Models and molecular mechanisms for trade-offs in the context of metabolism

Accumulating evidence for trade-offs involving metabolic traits has demonstrated their importance in the evolution of organisms. Metabolic models with different levels of complexity have already been considered when investigating mechanisms that explain various metabolic trade-offs. Here we provide a systematic review of modelling approaches that have been used to study and explain trade-offs between: (i) the kinetic properties of individual enzymes, (ii) rates of metabolic reactions, (iii) the rate and yield of metabolic pathways and networks, (iv) different metabolic objectives in single organisms and in metabolic communities, and (v) metabolic concentrations. In providing insights into the mechanisms underlying these five types of metabolic trade-offs obtained from constraint-based metabolic modelling, we emphasize the relationship of metabolic trade-offs to the classical black box Y-model that provides a conceptual explanation for resource acquisition-allocation trade-offs. In addition, we identify several pressing concerns and offer a perspective for future research in the identification and manipulation of metabolic trade-offs by relying on the toolbox provided by constraint-based metabolic modelling for single organisms and microbial communities.

What makes a reaction network "chemical"?

BACKGROUND

Reaction networks (RNs) comprise a set X of species and a set [Formula: see text] of reactions [Formula: see text], each converting a multiset of educts [Formula: see text] into a multiset [Formula: see text] of products. RNs are equivalent to directed hypergraphs. However, not all RNs necessarily admit a chemical interpretation. Instead, they might contradict fundamental principles of physics such as the conservation of energy and mass or the reversibility of chemical reactions. The consequences of these necessary conditions for the stoichiometric matrix [Formula: see text] have been discussed extensively in the chemical literature. Here, we provide sufficient conditions for [Formula: see text] that guarantee the interpretation of RNs in terms of balanced sum formulas and structural formulas, respectively.

RESULTS

Chemically plausible RNs allow neither a perpetuum mobile, i.e., a "futile cycle" of reactions with non-vanishing energy production, nor the creation or annihilation of mass. Such RNs are said to be thermodynamically sound and conservative. For finite RNs, both conditions can be expressed equivalently as properties of the stoichiometric matrix [Formula: see text]. The first condition is vacuous for reversible networks, but it excludes irreversible futile cycles and-in a stricter sense-futile cycles that even contain an irreversible reaction. The second condition is equivalent to the existence of a strictly positive reaction invariant. It is also sufficient for the existence of a realization in terms of sum formulas, obeying conservation of "atoms". In particular, these realizations can be chosen such that any two species have distinct sum formulas, unless [Formula: see text] implies that they are "obligatory isomers". In terms of structural formulas, every compound is a labeled multigraph, in essence a Lewis formula, and reactions comprise only a rearrangement of bonds such that the total bond order is preserved. In particular, for every conservative RN, there exists a Lewis realization, in which any two compounds are realized by pairwisely distinct multigraphs. Finally, we show that, in general, there are infinitely many realizations for a given conservative RN.

CONCLUSIONS

"Chemical" RNs are directed hypergraphs with a stoichiometric matrix [Formula: see text] whose left kernel contains a strictly positive vector and whose right kernel does not contain a futile cycle involving an irreversible reaction. This simple characterization also provides a concise specification of random models for chemical RNs that additionally constrain [Formula: see text] by rank, sparsity, or distribution of the non-zero entries. Furthermore, it suggests several interesting avenues for future research, in particular, concerning alternative representations of reaction networks and infinite chemical universes.

ODEbase: a repository of ODE systems for systems biology

Summary

Recently, symbolic computation and computer algebra systems have been successfully applied in systems biology, especially in chemical reaction network theory. One advantage of symbolic computation is its potential for qualitative answers to biological questions. Qualitative methods analyze dynamical input systems as formal objects, in contrast to investigating only part of the state space, as is the case with numerical simulation. However, corresponding tools and libraries have a different set of requirements for their input data than their numerical counterparts. A common format used in mathematical modeling of biological processes is Systems Biology Markup Language (SBML). We illustrate that the use of SBML data in symbolic computation requires significant pre-processing, incorporating external biological and mathematical expertise. ODEbase provides suitable input data derived from established existing biomodels, covering in particular the BioModels database.

Availability and implementation

ODEbase is available free of charge at https://odebase.org.

