Use of implicit methods from general sensitivity theory to develop a systematic approach to metabolic control. II. Complex systems

In the accompanying paper (Cascante et al., this issue) we have used general sensitivity theory to develop a matrix algebra that, in the case of sequential reactions, directly relates global and local properties of a given system. In complex biochemical systems this direct relationship is not possible due to the existence of linear dependencies among fluxes and among metabolite concentrations (conserved aggregate concentrations in BST or moiety-conserved concentrations in MCT). In this paper our matrix algebra is applied to conserved cycles and branched pathways, and it is shown that with minor modifications it again relates global properties to the local properties of the enzymes in the system. In the case of conserved cycles, elasticities become modified due to the existence of linear dependencies among the concentration variables in the cycle. In branched pathways, new matrix elements involving ratios of fluxes appear. With these modifications, one can show that the so-called theorems of metabolic control theory specific to these types of pathways are special cases of more general relationships. Rules for the construction of matrices relating global and local properties are given that apply to an arbitrary system of cycles and branches. The implicit approach developed in these papers, which is a generalization of that used in MCT, allows one to make more direct comparisons with the general explicit approach originally developed in BST.     

Metabolic control and its analysis. Additional relationships between elasticities and control coefficients

Existing theorems from the analysis of metabolic control have been taken and embedded in a simple matrix algebra procedure for calculating the flux control coefficients of enzymes (formerly known as sensitivities) in a metabolic pathway from their kinetic properties (their elasticities). New theorems governing the flux control coefficients of branched pathways and substrate cycles have been derived to allow the procedure to be applied to complex pathway configurations. Modifications to the elasticity terms used in the equations have been theoretically justified so that the method remains valid for pathways with conserved metabolites (for example, the adenine nucleotide pool or the intermediates of a catalytic cycle such as the tricarboxylic acid cycle) or with pools of metabolites kept very near to equilibrium by very rapid reactions. The matrix equations generated using these theorems and relationships may be solved algebraically or numerically. Algebraic solutions have been used to determine the factors responsible for the degree of amplification of flux control coefficients by substrate cycles and to show that it is possible to derive expressions for the elasticities of a group of enzymes.     

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.     

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.     

Theory of metabolism regulation: a complete system of equations for regulation coefficients

Basic quantitative parameters of control in a metabolic system are considered: control coefficients of enzymes with respect to metabolic fluxes and concentrations, and in the case when there are conservation laws, the response coefficients of metabolic fluxes and concentrations to changes in the conserved sums of metabolite concentrations (e. g. conserved moieties). Relationships are obtained which generalize the well known connectivity relations for the case of metabolites binding by conservation laws. Additional relationships are obtained which complement the set of connectivity relations up to the complete system of equations for determining all the control coefficients. The control coefficients are expressed through the enzyme elasticity coefficients, steady state metabolic fluxes and concentrations. Formulas are derived which express response coefficients of flux and concentrations through the enzyme control and elasticity coefficients and metabolite concentrations.     

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.     

On the stability of metabolic cycles

We investigate the stability properties of two different classes of metabolic cycles using a combination of analytical and computational methods. Using principles from structural kinetic modeling (SKM), we show that the stability of metabolic networks with certain structural regularities can be studied using a combination of analytical and computational techniques. We then apply these techniques to a class of single input, single output metabolic cycles, and find that the cycles are stable under all conditions tested. Next, we extend our analysis to a small autocatalytic cycle, and determine parameter regimes within which the cycle is very likely to be stable. We demonstrate that analytical methods can be used to understand the relationship between kinetic parameters and stability, and that results from these analytical methods can be confirmed with computational experiments. In addition, our results suggest that elevated metabolite concentrations and certain crucial saturation parameters can strongly affect the stability of the entire metabolic cycle. We discuss our results in light of the possibility that evolutionary forces may select for metabolic network topologies with a high intrinsic probability of being stable. Furthermore, our conclusions support the hypothesis that certain types of metabolic cycles may have played a role in the development of primitive metabolism despite the absence of regulatory mechanisms.     

