Increasing the flux in metabolic pathways: A metabolic control analysis perspective

The problems of engineering increased flux in metabolic pathways are analyzed in terms of the understanding provided by metabolic control analysis. Over-expression of a single enzyme is unlikely to be effective unless it is known to have a high flux control coefficient, which can be used as an approximate predictive tool. This is likely to rule out enzymes subject to feedback inhibition, because it transfers control downstream from the inhibited enzyme to the enzymes utilizing the feedback metabolite. Although abolishing feedback inhibition can restore flux control to an enzyme, it is also likely to cause large increases in the concentrations of metabolic intermediates. Simultaneous and coordinated over-expression of most of the enzymes in a pathway can, in principle, produce substantial flux increases without changes in metabolite levels, though technically it may be difficult to achieve. It is, however, closer to the method used by cells to change flux levels, where coordinated changes in the level of activity of pathway enzymes are the norm. Another option is to increase the demand for the pathway product, perhaps by increasing its rate of excretion or removal. Copyright 1998 John Wiley & Sons, Inc.     

Irreversibility in unbranched pathways: preferred positions based on regulatory considerations

It has been observed experimentally that most unbranched biosynthetic pathways have irreversible reactions near their beginning, many times at the first step. If there were no functional reasons for this fact, then one would expect irreversible reactions to be equally distributed among all positions in such pathways. Since this is not the case, we have attempted to identify functional consequences of having an irreversible reaction early in the pathway. We systematically varied the position of the irreversible reaction in model pathways and compared the resulting systemic behavior according to several criteria for functional effectiveness, using the method of mathematically controlled comparisons. This technique minimizes extraneous differences in systemic behavior and identifies those that are fundamental. Our results show that a pathway with an irreversible reaction located at the first step, and with all other reactions reversible, is on average better than an otherwise equivalent pathway with all reactions reversible, which in turn is on average better than an otherwise equivalent pathway with an irreversible reaction located at any step other than the first. Pathways with an irreversible first reaction and low concentrations of intermediates (one of the primary criteria for functional effectiveness) exhibit the following profile when compared to fully reversible pathways: changes in the concentration of intermediates in response to changes in the level of initial substrate are equally low, the robustness of the intermediate concentrations and of the flux is similar, the margins of stability are similar, flux is more responsive to changes in demand for end product, intermediate concentrations are less responsive to changes in demand for end product, and transient times are shorter. These results provide a functional rationale for the positioning of irreversible reactions at the beginning of unbranched biosynthetic pathways.     

Matrix method for determining steps most rate-limiting to metabolic fluxes in biotechnological processes

The metabolic control theory developed by Kacser, Burns, Heinrich, and Rapoport is briefly outlined, extended, and transformed so as optimally to address some biotechnological questions. The extensions include (i) a new theorem that relates the control of metabolite concentrations by enzyme activities to flux ratios at branches in metabolic pathways; (ii) a new theorem that does the same for the control of the distribution of the flux over two branches; (iii) a method that expresses these controls into properties (the so-called elasticity coefficients) of the enzymes in the pathway; and (iv) a theorem that relates the effects of changes in metabolite concentrations on reaction rates to the effects of changes in enzyme properties on the same rates. Matrix equations relating the flux control and concentration control coefficients to the elasticity coefficients of enzymes in simple linear and branched pathways incorporating feedback are given, together with their general solutions and a numerical example. These equations allow one to develop rigorous criteria by which to decide the optimal strategy for the improvement of a microbial process. We show how this could be used in deciding which property of which enzyme should be changed in order to obtain the maximal concentration of a metabolite or the maximal metabolic flux.     

