Equivalence of branched and unbranched Michaelian pathways concerning periodic signal transmission

Sinusoidal oscillation transmission through branched metabolic pathways is studied. Two systems are analyzed, which are composed of two convergent reaction branches and differ in the length of one of them. Linear kinetics is assumed first. Michaelis-Menten enzymes are then considered by using previous results that suggest their behavior with respect to propagation of oscillations is close to linearity around the mean input flux. As a result, there exist ways to modulate the activity of the enzymes so that propagation is equivalent for branched and specific unbranched pathways. Cells may have taken advantage of such a possibility in cases where oscillations have a biological role.     

Transition time control analysis of a glycolytic system under different glucose concentrations. Control of transition time versus control of flux

Control and Response Coefficients of transition time have been determined in a rat liver glycolytic system under different glucose concentrations. Results have been compared with the Flux Control and Flux Response Coefficients measured in the same conditions, showing that transition time and flux are different responses of the system, subject to different regulation and control. Control Coefficients of flux and transition time show a very different profile in each condition of glucose concentration assayed. Ratio of Flux Control coefficients of glucokinase over phosphofructokinase at 5 and 20 mM glucose concentration changes from 3.2 to 0.5, while the same ratio in the case of Transition Time Control Coefficients moves from 0.6 to 0.93. Moreover, the absolute values of Transition Time Control Coefficients in glycolytic conditions are one order of magnitude bigger than in gluconeogenic conditions. Values of Response Coefficients also show that the transition time has a bigger sensitivity to changes in glucose concentration than the flux in all conditions assayed, but particularly in glycolytic ones.     

Application of electrical analogues for control analysis of simple metabolic pathways.

I have used electrical analogues for calculating the Flux Control Coefficients of metabolic pathways. An analogue circuit consists of resistances that are connected in series (or parallel) with a voltage (or current) source. In constructing the analogues, each of the enzymes in the pathway is associated with a resistance whose magnitude depends on the Elasticity Coefficients of the enzymes. These circuits can be designed in a heuristic fashion directly from the configuration of the pathway, without the necessity of writing down the governing equations with the use of Summation and Connectivity Theorems. The Flux Control Coefficients of the enzymes are represented by voltages across (or currents through) the resistances and are determined by an application of Ohm's Law. Results are given for (a) a simple linear pathway without feedback or feedforward regulation, and (b) a linear pathway with feedback inhibition. The analogue circuits are also convenient for assessing the relative importance of the various enzymes in flux control, and for simplifying the structure of a given pathway.     

Study of the flux and transition time control coefficient profiles in a metabolic system in vitro and the effect of an external stimulator.

Control of flux and transition time was investigated with a reconstructed rabbit muscle glycolytic system in vitro as an experimental model. The results show agreement with the summation property for the Flux Control Coefficients [Kacser & Burns (1973) Symp. Soc. Exp. Biol. 27, 65-104; Heinrich & Rapoport (1974) Eur. J. Biochem. 42, 89-95]. Control of flux is almost exclusively located at the hexokinase- and phosphofructokinase-catalysed steps, whereas control of transition time is distributed more evenly between the enzymes of the system. The summation value of the Transition Time Control Coefficients is near to -1, suggesting the existence of another Summation Theorem besides that already stated for Flux Control Coefficients. Finally, we study the effect of an external stimulator of the system (fructose 2,6-bisphosphate) on the Control Coefficient profiles. The effect appears to be greater on the Transition Time Control Coefficient distribution than on the Flux Control Coefficients.     

Control, regulation and thermodynamics of free-energy transduction

The quantitative formalism called Metabolic Control Theory makes it possible to be precise in discussions of metabolic control. To illustrate this, I will mention 2 experimental systems where free energy is converted from one form to another, i.e., bacteriorhodopsin liposomes and mitochondrial oxidative phosphorylation. More specifically I shall discuss how the distribution of the control of fluxes, concentrations and potentials, among the various enzymes (catalysts) in these systems has been measured and how this distribution can be understood in terms of the enzyme properties. From the outset, Metabolic Control Theory was valid for branched metabolic pathways with non-linear kinetics. Yet, it seemed to be limited to metabolic pathways without enzyme-enzyme interactions and to steady states. It is now clear that these limitations were apparent only and recent extensions to Metabolic Control Theory deal explicitly with enzyme-enzyme interaction and with transient-time analysis. Other limitations are inherent. For instance, Metabolic Control Theory pays for its clarity and exactness by being limited to small modulations. Mosaic Non Equilibrium Thermodynamics and Biochemical System Analysis are formalisms that deal with larger changes, at the cost of accuracy and exactness.     

