Combining pathway analysis with flux balance analysis for the comprehensive study of metabolic systems

The elucidation of organism-scale metabolic networks necessitates the development of integrative methods to analyze and interpret the systemic properties of cellular metabolism. A shift in emphasis from single metabolic reactions to systemically defined pathways is one consequence of such an integrative analysis of metabolic systems. The constraints of systemic stoichiometry, and limited thermodynamics have led to the definition of the flux space within the context of convex analysis. The flux space of the metabolic system, containing all allowable flux distributions, is constrained to a convex polyhedral cone in a high-dimensional space. From metabolic pathway analysis, the edges of the high-dimensional flux cone are vectors that correspond to systemically defined "extreme pathways" spanning the capabilities of the system. The addition of maximum flux capacities of individual metabolic reactions serves to further constrain the flux space and has led to the development of flux balance analysis using linear optimization to calculate optimal flux distributions. Here we provide the precise theoretical connections between pathway analysis and flux balance analysis allowing for their combined application to study integrated metabolic function. Shifts in metabolic behavior are calculated using linear optimization and are then interpreted using the extreme pathways to demonstrate the concept of pathway utilization. Changes to the reaction network, such as the removal of a reaction, can lead to the generation of suboptimal phenotypes that can be directly attributed to the loss of pathway function and capabilities. Optimal growth phenotypes are calculated as a function of environmental variables, such as the availability of substrate and oxygen, leading to the definition of phenotypic phase planes. It is illustrated how optimality properties of the computed flux distributions can be interpreted in terms of the extreme pathways. Together these developments are applied to an example network and to core metabolism of Escherichia coli demonstrating the connections between the extreme pathways, optimal flux distributions, and phenotypic phase planes. The consequences of changing environmental and internal conditions of the network are examined for growth on glucose and succinate in the face of a variety of gene deletions. The convergence of the calculation of optimal phenotypes through linear programming and the definition of extreme pathways establishes a different perspective for the understanding of how a defined metabolic network is best used under different environmental and internal conditions or, in other words, a pathway basis for the interpretation of the metabolic reaction norm.     

Candidate states of Helicobacter pylori's genome-scale metabolic network upon application of "loop law" thermodynamic constraints

Constraint-based modeling has proven to be a useful tool in the analysis of biochemical networks. To date, most studies in this field have focused on the use of linear constraints, resulting from mass balance and capacity constraints, which lead to the definition of convex solution spaces. One additional constraint arising out of thermodynamics is known as the "loop law" for reaction fluxes, which states that the net flux around a closed biochemical loop must be zero because no net thermodynamic driving force exists. The imposition of the loop-law can lead to nonconvex solution spaces making the analysis of the consequences of its imposition challenging. A four-step approach is developed here to apply the loop-law to study metabolic network properties: 1), determine linear equality constraints that are necessary (but not necessarily sufficient) for thermodynamic feasibility; 2), tighten V(max) and V(min) constraints to enclose the remaining nonconvex space; 3), uniformly sample the convex space that encloses the nonconvex space using standard Monte Carlo techniques; and 4), eliminate from the resulting set all solutions that violate the loop-law, leaving a subset of steady-state solutions. This subset of solutions represents a uniform random sample of the space that is defined by the additional imposition of the loop-law. This approach is used to evaluate the effect of imposing the loop-law on predicted candidate states of the genome-scale metabolic network of Helicobacter pylori.     

Analysis of Metabolic Capabilities Using Singular Value Decomposition of Extreme Pathway Matrices

It is now possible to construct genome-scale metabolic networks for particular microorganisms. Extreme pathway analysis is a useful method for analyzing the phenotypic capabilities of these networks. Many extreme pathways are needed to fully describe the functional capabilities of genome-scale metabolic networks, and therefore, a need exists to develop methods to study these large sets of extreme pathways. Singular value decomposition (SVD) of matrices of extreme pathways was used to develop a conceptual framework for the interpretation of large sets of extreme pathways and the steady-state flux solution space they define. The key results of this study were: 1), convex steady-state solution cones describing the potential functions of biochemical networks can be studied using the modes generated by SVD; 2), Helicobacter pylori has a more rigid metabolic network (i.e., a lower dimensional solution space and a more dominant first singular value) than Haemophilus influenzae for the production of amino acids; and 3), SVD allows for direct comparison of different solution cones resulting from the production of different amino acids. SVD was used to identify key network branch points that may identify key control points for regulation. Therefore, SVD of matrices of extreme pathways has proved to be a useful method for analyzing the steady-state solution space of genome-scale metabolic networks.     

