Integrating Metabolic Networks into Hybrid Bioprocess Models

The optimization and control of bioprocesses require robust in silico models that can accurately capture the complex and dynamic behavior of living cells. While hybrid models that combine machine learning with mechanistic equations have emerged as a powerful tools, they often require relatively large datasets and might yield inconsistent predictions that violate the stoichiometry of metabolism.

In this study, we introduce FBA-Hyb , a multi-scale hybrid modeling framework that tightly integrates genome-scale metabolic networks via flux balance analysis (FBA) into its architecture. In our FBA-Hyb framework, artificial neural networks predict key FBA inputs (substrate uptake rates and cellular objectives) while a surrogate FBA module translates them into the metabolic fluxes that govern the bioprocess. A key novelty is that the FBA optimization step is replaced by a surrogate generated with symbolic regression, which encapsulates the FBA model into a compact analytical expression. This allows easy backpropagation through the integration of the neural controlled differential equationbased FBA-Hyb bioprocess model.

We validated FBA-Hyb against a standard hybrid model ( Std-Hyb ) using two Escherichia coli fedbatch case studies. In the first study, FBA-Hyb achieved a 42 % average improvement in predictive accuracy ( R ² ) during a leave-one-process-out cross validation. Crucially, FBA-Hyb maintains strict stoichiometric feasibility even during extrapolation. Meanwhile, an alternative approach based on standard architectures leads to stoichiometrically inconsistent solutions in 22 % of the cases analyzed. In the second case study, we demonstrate how FBA-Hyb effectively simulates unmeasured chemical species and discovers a metabolic shift in sulfate-limited regimes during bioprocessing.

By providing a modular, biologically consistent, and computationally efficient architecture, FBA-Hyb offers a robust foundation for the next generation of bioprocess models and sustainable process optimization.

Graphical Abstract

Highlights

• FBA-Hyb integrates flux balance analysis (FBA) into hybrid bioprocess models.

• Symbolic regression discovers a simple closed-form FBA surrogate model.

• The FBA surrogate ensures accurate reaction stoichiometry.

• A neural network predicting the FBA objective keeps the model flexible.

• FBA-Hyb has superior capabilities and accuracy compared to the current standard.