Multi-criteria optimization of regulation in metabolic networks

Determining the regulation of metabolic networks at genome scale is a hard task. It has been hypothesized that biochemical pathways and metabolic networks might have undergone an evolutionary process of optimization with respect to several criteria over time. In this contribution, a multi-criteria approach has been used to optimize parameters for the allosteric regulation of enzymes in a model of a metabolic substrate-cycle. This has been carried out by calculating the Pareto set of optimal solutions according to two objectives: the proper direction of flux in a metabolic cycle and the energetic cost of applying the set of parameters. Different Pareto fronts have been calculated for eight different "environments" (specific time courses of end product concentrations). For each resulting front the so-called knee point is identified, which can be considered a preferred trade-off solution. Interestingly, the optimal control parameters corresponding to each of these points also lead to optimal behaviour in all the other environments. By calculating the average of the different parameter sets for the knee solutions more frequently found, a final and optimal consensus set of parameters can be obtained, which is an indication on the existence of a universal regulation mechanism for this system.The implications from such a universal regulatory switch are discussed in the framework of large metabolic networks.     

Mathematical modelling of dynamics and control in metabolic networks. II. Simple dimeric enzymes

The dynamics of enzyme cooperativity are examined by studying a homotropic dimeric enzyme with identical reaction sites, both of which follow irreversible Michaelis-Menten kinetics. The problem is approached via scaling and linearization of the governing mass action kinetic equations. Homotropic interaction between the two sites are found to depend on three dimensionless groups, two for the substrate binding step and one for the chemical transformation. The interaction between the two reaction sites is shown capable of producing dynamic behavior qualitatively different from that of a simple Michaelis-Menten system; when the two sites interact to increase enzymatic activity over that of two independent monomeric enzymes (positive cooperativity) damped oscillatory behavior is possible, and for negative cooperativity in the chemical transformation step a multiplicity of steady states can occur, with one state unstable and leading to runaway behavior. Linear analysis gives significant insight into system dynamics, and their parametric sensitivity, and a way to identify regions of the parameter space where the approximate quasi-stationary and quasi-equilibrium analyses are appropriate.     

Mathematical modelling of dynamics and control in metabolic networks. I. On Michaelis-Menten kinetics

As a starting point for modeling of metabolic networks this paper considers the simple Michaelis-Menten reaction mechanism. After the elimination of diffusional effects a mathematically intractable mass action kinetic model is obtained. The properties of this model are explored via scaling and linearization. The scaling is carried out such that kinetic properties, concentration parameters and external influences are clearly separated. We then try to obtain reasonable estimates for values of the dimensionless groups and examine the dynamic properties of the model over this part of the parameter space. Linear analysis is found to give excellent insight into reaction dynamics and it also gives a forum for understanding and justifying the two commonly used quasi-stationary and quasi-equilibrium analyses. The first finding is that there are two separate time scales inherent in the model existing over most of the parameter space, and in particular over the regions of importance here. Full modal analysis gives a new interpretation of quasi-stationary analysis, and its extension via singular perturbation theory, and a rationalization of the quasi-equilibrium approximation. The new interpretation of the quasi-steady state assumption is that the applicability is intimately related to dynamic interactions between the concentration variables rather than the traditional notion that a quasi-stationary state is reached, after a short transient period, where the rates of formation and decomposition of the enzyme intermediate are approximately equal. The modal analysis reveals that the generally used criterion for the applicability of quasi-stationary analysis that total enzyme concentration must be much less than total substrate concentration, et much less than St, is incomplete and that the criterion et much less than Km much less than St (Km is the well known Michaelis constant) is the appropriate one. The first inequality (et much less than Km) guarantees agreement over the longer time scale leading to quasi-stationary behavior or the applicability of the zeroth order outer singular perturbation solution but the second half of the criterion (Km much less than St) justifies zeroth order inner singular perturbation solution where the substrate concentration is assumed to be invariant. Furthermore linear analysis shows that when a fast mode representing the binding of substrate to the enzyme is fast it can be relaxed leading to the quasi-equilibrium assumption. The influence of the dimensionless groups is ascertained by integrating the equations numerically, and the predictions made by the linear analysis are found to be accurate.(ABSTRACT TRUNCATED AT 400 WORDS)     

Flux modules in metabolic networks

The huge number of elementary flux modes in genome-scale metabolic networks makes analysis based on elementary flux modes intrinsically difficult. However, it has been shown that the elementary flux modes with optimal yield often contain highly redundant information. The set of optimal-yield elementary flux modes can be compressed using modules. Up to now, this compression was only possible by first enumerating the whole set of all optimal-yield elementary flux modes. We present a direct method for computing modules of the thermodynamically constrained optimal flux space of a metabolic network. This method can be used to decompose the set of optimal-yield elementary flux modes in a modular way and to speed up their computation. In addition, it provides a new form of coupling information that is not obtained by classical flux coupling analysis. We illustrate our approach on a set of model organisms.     

