Use of implicit methods from general sensitivity theory to develop a systematic approach to metabolic control. I. Unbranched pathways

It is shown that metabolic control theory (MCT), is its present form, is a particular case of general sensitivity theory, which studies the effects of parameter variations on the behavior of dynamic systems. It has been shown that metabolic control theory is obtained from this more general theory for the particular case of steady-state and linear relationships between velocities and enzyme concentrations. In such conditions the relationships between elasticities and flux control coefficients are easily obtained. These relationships are in the form of a matrix product constructed in a priori form. Relationships between combined response coefficients and concentration control coefficients are presented. The use of implicit methodology from general sensitivity theory provides a generalization of MCT, which is applied to unbranched pathways. For this particular case, provided the matrices have been properly constructed, the matrix of global properties (flux and concentration control coefficients) can be obtained by inversion of the matrix of local properties (elasticities). The theorems of MCT (concentration summation, flux summation, flux connectivity, and concentration connectivity) applicable for unbranched pathways are directly obtained by inspection of the matrix product. With these results, the present theoretical basis of MCT is extended with a more structured framework that allows a wider range of application. The results make clearer the relatedness of MCT to the more general approach provided by biochemical systems theory (BST).     

The matrix method of metabolic control analysis: its validity for complex pathway structures

The sensitivities of the variables of a metabolic system (such as fluxes and concentrations) to variations in enzyme concentration are expressed in metabolic control analysis as control coefficients. The matrix method is a system of writing matrix equations that generate expressions for the control coefficients in terms of the characteristics of the components (principally the enzymes). Previously, the matrix method has been considered in terms of simple pathway structures; here we justify its applicability to complex pathways, such as those with multiple branches. It is shown that this requires modification of the branch point relationship to take account of changes of flux along the limbs of the branch and of stoichiometric factors. The method of deriving the flux control coefficients with respect to different fluxes in the system is extended to cope with these circumstances.     

Control of the metabolic flux in a system with high enzyme concentrations and moiety-conserved cycles. The sum of the flux control coefficients can drop significantly below unity.

In a number of metabolic pathways enzyme concentrations are comparable to those of substrates. Recently it has been shown that many statements of the 'classical' metabolic control theory are violated if such a system contains a moiety-conserved cycle. For arbitrary pathways we have found: (a) the equation connecting coefficients CEiJ (obtained by varying the Ei concentration) and CviJ (obtained by varying the kicat), and (b) modified summation equations. The sum of the enzyme control coefficients (equal to unity under the 'classical' theory) appears always to be below unity in the systems considered. The relationships revealed were illustrated by a numerical example where the sum of coefficients CEiJ reached negative values. A method for experimental measurements of the above coefficients is proposed.     

Getting to the inside of cells using metabolic control analysis

Metabolic control analysis can relate control properties of an intact system to kinetic properties (elasticity coefficients) of the enzymes within that system. The method formulating the former as matrix inverse of the latter is elaborated here for the general case and founded in standard metabolic control theory. Then a method is developed that accomplishes the reverse: it is shown that a matrix containing all elasticity coefficients and information concerning the pathway structure equals the inverse of a matrix containing flux and concentration control coefficients. As a consequence, by measuring the control properties of an intact system, one is able to deduce its in situ pathway structure and enzyme kinetic properties: This solves the ever-present question of whether the kinetic properties of enzymes in their isolated state differ from those under the conditions prevailing in the cell.     

Summation theorems for flux and concentration control coefficients of dynamic systems

Metabolic control analysis (MCA) was developed to quantify how system variables are affected by parameter variations in a system. In addition, MCA can express the global properties of a system in terms of the individual catalytic steps, using connectivity and summation theorems to link the control coefficients to the elasticity coefficients. MCA was originally developed for steady-state analysis and not all summation theorems have been derived for dynamic systems. A method to determine time-dependent flux and concentration control coefficients for dynamic systems by expressing the time domain as a function of percentage progression through any arbitrary fixed interval of time is reported. Time-dependent flux and concentration control coefficients of dynamic systems, provided that they are evaluated in this novel way, obey the same summation theorems as steady-state flux and concentration control coefficients, respectively.     

