Formulation of a Universal First-Order Rate Constant for Enzymatic Reactions

It is a common practice to employ k _cat[E]₀/K _m as a first-order rate constant for the analysis of an enzymatic reaction, where [E]₀ is the total enzyme concentration. I describe in this report a serious shortcoming in analyzing enzymatic reactions when k _cat[E]₀/K _m is employed and show that k _cat[E]₀/K _m can only be applied under very limited conditions. I consequently propose the use of a more universal first-order rate constant, k _cat[ES]_K/[S]₀, where [ES]_K is the initial equilibrium concentration of the ES-complex derived from [E]₀, [S]₀ and K _m. Employing k _cat[ES]_K/[S]₀ as the first-order rate constant enables all enzymatic reactions to be reasonably simulated under a wide range of conditions, and the catalytic and binding contributions to the rate constant of any enzyme can be determined under any and all conditions.     

Heavy Metal Ions Inhibition of Jack Bean Urease: Potential for Rapid Contaminant Probing

The kinetics of heavy metal ions inhibition of jack bean urease was studied by progress curve analysis in a reaction system without enzyme-inhibitor preincubation. The inhibition was found to be biphasic with an initial, small inhibitory phase changing over the time course of 5–10?min into a final linear steady state with a lower velocity. This time-dependent pattern was best described by mechanism B of slow-binding inhibition, involving the rapid formation of an EI complex that subsequently undergoes slow conversion to a more stable EI* complex. The kinetic parameters of the process, the inhibition constants Ki and K_i^* and the forward k5 and reverse k6 rate constants for the conversion, were evaluated from the reaction progress curves by nonlinear regression treatment. Based on the values of the overall inhibition constant K_i^*, the heavy metal ions were found to inhibit urease in the following decreasing order: Hg2+ >?Cu2+ >?Zn2+ >?Cd2+ >?Ni2+ >?Pb2+ >?Co2+ >?Fe3+ >?As3+. With the K_i^* values as low as 1.9?nM for Hg2+ and 7.1?nM for Cu2+, 100–1000 times lower than those of the other ions, urease may be utilized as a bioindicator of the trace levels of these ions in environmental monitoring, bioprocess control or pharmaceutical analysis.     

Product formation during batch fermentation with recombinant Escherichia coli containing a runaway plasmid

The product formation during batch fermentation with recombinant E. coli containing a runaway replication plasmid has been examined. Theoretical modelling is combined with experimental work to study the effect of operating conditions. In particular the influence of induction profile has been investigated. High sensitivity to operating conditions is observed, and both model and experimental data illustrate the presence of very narrow limits for an optimal induction profile.List of Symbols f _i function for allocation of energy to the i'th reaction in the one substrate model - g _i function for allocation of energy to the i'th reaction in the two substrate model - h function for inhibition by plasmid material - K _i (h^–1) kinetic rate constant for the i'th reaction - k _i (g/l) saturation constants - K _p (g P/g biomass) saturation constant for recombinant protein synthesis - K _s (g/l) inhibition constant of glucose on acetate metabolism - K _p,i (g P/g biomass) inhibition constant of plasmid material on cellular activity - p (g/l) extracellular acetic acid concentration - r _i (h^–1) specific rate of i'th reaction - s (g/l) extracellular glucose concentration - X _i (g i/g biomass) intracellular concentration of the i'th component - _ij stoichiometric coefficients for the i'th metabolic product in the j'th reaction - _ij stochiometric coefficients for the i'th component in the biotic phase in the j'th reaction - _i relative allocation of energy to the i'th reaction with growth on acetate compared with growth on glucose     

Multiple intermediates in the reaction of bovine beta-trypsin with bovine pancreatic trypsin inhibitor (Kunitz)

