An interval approach for dealing with flux distributions and elementary modes activity patterns

This work introduces the use of an interval representation of fluxes. This representation can be useful in two common situations: (a) when fluxes are uncertain due to the lack of accurate measurements and (b) when the flux distribution is partially unknown. In addition, the interval representation can be used for other purposes such as dealing with inconsistency or representing a range of behaviour. Two main problems are addressed. On the one hand, the translation of a metabolic flux distribution into an elementary modes or extreme pathways activity pattern is analysed. In general, there is not a unique solution for this problem but a range of solutions. To represent the whole solution region in an easy way, it is possible to compute the alpha-spectrum (i.e., the range of possible values for each elementary mode or extreme pathway activity). Herein, a method is proposed which, based on the interval representation of fluxes, makes it possible to compute the alpha-spectrum from an uncertain or even partially unknown flux distribution. On the other hand, the concept of the flux-spectrum is introduced as a variant of the metabolic flux analysis methodology that presents some advantages: applicable when measurements are insufficient (underdetermined case), integration of uncertain measurements, inclusion of irreversibility constraints and an alternative procedure to deal with inconsistency. Frequently, when applying metabolic flux analysis the available measurements are insufficient and/or uncertain and the complete flux distribution cannot be uniquely calculated. The method proposed here allows the determination of the ranges of possible values for each non-calculable flux, resulting in a flux region called flux-spectrum. In order to illustrate the proposed methods, the example of the metabolic network of CHO cells cultivated in stirred flasks is used.     

METATOOL: for studying metabolic networks

MOTIVATION: To reconstruct metabolic pathways from biochemical and/or genome sequence data, the stoichiometric and thermodynamic feasibility of the pathways has to be tested. This is achieved by characterizing the admissible region of flux distributions in steady state. This region is spanned by what can be called a convex basis. The concept of 'elementary flux modes' provides a mathematical tool to define all metabolic routes that are feasible in a given metabolic network. In addition, we define 'enzyme subsets' to be groups of enzymes that operate together in fixed flux proportions in all steady states of the system. RESULTS: Algorithms for computing the convex basis and elementary modes developed earlier are briefly reviewed. A newly developed algorithm for detecting all enzyme subsets in a given network is presented. All of these algorithms have been implemented in a novel computer program named METATOOL, whose features are outlined here. The algorithms are illustrated by an example taken from sugar metabolism. AVAILABILITY: METATOOL is available from ftp://bmsdarwin.brookes.ac. uk/pub/software/ibmpc/metatool. SUPPLEMENTARY INFORMATION: http://www. biologie.hu-berlin.de/biophysics/Theory/tpfeiffer/metatoo l.html     

Combining pathway analysis with flux balance analysis for the comprehensive study of metabolic systems

The elucidation of organism-scale metabolic networks necessitates the development of integrative methods to analyze and interpret the systemic properties of cellular metabolism. A shift in emphasis from single metabolic reactions to systemically defined pathways is one consequence of such an integrative analysis of metabolic systems. The constraints of systemic stoichiometry, and limited thermodynamics have led to the definition of the flux space within the context of convex analysis. The flux space of the metabolic system, containing all allowable flux distributions, is constrained to a convex polyhedral cone in a high-dimensional space. From metabolic pathway analysis, the edges of the high-dimensional flux cone are vectors that correspond to systemically defined "extreme pathways" spanning the capabilities of the system. The addition of maximum flux capacities of individual metabolic reactions serves to further constrain the flux space and has led to the development of flux balance analysis using linear optimization to calculate optimal flux distributions. Here we provide the precise theoretical connections between pathway analysis and flux balance analysis allowing for their combined application to study integrated metabolic function. Shifts in metabolic behavior are calculated using linear optimization and are then interpreted using the extreme pathways to demonstrate the concept of pathway utilization. Changes to the reaction network, such as the removal of a reaction, can lead to the generation of suboptimal phenotypes that can be directly attributed to the loss of pathway function and capabilities. Optimal growth phenotypes are calculated as a function of environmental variables, such as the availability of substrate and oxygen, leading to the definition of phenotypic phase planes. It is illustrated how optimality properties of the computed flux distributions can be interpreted in terms of the extreme pathways. Together these developments are applied to an example network and to core metabolism of Escherichia coli demonstrating the connections between the extreme pathways, optimal flux distributions, and phenotypic phase planes. The consequences of changing environmental and internal conditions of the network are examined for growth on glucose and succinate in the face of a variety of gene deletions. The convergence of the calculation of optimal phenotypes through linear programming and the definition of extreme pathways establishes a different perspective for the understanding of how a defined metabolic network is best used under different environmental and internal conditions or, in other words, a pathway basis for the interpretation of the metabolic reaction norm.     