Computational quantification of global effects induced by mutations and drugs in signaling networks of colorectal cancer cells

Colorectal cancer (CRC) is one of the most deadly and commonly diagnosed tumors worldwide. Several genes are involved in its development and progression. The most frequent mutations concern APC, KRAS, SMAD4, and TP53 genes, suggesting that CRC relies on the concomitant alteration of the related pathways. However, with classic molecular approaches, it is not easy to simultaneously analyze the interconnections between these pathways. To overcome this limitation, recently these pathways have been included in a huge chemical reaction network (CRN) describing how information sensed from the environment by growth factors is processed by healthy colorectal cells. Starting from this CRN, we propose a computational model which simulates the effects induced by single or multiple concurrent mutations on the global signaling network. The model has been tested in three scenarios. First, we have quantified the changes induced on the concentration of the proteins of the network by a mutation in APC, KRAS, SMAD4, or TP53. Second, we have computed the changes in the concentration of p53 induced by up to two concurrent mutations affecting proteins upstreams in the network. Third, we have considered a mutated cell affected by a gain of function of KRAS, and we have simulated the action of Dabrafenib, showing that the proposed model can be used to determine the most effective amount of drug to be delivered to the cell. In general, the proposed approach displays several advantages, in that it allows to quantify the alteration in the concentration of the proteins resulting from a single or multiple given mutations. Moreover, simulations of the global signaling network of CRC may be used to identify new therapeutic targets, or to disclose unexpected interactions between the involved pathways.

Gain and loss of function mutations in biological chemical reaction networks: a mathematical model with application to colorectal cancer cells

This paper studies a system of Ordinary Differential Equations modeling a chemical reaction network and derives from it a simulation tool mimicking Loss of Function and Gain of Function mutations found in cancer cells. More specifically, from a theoretical perspective, our approach focuses on the determination of moiety conservation laws for the system and their relation with the corresponding stoichiometric surfaces. Then we show that Loss of Function mutations can be implemented in the model via modification of the initial conditions in the system, while Gain of Function mutations can be implemented by eliminating specific reactions. Finally, the model is utilized to examine in detail the G1-S phase of a colorectal cancer cell.

NetRed, an algorithm to reduce genome-scale metabolic networks and facilitate the analysis of flux predictions

Flux balance analysis (FBA) of large, genome-scale stoichiometric models (GSMs) is a powerful and popular method to predict cell-wide metabolic activity. FBA typically generates a flux vector containing O(1,000) fluxes. The interpretation of such a flux vector is difficult, even for expert users, because of the large size and complex topology of the underlying metabolic network. This interpretation could be simplified by condensing the network to a reduced, yet fully representative version. Toward this goal we report NetRed, an algorithm that systematically reduces a stoichiometric matrix and a corresponding flux vector to a more easily interpretable form. The reduction offered by NetRed is transparent because it relies purely on matrix algebra and not on optimization. Uniquely, it involves zero information loss; therefore, the original unreduced network can be easily recovered from the reduced network. The inputs to NetRed are (i) a stoichiometric matrix, (ii) a flux vector with numerical flux values, and (iii) a list of "protected" metabolites recommended by the user to remain in the reduced network. NetRed outputs a reduced metabolic network containing a reduced number of metabolites, of which the protected metabolites are a subset. The algorithm also generates a corresponding reduced flux vector. Due to its simplified presentation and easier interpretability, the reduced network allows the user to quickly find fluxes through metabolites and reaction modes or pathways of interest. In this manuscript, we first demonstrate NetRed on a simple network consisting of glycolysis and the pentose phosphate pathway (PPP), wherein NetRed reduced the PPP to a single net reaction. We followed this with applications of NetRed to E. coli and yeast GSMs. NetRed reduced the size of an E. coli GSM by 20- to 30-fold and enabled a comprehensive comparison of aerobic and anaerobic metabolism. The application of NetRed to a yeast GSM allowed for easy mechanistic interpretation of a double-gene knockout that rerouted flux toward dihydroartemisinic acid. When applied to an E. coli strain engineered for enhanced valine production, NetRed allowed for a holistic interpretation of the metabolic rerouting resulting from multiple genetic interventions.

