Translating biochemical network models between different kinetic formats

Mechanistic biochemical network models describe the dynamics of intracellular metabolite pools in terms of substance concentrations, stoichiometry and reaction kinetics. Data from stimulus response experiments are currently the most informative source for in-vivo parameter estimation in such models. However, only a part of the parameters of classical enzyme kinetic models can usually be estimated from typical stimulus response data. For this reason, several alternative kinetic formats using different “languages” (e.g. linear, power laws, linlog, generic and convenience) have been proposed to reduce the model complexity. The present contribution takes a rigorous “multi-lingual” approach to data evaluation by translating biochemical network models from one kinetic format into another. For this purpose, a new high-performance algorithm has been developed and tested. Starting with a given model, it replaces as many kinetic terms as possible by alternative expressions while still reproducing the experimental data. Application of the algorithm to a published model for Escherichia coli's sugar metabolism demonstrates the power of the new method. It is shown that model translation is a powerful tool to investigate the information content of stimulus response data and the predictive power of models. Moreover, the local and global approximation capabilities of the models are elucidated and some pitfalls of traditional single model approaches to data evaluation are revealed.     

Optimal re-design of primary metabolism in Escherichia coli using linlog kinetics

This paper examines the validity of the linlog approach, which was recently developed in our laboratory, by comparison of two different kinetic models for the metabolic network of Escherichia coli. The first model is a complete mechanistic model; the second is an approximative model in which linlog kinetics are applied. The parameters of the linlog model (elasticities) are derived from the mechanistic model. Three different optimization cases are examined. In all cases, the objective is to calculate the enzyme levels that maximize a certain flux while keeping the total amount of enzyme constant and preventing large changes of metabolite concentrations. For an average variation of metabolite levels of 10% and individual changes of a factor 2, the predicted enzyme levels, metabolite concentrations and fluxes of both models are highly similar. This similarity holds for changes in enzyme level of a factor 4-6 and for changes in fluxes up to a factor 6. In all three cases, the predicted optimal enzyme levels could neither have been found by intuition-based approaches, nor on basis of flux control coefficients. This demonstrates that kinetic models are essential tools in Metabolic Engineering. In this respect, the linlog approach is a valuable extension of MCA, since it allows construction of kinetic models, based on MCA parameters, that can be used for constrained optimization problems and are valid for large changes of metabolite and enzyme levels.     

Modelling, steady state analysis and optimization of the catalytic efficiency of the triosephosphate isomerase

In the present work we have modelled and optimized the reaction mechanism of the triose phosphate isomerase (TIM) enzyme (E.C. 5.3.1.1). For this purpose we have used an approach that combines the S-system representation within the power law formalism and linear programming techniques. By this means we have explored those rate constants whose alterations are likely to improve the catalytic efficiency of the enzyme and investigated the available room for optimization in different metabolic conditions. The role and plausibility of the different types of mutations on the evolution of this enzyme have also been considered. Steady state sensitivity analysis was carried out and a new set of aggregated logarithmic gains was defined in order to quantify the responses of the system to changes in groups of rate constants that could be explained in terms of mutations affecting the catalytic properties of the enzyme. Evaluation of these logarithmic gains at different levels of saturation and disequilibrium ratios enabled us to reach conclusions about the meaning and role of the diffusion limitation terms. The catalytic efficiency of the monoenzymatic system was optimized through changes in the kinetic rate constants within different sets of restrictions ranging from thermodynamic or kinetic to evolutionary ones. Results showed that, at very different conditions, there is still room for improvement in the TIM enzyme. Thus, in a wide range of metabolically significant values of the disequilibrium ratio there is a minimal variation in the optimal profile that yields 2.1 times the velocity of the basal states. Though most of this increase is accounted for by the increase of the second order constants (that could have already reached a theoretical maximum) significant increases (10%) in catalytic efficiencies are obtained by changes of the internal steps only. Besides these new findings our optimization approach has been able to reproduce results obtained with other approaches.     

A new framework for the estimation of control parameters in metabolic pathways using lin-log kinetics.

