Computer-aided synthesis of biochemical pathways

Towards scaling elementary flux mode computation

While elementary flux mode (EFM) analysis is now recognized as a cornerstone computational technique for cellular pathway analysis and engineering, EFM application to genome-scale models remains computationally prohibitive. This article provides a review of aspects of EFM computation that elucidates bottlenecks in scaling EFM computation. First, algorithms for computing EFMs are reviewed. Next, the impact of redundant constraints, sensitivity to constraint ordering and network compression are evaluated. Then, the advantages and limitations of recent parallelization and GPU-based efforts are highlighted. The article then reviews alternative pathway analysis approaches that aim to reduce the EFM solution space. Despite advances in EFM computation, our review concludes that continued scaling of EFM computation is necessary to apply EFM to genome-scale models. Further, our review concludes that pathway analysis methods that target specific pathway properties can provide powerful alternatives to EFM analysis.

gEFM: An Algorithm for Computing Elementary Flux Modes Using Graph Traversal

Computational methods to engineer cellular metabolism promise to play a critical role in producing pharmaceutical, repairing defective genes, destroying cancer cells, and generating biofuels. Elementary Flux Mode (EFM) analysis is one such powerful technique that has elucidated cell growth and regulation, predicted product yield, and analyzed network robustness. EFM analysis, however, is a computationally daunting task because it requires the enumeration of all independent and stoichiometrically balanced pathways within a cellular network. We present in this paper an EFM enumeration algorithm, termed graphical EFM or gEFM. The algorithm is based on graph traversal, an approach previously assumed unsuitable for enumerating EFMs. The approach is derived from a pathway synthesis method proposed by Mavrovouniotis et al. The algorithm is described and proved correct. We apply gEFM to several networks and report runtimes in comparison with other EFM computation tools. We show how gEFM benefits from network compression. Like other EFM computational techniques, gEFM is sensitive to constraint ordering; however, we are able to demonstrate that knowledge of the underlying network structure leads to better constraint ordering. gEFM is shown to be competitive with state-of-the-art EFM computational techniques for several networks, but less so for networks with a larger number of EFMs.

Monte-Carlo modeling of the central carbon metabolism of Lactococcus lactis: insights into metabolic regulation

Metabolic pathways are complex dynamic systems whose response to perturbations and environmental challenges are governed by multiple interdependencies between enzyme properties, reactions rates, and substrate levels. Understanding the dynamics arising from such a network can be greatly enhanced by the construction of a computational model that embodies the properties of the respective system. Such models aim to incorporate mechanistic details of cellular interactions to mimic the temporal behavior of the biochemical reaction system and usually require substantial knowledge of kinetic parameters to allow meaningful conclusions. Several approaches have been suggested to overcome the severe data requirements of kinetic modeling, including the use of approximative kinetics and Monte-Carlo sampling of reaction parameters. In this work, we employ a probabilistic approach to study the response of a complex metabolic system, the central metabolism of the lactic acid bacterium Lactococcus lactis, subject to perturbations and brief periods of starvation. Supplementing existing methodologies, we show that it is possible to acquire a detailed understanding of the control properties of a corresponding metabolic pathway model that is directly based on experimental observations. In particular, we delineate the role of enzymatic regulation to maintain metabolic stability and metabolic recovery after periods of starvation. It is shown that the feedforward activation of the pyruvate kinase by fructose-1,6-bisphosphate qualitatively alters the bifurcation structure of the corresponding pathway model, indicating a crucial role of enzymatic regulation to prevent metabolic collapse for low external concentrations of glucose. We argue that similar probabilistic methodologies will help our understanding of dynamic properties of small-, medium- and large-scale metabolic networks models.

An algorithm for rapid computational construction of metabolic networks: a cholesterol biosynthesis example

Alternative pathways of metabolic networks represent the escape routes that can reduce drug efficacy and can cause severe adverse effects. In this paper we introduce a mathematical algorithm and a coding system for rapid computational construction of metabolic networks. The initial data for the algorithm are the source substrate code and the enzyme/metabolite interaction tables. The major strength of the algorithm is the adaptive coding system of the enzyme-substrate interactions. A reverse application of the algorithm is also possible, when optimisation algorithm is used to compute the enzyme/metabolite rules from the reference network structure. The coding system is user-defined and must be adapted to the studied problem. The algorithm is most effective for computation of networks that consist of metabolites with similar molecular structures. The computation of the cholesterol biosynthesis metabolic network suggests that 89 intermediates can theoretically be formed between lanosterol and cholesterol, only 20 are presently considered as cholesterol intermediates. Alternative metabolites may represent links with other metabolic networks both as precursors and metabolites of cholesterol. A possible cholesterol-by-pass pathway to bile acids metabolism through cholestanol is suggested.