Extreme pathway analysis of human red blood cell metabolism

The development of high-throughput technologies and the resulting large-scale data sets have necessitated a systems approach to the analysis of metabolic networks. One way to approach the issue of complex metabolic function is through the calculation and interpretation of extreme pathways. Extreme pathways are a mathematically defined set of generating vectors that describe the conical steady-state solution space for flux distributions through an entire metabolic network. Herein, the extreme pathways of the well-characterized human red blood cell metabolic network were calculated and interpreted in a biochemical and physiological context. These extreme pathways were divided into groups based on such criteria as their cofactor and by-product production, and carbon inputs including those that 1) convert glucose to pyruvate; 2) interchange pyruvate and lactate; 3) produce 2,3-diphosphoglycerate that binds to hemoglobin; 4) convert inosine to pyruvate; 5) induce a change in the total adenosine pool; and 6) dissipate ATP. Additionally, results from a full kinetic model of red blood cell metabolism were predicted based solely on an interpretation of the extreme pathway structure. The extreme pathways for the red blood cell thus give a concise representation of red blood cell metabolism and a way to interpret its metabolic physiology.     

Monte Carlo sampling can be used to determine the size and shape of the steady-state flux space

Constraint-based modeling results in a convex polytope that defines a solution space containing all possible steady-state flux distributions. The properties of this polytope have been studied extensively using linear programming to find the optimal flux distribution under various optimality conditions and convex analysis to define its extreme pathways (edges) and elementary modes. The work presented herein further studies the steady-state flux space by defining its hyper-volume. In low dimensions (i.e. for small sample networks), exact volume calculation algorithms were used. However, due to the #P-hard nature of the vertex enumeration and volume calculation problem in high dimensions, random Monte Carlo sampling was used to characterize the relative size of the solution space of the human red blood cell metabolic network. Distributions of the steady-state flux levels for each reaction in the metabolic network were generated to show the range of flux values for each reaction in the polytope. These results give insight into the shape of the high-dimensional solution space. The value of measuring uptake and secretion rates in shrinking the steady-state flux solution space is illustrated through singular value decomposition of the randomly sampled points. The V(max) of various reactions in the network are varied to determine the sensitivity of the solution space to the maximum capacity constraints. The methods developed in this study are suitable for testing the implication of additional constraints on a metabolic network system and can be used to explore the effects of single nucleotide polymorphisms (SNPs) on network capabilities.     

多约束代谢网络模型的研究进展

基于约束的基因组尺度代谢网络模型(genome-scale metabolic models,GEMs)分析已被广泛应用于代谢表型的预测.而实际细胞中代谢速率除计量学约束外,还受到酶资源可用性和反应热力学可行性等其他因素影响,在GEMs中整合酶资源约束或者热力学约束构建多约束代谢网络模型可以进一步缩小优化解空间,提升细...     

Integrating thermodynamic and enzymatic constraints into genome-scale metabolic models

Stoichiometric genome-scale metabolic network models (GEMs) have been widely used to predict metabolic phenotypes. In addition to stoichiometric ratios, other constraints such as enzyme availability and thermodynamic feasibility can also limit the phenotype solution space. Extended GEM models considering either enzymatic or thermodynamic constraints have been shown to improve prediction accuracy. In this paper, we propose a novel method that integrates both enzymatic and thermodynamic constraints in a single Pyomo modeling framework (ETGEMs). We applied this method to construct the EcoETM (E. coli metabolic model with enzymatic and thermodynamic constraints). Using this model, we calculated the optimal pathways for cellular growth and the production of 22 metabolites. When comparing the results with those of iML1515 and models with one of the two constraints, we observed that many thermodynamically unfavorable and/or high enzyme cost pathways were excluded from EcoETM. For example, the synthesis pathway of carbamoyl-phosphate (Cbp) from iML1515 is both thermodynamically unfavorable and enzymatically costly. After introducing the new constraints, the production pathways and yields of several Cbp-derived products (e.g. L-arginine, orotate) calculated using EcoETM were more realistic. The results of this study demonstrate the great application potential of metabolic models with multiple constraints for pathway analysis and phenotype prediction.     