The control properties of biochemical pathways can be described by control coefficients and elasticities, as defined in the framework of metabolic control analysis. The determination of these parameters using the traditional metabolic control analysis relationships is, however, limited by experimental difficulties (e.g. realizing and measuring small changes in biological systems) and lack of appropriate mathematical procedures (e.g. when the more practical large changes are made). In this paper, the recently developed lin-log approach is proposed to avoid the above-mentioned problems and is applied to estimate control parameters from measurements obtained in steady state experiments. The lin-log approach employs approximative linear-logarithmic kinetics parameterized by elasticities and provides analytical solutions for fluxes and metabolite concentrations when large changes are made. Published flux and metabolite concentration data are used, obtained from a reconstructed section of glycolysis converting 3-phosphoglycerate to pyruvate [Giersch, C. (1995) Eur. J. Biochem. 227, 194-201]. With the lin-log approach, all data from different experiments can be combined to give realistic elasticity and flux control coefficient estimates by linear regression. Despite the large changes, a good agreement of fluxes and metabolite concentrations is obtained between the measured and calculated values according to the lin-log model. Furthermore, it is shown that the lin-log approach allows a rigorous statistical evaluation to identify the optimal reference state and the optimal model structure assumption. In conclusion, the lin-log approach addresses practical problems encountered in the traditional metabolic control analysis-based methods by introducing suitable nonlinear kinetics, thus providing a novel framework with improved procedures for the estimation of elasticities and control parameters from large perturbation experiments.     

Interdependence between cooperativity and control coefficients

Influence of the concentration of internal metabolites on the control coefficient (defined as fractional change in flux per fractional change in enzyme activity) and regulatory properties of a given enzyme have been studied theoretically using a cyclic model of three enzymes. This model is useful to investigate the properties of the flux control coefficient for an enzyme following different rate equations. Enzymes can have high or low values of control coefficient irrespective of the type of kinetic equation, but the results obtained show that the sensitivity of these values to substrate variations is strongly dependent on its rate equation. These results help identify which kinetic equation allows the best control of a given metabolic pathway. These results have been applied to the purine nucleotide cycle. It is demonstrated that the best control of the cycle is reached when the irreversible reaction catalyzed by AMP deaminase follows a rate law that corresponds to a rational function of 2:2 degree with respect to AMP concentration.     

Alternative kinetic laws to describe the turnover of the microbial biomass

Many models that describe the turnover of the microbial biomass in soil use either first order kinetics where the rate of turnover is directly proportional to the microbial mass, or a variant of the Michaelis-Menten law that describes enzyme kinetics. To account for the different rates of microbial turnover observed at different times after the addition of substrate, some authors have suggested the existence of more than one pool of biomass. Each pool obeys the same kinetic law but with a different rate. In other experiments a disproportionately large increase in the turnover of native organisms has been observed relative to the amount of fresh substrate added. A change in the kinetic law describing the turnover of organisms can account for these observations and yet retain the simplicity of a single pool of micro-organisms. However where multiple pools of organisms are justified a mixed kinetic law with both first and second order terms may be more appropriate; in other words one pool of micro-organisms but two rate constants. The advantage of retaining a single pool of microbial biomass is that models may more readily be constructed in relation to the routine measurements of total microbial mass.     

Diffusion-dependent rates for the hydrolysis reaction catalyzed by glyoxalase II from rat erythrocytes

Glyoxalase II from rat erythrocytes is a near optimal catalyst for the hydrolysis of S-D-lactoylglutathione in the sense that the magnitude of kcat/Km is limited, in large part, by the rate constant for diffusion-controlled encounter between substrate and active site. The experimental basis for this conclusion is derived from the dependencies of the kinetic properties of the enzyme on solution viscosity (pH 7, Ic = 0.1 M, 25 degrees C). When sucrose is used as a viscogenic agent, kcat/Km for S-D-lactoylglutathione (8.8 x 10(5) M-1 s-1) decreases markedly with increasing solution viscosity. This effect appears not to be due to a sucrose-induced change in the intrinsic kinetic properties of the enzyme, since kcat/Km for the slow substrate S-acetylglutathione (3.7 x 10(4) M-1 s-1) is nearly independent of solution viscosity. Quantitative treatment of the data using Stoke's law indicates that the rate of hydrolysis of S-D-lactoylglutathione will be approximately 50% diffusion limited when [substrate] much less than Km; the encounter complex between enzyme and substrate partitions nearly equally between product formation and dissociation to form free enzyme and substrate. The same conclusion is reached when glycerol is used as a viscogenic agent, once the apparent activation effect of glycerol on the intrinsic activity of the enzyme is taken into account. Finally, the rate of formation of the encounter complex between substrate and active site may be governed to a significant extent by charge-charge interactions.(ABSTRACT TRUNCATED AT 250 WORDS)     

In vivo kinetics of primary metabolism in Saccharomyces cerevisiae studied through prolonged chemostat cultivation