Bond graph modelling of the cardiac action potential: implications for drift and non-unique steady states

Mathematical models of cardiac action potentials have become increasingly important in the study of heart disease and pharmacology, but concerns linger over their robustness during long periods of simulation, in particular due to issues such as model drift and non-unique steady states. Previous studies have linked these to violation of conservation laws, but only explored those issues with respect to charge conservation in specific models. Here, we propose a general and systematic method of identifying conservation laws hidden in models of cardiac electrophysiology by using bond graphs, and develop a bond graph model of the cardiac action potential to study long-term behaviour. Bond graphs provide an explicit energy-based framework for modelling physical systems, which makes them well suited for examining conservation within electrophysiological models. We find that the charge conservation laws derived in previous studies are examples of the more general concept of a 'conserved moiety'. Conserved moieties explain model drift and non-unique steady states, generalizing the results from previous studies. The bond graph approach provides a rigorous method to check for drift and non-unique steady states in a wide range of cardiac action potential models, and can be extended to examine behaviours of other excitable systems.

On the Relation between Chemical Oscillations and Self-Replication

One proposed scenario for the emergence of biochemical oscillations is that they may have provided the basic mechanism behind cellular self-replication by growth and division. However, alternative scenarios not requiring any chemical oscillation have also been proposed. Each of the various protocell models proposed to support one or another scenario comes with its own set of specific assumptions, which makes it difficult to ascertain whether chemical oscillations are required or not for cellular self-replication. This article compares these two cases within a single whole-cell model framework. This model relies upon a membrane embedding a chemical reaction network (CRN) synthesizing all the cellular constituents, including the membrane, by feeding from an external nutrient. Assuming the osmolarity is kept constant, the system dynamics are governed by a set of nonlinear differential equations coupling the chemical concentrations and the surface-area-to-volume ratio. The resulting asymptotic trajectories are used to determine the cellular shape by minimizing the membrane bending energy (within an approximate predefined family of shapes). While the stationary case can be handled quite generally, the oscillatory one is investigated using a simple oscillating CRN example, which is used to identify features that are expected to hold for any network. It is found that cellular self-replication can be reached with or without chemical oscillations, and that a requirement common to both stationary and oscillatory cases is that a minimum spontaneous curvature of the membrane is required for the cell to divide once its area and volume are both doubled. The oscillatory case can result in a greater variety of cellular shape trajectories but raises additional constraints for cellular division and self-replication: (i) the ratio of doubling time to oscillation period should be an integer, and (ii) if the oscillation amplitude is sufficiently high, then the spontaneous curvature must be below a maximum value to avoid early division before the end of the cycle. Because of these additional stringent constraints, it is likely that early protocells did not rely upon chemical oscillations. Biochemical oscillations typical of modern evolved cells may have emerged later through evolution for other reasons (e.g., metabolic advantage) and must have required additional feedback mechanisms for such a self-replicating system to be robust against even slight environmental variations (e.g., temperature fluctuations).

Identification of Conserved Moieties in Metabolic Networks by Graph Theoretical Analysis of Atom Transition Networks

Conserved moieties are groups of atoms that remain intact in all reactions of a metabolic network. Identification of conserved moieties gives insight into the structure and function of metabolic networks and facilitates metabolic modelling. All moiety conservation relations can be represented as nonnegative integer vectors in the left null space of the stoichiometric matrix corresponding to a biochemical network. Algorithms exist to compute such vectors based only on reaction stoichiometry but their computational complexity has limited their application to relatively small metabolic networks. Moreover, the vectors returned by existing algorithms do not, in general, represent conservation of a specific moiety with a defined atomic structure. Here, we show that identification of conserved moieties requires data on reaction atom mappings in addition to stoichiometry. We present a novel method to identify conserved moieties in metabolic networks by graph theoretical analysis of their underlying atom transition networks. Our method returns the exact group of atoms belonging to each conserved moiety as well as the corresponding vector in the left null space of the stoichiometric matrix. It can be implemented as a pipeline of polynomial time algorithms. Our implementation completes in under five minutes on a metabolic network with more than 4,000 mass balanced reactions. The scalability of the method enables extension of existing applications for moiety conservation relations to genome-scale metabolic networks. We also give examples of new applications made possible by elucidating the atomic structure of conserved moieties.

Conserved moieties are transferred between metabolites in internal reactions of a metabolic network but are not synthesised, degraded or exchanged with the environment. The total amount of a conserved moiety in the metabolic network is therefore constant over time. Metabolites that share a conserved moiety have interdependent concentrations because their total amount is constant. Identification of conserved moieties results in a concise description of all concentration dependencies in a metabolic network. The problem of identifying conserved moieties has previously been formulated in terms of the stoichiometry of metabolic reactions. Methods based on this formulation are computationally intractable for large networks. We show that reaction stoichiometry alone gives insufficient information to identify conserved moieties. By first incorporating additional data on the fate of atoms in metabolic reactions, we developed and implemented a computationally tractable algorithm to identify conserved moieties and their atomic structure.