Discovering drug-drug interactions: a text-mining and reasoning approach based on properties of drug metabolism

MOTIVATION

Identifying drug-drug interactions (DDIs) is a critical process in drug administration and drug development. Clinical support tools often provide comprehensive lists of DDIs, but they usually lack the supporting scientific evidences and different tools can return inconsistent results. In this article, we propose a novel approach that integrates text mining and automated reasoning to derive DDIs. Through the extraction of various facts of drug metabolism, not only the DDIs that are explicitly mentioned in text can be extracted but also the potential interactions that can be inferred by reasoning.

RESULTS

Our approach was able to find several potential DDIs that are not present in DrugBank. We manually evaluated these interactions based on their supporting evidences, and our analysis revealed that 81.3% of these interactions are determined to be correct. This suggests that our approach can uncover potential DDIs with scientific evidences explaining the mechanism of the interactions.

OptForce: an optimization procedure for identifying all genetic manipulations leading to targeted overproductions

Computational procedures for predicting metabolic interventions leading to the overproduction of biochemicals in microbial strains are widely in use. However, these methods rely on surrogate biological objectives (e.g., maximize growth rate or minimize metabolic adjustments) and do not make use of flux measurements often available for the wild-type strain. In this work, we introduce the OptForce procedure that identifies all possible engineering interventions by classifying reactions in the metabolic model depending upon whether their flux values must increase, decrease or become equal to zero to meet a pre-specified overproduction target. We hierarchically apply this classification rule for pairs, triples, quadruples, etc. of reactions. This leads to the identification of a sufficient and non-redundant set of fluxes that must change (i.e., MUST set) to meet a pre-specified overproduction target. Starting with this set we subsequently extract a minimal set of fluxes that must actively be forced through genetic manipulations (i.e., FORCE set) to ensure that all fluxes in the network are consistent with the overproduction objective. We demonstrate our OptForce framework for succinate production in Escherichia coli using the most recent in silico E. coli model, iAF1260. The method not only recapitulates existing engineering strategies but also reveals non-intuitive ones that boost succinate production by performing coordinated changes on pathways distant from the last steps of succinate synthesis.

Over the past few years, there has been an unprecedented increase in the use of microorganisms for the production of biofuels, industrial chemicals and pharmaceutical precursors. In this regard, biotechnologists are confronted with the challenge to efficiently convert biomass and other renewable resources into useful biochemicals. With the advent of organism-specific mathematical models of metabolism, scientists have used computations to identify genetic modifications that maximize the yield of a desired product. In this paper, we introduce OptForce, an algorithm that identifies all possible metabolic interventions that lead to the overproduction of a biochemical of interest. Unlike existing techniques, OptForce does not rely on the maximization of a fitness function to predict metabolic fluxes. Instead, OptForce contrasts the metabolic flux patterns observed in an initial strain and a strain overproducing the chemical at the target yield. The essence of this procedure is the identification of all coordinated reaction modifications that force the network towards the overproduction target. We used OptForce to predict metabolic interventions for succinate overproduction in Escherichia coli. The results described in this paper not only uncover existing strain designs for succinate production but also elucidate new ones that can be experimentally explored.

Computing the shortest elementary flux modes in genome-scale metabolic networks

MOTIVATION

Elementary flux modes (EFMs) represent a key concept to analyze metabolic networks from a pathway-oriented perspective. In spite of considerable work in this field, the computation of the full set of elementary flux modes in large-scale metabolic networks still constitutes a challenging issue due to its underlying combinatorial complexity.