In this study, prolonged chemostat cultivation is applied to investigate in vivo enzyme kinetics of Saccharomyces cerevisiae. S. cerevisiae was grown in carbon-limited aerobic chemostats for 70-95 generations, during which multiple steady states were observed, characterized by constant intracellular fluxes but significant changes in intracellular metabolite concentrations and enzyme capacities. We provide evidence for two relevant kinetic mechanisms for sustaining constant fluxes: in vivo near-equilibrium of reversible reactions and tight regulation of irreversible reactions by coordinated changes of metabolic effectors. Using linear-logarithmic kinetics, we illustrate that these multiple steady-state measurements provide linear constraints between elasticity parameters instead of their absolute values. Upon perturbation by a glucose pulse, glucose uptake and ethanol excretion in prolonged cultures were remarkably lower, compared to a reference culture perturbed at 10 generations. Metabolome measurements during the transient indicate that the differences might be due to a reduced ATP regeneration capacity in prolonged cultures.     

Clinical Use of Aromatase Inhibitors: Current Data and Future Perspectives

Abstract

A systematic procedure for the kinetic study of irreversible inhibition when the enzyme is consumed in the reaction which it catalyses, has been developed and analysed. Whereas in most reactions the enzymes are regenerated after each catalytic event and serve as reusable transacting effectors, in the consumed enzymes each catalytic center participates only once and there is no enzyme turnover. A systematic kinetic analysis of irreversible inhibition of these enzyme reactions is presented. Based on the algebraic criteria proposed in this work, it should be possible to evaluate either the mechanism of inhibition (complexing or non-complexing), or the type of inhibition (competitive, non-competitive, uncompetitive, mixed non-competitive). In addition, all kinetic constants involved in each case could be calculated. An experimental application of this analysis is also presented, concerning peptide bond formation in vitro. Using the puromycin reaction, which is a model reaction for the study of peptide bond formation in vitro and which follows the same kinetic law as the enzymes under study, we have found that: (i) the antibiotic spiramycin inhibits the puromycin reaction as a competitive irreversible inhibitor in a one step mechanism with an association rate constant equal to 1.3 × 10⁴M^-1s^-1 and, (ii) hydroxylamine inhibits the same reaction as an irreversible non-competitive inhibitor also in a one step mechanism with a rate constant equal to 1.6 × 10^-3 M^-1s^-1.     

Mechanistic origin of the kinetic cooperativity of hexokinase type L1 from wheat germ

A theoretical analysis is presented which shows that initial velocity data for hexokinase L1 catalysis of glucose phosphorylation by MgATP cannot be reconciled with the observed rate of the 'mnemonical' conformational transition which has been proposed to account for the kinetic cooperativity of the enzyme. The basic kinetic properties of hexokinase L1 and other allegedly 'mnemonical' enzymes appear to be fully consistent with an ordered ternary-complex mechanism in which the leading substrate participates in abortive-complex formation. It is concluded that, so far, no enzyme displaying kinetic cooperativity has been convincingly demonstrated to operate by a 'mnemonical' type of reaction mechanism.     

Implications of the integrated rate law for the reactions of Paracoccus denitrificans nitrite reductase.

The integrated rate law for the reaction of the nitrite reductase of Paracoccus denitrificans, a cytochrome cd, has been established for turnover assays using donor ferrocytochromes c and either nitrite or molecular oxygen as the ultimate acceptor. The time course for the concentration of ferrocytochrome follows the law: formula: (see text), where S is the concentration of donor ferrocytochrome c, So is the initial concentration, t is time, and u1, u2, and u3 are empirical parameters that are constant for a given experiment but depend upon the initial substrate concentration. In particular, all the u1 increase with decreasing initial ferrocytochrome concentration. Saturation of reaction rates at high donor ferrocytochrome concentrations was not observed. The parameter u1 was proportional to the enzyme concentration while u2 and u3 were not. The form of the integrated rate law and the behavior of the u1 impose severe restrictions on possible kinetic schemes for the activity of the enzyme. Contemporary mechanisms that have been proposed for mitochondrial oxidase aa3 are examined and found to be inadequate to explain the reactivity of cytochrome cd. The simplest interpretations of the cytochrome cd data suggest that the enzyme does not bind the ferri and ferro forms of donor cytochromes c with equal affinity and that the enzyme is subject to inhibition by a product of reaction. Eucaryotic horse cytochrome c reacts with the Paracoccus cytochrome cd with 77% of the activity when Paracoccus cytochrome c550 is used as the electron donor.     