The kinetics of the formation of the complex between bovine β-trypsin and the bovine basic pancreatic trypsin inhibitor (BPTI) was investigated using three different signals: the displacement of proflavine, the optical density changes in the UV region, and the loss of the enzymatic activity. For the three different signals, with inhibitor in excess over bovine β-trypsin ([BPTI] ≥ 5 × [bovine β-trypsin]), the time course of the reaction corresponds to a pseudo-first-order process. The concentration dependence of the rate is second order at low BPTI concentrations and tends to first order at high inhibitor concentrations. This behavior may be explained by relatively rapid preequilibria followed by limiting first-order processes according to The values of K_i, k_+i, and k_(on)i ( = k_+i/K_i) have been determined for the different reactions at three pH values: 6.80, 4.80, and 3.50. The kinetic parameters differ widely for the processes reflected by the various signals; the difference increases upon lowering pH. The results indicate that the formation of the bovine β-trypsin–BPTI complex is not an all-or-nothing process, but involves several intermediates corresponding to discrete reaction steps, which are differently affected by ionization processes.     

Purification, characterization and inhibition of sterol C24-methyltransferase from Candida albicans

Solubilized sterol C24-methyltransferase (24-SMT) was purified to homogeneity from a cell extract of the yeast Candida albicans (Ca) by anion exchange chromatography, gel permeation chromatography and fast performance liquid chromatography using a Mono Q column. The purified enzyme has an apparent molecular mass of 178 kDa on gel permeation chromatography and 43 kDa on SDS/PAGE, indicating that it is composed of four identical subunits. The substrate requirement of the native enzyme has an optimal specificity for zymosterol with associated kinetic constants of K_m 50 μM and k_cat of 0.01 s⁻¹. The product of the enzyme incubated with zymosterol was fecosterol. Inhibition of the catalyst was observed with substrate analogs designed as transition state analogs (25-azalanosterol, K_i = 54 nM and 24 (R,S),25-epiminolanosterol, K_i = 11 nM) or as mechanism-based inactivators (26,27-dehydrozymosterol, K_i 9 μM) and k_inact = 0.03 min⁻¹) of the C24-methylation reaction. Product analogs ergosterol and fecosterol, but neither cholesterol nor sitosterol, inhibited activity affording K_i values of 20 and 72 μM, respectively. Ammonium and thia analogs of the intermediates of the sterol C24-methyl reaction sequence were effective growth inhibitors exhibiting IC₅₀ values that ranged from 3 to 20 μM.     

Kinetic Analysis of the Noncompetitive Inhibition of the Lignin-Peroxidase-catalyzed Reaction by Oxalic Acid

Oxalic acid was found to inhibit noncompetitively the Cα-Cβ bond cleavage of veratrylglycerol catalyzed by a lignin peroxidase (LiP) isozyme of the white-rot fungus P. chrysosporium. With greater amounts of oxalic acid in the LiP system, the substrate was not converted to veratraldehyde but was almost all recovered. Oxalic acid was shown to be decomposed to CO₂ during the enzymatic reaction. The results clearly indicate that oxalic acid reduced the cation radical intermediate formed in the reaction back to the substrate to block the production of veratraldehyde. A novel equation has been derived to explain the mechanism for this unique non-competitive inhibition that is different from the classical noncompetitive one. The inhibition constant K_i obtained here, which is different from the classical inhibition constant K_i, is defined as the ratio of the rate constant (k_p) for product formation to the rate constant (k_i) for the reduction of the cation radical to the substrate.     