Control of glycolysis in rat liver by glucokinase and phosphofructokinase: influence of glucose concentration

Summary Control of glucose metabolism in rat liver under different glucose concentrations was studied. Flux Control Coefficients of glucokinase, glucose 6-phosphate isomerase and phosphofructokinase were determined by the shortening and enzyme titration method. Results obtained show that glucose concentration in liver can play an important role in control of liver glycolysis by enhancing the Flux Control Coefficient of phosphofructokinase. Possible physiological significance of this fact is discussed.     

Integrating tracer-based metabolomics data and metabolic fluxes in a linear fashion via Elementary Carbon Modes

Constraints-based modeling is an emergent area in Systems Biology that includes an increasing set of methods for the analysis of metabolic networks. In order to refine its predictions, the development of novel methods integrating high-throughput experimental data is currently a key challenge in the field. In this paper, we present a novel set of constraints that integrate tracer-based metabolomics data from Isotope Labeling Experiments and metabolic fluxes in a linear fashion. These constraints are based on Elementary Carbon Modes (ECMs), a recently developed concept that generalizes Elementary Flux Modes at the carbon level. To illustrate the effect of our ECMs-based constraints, a Flux Variability Analysis approach was applied to a previously published metabolic network involving the main pathways in the metabolism of glucose. The addition of our ECMs-based constraints substantially reduced the under-determination resulting from a standard application of Flux Variability Analysis, which shows a clear progress over the state of the art. In addition, our approach is adjusted to deal with combinatorial explosion of ECMs in genome-scale metabolic networks. This extension was applied to infer the maximum biosynthetic capacity of non-essential amino acids in human metabolism. Finally, as linearity is the hallmark of our approach, its importance is discussed at a methodological, computational and theoretical level and illustrated with a practical application in the field of Isotope Labeling Experiments.     

A novel galactosyl-binding lectin from the plasma of the blood clam,Anadara granosa (L) and a study of its combining site

Control Analysis has been carried out in the first steps of a rat liver glycolytic system. Attention has been focused on the effect of several glucose concentrations on the control, particularly regarding the role of glucokinase. From kinetic studies of the whole metabolic system we have obtained information on the flux variation under different glucose concentrations. This information together with the kinetics of glucokinase has allowed us to calculate Flux Control and Elasticity Coefficients for glucokinase and the Response Coefficient of the system with respect to glucose. The changes in of the value of Flux Control Coefficients demonstrates that in conditions of low glucose concentration, glucokinase is the main enzyme in controlling the flux through the pathway, but at high glucose concentration the control moves to phosphofructokinase. Next, we have compared our results with those obtained with the shortening and titration method, previously described (Torres, N.V., Mateo, F., Mélendez-Hevia, E. and Kacser, H., (1986) Biochem. J. 234, 169–174; Torres, N.V. and Meléndez-Hevia, E. 1991. Molec. Cell. Biochem. 101, 1–10). Furthermore, from knowledge of the enzyme kinetics of the system we have been able to build a model of the pathway that allows us computer similation of its behavior and calculation of the Flux Control Coefficient profile at different glucose concentrations. By the three methods the results correlate, supporting the use of the pathway substrate as external modulator of the metabolic system as a tool for practical application of Control Analysis.     