Topological analysis of metabolic control

A topological approach is presented for the analysis of control and regulation in metabolic pathways. In this approach, the control structure of a metabolic pathway is represented by a weighted directed graph. From an inspection of the topology of the graph, the control coefficients of the enzymes are evaluated in a heuristic manner in terms of the enzyme elasticities. The major advantage of the topological approach is that it provides a visual framework for (1) calculating the control coefficients of the enzymes, (2) analyzing the cause-effect relationships of the individual enzymes, (3) assessing the relative importance of the enzymes in metabolic regulation, and (4) simplifying the structure of a given pathway, from a regulatory viewpoint. Results are obtained for (a) an unbranched pathway in the absence of feedback the feedforward regulation and (b) an unbranched pathway with feedback inhibition. Our formulation is based on the metabolic control theory of Kacser and Burns (1973) and Heinrich and Rapoport (1974).     

Effects of product inhibition in metabolic pathways: stability and control

The effect of product inhibition in metabolic pathways is examined using (a) an unbranched pathway in the absence of endproduct inhibition and (b) an unbranched pathway with endproduct inhibition. It is shown that product inhibition may be considered an alternative mechanism to endproduct inhibition for reducing the overall logarithmic gain of an unregulated pathway. When product inhibition and endproduct inhibition are both present, they act in concert with each other to lower the overall logarithmic gain and alleviate parameter sensitivities. Product inhibition is also found to exert a stabilizing influence that competes with the destabilizing effect of endproduct inhibition in controlling the dynamic behavior.     

Control analysis of mammalian serine biosynthesis. Feedback inhibition on the final step.

The flux of serine biosynthesis in the liver of the normal rabbit, and of the rat on a low protein diet, is most sensitive to the activity of phosphoserine phosphatase (flux control coefficient up to 0.97), the last of the three enzymes in the pathway after it branches from glycolysis. The concentration of the pathway product, serine, has a strong controlling influence on the flux (response coefficient up to -0.64) through feedback inhibition at this step. The pathway is therefore controlled primarily by the demand for serine rather than the supply of the pathway precursor, 3-phosphoglycerate. Under conditions where there is a lower biosynthetic flux, the flux control coefficients of the first two enzymes of the pathway are increased, and are probably dominant in the rat on a normal diet. In rabbit liver, when ethanol is used to inhibit serine biosynthesis, control can be distributed between the three enzymes, even though the reactions catalysed by the first two remain close to equilibrium. Apart from their intrinsic value in aiding the understanding of the regulation of mammalian serine metabolism, our findings illustrate the danger of assuming that there are invariant design principles in the regulation of metabolic pathways, such as feedback control on the first step after a branch.     

Coherent and robust modulation of a metabolic network by cytoskeletal organization and dynamics

In order to investigate the influence of cytoskeletal organization and dynamics on cellular biochemistry, a mathematical model was formulated based on our own experimental evidence. The model couples microtubular protein (MTP) dynamics to the glycolytic pathway and its branches: the Krebs cycle, ethanolic fermentation, and the pentose phosphate (PP) pathway. Results show that the flux through glycolysis coherently and coordinately increases or decreases with increased or decreased levels of polymerized MTP, respectively. The rates of individual enzymatic steps and metabolite concentrations change with the polymeric status of MTP throughout the metabolic network. Negative control is exerted by the PP pathway on the glycolytic flux, and the extent of inhibition depends inversely on the polymerization state of MTP, i.e. a high degree of polymerization relieves the negative control. The stability of the model's steady state dynamics for a wide range of variation of metabolic parameters increased with the degree of polymerized MTP. The findings indicate that the organization of the cytoskeleton bestows coherence and robustness to the coordination of cellular metabolism.     

Optimal design of feedback control by inhibition

The local stability of unbranched biosynthetic pathways is examined by mathematical analysis and computer simulation using a novel nonlinear formalism that appears to accurately describe biochemical systems. Four factors affecting the stability are examined: strength of feedback inhibition, equalization of the values among the corresponding kinetic parameters for the reactions of the pathway, pathway length, and alternative patterns of feedback interactions. The strength of inhibition and the pattern of feedback interactions are important determinants of steady-state behavior. The simple pattern of end-product inhibition in unbranched pathways may have evolved because it optimizes the steady-state behavior and is temporally most responsive to change. Stability in these simple systems is achieved by shortening pathway length either physically or, in the case of necessarily long pathways, kinetically by a wide devergence in the values of the corresponding kinetic parameters for the reactions of the pathway. These conclusions are discussed in the light of available experimental evidence.     