Control analysis of biochemical pathways: a novel procedure for calculating control coefficients, and an additional theorem for branched pathways

A novel method for calculating control coefficients of individual enzymes on fluxes and concentrations in metabolic pathways is presented. This method is derived by applying the theorem on implicit functions to the equations defining the steady state metabolite concentrations; it allows verification of the existing summation theorems and connectivity relations, and leads to a novel theorem for flux control coefficients in branched pathways. The method and the novel theorem are illustrated by several examples.     

鸟苷产生菌的代谢途径分析

代谢工程要解决的主要问题就是改变某些途径中的碳架物质流量或改变碳架物质流在不同途径中的流量分布,其目标就是修饰初级代谢,将碳架物质流导入目的产物的载流途径以获得产物的最大转化率。利用途径分析方法对枯草芽孢杆菌生产鸟苷的途径进行了分析,建立了3种基础模型,鸟苷理论摩尔产率分别是0．625、0．75和0．667,确定了枯草芽孢杆菌生产鸟苷的最佳途径的通量分布。     

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.     

Quantitative approaches for analysing fluxes through plant metabolic networks using NMR and stable isotope labelling

The quantitative analysis of metabolic networks is a prerequisite for understanding the integration and regulation of plant metabolism and for devising rational approaches for manipulating resource allocation in plants. The analysis of steady state stable isotope labelling experiments using nuclear magnetic resonance (NMR) spectroscopy has developed into a powerful method for determining these fluxes in micro-organisms and its application to heterotrophic plant metabolism is increasing. After an introductory discussion of the well known role of stable isotopes in pathway delineation, the review considers their application to metabolic flux analysis in plants. These applications are divided into two groups – small scale analyses of fluxes through particular pathways and large scale analyses of multiple fluxes through metabolic networks – and the problems caused by the complexity of intermediary metabolism in plants are discussed. It is concluded that metabolic flux analysis provides a powerful method for defining the metabolic phenotype of wild type, mutant and transgenic plants and that its development should be pursued.     

The genome-scale metabolic extreme pathway structure in Haemophilus influenzae shows significant network redundancy

Genome-scale metabolic networks can be characterized by a set of systemically independent and unique extreme pathways. These extreme pathways span a convex, high-dimensional space that circumscribes all potential steady-state flux distributions achievable by the defined metabolic network. Genome-scale extreme pathways associated with the production of non-essential amino acids in Haemophilus influenzae were computed. They offer valuable insight into the functioning of its metabolic network. Three key results were obtained. First, there were multiple internal flux maps corresponding to externally indistinguishable states. It was shown that there was an average of 37 internal states per unique exchange flux vector in H. influenzae when the network was used to produce a single amino acid while allowing carbon dioxide and acetate as carbon sinks. With the inclusion of succinate as an additional output, this ratio increased to 52, a 40% increase. Second, an analysis of the carbon fates illustrated that the extreme pathways were non-uniformly distributed across the carbon fate spectrum. In the detailed case study, 45% of the distinct carbon fate values associated with lysine production represented 85% of the extreme pathways. Third, this distribution fell between distinct systemic constraints. For lysine production, the carbon fate values that represented 85% of the pathways described above corresponded to only 2 distinct ratios of 1:1 and 4:1 between carbon dioxide and acetate. The present study analysed single outputs from one organism, and provides a start to genome-scale extreme pathways studies. These emergent system-level characterizations show the significance of metabolic extreme pathway analysis at the genome-scale.     