Metabolic flux analysis and visualization

One of the ultimate goals of systems biology research is to obtain a comprehensive understanding of the control mechanisms of complex cellular metabolisms. Metabolic Flux Analysis (MFA) is a important method for the quantitative estimation of intracellular metabolic flows through metabolic pathways and the elucidation of cellular physiology. The primary challenge in the use of MFA is that many biological networks are underdetermined systems; it is therefore difficult to narrow down the solution space from the stoichiometric constraints alone. In this tutorial, we present an overview of Flux Balance Analysis (FBA) and (13)C-Metabolic Flux Analysis ((13)C-MFA), both of which are frequently used to solve such underdetermined systems, and we demonstrate FBA and (13)C-MFA using the genome-scale model and the central carbon metabolism model, respectively. Furthermore, because such comprehensive study of intracellular fluxes is inherently complex, we subsequently introduce various pathway mapping and visualization tools to facilitate understanding of these data in the context of the pathways. Specific visualization of MFA results using the BioCyc Omics Viewer and Pathway Projector are shown as illustrative examples.     

An algorithm for efficient identification of branched metabolic pathways

This article presents a new graph-based algorithm for identifying branched metabolic pathways in multi-genome scale metabolic data. The term branched is used to refer to metabolic pathways between compounds that consist of multiple pathways that interact biochemically. A branched pathway may produce a target compound through a combination of linear pathways that split compounds into smaller ones, work in parallel with many compounds, and join compounds into larger ones. While branched metabolic pathways predominate in metabolic networks, most previous work has focused on identifying linear metabolic pathways. The ability to automatically identify branched pathways is important in applications that require a deeper understanding of metabolism, such as metabolic engineering and drug target identification. The algorithm presented in this article utilizes explicit atom tracking to identify linear metabolic pathways and then merges them together into branched metabolic pathways. We provide results on several well-characterized metabolic pathways that demonstrate that the new merging approach can efficiently find biologically relevant branched metabolic pathways.     

Flux balance analysis of photoautotrophic metabolism

Photosynthesis is the principal process responsible for fixation of inorganic carbon dioxide into organic molecules with sunlight as the energy source. Potentially, many chemicals could be inexpensively produced by photosynthetic organisms. Mathematical modeling of photoautotrophic metabolism is therefore important to evaluate maximum theoretical product yields and to deeply understand the interactions between biochemical energy, carbon fixation, and assimilation pathways. Flux balance analysis based on linear programming is applied to photoautotrophic metabolism. The stoichiometric network of a model photosynthetic prokaryote, Synechocystis sp. PCC 6803, has been reconstructed from genomic data and biochemical literature and coupled with a model of the photophosphorylation processes. Flux map topologies for the hetero-, auto-, and mixotrophic modes of metabolism under conditions of optimal growth were determined and compared. The roles of important metabolic reactions such as the glyoxylate shunt and the transhydrogenase reaction were analyzed. We also theoretically evaluated the effect of gene deletions or additions on biomass yield and metabolic flux distributions.     

Effect of genetic convergence on phylogenetic inference

Phylogenetic reconstructions are a major component of many studies in evolutionary biology, but their accuracy can be reduced under certain conditions. Recent studies showed that the convergent evolution of some phenotypes resulted from recurrent amino acid substitutions in genes belonging to distant lineages. It has been suggested that these convergent substitutions could bias phylogenetic reconstruction toward grouping convergent phenotypes together, but such an effect has never been appropriately tested. We used computer simulations to determine the effect of convergent substitutions on the accuracy of phylogenetic inference. We show that, in some realistic conditions, even a relatively small proportion of convergent codons can strongly bias phylogenetic reconstruction, especially when amino acid sequences are used as characters. The strength of this bias does not depend on the reconstruction method but varies as a function of how much divergence had occurred among the lineages prior to any episodes of convergent substitutions. While the occurrence of this bias is difficult to predict, the risk of spurious groupings is strongly decreased by considering only 3rd codon positions, which are less subject to selection, as long as saturation problems are not present. Therefore, we recommend that, whenever possible, topologies obtained with amino acid sequences and 3rd codon positions be compared to identify potential phylogenetic biases and avoid evolutionarily misleading conclusions.     

