Control analysis of systems having two steps catalyzed by the same protein molecule in unbranched chains

The analysis of the control of a metabolic pathway having an enzyme catalyzing two different reactions (or a protein displaying two different activities) has been performed. For such systems although the summation theorems are valid, the flux and concentration connectivity theorems of the metabolic control analysis are not valid. Another general relationship of control analysis is shown to be more widely obeyed and holds in these systems. An exemplary case, where the enzyme catalyzes two irreversible reactions, demonstrates that the level of one internal intermediate is constant, i.e. it does not depend upon the independent variables of the system.     

Modeling cellulase kinetics on lignocellulosic substrates

The enzymatic hydrolysis of cellulose to glucose by cellulases is one of the major steps involved in the conversion of lignocellulosic biomass to yield biofuel. This hydrolysis by cellulases, a heterogeneous reaction, currently suffers from some major limitations, most importantly a dramatic rate slowdown at high degrees of conversion. To render the process economically viable, increases in hydrolysis rates and yields are necessary and require improvement both in enzymes (via protein engineering) and processing, i.e. optimization of reaction conditions, reactor design, enzyme and substrate cocktail compositions, enzyme recycling and recovery strategies. Advances in both areas in turn strongly depend on the progress in the accurate quantification of substrate–enzyme interactions and causes for the rate slowdown. The past five years have seen a significant increase in the number of studies on the kinetics of the enzymatic hydrolysis of cellulose. This review provides an overview of the models published thus far, classifies and tabulates these models, and presents an analysis of their basic assumptions. While the exact mechanism of cellulases on lignocellulosic biomass is not completely understood yet, models in the literature have elucidated various factors affecting the enzymatic rates and activities. Different assumptions regarding rate-limiting factors and basic substrate–enzyme interactions were employed to develop and validate these models. However, the models need to be further tested against additional experimental data to validate or disprove any underlying hypothesis. It should also provide better insight on additional parameters required in the case that more substrate and enzyme properties are to be included in a model.     

Characteristics of unbuffered gel-immobilized urease particles. II. Overall rate of reaction

The overall rates of reaction of unbuffered gel-immobilized urease particles have been investigated with the aid of a packed-bed differential recycle reactor. Both substrate and enzyme concentrations have received attention. Cylindrical gel particles contained within impermeable tubelets were used to provide the physical strength necessary for the packed-bed arrangement and a one dimensional diffusion path to aid understanding of the complex interactions between substrate and product diffusion, and their effect on the reactions taking place. The experimental data have been interpreted with the aid of an enzyme rate equation (ERE) which relates the free solution characteristics of the enzyme to the conditions within a diffusion limited particle. The internal hydrogen ion profiles have been accommodated by a lumped parameter, the apparent pH (pH_app). Two methods have been suggested for the calculation of pH_app and the loss of activity on particle preparation, these methods are based on the use of the ERE in conjunction with experimental data.     

Equilibrium and water proton relaxation rate enhancement properties of formyltetrahydrofolate synthetase-manganous ion-substrate complexes.

Binding of Mn(pi)-nucleotide complexes to the enzyme formyltertrahydrofolate synthetase (EC 6.3.4.3) from Clostridium cylindrosporum has been examined in the presence and absence of other substrates by solvent proton relaxation mearurements. MnADP and MnATP form ternary complexes with the enzyme with highly enhanced proton relaxation rates for water. The enhancement parameters, epsilont, for the MnADP and MnATP ternary complexes are 19.8 and 12.5, respectively at 24.3 MHZ and 25 degrees. Titration curves with constant total concentrations of enzyme and Mn(pi) with variable nucleotide concentration are similar to those observed in similar titrations with the endp and MnATP are 175 muM and 64 muM, respectively at 25 degrees. Addition of tetrahydrofolate to solutions of the MnADP OR MnATP ternary complexes lowers the observed relaxation enhancement markedly. An analysis of titration curves with constant total concentrations of enzyme, Mn(pi), and nucleotide with variable tetrahydrofolate concentration gives the dissociation constant for tetrahydrofolate from the respective quaternary complexes. The affinity of the enzyme for tetrahydrofolate is increased 6-fold when MnADP is present at the active site whereas a 3-fold increase is observed with MnATP present. Furthermore, there is a 20-fold increase in the enzyme's affinity for tetrahydrofolate when both MnADP and the third substrate, formate, are present. The observed relaxation rate of water for solutions of the complex, enzyme-MnADP-tetrahydrofolate-formate, is deenhanced with respect to the rate observed for the simple aquo-Mn(pi) solution. Addition of nitrate to solutions of the above complex increases the affinity of the enzyme for tetrahydrofolate and MnADP by an additional factor of 5 and lowers the relaxation rate further to a value which approaches that for solutions of the enzyme and substrates which lack the paramagnetic cation.     