From Metabolomics to Fluxomics: A Computational Procedure to Translate Metabolite Profiles into Metabolic Fluxes

We describe a believed-novel procedure for translating metabolite profiles (metabolome) into the set of metabolic fluxes (fluxome) from which they originated. Methodologically, computational modeling is integrated with an analytical platform comprising linear optimization, continuation and dynamic analyses, and metabolic control. The procedure was tested with metabolite profiles obtained from ex vivo mice Langendorff-heart preparations perfused with glucose. The metabolic profiles were analyzed using a detailed kinetic model of the glucose catabolic pathways including glycolysis, pentose phosphate (PP), glycogenolysis, and polyols to translate the glucose metabolome of the heart into the fluxome. After optimization, the ability of the model to simulate the initial metabolite profile was confirmed, and metabolic fluxes as well as the structure of control and regulation of the glucose catabolic network could be calculated. We show that the step catalyzed by phosphofructokinase together with ATP demand and glycogenolysis exert the highest control on the glycolytic flux. The negative flux control exerted by phosphofructokinase on the PP and polyol pathways revealed that the extent of glycolytic flux directly affects flux redirection through these pathways, i.e., the higher the glycolytic flux the lower the PP and polyols. This believed-novel methodological approach represents a step forward that may help in designing therapeutic strategies targeted to diagnose, prevent, and treat metabolic diseases.     

Achieving Optimal Growth through Product Feedback Inhibition in Metabolism

Recent evidence suggests that the metabolism of some organisms, such as Escherichia coli, is remarkably efficient, producing close to the maximum amount of biomass per unit of nutrient consumed. This observation raises the question of what regulatory mechanisms enable such efficiency. Here, we propose that simple product-feedback inhibition by itself is capable of leading to such optimality. We analyze several representative metabolic modules—starting from a linear pathway and advancing to a bidirectional pathway and metabolic cycle, and finally to integration of two different nutrient inputs. In each case, our mathematical analysis shows that product-feedback inhibition is not only homeostatic but also, with appropriate feedback connections, can minimize futile cycling and optimize fluxes. However, the effectiveness of simple product-feedback inhibition comes at the cost of high levels of some metabolite pools, potentially associated with toxicity and osmotic imbalance. These large metabolite pool sizes can be restricted if feedback inhibition is ultrasensitive. Indeed, the multi-layer regulation of metabolism by control of enzyme expression, enzyme covalent modification, and allostery is expected to result in such ultrasensitive feedbacks. To experimentally test whether the qualitative predictions from our analysis of feedback inhibition apply to metabolic modules beyond linear pathways, we examine the case of nitrogen assimilation in E. coli, which involves both nutrient integration and a metabolic cycle. We find that the feedback regulation scheme suggested by our mathematical analysis closely aligns with the actual regulation of the network and is sufficient to explain much of the dynamical behavior of relevant metabolite pool sizes in nutrient-switching experiments.     

Defining control coefficients in non-ideal metabolic pathways

The extent to which an enzyme controls a flux has been defined as the effect on that flux of a small modulation of the activity of that enzyme divided by the magnitude of the modulation. We here show that in pathways with metabolic channelling or high enzyme concentrations and conserved moieties involving both enzymic and non-enzymic species, this definition is ambiguous; the magnitude of the corresponding flux control coefficient depends on how the enzyme activity is modulated. This is illustrated with two models of biochemically relevant pathways, one in which dynamic metabolite channelling plays a role, and one with a moiety-conserved cycle. To avoid such ambiguity, we view biochemical pathways in a more detailed manner, i.e., as a network of elemental steps. We define 'elemental control coefficients' in terms of the effect on a flux of an equal modulation of the forward and reverse rate constant of any such elemental step (which may correspond to transitions between enzyme states). This elemental control coefficient is independent of the method of modulation. We show how metabolic control analysis can proceed when formulated in terms of the elemental control coefficients and how the traditional control coefficients are related to these elemental control coefficients. An 'impact' control coefficient is defined which quantifies the effect of an activation of all elemental processes in which an enzyme is involved. It equals the sum of the corresponding elemental control coefficients. In ideal metabolic pathways this impact control coefficient reduces to the traditional flux control coefficient. Differences between the traditional control coefficients are indicative of non-ideality of a metabolic pathway, i.e. of channelling or high enzyme concentrations.     