Updated model of the batch acetone-butanol fermentation

Summary The recent models of the Acetone-Butanol fermentation did not adequately describe the culture inhibition by the accumulating metabolites and were unable to simulate the acidogenic culture dynamics at elevated pH levels. The present updated modification of the model features a generalised inhibition term and a pH dependent terms for intracellular conversion of undissociated acids into solvent products. The culture dynamics predictions by the developed model compared well with experimental results from an unconventional acidogenic fermentation ofC. acetobutylicum.Nomenclature A acetone concentration in the fermentation broth, [g/L] - AA total concentration of dissociated and undissociated acetic acid, [g/L] - AA _undiss concentration of undissociated acetic acid, [g/L] - APS Absolute Parameter Sensitivity - AT acetoin concentration in the fermentation broth, [g/L] - B butanol concentration in the fermentation broth, [g/L] - BA total concentration of dissociated and undissociated butyric acid, [g/L] - BA _undiss concentration of undissociated butyric acid, [g/L] - E ethanol concentration in the fermentation broth, [g/L] - f(T) inhibition function as defined in Equation (2) - k ₁ constant in Equation (4), [g substrate/g biomass] - k ₂ constant in Equation (4), [g substrate/(g biomass.h)] - k ₁ constant in Equation (5), [g substrate/(g biomass] - k ₂ constant in Equation (5), [g substrate/(g biomass.h)] - k ₃ constant in Equation (6), [g butyric acid/g substrate] - k ₄ constant in Equation (6), [g butyric acid/(g biomass.h)] - k ₅ constant in Equation (7), [g butanol/g substrate] - k ₆ constant in Equation (8), [g acetic acid/g substrate] - k ₇ constant in Equation (8), [g acetic acid/(g biomass.h)] - k ₈ constant in Equation (9), [g acetone/g substrate] - k ₉ constant in Equation (10), [g ethanol/g substrate] - k ₁₀ constant in Equation (11), [g acetoin/g substrate] - k ₁₁ constant in Equation (12), [g lactic acid/g substrate] - K _I Inhibition constant, [g inhibitory products/L] - ke maintenance energy requirement for the cell, [g substrate/(g biomass.h)] - K _AA acetic acid saturation constant, [g acetic acid/L] - K _BA butyric acid saturation constant, [g butyric acid/L] - K _S Monod's saturation constant, [g substrate/L] - LA lactic acid concentration in the fermentation broth, [g/L] - m _i ,n _i constants in Equation (14) - n empirical constant, dependent on degree of inhibition. - P concentration of inhibitory products (B+BA+AA), [g/L] - P _max maximum value of product concentration to inhibit the fermentation, [g/L] - pK_a equilibrium constant - r _A rate of acetone production, [g acetone/L.h] - r _AA rate of acetic acid production, [g acetic acid/L.h] - r _AT rate of acetoin production, [g acetoin/L.h] - r _B rate of butanol production, [g butanol/L.h] - r _BA rate of butyric acid production, [g butyric acid/L.h] - r _E rate of ethanol production, [g ethanol/L.h] - RPS Relative Parameter Sensitivity - r _LA rate of lactic acid production, [g lactic acid/L.h] - r _S dS/dt=total substrate consumption rate, [g substrate/L.h] - r _S substrate utilization rate, [g substrate/L.h] - S substrate concentration in the fermentation broth, [g substrate/L] - S ₀ initial substrate concentration, [substrate/L] - t time, [h] - X biomass concentration, [g/L] - Y _X yield of biomass with respect to substrate, [g biomass/g substrate] - Y _{P i} yield of metabolic product with respect to substrate, [g product/g substrate] Derivatives dX/dt rate of biomass production, [g biomass/L.h] - dP _i /dt rate of product formation, [g product/L.h] Greek letters specific growth rate of the culture, [h^–1] - I specific growth rate of the culture in the presence of the inhibitory products, [h^–1] - µ_max maximum specific growth rate of the culture, [h^–1]     

A Model for the Unusual Kinetics of Thermal Denaturation of Rubredoxin

The thermal denaturation of the simple, redox-active iron protein rubredoxin is characterized by a slow, irreversible decay of the characteristic red color of the iron center at elevated temperatures in the presence of oxygen at pH 7.8. The denaturation rate is essentially constant and the time period for complete bleaching is nearly independent of protein concentration. These two characteristics of the kinetics can be fit by a simple self-catalyzed kinetics model consisting of the combination of a first-order decay and catalysis by some product of that decay, i.e., dP/dt=k ₁[A]+(k ₂[P][A])/(K _m+[A]), where A is native rubredoxin, P, is unspecified product, k ₁ is a first-order rate constant, and k ₂ and K _m are the catalytic constants. In order for the second term to be of this simple form over the full course of a decay, the model must include the condition that the reaction is effectively irreversible. This model has properties which suggest other biological roles in regulation (changes in k ₁ or k ₂ can dramatically modulate the kinetics), in timing (titer-independent fixed reaction time), and in self-activation reactions. At one extreme (k₁ k₂) the kinetics becomes exponential, but at the other extreme (k₂ k₁) they show a dramatic and rapid terminal increase after a lag period. Some obvious possible roles in the kinetics of programmed cell death, prion disease, and protease autoactivation are discussed.     