RESULTS

In this article, we illustrate that the full set of EFMs can be enumerated in increasing order of number of reactions via integer linear programming. In this light, we present a novel procedure to efficiently determine the K-shortest EFMs in large-scale metabolic networks. Our method was applied to find the K-shortest EFMs that produce lysine in the genome-scale metabolic networks of Escherichia coli and Corynebacterium glutamicum. A detailed analysis of the biological significance of the K-shortest EFMs was conducted, finding that glucose catabolism, ammonium assimilation, lysine anabolism and cofactor balancing were correctly predicted. The work presented here represents an important step forward in the analysis and computation of EFMs for large-scale metabolic networks, where traditional methods fail for networks of even moderate size.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Elementary mode analysis: a useful metabolic pathway analysis tool for characterizing cellular metabolism

Elementary mode analysis is a useful metabolic pathway analysis tool to identify the structure of a metabolic network that links the cellular phenotype to the corresponding genotype. The analysis can decompose the intricate metabolic network comprised of highly interconnected reactions into uniquely organized pathways. These pathways consisting of a minimal set of enzymes that can support steady state operation of cellular metabolism represent independent cellular physiological states. Such pathway definition provides a rigorous basis to systematically characterize cellular phenotypes, metabolic network regulation, robustness, and fragility that facilitate understanding of cell physiology and implementation of metabolic engineering strategies. This mini-review aims to overview the development and application of elementary mode analysis as a metabolic pathway analysis tool in studying cell physiology and as a basis of metabolic engineering.

Algebraic method for the analysis of signaling crosstalk

A unified mathematical description that expresses the characteristics of whole systems is necessary for an understanding of signal transduction cascades. In this study we explore an algebraic method, named extreme signaling flow, enhanced from the concept of extreme pathway, to analyze signal transduction systems. This method enables us to represent the long-term potentiation (LTP) and the long-term depression (LTD) of hippocampal neuronal plasticity in an integrated simulation model. The model is validated by comparing the results of redundancy, reaction participation, and in silico knockout analysis with biological knowledge available from the literature. The following properties are assumed in these computational analyses: (1) LTP is fault-tolerant under network modification, (2) protein kinase C and MAPK have numerous routes to LTP induction, (3) calcium-calmodulin kinase II has a few routes to LTP induction, and (4) calcineurin has many routes to LTD induction. These results demonstrate that our approach produces an integrated framework for analyzing properties of large-scale systems with complicated signal transduction.

Use of pathway analysis and genome context methods for functional genomics of Mycoplasma pneumoniae nucleotide metabolism

Elementary modes analysis allows one to reveal whether a set of known enzymes is sufficient to sustain functionality of the cell. Moreover, it is helpful in detecting missing reactions and predicting which enzymes could fill these gaps. Here, we perform a comprehensive elementary modes analysis and a genomic context analysis of Mycoplasma pneumoniae nucleotide metabolism, and search for new enzyme activities. The purine and pyrimidine networks are reconstructed by assembling enzymes annotated in the genome or found experimentally. We show that these reaction sets are sufficient for enabling synthesis of DNA and RNA in M. pneumoniae. Special focus is on the key modes for growth. Moreover, we make an educated guess on the nutritional requirements of this micro-organism. For the case that M. pneumoniae does not require adenine as a substrate, we suggest adenylosuccinate synthetase (EC 6.3.4.4), adenylosuccinate lyase (EC 4.3.2.2) and GMP reductase (EC 1.7.1.7) to be operative. GMP reductase activity is putatively assigned to the NRDI_MYCPN gene on the basis of the genomic context analysis. For the pyrimidine network, we suggest CTP synthase (EC 6.3.4.2) to be active. Further experiments on the nutritional requirements are needed to make a decision. Pyrimidine metabolism appears to be more appropriate as a drug target than purine metabolism since it shows lower plasticity.

Metabolic flux estimation--a self-adaptive evolutionary algorithm with singular value decomposition

Metabolic flux analysis is important for metabolic system regulation and intracellular pathway identification. A popular approach for intracellular flux estimation involves using 13C tracer experiments to label states that can be measured by nuclear magnetic resonance spectrometry or gas chromatography mass spectrometry. However, the bilinear balance equations derived from 13C tracer experiments and the noisy measurements require a nonlinear optimization approach to obtain the optimal solution. In this paper, the flux quantification problem is formulated as an error-minimization problem with equality and inequality constraints through the 13C balance and stoichiometric equations. The stoichiometric constraints are transformed to a null space by singular value decomposition. Self-adaptive evolutionary algorithms are then introduced for flux quantification. The performance of the evolutionary algorithm is compared with ordinary least squares estimation by the simulation of the central pentose phosphate pathway. The proposed algorithm is also applied to the central metabolism of Corynebacterium glutamicum under lysine-producing conditions. A comparison between the results from the proposed algorithm and data from the literature is given. The complexity of a metabolic system with bidirectional reactions is also investigated by analyzing the fluctuations in the flux estimates when available measurements are varied.