Metabolic Engineering with power-law and linear-logarithmic systems

Metabolic Engineering aims to improve the performance of biotechnological processes through rational manipulation rather than random mutagenesis of the organisms involved. Such a strategy can only succeed when a mathematical model of the target process is available. Simplifying assumptions are often needed to cope with the complexity of such models in an efficient way, and the choice of such assumptions often leads to models that fall within a certain structural template or formalism. The most popular formalisms can be grouped in two categories: power-law and linear-logarithmic. As optimization and analysis of a model strongly depends on its structure, most methods in Metabolic Engineering have been defined within a given formalism and never used in any other.In this work, the four most commonly used formalisms (two power-law and two linear-logarithmic) are placed in a common framework defined within Biochemical Systems Theory. This framework defines every model as matrix equations in terms of the same parameters, enabling the formulation of a common steady state analysis and providing means for translating models and methods from one formalism to another. Several Metabolic Engineering methods are analysed here and shown to be variants of a single equation. Particularly, two problem solving philosophies are compared: the application of the design equation and the solution of constrained optimization problems. Generalizing the design equation to all the formalisms shows it to be interchangeable with the direct solution of the rate law in matrix form. Furthermore, optimization approaches are concluded to be preferable since they speed the exploration of the feasible space, implement a better specification of the problem and exclude unrealistic results.Beyond consolidating existing knowledge and enabling comparison, the systematic approach adopted here can fill the gaps between the different methods and combine their strengths.     

Monte-Carlo Modeling of the Central Carbon Metabolism of Lactococcus lactis: Insights into Metabolic Regulation

Metabolic pathways are complex dynamic systems whose response to perturbations and environmental challenges are governed by multiple interdependencies between enzyme properties, reactions rates, and substrate levels. Understanding the dynamics arising from such a network can be greatly enhanced by the construction of a computational model that embodies the properties of the respective system. Such models aim to incorporate mechanistic details of cellular interactions to mimic the temporal behavior of the biochemical reaction system and usually require substantial knowledge of kinetic parameters to allow meaningful conclusions. Several approaches have been suggested to overcome the severe data requirements of kinetic modeling, including the use of approximative kinetics and Monte-Carlo sampling of reaction parameters. In this work, we employ a probabilistic approach to study the response of a complex metabolic system, the central metabolism of the lactic acid bacterium Lactococcus lactis, subject to perturbations and brief periods of starvation. Supplementing existing methodologies, we show that it is possible to acquire a detailed understanding of the control properties of a corresponding metabolic pathway model that is directly based on experimental observations. In particular, we delineate the role of enzymatic regulation to maintain metabolic stability and metabolic recovery after periods of starvation. It is shown that the feedforward activation of the pyruvate kinase by fructose-1,6-bisphosphate qualitatively alters the bifurcation structure of the corresponding pathway model, indicating a crucial role of enzymatic regulation to prevent metabolic collapse for low external concentrations of glucose. We argue that similar probabilistic methodologies will help our understanding of dynamic properties of small-, medium- and large-scale metabolic networks models.     

Steady-state analysis of metabolic pathways: comparing the double modulation method and the lin-log approach

The increasing interest in studying enzyme kinetics under in vivo conditions requires practical methods to estimate control parameters from experimental data. In contrast to currently established approaches of dynamic modelling, this paper addresses the steady-state analysis of metabolic pathways. Within the framework of metabolic control analysis (MCA), elasticity coefficients are used to describe the control properties of a local enzyme reaction. The double modulation method is one of the first experimental approaches to estimate elasticity coefficients from measurements of steady-state flux rates and metabolite concentrations. We propose a generalized form of the double modulation method and compare it to the recently developed linear-logarithmic approach.     

Some properties of an alcohol dehydrogenase partially purified from baker''s yeast grown without added zinc.

Alcohol dehydrogenase was partially purified from yeast (Saccharomyces cerevisiae) grown in the presence of 20 muM-MnSO4 without added Zn2+ and from yeast grown in the presence of 1.8 muM-MnSO4. The enzyme from yeast grown with added Zn2+ has the same properties as the crystalline enzyme from commercial supplies of baker's yeast. The enzyme from yeast grown without added An2+ has quite different properties. It has a mol.wt. in the region of 72000 and an S 20 w of 5.8S. The values can be compared with a mol.wt. of 141000 and an S 20 w of 7.6S for the crystalline enzyme. ADP-ribose, a common impurity in commercial samples of NAD+, is a potent competitive inhibitor of the new enzyme (K1 = 0.5 muM), but is not so for the crystalline enzyme. The observed maximum rate of ethanol oxidation at pH 7.05 and 25 degrees C was decreased 12-fold by the presence of 0.06 mol of inhibitor/mol of NAD+ when using the enzyme from Zn2+-deficient yeast, but with crystalline enzyme the maximum rate was essentially unchanged by this concentration of inhibitor. The kinetic characteristics for the two enzymes with ethanol, butan-1-ol, acetaldehyde and butyraldehyde as substrates are markedly different. These kinetic differences are discussed in relation to the mechanism of catalysis for the enzyme from Zn2+-deficient yeast.     