Flow injection chemiluminescence study of acridinium ester stability and kinetics of decomposition

Decomposition of phenyl acridinium-9-carboxylate is monitored using electrogenerated chemiluminescence in a flow system. The formation of the pseudobase from the acridinium ester [AE] is described by rate = k′₁[AE] + k″₁[AE][OH^?]^0.5, where k′₁ = 0.020 ± 0.006 s^?1 and k″₁ = 2.1 ± 0.8 (L/mol)^?0.5 s^?1. Irreversible decomposition of the pseudobase is described by rate = k′₂[AE][OH^?], where k′₂ = 20.1 ± 3.8 (L/mol s). These kinetic equations, plus measurement of variation in emission intensity for constant acridinium ester concentration, are used to predict the resulting emission intensity v. pH behaviour given various contact times (in the 0.25 to 25 s range) for the acridinium ester to be in an alkaline solution prior to initiation of the chemiluminescence reaction.     

Mechanism and role of furosemide-sensitive K+ transport in L cells: A genetic approach

Summary As part of a genetic study of the mechanisms for cation transport in cultured mammalian cells, two mouse fibroblastic cell lines have been compared with respect to unidirectional⁴²K⁺ influx. The cell lines areLM(TK ^–) andLTK-5, a mutant selected fromLM(TK ^–) by the ability to grow in medium containing 0.2mm K⁺. In both cell lines, the overall influx can be resolved into three components: (i) a ouabain- and vanadate-sensitive component ( ⁱ M_K ^f), presumably the Na/K pump, which is a saturable function of extracellular K⁺ with aK _1/2 of 1.3mm; (ii) a furosemide-sensitive component ( ⁱ M_k ^fx), also a saturable function of extracellular K⁺, with aK _1/2 of 6mm; and (iii) a diffusional component ( ⁱ M_k ^d); which is a linear function of extracellular K⁺.By several independent criteria, ⁱ M_k ^o and ⁱ M_k ^f appear to be distinct transport processes. First, as indicated above, they can be separated with the use of inhibitors. In addition, they can be separated genetically, since theLTK-5 mutant shows a threefold elevation in ⁱ M_k ^f with no change in ⁱ M_k ^o. And finally, extracellular Na⁺ has no effect on ⁱ M_k ^o, but stimulates ⁱ M_k ^f, a result consistent with the notion that ⁱ M_k ^f influx occurs by Na–K cotransport.Further experiments were directed towards understanding the nature of theLTK-5 mutation and the physiological role of ⁱ M_k ^f. LTK-5 differs from the parental cell line, not only in having an increased ⁱ M_k ^f, but also in having a large cell volume, a slow maximal growth rate, and an ability to grow at 0.2mm K⁺. The most straightforward interpretation — that the increased ⁱ M_k ^f is itself responsible—is unlikely since the addition of furosemide to the growth medium had no effect upon the growth rate or cell volume of the mutant at either normal or low extracellular K⁺ concentrations. It did, however, render the parent capable of growth at 0.2mm K⁺. Possible interpretations are discussed.     

Reactions of the amino groups of RNAse A: IV. The effects of protein concentration

A study has been made of the effect of ribonuclease (RNAse) concentration on the properties of the amino groups. The biphasic dependence of pK on pH which has been established (Goldfarb and Martin, Bioorg. Chem.5, 137 (1976)). for 5 μM solution of RNAse also have been shown to occur for 50 μM solutions. In the lower pH range (7.5–8.5) the values of pK obtained with 50 μM solutions were similar to those obtained with 5 μM solutions (pK = 7.5) but the intrinsic constants were smaller. In the higher pH range (8.5–10) the pKs in the more concentrated solutions were larger than those found at the smaller concentration and the intrinsic constants were generally smaller. A quantitative study of the concentration vs k_i relation at pH 7.5 indicated a sigmoid relationship for all of the subsets with a constant maximum value equal to, and less than that at 5 μM RNAse and a constant minimum value above that at 20 μM. Parallel studies with oxidized RNAse gave parallel, although not identical, results from which it is proposed that the concentration effect does not arise totally from the three-dimensional structure of native RNAse.     

