EINN: An enzyme-informed neural network guided by an enzyme-constrained genome-scale metabolic model

Predicting cellular metabolism from molecular data is a central challenge in systems biology, with direct implications for engineering microbial cell factories. Genome-scale metabolic models (GEMs) provide a mechanistic foundation for such predictions, but their accuracy is limited by an inability to account for the finite catalytic capacity of the proteome. Enzyme-constrained GEMs (ecGEMs) address this by incorporating enzyme turnover numbers and proteome allocation constraints. Hybrid neural-mechanistic models like Artificial Metabolic Network (AMN) and Metabolic-Informed Neural Network (MINN) have sought to combine the flexibility of machine learning with the structure of GEMs, yet they rely on conventional, unconstrained metabolic networks, allowing flux predictions to violate enzyme capacity constraints and pushing models toward biologically unrealistic solution spaces. Here, we introduce the Enzyme-Informed Neural Network (EINN), a conceptual framework which relies on ecGEMs to constrain the neural network. We implement two variants of EINN, ecAMN and ecMINN, using Escherichia coli (E. coli) ecGEMs built with the GECKO 3.0 toolbox, and systematically evaluate their performance. Compared to their unconstrained counterparts, ecAMN significantly improves training stability and predictive accuracy by eliminating convergence to poor local minima, while ecMINN reduces overfitting and achieves lower error rates through mechanistic integration of proteomic data as flux bounds. Together, these results show that constraining the neural network's solution space with enzymatic capacity limits is a more effective hybrid modeling foundation than relying on conventional GEMs alone.