The effects of alternate optimal solutions in constraint-based genome-scale metabolic models☆
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.
Section snippets
Linear programming based analysis
A metabolic network is constrained by the imposition of stoichiometric constraints that correspond to the mass balance around each metabolite. When the metabolic network has a steady state distribution of fluxes the constraints are described by a system of linear equations as shown below:where S is the m×n stoichiometric matrix of all the reactions in the metabolic network, m is the number of metabolites, n is the number of fluxes (reaction rates), and v is the flux vector of the
Calculation of flux variability due to alternate optima
LP problems can have multiple solutions that have the exact same optimal value for the objective function and satisfy all of the constraints. In order to investigate the effects of these alternate optima, approaches must be developed to calculate the multiplicity of solutions. Previously, a study of the multiple flux distributions that satisfy the stoichiometric and capacity constraints and have the same objective function has been presented in Lee et al. (2000). In that study, all the
Effect of flux variability on QP-based analysis
The existence of alternate optimal solutions will have an effect on QP-based analyses. In QP-based approaches a reference flux distribution is selected that is often derived from an LP optimization problem. The selection of this reference flux distribution can be complicated by the presence of alternate optima as discussed in the previous section. Thus while the flux distribution calculated in the QP-based approach is unique, the reference state may not be unique leading to a different result
Effect of sub-optimal flux distributions on LP & QP-based analysis
To this point, flux variation has been characterized under the assumption that the optimal solution is attained by the metabolic network. However, metabolic systems may not necessarily operate at the full optimal state, but may be operating in near optimal or sub-optimal modes. Unlike optimization of engineering processes where the tolerances in the value of the objective function are small, biological systems can be expected to have tolerances that are larger and thus the objective values can
Identification of equivalent pathways
Often times there are many different routes through a metabolic network that can provide the same net stoichiometric conversion of substrates into products (Price et al., 2002b). The existence of alternate optimal solutions motivates the identification of these different routes that may exist in a network. These different pathways can be termed as equivalent pathways and are a condition-independent or structural property of the network. The complete set of these equivalent pathways can be found
Discussion
Biological systems often contain redundancies that contribute significantly to their robustness. Herein, the redundancies in the metabolic network that lead to alternate optimal flux distributions and hence contribute to robustness in the context of optimal growth and flux variability are analyzed. An approach to characterize and study the variability of the fluxes in alternate optimal solutions is presented. This approach for the characterization of the alternate optima is demonstrated using
Acknowledgements
Authors would like to acknowledge Sharon Wiback, Stephen Fong, Bernhard Palsson, Daniel Segre, Brian Gates and George Church for valuable comments on this manuscript. Financial support for this work provided by the Department of Energy (Grant No: DE-FG03-01-01ER25499/A000) is gratefully acknowledged.
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Supplementary data associated with this article can be found, in the online version