Analytical solution for a hybrid Logistic‐Monod cell growth model in batch and continuous stirred tank reactor culture

Abstract

Monod and Logistic growth models have been widely used as basic equations to describe cell growth in bioprocess engineering. In the case of the Monod equation, the specific growth rate is governed by a limiting nutrient, with the mathematical form similar to the Michaelis-Menten equation. In the case of the Logistic equation, the specific growth rate is determined by the carrying capacity of the system, which could be growth-inhibiting factors (i.e., toxic chemical accumulation) other than the nutrient level. Both equations have been found valuable to guide us build unstructured kinetic models to analyze the fermentation process and understand cell physiology. In this work, we present a hybrid Logistic-Monod growth model, which accounts for multiple growth-dependent factors including both the limiting nutrient and the carrying capacity of the system. Coupled with substrate consumption and yield coefficient, we present the analytical solutions for this hybrid Logistic-Monod model in both batch and continuous stirred tank reactor (CSTR) culture. Under high biomass yield (Y x/s ) conditions, the analytical solution for this hybrid model is approaching to the Logistic equation; under low biomass yield condition, the analytical solution for this hybrid model converges to the Monod equation. This hybrid Logistic-Monod equation represents the cell growth transition from substrate-limiting condition to growth-inhibiting condition, which could be adopted to accurately describe the multi-phases of cell growth and may facilitate kinetic model construction, bioprocess optimization, and scale-up in industrial biotechnology. K E Y W O R D S analytical solution, batch and CSTR culture, cell growth, logistic growth, monod equation, hybrid model

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“…The steady state solution of species A or B decreases as we increase the dilution rate. This trend is consistent with the population concentration as observed when a single species is cultivated in the chemostat [29].…”

“…The intriguing solutions of the discrete Logistic model were discovered to have complex dynamic behaviors, ranging from stable points, to bifurcating stable cycles, to chaotic fluctuations, depending on the initial parameter conditions [28]. Both Monod and Logistic equations benefit us to quantitatively understand microbial interaction kinetics [29].…”

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“…The steady-state solution of species A or B decreases as we increase the dilution rate. This trend is consistent with the population concentration as observed when a single species is cultivated in the chemostat (Xu, 2019).…”

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