Advantages and disadvantages of aggregating fluxes into synthetic and degradative fluxes when modelling metabolic pathways.

It is now widely accepted that mathematical models are needed to predict the behaviour of complex metabolic networks in the cell, in order to have a rational basis for planning metabolic engineering with biotechnological or therapeutical purposes. The great complexity of metabolic networks makes it crucial to simplify them for analysis, but without violating key principles of stoichiometry or thermodynamics. We show here, however, that models for branched complex systems are sometimes obtained that violate the stoichiometry of fluxes at branch points and as a result give unrealistic metabolite concentrations at the steady state. This problem is especially important when models are constructed with the S-system form of biochemical systems theory. However, the same violation of stoichiometry can occur in metabolic control analysis if control coefficients are assumed to be constant when trying to predict the effects of large changes. We derive the appropriate matrix equations to analyse this type of problem systematically and to assess its extent in any given model.     

Detection of elementary flux modes in biochemical networks: a promising tool for pathway analysis and metabolic engineering

Rational metabolic engineering requires powerful theoretical methods such as pathway analysis, in which the topology of metabolic networks is considered. All metabolic capabilities in steady states are composed of elementary flux modes, which are minimal sets of enzymes that can each generate valid steady states. The modes of the fructose-2,6-bisphosphate cycle, the combined tricarboxylic-acid-glyoxylate-shunt system and tryptophan synthesis are used here for illustration. This approach can be used for many biotechnological applications such as increasing the yield of a product, channelling a product into desired pathways and in functional reconstruction from genomic data.     

Integration of temporal analysis and control analysis of metabolic systems.

A theory is developed that integrates approaches to the analysis of pathway transient response and metabolic control analysis. A Temporal Control Coefficient is defined that is a measure of the system's transient response to modulation of enzyme activity or concentration. The approach allows for the analysis of the establishment of a steady state from rest, of the system's 'agility' of response to minor perturbations of a pre-existing steady state and of the macroscopic transition between steady states. In the last-mentioned case it is shown that, like the transient time itself, the control of transient response retains the property of independence from the mechanism of the transition. In consequence, the Temporal Control Coefficient can be defined in terms of the control properties of the initial and final states alone without reference to the mechanism of transition. A summation property is shown to apply to the Temporal Control Coefficients in each case. Connectivity relationships between elasticities and Temporal Control Coefficients are also established.     

Controllability of non-linear biochemical systems

Mathematical methods of biochemical pathway analysis are rapidly maturing to a point where it is possible to provide objective rationale for the natural design of metabolic systems and where it is becoming feasible to manipulate these systems based on model predictions, for instance, with the goal of optimizing the yield of a desired microbial product. So far, theory-based metabolic optimization techniques have mostly been applied to steady-state conditions or the minimization of transition time, using either linear stoichiometric models or fully kinetic models within biochemical systems theory (BST). This article addresses the related problem of controllability, where the task is to steer a non-linear biochemical system, within a given time period, from an initial state to some target state, which may or may not be a steady state. For this purpose, BST models in S-system form are transformed into affine non-linear control systems, which are subjected to an exact feedback linearization that permits controllability through independent variables. The method is exemplified with a small glycolytic-glycogenolytic pathway that had been analyzed previously by several other authors in different contexts.     

In vivo 13C NMR determines metabolic fluxes and steady state in linseed embryos

The dynamics of developing linseed embryo metabolism was investigated using (13)C-labelling experiments where the real-time kinetics of label incorporation into metabolites was monitored in situ using in vivo NMR. The approach took advantage of the occurrence in this plant tissue of large metabolite pools - such as sucrose or lipids - to provide direct and quantitative measurement of the evolution of the labelling state within central metabolism. As a pre-requisite for the use of steady state flux measurements it was shown that isotopic steady state was reached within 3 h at the level of central intermediates whereas it took a further 6h for the sucrose pool. Complete isotopic and metabolic steady state took 18 h to be reached. The data collected during the transient state where label was equilibrated but the metabolic steady state was incomplete, enabled the rates of lipid and sucrose synthesis to be measured in situ on the same sample. This approach is suitable to get a direct assessment of metabolic time-scales within living plant tissues and provides a valuable complement to steady state flux determinations.     

