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

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.