Control involving metabolism and gene expression

Control of DNA supercoiling by the free-energy of hydrolysis of ATP that involves gene expression is analyzed in terms of three levels of unconnected metabolic pathways. These are synthesis and breakdown of topoisomerase mRNAs, synthesis and breakdown of topoisomerase proteins and supercoiling and relaxation of DNA. The so-called square-matrix method previously developed for the control of metabolic pathways, is extended to deal with this hierarchical control system. It turns out that also in this case, the matrix of control coefficients is equal to the inverse of the so-called elasticity matrix, which contains all relevant elasticity coefficients as well as information about the structure and connectedness of the pathways involved. For a simpler case of a hierarchy of two systems, we demonstrate that the explicit matrix inversion method may be replaced by an implicit method in which the regulatory effects that run through the other level are described by an additional elasticy coefficient which may then be treated as if local.     

Experimental determination of flux control distribution in biochemical systems: in vitro model to analyze transient metabolite concentrations

Determination of the control coefficients allows the identification of rate-controlling steps in a reaction system. However, the measurement of the flux control coefficients in a biochemical system is not a trivial task, except for some special cases. We have developed a theoretical basis for the direct determination of these coefficients from dynamic responses. In order to show the validity of this methodology experimentally, the dynamic approach is applied to an in vitro reconstituted partial glycolytic pathway to determine the flux control coefficients of hexokinase and phosphofructokinase. It is shown that the dynamic approach gives consistent results, which agree well with values obtained by the direct enzyme titration method. The detailed procedure and potential applications to other systems, such as immobilized enzyme or cell reactors, are discussed. (c) 1993 Wiley & Sons, Inc.     

Optimization-based metabolic control analysis

In this work, a novel optimization-based metabolic control analysis (OMCA) method is introduced for reducing data requirement for metabolic control analysis (MCA). It is postulated that using the optimal control approach, the fluxes in a metabolic network are correlated to metabolite concentrations and enzyme activities as a state-feedback control system that is optimal with respect to a homeostasis objective. It is then shown that the optimal feedback gains are directly related to the elasticity coefficients (ECs) of MCA. This approach requires determination of the relative "importance" of metabolites and fluxes for the system, which is possible with significantly reduced experimental data, as compared with typical MCA requirements. The OMCA approach is applied to a top-down control model of glycolysis in hepatocytes. It is statistically demonstrated that the OMCA model is capable of predicting the ECs observed experimentally with few exceptions. Further, an OMCA-based model reconciliation study shows that the modification of four assumed stoichiometric coefficients in the model can explain most of the discrepancies, with the exception of elasticities with respect to the NADH/NAD ratio.     

Bidirectional reaction steps in metabolic networks: I. Modeling and simulation of carbon isotope labeling experiments

The extension of metabolite balancing with carbon labeling experiments, as described by Marx et al. (Biotechnol. Bioeng. 49: 11-29), results in a much more detailed stationary metabolic flux analysis. As opposed to basic metabolite flux balancing alone, this method enables both flux directions of bidirectional reaction steps to be quantitated. However, the mathematical treatment of carbon labeling systems is much more complicated, because it requires the solution of numerous balance equations that are bilinear with respect to fluxes and fractional labeling. In this study, a universal modeling framework is presented for describing the metabolite and carbon atom flux in a metabolic network. Bidirectional reaction steps are extensively treated and their impact on the system's labeling state is investigated. Various kinds of modeling assumptions, as usually made for metabolic fluxes, are expressed by linear constraint equations. A numerical algorithm for the solution of the resulting linear constrained set of nonlinear equations is developed. The numerical stability problems caused by large bidirectional fluxes are solved by a specially developed transformation method. Finally, the simulation of carbon labeling experiments is facilitated by a flexible software tool for network synthesis. An illustrative simulation study on flux identifiability from available flux and labeling measurements in the cyclic pentose phosphate pathway of a recombinant strain of Zymomonas mobilis concludes this contribution. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 55: 101-117, 1997.     