Analysis of valine uptake by Commelina mesophyll cells in a biphasic active and a diffusional component

Valine uptake by isolated Commelina benghalensis L. mesophyll cells was measured over a wide concentration range (10^-6–4·10^-2 mol l^-1). The uptake data were subjected to iterative fitting. Experiments with carbonyl cyanide mchlorophenyl hydrazone (CCCP), diethylstilbestrol (DES), and p-chloromercuriphenylsulphonic acid (PCMBS) provided evidence that the biphasic uptake kinetics of valine consists of a diffusional component and a biphasic active uptake. The data from the control experiments, were also best fitted to one diffusional component and two Michaelis-Menten systems. The presence of two carrier systems in the plasmalemma, however, was considered to be virtual for the following reasons: (1) Both phases of active uptake were equally decreased by high concentrations of K⁺-ions. (2) Fusicoccin stimulated the active uptake in both phases to the same extent. (3) Inhibitors of the proton-driven uptake (CCCP, DES, PCMBS) similarly inhibited the active uptake at all concentrations. (4) The active uptake equally responded in both phases to changes in the pH. (5) Light also promoted the active uptake over the whole concentration range. These results strongly indicate that, despite its biphasic character, the active uptake is due to one proton-driven carrier system.Abbreviations CCCP carbonyl cyanide m-chlorophenyl hydrazone - DES diethylstilbestrol - FC fusicoccin - MES 2-(N-morpholino)ethanesulphonic acid monohydrate - PCMBS p-chloromercuriphenylsulphonic acid - v uptake velocity - S substrate concentration - K _m1 and K _m2 Michaelis constants of the apparent high- and low-affinity system, respectively - V _m1 and V _m2 maximal uptake velocities of the apparent high- and low-affinity system - k linear uptake constant     

Purification and characterization of a low molecular weight cysteine proteinase inhibitor from bovine muscle

A low-M_r tight binding proteinase inhibitor was purified from bovine muscle by alkaline denaturation of cysteine proteinases, gel filtration on Sexphadex G-75 and affinity chromatography on carboxymethyl-papain-Sepharose. Chromatofocusing separated three isoforms which are similar in their M_r of about 14 000, their stability with heating at 80°C and their inhibitory activity towards cathepsin H, cathepsin B and papain. The equilibrium constants (K_i) were determined for these three cysteine proteinases but for cathepsin H, association (k_ass) and dissociation (k_diss) rate constants were also evaluated. K_i values of 56 nM and 8.4 nM were found for cathepsin B and cathepsin H, respectively. For papain, K_i was in the range of 0.1–1 nM. The kinetic features of enzyme-inhibitor binding suggest a possible role for this low-M_r protein inhibitor in controlling ‘in vivo’ cathepsin H proteolytic activity. With regard to cathepsin B, such a physiological role was less evident.     

Sensitivity analysis of fed-batch fermentation for the production of penicillin G

Summary The mixed metabolism model of Nicolaiet al. (1991) has been used to analyse the sensitivity of fed-batch fermentation for penicillin G with respect to four parameters. The cases of fully endogenous, fully maintenance and mixed metabolisms have been studied. Sensitivities with respect to two parameters,K _x andK _i, were much larger than for the other two. Differences between the three kinds of metabolism were sharpest forK _p. The implications of both observations are discussed.     

Kinetic behavior of penicillin acylase immobilized on acrylic carrier

The usefulness of Lilly's kinetic equation to describe penicillin G hydrolysis performed by immobilized penicillin acylase onto the acrylic carrier has been shown. Based on the experimental results characteristic kinetic constants have been estimated. The effect of noncompetitive inhibition of 6-amino penicillanic acid has not been found. Five components of reaction resistance have been defined. These components were also estimated for the reaction of the native enzyme as well as the Boehringer preparation.List of Symbols C _E g/m³ enzyme concentration - C _P,C _Q mol/m³ product concentrations - C _S mol/m³ substrate concentration - C _SO mol/m³ initial substrate concentration - K _A mol/m³ constant which defines the affinity of a substrate to the enzyme - K _iS mol/m³ substrate inhibitory constant - K _iP mol/m³ PhAA inhibitory constant - K _iQ mol/m³ 6-APA inhibitory constant - k ₃ mol/g/min constant rate of dissociation of the active complex - R(1) concentrational component of reaction resistance - R(2) resistance component derived from substrate affinity - R(3) resistance component due to the inhibition of the enzyme by substrate - R(4) resistance component due to the inhibition of the enzyme by PhAA - R(5) resistance component due to inhibition of the enzyme by 6-APA - r = dC_s/dt mol/m³ min rate of reaction - t min reaction time - (i) relative resistance of reaction     

