Modeling the asymmetric metabolic shifts in poly-γ-glutamic acid batch fermentation using a non-linear Boltzmann...

A four-parameter Boltzmann empirical model was evaluated as an alternative to the classical Logistic-Luedeking-Piret (L-LP) kinetic framework for describing batch fermentation of poly-γ-glutamic acid (γ-PGA) by Bacillus subtilis natto. To characterise the classical model's failure to reproduce the post-peak decline in γ-PGA concentration observed under carbon-depleted conditions, a first-order product degradation term was incorporated into the L-LP framework as a diagnostic extension. Across 36 h of batch fermentation, bacterial growth followed a typical sigmoidal curve, reaching a maximum specific growth rate of 0.7362 h⁻¹ and a maximum biomass concentration of 1.82 g/L (dry cell weight). γ-PGA production peaked at 18.76 g/L and then declined modestly as glucose became depleted. The Boltzmann model matched the Logistic model in describing biomass accumulation (R² ≈ 0.99) but fit the γ-PGA and glucose kinetics substantially better, lowering RMSE by approximately 10%, 45% and 47% for biomass, γ-PGA and glucose respectively relative to the classical model; information-criterion comparisons favoured the Boltzmann description further (ΔAIC = 11.72 for γ-PGA and 12.70 for glucose). The Boltzmann model reproduced the near-plateau phase of γ-PGA accumulation more accurately than the classical model, although it did not fully capture the slight late-stage decrease. Together, these results establish a practical empirical kinetic framework for γ-PGA batch fermentation, offering a quantitative basis for subsequent process optimisation and fed-batch scale-up studies.