Dynamic stability of steady states and static stabilization in unbranched metabolic pathways

The paper is concerned with the conditions of dynamic (asymptotic) stability of steady states in unbranched metabolic pathways. The stationary flux in such pathways is generally determined by the concentration of the end product due to the effector action of this product on the reactions proceeding in its synthetic pathway. The delay in feedback circuits causes violation of dynamic stability at large static stabilization factors. A method permitting analytic estimation of the critical stabilization factor is suggested. Sufficient and necessary conditions for asymptotic stability of the steady state in the general case of the pathway with a single feedback loop have been established. Mechanisms for maintenance of the steady state asymptotic stability at large static stabilization factors are studied. It has been shown that the range of dynamic stability can be widened greatly, if the pathway contains one or two reactions (but not more) of relatively small effective rate constants. Short strong negative feedback is also found to extend considerably the range of dynamic stability of the pathway. The feedback is more effective if it acts on the reaction with small effective rate constant.     

Complementary identification of multiple flux distributions and multiple metabolic pathways

Cell robustness and complexity have been recognized as unique features of biological systems. Such robustness and complexity of metabolic-reaction systems can be explored by discovering, or identifying, the multiple flux distributions (MFD) and redundant pathways that lead to a given external state; however, this is exceedingly cumbersome to accomplish. It is, therefore, highly desirable to establish an effective computational method for their identification, which, in turn, gives rise to a novel insight into the cellular function. An effective approach is proposed for complementarily identifying MFD in metabolic flux analysis and multiple metabolic pathways (MMP) in structural pathway analysis. This approach judiciously integrates flux balance analysis (FBA) based on linear programming and the graph-theoretic method for determining reaction pathways. A single metabolic pathway, with the concomitant flux distribution and the overall reaction manifesting itself as the desired phenotype under some environmental conditions, is determined by FBA from the initial candidate sequence of metabolic reactions. Subsequently, the graph-theoretic method recovers all feasible MMP and the corresponding MFD. The approach's efficacy is demonstrated by applying it to the in silico Escherichia coli model under various culture conditions. The resultant MMP and MFD attaining a unique external state reveal the surprising adaptability and robustness of the intricate cellular network as a key to cell survival against environmental or genetic changes. These results indicate that the proposed approach would be useful in facilitating drug discovery.     

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).     

Optimal design of feedback control by inhibition

Summary A recently developed, nonlinear method of systems analysis has been used to compare alternative patterns of control by feedback inhibition in otherwise equivalent unbranched biosynthetic pathways. The steady state performance of the simple case with end-product inhibition at the first physiologically important step is optimal with respect to the following criteria: the ability to (i) meet an increased demand for the end-product, (ii) meet this increased demand with limited accumulation of the intermediates, (iii) respond to an increased supply of the initial substrate, and (iv) limit the accumulation of the intermediates while responding to the increased availability of the initial substrate. The importance of these properties for the selection of feedback patterns in biosynthetic pathways is discussed.     

Deregulation of Feedback Inhibition of Phosphoenolpyruvate Carboxylase for Improved Lysine Production in Corynebacterium glutamicum

Allosteric regulation of phosphoenolpyruvate carboxylase (PEPC) controls the metabolic flux distribution of anaplerotic pathways. In this study, the feedback inhibition of Corynebacterium glutamicum PEPC was rationally deregulated, and its effect on metabolic flux redistribution was evaluated. Based on rational protein design, six PEPC mutants were designed, and all of them showed significantly reduced sensitivity toward aspartate and malate inhibition. Introducing one of the point mutations (N917G) into the ppc gene, encoding PEPC of the lysine-producing strain C. glutamicum LC298, resulted in ∼37% improved lysine production. In vitro enzyme assays and ¹³C-based metabolic flux analysis showed ca. 20 and 30% increases in the PEPC activity and corresponding flux, respectively, in the mutant strain. Higher demand for NADPH in the mutant strain increased the flux toward pentose phosphate pathway, which increased the supply of NADPH for enhanced lysine production. The present study highlights the importance of allosteric regulation on the flux control of central metabolism. The strategy described here can also be implemented to improve other oxaloacetate-derived products.     