Species Differences in the Selective Inhibition of Monoamine Oxidase (1-methyl-2-phenylethyl)hydrazine and its Potentiation by Cyanide

The potentiating effects of cyanide on the inhibition of rat liver mitochondrial monoamine oxidase-A & B and of ox liver mitochondrial MAO-B by pheniprazine [(1-methyl-2-phenylethyl)hydrazine] has been studied. Pheniprazine was shown to behave as a mechanism-based MAO inhibitor. For rat liver MAO-B, the initial non-covalent step was characterized by dissociation constant (K _i) of 2450 nM and the first-order rate constant (k ₊₂) for the covalent adduct formation was 0.16 min⁻¹. As a reversible inhibitor it was selective towards rat liver MAO-A (K _i = 420 nM) but the rate of irreversible inhibition of that enzyme was considerably slower (k ₊₂ = 0.06 min⁻¹). MAO-B from ox liver more closely resembled MAO-A from the rat in sensitivity to reversible inhibition by pheniprazine (K _i = 450 nm) but it was closer to rat liver MAO-B in rate of irreversible inhibition (k ₊₂ = 0.29 min⁻¹). The K _i values were significantly decreased in the presence of KCN but there was little effect on the k ₊₂ values. However, sensitivities of the different enzymes to KCN varied widely and considerably higher concentrations of KCN were required for this effect to be apparent with the rat liver mitochondrial MAO-A than with MAO-B from rat and ox liver. The kinetic behaviour of cyanide activation was consistent with partial (non-essential) competitive activation in all cases. Special issue dedicated to Dr. Moussa Youdim.     

Integral kinetics of the cleavage of maltose catalysed by α-glucosidase fromSaccharomyces carlsbergensis

Summary A new, sensitive and continuous assay for -glucosidase is described exploiting the different angles of rotation for the substrate maltose and the product glucose. Kinetic experiments revealed a very pronounced product inhibition of -glucosidase fromSaccharomyces carlsbergensis with a K_i of 4.85·10^–3 M for glucose.The K_M of maltose was found to be 37.8·10^–3 M. Taking these values, an integral kinetic curve for the enzymatic hydrolysis of maltose was calculated, which is shown to fit the experimental data.Symbols used k₁ (min^–1) pseudo first-order rate constant (for enzymatic cleavage) - k₂ (min^–1) rate constant (for mutarotation reaction) - I, P (mol/1) inhibitor (product) concentration - k_i (mmol/1) inhibitor constant - K_M (mmol/l) Michaelis constant - [M] ₅₈₉ ³⁰ (degree/m · l/mol) molecular rotation at 30°C and 589 nm - s (mmol/l) substrate concentration - R (mmol/mg · min) reaction rate - V_max (mmol/mg · min) maximal rate - U (mol/min) activity unit (here at 30°C and pH=6.8) Indices O initial value - max maximal value     

Determination of the optimum operating time for batch isothermal performance of enzyme-catalyzed multisubstrate reactions

