are adsorbed on the glyceride monolayer. After a study of the stirring procedure, two series of systematic experiments were done: a) the bulk concentration of the enzyme was varied with different constant surface concentrations of the substrate, and b) the surface concentration of the substrate was varied with different constant bulk concentrations of the enzyme. The influence of the surface concentration of substrate on a) the rate of lipolysis, V,; b) the enzyme activity, a,; and c) the enzyme adsorption, Ze, were each determined by a different procedure. The values obtained verify the enzymic activity equation (a = V/Ze). The roles of other factors (Ca2+ ions, and pH) which govern the adsorption of the enzyme and its specific activity were also studied in preliminary experiments.     

Constant-ratio alternative substrate approach for differentiation of enzyme mechanisms

A new kinetic approach using alternative substrates as a tool for studying enzyme mechanisms is described. In this method the substrate to alternative substrate ratio is maintained constant and the common product (or summation of product analogs) is measured. The double-reciprocal plots so obtained at several constant ratios generate different patterns for various mechanisms, thus permitting a choice of kinetic model. In some cases, secondary intercept plots are utilized as a diagnostic aid. Another feature of this approach is that most of the resultant plots are linear. The graphical patterns for four cases of two-substrate, two-product reactions are presented as examples. These patterns allow one to differentiate several mechanisms which are not distinguishable by conventional alternative substrate, competitive inhibitor, or product inhibition studies alone. When used in combination with other methods, various mechanisms involving isomerization and abortive complex formation can be differentiated even if only one alternative substrate is available.     

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.     

Measurement of key metabolic enzyme activities in mammalian cells using rapid and sensitive microplate‐based assays

Sensitive microplate‐based assays to determine low levels of key enzyme activities in mammalian cells are presented. The enzyme platform consists of four cycling assays to measure the activity of 28 enzymes involved in central carbon and glutamine metabolism. The sensitivity limit of all cycling assays was between 0.025 and 0.4 nmol product. For the detection of glutaminase activity, a new glutamate cycle system involving the enzymes glutamate dehydrogenase and aspartate transaminase was established. The relative standard deviation of the method was found to be 1.7% with a limit of detection of 8.2 pmol and a limit of quantitation of 24.8 pmol. Hence, cell extracts could be highly diluted to reduce interferences caused by other components in the extract, which in addition minimized underestimates or overestimates of actual enzyme activities. Since substrate concentrations could be maintained at a nearly constant level throughout the assay product accumulation during the reaction was low, which minimized product inhibition. As an example, the enzyme platform was used to investigate maximum enzyme activities of stationary‐phase MDCK cells grown in serum‐containing GMEM medium as typically used in influenza vaccine production. Biotechnol. Bioeng. 2010;107: 566–581. © 2010 Wiley Periodicals, Inc.     

About and beyond the Henri-Michaelis-Menten rate equation for single-substrate enzyme kinetics

For more than a century the simple single-substrate enzyme kinetics model and related Henri-Michaelis-Menten (HMM) rate equation have been thoroughly explored in various directions. In the present paper we are concerned with a possible generalization of this rate equation recently proposed by F. Kargi (BBRC 382 (2009) 157-159), which is assumed to be valid both in the case that the total substrate or enzyme is in excess and the quasi-steady-state is achieved. We demonstrate that this generalization is grossly inadequate and propose another generalization based on application of the quasi-steady-state condition and conservation equations for both enzyme and substrate. The standard HMM equation is derived by (a) assuming the quasi-steady-state condition, (b) applying the conservation equation only for the enzyme, and (c) assuming that the substrate concentration at quasi-steady-state can be approximated by the total substrate concentration [S](0). In our formula the rate is already expressed through [S](0), and we only assume that when quasi-steady-state is achieved the amount of product formed is negligible compared to [S](0). Numerical simulations show that our formula is generally more accurate than the HMM formula and also can provide a good approximation when the enzyme is in excess, which is not the case for the HMM formula. We show that the HMM formula can be derived from our expression by further assuming that the total enzyme concentration is negligible compared to [S](0).     