The efficiency of reactant site sampling in network-free simulation of rule-based models for biochemical systems

Rule-based models, which are typically formulated to represent cell signaling systems, can now be simulated via various network-free simulation methods. In a network-free method, reaction rates are calculated for rules that characterize molecular interactions, and these rule rates, which each correspond to the cumulative rate of all reactions implied by a rule, are used to perform a stochastic simulation of reaction kinetics. Network-free methods, which can be viewed as generalizations of Gillespie's method, are so named because these methods do not require that a list of individual reactions implied by a set of rules be explicitly generated, which is a requirement of other methods for simulating rule-based models. This requirement is impractical for rule sets that imply large reaction networks (i.e. long lists of individual reactions), as reaction network generation is expensive. Here, we compare the network-free simulation methods implemented in RuleMonkey and NFsim, general-purpose software tools for simulating rule-based models encoded in the BioNetGen language. The method implemented in NFsim uses rejection sampling to correct overestimates of rule rates, which introduces null events (i.e. time steps that do not change the state of the system being simulated). The method implemented in RuleMonkey uses iterative updates to track rule rates exactly, which avoids null events. To ensure a fair comparison of the two methods, we developed implementations of the rejection and rejection-free methods specific to a particular class of kinetic models for multivalent ligand-receptor interactions. These implementations were written with the intention of making them as much alike as possible, minimizing the contribution of irrelevant coding differences to efficiency differences. Simulation results show that performance of the rejection method is equal to or better than that of the rejection-free method over wide parameter ranges. However, when parameter values are such that ligand-induced aggregation of receptors yields a large connected receptor cluster, the rejection-free method is more efficient.     

Enzyme-enzyme interactions and control analysis. 2. The case of non-independence: heterologous associations

The association of different enzymes into a complex may induce changes in the kinetic parameters of its component enzymes. This implies that they cannot be treated as independent catalysts. It will affect the formulations and theorems of control analysis and necessitates the introduction of additional elasticities reflecting the effect of one enzyme on the rate of another. We show how this is achieved as an extension of the classical treatment. We present modified summation and connectivity theorems incorporating both homologous and heterologous interactions. The case of channelling of metabolites in such complexes is considered and an experimental method for its detection is suggested.     

Modular metabolic control analysis of large responses

Deciphering the laws that govern metabolic responses of complex systems is essential to understand physiological functioning, pathological conditions and the outcome of experimental manipulations of intact cells. To this aim, a theoretical and experimental sensitivity analysis, called modular metabolic control analysis (MMCA), was proposed. This field was previously developed under the assumptions of infinitesimal changes and/or proportionality between parameters and rates, which are usually not fulfilled in vivo. Here we develop a general MMCA for two modules, not relying on those assumptions. Control coefficients and elasticity coefficients for large changes are defined. These are subject to constraints: summation and response theorems, and relationships that allow calculating control from elasticity coefficients. We show how to determine the coefficients from top-down experiments, measuring the rates of the isolated modules as a function of the linking intermediate (there is no need to change parameters inside the modules). The novel formalism is applied to data of two experimental studies from the literature. In one of these, 40% increase in the activity of the supply module results in less than 4% increase in flux, while infinitesimal MMCA predicts more than 30% increase in flux. In addition, it is not possible to increase the flux by manipulating the activity of demand. The impossibility of increasing the flux by changing the activity of a single module is due to an abrupt decrease of the control of the modules when their corresponding activities are increased. In these cases, the infinitesimal approach can give highly erroneous predictions.     

