Robust generation of elementary flux modes

Abstract

Elementary flux modes (EFMs) are vectors defined from a metabolic reaction network, giving the connections between substrates and products. EFMs-based metabolic flux analysis (MFA) estimates the flux over each EFM from external flux measurements through least-squares data fitting. In previous work we presented an optimization method of column generation type that facilitates EFMs-based MFA when the metabolic reaction network is so large that enumerating all EFMs is prohibitive. In this work we extend this model by including errors on measurements in a robust optimization framework. In the robust optimization problem, the least-squares data fitting is minimized subject to the error on each metabolite being as unfavourable as it can be, within a given interval. In general, inclusion of robustness may make the optimization problem significantly harder. However, we show that in our case the robust problem can be stated as a convex quadratic programming problem, i.e., of the same form as the original non-robust problem. Additionally, we demonstrate that the column-generation technique of the non-robust problem can be extended also to the robust problem. Furthermore, the option to indicate intervals on metabolites that are not measured is introduced in this column generation framework. The effect of including robustness in the model is evaluated in a case-study, which indicated that the solutions of our non-robust problems are in fact near-optimal also when robustness is considered. On the other hand, the addition of intervals on unmeasured metabolites resulted in a change of optimal solution. Implying that the inclusion of intervals on unmeasured metabolites is more important than the direct consideration of measurement errors, this despite up to 20% errors.