Metabolic network sensitivity analysis

A linear sensitivity analysis of metabolic regulation in nonsteady states is described. This treatment considers the effects of enzymatic and nonenzymatic reactions and spontaneous rapid equilibria. Sensitivity coefficients summarizing the influence of metabolite concentrations on reaction rates and pathway net flux are defined, as are sensitivity coefficients summarizing the effects of enzymes on metabolite concentrations and net flux. The sensitivity analysis is implemented in an easily used set of computer programs. A four-enzyme test model was shown to be resistant to intuitive interpretation. Sensitivity analysis showed a shift of control from the end of the enzymic sequence to the beginning of the sequence with changing metabolic state. The homeostatic behavior of the test system was shown to depend on the nonenzymatic reactions as well as on the enzymes. Under certain conditions metabolic regulation is shared so intimately among enzymes and spontaneous reactions that separation of their effects is impossible.     

The physiology of weak selection

Reaction rates in metabolic pathways typically exhibit a kind of diminishing returns in which small variations in the activities of the individual enzymes have very little effect on overall flux. These effects are measured by the control coefficients of the enzymes, and most systems are governed by the summation theorem stating that all control coefficients must sum to unity. One implication is that complex systems will not usually contain single rate limiting steps, but rather be controlled to a greater or lesser extent by many enzymes, each exerting relatively small control. Wright understood this principle in 1934 and used it for his physiological theory of dominance. With respect to small variations in enzyme activity, the principle implies that many small variations should have only mild effects on fitness. Analysis of nucleotide polymorphisms in the genes for glucose-6-phosphate dehydrogenase and alkaline phosphatase in Escherichia coli implies that most amino acid replacements are harmful, and that the average selection coefficient against amino acid replacements that are polymorphic in natural populations is 1 x 10(-7) to 5 x 10(-7). In experiments to determine the a priori distribution of selection coefficients among random amino acid replacements, 25 replacements in beta-galactosidase were created by genetic means, and 22 of these produced selective effects too small to be detected in chemostat competition experiments (s less than 0.004 per generation).     

A MINE Alternative to D-Optimal Designs for the Linear Model

Doing large-scale genomics experiments can be expensive, and so experimenters want to get the most information out of each experiment. To this end the Maximally Informative Next Experiment (MINE) criterion for experimental design was developed. Here we explore this idea in a simplified context, the linear model. Four variations of the MINE method for the linear model were created: MINE-like, MINE, MINE with random orthonormal basis, and MINE with random rotation. Each method varies in how it maximizes the MINE criterion. Theorem 1 establishes sufficient conditions for the maximization of the MINE criterion under the linear model. Theorem 2 establishes when the MINE criterion is equivalent to the classic design criterion of D-optimality. By simulation under the linear model, we establish that the MINE with random orthonormal basis and MINE with random rotation are faster to discover the true linear relation with regression coefficients and observations when . We also establish in simulations with , , and 1000 replicates that these two variations of MINE also display a lower false positive rate than the MINE-like method and additionally, for a majority of the experiments, for the MINE method.     

Comparisons of flux control exerted by mitochondrial outer-membrane carnitine palmitoyltransferase over ketogenesis in hepatocytes and mitochondria isolated from suckling or adult rats.

