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

Abstract

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 highdimensional 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.

Computational Methods to Interpret and Integrate Metabolomic Data

“…Using FBA, in silico studies on the systemic properties of the Haemophilus influenzae and E. coli (Edwards & Palsson, 2000) metabolic networks have been completed. Under various substrate conditions, Schilling et al (2000a) explored the metabolic capabilities and predicted functions of a sub-system of the E. coli using FBA.…”

In Silicoanalysis of perturbed steroidogenesis and gonad growth in fathead minnows (P. promelas) exposed to 17α-ethynylestradiol

“…Such transformation allowed the in silico model to be represented as a matrix-vector product (s Á ¼ 0) whose solution was the non-trivial set of coefficients for the 1 Â n flux vector '' [Schilling et al 2000]. The solution to such a system resided within the null space spanned by nonnegative basis vectors of the S-matrix which created a polyhedral 'cone' encompassing feasible solutions to the in silico steroid model [Schilling and Palsson 1998;Schilling et al 1999].…”

“…Bounds on reversible reactions (especially for cofactors) were set to À1,000 (for i ) and 1,000 fmol/mg/h (for i ). In order to maintain metabolite mass-balances, exchange fluxes for individual steroid metabolites participating in the biomass reaction were nulled so as to allow a single output for these metabolites from the model (i.e., to produce biomass) and avoid imposition of competing export demands [Schilling et al 2000].…”

Analysis and Engineering of Metabolic Pathway Fluxes in Corynebacterium glutamicum

“…The stoichiometric data and models available provide the basis for various in silico applications differing in the methodology and algorithms used. Major contributions for C. glutamicum have been obtained from flux balancing [66] and elementary (flux) mode analysis [67]. In short, flux balance analysis is a constraint based method to analyze stoichiometric networks.…”