Frontal affinity chromatography coupled to mass spectrometry for screening mixtures of enzyme inhibitors.

Frontal affinity chromatography coupled online to mass spectrometry (FAC/MS) has previously been used to estimate binding constants for individual protein ligands present in mixtures of compounds. In this study FAC/MS is used to determine enzyme substrate kinetic parameters and binding constants for enzyme inhibitors. Recombinant human N-acetylglucosaminyltransferase V was biotinylated and adsorbed onto immobilized streptavidin in a microcolumn (20 microL). The enzyme was shown to be catalytically competent transferring GlcNAc from the donor UDP-GlcNAc to beta-d-GlcpNAc-(1-->2)-alpha-d-Manp-(1-->6)-beta-d-Glcp-OR acceptor giving beta-d-GlcpNAc-(1-->2)-[beta-d-GlcpNAc-(1-->6)]-alpha-d-Manp-(1-->6)-beta-d-Glcp-OR as the reaction product. The kinetic parameters K(m) and V(max) for the immobilized enzyme could be determined by FAC/MS and were comparable to those measured in solution. Analysis of a mixture of eight trisaccharide analogs in a single run yielded K(d) values for each of the eight compounds ranging from 0.3 to 36 microM. These K(d) values were 2 to 10 times lower than the inhibition constants, K(I)'s, determined in solution using a standard radiochemical assay. However, the ranking order of K(d)'s was the same as the ranking of K(I) values. FAC/MS assays can therefore be employed for the rapid estimation of inhibitor K(d) values making it a valuable tool for enzyme inhibitor evaluations.     

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.     

Enzyme--substrate interaction in lipid monolayers. III. A study of the variation of the surface concentration with lipolysis.

Monolayers of a diacylglycerol were submitted to the action of lipase, keeping the area constant. The variation of lipase, keeping the area constant. The variation of the surface concentration gamma of the substrate with time was derived from the recorded reduction of the surface pressure pi (the isotherm of the monolayer being previously established). The rate -d gamma/dt was determined both as a function of the surface concentration gamma of the substrate and as a function of the bulk concentration C of the enzyme in the underlying solution. The rate depends on the quantity of enzyme ze adsorbed on the monolayer and on the enzymatic specific activity alpha of these adsorbed enzyme molecules. Both ze and alpha vary with gamma. The two variations have been quantitatively dissociated. The curves of ze and of alpha as functions of gamma coincide with those previously established in the study of hydrolysis under constant surface pressure.     

Enzyme kinetic assays with surface plasmon resonance (BIAcore) based on competition between enzyme and creatinine antibody

A procedure is described which allows the characterization of enzyme by a hybrid approach using an enzyme and an antibody. The presented method is related to the affinity determination of antibodies by the 'affinity in solution' procedure for BlAcore. The antibody is used as an indicator for the concentration of substrate, which is also the antigen. A mixture of enzyme, substrate and antibody is incubated, and an aliquot of this solution is injected periodically into a flowcell containing immobilized substrate, which is bound by the antibody, but not cleaved by the enzyme. The chosen initial concentration of substrate inhibits the binding of antibody to the immobilized substrate by 90%. During the enzymatic reaction, increased amounts of antibody bind to the surface, as the substrate concentration is decreased. With this method, the cleavage of creatinine with creatinine iminohydrolase (6 mU/ml) was monitored for up to 11 h. A recently developed monoclonal antibody against creatinine was used as the indicating protein. For the calculation of enzyme activity, the signals were compared with a calibration curve for inhibition of antibody binding to the chip by creatinine in solution.     

Kinetic properties of enzyme populations in vivo: alkaline phosphatase of the Escherichia coli periplasm.