Understanding the roadmap of metabolism by pathway analysis

The theoretical investigation of the structure of metabolic systems has recently attracted increasing interest. In this chapter, the basic concepts of metabolic pathway analysis are described and various applications are outlined. In particular, the concepts of nullspace and elementary flux modes are explained. The presentation is illustrated by a simple example from tyrosine metabolism and a system describing lysine production in Corynebacterium glutamicum. The latter system gives rise to 37 elementary modes, 36 of which produce lysine with different molar yields. The examples illustrate that metabolic pathway analysis is a useful tool for better understanding the complex architecture of intracellular metabolism, for determining the pathways on which the molar conversion yield of a substrate-product pair under study is maximal, and for assigning functions to orphan genes (functional genomics). Moreover, problems emerging in the modeling of large networks are discussed. An outlook on current trends in the field concludes the chapter.

Recovering metabolic pathways via optimization

A metabolic pathway is a coherent set of enzyme catalysed biochemical reactions by which a living organism transforms an initial (source) compound into a final (target) compound. Some of the different metabolic pathways adopted within organisms have been experimentally determined. In this paper, we show that a number of experimentally determined metabolic pathways can be recovered by a mathematical optimization model.

Analyzing Biochemical Pathways Using Neural Networks and Genetic Algorithms

The analysis of biochemical pathways has recently gained much interest as these are the processes that keep us alive. A deeper understanding of biochemical reactions must analyze them at atomic resolution. In order to achieve that we have developed a reaction database with the information on the well known Biochemical Pathways wall chart. Based on that, 3D models of the substrates and intermediates of biochemical reactions can be built. It is shown how this information can be used for searching for inhibitors of enzyme catalyzed reactions by superimposition of 3D structures with a genetic algorithm. Physicochemical properties of the bonds directly involved in the reaction event allow a classification of these enzyme catalyzed reactions by self-organizing neural networks. This classification is compared with the enzyme code (EC) classification.

Constrained optimization of L-lysine production based on metabolic flux using a mathematical programming method

Constrained optimization for microbial fermentation was studied. For optimization, we used not the maximum principle but a nonlinear programming method because of the need to consider many metabolic reactions. In the case of L-lysine fermentation, the optimization problem in L-lysine production was formulated as a nonlinear programming problem. In general, the state equations based on material balances are represented as differential equations, but such equations which are dependent on time can not be applied to a nonlinear programming problem. Therefore, the state equations were made discrete in a time base, and a new single vector which is not dependent on time was substituted. From these formulae, the objective function and the constraints using nonlinear programming problem were defined as the amount of L-lysine produced, and as a metabolic reaction model and empirical equations, respectively. Computer program was developed to solve this constrained nonlinear programming problem. The applied algorithm of the computer programming was a sequential quadratic programming method (SQP method). When the constrained nonlinear programming problem is solved using the SQP method, the maximum amount of L-lysine produced and the optimal feeding rate of L-threonine could be calculated. From the calculated results, it was clear that introduction of the equality and inequality constraints was easy. L-Lysine at a concentration up to 75.3 g/l could be produced when the fermentation was carried out under optimal conditions.

Inferring meaningful pathways in weighted metabolic networks

An approach is presented for computing meaningful pathways in the network of small molecule metabolism comprising the chemical reactions characterized in all organisms. The metabolic network is described as a weighted graph in which all the compounds are included, but each compound is assigned a weight equal to the number of reactions in which it participates. Path finding is performed in this graph by searching for one or more paths with lowest weight. Performance is evaluated systematically by computing paths between the first and last reactions in annotated metabolic pathways, and comparing the intermediate reactions in the computed pathways to those in the annotated ones. For the sake of comparison, paths are computed also in the un-weighted raw (all compounds and reactions) and filtered (highly connected pool metabolites removed) metabolic graphs, respectively. The correspondence between the computed and annotated pathways is very poor (<30%) in the raw graph; increasing to approximately 65% in the filtered graph; reaching approximately 85% in the weighted graph. Considering the best-matching path among the five lightest paths increases the correspondence to 92%, on average. We then show that the average distance between pairs of metabolites is significantly larger in the weighted graph than in the raw unfiltered graph, suggesting that the small-world properties previously reported for metabolic networks probably result from irrelevant shortcuts through pool metabolites. In addition, we provide evidence that the length of the shortest path in the weighted graph represents a valid measure of the "metabolic distance" between enzymes. We suggest that the success of our simplistic approach is rooted in the high degree of specificity of the reactions in metabolic pathways, presumably reflecting thermodynamic constraints operating in these pathways. We expect our approach to find useful applications in inferring metabolic pathways in newly sequenced genomes.