On the analysis of futile cycles in metabolism

So-called futile cycles in cellular metabolism consist of paired opposing reactions that, if simultaneously operant, act only to degrade free energy of ATP to heat. Previous considerations of the behavior of such substrate cycles have indicated their possible usefulness in regulating flux along metabolic pathways, but such analyses have treated the cycles in isolation, i.e. without taking into account the effects of enzymatic inputs to and outputs from the cycle. We here develop models of three typical substrate cycles that include enzymatic inputs to and outputs from the cycle and allow the enzymes of the cycles per se to be subject to a variety of allosteric modulations. The non-linear equations which describe these models were solved by an iterative procedure for sets of parameter values of metabolic interest. The results, when analyzed using appropriate definitions of regulatory sensitivity and energetic futility, demonstrate that the effects of the enzymes leading into and out of the cycle may cause profound changes in the operation of the substrate cycle and therefore may not be ignored. We find that the structural differences among the three cycles considered here result in corresponding functional differences. Our results suggest that (1) the fructose-6-P/fructose-1,6-di-P cycle acts effectively to gate bidirectional flux, but doesn't appreciably enhance regulation of unidirectional flux, (2) the glucose/glucose-6-P cycle is well suited to perform a homeostatic function and to adjust the set points for these two metabolites, and (3) the cycle at the pyruvate crossroads functions largely as a complex switch box that directs metabolic flow towards gluconeogenesis or glycolysis not only in response to inputs of or requirements for oxaloacetate, pyruvate, and phosphoenolpyruvate, but also in response to the combined action of allosteric modulators on the individual enzymes of this substrate cycle.     

Chemical basis of metabolic network organization

Although the metabolic networks of the three domains of life consist of different constituents and metabolic pathways, they exhibit the same scale-free organization. This phenomenon has been hypothetically explained by preferential attachment principle that the new-recruited metabolites attach preferentially to those that are already well connected. However, since metabolites are usually small molecules and metabolic processes are basically chemical reactions, we speculate that the metabolic network organization may have a chemical basis. In this paper, chemoinformatic analyses on metabolic networks of Kyoto Encyclopedia of Genes and Genomes (KEGG), Escherichia coli and Saccharomyces cerevisiae were performed. It was found that there exist qualitative and quantitative correlations between network topology and chemical properties of metabolites. The metabolites with larger degrees of connectivity (hubs) are of relatively stronger polarity. This suggests that metabolic networks are chemically organized to a certain extent, which was further elucidated in terms of high concentrations required by metabolic hubs to drive a variety of reactions. This finding not only provides a chemical explanation to the preferential attachment principle for metabolic network expansion, but also has important implications for metabolic network design and metabolite concentration prediction.     

Metabolic futile cycles and their functions: a systems analysis of energy and control

It has long been hypothesised that futile cycles in cellular metabolism are involved in the regulation of biochemical pathways. Following the work of Newsholme and Crabtree, a quantitative theory was developed for this idea based on open-system thermodynamics and metabolic control analysis. It is shown that the stoichiometric sensitivity of an intermediary metabolite concentration with respect to changes in steady-state flux is governed by the effective equilibrium constant of the intermediate formation, and the equilibrium can be regulated by a futile cycle. The direction of the shift in the effective equilibrium constant depends on the direction of operation of the futile cycle. High stoichiometric sensitivity corresponds to ultrasensitivity of an intermediate concentration to net flow through a pathway; low stoichiometric sensitivity corresponds to super-robustness of concentration with respect to changes in flux. Both cases potentially play important roles in metabolic regulation. Futile cycles actively shift the effective equilibrium by expending energy; the magnitude of changes in effective equilibria and sensitivities is a function of the amount of energy used by a futile cycle. This proposed mechanism for control by futile cycles works remarkably similar to kinetic proofreading in biosynthesis. The sensitivity of the system is also intimately related to the rate of concentration fluctuations of intermediate metabolites. The possibility of different roles for the two major mechanisms within cellular biochemical regulation, namely reversible chemical modifications via futile cycles and shifting equilibrium by macromolecular binding, are discussed.     