A cometabilic kinetics model incorporating enzyme inhbition, inactivation, and recovery: I. Model development, analysis, and testing

Cometabolic biodegradation prcesses are important for bioremediation of hazardous waste sites. However, these proceeses are not well understood and have not been modeled thoroughly. Traditional Michaelis-Menten kinetics models often are used, but toxic effects and bacterial responses to toxicity may cause changes in enzyme levels, rendering such models inappropriate. In this article, a conceptual and mathematical model of cometabolic enzyme kinetics i described. Model derivation is based on enzyme/growth-substrate/nongrowth-substrate interaction and incorporates enzyme inhibition (caused by the presence of a cometabolic compound), inactivation (resulting from toxicity of a cometabolic product), and recovery (associated with bacterial synthesis of new enbzyme in response to inactivation). The mathematical model consists of a system of two, nonlinear ordinary differential equations that can be solved implicitly using numerical methods, providing estimates of model parameters. Model analysis shows that growth substraate adn nongrowth substrate oxidation rates are related by a dimensionless constant. Reliability of tehy model solution prcedure is verifiedl by abnalyzing data ses, containing random error, from simulated experimentss with trichhloroethyylene (TCE) degradation by ammonia-oxidizing bacterialunder various conditions. Estimation of the recovery rate contant is deterimined to be sensitive to intial TCE concentration. Model assumptions are evaluated in a companion article using data from TCE degradation experiments with amoniaoxidizing bacteria. (c) 1995 John Wiley & Sons, Inc.     

Steady-state parameters of an enzyme from n.m.r. spin transfer with thermal variation.

N.m.r. spin-exchange analysis of enzymic reactions at chemical equilibrium is akin to radioactive-tracer-exchange analysis; unidirectional flux rates are obtained for the overall reaction. These data, by themselves, are not sufficient to define the values of all the individual rate constants or steady-state parameters. However, it is shown that, by measuring the dependence of the exchange rate constants on solute concentration and temperature, the individual rate constants, and hence the steady-state parameters, can be obtained for a simple enzyme system.     

Kinetics of suicide substrates. Practical procedures for determining parameters.

Many clinically important or mechanistically interesting inhibitors react with enzymes by a branched pathway in which inactivation of the enzyme and formation of product are competing reactions. The steady-state kinetics for this pathway [Waley (1980) Biochem. J. 185, 771-773] gave equations for progress curves that were cumbersome. A convenient linear plot is now described. The time (t1/2) for 50% inactivation of the enzyme (this is also the time for 50% formation of product), or for 50% loss of substrate, is measured in a series of experiments in which the concentration of inhibitor, [I]0, is varied; in these experiments the ratio of the concentration of enzyme to the concentration of inhibitor is kept fixed. Then a plot of [I]0 X t1/2 against [I]0 is linear, and the kinetic parameters can be found from the slope and intercept. Furthermore, simplifications of the equations for progress curves are described that are valid when the concentration of inhibitors is high, or is low, or when the extent of reaction is low. The use of simulated data has shown that the recommended methods are not unduly sensitive to experimental error.     

Analysis of ground-state and transition-state effects in enzyme catalysis.

"The entire and sole source of catalytic power is the stabilization of the transition state; reactant-state interactions are by nature inhibitory and only waste catalytic power". So reads a literature quote expressing the current view on enzyme catalysis proposed by Pauling over 40 years ago. Its validity is now examined by means of a "split-site" model in which an active site is subdivided into a region of binding and a region of reaction. Analysis of the resulting free energy levels clarifies several points of confusion regarding the nature of enzyme catalysis, including why enzyme/substrate complexes form if, indeed, they only "waste catalytic power". Circumstances are defined in which an evolving enzyme can both lower Km (i.e., enhance substrate binding) and improve the forward catalytic rate while never meddling with the transition structure at the reactive site. It is argued that this process is most advantageously viewed as a substrate destabilization embodying "conserved" interactions at the binding region. Classical transition-state stabilization and an "anti-Pauling" effect are both capable of inducing rate accelerations. In certain circumstances, the latter can predominate as it does with many enzyme-like intramolecular reactions. Behavioral modes discussed herein are applicable to the chemistry of catalytic host/guest and enzyme systems.     

Control analysis of time-dependent metabolic systems

Metabolic Control Analysis is extended to time dependent systems. It is assumed that the time derivative of the metabolite concentrations can be written as a linear combination of rate laws, each one of first order with respect to the corresponding enzyme concentration. The definitions of the control and elasticity coefficients are extended, and a new type of coefficient ("time coefficient", "T") is defined. First, we prove that simultaneous changes in all enzyme concentrations by the same arbitrary factor, is equivalent to a change in the time scale. When infinitesimal changes are considered, these arguments lead to the derivation of general summation theorems that link control and time coefficients. The comparison of two systems with identical rates, that only differ in one metabolite concentration, leads to a method for the construction of general connectivity theorems, that relate control and elasticity coefficients. A mathematical proof in matrix form, of the summation and connectivity relationships, for time dependent systems is given. Those relationships allow one to express the control coefficients in terms of the elasticity and time coefficients for the case of unbranched pathway.     