A transfer-function representation for regulatory responses of a controlled metabolic pathway

A transfer-function representation for the response of a controlled metabolic pathway to the changes in influx and efflux rates of metabolites is formulated to describe analytically and approximately the regulatory behavior of the pathway around a steady state. The pathway model analysed is an open and homogeneous system which consists of two consecutive enzymatic reactions catalyzed by an allosteric enzyme of Monod-Wyman-Changeux (MWC) dimeric model and a Michaelis-Menten-type enzyme, respectively, and undergoes the feedback inhibition by the end product. The rate equation for the system (a system of ordinary differential equations) is linearized about a steady state, so that the responses of the reaction rates to the changes in influx rate of the substrate and efflux rate of the end product are expressed in a form of transfer function. The formulation leads to the transfer function for the response of production rate of the end product to the change in its efflux rate to clarify the regulatory response of feedback mechanism in controlled metabolic pathways. The relationship among the chemical species in the system at steady stete also supports a reasonable assumption that the regulatory mechanisms in metabolic pathways are to control the production of end product against the change in its demand from the cellular environments.     

A metabolic control analysis of kinetic controls in ATP free energy metabolism in contracting skeletal muscle

A system analysis ofATP free energy metabolism in skeletal muscle was made using theprinciples of metabolic control theory. We developed a network model ofATP free energy metabolism in muscle consisting of actomyosin ATPase,sarcoplasmic reticulum (SR) Ca²⁺-ATPase, and mitochondria.These components were sufficient to capture the major aspects of theregulation of the cytosolic ATP-to-ADP concentration ratio (ATP/ADP) inmuscle contraction and had inherent homeostatic properties regulatingthis free energy potential. As input for the analysis, we used ATPmetabolic flux and the cytosolic ATP/ADP at steady state at sixcontraction frequencies between 0 and 2 Hz measured in human forearmflexor muscle by ³¹P-NMR spectroscopy. We used themathematical formalism of metabolic control theory to analyze thedistribution of fractional kinetic control of ATPase flux and theATP/ADP in the network at steady state among the components over thisexperimental range and an extrapolated range of stimulation frequencies(up to 10 Hz). The control analysis showed that the contractileactomyosin ATPase has dominant kinetic control of ATP flux in forearmflexor muscle over the 0- to 1.6-Hz range of contraction frequenciesthat resulted in steady states, as determined by ³¹P-NMR.However, flux control begins to shift toward mitochondria at >1 Hz.This inversion of flux control from ATP demand to ATP supply controlhierarchy progressed as the contraction frequency increased past 2 Hzand was nearly complete at 10 Hz. The functional significance of thisresult is that, at steady state, ATP free energy consumption cannotoutstrip the ATP free energy supply. Therefore, this reduced,three-component muscle ATPase system is inherently homeostatic.

     

In vitro control analysis of an enzyme system: Experimental and analytical developments

In this paper we describe a flow-through system for reconstituting parts of metabolism from purified enzymes. This involves pumping continuously into a reaction chamber, fresh enzymes and reagents so that metabolic reactions occur in the chamber. The waste products leave the chamber via the outflow so that a steady state can be setup. The system we chose consisted of a single enzyme, lactate dehydrogenase. This enzyme was chosen because it consumes NADH in the chamber which could be monitored spectrophotometrically. The aim of the work was to investigate whether a steady state could be achieved in the flow system and whether a metabolic control analysis could be done. We measured two control coefficients, C_LDH and C_pump for the enzyme flux and NADH concentration and confirmed that the summation theorem applied to this system. The advantage of a flow-through system is that the titrations necessary to estimate the control coefficients can be easily and precisely controlled; this means that accurate estimates for the control coefficients can be obtained. In the paper, we discuss some statistical aspects of the data analysis and some possible applications of the technique, including a method to determine the presence of metabolic channelling between two different enzymes.     

