Reduced models of networks of coupled enzymatic reactions

The Michaelis-Menten equation has played a central role in our understanding of biochemical processes. It has long been understood how this equation approximates the dynamics of irreversible enzymatic reactions. However, a similar approximation in the case of networks, where the product of one reaction can act as an enzyme in another, has not been fully developed. Here we rigorously derive such an approximation in a class of coupled enzymatic networks where the individual interactions are of Michaelis-Menten type. We show that the sufficient conditions for the validity of the total quasi-steady state assumption (tQSSA), obtained in a single protein case by Borghans, de Boer and Segel can be extended to sufficient conditions for the validity of the tQSSA in a large class of enzymatic networks. Secondly, we derive reduced equations that approximate the network's dynamics and involve only protein concentrations. This significantly reduces the number of equations necessary to model such systems. We prove the validity of this approximation using geometric singular perturbation theory and results about matrix differentiation. The ideas used in deriving the approximating equations are quite general, and can be used to systematize other model reductions.     

Michaelis-Menten kinetics at high enzyme concentrations

The total quasi-steady state approximation (tQSSA) for the irreversible Michaelis-Menten scheme is derived in a consistent manner. It is found that self-consistency of the initial transient guarantees the uniform validity of the tQSSA, but does not guarantee the validity of the linearization in the original derivation of Borghans et al. (1996, Bull. Math. Biol., 58, 43–63). Moreover, the present rederivation yielded the noteworthy result that the tQSSA is at least roughly valid for any substrate and enzyme concentrations. This reinforces and extends the original assertion that the parameter domain for which the tQSSA is valid overlaps the domain of validity of the standard quasi-steady state approximation and includes the limit of high enzyme concentrations. The criteria for the uniform validity of the original (linearized) tQSSA are corrected, and are used to derive approximate solutions that are uniformly valid in time. These approximations overlap and extend the domains of validity of the standard and reverse quasi-steady state approximations.     

The total quasi-steady-state approximation is valid for reversible enzyme kinetics

The Briggs-Haldane approximation of the irreversible Michaelis-Menten scheme of enzyme kinetics is cited in virtually every biochemistry textbook and is widely considered the classic example of a quasi-steady-state approximation. Though of similar importance, the reversible Michaelis-Menten scheme is not as well characterized. This is a serious limitation since even enzymatic reactions that go to completion may be reversible. The current work derives a total quasi-steady-state approximation (tQSSA) for the reversible Michaelis-Menten and delineates its validity domain. The tQSSA allows the derivation of uniformly valid approximations for the limit of low enzyme concentrations, ET相似文献   

Investigations of reaction kinetics for immobilized enzymes--identification of parameters in the presence of diffusion limitation

A method is proposed for identification of kinetic parameters when diffusion of substrates is limiting in reactions catalyzed by immobilized enzymes. This method overcomes conventional sequential procedures, which assume immobilization does not affect the conformation of the enzyme and, thus, consider intrinsic and inherent kinetics to be the same. The coupled equations describing intraparticle mass transport are solved simultaneously using numerical methods and are used for direct estimation of kinetic parameters by fitting modeling results to time-course measurements in a stirred tank reactor. While most traditional procedures were based on Michaelis-Menten kinetics, the method presented here is applicable to more complex kinetic mechanisms involving multiple state variables, such as ping-pong bi-bi. The method is applied to the kinetic resolution of (R/S)-1-methoxy-2-propanol with vinyl acetate catalyzed by Candida antarctica lipase B. A mathematical model is developed consisting of irreversible ping-pong bi-bi kinetics, including competitive inhibition of both enantiomers. The kinetic model, which fits to experimental data over a wide range of both substrates (5-95%) and temperatures (5-56 degrees C), is used for simulations to study typical behavior of immobilized enzyme systems.     

Stochastic simulation of enzyme-catalyzed reactions with disparate timescales

Many physiological characteristics of living cells are regulated by protein interaction networks. Because the total numbers of these protein species can be small, molecular noise can have significant effects on the dynamical properties of a regulatory network. Computing these stochastic effects is made difficult by the large timescale separations typical of protein interactions (e.g., complex formation may occur in fractions of a second, whereas catalytic conversions may take minutes). Exact stochastic simulation may be very inefficient under these circumstances, and methods for speeding up the simulation without sacrificing accuracy have been widely studied. We show that the “total quasi-steady-state approximation” for enzyme-catalyzed reactions provides a useful framework for efficient and accurate stochastic simulations. The method is applied to three examples: a simple enzyme-catalyzed reaction where enzyme and substrate have comparable abundances, a Goldbeter-Koshland switch, where a kinase and phosphatase regulate the phosphorylation state of a common substrate, and coupled Goldbeter-Koshland switches that exhibit bistability. Simulations based on the total quasi-steady-state approximation accurately capture the steady-state probability distributions of all components of these reaction networks. In many respects, the approximation also faithfully reproduces time-dependent aspects of the fluctuations. The method is accurate even under conditions of poor timescale separation.     

