Analysis of metabolic capabilities using singular value decomposition of extreme pathway matrices - PubMed (original) (raw)

Analysis of metabolic capabilities using singular value decomposition of extreme pathway matrices

Nathan D Price et al. Biophys J. 2003 Feb.

Abstract

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.

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Figures

FIGURE 1

FIGURE 1

Schematic of a biochemical reaction network and its convex, steady-state solution cone. Extreme pathway analysis generates a set of vectors that define a convex solution space. This solution space circumscribes all possible flux distributions, thus the phenotypes, of the biochemical reaction network.

FIGURE 2

FIGURE 2

The singular value decomposition of the extreme pathway matrix.

FIGURE 3

FIGURE 3

Conceptual framework for application of singular value decomposition to extreme pathway analysis. In panel A, a convex representation of the cone defined by the extreme pathways is illustrated. The first mode from SVD analysis represents the principal direction of the cone. In panel B, the effect of symmetry of the convex cone is demonstrated. The areas shown represent cross sections of the multidimensional cone. The extreme pathways are represented by gray circles on the edges of the cone. The first mode is represented by a black circle inside the space. With “soft edges” in a cone, the dominant mode is “pulled” toward this region of the space. In panel C, we can see how the second and third principal modes characterize the convex cone. In the plane perpendicular to the first mode, the second and third modes characterize the variance in mutually perpendicular directions. Movement along the second mode allows for the most control of the space characterized by the first mode. In panel D, the effect of “soft edges” on the directions of the second and third mode is demonstrated.

FIGURE 4

FIGURE 4

Singular value decomposition of sample reaction networks. The SVD of these three reaction networks demonstrates how the SVD characterizes general properties of the solution cone, defining the phenotypic possibilities of the metabolic networks. In panel A, we have the simplest case—the linear chain of reactions has simply one extreme pathway, and consequently the first mode characterizes 100% of the extreme pathway matrix. This cone can be visualized as a simple vector, the “narrowest” possible case. In panel B, the system has four extreme pathways. Its first mode characterizes 47% of the total solution space. In panel C, the system has six extreme pathways. The first mode characterizes only 39% of the system. The first modes represent valid biochemical pathways in the network. The black and gray arrows in subsequent modes represent increased and decreased flux levels, respectively. The widths of all of the arrows in the representations of the modes are proportional to the flux through the corresponding reaction.

FIGURE 5

FIGURE 5

Cumulative fractional contributions for the singular value decompositions of the extreme pathway matrices in H. influenzae and H. pylori. The cumulative fractional contribution is defined as the sum of the first n fractional singular values (reported as a percent). This value represents the contribution of the first n modes to the overall description of the system. The rank of the respective extreme pathway matrix is shown for nonessential amino acids. The _S_crit value is the number of singular values that account for ≥95% of the variance in the matrices. Entries with “- - -” correspond to essential amino acids.

FIGURE 6

FIGURE 6

Key branch points in H. influenzae metabolic network for the synthesis of alanine and histidine. In panel A, there are 18 extreme pathways represented, each with the maximum yield of alanine. A key branch point in mode 2 of the metabolic network for alanine synthesis is shown in panel B. In panel C, there are 64 extreme pathways represented, each with the maximum histidine yield. A key branch point in mode 11 of the metabolic network for histidine synthesis is shown in panel D. Black and gray fluxes change in opposite directions. Dashed lines correspond to reactions not shown. Indicated metabolites are the following: E4P, erythrose 4-phosphate; F6P, fructose 6-phosphate; FDP, fructose-1,6-diphosphate; FRU, fructose; OA, oxaloacetate; PRPP, phosphoribosyl pyrophosphate; PYR, pyruvate; R5P, ribose 5-phosphate; RL5P, D-ribulose 5-phosphate; S7P, sedo-heptulose 7-phosphate; T3P1, glyceraldehyde 3-phosphate; T3P2, dihydroxyacetone phosphate; X5P, D-xylulose-5-phosphate. Reactions are catalyzed by the following enzymes: fba, fructose-1,6-bisphosphatate aldolase; fbp, fructose-1,6-bisphosphatase; pfkA, phosphofructokinase; prsA, phosphoribosyl pyrophosphate synthase; rpiA, ribose-5-phosphate isomerase A; talB, transaldolase B; tktA, transketolase; tpi, triosphophate isomerase.

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