Interplay between Constraints, Objectives, and Optimality for Genome-Scale Stoichiometric Models

High-throughput data generation and genome-scale stoichiometric models have greatly facilitated the comprehensive study of metabolic networks. The computation of all feasible metabolic routes with these models, given stoichiometric, thermodynamic, and steady-state constraints, provides important insights into the metabolic capacities of a cell. How the feasible metabolic routes emerge from the interplay between flux constraints, optimality objectives, and the entire metabolic network of a cell is, however, only partially understood. We show how optimal metabolic routes, resulting from flux balance analysis computations, arise out of elementary flux modes, constraints, and optimization objectives. We illustrate our findings with a genome-scale stoichiometric model of Escherichia coli metabolism. In the case of one flux constraint, all feasible optimal flux routes can be derived from elementary flux modes alone. We found up to 120 million of such optimal elementary flux modes. We introduce a new computational method to compute the corner points of the optimal solution space fast and efficiently. Optimal flux routes no longer depend exclusively on elementary flux modes when we impose additional constraints; new optimal metabolic routes arise out of combinations of elementary flux modes. The solution space of feasible metabolic routes shrinks enormously when additional objectives---e.g. those related to pathway expression costs or pathway length---are introduced. In many cases, only a single metabolic route remains that is both feasible and optimal. This paper contributes to reaching a complete topological understanding of the metabolic capacity of organisms in terms of metabolic flux routes, one that is most natural to biochemists and biotechnologists studying and engineering metabolism.高通量数据生成与基因组规模化学计量模型（genome-scale stoichiometric models）极大推动了代谢网络（metabolic networks）的系统性研究。借助此类模型，在化学计量约束、热力学约束及稳态约束的限定下计算所有可行代谢通路，可为解析细胞的代谢能力提供重要理论见解。然而，可行代谢通路如何由通量约束（flux constraints）、最优性目标（optimality objectives）与细胞完整代谢网络之间的相互作用催生，这一问题尚未完全明晰。本研究阐明了由通量平衡分析（flux balance analysis）计算得到的最优代谢通路，如何从基本通量模式（elementary flux modes）、约束条件与优化目标中衍生而来。我们以大肠杆菌（Escherichia coli）代谢的基因组规模化学计量模型为例，对研究发现进行了演示。当仅存在单一通量约束时，所有可行的最优通量通路均可仅通过基本通量模式推导得出。本研究共发现多达1.2亿条此类最优基本通量模式。我们提出了一种新型计算方法，可快速高效地计算最优解空间（optimal solution space）的顶点。当引入额外约束时，最优通量通路不再仅依赖于基本通量模式；新的最优代谢通路可通过基本通量模式的组合产生。当引入额外优化目标（例如与通路表达成本（pathway expression costs）或通路长度（pathway length）相关的目标）时，可行代谢通路的解空间会大幅收缩。在多数情形下，仅存一条同时满足可行性与最优性的代谢通路。本研究有助于从代谢通量通路的角度，实现对生物体代谢能力的完整拓扑学认知——这一认知视角对于从事代谢研究与代谢工程的生物化学家（biochemists）与生物技术专家（biotechnologists）而言最为自然贴合。