Studies were conducted to determine the role that diffusion may play in the in vivo kinetics of the Escherichia coli periplasmic enzyme, alkaline phosphatase (AP, encoded by the gene pho A). Passive diffusion of solutes, from solution into the periplasm, is thought to occur mainly through porins in the outer membrane. The outer membrane therefore serves as a diffusion barrier separating a population of periplasmic enzymes from bulk substrate. E. coli strains containing a plasmid with the pho A gene linked to the lac promoter were used in this study in order to vary the amount of enzyme per cell. Alkaline phosphatase assays were conducted with intact cells, and the substrate concentration at half-maximum velocity (normally the Km for the enzyme) was determined as a function of enzyme concentration per cell. The results showed that diffusion of substrate to the enzyme caused as much as a 1000-fold change in this parameter, compared to that of purified enzyme. This suggested that diffusion was the rate-limiting step of the enzymatic reaction in these cells. In agreement with this type of reaction, Eadie-Hofstee and Lineweaver-Burk plots were not linear. At their extremes, these plots represented two types of kinetics. At high substrate concentration, equilibrium of substrate between bulk solution and the periplasm was achieved, and the kinetic properties conformed to Michaelis-Menten. At low substrate concentrations, there were a large number of free (unbound) enzymes, and each substrate molecule that entered the periplasm, through the diffusion barrier, resulted in product formation.(ABSTRACT TRUNCATED AT 250 WORDS)     

Kinetics of wheat germ lipoxygenase adsorbed to hydrophobic surfaces

Lipoxygenase binds to Teflon or epoxy coated surfaces, presumably through a hydrophobic interaction. The kinetics of bound enzyme differ from the kinetics of free enzyme both in the effect of substrate concentration on velocity and in the dependence of the induction time on substrate concentration. The binding site for the product hydroperoxide appears to be masked when the enzyme is bound to hydrophobic surfaces. In vivo kinetic behavior of the enzyme may be more closely approximated by that of adsorbed enzyme than by free enzyme.     

High-performance liquid chromatography determination of enzyme activities in the presence of small amounts of product

Enzymes can be assayed by HPLC by calculating the amount of substrate(s) left over, or product formed, through the peak area ratios with a suitable internal standard. However, sometimes the substrates used are contaminated with small amounts of products and this can lead to errors in the determination of the enzyme activity. A method for a HPLC test of such enzymes, which prevents eventual errors, uses the ratio substrate/product at time zero as internal standard and the kinetics can be followed with the aid of a simple mathematical equation. This approach was applied to the determination of the activities of papain, urokinase, NAD glycohydrolase, and pyruvate kinase samples and it was compared with the data obtained by the internal standard method, giving reproducible results in all cases.     

Flow analysis for determination of paraoxon with use of immobilized acetylcholinesterase reactor and new type of chemiluminescent reaction

A highly sensitive flow analysis method for determination of acetylcholinesterase (AChE) inhibitors like organophosphorous pesticides using a new chemiluminescent reaction was developed and optimized. This method is fast, sensitive, and cheap, because it requires only one enzyme and its substrate. The system incorporates a reactor with immobilized AChE on controlled pore glass (CPG) and a chemiluminometric detector. Variations in enzyme activity due to inhibition are measured from the changes of concentrations of thiocholine produced when the substrate (acetylthiocholine chloride) is pumped before and after the passage of the solution containing the pesticide through the immobilized AChE reactor. Thiocholine is determined by a new chemiluminescent reaction with luminol in the presence of potassium ferricyanide. The percentage inhibition of enzyme activity is correlated to the pesticide concentration. The inhibited enzyme is reactivated by 10 mM pyridine-2-aldoxime methiodide (2-PAM). The experimental conditions were first optimized for activity determination of the effect of pH, flow rates, and Tris concentrations. For the measurement of AChE inhibition, the appropriate concentration of the substrate is selected such that the rate of noninhibited reaction can be considered unchanged and could be used as a reference. For optimization of experimental conditions for inhibition, several parameters of the system are studied and discussed: flow rate, enzyme-pesticide contact time, luminol concentration, ferricyanide concentration, 2-PAM concentration, and configuration of the FIA manifold. Paraoxon, an organophosphorous pesticide was tested. For an inhibition time of 10 min the calibration graph is linear from 0.1 to 1 ppm paraoxon with a relative standard deviation (n = 5) of 4.6% at 0.5 ppm. For an inhibition time of 30 min the calibration graph is linear from 25 to 250 ppb paraoxon.