Biochemical Engineering Fundamentals, Part 1 - Bailey J., Ollis D. 1989

Kinetics of substrate utilization, metabolite production, and biomass formation in cell cultures
Kinetics of metabolite production
Chemically structured models for the kinetics of cellular product formation

Compared to studying the growth kinetics of individual Cells and Cell populations, relatively little work has been devoted to developing structured models within the context of metabolic product formation kinetics. However, it can be expected that the advancement of structured models for cell growth will soon necessitate The Development of A number of such models. In this section, we examine an interesting process involving The formation of one of many secondary metabolites, using it as a case study to introduce several important aspects of microbiological antibiotic synthesis processes, as well as novel approaches to modeling their kinetics.

The synthesis of several microbial Antibiotics and other secondary metabolites is inhibited by high concentrations of phosphate ions. Since phosphate is strictly essential for cell growth, there must evidently be some optimal concentration level. Indeed, such an optimal initial phosphate concentration level has been observed, for example, during The Biosynthesis of Alkaloids by Claviceps purpurea (Fig. 7.32).

The experimental data presented in Fig. 7.32 are well-modeled by the following kinetic equations for batch culture cell growth and metabolic product formation [26].

Cell growth:

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Phosphate concentration in the medium:

Intracellular phosphate concentration:

Alkaloid formation:

In this model, the specific growth rate is described by the Teissier equation, in which one of the independent variables is the intracellular phosphate concentration p_i. (This variable represents the mass of KH₂PO₄ in grams contained in 1 g of biomass and reflects cell composition. All other variables in this model describing system components are measured in grams per liter of culture.) The term k₂x² represents The rate of cell lysis.

FIG. 7.32. Time course of biomass density (BD) of C. purpurea (●, g dry weight per liter) and total produced alkaloids (TA) (○, mg/L) at various initial phosphate concentrations. Curves represent the corresponding calculated data obtained using the mathematical models described in the text. [Reprinted with permission from: Pažoutová S., Votruba J., Řeháček Z., A Mathematical Model of Growth and Alkaloid Production in the Submerged Culture of Claviceps purpurea; Biotech. Bioeng., 23, 2837 (1981).]

As a result of lysis, phosphate is released into the medium; as indicated by the second term on the right-hand side of equation (7.95), its amount is proportional to the phosphate content in The Cell mass Y_p/x and the intracellular phosphate concentration p_i. The first term of the same equation describes The Active Transport of phosphate into the cell; The kinetics of this process are characterized by saturation behavior. The material balance equation for intracellular phosphate [Equation (7.96)] is a variation of Equation (7.67); here it is taken into account that its rate of formation equals the difference between the rate of phosphate transport into the cell and the rate of phosphate incorporation into cellular components. In the product formation rate equation, inhibition by phosphate is accounted for in the same manner as this effect was previously incorporated when modeling phosphatase enzyme repression.

Table 7.5. Parameter values in the structured model of alkaloid biosynthesis kinetics in C. purpurea^a

^a Reprinted with permission from: Pažoutová S., Votruba J., Řeháček Z., A Mathematical Model of Growth and Alkaloid Production in the Submerged Culture of Claviceps purpurea, Biotech. Bioeng., 23, 2837 (1981).

The solid curves in Fig. 7.32 were calculated using the described model, and the corresponding parameter values are listed in Table 7.5. This structured model accurately captures the patterns of biomass and alkaloid mass variation over time, including the presence of a maximum on the total alkaloid accumulation curve; however, its accuracy is insufficiently high at high phosphate concentrations. Specifically, this model demonstrated that the maximum total alkaloid yield is produced at an initial phosphate concentration in the medium of 0.17 g/L. Mathematical simulation of various phosphate feeding strategies (including pulsed phosphate additions at different process phases) showed that this approach cannot achieve a significant increase in alkaloid yield compared to a standard batch process with an optimal initial phosphate concentration in the medium. In other cases, which we will examine in subsequent chapters, fed-batch processes prove to be very useful.