Multiplicity and stability analysis of microorganisms in continuous culture: effects of metabolic overflow and growth inhibition

Metabolic overflow (enhanced uptake of substrate and secretion of intermediates) is a phenomenon often observed for cells grown under substrate excess. Growth inhibition by substrate and/or product is also normally found for this kind of culture. An effort is made in this work to analyze the dynamic behavior of a continuous culture subject to metabolic overflow and growth inhibition by substrate and/or product. Analysis of a model system shows that in a certain range of operating conditions three nonwashout steady state solutions are possible. Local stability analysis indicates that only two of them are stable thus leading to multiplicity and hysteresis. Further analysis of the intrinsic effects of different terms describing the metabolic overflow and growth inhibitions reveals that for the model system and the parameters considered, the combined effects of product inhibition and an enhanced formation rate of product under substrate excess cause the multiplicity and hysteresis. Growth inhibition by substrate and/or an enhanced substrate uptake appear not to be necessary conditions. The combined effects of enhanced product formation and product inhibition can also lead to unusual dynamic behavior such as a prolonged time period to reach a steady state, oscillatory transition from one steady state to another, and sustained oscillations. Using the occurrence of multiplicity and oscillation as criteria, the operating regime of a continuous culture can be divided into four domains: one with multiplicity and oscillation, one with unique steady state but possible oscillatory behavior, the other two with unique and stable steady state. The model predictions are in accordance with recent experimental results. The results presented in this work may be used as guidelines for choosing proper operating conditions of similar culture systems to avoid undesired instability and multiplicity. Copyright 1998 John Wiley & Sons, Inc.     

Optimization of regulatory architectures in metabolic reaction networks

Successful biotechnological applications, such as amino acid production, have demonstrated significant improvement in bioprocess performance by genetic modifications of metabolic control architectures and enzyme expression levels. However, the stoichiometric complexity of metabolic pathways, along with their strongly nonlinear nature and regulatory coupling, necessitates the use of structured kinetic models to direct experimental applications and aid in quantitative understanding of cellular bioprocesses. A novel optimization problem is introduced here, the objective of which is to identify changes in the regulatory characteristics of pertinent enzymes and in their cellular content which should be implemented to optimize a particular metabolic process. The mathematical representation of the metabolic reaction networks used is the S-system representation, which at steady state is characterized by linear equations. Exploiting the linearity of the representation, we formulated the optimization problem as a mixed-integer linear programming (MILP) problem. This formulation allows the consideration of a regulatory superstructure that contains all alternative regulatory structures that can be considered for a given pathway. The proposed approach is developed and illustrated using a simple linear pathway. Application of the framework on a complicated pathway-namely, the xanthine monophosphate (XMP) and guanosine monophosphate (GMP) synthesis pathway-identified the modification of the regulatory architecture that, along with changes in enzyme expression levels, can increase the XMP and GMP concentration by over 114 times the reference value, which is 50 times more than could be achieved by changes in enzyme expression levels only. (c) 1996 John Wiley & Sons, Inc.     

Towards a metabolic and isotopic steady state in CHO batch cultures for reliable isotope-based metabolic profiling

Attaining metabolic and isotopic balanced growth is one critical condition for physiological studies using isotope-labeled tracers, but is very difficult to obtain in batch culture due to the extensive metabolite exchange with the surrounding medium and related physiological changes. In the present study, we investigated metabolic and isotopic behavior of CHO cells in differently designed media. We observed that the assumption of balanced cell growth cannot be justified in batch culture of CHO cells directly using conventional, commercially available media. By systematically redesigning media composition and characterizing metabolic steady state based on mass balances and measurement of labeling dynamics, we achieved balanced cell growth for the main cellular substrates in CHO cells. This was done in a step-by-step analysis of growth and primary metabolism of CHO cells with the use of [U-¹³C]glucose feeding and adjusting concentrations of amino acids in the growth medium. The optimized media obtained at the end of the study provide balanced growth and isotopic steady state or at least asymptotic steady state. As a result, we established a platform to conduct isotope-based physiological studies of mammalian systems more reliably and therefore well suited for later use in metabolic profiling of mammalian systems such as ¹³C-labeled metabolic flux analysis.     

