Oddsdóttir Hildur Æsa, Hagrot Erika, Chotteau Véronique, Forsgren Anders
Department of Mathematics, Optimization and Systems Theory, KTH Royal Institute of Technology, Stockholm SE-100 44, Sweden.
Division of Industrial Biotechnology/Bioprocess Design, KTH Royal Institute of Technology, Albanova Center, Stockholm SE-106 91, Sweden.
Math Biosci. 2016 Mar;273:45-56. doi: 10.1016/j.mbs.2015.12.009. Epub 2015 Dec 31.
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. The measurements used in the data fitting are subject to errors. A robust optimization problem includes information on errors and gives a way to examine the sensitivity of the solution of the EFMs-based MFA to these errors. In general, formulating a robust optimization problem may make the problem significantly harder. We show that in the case of the EFMs-based MFA, when the errors are only in measurements and bounded by an interval, the robust problem can be stated as a convex quadratic programming (QP) problem. We have previously shown how the data fitting problem may be solved in a column-generation framework. In this paper, we show how column generation may be applied also to the robust problem, thereby avoiding explicit enumeration of EFMs. Furthermore, the option to indicate intervals on metabolites that are not measured is introduced in this column generation framework. The robustness of the data is evaluated in a case-study, which indicates that the solutions of our non-robust problems are in fact near-optimal also when robustness is considered, implying that the errors in measurement do not have a large impact on the optimal solution. Furthermore, we showed that the addition of intervals on unmeasured metabolites resulted in a change in the optimal solution.
基本通量模式（EFMs）是从代谢反应网络定义的向量，给出了底物和产物之间的联系。基于EFMs的代谢通量分析（MFA）通过最小二乘数据拟合从外部通量测量估计每个EFMs上的通量。数据拟合中使用的测量存在误差。一个鲁棒优化问题包含误差信息，并给出了一种方法来检验基于EFMs的MFA的解对这些误差的敏感性。一般来说，制定一个鲁棒优化问题可能会使问题显著变难。我们表明，在基于EFMs的MFA的情况下，当误差仅存在于测量中且由一个区间界定，鲁棒问题可以表述为一个凸二次规划（QP）问题。我们之前已经展示了如何在列生成框架中解决数据拟合问题。在本文中，我们展示了列生成也可以应用于鲁棒问题，从而避免对EFMs进行显式枚举。此外，在这个列生成框架中引入了对未测量代谢物指示区间的选项。在一个案例研究中评估了数据的鲁棒性，这表明当考虑鲁棒性时，我们的非鲁棒问题的解实际上也是接近最优的，这意味着测量误差对最优解没有很大影响。此外，我们表明在未测量代谢物上添加区间会导致最优解发生变化。