Exploring the gap between dynamic and constraint-based models of metabolism

Systems biology provides new approaches for metabolic engineering through the development of models and methods for simulation and optimization of microbial metabolism. Here we explore the relationship between two modeling frameworks in common use namely, dynamic models with kinetic rate laws and constraint-based flux models. We compare and analyze dynamic and constraint-based formulations of the same model of the central carbon metabolism of Escherichia coli. Our results show that, if unconstrained, the space of steady states described by both formulations is the same. However, the imposition of parameter-range constraints can be mapped into kinetically feasible regions of the solution space for the dynamic formulation that is not readily transferable to the constraint-based formulation. Therefore, with partial kinetic parameter knowledge, dynamic models can be used to generate constraints that reduce the solution space below that identified by constraint-based models, eliminating infeasible solutions and increasing the accuracy of simulation and optimization methods.     

The genome-scale metabolic extreme pathway structure in Haemophilus influenzae shows significant network redundancy

Genome-scale metabolic networks can be characterized by a set of systemically independent and unique extreme pathways. These extreme pathways span a convex, high-dimensional space that circumscribes all potential steady-state flux distributions achievable by the defined metabolic network. Genome-scale extreme pathways associated with the production of non-essential amino acids in Haemophilus influenzae were computed. They offer valuable insight into the functioning of its metabolic network. Three key results were obtained. First, there were multiple internal flux maps corresponding to externally indistinguishable states. It was shown that there was an average of 37 internal states per unique exchange flux vector in H. influenzae when the network was used to produce a single amino acid while allowing carbon dioxide and acetate as carbon sinks. With the inclusion of succinate as an additional output, this ratio increased to 52, a 40% increase. Second, an analysis of the carbon fates illustrated that the extreme pathways were non-uniformly distributed across the carbon fate spectrum. In the detailed case study, 45% of the distinct carbon fate values associated with lysine production represented 85% of the extreme pathways. Third, this distribution fell between distinct systemic constraints. For lysine production, the carbon fate values that represented 85% of the pathways described above corresponded to only 2 distinct ratios of 1:1 and 4:1 between carbon dioxide and acetate. The present study analysed single outputs from one organism, and provides a start to genome-scale extreme pathways studies. These emergent system-level characterizations show the significance of metabolic extreme pathway analysis at the genome-scale.     

Theory for the systemic definition of metabolic pathways and their use in interpreting metabolic function from a pathway-oriented perspective

Cellular metabolism is most often described and interpreted in terms of the biochemical reactions that make up the metabolic network. Genomics is providing near complete information regarding the genes/gene products participating in cellular metabolism for a growing number of organisms. As the true functional units of metabolic systems are its pathways, the time has arrived to define metabolic pathways in the context of whole-cell metabolism for the analysis of the structural design and capabilities of the metabolic network. In this study, we present the theoretical foundations for the identification of the unique set of systemically independent biochemical pathways, termed extreme pathways, based on system stochiometry and limited thermodynamics. These pathways represent the edges of the steady-state flux cone derived from convex analysis, and they can be used to represent any flux distribution achievable by the metabolic network. An algorithm is presented to determine the set of extreme pathways for a system of any complexity and a classification scheme is introduced for the characterization of these pathways. The property of systemic independence is discussed along with its implications for issues related to metabolic regulation and the evolution of cellular metabolic networks. The underlying pathway structure that is determined from the set of extreme pathways now provides us with the ability to analyse, interpret, and perhaps predict metabolic function from a pathway-based perspective in addition to the traditional reaction-based perspective. The algorithm and classification scheme developed can be used to describe the pathway structure in annotated genomes to explore the capabilities of an organism.     

Metabolic capabilities of Escherichia coli: I. synthesis of biosynthetic precursors and cofactors

Metabolism of living cells converts substrates into metabolic energy, redox potential and metabolic end products that are essential to maintain cellular function. The flux distribution among the various biochemical pathways is determined by the kinetic properties of enzymes which are subject to strict regulatory control. Simulation of metabolic behavior therefore requires the complete knowledge of biochemical pathways, enzyme kinetics as well as their regulation. Unfortunately, complete kinetic and regulatory information is not available for microbial cells, thus preventing accurate dynamic simulation of their metabolic behavior. However, it is possible to define wider limits on metabolic behavior based solely on flux balances of biochemical pathways. We present here comprehensive information about the catabolic pathways of the bacterium Escherichia coli. Using this biochemical database, we formulate a stoichiometric model of the bacterial network of fueling reactions. After logical structural reduction, the network consists of 53 metabolic fluxes and 30 metabolites. The solution space of this under-determined system of equations presents the bounds of metabolic flux distribution that the bacterial cell can achieve. We use specific objective functions and linear optimization to investigate the capability of E. coli catabolism to maximally produce the 12 biosynthetic precursors and three key cofactors within this solution space. For the three cofactors, the maximum yields are calculated to be 18.67 ATP, 11.6 NADH and 11 NADPH per glucose molecule, respectively. The yields of NADH and NADPH are less than 12 owing to the energy costs of importing glucose. These constraints are made explicit by the interpretation of shadow prices. The optimal yields of the 12 biosynthetic precursors are computed. Four of the 12 precursors (3-phosphoglycerate, phosphoenolpyruvate, pyruvate and oxaloacetate) can be made by E. coli with complete carbon conversion. Conversely, none of the sugar monophosphates can be made with 100% carbon conversion and analysis of the shadow prices reveals that this conversion is constrained by the energy cost of importing glucose. Three of the 12 precursors (acetyl-coA, α-ketoglutarate, and succinyl-coA) cannot be made with full carbon conversion owing to stoichiometric constraints; there is no route to these compounds without carrying out a decarboxylation reaction. Metabolite flux balances and linear optimization have thus been used to determine the catabolic capabilities of E. coli .     