Systems biology towards life in silico: mathematics of the control of living cells

Systems Biology is the science that aims to understand how biological function absent from macromolecules in isolation, arises when they are components of their system. Dedicated to the memory of Reinhart Heinrich, this paper discusses the origin and evolution of the new part of systems biology that relates to metabolic and signal-transduction pathways and extends mathematical biology so as to address postgenomic experimental reality. Various approaches to modeling the dynamics generated by metabolic and signal-transduction pathways are compared. The silicon cell approach aims to describe the intracellular network of interest precisely, by numerically integrating the precise rate equations that characterize the ways macromolecules’ interact with each other. The non-equilibrium thermodynamic or ‘lin–log’ approach approximates the enzyme rate equations in terms of linear functions of the logarithms of the concentrations. Biochemical Systems Analysis approximates in terms of power laws. Importantly all these approaches link system behavior to molecular interaction properties. The latter two do this less precisely but enable analytical solutions. By limiting the questions asked, to optimal flux patterns, or to control of fluxes and concentrations around the (patho)physiological state, Flux Balance Analysis and Metabolic/Hierarchical Control Analysis again enable analytical solutions. Both the silicon cell approach and Metabolic/Hierarchical Control Analysis are able to highlight where and how system function derives from molecular interactions. The latter approach has also discovered a set of fundamental principles underlying the control of biological systems. The new law that relates concentration control to control by time is illustrated for an important signal transduction pathway, i.e. nuclear hormone receptor signaling such as relevant to bone formation. It is envisaged that there is much more Mathematical Biology to be discovered in the area between molecules and Life.     

An extended dynamic model of oxidative phosphorylation.

The presented model based on an earlier one (Korzeniewski, B. and Froncisz, W. (1989) Studia Biophys. 132, 173-187) simulates concentration changes in time of chemical compounds and thermodynamic forces during respiration of cell suspension in a closed chamber. A set of differential equations solved numerically describes the utilization of oxygen up to anaerobiosis and the behaviour of the system after a sudden pulse of oxygen. Flux control coefficients for most important reactions (enzymes) of oxidative phosphorylation were calculated. A good qualitative and (when a direct comparison is possible) quantitative agreement with experimental results can be observed. The following conclusions can be drawn from the simulation: (1) Wilson's steady state model is not in contradiction with sharing of the control over the respiration between some steps and displacement of the ATP/ADP carrier from equilibrium. (2) The overshoot characteristics of the delta microH+ time-course after reoxygenation can be explained without using the lag-phase kinetics of ATP-synthetase. (3) A 'hot region' (sharp changes of many parameters) can be distinguished when the oxygen concentration approaches zero; only cytochrome oxidase is clearly sensitive on oxygen concentration in all its range. (4) Control over oxidative phosphorylation is shared mainly between inputs of the system (ATP utilization and substrate dehydrogenation) and the proton leak.     

Decreased-activity mutants of phosphoglucose isomerase in the cytosol and chloroplast of Clarkia xantiana. Impact on mass-action ratios and fluxes to sucrose and starch, and estimation of Flux Control Coefficients and Elasticity Coefficients.

1. Subcellular-compartment-specific decreased-activity mutants of phosphoglucose isomerase in Clarkia xantiana were used to analyse the control of sucrose and starch synthesis during photosynthesis. Mutants were available in which the plastid phosphoglucose isomerase complement is decreased to 75% or 50% of the wild-type level, and the cytosol complement to 64%, 36% or 18% of the wild-type level. 2. The effects on the [product]/[substrate] ratio and on fluxes to sucrose or starch and the rate of photosynthesis were studied with the use of saturating or limiting light intensity to impose a high or low flux through these pathways. 3. Removal of a small fraction of either phosphoglucose isomerase leads to a significant shift of the [product]/[substrate] ratio away, from equilibrium. We conclude that there is no 'excess' of enzyme over that needed to maintain its reactants reasonably close to equilibrium. 4. Decreased phosphoglucose isomerase activity can also alter the fluxes to starch or sucrose. However, the effect on flux does not correlate with the extent of disequilibrium, and also varies depending on the subcellular compartment and on the conditions. 5. The results were used to estimate Flux Control Coefficients for the chloroplast and cytosolic phosphoglucose isomerases. The chloroplast isoenzyme exerts control on the rate of starch synthesis and on photosynthesis in saturating light intensity and CO2, but not at low light intensity. The cytosolic enzyme only exerts significant control when its complement is decreased 3-5-fold, and differs from the plastid isoenzyme in exerting more control in low light intensity. It has a positive Control Coefficient for sucrose synthesis, and a negative Control Coefficient for starch synthesis. 6. The Elasticity Coefficients in vivo of the cytosolic phosphoglucose isomerase were estimated to lie between 5 and 8 in the wild-type. They decrease in mutants with a lowered complement of cytosolic phosphoglucose isomerase. 7. The implications of these results for regulation and for evolution are discussed.     