Complementary identification of multiple flux distributions and multiple metabolic pathways

Cell robustness and complexity have been recognized as unique features of biological systems. Such robustness and complexity of metabolic-reaction systems can be explored by discovering, or identifying, the multiple flux distributions (MFD) and redundant pathways that lead to a given external state; however, this is exceedingly cumbersome to accomplish. It is, therefore, highly desirable to establish an effective computational method for their identification, which, in turn, gives rise to a novel insight into the cellular function. An effective approach is proposed for complementarily identifying MFD in metabolic flux analysis and multiple metabolic pathways (MMP) in structural pathway analysis. This approach judiciously integrates flux balance analysis (FBA) based on linear programming and the graph-theoretic method for determining reaction pathways. A single metabolic pathway, with the concomitant flux distribution and the overall reaction manifesting itself as the desired phenotype under some environmental conditions, is determined by FBA from the initial candidate sequence of metabolic reactions. Subsequently, the graph-theoretic method recovers all feasible MMP and the corresponding MFD. The approach's efficacy is demonstrated by applying it to the in silico Escherichia coli model under various culture conditions. The resultant MMP and MFD attaining a unique external state reveal the surprising adaptability and robustness of the intricate cellular network as a key to cell survival against environmental or genetic changes. These results indicate that the proposed approach would be useful in facilitating drug discovery.

OptStrain: a computational framework for redesign of microbial production systems

This paper introduces the hierarchical computational framework OptStrain aimed at guiding pathway modifications, through reaction additions and deletions, of microbial networks for the overproduction of targeted compounds. These compounds may range from electrons or hydrogen in biofuel cell and environmental applications to complex drug precursor molecules. A comprehensive database of biotransformations, referred to as the Universal database (with >5700 reactions), is compiled and regularly updated by downloading and curating reactions from multiple biopathway database sources. Combinatorial optimization is then used to elucidate the set(s) of non-native functionalities, extracted from this Universal database, to add to the examined production host for enabling the desired product formation. Subsequently, competing functionalities that divert flux away from the targeted product are identified and removed to ensure higher product yields coupled with growth. This work represents an advancement over earlier efforts by establishing an integrated computational framework capable of constructing stoichiometrically balanced pathways, imposing maximum product yield requirements, pinpointing the optimal substrate(s), and evaluating different microbial hosts. The range and utility of OptStrain are demonstrated by addressing two very different product molecules. The hydrogen case study pinpoints reaction elimination strategies for improving hydrogen yields using two different substrates for three separate production hosts. In contrast, the vanillin study primarily showcases which non-native pathways need to be added into Escherichia coli. In summary, OptStrain provides a useful tool to aid microbial strain design and, more importantly, it establishes an integrated framework to accommodate future modeling developments.

Netscan: a procedure for generating reaction networks by size

In this paper, we describe an algorithm which can be used to generate the collection of networks, in order of increasing size, that are compatible with a list of chemical reactions and that satisfy a constraint. Our algorithm has been encoded and the software, called Netscan, can be freely downloaded from ftp://ftp.stat.ubc.ca/pub/riffraff/Netscanfiles, along with a manual, for general scientific use. Our algorithm may require pre-processing to ensure that the quantities it acts on are physically relevant and because it outputs sets of reactions, which we call canonical networks, that must be elaborated into full networks.

Thermodynamic constraints for biochemical networks

The constraint-based approach to analysis of biochemical systems has emerged as a useful tool for rational metabolic engineering. Flux balance analysis (FBA) is based on the constraint of mass conservation; energy balance analysis (EBA) is based on non-equilibrium thermodynamics. The power of these approaches lies in the fact that the constraints are based on physical laws, and do not make use of unknown parameters. Here, we show that the network structure (i.e. the stoichiometric matrix) alone provides a system of constraints on the fluxes in a biochemical network which are feasible according to both mass balance and the laws of thermodynamics. A realistic example shows that these constraints can be sufficient for deriving unambiguous, biologically meaningful results. The thermodynamic constraints are obtained by comparing of the sign pattern of the flux vector to the sign patterns of the cycles of the internal cycle space via connection between stoichiometric network theory (SNT) and the mathematical theory of oriented matroids.