Necessary and sufficient conditions for protocell growth

We consider a generic protocell model consisting of any conservative chemical reaction network embedded within a membrane. The membrane results from the self-assembly of a membrane precursor and is semi-permeable to some nutrients. Nutrients are metabolized into all other species including the membrane precursor, and the membrane grows in area and the protocell in volume. Faithful replication through cell growth and division requires a doubling of both cell volume and surface area every division time (thus leading to a periodic surface area-to-volume ratio) and also requires periodic concentrations of the cell constituents. Building upon these basic considerations, we prove necessary and sufficient conditions pertaining to the chemical reaction network for such a regime to be met. A simple necessary condition is that every moiety must be fed. A stronger necessary condition implies that every siphon must be either fed, or connected to species outside the siphon through a pass reaction capable of transferring net positive mass into the siphon. And in the case of nutrient uptake through passive diffusion and of constant surface area-to-volume ratio, a sufficient condition for the existence of a fixed point is that every siphon be fed. These necessary and sufficient conditions hold for any chemical reaction kinetics, membrane parameters or nutrient flux diffusion constants.

Filamentation as primitive growth mode?

Osmotic pressure influences cellular shape. In a growing cell, chemical reactions and dilution induce changes in osmolarity, which in turn influence the cellular shape. Using a protocell model relying upon random conservative chemical reaction networks with arbitrary stoichiometry, we find that when the membrane is so flexible that its shape adjusts itself quasi-instantaneously to balance the osmotic pressure, the protocell either grows filamentous or fails to grow. This behavior is consistent with a mathematical proof. This suggests that filamentation may be a primitive growth mode resulting from the simple physical property of balanced osmotic pressure. We also find that growth is favored if some chemical species are only present inside the protocell, but not in the outside growth medium. Such an insulation requires specific chemical schemes. Modern evolved cells such as E. coli meet these requirements through active transport mechanisms such as the phosphotransferase system.

Chemical Schemes for Maintaining Different Compositions Across a Semi-permeable Membrane with Application to Proto-cells

Osmotic pressure arising from a higher total chemical concentration inside proto-cells is thought to have played a role in the emergence and selection of self-replicating proto-cells. We present two chemical schemes through which different equilibrium compositions can coexist on each side of a semi-permeable membrane. The first scheme relies upon the concept of moieties and associated number of degrees of freedom. The second scheme relies upon the concept of siphons and of pass reaction capable of transferring matter from outside a siphon into it. Using simple example reaction networks, we show that both schemes are compatible with stationary proto-cell growth with up-concentration, but suffer from shortcomings. To alleviate these we propose a third scheme derived from the second one by having the pass reaction catalyzed by the membrane surface instead of occurring in bulk solution. This may have proven an intermediate step before having the pass reaction occurring only when the nutrient crosses the membrane. This suggests an evolutionary path for the emergence of active transport.

Identifying all moiety conservation laws in genome-scale metabolic networks

The stoichiometry of a metabolic network gives rise to a set of conservation laws for the aggregate level of specific pools of metabolites, which, on one hand, pose dynamical constraints that cross-link the variations of metabolite concentrations and, on the other, provide key insight into a cell's metabolic production capabilities. When the conserved quantity identifies with a chemical moiety, extracting all such conservation laws from the stoichiometry amounts to finding all non-negative integer solutions of a linear system, a programming problem known to be NP-hard. We present an efficient strategy to compute the complete set of integer conservation laws of a genome-scale stoichiometric matrix, also providing a certificate for correctness and maximality of the solution. Our method is deployed for the analysis of moiety conservation relationships in two large-scale reconstructions of the metabolism of the bacterium E. coli, in six tissue-specific human metabolic networks, and, finally, in the human reactome as a whole, revealing that bacterial metabolism could be evolutionarily designed to cover broader production spectra than human metabolism. Convergence to the full set of moiety conservation laws in each case is achieved in extremely reduced computing times. In addition, we uncover a scaling relation that links the size of the independent pool basis to the number of metabolites, for which we present an analytical explanation.