Substrate-velocity relationships for the Trichoderma viride cellulase-catalyzed hydrolysis of cellulose.

The influence of substrate and enzyme concentrations on the rate of saccharification of two defined insoluble cellulose substrates, Avicel (FMC Corp., Philadelphia, Pa.) and Solka-Floc (James River Co., Berlin, N.H.), by the cellulase enzyme system of Trichoderma viride was evaluated. In the assays, enzyme concentrations ranging from 0.004 to 0.016 IU/ml and substrate concentrations up to 10% (wt/vol) were used. Analysis by initial velocity methods found the maximum velocity of saccharification to be nearly equivalent for the two substrates and the Km for the two substrates to be of a similar magnitude, i.e., 0.20% (wt/vol) for Solka-Floc and 0.63% (wt/vol) for Avicel. Studies in which relatively high substrate concentrations (greater than 15 times the Km) were used demonstrated that the enzyme exhibited very different apparent substrate inhibition properties for the two substrates. The rate of saccharification of Avicel at relatively high substrate concentrations was up to 35% lower than the maximum rate which was observed at lower substrate concentrations. The Avicel concentration corresponding to the maximum rate of saccharification was dependent on the enzyme concentration. In contrast to the results with Avicel, the enzyme did not exhibit substrate inhibition with the Solka-Floc substrate. Potential differences in the degree of substrate inhibition with different substrates, as reported here, are particularly relevant to the experimental design of comparative studies.     

Slow-binding inhibition of NAD+ glycohydrolase by arabino analogues of beta-NAD.

Modifications at the 2'-position of the nicotinamide-ribosyl moiety influence dramatically the nature of the interactions of the modified beta-NAD+ with calf spleen NAD+ glycohydrolase (EC 3.2.2.6), an enzyme that cleaves the nicotinamide-ribose bound in NAD(P)+. Nicotinamide arabinoside adenine dinucleotide (ara-NAD+) and nicotinamide 2'-deoxy-2'-fluoroarabinoside adenine dinucleotide (araF-NAD+) are not hydrolyzed at measurable rates and are the first documented examples of reversible slow binding inhibitors of this class of enzyme. The kinetic data obtained are consistent with both slow kon and koff rate constants in the formation of an enzyme-inhibitor complex, i.e. the association rate constants are about 10(4) and 10(6) slower than diffusion rates, respectively, for araF-NAD+ and ara-NAD+, and the half-life of the complex is about 3-10 min for both analogues. The kinetic model does not account for a slow turnover of an ADP-ribosyl-enzyme intermediary complex. AraF-NAD+ is one of the most potent inhibitors described for NAD+ glycohydrolase.     

Affinity labeling of pig lung glutathione S-transferase pi by 4-(fluorosulfonyl)benzoic acid

The compound 4-(fluorosulfonyl)benzoic acid (4-FSB) functions as an affinity label of the dimeric pig lung pi class glutathione S-transferase yielding a completely inactive enzyme. Protection against inactivation is provided by glutathione-based ligands, suggesting that the reaction target is near or part of the glutathione binding site. Radioactive 4-FSB is incorporated to the extent of 1 mol per mole of enzyme subunit. Peptide mapping revealed that 4-FSB reacts with two tyrosine residues in the ratio 69% Tyr7 and 31% Tyr106. The ratio is not changed by the addition of ligands. The results suggest that only one of the tyrosine residues can be labeled in the active site of a given subunit; i.e., reactions with Tyr7 and Tyr106 are mutually exclusive. We propose that the difference in labeling of these tyrosine residues is related to their pKa values, with Tyr7 exhibiting the lower pKa. The modified enzyme no longer binds to a S-hexylglutathione-agarose affinity column, even when only one of the active sites contains 4-FSB; these results may reflect interaction between the subunits. We conclude that Tyr7 and Tyr106 of the pig lung class pi glutathione S-transferase are important for function and are located at or close to the substrate binding site of the enzyme.     

Leukotriene C synthase in mouse mastocytoma cells. An enzyme distinct from cytosolic and microsomal glutathione transferases.