Control analysis of metabolic networks. 2. Total differentials and general formulation of the connectivity relations

The mathematical background of the connectivity relations of metabolic control theory is analysed. The connectivity relations are shown to reflect general properties of total differentials of reaction rate vi, flux J, and metabolite concentration Xj. Connectivity relations hold for any metabolic network in which all vi are homogeneous functions of enzyme concentration Ei. This notion allows established algebraic methods to be used for the formulation of connectivity relations for metabolic systems in which numerous constraints are imposed on metabolite concentrations. A general procedure to derive connectivity relations for such metabolic systems is given. To encourage a broader audience to apply control theory to physiological systems, an easy-to-use graphical procedure is derived for formulating connectivity relations for biochemical systems in which no metabolite is involved in more than one constraint.     

Complexity of vitamin E metabolism

Bioavailability of vitamin E is influenced by several factors, most are highlighted in this review. While gender, age and genetic constitution influence vitamin E bioavailability but cannot be modified, life-style and intake of vitamin E can be. Numerous factors must be taken into account however, i.e., when vitamin E is orally administrated, the food matrix may contain competing nutrients. The complex metabolic processes comprise intestinal absorption, vascular transport, hepatic sorting by intracellular binding proteins, such as the significant α-tocopherol-transfer protein, and hepatic metabolism. The coordinated changes involved in the hepatic metabolism of vitamin E provide an effective physiological pathway to protect tissues against the excessive accumulation of, in particular, non-α-tocopherol forms. Metabolism of vitamin E begins with one cycle of CYP4F2/CYP3A4-dependent ω-hydroxylation followed by five cycles of subsequent β-oxidation, and forms the water-soluble end-product carboxyethylhydroxychroman. All known hepatic metabolites can be conjugated and are excreted, depending on the length of their sidechain, either via urine or feces. The physiological handling of vitamin E underlies kinetics which vary between the different vitamin E forms. Here, saturation of the side-chain and also substitution of the chromanol ring system are important. Most of the metabolic reactions and processes that are involved with vitamin E are also shared by other fat soluble vitamins. Influencing interactions with other nutrients such as vitamin K or pharmaceuticals are also covered by this review. All these processes modulate the formation of vitamin E metabolites and their concentrations in tissues and body fluids. Differences in metabolism might be responsible for the discrepancies that have been observed in studies performed in vivo and in vitro using vitamin E as a supplement or nutrient. To evaluate individual vitamin E status, the analytical procedures used for detecting and quantifying vitamin E and its metabolites are crucial. The latest methods in analytics are presented.     

Though with constraints imposed by endosymbiosis,preferential attachment is still a plausible mechanism responsible for the evolution of the chloroplast metabolic network

Chloroplasts evolved as a result of endosymbiosis, during which sophisticated mechanisms evolved to translocate nucleus‐encoded plastid‐targeted enzymes into the chloroplast to form the chloroplast metabolic network. Given the constraints and complexity of endosymbiosis, will preferential attachment still be a plausible mechanism for chloroplast metabolic network evolution? We answer this question by analysing the metabolic network properties of the chloroplast and a cyanobacterium, Synechococcus sp. WH8102 (syw). First, we found that enzymes related to more ancient pathways are more connected, and synthetases have the highest connectivity. Most of the enzymes shared by the two densest cores between the chloroplast and syw are synthetases. Second, the highly conserved functional modules mainly consist of highly connected enzymes. Finally, isozymes and enzymes from endosymbiotic gene transfer (EGT) were distributed mainly in conserved modules and showed higher connectivity than nonisozymes or non‐EGT enzymes. These results suggest that even with severe evolutionary constraints imposed by endosymbiosis, preferential attachment is still a plausible mechanism responsible for the evolution of the chloroplast metabolic network. However, the current analysis may not completely differentiate whether the chloroplast network properties reflect the evolution of the chloroplast network through preferential attachment or has been inherited from its cyanobacterial ancestor. To fully differentiate these two possibilities, further analyses of the metabolic network structure properties of organisms at various intermediate evolutionary stages between cyanobacteria and the chloroplast are needed.     