Responses to light intensity in a genome-scale model of rice metabolism

We describe the construction and analysis of a genome-scale metabolic model representing a developing leaf cell of rice (Oryza sativa) primarily derived from the annotations in the RiceCyc database. We used flux balance analysis to determine that the model represents a network capable of producing biomass precursors (amino acids, nucleotides, lipid, starch, cellulose, and lignin) in experimentally reported proportions, using carbon dioxide as the sole carbon source. We then repeated the analysis over a range of photon flux values to examine responses in the solutions. The resulting flux distributions show that (1) redox shuttles between the chloroplast, cytosol, and mitochondrion may play a significant role at low light levels, (2) photorespiration can act to dissipate excess energy at high light levels, and (3) the role of mitochondrial metabolism is likely to vary considerably according to the balance between energy demand and availability. It is notable that these organelle interactions, consistent with many experimental observations, arise solely as a result of the need for mass and energy balancing without any explicit assumptions concerning kinetic or other regulatory mechanisms.

Detecting structural invariants in biological reaction networks

The detection and analysis of structural invariants in cellular reaction networks is of central importance to achieve a more comprehensive understanding of metabolism. In this work, we review different kinds of structural invariants in reaction networks and their Petri net-based representation. In particular, we discuss invariants that can be obtained from the left and right null spaces of the stoichiometric matrix which correspond to conserved moieties (P-invariants) and elementary flux modes (EFMs, minimal T-invariants). While conserved moieties can be used to detect stoichiometric inconsistencies in reaction networks, EFMs correspond to a mathematically rigorous definition of the concept of a biochemical pathway. As outlined here, EFMs allow to devise strategies for strain improvement, to assess the robustness of metabolic networks subject to perturbations, and to analyze the information flow in regulatory and signaling networks. Another important aspect addressed by this review is the limitation of metabolic pathway analysis using EFMs to small or medium-scale reaction networks. We discuss two recently introduced approaches to circumvent these limitations. The first is an algorithm to enumerate a subset of EFMs in genome-scale metabolic networks starting from the EFM with the least number of reactions. The second approach, elementary flux pattern analysis, allows to analyze pathways through specific subsystems of genome-scale metabolic networks. In contrast to EFMs, elementary flux patterns much more accurately reflect the metabolic capabilities of a subsystem of metabolism as well as its integration into the entire system.

Systems approaches to modelling pathways and networks

It has become commonly accepted that systems approaches to biology are of outstanding importance to gain understanding from the vast amount of data which is presently being generated by advancing high-throughput technologies. The diversity of methods to model pathways and networks has significantly expanded over the past two decades. Modern and traditional approaches are equally important and recent activities aim at integrating the advantages of both. While traditional methods, based on differential equations, are useful to study the dynamics of small systems, modern constraint-based models can be applied to genome-scale systems, but are not able to capture dynamic features. Integrating different approaches is important to develop consistent theoretical descriptions encompassing various scales of biological information. The rapid progress of the field of theoretical systems biology, however, demonstrates how our fundamental theoretical understanding of biology is gaining momentum. The scientific community has apparently accepted the challenge to truly understand the principles of life.

Building and analysing genome-scale metabolic models

Reconstructing a model of the metabolic network of an organism from its annotated genome sequence would seem, at first sight, to be one of the most straightforward tasks in functional genomics, even if the various data sources required were never designed with this application in mind. The number of genome-scale metabolic models is, however, lagging far behind the number of sequenced genomes and is likely to continue to do so unless the model-building process can be accelerated. Two aspects that could usefully be improved are the ability to find the sources of error in a nascent model rapidly, and the generation of tenable hypotheses concerning solutions that would improve a model. We will illustrate these issues with approaches we have developed in the course of building metabolic models of Streptococcus agalactiae and Arabidopsis thaliana.

Graph-theoretic methods for the analysis of chemical and biochemical networks. I. Multistability and oscillations in ordinary differential equation models

A chemical mechanism is a model of a chemical reaction network consisting of a set of elementary reactions that express how molecules react with each other. In classical mass-action kinetics, a mechanism implies a set of ordinary differential equations (ODEs) which govern the time evolution of the concentrations. In this article, ODE models of chemical kinetics that have the potential for multiple positive equilibria or oscillations are studied. We begin by considering some methods of stability analysis based on the digraph of the Jacobian matrix. We then prove two theorems originally given by A. N. Ivanova which correlate the bifurcation structure of a mass-action model to the properties of a bipartite graph with nodes representing chemical species and reactions. We provide several examples of the application of these theorems.

