The effects of alternate optimal solutions in constraint-based genome-scale metabolic models

Abstract

Genome-scale constraint-based models of several organisms have now been constructed and are being used for model driven research. A key issue that may arise in the use of such models is the existence of alternate optimal solutions wherein the same maximal objective (e.g., growth rate) can be achieved through different flux distributions. Herein, we investigate the effects that alternate optimal solutions may have on the predicted range of flux values calculated using currently practiced linear (LP) and quadratic programming (QP) methods. An efficient LP-based strategy is described to calculate the range of flux variability that can be present in order to achieve optimal as well as suboptimal objective states. Sample results are provided for growth predictions of E. coli using glucose, acetate, and lactate as carbon substrates. These results demonstrate the extent of flux variability to be highly dependent on environmental conditions and network composition. In addition we examined the impact of alternate optima for growth under gene knockout conditions as calculated using QP-based methods. It was observed that calculations using QP-based methods can show significant variation in growth rate if the flux variability among alternate optima is high. The underlying biological significance and general source of such flux variability is further investigated through the identification of redundancies in the network (equivalent reaction sets) that lead to alternate solutions. Collectively, these results illustrate the variability inherent in metabolic flux distributions and the possible implications of this heterogeneity for constraint-based modeling approaches. These methods also provide an efficient and robust method to calculate the range of flux distributions that can be derived from quantitative fermentation data.

Introduction

Development of high throughput technologies for probing various biological processes in an organism at the gene, protein, and metabolite levels has resulted in the generation of large amounts of information. This influx of information has motivated the development of several quantitative approaches to analyze biological systems and interpret the large-scale data sets. The importance of such computational approaches for enhancing the understanding of biological systems and for the generation of experimentally verifiably hypotheses has been recognized (Endy and Brent, 2001; Kitano, 2002). One of the well studied areas is the analysis of metabolic networks (Reich and Selkov, 1981; Fell, 1996; Heinrich and Schuster, 1996; Stephanopoulos et al., 1998; Varner and Ramkrishna, 1999; Fell, 1996). Among the quantitative approaches that exist for the analysis of metabolic networks, constraint-based modeling has attracted attention due to its ability to analyze genome-scale metabolic networks while using very few model parameters (Bailey, 2001; Palsson, 2000).

Constraint-based modeling involves the application of a series of constraints arising from the consideration of stoichiometry, thermodynamics, flux capacity, and regulatory restraints under which reactions operate in a metabolic network (Varma and Palsson, 1994; Price et al., 2003; Edwards et al., 2002; Bonarius et al., 1997). These constraints serve to limit the range of attainable flux distributions or metabolic phenotypes that can be achieved by an organism. Applying mass balancing approaches to a metabolic network leads to a convex mathematical representation of linear equations and inequalities that define an underdetermined system. One approach that is commonly used to explore the metabolic capabilities defined by these constraints is linear programming (LP). In this approach suitable metabolic objectives are posed for the system to maximize or minimize, yielding optimal flux distributions. Typical examples of objective functions include maximization of ATP synthesis, minimization of substrate utilization, and maximization of growth rate.

Linear programming problems with growth rate maximization are commonly used to investigate the metabolic network of microorganisms (Edwards and Palsson, 2000; Edwards and Palsson, 1999; Schilling et al., 2002). Recently, the ability of the constraint-based models to predict the growth and by-product secretion patterns on various substrates for E. coli has been experimentally validated (Edwards et al., 2001), along with the ability to predict the outcome of adaptive evolution of suboptimal laboratory strains (Ibarra et al., 2002). In addition to the use of LP and other related approaches based on convexity principles (Schilling et al., 1999), the constraint-based modeling framework has been extended to include energy balance constraints (Beard et al., 2002) and dynamic rate of change of flux constraints (Mahadevan et al., 2002). Recent advances in the constraint-based approach have seen alternative optimization procedures developed to enable the exploration of an extended range of metabolic function including the use of mixed integer linear programming (MILP) (Burgard et al., 2001; Lee et al., 2000; Hatzimanikatis et al., 1996). Recently a QP-based approach (termed minimization of metabolic adjustment—MOMA) was developed to examine the growth characteristics of mutant strains, in which specific genes have been deleted (Segre et al., 2002).

In every approach that is used to explore the solution space defined by the variety of metabolic constraints a flux distribution is determined that describes the flow of mass and energy through the set of reactions comprising the network. While these flux distributions are at times unique, in the majority of cases they are non-unique particularly with reference to how the network can achieve various objectives. For any given optimal flux distribution there may exist alternate optimal solutions that characterize a region in the flux space that generates the same objective value (e.g., growth rate) via different flux distribution patterns. These represent multiple solutions that are feasible based on the LP formulation and could represent biologically meaningful solutions. The source of these alternate optima and non-unique solutions lies in the biological design of metabolism and the inherent redundancies built into the reaction network. Very few publications with the exception of Lee et al. (Phalakornkule et al., 2001; Lee et al., 2000) have looked at the characterization and significance of alternate optimal flux distributions in metabolic systems.

In this manuscript we address the topic of redundancy and flux variability in metabolic systems. We focus on the impact of such redundancies on current LP and QP solution methods used within the constraint-based approach to modeling. We analyze the effect of the alternate optimal solutions on the predictions of the mutant growth rates made using the QP-based analysis, where a reference flux is chosen based on a previous LP solution. In addition we characterize the range and variability of flux values under conditions (growth on glucose, lactate, and acetate) where alternate optimal solutions exist using a genome-scale model of E. coli as our example system, and assess the underlying biological significance. Furthermore we present an algorithm for the identification of redundancies in metabolic networks that give rise to the existence of alternate optimal solutions. These efforts demonstrate the biological source of variability inherent in metabolic flux distributions and the modeling considerations that must be made due to such functional heterogeneity.