Theoretical approach to the steady-state kinetics of a bi-substrate acyl-transfer enzyme reaction that follows a hydrolysable-acyl-enzyme-based mechanism. Application to the study of lysophosphatidylcholine:lysophosphatidylcholine acyltransferase from rabbit lung.

A kinetic model is proposed for catalysis by an enzyme that has several special characteristics: (i) it catalyses an acyl-transfer bi-substrate reaction between two identical molecules of substrate, (ii) the substrate is an amphiphilic molecule that can be present in two physical forms, namely monomers and micelles, and (iii) the reaction progresses through an acyl-enzyme-based mechanism and the covalent intermediate can react also with water to yield a secondary hydrolytic reaction. The theoretical kinetic equations for both reactions were deduced according to steady-state assumptions and the theoretical plots were predicted. The experimental kinetics of lysophosphatidylcholine:lysophosphatidylcholine acyltransferase from rabbit lung fitted the proposed equations with great accuracy. Also, kinetics of inhibition by products behaved as expected. It was concluded that the competition between two nucleophiles for the covalent acyl-enzyme intermediate, and not a different enzyme action depending on the physical state of the substrate, is responsible for the differences in kinetic pattern for the two activities of the enzyme. This conclusion, together with the fact that the kinetic equation for the transacylation is quadratic, generates a 'hysteretic' pattern that can provide the basis of self-regulatory properties for enzymes to which this model could be applied.     

Kinetic evidence for the presence of two forms of M2-type pyruvate kinase in rat small intestine

Some kinetic properties of pyruvate kinase from rat small intestine have been investigated. The relative insensitivity of the enzyme to ATP inhibition and the amino acid inhibition pattern allows the conclusion that intestinal pyruvate kinase belongs to the M₂-type. The pyruvate kinase activity as a function of the phosphoenol pyruvate concentration is characterized by two different n values. The activity correlating with the low n value is stimulated by Fru-1,6-P₂, whereas the activity at higher phosphoenol pyruvate concentrations is not influenced by this glycolytic intermediate. These results, together with the partial relief of the amino acid inhibition by Fru-1,6-P₂, show that two forms of the enzyme are present with different kinetic properties. The metabolic implication of the kinetic properties of pyruvate kinase for rat small intestine is discussed.     

Shear breakage of nylon membrane microcapsules in a turbine reactor

The breakage of nylon membrane microcapsules is proposed as a new method to study and quantify shear effects in biological systems. A critique of this method shows that a narrower particle size distribution may be an important improvement in the breakage study as well as breakage control in many bioreactor and biotechnological applications. In a turbine reactor, it was shown that the primary process which determines the microcapsule breakage is the shear effect. The breakage kinetics are first order with regard to the microcapsule concentration. The breakage kinetic constant was ob served to be dependent on the temperature and the particle size, and proportional to the average shear rate and the third power of the turbine angular velocity. Decrease of the breakage kinetic constant with temperature can be explained by a decrease of fluid viscosity and a change in nylon membrane properties. An increase in the breakage kinetic constant with the microcapsule diameter can be due to a lowering of internal pressure and a reduction of the membrane resistance with size. Proportionality between the breakage kinetic constant and the shear rate shows that shear is the main process which leads to microcapsule breakage. The additional intervention in the shear rate expression of the turbine angular speed in the form of the turbine and particle velocities, results in the dependence of the breakage kinetic constant on the third power of the angular velocity.     

An analysis of the flow of a power law fluid due to ciliary motion in an infinite channel

The propulsion mechanics of cilia-induced flow is studied through a mathematical model. The problem of two-dimensional motion of a power law fluid inside a channel with ciliated walls is considered. The characteristics of ciliary systems are determined by the dominance of viscous effects over inertial effects using the long-wavelength approximation. Solutions for the longitudinal, transverse, and resultant velocities are obtained. The pressure gradient and volume flow rate for different values of the power law index are also calculated. The flow properties for the power law fluid are determined as a function of the cilia and metachronal wave velocity. The viscous and power law fluid are compared and discussed graphically.