Extreme pathway analysis of human red blood cell metabolism

The development of high-throughput technologies and the resulting large-scale data sets have necessitated a systems approach to the analysis of metabolic networks. One way to approach the issue of complex metabolic function is through the calculation and interpretation of extreme pathways. Extreme pathways are a mathematically defined set of generating vectors that describe the conical steady-state solution space for flux distributions through an entire metabolic network. Herein, the extreme pathways of the well-characterized human red blood cell metabolic network were calculated and interpreted in a biochemical and physiological context. These extreme pathways were divided into groups based on such criteria as their cofactor and by-product production, and carbon inputs including those that 1) convert glucose to pyruvate; 2) interchange pyruvate and lactate; 3) produce 2,3-diphosphoglycerate that binds to hemoglobin; 4) convert inosine to pyruvate; 5) induce a change in the total adenosine pool; and 6) dissipate ATP. Additionally, results from a full kinetic model of red blood cell metabolism were predicted based solely on an interpretation of the extreme pathway structure. The extreme pathways for the red blood cell thus give a concise representation of red blood cell metabolism and a way to interpret its metabolic physiology.     

Stoichiometric model and metabolic flux analysis for Leptospirillum ferrooxidans

A metabolic model for Leptospirillum ferrooxidans was developed based on the genomic information of an analogous iron oxidizing bacteria and on the pathways of ferrous iron oxidation, nitrogen and CO₂ assimilation based on experimental evidence for L. ferrooxidans found in the literature. From this metabolic reconstruction, a stoichiometric model was built, which includes 86 reactions describing the main catabolic and anabolic aspects of its metabolism. The model obtained has 2 degrees of freedom, so two external fluxes were estimated to achieve a determined and observable system. By using the external oxygen consumption rate and the generation flux biomass as input data, a metabolic flux map with a distribution of internal fluxes was obtained. The results obtained were verified with experimental data from the literature, achieving a very good prediction of the metabolic behavior of this bacterium at steady state. Biotechnol. Bioeng. 2010;107:696–706. © 2010 Wiley Periodicals, Inc.     

Theory for the systemic definition of metabolic pathways and their use in interpreting metabolic function from a pathway-oriented perspective

Cellular metabolism is most often described and interpreted in terms of the biochemical reactions that make up the metabolic network. Genomics is providing near complete information regarding the genes/gene products participating in cellular metabolism for a growing number of organisms. As the true functional units of metabolic systems are its pathways, the time has arrived to define metabolic pathways in the context of whole-cell metabolism for the analysis of the structural design and capabilities of the metabolic network. In this study, we present the theoretical foundations for the identification of the unique set of systemically independent biochemical pathways, termed extreme pathways, based on system stochiometry and limited thermodynamics. These pathways represent the edges of the steady-state flux cone derived from convex analysis, and they can be used to represent any flux distribution achievable by the metabolic network. An algorithm is presented to determine the set of extreme pathways for a system of any complexity and a classification scheme is introduced for the characterization of these pathways. The property of systemic independence is discussed along with its implications for issues related to metabolic regulation and the evolution of cellular metabolic networks. The underlying pathway structure that is determined from the set of extreme pathways now provides us with the ability to analyse, interpret, and perhaps predict metabolic function from a pathway-based perspective in addition to the traditional reaction-based perspective. The algorithm and classification scheme developed can be used to describe the pathway structure in annotated genomes to explore the capabilities of an organism.     