In silico atomic tracing by substrate-product relationships in Escherichia coli intermediary metabolism

We present a software system that computationally reproduces biochemical radioisotope-tracer experiments. It consists of three main components: A mapping database of substrate-product atomic correspondents derived from known reaction formulas, a tracing engine that can compute all pathways between two given compounds by using the mapping database, and a graphical user interface. As the system can facilitate the display of all possible pathways between any two compounds and the tracing of every single carbon, nitrogen, or sulfur atom in the metabolism, it complements and bridges other metabolic databases and simulations on fixed models.

Reconstructing metabolic flux vectors from extreme pathways: defining the alpha-spectrum

The move towards genome-scale analysis of cellular functions has necessitated the development of analytical (in silico) methods to understand such large and complex biochemical reaction networks. One such method is extreme pathway analysis that uses stoichiometry and thermodynamic irreversibly to define mathematically unique, systemic metabolic pathways. These extreme pathways form the edges of a high-dimensional convex cone in the flux space that contains all the attainable steady state solutions, or flux distributions, for the metabolic network. By definition, any steady state flux distribution can be described as a nonnegative linear combination of the extreme pathways. To date, much effort has been focused on calculating, defining, and understanding these extreme pathways. However, little work has been performed to determine how these extreme pathways contribute to a given steady state flux distribution. This study represents an initial effort aimed at defining how physiological steady state solutions can be reconstructed from a network's extreme pathways. In general, there is not a unique set of nonnegative weightings on the extreme pathways that produce a given steady state flux distribution but rather a range of possible values. This range can be determined using linear optimization to maximize and minimize the weightings of a particular extreme pathway in the reconstruction, resulting in what we have termed the alpha-spectrum. The alpha-spectrum defines which extreme pathways can and cannot be included in the reconstruction of a given steady state flux distribution and to what extent they individually contribute to the reconstruction. It is shown that accounting for transcriptional regulatory constraints can considerably shrink the alpha-spectrum. The alpha-spectrum is computed and interpreted for two cases; first, optimal states of a skeleton representation of core metabolism that include transcriptional regulation, and second for human red blood cell metabolism under various physiological, non-optimal conditions.

A computational model for the identification of biochemical pathways in the krebs cycle

We have applied an algorithmic methodology which provably decomposes any complex network into a complete family of principal subcircuits to study the minimal circuits that describe the Krebs cycle. Every operational behavior that the network is capable of exhibiting can be represented by some combination of these principal subcircuits and this computational decomposition is linearly efficient. We have developed a computational model that can be applied to biochemical reaction systems which accurately renders pathways of such reactions via directed hypergraphs (Petri nets). We have applied the model to the citric acid cycle (Krebs cycle). The Krebs cycle, which oxidizes the acetyl group of acetyl CoA to CO(2) and reduces NAD and FAD to NADH and FADH(2), is a complex interacting set of nine subreaction networks. The Krebs cycle was selected because of its familiarity to the biological community and because it exhibits enough complexity to be interesting in order to introduce this novel analytic approach. This study validates the algorithmic methodology for the identification of significant biochemical signaling subcircuits, based solely upon the mathematical model and not upon prior biological knowledge. The utility of the algebraic-combinatorial model for identifying the complete set of biochemical subcircuits as a data set is demonstrated for this important metabolic process.

A method for classifying metabolites in topological pathway analyses based on minimization of pathway number

Metabolic pathway analysis based on the concept of elementary flux mode is a valuable tool for reconstruction of bacterial metabolisms and in predicting optimal conversion yields in biotechnology. However, pathway analysis of large and highly entangled metabolic networks meets the problem of combinatorial explosion of possible routes across the networks. Here we propose a method for coping with this problem by suitably classifying metabolites as external or internal. External metabolites are considered to have buffered concentrations while internal metabolites have to fulfil a balance condition at steady state. For many substances such as nutrients and excreted products, there are biochemical reasons to classify them as external. In addition, other substances (especially at central branching points) can operationally be considered external in order to avoid combinatorial explosion. We suggest to find such a classification of metabolites that minimizes the number of elementary flux modes (pathways). This is motivated by the objectives of finding such a description of the system that reduces as much as possible the amount of necessary data and of removing the ambiguity and arbitrariness in the classification of metabolites in an automated, systematic way. For networks of moderate size, the solution to this combinatorial minimization problem can be found by exhaustive search. To tackle also larger systems, a stochastic optimization program based on the Metropolis algorithm was developed. Both methods are applied, for illustration, to several reaction schemes including a larger network representing glutathione metabolism.