Rapid determination of kinetic constants by competitive spectrophotometry

The use of competitive spectrophotometry to measure kinetic constants for enzyme-catalyzed reactions is described. The equation for the progress curve characterizing the kinetic behavior of an enzyme acting simultaneously on two alternative substrates is derived. By the addition of a competition term to the integrated Michaelis-Menten equation, the kinetic constants of an alternative substrate can be evaluated by measuring the competition with a substrate of known kinetic constants in a single experiment. Studies are presented involving the enzymes leucine aminopeptidase (LAP) and carboxypeptidase A (CPA). The results obtained with LAP and CPA showed that the kinetic constants determined using competitive spectrophotometry were in agreement with values cited in the literature or with values determined by single substrate enzyme kinetics.     

Multifractality in intracellular enzymatic reactions

Enzymatic kinetics adjust well to the Michaelis-Menten paradigm in homogeneous media with dilute, perfectly mixed reactants. These conditions are quite different from the highly structured cell plasm, so applications of the classic kinetics theory to this environment are rather limited. Cytoplasmic structure produces molecular crowding and anomalous diffusion of substances, modifying the mass action kinetic laws. The reaction coefficients are no longer constant but time-variant, as stated in the fractal kinetics theory. Fractal kinetics assumes that enzymatic reactions on such heterogeneous media occur within a non-Euclidian space characterized by a certain fractal dimension, this fractal dimension gives the dependence on time of the kinetic coefficients. In this work, stochastic simulations of enzymatic reactions under molecular crowding have been completed, and kinetic coefficients for the reactions, including the Michaelis-Menten parameter KM, were calculated. The simulations results led us to confirm the time dependence of michaelian kinetic parameter for the enzymatic catalysis. Besides, other chaos related phenomena were pointed out from the obtained KM time series, such as the emergence of strange attractors and multifractality.     

Kinetic constraints for formation of steady states in biochemical networks

The constraint-based analysis has emerged as a useful tool for analysis of biochemical networks. This work introduces the concept of kinetic constraints. It is shown that maximal reaction rates are appropriate constraints only for isolated enzymatic reactions. For biochemical networks, it is revealed that constraints for formation of a steady state require specific relationships between maximal reaction rates of all enzymes. The constraints for a branched network are significantly different from those for a cyclic network. Moreover, the constraints do not require Michaelis-Menten constants for most enzymes, and they only require the constants for the enzymes at the branching or cyclic point. Reversibility of reactions at system boundary or branching point may significantly impact on kinetic constraints. When enzymes are regulated, regulations may impose severe kinetic constraints for the formation of steady states. As the complexity of a network increases, kinetic constraints become more severe. In addition, it is demonstrated that kinetic constraints for networks with co-regulation can be analyzed using the approach. In general, co-regulation enhances the constraints and therefore larger fluctuations in fluxes can be accommodated in the networks with co-regulation. As a first example of the application, we derive the kinetic constraints for an actual network that describes sucrose accumulation in the sugar cane culm, and confirm their validity using numerical simulations.     

Introducing total substrates simplifies theoretical analysis at non-negligible enzyme concentrations: pseudo first-order kinetics and the loss of zero-order ultrasensitivity

The Briggs–Haldane standard quasi-steady state approximation and the resulting rate expressions for enzyme driven biochemical reactions provide crucial theoretical insight compared to the full set of equations describing the reactions, mainly because it reduces the number of variables and equations. When the enzyme is in excess of the substrate, a significant amount of substrate can be bound in intermediate complexes, so-called substrate sequestration. The standard quasi-steady state approximation is known to fail under such conditions, a main reason being that it neglects these intermediate complexes. Introducing total substrates, i.e., the sums of substrates and intermediate complexes, provides a similar reduction of the number of variables to consider but without neglecting the contribution from intermediate complexes. The present theoretical study illustrates the usefulness of such simplifications for the understanding of biochemical reaction schemes. We show how introducing the total substrates allows a simple analytical treatment of the relevance of significant enzyme concentrations for pseudo first-order kinetics and reconciles two proposed criteria for the validity of the pseudo first-order approximation. In addition, we show how the loss of zero-order ultrasensitivity in covalent modification cycles can be analyzed, in particular that approaches such as metabolic control analysis are immediately applicable to scenarios described by the total substrates with enzyme concentrations higher than or comparable to the substrate concentrations. A simple criterion which excludes the possibility of zero-order ultrasensitivity is presented.     

