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.

Using metabolic flux data to further constrain the metabolic solution space and predict internal flux patterns: the Escherichia coli spectrum.

Constraint-based metabolic modeling has been used to capture the genome-scale, systems properties of an organism's metabolism. The first generation of these models has been built on annotated gene sequence. To further this field, we now need to develop methods to incorporate additional "omic" data types including transcriptomics, metabolomics, and fluxomics to further facilitate the construction, validation, and predictive capabilities of these models. The work herein combines metabolic flux data with an in silico model of central metabolism of Escherichia coli for model centric integration of the flux data. The extreme pathways for this network, which define the allowable solution space for all possible flux distributions, are analyzed using the alpha-spectrum. The alpha-spectrum determines which extreme pathways can and cannot contribute to the metabolic flux distribution for a given condition and gives the allowable range of weightings on each extreme pathway that can contribute. Since many extreme pathways cannot be used under certain conditions, the result is a "condition-specific" solution space that is a subset of the original solution space. The alpha-spectrum results are used to create a "condition-specific" extreme pathway matrix that can be analyzed using singular value decomposition (SVD). The first mode of the SVD analysis characterizes the solution space for a given condition. We show that SVD analysis of the alpha-spectrum extreme pathway matrix that incorporates measured uptake and byproduct secretion rates, can predict internal flux trends for different experimental conditions. These predicted internal flux trends are, in general, consistent with the flux trends measured using experimental metabolic flux analysis techniques.

Network-based analysis of metabolic regulation in the human red blood cell.

Reconstruction of cell-scale metabolic networks is now possible. A description of allowable metabolic network functions can be obtained using extreme pathways, which are the convex basis vectors of the solution space containing all steady state flux distributions. However, only a portion of these allowable network functions are physiologically possible due to kinetic and regulatory constraints. Methods are now needed that enable us to take a defined metabolic network and deduce candidate regulatory structures that control the selection of these physiologically relevant states. One such approach is the singular value decomposition (SVD) of extreme pathway matrices (P), which allows for the characterization of steady state solution spaces. Eigenpathways, which are the left singular vectors from the SVD of P, can be described and categorized by their biochemical function. SVD of P for the human red blood cell showed that the first five eigenpathways, out of a total of 23, effectively characterize all the relevant physiological states of red blood cell metabolism calculated with a detailed kinetic model. Thus, with five degrees of freedom the magnitude and nature of the regulatory needs are defined. Additionally, the dominant features of these first five eigenpathways described key metabolic splits that are indeed regulated in the human red blood cell. The extreme pathway matrix is derived directly from network topology and only knowledge of Vmax values is needed to reach these conclusions. Thus, we have implemented a network-based analysis of regulation that complements the study of individual regulatory events. This topological approach may provide candidate regulatory structures for metabolic networks with known stoichiometry but poorly characterized regulation.

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.

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.

In silico model-driven assessment of the effects of single nucleotide polymorphisms (SNPs) on human red blood cell metabolism.

The completion of the human genome project and the construction of single nucleotide polymorphism (SNP) maps have lead to significant efforts to find SNPs that can be linked to pathophysiology. In silico models of complete biochemical reaction networks relate a cell's individual reactions to the function of the entire network. Sequence variations can in turn be related to kinetic properties of individual enzymes, thus allowing an in silico model-driven assessment of the effects of defined SNPs on overall cellular functions. This process is applied to defined SNPs in two key enzymes of human red blood cell metabolism: glucose-6-phosphate dehydrogenase and pyruvate kinase. The results demonstrate the utility of in silico models in providing insight into differences between red cell function in patients with chronic and nonchronic anemia. In silico models of complex cellular processes are thus likely to aid in defining and understanding key SNPs in human pathophysiology.

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.