Metabolic pathways in the post-genome era

Metabolic pathways are a central paradigm in biology. Historically, they have been defined on the basis of their step-by-step discovery. However, the genome-scale metabolic networks now being reconstructed from annotation of genome sequences demand new network-based definitions of pathways to facilitate analysis of their capabilities and functions, such as metabolic versatility and robustness, and optimal growth rates. This demand has led to the development of a new mathematically based analysis of complex, metabolic networks that enumerates all their unique pathways that take into account all requirements for cofactors and byproducts. Applications include the design of engineered biological systems, the generation of testable hypotheses regarding network structure and function, and the elucidation of properties that can not be described by simple descriptions of individual components (such as product yield, network robustness, correlated reactions and predictions of minimal media). Recently, these properties have also been studied in genome-scale networks. Thus, network-based pathways are emerging as an important paradigm for analysis of biological systems.

Extreme pathway lengths and reaction participation in genome-scale metabolic networks

Extreme pathways are a unique and minimal set of vectors that completely characterize the steady-state capabilities of genome-scale metabolic networks. A framework is provided to mathematically characterize extreme pathway length and to study how individual reactions participate in the extreme pathway structure of a network. The length of an extreme pathway is the number of reactions that comprise it. Reaction participation is the percentage of extreme pathways that utilize a given reaction. These properties were computed for the production of individual amino acids and protein production in Helicobacter pylori and individual amino acid production in Haemophilus influenzae. Reaction participation classifies the reactions into groups that are always, sometimes, or never utilized for the production of a target product. The utilized reactions can be further grouped into correlated subsets of reactions, some of which are non-obvious, and which may, in turn, suggest regulatory structure. The length of the extreme pathways did not correlate with product yield or chemical complexity. The distributions of extreme pathway lengths in H. pylori were also very different from those in H. influenzae, showing a distinct systemic difference between the two organisms, despite overall similar metabolic networks. Reaction participation and extreme pathway lengths thus serve to elucidate systemic biological features.

A functional genomics approach using metabolomics and in silico pathway analysis

In the field of functional genomics increasing effort is being undertaken to analyze the function of orphan genes using metabolome data. Improved analytical equipment allows screening simultaneously for a high number of metabolites. Such metabolite profiles are analyzed using multivariate data analysis techniques and changes in the genotype will in many cases lead to different metabolite profiles. Here, a theoretical framework that may be applied to identify the function of orphan genes is presented. The approach is based on a combination of metabolome analysis combined with in silico pathway analysis. Pathway analysis may be carried out using convex analysis and a change in the active pathway structure of deletion mutants expressed in a different metabolite profile may disclose the function or the functional class of an orphan gene. The concept is illustrated using a simplified model for growth of Saccharomyces cerevisiae.

Network genomics--a novel approach for the analysis of biological systems in the post-genomic era

Network Genomics studies genomics and proteomics foundations of cellular networks in biological systems. It complements systems biology in providing information on elements, their interaction and their functional interplay in cellular networks. The relationship between genomic and proteomic high-throughput technologies and computational methods are described, as well as several examples of specific network genomic application are presented.

Metabolic pathway analysis of a recombinant yeast for rational strain development

Elementary mode analysis has been used to study a metabolic pathway model of a recombinant Saccharomyces cerevisiae system that was genetically engineered to produce the bacterial storage compound poly-beta-hydroxybutyrate (PHB). The model includes biochemical reactions from the intermediary metabolism and takes into account cellular compartmentalization as well as the reversibility/irreversibility of the reactions. The reaction network connects the production and/or consumption of eight external metabolites including glucose, acetate, glycerol, ethanol, PHB, CO(2), succinate, and adenosine triphosphate (ATP). Elementary mode analysis of the wild-type S. cerevisiae system reveals 241 unique reaction combinations that balance the eight external metabolites. When the recombinant PHB pathway is included, and when the reaction model is altered to simulate the experimental conditions when PHB accumulates, the analysis reveals 20 unique elementary modes. Of these 20 modes, 7 produce PHB with the optimal mode having a theoretical PHB carbon yield of 0.67. Elementary mode analysis was also used to analyze the possible effects of biochemical network modifications and altered culturing conditions. When the natively absent ATP citrate-lyase activity is added to the recombinant reaction network, the number of unique modes increases from 20 to 496, with 314 of these modes producing PHB. With this topological modification, the maximum theoretical PHB carbon yield increases from 0.67 to 0.83. Adding a transhydrogenase reaction to the model also improves the theoretical conversion of substrate into PHB. The recombinant system with the transhydrogenase reaction but without the ATP citrate-lyase reaction has an increase in PHB carbon yield from 0.67 to 0.71. When the model includes both the ATP citrate-lyase reaction and the transhydrogenase reaction, the maximum theoretical carbon yield increases to 0.84. The reaction model was also used to explore the possibility of producing PHB under anaerobic conditions. In the absence of oxygen, the recombinant reaction network possesses two elementary modes capable of producing PHB. Interestingly, both modes also produce ethanol. Elementary mode analysis provides a means of deconstructing complex metabolic networks into their basic functional units. This information can be used for analyzing existing pathways and for the rational design of further modifications that could improve the system's conversion of substrate into product.