The primary aim of this paper was to calculate and report flux control coefficients for mitochondrial outer-membrane carnitine palmitoyltransferase (CPT I) over hepatic ketogenesis because its role in controlling this pathway during the neonatal period is of academic importance and immediate clinical relevance. Using hepatocytes isolated from suckling rats as our model system, we measured CPT I activity and carbon flux from palmitate to ketone bodies and to CO2 in the absence and presence of a range of concentrations of etomoxir. (This is converted in situ to etomoxir-CoA which is a specific inhibitor of the enzyme.) From these data we calculated the individual flux control coefficients for CPT I over ketogenesis, CO2 production and total carbon flux (0.51 +/- 0.03; -1.30 +/- 0.26; 0.55 +/- 0.07, respectively) and compared them with equivalent coefficients calculated by similar analyses [Drynan, L., Quant, P.A. & Zammit, V.A. (1996) Biochem. J. 317, 791-795] in hepatocytes isolated from adult rats (0.85 +/- 0.20; 0.23 +/- 0.06; 1.06 +/- 0.29). CPT I exerts significantly less control over ketogenesis in hepatocytes isolated from suckling rats than those from adult rats. In the suckling systems the flux control coefficients for CPT I over ketogenesis specifically and over total carbon flux (< 0.6) are not consistent with the enzyme being rate-limiting. Broadly similar results were obtained and conclusions drawn by reanalysis of previous data {from experiments in mitochondria isolated from suckling or adult rats [Krauss, S., Lascelles, C.V., Zammit, V.A. & Quant, P.A. (1996) Biochem. J. 319, 427-433]} using a different approach of control analysis, although it is not strictly valid to compare flux control coefficients from different systems. Our overall conclusion is that flux control coefficients for CPT I over oxidative fluxes from palmitate (or palmitoyl-CoA) differ markedly according to (a) the metabolic state, (b) the stage of development, (c) the specific pathway studied and (d) the model system.     

Canonical sensitivities: a useful tool to deal with large perturbations in metabolic network modeling

The dynamic range of metabolic models can be extended to deal with large perturbations by introducing the related concepts of "generalized" kinetic order and "canonical" sensitivities. Generalized kinetic orders are built as a well-defined non linear combination of the canonical sensitivities coefficients, which in turn are obtained by a least-squares regression on central composite factorial design data. In a such way, the whole domain of the operating variables is mapped without need to determine locally neither the first nor the second order model derivatives. The method was validated through numerical simulations, its predictions being compared with those coming from a Michaelis-Menten formalism taken as reference. In parallel, two variants of the Power-law formalism (S-system, least-squares GMA) also were tested. The canonical sensitivities method produced the widest range to predict metabolite concentrations and metabolic fluxes at the steady states. In addition, the variation pattern for the logarithmic gains and for the characteristic eigenvalues have been accurately determined from a unique overall model, being both required to make realistic analysis in metabolic engineering. The achieved information also can be expressed in terms of those typical coefficients derived from the Metabolic Control Analysis (MCA). Even if current first order Power-law or MCA formalisms were used, the canonical sensitivities approach provides a significant advantage, since complete sets of homologous, accurate, locally valid metabolic coefficients can be simultaneously recovered from the array proposed, being representative of the whole range of the operating variables instead of a unique nominal condition as is usual.     

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.     

Control of molecular transformations in polyenzyme systems: quantitative theory of the regulation of metabolism

An attempt of a comprehensive treatment of the theory of metabolic control is presented. The introductory section giving an outline of the early development of the theory, is followed by definitions quantifying the control in the metabolic system. By means of the perturbation method the complete system of equations is obtained which allows one to express all the enzyme control coefficients ("global" coefficients) through the elasticity coefficients characterizing kinetic properties of individual enzymes ("local" coefficients) and through the steady-state values of metabolic fluxes and concentrations. It is shown how connectivity relations between global and local coefficients should be modified when conserved sums of intermediates are present in the system. A new theorem is derived, it allows one to express the global response of the system to any change in the external parameter (such as external effector concentration, or temperature, pH, ionic strength, ets.) through the control coefficients and local responses of individual reaction steps. Explicit formulas are derived for response coefficients of the fluxes and concentrations to changes in the conserved sums of intermediates, which express the values of these global coefficients through the control and elasticity coefficients of enzymes and steady-state pools. The results obtained comprise as a special case all the results published so far in the literature.