Kinetic modelling of plant metabolic pathways

This paper provides a review of kinetic modelling of plant metabolic pathways as a tool for analysing their control and regulation. An overview of different modelling strategies is presented, starting with those approaches that only require a knowledge of the network stoichiometry; these are referred to as structural. Flux-balance analysis, metabolic flux analysis using isotope labelling, and elementary mode analysis are briefly mentioned as three representative examples. The main focus of this paper, however, is a discussion of kinetic modelling, which requires, in addition to the stoichiometry, a knowledge of the kinetic properties of the constituent pathway enzymes. The different types of kinetic modelling analysis, namely time-course simulation, steady-state analysis, and metabolic control analysis, are explained in some detail. An overview is presented of strategies for obtaining model parameters, as well as software tools available for simulation of such models. The kinetic modelling approach is exemplified with discussion of three models from the general plant physiology literature. With the aid of kinetic modelling it is possible to perform a control analysis of a plant metabolic system, to identify potential targets for biotechnological manipulation, as well as to ascertain the regulatory importance of different enzymes (including isoforms of the same enzyme) in a pathway. Finally, a framework is presented for extending metabolic models to the whole-plant scale by linking biochemical reactions with diffusion and advective flow through the phloem. Future challenges include explicit modelling of subcellular compartments, as well as the integration of kinetic models on the different levels of the cellular and organizational hierarchy.     

Analysis of pharmacokinetic parameters for assessment of dextromethorphan metabolic phenotypes

In this study, the metabolic ratios of dextromethorphan to dextrorphan (DM/DX) in plasma were calculated at steady state after administering 2 dosage forms (Medicon) and Detusiv) of DM with different release rates. The urinary metabolic ratio for each subject was also determined based on the total drug concentration in the urine. An analysis of pharmacokinetic parameters for determining the DM metabolic phenotype was conducted. Results demonstrate that double logarithmic correlations between the metabolic ratios based on pharmacokinetic parameters of either AUC(0-tau,ss), C(max,ss), C(min,ss), or C(ave,ss) for Medicon and Detusiv and the urinary metabolic ratios were all significant. Probit plots of the metabolic ratios based on these pharmacokinetic parameters revealed 2 clusters of distribution, representing extensive and intermediate metabolizers. An antimode of 2.0 for total drug based on these pharmacokinetic parameters was determined and correspondingly referred to an antimode of 0.02 for the urinary metabolic ratio to delineate extensive and intermediate metabolizers. This model was also verified to be appropriate when using total plasma concentrations of DM and DX at any time during the period of the dosing interval at steady state to calculate the metabolic ratio for identifying extensive and intermediate metabolizers. Therefore, the metabolic ratio based on the pharmacokinetic parameters of either AUC(0-tau,ss), C(max,ss), C(min,ss), or C(ave,ss) and plasma concentrations of DM and DX in a single blood sample at steady state are proposed as an alternative way to identify phenotypes of CYP2D6.     

Basic studies on metabolic steady state. Incompletely mixed systems

This paper is the second of a pair dealing with some mathematical properties of metabolic steady state. An investigator wishing to compute the rate of appearance and/or disappearance of a metabolite in steady state within an intact biological system will usually appeal to a method involving radioactive tracers. It is shown that while the investigator’s choice of the mode of tracer administration (constant infusion or single injection) is largely arbitrary, the mathematical interpretation of the results may depend upon the presence or absence of gradients in certain of the variables of the system. The latter will be the case if the system is sampled at a point within the distribution space of the metabolite which is not a source point but is otherwise arbitrary. In order to deduce a formula which gives the required rate, he must have knowledge of the gradient of concentration of the traced substance, and sometimes of the gradient of specific activity.     

Temporal analysis of metabolic systems and its application to metabolite channelling

When a metabolic system undergoes a transition between steady states, the lag or transition time of the system is determined by the aggregated lifetimes of the metabolite pools. This allows the transition time, and hence the temporal responsiveness of the system, to be estimated from a knowledge of the starting and finishing steady states and obviates the need for dynamic measurements. The analysis of temporal response in metabolic systems may be integrated with the general field of metabolic control analysis by the definition of a temporal control coefficient (C) in terms of flux and concentration control coefficients. The temporal control coefficient exhibits summation and other properties analogous to the flux and concentration control coefficients. For systems in which static metabolite channels exits, the major kinetic advantage of channelling is a reduction in pool sizes and, as a result, a more rapid system response reflected in a reduced transition time. The extent of the channelling advantage may therefore be assessed from a knowledge of the system transition time. This reveals that no channelling advantage is achieved at high enzyme concentration (i.e., comparable to K_m) or, in the case of ‘leaky’ channels, where rapid equilibrium kinetic mechanisms obtain. In the case of a perfect channel with no leakage and direct transfer of metabolite between adjacent enzyme active sites, the transition time is minimized and equal to the lifetime of the enzyme–substrate complex.     