Determination of redundancy and systems properties of the metabolic network of Helicobacter pylori using genome-scale extreme pathway analysis

The capabilities of genome-scale metabolic networks can be described through the determination of a set of systemically independent and unique flux maps called extreme pathways. The first study of genome-scale extreme pathways for the simultaneous formation of all nonessential amino acids or ribonucleotides in Helicobacter pylori is presented. Three key results were obtained. First, the extreme pathways for the production of individual amino acids in H. pylori showed far fewer internal states per external state than previously found in Haemophilus influenzae, indicating a more rigid metabolic network. Second, the degree of pathway redundancy in H. pylori was essentially the same for the production of individual amino acids and linked amino acid sets, but was approximately twice that of the production of the ribonucleotides. Third, the metabolic network of H. pylori was unable to achieve extensive conversion of amino acids consumed to the set of either nonessential amino acids or ribonucleotides and thus diverted a large portion of its nitrogen to ammonia production, a potentially important result for pH regulation in its acidic habitat. Genome-scale extreme pathways elucidate emergent system-wide properties. Extreme pathway analysis is emerging as a potentially important method to analyze the link between the metabolic genotype and its phenotypes.

Calculability analysis in underdetermined metabolic networks illustrated by a model of the central metabolism in purple nonsulfur bacteria

Metabolite balancing has turned out to be a powerful computational tool in metabolic engineering. However, the linear equation systems occurring in this analysis are often underdetermined. If it is difficult or impossible to find the missing constraints, it is nevertheless feasible in some cases to determine the values of a subset of the unknown rates. Here, a procedure for finding out which reaction rates can be uniquely calculated in underdetermined metabolic networks and computing these rates is given. The method is based on the null space to the stoichiometry matrix corresponding to the reactions with unknown rates. It is shown that this method is considerably easier to handle than an algorithm given previously (Van der Heijden et al., 1994a). Furthermore, a useful elementary representation of the null space is presented which is closely related with the elementary flux modes. This unique representation is central to a more general approach to observability/calculability analysis. In particular, it allows one to find, in an easy way, those sets of measurable rates that enable a calculation of a certain unknown rate. Besides, rates which are never calculable by metabolite balancing may be easily detected by this method. The applicability of these methods is illustrated by a model of the central metabolism in purple nonsulfur bacteria. The photoheterotrophic growth of these representatives of anoxygenic photosynthetic bacteria is stoichiometrically analyzed. Interesting metabolic constraints caused by the necessary balancing of NADPH can be detected in a highly underdetermined system. This is, to our knowledge, the first application of stoichiometric analysis to the metabolic network in this bacteria group using metabolite balancing techniques. A new software tool, the FluxAnalyzer, is introduced. It allows quantitative and structural analysis of metabolic networks in a graphical user interface.

Modeling and simulation: tools for metabolic engineering

Mathematical modeling is one of the key methodologies of metabolic engineering. Based on a given metabolic model different computational tools for the simulation, data evaluation, systems analysis, prediction, design and optimization of metabolic systems have been developed. The currently used metabolic modeling approaches can be subdivided into structural models, stoichiometric models, carbon flux models, stationary and nonstationary mechanistic models and models with gene regulation. However, the power of a model strongly depends on its basic modeling assumptions, the simplifications made and the data sources used. Model validation turns out to be particularly difficult for metabolic systems. The different modeling approaches are critically reviewed with respect to their potential and benefits for the metabolic engineering cycle. Several tools that have emerged from the different modeling approaches including structural pathway synthesis, stoichiometric pathway analysis, metabolic flux analysis, metabolic control analysis, optimization of regulatory architectures and the evaluation of rapid sampling experiments are discussed.