植物生活史型定量划分及其权重配置方法——以四棱豆生活史型划分为例

植物生活史型定量划分方法研究是植物生活史型研究的重要内容。现有的生活史型定量划分方法是基于主成分分析法(PCA)建立起来的,未考虑性状指标之间的相互影响,因此需要探索适用于"网状"结构指标体系的植物生活史型划分新方法。根据植物生活史型划分指标的层次性特点,以攀援型和矮生型四棱豆(Psophocarpus tetragonolobus)为例,分别采用主成分分析法(PCA)、层次分析法(AHP)和网络层次分析法(ANP)对性状指标进行权重配置,计算性状指标的综合得分和生活史型划分参数,结果如下:与ANP相比,PCA计算的V型(营养生长型)参数值偏低(x在0.39以下),S型(有性生殖型)参数值偏高(z在0.453以上);AHP计算的V型参数值偏高(x在0.614以上),S型参数值偏低(z在0.088以下);3种方法计算的生活史型划分参数差异明显。由于PCA、AHP均要求性状指标之间相互独立,不能排除性状指标之间的关联,因此基于PCA、AHP的四棱豆生活史型划分结果均出现了偏差,表明性状指标之间的相关性影响了生活史型划分结果。ANP的指标体系为"网状"结构,其控制层、网络层各个指标之间均存在关联。构建ANP的判断矩阵时提取了性状指标的相关矩阵信息,权重配置反映了性状指标之间的相关关系。基于ANP方法对攀援型和矮生型四棱豆生活史型的划分结果分别为V0.517C0.327S0.156和V0.416C0.43S0.154。当性状指标之间的相关不显著时,可采用PCA和AHP法分配权重;当性状指标之间存在显著相关时,采用ANP法进行权重配置更为恰当。综上所述,基于ANP的植物生活史型划分方法解决了性状指标之间相互影响问题,为植物生活史型定量研究提供了新的有效方法。     

The connectivity structure,giant strong component and centrality of metabolic networks

MOTIVATION: Structural and functional analysis of genome-based large-scale metabolic networks is important for understanding the design principles and regulation of the metabolism at a system level. The metabolic network is conventionally considered to be highly integrated and very complex. A rational reduction of the metabolic network to its core structure and a deeper understanding of its functional modules are important. RESULTS: In this work, we show that the metabolites in a metabolic network are far from fully connected. A connectivity structure consisting of four major subsets of metabolites and reactions, i.e. a fully connected sub-network, a substrate subset, a product subset and an isolated subset is found to exist in metabolic networks of 65 fully sequenced organisms. The largest fully connected part of a metabolic network, called 'the giant strong component (GSC)', represents the most complicated part and the core of the network and has the feature of scale-free networks. The average path length of the whole network is primarily determined by that of the GSC. For most of the organisms, GSC normally contains less than one-third of the nodes of the network. This connectivity structure is very similar to the 'bow-tie' structure of World Wide Web. Our results indicate that the bow-tie structure may be common for large-scale directed networks. More importantly, the uncovered structure feature makes a structural and functional analysis of large-scale metabolic network more amenable. As shown in this work, comparing the closeness centrality of the nodes in the GSC can identify the most central metabolites of a metabolic network. To quantitatively characterize the overall connection structure of the GSC we introduced the term 'overall closeness centralization index (OCCI)'. OCCI correlates well with the average path length of the GSC and is a useful parameter for a system-level comparison of metabolic networks of different organisms. SUPPLEMENTARY INFORMATION: http://genome.gbf.de/bioinformatics/     