This communication consists of a mathematical analysis encompassing the maximization of the average rate of monomer production in a batch reactor performing an enzymatic reaction in a system consisting of a multiplicity of polymeric substrates which compete with one another for the active site of a soluble enzyme, under the assumption that the form of the rate expression is consistent with the Michaelis-Menten mechanism. The general form for the functional dependence of the various substrate concentrations on time is obtained in dimensionless form using matrix terminology; the optimum batch time is found for a simpler situation and the effect of various process and system variables thereon is discussed. The reasoning developed here emphasizes, in a quantitative fashion, the fact that the commonly used lumped substrate approaches lead to nonconservative decisions in industrial practice, and hence should be avoided when searching for trustworthy estimates of optimum operation.List of Symbols O 1/s row vector of zeros - a 1/s row vector of rate constants k _i(i = 2,...,N) - A 1/s matrix of rate constants k _i and k_–i (i=2,...,N) - b 1/s row vector of rate constant k ₂ and zeros - C mol/m³ molar concentration of S - C mol/m³ vector of molar concentrations of C _i (i=0, 1, 2, ..., N) - C ₀ mol/m³ column vector of initial molar concentrations of C _i(i=0, 1, 2,.., N) - C _–01 mol/m³ column vector of initial molar concentrations of C _i(i=2,..., N) - C _{E, tot} mol/m³ total molar concentration of enzyme molecules - C _i mol/m³ molar concentration of S _i (i=0,1,2,...,N) - C _{i, o} mol/m³ initial molar concentration of S _i(i=0, 1, 2, ..., N) - E enzyme molecule - I identity matrix - K 1/s matrix of lumped rate constants - k _i 1/s pseudo-first order lumped rate constant associated with the formation of S _i -1 (i=1, 2, ...,N) - k _{cat, i} 1/s first order rate constant associated with the formation of S _i-1 (i=1, 2, ..., N) - K _m mol/m³ Michaelis-Menten constant - L number of distinct eigenvalues - M _i multiplicity of the i-th eigenvalue - N maximum number of monomer residues in a single polymeric molecule - r ₁ mol/m³ s rate of formation of S ₀ - r _i mol/m³ s rate of release of S _i -1 - r _opt maximum average dimensionless rate of production of monomer S₀ - S lumped, pseudo substrate - S₁ inert moiety - S _i substrate containing i monomer residues, each labile to detachment as - S₀ by enzymatic action (i=1,2,...,N) - t s time elapsed since startup of batch reaction - t _lag s time interval required for cleaning, loading, and unloading the batch reactor - t _opt s time interval leading to the maximum average rate of monomer production - v _ij s^1-j eigenvectors associated with eigenvalue imi (i=1, 2, ..., L; j =1, 2, ..., M_i) Greek Symbols _ij mol/m³ arbitrary constant associated with eigenvalue _i (i=1, 2, ..., L; j=1, 2, ..., M _i ) - 1/s generic eigenvalue - _i 1/s i-th eigenvalue     

Ligand Binding to the FA3-FA4 Cleft Inhibits the Esterase-Like Activity of Human Serum Albumin

The hydrolysis of 4-nitrophenyl esters of hexanoate (NphOHe) and decanoate (NphODe) by human serum albumin (HSA) at Tyr411, located at the FA3-FA4 site, has been investigated between pH 5.8 and 9.5, at 22.0°C. Values of K _s, k ₊₂, and k ₊₂/K _s obtained at [HSA] ≥ 5×[NphOXx] and [NphOXx] ≥ 5×[HSA] (Xx is NphOHe or NphODe) match very well each other; moreover, the deacylation step turns out to be the rate limiting step in catalysis (i.e., k ₊₃ << k ₊₂). The pH dependence of the kinetic parameters for the hydrolysis of NphOHe and NphODe can be described by the acidic pK _a-shift of a single amino acid residue, which varies from 8.9 in the free HSA to 7.6 and 7.0 in the HSA:NphOHe and HSA:NphODe complex, respectively; the pK>_a-shift appears to be correlated to the length of the fatty acid tail of the substrate. The inhibition of the HSA-Tyr411-catalyzed hydrolysis of NphOHe, NphODe, and 4-nitrophenyl myristate (NphOMy) by five inhibitors (i.e., diazepam, diflunisal, ibuprofen, 3-indoxyl-sulfate, and propofol) has been investigated at pH 7.5 and 22.0°C, resulting competitive. The affinity of diazepam, diflunisal, ibuprofen, 3-indoxyl-sulfate, and propofol for HSA reflects the selectivity of the FA3-FA4 cleft. Under conditions where Tyr411 is not acylated, the molar fraction of diazepam, diflunisal, ibuprofen, and 3-indoxyl-sulfate bound to HSA is higher than 0.9 whereas the molar fraction of propofol bound to HSA is ca. 0.5.     