Production and engineering of terpenoids in plant cell culture

Terpenoids are a diverse class of natural products that have many functions in the plant kingdom and in human health and nutrition. Their chemical diversity has led to the discovery of over 40,000 different structures, with several classes serving as important pharmaceutical agents, including the anticancer agents paclitaxel (Taxol) and terpenoid-derived indole alkaloids. Many terpenoid compounds are found in low yield from natural sources, so plant cell cultures have been investigated as an alternate production strategy. Metabolic engineering of whole plants and plant cell cultures is an effective tool to both increase terpenoid yield and alter terpenoid distribution for desired properties such as enhanced flavor, fragrance or color. Recent advances in defining terpenoid metabolic pathways, particularly in secondary metabolism, enhanced knowledge concerning regulation of terpenoid accumulation, and application of emerging plant systems biology approaches, have enabled metabolic engineering of terpenoid production. This paper reviews the current state of knowledge of terpenoid metabolism, with a special focus on production of important pharmaceutically active secondary metabolic terpenoids in plant cell cultures. Strategies for defining pathways and uncovering rate-influencing steps in global metabolism, and applying this information for successful terpenoid metabolic engineering, are emphasized.     

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.     

Multiplicity of Steady States in Glycolysis and Shift of Metabolic State in Cultured Mammalian Cells

Cultured mammalian cells exhibit elevated glycolysis flux and high lactate production. In the industrial bioprocesses for biotherapeutic protein production, glucose is supplemented to the culture medium to sustain continued cell growth resulting in the accumulation of lactate to high levels. In such fed-batch cultures, sometimes a metabolic shift from a state of high glycolysis flux and high lactate production to a state of low glycolysis flux and low lactate production or even lactate consumption is observed. While in other cases with very similar culture conditions, the same cell line and medium, cells continue to produce lactate. A metabolic shift to lactate consumption has been correlated to the productivity of the process. Cultures that exhibited the metabolic shift to lactate consumption had higher titers than those which didn’t. However, the cues that trigger the metabolic shift to lactate consumption state (or low lactate production state) are yet to be identified. Metabolic control of cells is tightly linked to growth control through signaling pathways such as the AKT pathway. We have previously shown that the glycolysis of proliferating cells can exhibit bistability with well-segregated high flux and low flux states. Low lactate production (or lactate consumption) is possible only at a low glycolysis flux state. In this study, we use mathematical modeling to demonstrate that lactate inhibition together with AKT regulation on glycolysis enzymes can profoundly influence the bistable behavior, resulting in a complex steady-state topology. The transition from the high flux state to the low flux state can only occur in certain regions of the steady state topology, and therefore the metabolic fate of the cells depends on their metabolic trajectory encountering the region that allows such a metabolic state switch. Insights from such switch behavior present us with new means to control the metabolism of mammalian cells in fed-batch cultures.     

Self-Organization of Plant Vascular Systems: Claims and Counter-Claims about the Flux-Based Auxin Transport Model

The plant hormone auxin plays a central role in growth and morphogenesis. In shoot apical meristems, auxin flux is polarized through its interplay with PIN proteins. Concentration-based mathematical models of the flux can explain some aspects of phyllotaxis for the L1 surface layer, where auxin accumulation points act as sinks and develop into primordia. The picture differs in the interior of the meristem, where the primordia act as auxin sources, leading to the initiation of the vascular system. Self-organization of the auxin flux involves large numbers of molecules and is difficult to treat by intuitive reasoning alone; mathematical models are therefore vital to understand these phenomena. We consider a leading computational model based on the so-called flux hypothesis. This model has been criticized and extended in various ways. One of the basic counter-arguments is that simulations yield auxin concentrations inside canals that are lower than those seen experimentally. Contrary to what is claimed in the literature, we show that the model can lead to higher concentrations within canals for significant parameter regimes. We then study the model in the usual case where the response function Φ defining the model is quadratic and unbounded, and show that the steady state vascular patterns are formed of loopless directed trees. Moreover, we show that PIN concentrations can diverge in finite time, thus explaining why previous simulation studies introduced cut-offs which force the system to have bounded PIN concentrations. Hence, contrary to previous claims, extreme PIN concentrations are not due to numerical problems but are intrinsic to the model. On the other hand, we show that PIN concentrations remain bounded for bounded Φ, and simulations show that in this case, loops can emerge at steady state.