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.

rate is proportional to the nutrient level. Under unlimited nutrient conditions (S → +∞), cells could reach their maximal growth potential (the cell is therefore saturated by the substrate) and follow zerothorder kinetics. The specific growth rate follows a monotonically increasing pattern as we increase the concentration of the limiting nutrient (S). (2)), was first introduced by the UK sociologist Thomas Malthus to describe the "the law of population growth" at the end of 18th century. This model later was formulated and derived by the Belgian mathematician Pierre François Verhulst to describe the self-limiting growth of a biological population in 1838. With little self-limiting factor (X → 0), the population attains the maximal grow rate (μ max ). As the cell growth, the population starts inhibiting themselves (could be considered as a negative auto-regulation loop). With sufficient self-limiting factors (X → X m ), the population reaches the carrying capacity of the system and the growth rate approaches to zero.

The Logistic equation (Equation
The specific growth rate follows a linearly decreasing pattern as the cell population (X) increases.
Both Monod and Logistic model have been used extensively to analyze the fermentation process and study microbial consortia interactions. For example, an expanded form of the Monod equation was proposed to account for product, cell, and substrate inhibitions (Han & Levenspiel, 1988;Levenspiel, 1980;Luong, 1987). 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). A squareroot boundary between cell growth rate and biomass yield has been proposed (Wong, Tran, & Liao, 2009). Coupled Monod equations were applied to describe the complicated predatorprey (oscillatory) relationship between Dictyostelium discoideum and Escherichia coli in Chemostat (Tsuchiya, Drake, Jost, & Fredrickson, 1972). Much earlier than the Monod equation, the Logistic growth was used by the American biophysicist Alfred J.
Lotka and the Italian mathematician Vito Volterra to describe the famous Lotka-Volterra predator-prey ecological model (Lotka, 1926;Volterra, 1926). More interestingly, the solutions of the discrete Logistic growth model were elegantly analyzed by the Australian ecologist Robert May (Baron May of Oxford) in the early 1970s. It was discovered that complex dynamic behaviors could arise from this simple Logistic equation, ranging from stable points to bifurcating stable cycles, to chaotic fluctuations, all depending on the initial parameter conditions (May, 1976). Both models benefit us to analyze the microbial process and explore unknown biological phenomena.
To account for both substrate-limiting and self-inhibiting factors, herein we propose a hybrid Logistic-Monod model (Equation (3) (4)) and yield coefficient (Y x/s ), the implicit form of the analytical solution for cell growth (X, Equation (5)) and substrate (S, Equation (6)) could be easily solved by separation of variables or Laplace transformation.
A typical Monod-type kinetics was plotted for batch culture ( Figure 1a). It should be noted that the initial conditions are prescribed as S = S 0 and X = X 0 at the beginning of cultivation (t = 0).
In the case for Logistic model, we could also arrive the analytical solutions for cell growth (X, Equation (7)) and substrate (S, Equation (8) substrate consumption kinetics (Equation (4)). It should be noted that cell growth is independent of substrate consumption in the Logistic model, but the substrate will deplete proportionally with cell growth (Figure 1b). Due to the simplicity of the Logistic equation, we could arrive at the explicit solution for cell growth (X) and substrate (S).
Similarly, by coupling Equations (3) with (4), the implicit solutions for the hybrid Logistic-Monod equations (Equation (3)) could be derived analytically with the aid of the symbolic computation package of MATLAB. This hybrid Logistic-Monod model (Equation (3)) retains the elementary differential equation norm and should be solved analytically by either separation of variables or Laplace transformation, despite the derivation process will be trivial. The exact solution for cell growth (X, Equation (9)) and substrate (S, Equation (10) We next will explore the steady-state solutions of three growth models in CSTR culture. Based on mass balance and the substrate concentration in the feeding stream (S F ), we could list the mass balance for cell growth (Equation (11)) and substrate consumption (Equation (12)). When the CSTR mass balance equations (Equations (11) and (12)) are coupled with the Monod growth kinetics (Equation (1)), it is easy to arrive the steady-state substrate and cell concentration in the CSTR (Equations (13) and (14)), which has been widely taught in Biochemical engineering or Bioprocess engineering textbooks. As the dilution rate increases, the substrate concentration increases with decreasing cell concentration at the outlet flow of the CSTR (Figure 2a). Similarly, when the mass balance equations (Equations (11) and (12)) are coupled with the Logistic growth kinetics (Equation (2)), the steady-state solutions for substrate and biomass could be derived analytically (Equations (15) and (16)). As the dilution rate increases, the substrate concentration linearly increases accompanying with proportionally decreased cell concentration at the outlet flow of the CSTR (Figure 2b). Finally, for the hybrid Logistic-Monod model, we could also derive the steady-state solutions for the substrate and biomass concentration (Equations (17) and (18), Figure 2c), when the CSTR mass balance equations (Equations (11) and (12)) are coupled with the hybrid Logistic-Monod model (Equation (3)). Plotting all three models together (Figure 2

for both the batch and CSTR
culture, assuming that all the substrate could be converted to biomass. When biomass is the only product, the optimal dilution rate (D opt ) and the washout dilution rate (D w ) could also be analytically derived. Operation under D opt will maximize biomass produtivity (P = DX), and D w is the maximal dilution rate that engineer could possibly run the CSTR system (biomass will be washed out under D w ).     Matlab implicit function fplot was used to draw most of the solutions for Figure 1 and Figure 2. Matlab code has been compiled into a supplementary file, has been uploaded to the journal website (Table 1 and 2).