Metabolite coupling in genome-scale metabolic networks

Background  

Biochemically detailed stoichiometric matrices have now been reconstructed for various bacteria, yeast, and for the human cardiac mitochondrion based on genomic and proteomic data. These networks have been manually curated based on legacy data and elementally and charge balanced. Comparative analysis of these well curated networks is now possible. Pairs of metabolites often appear together in several network reactions, linking them topologically. This co-occurrence of pairs of metabolites in metabolic reactions is termed herein "metabolite coupling." These metabolite pairs can be directly computed from the stoichiometric matrix, S. Metabolite coupling is derived from the matrix ŜŜ ^T, whose off-diagonal elements indicate the number of reactions in which any two metabolites participate together, where Ŝ is the binary form of S.     

Genotype networks in metabolic reaction spaces

Background  

A metabolic genotype comprises all chemical reactions an organism can catalyze via enzymes encoded in its genome. A genotype is viable in a given environment if it is capable of producing all biomass components the organism needs to survive and reproduce. Previous work has focused on the properties of individual genotypes while little is known about how genome-scale metabolic networks with a given function can vary in their reaction content.     

Essentiality and damage in metabolic networks

Understanding the architecture of physiological functions from annotated genome sequences is a major task for postgenomic biology. From the annotated genome sequence of the microbe Escherichia coli, we propose a general quantitative definition of enzyme importance in a metabolic network. Using a graph analysis of its metabolism, we relate the extent of the topological damage generated in the metabolic network by the deletion of an enzyme to the experimentally determined viability of the organism in the absence of that enzyme. We show that the network is robust and that the extent of the damage relates to enzyme importance. We predict that a large fraction (91%) of enzymes causes little damage when removed, while a small group (9%) can cause serious damage. Experimental results confirm that this group contains the majority of essential enzymes. The results may reveal a universal property of metabolic networks.     

Predicting synthetic rescues in metabolic networks

An important goal of medical research is to develop methods to recover the loss of cellular function due to mutations and other defects. Many approaches based on gene therapy aim to repair the defective gene or to insert genes with compensatory function. Here, we propose an alternative, network‐based strategy that aims to restore biological function by forcing the cell to either bypass the functions affected by the defective gene, or to compensate for the lost function. Focusing on the metabolism of single‐cell organisms, we computationally study mutants that lack an essential enzyme, and thus are unable to grow or have a significantly reduced growth rate. We show that several of these mutants can be turned into viable organisms through additional gene deletions that restore their growth rate. In a rather counterintuitive fashion, this is achieved via additional damage to the metabolic network. Using flux balance‐based approaches, we identify a number of synthetically viable gene pairs, in which the removal of one enzyme‐encoding gene results in a non‐viable phenotype, while the deletion of a second enzyme‐encoding gene rescues the organism. The systematic network‐based identification of compensatory rescue effects may open new avenues for genetic interventions.     

Structural correlations in bacterial metabolic networks

Background  

Evolution of metabolism occurs through the acquisition and loss of genes whose products acts as enzymes in metabolic reactions, and from a presumably simple primordial metabolism the organisms living today have evolved complex and highly variable metabolisms. We have studied this phenomenon by comparing the metabolic networks of 134 bacterial species with known phylogenetic relationships, and by studying a neutral model of metabolic network evolution.     

Understanding flux in plant metabolic networks

The revolutionary growth in our ability to identify the 'parts list' of cellular infrastructure in plants in detail, and to alter it with precision, challenges us to develop methods to quantify how these parts function. For components of metabolism, this means mapping fluxes at the level of metabolic networks. Advances in experimental, analytical and software tools for metabolic flux analysis now allow maps of the fluxes through central metabolism to be obtained from the results of stable-isotope-labeling experiments. Such maps have led to notable successes in understanding and engineering metabolic function in microorganisms. Recent studies in plants are giving insight into particular fluxes, such as those of the pentose phosphate pathway, and into general phenomena, such as substrate- or futile-cycles and compartmentation. The importance of experimental design and statistical analysis have been illustrated, and analyses of fluxes in heterotrophic plant tissues have been carried out recently.     

Visualizing regulatory interactions in metabolic networks

Background  

Direct visualization of data sets in the context of biochemical network drawings is one of the most appealing approaches in the field of data evaluation within systems biology. One important type of information that is very helpful in interpreting and understanding metabolic networks has been overlooked so far. Here we focus on the representation of this type of information given by the strength of regulatory interactions between metabolite pools and reaction steps.     

Reducing complexity in metabolic networks: making metabolic meshes manageable

The dynamics of complex systems can be effectively analyzed by judicious use of intrinsic time constants. Order of magnitude estimation based on time constants has been used successfully to examine the dynamic behavior of complicated processes. The main goal of this paper is to introduce this approach to the analysis of complex metabolic systems. Time constants and dynamic modes of motion are defined within the context of well-established linear algebra. The order of magnitude estimation is then introduced into the systemic framework. The main goals of the analysis are: to provide improved understanding of biochemical dynamics and their physiological significance, and to yield reduced dynamic models that are physiologically realistic but tractable for practical use.