Flux estimation using isotopic tracers: common ground for metabolic physiology and metabolic engineering

Metabolic physiologists and metabolic engineers share the need to estimate flux. However, the physiologist often works with systems that do not maintain steady state for long. Many sites cannot be sampled, and calculating mass and isotopic balance for the entire system may not be feasible. To deal with these constraints, metabolic physiologists have developed specialized isotopic techniques that may be unfamiliar to metabolic engineers. A selection of these techniques is presented here, not because it is anticipated that they would be used by engineers exactly as in the physiologist's setting, but because they illustrate novel applications of tracer methodology. Creative engineers may find new adaptations of these tools in metabolic engineering and opportunities to increase redundancy. Physiologists, entering into a dialog with engineers, may see more clearly the potential of comprehensive models and revisit the impediments to a more complete analysis of human metabolic systems.     

Interpreting metabolic complexity via isotope-assisted metabolic flux analysis

Isotope-assisted metabolic flux analysis (iMFA) is a powerful method to mathematically determine the metabolic fluxome from experimental isotope labeling data and a metabolic network model. While iMFA was originally developed for industrial biotechnological applications, it is increasingly used to analyze eukaryotic cell metabolism in physiological and pathological states. In this review, we explain how iMFA estimates the intracellular fluxome, including data and network model (inputs), the optimization-based data fitting (process), and the flux map (output). We then describe how iMFA enables analysis of metabolic complexities and discovery of metabolic pathways. Our goal is to expand the use of iMFA in metabolism research, which is essential to maximizing the impact of metabolic experiments and continuing to advance iMFA and biocomputational techniques.     

Dynamic metabolic flux analysis (DMFA): a framework for determining fluxes at metabolic non-steady state

Metabolic flux analysis (MFA) is a key tool for measuring in vivo metabolic fluxes in systems at metabolic steady state. Here, we present a new method for dynamic metabolic flux analysis (DMFA) of systems that are not at metabolic steady state. The advantages of our DMFA method are: (1) time-series of metabolite concentration data can be applied directly for estimating dynamic fluxes, making data smoothing and estimation of average extracellular rates unnecessary; (2) flux estimation is achieved without integration of ODEs, or iterations; (3) characteristic metabolic phases in the fermentation data are identified automatically by the algorithm, rather than selected manually/arbitrarily. We demonstrate the application of the new DMFA framework in three example systems. First, we evaluated the performance of DMFA in a simple three-reaction model in terms of accuracy, precision and flux observability. Next, we analyzed a commercial glucose-limited fed-batch process for 1,3-propanediol production. The DMFA method accurately captured the dynamic behavior of the fed-batch fermentation and identified characteristic metabolic phases. Lastly, we demonstrate that DMFA can be used without any assumed metabolic network model for data reconciliation and detection of gross measurement errors using carbon and electron balances as constraints.     

Design of large metabolic responses. Constraints and sensitivity analysis

Metabolic control analysis (Kacser & Burns (1973). Symp. Soc. Exp. Biol.27, 65-104; Heinrich & Rapoport (1974). Eur. J. Biochem.42, 89-95) has been extensively used to describe the response of metabolic concentrations and fluxes to small (infinitesimal) changes in enzyme concentrations and effectors. Similarly, metabolic control design (Acerenza (1993). J. theor. Biol.165, 63-85) has been proposed to design small metabolic responses. These approaches have the limitation that they were not devised to deal with large (non-infinitesimal) responses. Here we develop a strategy to design large changes in the metabolic variables. The only assumption made is that, for all the parameter values under consideration, the system has a unique stable steady state. The procedure renders the kinetic parameters of the rate equations that when embedded in the metabolic network produce the pattern of large changes in the steady-state variables that we aim to design. Structural and kinetic constraints impose restrictions on the type of responses that could be designed. We show that these conditions can be transformed into the language of mean-sensitivity coefficients and, as a consequence, a sensitivity analysis of large metabolic responses can be performed after the system has been designed. The mean-sensitivity coefficients fulfil conservation and summation relationships that in the limit reduce to the well-known theorems for infinitesimal changes. Finally, it is shown that the same procedure that was used to design metabolic responses and analyse their sensitivity properties can also be used to determine the values of kinetic parameters of the rate laws operating "in situ".