# Mono- and Polyauxic Growth Kinetics: A Semi-Mechanistic Framework for Complex Biological Dynamics

## TL;DR

This paper introduces a new mathematical framework for modeling microbial growth that better captures complex biological behaviors and improves bioprocess design.

## Contribution

A unified semi-mechanistic framework is proposed, integrating modified growth equations and robust regression techniques for accurate modeling of mono- and polyauxic growth.

## Key findings

- The framework successfully models complex multiphasic growth with interpretable parameters.
- Conventional single-phase models may miss important metabolic transitions in co-digestion systems.
- Optimization strategies and outlier removal improve model accuracy and reliability.

## Abstract

Kinetic modeling of microbial growth is essential for the design, optimization, and scale-up of industrial bioprocesses. Classical empirical models often lack biologically interpretable parameters or fail to capture complex multiphasic (polyauxic) behaviors, while fully mechanistic models are impractical for systems involving complex substrates and mixed cultures. This study proposes a unified mathematical framework that reformulates the canonical Boltzmann and Gompertz equations into semi-mechanistic forms, explicitly defining the maximum specific reaction rate and lag phase duration. Polyauxic growth is represented as a weighted sum of sigmoidal phases, subject to stringent constraints that ensure parameter identifiability, temporal consistency, and biological plausibility. The methodology integrates a workflow to address nonlinear regression in high-dimensional parameter spaces. A two-stage optimization strategy using Differential Evolution for global search followed by L-BFGS-B for local refinement avoid bias and heuristic parameter initialization. A Charbonnier loss function and the Robust Regression and Outlier Removal procedure are employed to identify and mitigate experimental outliers. Model parsimony is enforced using Akaike (AIC, AICc) and Bayesian (BIC) information criteria to select the optimal number of growth phases and avoid overparameterization. The framework was evaluated using experimental anaerobic digestion datasets, demonstrating that conventional single-phase models can obscure relevant metabolic transitions in co-digestion systems.