Network-level analysis of metabolic regulation in the human red blood cell using random sampling and singular value decomposition

Background  

Extreme pathways (ExPas) have been shown to be valuable for studying the functions and capabilities of metabolic networks through characterization of the null space of the stoichiometric matrix (S). Singular value decomposition (SVD) of the ExPa matrix P has previously been used to characterize the metabolic regulatory problem in the human red blood cell (hRBC) from a network perspective. The calculation of ExPas is NP-hard, and for genome-scale networks the computation of ExPas has proven to be infeasible. Therefore an alternative approach is needed to reveal regulatory properties of steady state solution spaces of genome-scale stoichiometric matrices.     

Assessment of the metabolic capabilities of Haemophilus influenzae Rd through a genome-scale pathway analysis

The annotated full DNA sequence is becoming available for a growing number of organisms. This information along with additional biochemical and strain-specific data can be used to define metabolic genotypes and reconstruct cellular metabolic networks. The first free-living organism for which the entire genomic sequence was established was Haemophilus influenzae. Its metabolic network is reconstructed herein and contains 461 reactions operating on 367 intracellular and 84 extracellular metabolites. With the metabolic reaction network established, it becomes necessary to determine its underlying pathway structure as defined by the set of extreme pathways. The H. influenzae metabolic network was subdivided into six subsystems and the extreme pathways determined for each subsystem based on stoichiometric, thermodynamic, and systems-specific constraints. Positive linear combinations of these pathways can be taken to determine the extreme pathways for the complete system. Since these pathways span the capabilities of the full system, they could be used to address a number of important physiological questions. First, they were used to reconcile and curate the sequence annotation by identifying reactions whose function was not supported in any of the extreme pathways. Second, they were used to predict gene products that should be co-regulated and perhaps co-expressed. Third, they were used to determine the composition of the minimal substrate requirements needed to support the production of 51 required metabolic products such as amino acids, nucleotides, phospholipids, etc. Fourth, sets of critical gene deletions from core metabolism were determined in the presence of the minimal substrate conditions and in more complete conditions reflecting the environmental niche of H. influenzae in the human host. In the former case, 11 genes were determined to be critical while six remained critical under the latter conditions. This study represents an important milestone in theoretical biology, namely the establishment of the first extreme pathway structure of a whole genome.     

Ab initio prediction of thermodynamically feasible reaction directions from biochemical network stoichiometry

Analysis of the stoichiometric structure of metabolic networks provides insights into the relationships between structure, function, and regulation of metabolic systems. Based on knowledge of only reaction stoichiometry, certain aspects of network functionality and robustness can be predicted. Current theories focus on breaking a metabolic network down into non-decomposable pathways able to operate in steady state. The physics underlying these theories is based on mass balance and the laws of thermodynamics. However, due to the inherent nonlinearity of the thermodynamic constraints on metabolic fluxes, computational analysis of large-scale biochemical systems can be expensive. In this study, it is shown how the feasible reaction directions may be determined by either computing the allowable ranges under the mass-balance and thermodynamic constraints or by analyzing the stoichiometric structure of the network. The computed reaction directions translate into a set of linear constraints necessary for thermodynamic feasibility. This set of necessary linear constraints is shown to be sufficient to guarantee feasibility in certain cases, thus translating the nonlinear thermodynamic constraints to linear. We show that for a reaction network of 44 internal reactions representing energy metabolism, the computed linear inequality constraints represent necessary and sufficient conditions for thermodynamic feasibility.