The reactions between cellobiose and cellobiose oxidase were investigated by stopped-flow spectrophotometry. Under anaerobic conditions rapid reduction of the associated flavin is followed by slower reduction of cytochrome b. The kinetic difference spectra are reported. The rate of flavin reduction depends on the cellobiose concentration (with an apparent second-order rate constant of approx. 10(5) M-1.s-1) but reaches a rate limit of approx. 20 s-1. In contrast, the rate of cytochrome b reduction decreases at high cellobiose concentrations. Kinetic titrations of the flavin and cytochrome b moieties yield the stoichiometries of the separate reactions, i.e. the number of moles of cellobiose needed to fully reduce 1 mole of each redox component. The rate constant for cytochrome b reduction, unlike that for flavin reduction, increased with enzyme concentration, prompting the conclusion that any given cytochrome b centre is reduced preferentially by flavin groups in different molecules rather than by its partner flavin within the same monomer. These data are discussed in the context of a scheme that rationalizes them and accounts for the overall stoichiometry in which three two-electron donors (cellobiose molecules) reduce two three-electron acceptors (the flavin-cytochrome b of cellobiose oxidase).     

Diffusion control in reversible enzyme reactions. Applications to carbonic anhydrase.

The limit to the possible rate of reversible enzymatic reactions set by the diffusional motion has been considered. It is found that not only the diffusion of the reactants to the enzyme but also the diffusion away of the products can be rate limiting. To avoid assumptions about the detailed nature of the enzyme only diffusion in the bulk aqueous medium is treated. By such an approach one obtains an upper limit to the possible rate. In the latter half of the paper the derived general equations are applied to the possible suggested reaction schemes for the enzyme carbonic anhydrase. It is found that a scheme involving HCO3- as substrate for the dehydration process and a direct reaction between buffer and enzyme is comsistent with the limits set by the diffusional motion, while several other possibilities can be ruled out.     

Kinetic analysis of enzyme reactions with slow-binding inhibition.

In this paper we present a general kinetic study of slow-binding inhibition processes, i.e. enzyme reactions that do not respond instantly to the presence of a competitive inhibitor. The analysis that we present is based on the equation that describes the formation of products with time in each case on the experimental progress curve. It is carried out under the condition of limiting enzyme concentration and allows the discrimination between the different cases of slow-binding inhibition. The mechanism in which the formation of complex enzyme-inhibitor is a single or two slow steps or follow a rapid equilibrium, has been considered. The corresponding explicit equations of each case have been obtained and checked by numerical integration. A kinetic data analysis to evaluate the corresponding kinetic parameters is suggested. We illustrate the method, numerically by computer simulation, of the reaction and present some numerical examples that demonstrate the applicability of our procedure.     

Kinetic Proofreading at Single Molecular Level: Aminoacylation of tRNAIle and the Role of Water as an Editor

Proofreading/editing in protein synthesis is essential for accurate translation of information from the genetic code. In this article we present a theoretical investigation of efficiency of a kinetic proofreading mechanism that employs hydrolysis of the wrong substrate as the discriminatory step in enzyme catalytic reactions. We consider aminoacylation of tRNA^Ile which is a crucial step in protein synthesis and for which experimental results are now available. We present an augmented kinetic scheme and then employ methods of stochastic simulation algorithm to obtain time dependent concentrations of different substances involved in the reaction and their rates of formation. We obtain the rates of product formation and ATP hydrolysis for both correct and wrong substrates (isoleucine and valine in our case, respectively), in single molecular enzyme as well as ensemble enzyme kinetics. The present theoretical scheme correctly reproduces (i) the amplitude of the discrimination factor in the overall rates between isoleucine and valine which is obtained as (1.8×10²).(4.33×10²) = 7.8×10⁴, (ii) the rates of ATP hydrolysis for both Ile and Val at different substrate concentrations in the aminoacylation of tRNA^Ile. The present study shows a non-michaelis type dependence of rate of reaction on tRNA^Ile concentration in case of valine. The overall editing in steady state is found to be independent of amino acid concentration. Interestingly, the computed ATP hydrolysis rate for valine at high substrate concentration is same as the rate of formation of Ile-tRNA^Ile whereas at intermediate substrate concentration the ATP hydrolysis rate is relatively low. We find that the presence of additional editing domain in class I editing enzyme makes the kinetic proofreading more efficient through enhanced hydrolysis of wrong product at the editing CP1 domain.