Kinetics of trypsin-catalyzed hydrolysis of some arginine-containing peptide methyl esters

The kinetics of trypsin-catalyzed hydrolysis (at pH 8.5) of methyl esters of some synthetic dipeptides containing the residues of both arginine and L(D)-p-fluorophenylalanine or L(D)-tyrosine has been studied. The digestion of Tos-(pF)Phe-Arg-OMe was shown as unfollowing the Michaelis-Menten kinetic since in the reaction course the substrate activation is observed and while the reaction product is the enzymatic process inhibitor. In contrast to this, the hydrolysis of other substrates studied, follows the normal Michaelis-Menten kinetic.     

Film diffusional influences on the kinetic parameters in packed-bed immobilized enzyme reactors

A number of studies on packed-bed immobilized enzyme reactors have shown the significant influence of the external film mass transfer resistance on the apparent kinetic parameters. Some of the earlier mathematical models using approximation techniques have attempted to explain the linearity of the S(0)x vs ln (1 - x) plots observed experimentally for systems obeying Michaelis-Menten kinetics. However, there has been no critical examination of the bounds of validity of the approximations used. Further, the situations where the above linearity is not valid have been examined neither conceptually nor quantitatively. The work presented here overcomes these drawbacks of the earlier analyses by approaching the problem from a different angle. Methods of evaluation of the intrinsic kinetic parameters under different experimental situations have been outlined and illustrated with several examples.     

The fractal architecture of cytoplasmic organization: Scaling,kinetics and emergence in metabolic networks

In this work, we highlight the links between fractals and scaling in cells and explore the kinetic consequences for biochemical reactions operating in fractal media. Based on the proposal that the cytoskeletal architecture is organized as a percolation lattice, with clusters emerging as fractal forms, the analysis of kinetics in percolation clusters is especially emphasized. A key consequence of this spatiotemporal cytoplasmic organization is that enzyme reactions following Michaelis-Menten or allosteric type kinetics exhibit higher rates in fractal media (for short times and at lower substrate concentrations) at the percolation threshold than in Euclidean media. As a result, considerably faster and higher amplification of enzymatic activity is obtained. Finally, we describe some of the properties bestowed by cytoskeletal organization and dynamics on metabolic networks.     

Characterization of glycine sarcosine N-methyltransferase and sarcosine dimethylglycine N-methyltransferase

Glycine betaine is accumulated in cells living in high salt concentrations to balance the osmotic pressure. Glycine sarcosine N-methyltransferase (GSMT) and sarcosine dimethylglycine N-methyltransferase (SDMT) of Ectothiorhodospira halochloris catalyze the threefold methylation of glycine to betaine, with S-adenosylmethionine acting as the methyl group donor. These methyltransferases were expressed in Escherichia coli and purified, and some of their enzymatic properties were characterized. Both enzymes had high substrate specificities and pH optima near the physiological pH. No evidence of cofactors was found. The enzymes showed Michaelis-Menten kinetics for their substrates. The apparent K(m) and V(max) values were determined for all substrates when the other substrate was present in saturating concentrations. Both enzymes were strongly inhibited by the reaction product S-adenosylhomocysteine. Betaine inhibited the methylation reactions only at high concentrations.     

Multi-enzyme kinetic analysis of glycolipid biosynthesis.

Gangliosides are acidic glycosphingolipids synthesized sequentially by a series of glycosyltransferases acting in parallel biosynthetic pathways. While most glycosyltransferases are highly specific, some, however, may catalyze equivalent steps in each pathway using different gangliosides as substrates (e.g. N-acetylgalactosaminyltransferase, sialyltransferase-IV). A multi-enzyme kinetic analysis was developed on the condition that serial enzymatic reactions operate below substrate saturation. A multi-enzyme kinetic analysis enabled a simultaneous calculation of the Vmax/Km value of each enzyme derived from the equilibrium concentration of the respective substrate. Substrate concentrations [S] were determined by radioactive labelling of gangliosides in intact cells with the precursor sugars [14C]galactose and [14C]glucosamine, followed by high-performance thin-layer chromatography and autoradiography of the radiolabelled glycolipids. On the basis of Michaelis-Menten kinetics, Vmax/Km values were derived from [S] by a system of linear equations. The procedure was used to analyze the development of the glycolipid composition during differentiation of rat gliomaxmurine neuroblastoma (NG108-15) cells. The Vmax/Km values calculated by multi-enzyme kinetic analysis were consistent with the kinetic data obtained with solubilized enzymes. Application of multi-enzyme kinetic analysis to published data on the correlation of enzyme activities with ganglioside levels in various cell lines and tissues indicated the validity of this method for analysis of the glycolipid biosynthesis, in particular, of its initial steps. On the basis of the kinetic analysis, it is suggested that the cell lines can be divided into two groups with respect to the substrate pools of GM3 used by sialyltransferase-II and N-acetylgalactosaminyltransferase-I. The first group encompasses the majority of the neuroblastoma cell lines and the embryonic rat brain where the two enzymes share a common pool of GM3. In the second group, the two enzymes do not compete for the same pool of GM3, indicating a different subcellular localization of CMP-NeuAc:GM3 alpha2-8-sialyltransferase and UDP-N-acetylgalactosaminyl:GM3 N-acetylgalactosaminyltransferase. In this study, the theory of a multi-enzyme kinetic analysis is discussed and its application to analysis of the glycolipid biosynthesis in neuroblastoma cells is demonstrated. A multi-enzyme kinetic analysis can be applied to other biosynthetic pathways and provides the advantage of analyzing kinetic data with intact cells or tissue samples.     