Optimal temperature and concentration profiles in a cascade of CSTR's performing Michaelis-Menten reactions with first order enzyme deactivation

A necessary condition is found for the intermediate temperatures and substrate concentrations in a series of CSTR's performing an enzyme-catalyzed reaction which leads to the minimum overall volume of the cascade for given initial and final temperatures and substrate concentrations. The reaction is assumed to occur in a single phase under steady state conditions. The common case of Michaelis-Menten kinetics coupled with first order deactivation of the enzyme is considered. This analysis shows that intermediate stream temperatures play as important a role as intermediate substrate concentrations when optimizing in the presence of nonisothermal conditions. The general procedure is applied to a practical example involving a series of two reactors with reasonable values for the relevant five operating parameters. These parameters are defined as dimensionless ratios involving activation energies (or enthalpy changes of reaction), preexponential factors, and initial temperature and substrate concentration. For negligible rate of deactivation, the qptimality condition corresponds to having the ratio of any two consecutive concentrations as a single-parameter increasing function of the previous ratio of consecutive concentrations.List of Symbols C _E,0 mol.m^–3 Initial concentration of active enzyme - C _E,i mol.m^–3 Concentration of active enzyme at the outlet of the i-th reactor - C _S,0 mol.m^–3 Initial concentration of substrate - C _S,i mol.m^–3 Concentration of substrate at the outlet of the i-th reactor - Da _i Damköhler number associated with the i-th reactor ((V _i.k_v,0.C_E,0)/(Q.C_S,0)) - Da _min Minimum value of the overall Damköhler number - Da _tot Overall Damköhler number - E _d J.mol^–1 Activation energy of the step of deactivation of the enzyme - E _m J.mol^–1 Standard enthalpy change of the step of binding of substrate to the enzyme - E _v J.mol^–1 Activation energy of the step of enzymatic transformation of substrate - i Integer variable - j Dummy integer variable - k Dummy integer variable - k _d,i s^–1 Kinetic constant associated with the deactivation of enzyme in the i-th reactor (k _d,o·exp{–E _d/(R.T _i}) - k _d,0 s^–1 Preexponential factor of the kinetic constant associated with the deactivation of the enzyme - K _m,i mol.m^–3 Equilibrium constant associated with the binding of substrate to the enzyme in the i-th reactor, (k _m,o·exp{–E _m}(R.T _i}) - K _m,0 mol.m^–3 Preexponential factor of the Michaelis-Menten constant associated with the binding of substrate to the enzyme - k _v,i s^–1 Kinetic constant associated with the transformation of the substrate by the enzyme in the i-th reactor (k _v,o·exp{–E _v/(R.T _i})) - k _v,0 s^–1 Preexponential factor of the kinetic constant associated with the transformation of the substrate by the enzyme - N Number of reactors in the series - Q m³.s^–1 Volumetric flow rate of reacting liquid through the reactor network - R J.K^–1.mol^–1 Ideal gas constant - T _i K Absolute temperature at the outlet of the i-th reactor - T ₀ K Initial absolute temperature - V _i m³ Volume of the i-th reactor - v _max mol.m^–3.s^–1 Maximum rate of reaction under saturation conditions of substrate - x _i Normalized concentration of substrate (C_S,i/C_{S, 0}) - x _i,opt Optimum value of the normalized concentration of substrate - y _i Dimensionless temperature (exp{–T ₀/T _i}) - y _i,opt Optimum value of the dimensionless temperature Greek Symbols Dimensionless preexponential factor associated with the Michaelis-Menten constant (K _m,0/C_s,0) - Dimensionless activation energy of the step of enzymatic transformation of substrate (E _v/R.T₀)) - Dimensionless standard enthalpy change of the step of binding of substrate to the enzyme (E _m/(R.T₀)) - Dimensionless activation energy of the step of deactivation of the enzyme (E _d/(R.T₀)) - Dimensionless deactivation preexponential factor ((k _d,0.C_S,0)/(k_v,0.C_E,0)