Age-regulated cycling metabolites are relevant for behavior

Circadian cycles of sleep:wake and gene expression change with age in all organisms examined. Metabolism is also under robust circadian regulation, but little is known about how metabolic cycles change with age and whether these contribute to the regulation of behavioral cycles. To address this gap, we compared cycling of metabolites in young and old Drosophila and found major age-related variations. A significant model separated the young metabolic profiles by circadian timepoint, but could not be defined for the old metabolic profiles due to the greater variation in this dataset. Of the 159 metabolites measured in fly heads, we found 17 that cycle by JTK analysis in young flies and 17 in aged. Only four metabolites overlapped in the two groups, suggesting that cycling metabolites are distinct in young and old animals. Among our top cyclers exclusive to young flies were components of the pentose phosphate pathway (PPP). As the PPP is important for buffering reactive oxygen species, and overexpression of glucose-6-phosphate dehydrogenase (G6PD), a key component of the PPP, was previously shown to extend lifespan in Drosophila, we asked if this manipulation also affects sleep:wake cycles. We found that overexpression in circadian clock neurons decreases sleep in association with an increase in cellular calcium and mitochondrial oxidation, suggesting that altering PPP activity affects neuronal activity. Our findings elucidate the importance of metabolic regulation in maintaining patterns of neural activity, and thereby sleep:wake cycles.     

Analysis of metabolic network based on conservation of molecular structure

Recent work has revealed much about chemical reactions inside hundreds of organisms as well as universal characteristics of metabolic networks, which shed light on the evolution of the networks. However, characteristics of individual metabolites have been neglected. For example, some carbohydrates have structures that are decomposed into small molecules by metabolic reactions, but coenzymes such as ATP are mostly preserved. Such differences in metabolite characteristics are important for understanding the universal characteristics of metabolic networks. To quantify the structure conservation of metabolites, we defined the "structure conservation index" (SCI) for each metabolite as the fraction of metabolite atoms restored to their original positions through metabolic reactions. As expected, coenzymes and coenzyme-like metabolites that have reaction loops in the network show a higher SCI. Using the index, we found that the sum of metabolic fluxes is negatively correlated with the structure preservation of metabolite. Also, we found that each reaction path around high SCI metabolites changes independently, while changes in reaction paths involving low SCI metabolites coincide through evolution processes. These correlations may provide a clue to universal properties of metabolic networks.     

Decomposition of metabolic network into functional modules based on the global connectivity structure of reaction graph

MOTIVATION: Metabolic networks are organized in a modular, hierarchical manner. Methods for a rational decomposition of the metabolic network into relatively independent functional subsets are essential to better understand the modularity and organization principle of a large-scale, genome-wide network. Network decomposition is also necessary for functional analysis of metabolism by pathway analysis methods that are often hampered by the problem of combinatorial explosion due to the complexity of metabolic network. Decomposition methods proposed in literature are mainly based on the connection degree of metabolites. To obtain a more reasonable decomposition, the global connectivity structure of metabolic networks should be taken into account. RESULTS: In this work, we use a reaction graph representation of a metabolic network for the identification of its global connectivity structure and for decomposition. A bow-tie connectivity structure similar to that previously discovered for metabolite graph is found also to exist in the reaction graph. Based on this bow-tie structure, a new decomposition method is proposed, which uses a distance definition derived from the path length between two reactions. An hierarchical classification tree is first constructed from the distance matrix among the reactions in the giant strong component of the bow-tie structure. These reactions are then grouped into different subsets based on the hierarchical tree. Reactions in the IN and OUT subsets of the bow-tie structure are subsequently placed in the corresponding subsets according to a 'majority rule'. Compared with the decomposition methods proposed in literature, ours is based on combined properties of the global network structure and local reaction connectivity rather than, primarily, on the connection degree of metabolites. The method is applied to decompose the metabolic network of Escherichia coli. Eleven subsets are obtained. More detailed investigations of the subsets show that reactions in the same subset are really functionally related. The rational decomposition of metabolic networks, and subsequent studies of the subsets, make it more amenable to understand the inherent organization and functionality of metabolic networks at the modular level. SUPPLEMENTARY INFORMATION: http://genome.gbf.de/bioinformatics/     

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).