Engineering metabolism and product formation in Corynebacterium glutamicum by coordinated gene overexpression

Single gene overexpression in product pathways such as lysine synthesis has often been employed in metabolic engineering efforts aiming at pathway flux amplification and metabolite overproduction. This approach is limited due to metabolic flux imbalances that often lead to unpredictable physiological responses and suboptimal metabolite productivity. This deficiency can be overcome by the coordinated overexpression of more than one flux controlling genes in a production pathway selected by considering their individual contributions on the cell physiology This concept is demonstrated by the simultaneous overexpression of pyruvate carboxylase and aspartate kinase, two key enzymes in central carbon metabolism and the lysine production pathway in Corynebacterium glutamicum. Contrary to expectations based on the importance of each of these two genes in lysine production, the monocistronic overexpression of either gene results in marginal changes in the overall lysine productivity due to either reduced cell growth or reduced lysine specific productivity. In contrast, the simultaneous amplification of the activities of the two enzymes yielded more than 250% increase of the lysine specific productivity in lactate minimal medium without affecting the growth rate or final cell density of the culture. These results demonstrate that significant flux amplification in complex pathways involving central carbon metabolism is possible through coordinated overexpression of more than one gene in the pathway. This can be achieved either by external, gene expression inducing, controls or controls responding to the physiological cellular state.     

A simple matrix method for determining flux control coefficients in complex pathways

Flux control coefficients express in quantitative terms the extent to which the steady state flux through a metabolic pathway is controlled by a particular parameter. Enzyme flux control coefficients can be calculated using matrix algebra methods which express the control coefficients in terms of parameters which can be determined experimentally (enzyme elasticities, flux ratios, metabolite ratios). This paper describes an algorithm based on a 'constraint' matrix which enables expressions for enzyme control coefficients to be written for pathways of any complexity.     

Using isotopomer path tracing to quantify metabolic fluxes in pathway models containing reversible reactions

As a more complete picture of the genetic and enzymatic composition of cells becomes available, there is a growing need to describe how cellular regulatory elements interact with the cellular environment to affect cell physiology. One means for describing intracellular regulatory mechanisms is concurrent measurement of multiple metabolic pathways and their interactions by metabolic flux analysis. Flux of carbon through a metabolic pathway responds to all cellular regulatory systems, including changes in enzyme and substrate concentrations, enzyme activation or inhibition, and ultimately genetic control. The extent to which metabolic flux analysis can describe cellular physiology depends on the number of pathways in the model and the quality of the data. Intracellular information is obtainable from isotopic tracer experiments, the most extensive being the determination of the isotopomer distribution, or specific labeling pattern, of intracellular metabolites. We present a rapid and novel solution method that determines the flux of carbon through complex pathway models using isotopomer data. This time-consuming problem was solved with the introduction of isotopomer path tracing, which drastically reduces the number of isotopomer variables to the number of isotopomers observed experimentally. We propose a partitioned solution method that takes advantage of the nearly linear relationship between fluxes and isotopomers. Whereas the stoichiometric matrix and the isotopomer matrix are invertible, simulated annealing and the Newton-Raphson method are used for the nonlinear components. Reversible reactions are described by a new parameter, the association factor, which scales hyperbolically with the rate of metabolite exchange. Automating the solution method permits a variety of models to be compared, thus enhancing the accuracy of results. A simplified example that contains all of the complexities of a comprehensive pathway model is presented. Copyright John Wiley & Sons, Inc.