A graph-theoretic method for detecting potential Turing bifurcations

The conditions for diffusion-driven (Turing) instabilities in systems with two reactive species are well known. General methods for detecting potential Turing bifurcations in larger reaction schemes are, on the other hand, not well developed. We prove a theorem for a graph-theoretic condition originally given by Volpert and Ivanova [Mathematical Modeling (Nauka, Moscow, 1987) (in Russian), p. 57] for Turing instabilities in a mass-action reaction-diffusion system involving n substances. The method is based on the representation of a reaction mechanism as a bipartite graph with two types of nodes representing chemical species and reactions, respectively. The condition for diffusion-driven instability is related to the existence of a structure in the graph known as a critical fragment. The technique is illustrated using a substrate-inhibited bifunctional enzyme mechanism which involves seven chemical species.

Systematic analysis of conservation relations in Escherichia coli genome-scale metabolic network reveals novel growth media

A biochemical species is called producible in a constraints-based metabolic model if a feasible steady-state flux configuration exists that sustains its nonzero concentration during growth. Extreme semipositive conservation relations (ESCRs) are the simplest semipositive linear combinations of species concentrations that are invariant to all metabolic flux configurations. In this article, we outline a fundamental relationship between the ESCRs of a metabolic network and the producibility of a biochemical species under a nutrient media. We exploit this relationship in an algorithm that systematically enumerates all minimal nutrient sets that render an objective species weakly producible (i.e., producible in the absence of thermodynamic constraints) through a simple traversal of ESCRs. We apply our results to a recent genome scale model of Escherichia coli metabolism, in which we traverse the 51 anhydrous ESCRs of the metabolic network to determine all 928 minimal aqueous nutrient media that render biomass weakly producible. Applying irreversibility constraints, we find 287 of these 928 nutrient sets to be thermodynamically feasible. We also find that an additional 365 of these nutrient sets are thermodynamically feasible in the presence of oxygen. Since biomass producibility is commonly used as a surrogate for growth in genome scale metabolic models, our results represent testable hypotheses of alternate growth media derived from in silico analysis of the E. coli genome scale metabolic network.

An improved algorithm for stoichiometric network analysis: theory and applications

MOTIVATION

Genome scale analysis of the metabolic network of a microorganism is a major challenge in bioinformatics. The combinatorial explosion, which occurs during the construction of elementary fluxes (non-redundant pathways) requires sophisticated and efficient algorithms to tackle the problem.

RESULTS

Mathematically, the calculation of elementary fluxes amounts to characterizing the space of solutions to a mixed system of linear equalities, given by the stoichiometry matrix, and linear inequalities, arising from the irreversibility of some or all of the reactions in the network. Previous approaches to this problem have iteratively solved for the equalities while satisfying the inequalities throughout the process. In an extension of previous work, here we consider the complementary approach and derive an algorithm which satisfies the inequalities one by one while staying in the space of solution of the equality constraints. Benchmarks on different subnetworks of the central carbon metabolism of Escherichia coli show that this new approach yields a significant reduction in the execution time of the calculation. This reduction arises since the odds that an intermediate elementary flux already fulfills an additional inequality are larger than when having to satisfy an additional equality constraint.

Elucidation and structural analysis of conserved pools for genome-scale metabolic reconstructions

In this article, we introduce metabolite concentration coupling analysis (MCCA) to study conservation relationships for metabolite concentrations in genome-scale metabolic networks. The analysis allows the global identification of subsets of metabolites whose concentrations are always coupled within common conserved pools. Also, the minimal conserved pool identification (MCPI) procedure is developed for elucidating conserved pools for targeted metabolites without computing the entire basis conservation relationships. The approaches are demonstrated on genome-scale metabolic reconstructions of Helicobacter pylori, Escherichia coli, and Saccharomyces cerevisiae. Despite significant differences in the size and complexity of the examined organism's models, we find that the concentrations of nearly all metabolites are coupled within a relatively small number of subsets. These correspond to the overall exchange of carbon molecules into and out of the networks, interconversion of energy and redox cofactors, and the transfer of nitrogen, sulfur, phosphate, coenzyme A, and acyl carrier protein moieties among metabolites. The presence of large conserved pools can be viewed as global biophysical barriers protecting cellular systems from stresses, maintaining coordinated interconversions between key metabolites, and providing an additional mode of global metabolic regulation. The developed approaches thus provide novel and versatile tools for elucidating coupling relationships between metabolite concentrations with implications in biotechnological and medical applications.

