Chapman Hester, Kratochvíl Miroslav, Ebenhöh Oliver, Wilken St Elmo
Institute of Quantitative and Theoretical Biology, Heinrich Heine University, Düsseldorf 40255, Germany.
Luxembourg Centre of Systems Biomedicine, University of Luxembourg, Esch-sur-Alzette L-4367, Luxembourg.
Bioinformatics. 2025 Jun 2;41(6). doi: 10.1093/bioinformatics/btaf287.
Sensitivity analysis is a useful tool to identify key parameters in metabolic models. It is typically only applied to the growth rate, disregarding the sensitivity of other solution variables to parameters. Further, sensitivity analysis of elementary flux modes could provide low-dimensional insights into optimal solutions, but they are not defined when a model is subject to inhomogeneous flux constraints, such as the frequently used ATP maintenance reaction.
We introduce optimal flux modes (OFMs), an analogue to elementary flux modes (EFMs), but specifically applied to optimal solutions of constraint-based models. Further, we prove that implicit differentiation can always be used to efficiently calculate the sensitivities of both whole-model solutions and OFM-based solutions to model parameters. This allows for fine-grained sensitivity analysis of the optimal solution, and investigation of how these parameters exert control on the optimal composition of OFMs. This novel framework is implemented in DifferentiableMetabolism.jl, a software package designed to efficiently differentiate solutions of constraint-based models. To demonstrate scalability, we differentiate solutions of 342 yeast models; additionally we show that sensitivities of specific subsystems can guide metabolic engineering. Applying our scheme to an Escherichia coli model, we find that OFM sensitivities predict the effect of knockout experiments on waste product accumulation. Sensitivity analysis of OFMs also provides key insights into metabolic changes resulting from parameter perturbations.
Software introduced here is available as open-source Julia packages DifferentiableMetabolism.jl (https://github.com/stelmo/DifferentiableMetabolism.jl) and ElementaryFluxModes.jl (https://github.com/HettieC/ElementaryFluxModes.jl), which both work on all major operating systems and computer architectures. Code to reproduce all results is available from https://github.com/HettieC/DifferentiableOFMPaper, and as an archive from https://doi.org/10.5281/zenodo.15183208.
敏感性分析是识别代谢模型中关键参数的有用工具。它通常仅应用于生长速率，而忽略了其他解变量对参数的敏感性。此外，基本通量模式的敏感性分析可以为最优解提供低维见解，但当模型受到非均匀通量约束（如常用的ATP维持反应）时，它们并未定义。
我们引入了最优通量模式（OFM），它类似于基本通量模式（EFM），但专门应用于基于约束的模型的最优解。此外，我们证明隐式微分总是可以用于有效计算全模型解和基于OFM的解对模型参数的敏感性。这允许对最优解进行细粒度的敏感性分析，并研究这些参数如何控制OFM的最优组成。这个新框架在DifferentiableMetabolism.jl中实现，这是一个旨在有效区分基于约束的模型的解的软件包。为了展示可扩展性，我们区分了342个酵母模型中的解；此外，我们表明特定子系统的敏感性可以指导代谢工程。将我们的方案应用于大肠杆菌模型，我们发现OFM敏感性预测了基因敲除实验对废物积累的影响。OFM的敏感性分析还为参数扰动引起的代谢变化提供了关键见解。
这里介绍的软件作为开源Julia包DifferentiableMetabolism.jl（https://github.com/stelmo/DifferentiableMetabolism.jl）和ElementaryFluxModes.jl（https://github.com/HettieC/ElementaryFluxModes.jl）提供，它们适用于所有主要操作系统和计算机架构。重现所有结果的代码可从https://github.com/HettieC/DifferentiableOFMPaper获得，也可作为存档从https://doi.org/10.5281/zenodo.15183208获得。
