# The dynamic growth of bacterial cultures: real-time Bayesian estimation of substrate uptake rates in fed-batch fermentations of E. coli

## TL;DR

This paper introduces a new Bayesian method to estimate substrate uptake rates in real-time during E. coli fermentations, improving bioprocess control.

## Contribution

A novel Bayesian estimator is introduced that explicitly accounts for substrate uptake dynamics and adaptability rates in bioprocess modeling.

## Key findings

- Defining substrate uptake rate as a state variable improves estimation accuracy in fed-batch fermentations.
- The Bayesian estimator maintains near-zero residuals between model predictions and actual bioprocess outputs.
- The method enables robust state and parameter estimation under uncertainty in dynamic bioprocess environments.

## Abstract

Accurate real-time estimation of system states and metabolic parameters is essential for effective bioprocess control. However, the dynamics of microbial adaptation—the rate at which a microorganism adapts to changes in the substrate concentration—is often overlooked, leading to early-stage plant-model mismatches and inaccurate estimation of relevant parameters, such as the biomass yield on carbon source (\documentclass[12pt]{minimal}
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\begin{document}$$Y_{XC}$$\end{document}) or the maximum substrate uptake rate (\documentclass[12pt]{minimal}
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\begin{document}$$q_S^{\text {max}}$$\end{document}). This work introduces a novel model-based observer for simultaneous state and parameter estimation that explicitly accounts for substrate uptake dynamics. By defining the substrate uptake rate (\documentclass[12pt]{minimal}
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\begin{document}$$q_S$$\end{document}) as a state variable and introducing a random variable (\documentclass[12pt]{minimal}
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\begin{document}$$\lambda$$\end{document}) to represent the biomass-specific substrate uptake adaptability rate, we construct a Bayesian estimator that allows proper determination of the states and parameters in fed-batch fermentations of E. coli while maintaining near-zero centered residuals between the plant output and the proposed model stoichiometry. This work advances methods for robust state and adaptive parameter estimation in dynamic bioprocess environments under uncertainty.