Chromokinetics of metabolic pathways

Some methods to study and intuitively understand steady-state flows in complicated metabolic pathways are discussed. For this purpose, a suitable decomposition of complex metabolic schemes into smaller subsystems is used. These independent subsystems are then interpreted as basic colors of a chromatic coloring scheme. The mixture of these basic colors allows an intuitive picture of how a steady state in a metabolic pathway can be understood. Furthermore, actions of drugs can be more easily investigated on this basis. An anaerobic variant of pyruvate metabolism in rat liver mitochondria is presented as a simple example. This experiment allows measurement of the percentage that each basic color contributes to the steady states resulting from different experimental conditions. Possible implementations of existing algorithms and rational design of new drugs are discussed. A MATHEMATICA program, based on a new algorithm for finding all basic colors of stoichiometric networks, is included.

Conservation analysis in biochemical networks: computational issues for software writers

Large scale genomic studies are generating significant amounts of data on the structure of cellular networks. This is in contrast to kinetic data, which is frequently absent, unreliable or fragmentary. There is, therefore, a desire by many in the community to investigate the potential rewards of analyzing the more readily available topological data. This brief review is concerned with a particular property of biological networks, namely structural conservations (e.g. moiety conserved cycles). There has been much discussion in the literature on these cycles but a review on the computational issues related to conserved cycles has been missing. This review is concerned with the detection and characterization of conservation relations in arbitrary networks and related issues, which impinge on simulation simulation software writers. This review will not address flux balance constraints or small-world type analyses in any significant detail.

Systemic metabolic reactions are obtained by singular value decomposition of genome-scale stoichiometric matrices

Genome-scale metabolic networks can be reconstructed. The systemic biochemical properties of these networks can now be studied. Here, genome-scale reconstructed metabolic networks were analysed using singular value decomposition (SVD). All the individual biochemical conversions contained in a reconstructed metabolic network are described by a stoichiometric matrix (S). SVD of S led to the definition of the underlying modes that characterize the overall biochemical conversions that take place in a network and rank-ordered their importance. The modes were shown to correspond to systemic biochemical reactions and they could be used to identify the groups and clusters of individual biochemical reactions that drive them. Comparative analysis of the Escherichia coli, Haemophilus influenzae, and Helicobacter pylori genome-scale metabolic networks showed that the four dominant modes in all three networks correspond to: (1) the conversion of ATP to ADP, (2) redox metabolism of NADP, (3) proton-motive force, and (4) inorganic phosphate metabolism. The sets of individual metabolic reactions deriving these systemic conversions, however, differed among the three organisms. Thus, we can now define systemic metabolic reactions, or eigen-reactions, for the study of systems biology of metabolism and have a basis for comparing the overall properties of genome-specific metabolic networks.

The convex basis of the left null space of the stoichiometric matrix leads to the definition of metabolically meaningful pools

The stoichiometric matrix, S, represents a mapping of reaction rate vectors into a space of concentration time derivatives. The left null space of the stoichiometric matrix contains the dynamic invariants: a combination of concentration variables, referred to as metabolic pools, whose total concentration does not change over time. By analogy to the traditional reaction map formed by S, a compound map can be derived from -S(T). The analogy to flux analysis of the (right) null space of S enables us to classify the metabolic pools into three categories: Type A that contains chemical elements and their combinations in the form of certain moieties, Type B that contains such moieties in addition to cofactors carrying such moieties that are internal to the network, and Type C that contains only the cofactors. A convex formulation of the basis for the left null space allows us to directly classify the metabolic pools into these three categories. Type B metabolic pools include conservation pools that form conjugates of moiety-occupied and moiety-vacant concentration states of metabolites and cofactors. Type B metabolic pools thus describe the various states of moiety exchange between the primary substrates and the cofactors that capture properties like energy and redox potential. The convex basis gives clear insight into this exchange for glycolytic pathway in human red blood cell, including the identification of high and low energy pools that form conjugates. Examples suggest that pool maps may be more appropriate for signaling pathways than flux maps. The analysis of the left null space of the stoichiometric matrix allows us to define the achievable states of the cell and their physiological relevance.

