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.
Highlights
In 1949, French microbiologist Dr Jacques Monod, provided a quantitative description between bacterial growth rate and the concentration of a limiting substrate.This equation (Equation (1)) takes the mathematical form of Michaelis–Menten equation, where the substrate saturation constant (Ks) and the maximal specific cell growth rate could be graphically determined by the Lineweaver–Burk double‐reciprocal plot (Lineweaver & Burk, 1934)
Under unlimited nutrient conditions (S → +∞), cells could reach their maximal growth potential and follow zeroth‐ order kinetics
When the Monod equation was coupled with the Luedeking–Piret equation (Robert Luedeking, 1959), analytical solutions for cell growth, substrate consumption, and product formation could be derived (Garnier & Gaillet, 2015)
Summary
In 1949, French microbiologist Dr Jacques Monod (who was a Nobel Laureate in 1965, best known for his discovery of Lac operon), provided a quantitative description between bacterial growth rate and the concentration of a limiting substrate (glucose; Monod, 1949).This equation (Equation (1)) takes the mathematical form of Michaelis–Menten equation, where the substrate saturation constant (Ks) and the maximal specific cell growth rate (μmax) could be graphically determined by the Lineweaver–Burk double‐reciprocal plot (Lineweaver & Burk, 1934). KEYWORDS analytical solution, batch and CSTR culture, cell growth, logistic growth, monod equation, hybrid model
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