Glycogen and its metabolism

Glycogen is a branched polymer of glucose which serves as a reservoir of glucose units. The two largest deposits in mammals are in the liver and skeletal muscle but many cells are capable synthesizing glycogen. Its accumulation and utilization are under elaborate controls involving primarily covalent phosphorylation and allosteric ligand binding. Both muscle and liver glycogen reserves are important for whole body glucose metabolism and their replenishment is linked hormonally to nutritional status. Control differs between muscle and liver in part due to the existence of different tissue-specific isoforms at key steps. Control of synthesis is shared between transport into the muscle and the step catalyzed by glycogen synthase. Breakdown of liver glycogen, as part of blood glucose homeostasis, is also in response to nutritional cues. Muscle glycogen serves only to fuel muscular activity and its utilization is controlled by muscle contraction and by catecholamines. Though the number of enzymes directly involved in the metabolism of glycogen is quite small, many more proteins act indirectly in a regulatory capacity. Defects in the basic metabolizing enzymes lead to severe consequences whereas, with some exceptions, mutations in the regulatory proteins appear to cause a more subtle phenotypic change.     

Metabolomic profiling and genomic analysis of wheat aneuploid lines to identify genes controlling biochemical pathways in mature grain

Metabolomics is becoming an increasingly important tool in plant genomics to decipher the function of genes controlling biochemical pathways responsible for trait variation. Although theoretical models can integrate genes and metabolites for trait variation, biological networks require validation using appropriate experimental genetic systems. In this study, we applied an untargeted metabolite analysis to mature grain of wheat homoeologous group 3 ditelosomic lines, selected compounds that showed significant variation between wheat lines Chinese Spring and at least one ditelosomic line, tracked the genes encoding enzymes of their biochemical pathway using the wheat genome survey sequence and determined the genetic components underlying metabolite variation. A total of 412 analytes were resolved in the wheat grain metabolome, and principal component analysis indicated significant differences in metabolite profiles between Chinese Spring and each ditelosomic lines. The grain metabolome identified 55 compounds positively matched against a mass spectral library where the majority showed significant differences between Chinese Spring and at least one ditelosomic line. Trehalose and branched‐chain amino acids were selected for detailed investigation, and it was expected that if genes encoding enzymes directly related to their biochemical pathways were located on homoeologous group 3 chromosomes, then corresponding ditelosomic lines would have a significant reduction in metabolites compared with Chinese Spring. Although a proportion showed a reduction, some lines showed significant increases in metabolites, indicating that genes directly and indirectly involved in biosynthetic pathways likely regulate the metabolome. Therefore, this study demonstrated that wheat aneuploid lines are suitable experimental genetic system to validate metabolomics–genomics networks.     

Kinetics of metabolic pathways. A system in vitro to study the control of flux.

A method for determining Control Coefficients is proposed for systems studied in vitro and applied to a model pathway. Rat liver extract, which converts glucose into glycerol 3-phosphate, was used with the addition to the incubation mixture of fructose-bisphosphate aldolase, triose-phosphate isomerase and glycerol-3-phosphate dehydrogenase as 'auxiliary' enzymes, which leaves all the control on the first three enzymes. The flux of the metabolic pathway was recorded by assaying NADH decay. Flux Control Coefficients (CJE) of hexokinase, glucose-6-phosphate isomerase and phosphofructokinase were calculated by titration of the system with increasing quantities of extraneous enzymes. It is shown that the summation property is fulfilled. The applicability of this procedure to study the control in any metabolic pathway is discussed. Possible relevance of the method to conditions in vivo and its limitations are considered.