Department of Chemical Engineering, The Pennsylvania State University, 112A Fenske Laboratory, University Park, Pennsylvania, 16802, USA.
Biotechnol Bioeng. 1997 Oct 20;56(2):145-61. doi: 10.1002/(SICI)1097-0290(19971020)56:2<145::AID-BIT4>3.0.CO;2-P.
The S-System formalism provides a popular, versatile and mathematically tractable representation of metabolic pathways. At steady-state, after a logarithmic transformation, the S-System representation reduces into a system of linear equations. Thus, the maximization of a particular metabolite concentration or a flux subject to physiological constraints can be expressed as a linear programming (LP) problem which can be solved explicitly and exactly for the optimum enzyme activities. So far, the quantitative effect of parametric/experimental uncertainty on the S-model predictions has been largely ignored. In this work, for the first time, the systematic quantitative description of modeling/experimental uncertainty is attempted by utilizing probability density distributions to model the uncertainty in assigning a unique value to system parameters. This probabilistic description of uncertainty renders both objective and physiological constraints stochastic, demanding a probabilistic description for the optimization of metabolic pathways. Based on notions from chance-constrained programming and statistics, a novel approach is introduced for transforming the original stochastic formulation into a deterministic one which can be solved with existing optimization algorithms. The proposed framework is applied to two metabolic pathways characterized with experimental and modeling uncertainty in the kinetic orders. The computational results indicate the tractability of the method and the significant role that modeling and experimental uncertainty may play in the optimization of networks of metabolic reactions. While optimization results ignoring uncertainty sometimes violate physiological constraints and may fail to correctly assess objective targets, the proposed framework provides quantitative answers to questions regarding how likely it is to achieve a particular metabolic objective without exceeding a prespecified probability of violating the physiological constraints. Trade-off curves between metabolic objectives, probabilities of meeting these objectives, and chances of satisfying the physiological constraints, provide a concise and systematic way to guide enzyme activity alterations to meet an objective in the face of modeling and experimental uncertainty. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 56: 145-161, 1997.
S 系统形式主义为代谢途径提供了一种流行、通用且易于数学处理的表示。在稳态下，经过对数变换后，S 系统表示简化为线性方程组。因此，可以将特定代谢物浓度或通量的最大化表示为一个线性规划 (LP) 问题，可以为最优酶活性显式且准确地求解。到目前为止，参数/实验不确定性对 S 模型预测的定量影响在很大程度上被忽略了。在这项工作中，首次尝试通过利用概率密度分布来模拟系统参数赋值的唯一值的不确定性，对建模/实验不确定性进行系统的定量描述。这种不确定性的概率描述使得目标和生理约束都是随机的，需要对代谢途径的优化进行概率描述。基于机会约束规划和统计学的概念，引入了一种新的方法，将原始随机公式转换为可以使用现有优化算法求解的确定性公式。所提出的框架应用于两个具有实验和建模不确定性的代谢途径，在动力学阶数中存在不确定性。计算结果表明该方法的可行性以及建模和实验不确定性在代谢反应网络优化中可能起的重要作用。虽然忽略不确定性的优化结果有时会违反生理约束，并且可能无法正确评估目标，但所提出的框架提供了有关在不超过指定违反生理约束概率的情况下实现特定代谢目标的可能性的定量答案。代谢目标之间的权衡曲线、达到这些目标的概率以及满足生理约束的机会，为在面临建模和实验不确定性时指导酶活性改变以实现目标提供了一种简洁而系统的方法。（c）1997 John Wiley & Sons, Inc. Biotechnol Bioeng 56: 145-161, 1997.