The role of stoichiometric analysis in studies of metabolism: an example

Stoichiometric analysis uses matrix algebra to deduce the constraints implicit in metabolic networks. When applied to simple networks, it can often give the impression of being an unnecessarily complicated way of arriving at information that is obvious from inspection, for example, that the sum of the concentrations of the adenine nucleotides is constant. Applied to a more complicated example, that of glycolysis in Trypanosoma brucei, it yields information that is far from obvious and may have importance for developing therapeutic ways of eliminating this parasite. Even in simplified form, the network contains nine reactions or transport steps involving 11 metabolites. This immediately shows that there must be at least two stoichiometric constraints, and indeed two can be recognized by inspection: conservation of adenine nucleotides and conservation of the two forms of NAD. There is, however, a third, which involves eight different phosphorylated intermediates in non-obvious combinations and is very difficult to recognize by inspection. It is also difficult to recognize by inspection that no fourth stoichiometric constraint exists. Gaussian elimination provides a systematic way of analysing a network in such a way that all the stoichiometric relationships that it contains emerge automatically.

Invariant manifold methods for metabolic model reduction

After the decay of transients, the behavior of a set of differential equations modeling a chemical or biochemical system generally rests on a low-dimensional surface which is an invariant manifold of the flow. If an equation for such a manifold can be obtained, the model has effectively been reduced to a smaller system of differential equations. Using perturbation methods, we show that the distinction between rapidly decaying and long-lived (slow) modes has a rigorous basis. We show how equations for attracting invariant (slow) manifolds can be constructed by a geometric approach based on functional equations derived directly from the differential equations. We apply these methods to two simple metabolic models. (c) 2001 American Institute of Physics.

Synergism analysis of biochemical systems. II. Tensor formulation and treatment of stoichiometric constraints

The previous paper outlined a conceptual and mathematical framework for synergism analysis of kinetic models. Though the formalism presented there is adequate for studying simple models, the analysis of large-scale models benefits from the more effective formulation achieved in this work. The present formulation is based on simple tensor operations and takes advantage of the analogy between the formalisms for synergism and log-synergism analysis presented before. Well-known relationships of first-order sensitivity analysis and new relationships for (log-)synergism coefficients of various steady-state properties are cast in the new formal setting. The formalism is then extended to models that are subject to constraints between variables, fluxes and/or parameters. This treatment, which generalizes Reder's concept of link matrices, is applied to networks that include moiety conservation cycles [C. Reder, Metabolic control theory: a structural approach, J. Theor. Biol. 135 (1988) 175]. It is also used to take advantage of flux conservation at steady-state to simplify synergism analysis. Issues of numerical effectiveness are briefly discussed, and the theory illustrated with the study of synergistic behaviour in the metabolism of reactive oxygen species and of a scheme of dynamic channelling.

The modelling of metabolic systems. Structure, control and optimality

This article gives an overview of recent developments in the modelling of the structure, control and optimality of metabolic networks. In particular, methods of algebraically analysing the topology of such networks are presented. By these methods, conservation relations and elementary modes of functioning (biochemical routes) can be detected. The principles of metabolic control analysis are outlined. Various recent extensions of this theory are presented, such as an analysis in terms of time dependent variables and modular analysis. Evolutionary optimisation principles are applied to explain the catalytic efficiency of single enzymes as well as the structural design of metabolic pathways. Special results concern the optimal distribution of ATP consuming and ATP producing reactions in glycolysis.

The definitions of metabolic control analysis revisited

Various definitions of coefficients in metabolic control analysis are examined with respect to their theoretical consistency and practical applicability. We suggest agreement upon a definition for control coefficients which is clearly distinct from that for response coefficients, in such a way that the former describe inherent properties of the metabolic system while the latter refer to the influence of special parameters. Advantages and drawbacks of using normalized or non-normalized control coefficients are studied. It is shown that normalized control coefficients have the advantage of being invariant to a different rescaling of the particular fluxes. We demonstrate that some problems are easier to tackle if the consistency of time-independent control coefficients with their time-dependent counterparts is taken into account. It is shown that the matrix of flux control coefficients is an indempotent matrix. This allows an interpretation in terms of the transduction of the effect of parameter perturbations. Several aspects of the experimental measurement of control coefficients are discussed, with special reference to the different definitions.