Studies on the citryl-CoA-dependent inhibition of citrate-synthase with source variants from baker's yeast, Escherichia coli and Sulfolobus solfataricus

1) Citrate synthase from pig heart has previously been shown to display complex kinetic characteristics in the reactions with citryl-CoA, resulting in inhibition. The synthase from another eukaryotic source, baker's yeast, yields the same complex kinetics. 2) Synthases from a Gram-negative prokaryote, E. coli, and from an archaebacterium, S. solfataricus, catalyse the reactions of citryl-CoA in kinetics of the Michaelis-Menten type. A comparison of the rates of citryl-CoA hydrolysis (V') and physiological reaction (V), determined with these enzymes, corresponds to ratios of V'/V approximately 1 and approximately 2, respectively. Thus, and for the first time, there is no reason left to doubt the intermediate formation of citryl-CoA in the physiological reaction. 3) The complex kinetics indicated under 1) are related to efficient formation of citrate from citryl-CoA-derived acetyl-CoA and oxaloacetate in the presence of NADH and malate dehydrogenase. These conditions are not met by the enzymes from E. coli, S. solfataricus and by proteolytically nicked synthase species from pig heart. All these enzyme variants have low affinities to either one or both of the physiological substrates. Consistent with earlier ideas, the results indicate that the inhibition mechanism is related to high affinities of the enzyme for both acetyl-CoA and oxaloacetate.     

Real time kinetics of restriction endonuclease cleavage monitored by fluorescence resonance energy transfer.

The kinetics of PaeR7 endonuclease-catalysed cleavage reactions of fluorophor-labeled oligonucleotide substrates have been examined using fluorescence resonance energy transfer (FRET). A series of duplex substrates were synthesized with an internal CTCGAG PaeR7 recognition site and donor (fluorescein) and acceptor (rhodamine) dyes conjugated to the opposing 5' termini. The time-dependent increase in donor fluorescence resulting from restriction cleavage of these substrates was continuously monitored and the initial rate data was fitted to the Michaelis-Menten equation. The steady state kinetic parameters for these substrates were in agreement with the rate constants obtained from a gel electrophoresis-based fixed time point assay using radiolabeled substrates. The FRET method provides a rapid continuous assay as well as high sensitivity and reproducibility. These features should make the technique useful for the study of DNA-cleaving enzymes.     

Modeling of uncertainties in biochemical reactions

Mathematical modeling is an indispensable tool for research and development in biotechnology and bioengineering. The formulation of kinetic models of biochemical networks depends on knowledge of the kinetic properties of the enzymes of the individual reactions. However, kinetic data acquired from experimental observations bring along uncertainties due to various experimental conditions and measurement methods. In this contribution, we propose a novel way to model the uncertainty in the enzyme kinetics and to predict quantitatively the responses of metabolic reactions to the changes in enzyme activities under uncertainty. The proposed methodology accounts explicitly for mechanistic properties of enzymes and physico‐chemical and thermodynamic constraints, and is based on formalism from systems theory and metabolic control analysis. We achieve this by observing that kinetic responses of metabolic reactions depend: (i) on the distribution of the enzymes among their free form and all reactive states; (ii) on the equilibrium displacements of the overall reaction and that of the individual enzymatic steps; and (iii) on the net fluxes through the enzyme. Relying on this observation, we develop a novel, efficient Monte Carlo sampling procedure to generate all states within a metabolic reaction that satisfy imposed constrains. Thus, we derive the statistics of the expected responses of the metabolic reactions to changes in enzyme levels and activities, in the levels of metabolites, and in the values of the kinetic parameters. We present aspects of the proposed framework through an example of the fundamental three‐step reversible enzymatic reaction mechanism. We demonstrate that the equilibrium displacements of the individual enzymatic steps have an important influence on kinetic responses of the enzyme. Furthermore, we derive the conditions that must be satisfied by a reversible three‐step enzymatic reaction operating far away from the equilibrium in order to respond to changes in metabolite levels according to the irreversible Michelis–Menten kinetics. The efficient sampling procedure allows easy, scalable, implementation of this methodology to modeling of large‐scale biochemical networks. Biotechnol. Bioeng. 2011;108: 413–423. © 2010 Wiley Periodicals, Inc.