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

Kinetics of substrate utilization, metabolite production, and biomass formation in cell cultures
Conclusion

As we mentioned in the Introduction to this chapter, the ultimate goal of studying Cell growth kinetics is to provide a quantitative mathematical Description of the combined effects of an Organism's genetic makeup and its environment on the rates of processes occurring within a cell population. Currently, we have a variety of specific models at our disposal: some account for minor genetic variations, others for slight fluctuations in environmental composition, and still others describe the heterogeneity of cell cultures. Given the limited applicability of each such model, biochemical engineers must first clearly define the intended scope of application for a given kinetic model and shape its form and depth accordingly.

Tremendous advancements in the fundamental biological sciences and computer technology will eventually enable the creation of more reliable, comprehensive, and mechanistic models of cell growth kinetics.

In this chapter, we examined the relationships among biomass growth, substrate utilization, and The formation of cellular metabolic products. Experimental data helped us elucidate the key characteristics of growth in cell populations and filamentous organisms. We also reviewed the Mathematical models of microbial population growth required for the analysis and design of technological processes involving them.

Ch. 9 is devoted to the design, analysis, and scaling of various batch and continuous bioreactors. In our Structure/133.html">Discussion of this topic, we will rely primarily on the mathematical expressions for process rates reviewed in this chapter.

Before proceeding with the analysis of bioreactors, we must examine The kinetics of yet another Class of phenomena—namely, transport processes in macroscopic systems. Here we refer to The transport of oxygen, chlorine, and other gases in liquids, which is utilized, for example, in aeration and chlorination. Such transport phenomena are fundamentally closely linked to the natural or forced convection processes characteristic of bioreactors. The latter, in turn, cannot be properly evaluated without accounting for the power consumption of the bioreactor. Transport phenomena must also be taken into consideration when scaling up bioreactors. Heat transfer plays a major role in sterilization and other biochemical engineering processes as well. We will explore all of these problems and several other issues in the following chapter.

Exercises

7.1. Kinetics of batch Fermentation. Natural grape juice contains almost all the nutrients required for Yeast growth. Perform the following laboratory experiment on fermentation.

a) Acquire or set up a simple apparatus for producing beer or wine. To produce wine, you will need grape juice (this experiment requires juice free of insoluble solids), sugar, yeast, a 3.8 L vessel, a small beaker, and a delivery tube connected to a Water seal.

b) Conducting the fermentation process. Suspend active yeast in 100 mL of warm boiled water; mix the thoroughness to ensure no clumps of yeast mass remain in the suspension. Place 0.95 L of grape juice and 1.35 kg of sugar into the fermenter (sterilized with boiling water or a sodium sulfite solution), and add distilled water (slightly above room Temperature) until the fermenter is three-quarters full. Ensure that enough headspace remains above the liquid to accommodate a small hydrometer, and arrange for the hydrometer to be introduced through the fermenter lid. Add the yeast suspension, tightly seal the vessel, and shake vigorously for 30 s. Immerse the hydrometer into the reaction mixture. Seal the fermenter with a single-hole stopper fitted with a delivery tube connected to a water seal; the water level in the seal should be approximately 1 cm higher than the opening of the delivery tube.

c) Monitor the progress of fermentation over time (for a week) by tracking: 1) the evolution rate of CO₂ at a constant height difference between the delivery tube and the water seal (devise a method to determine the volume of a single CO₂ bubble); 2) the hydrometer readings (make sure the system remains airtight).

d) Express the results graphically; discuss the graphs in relation to the apparent periods of cell growth and the relationship between the rates of ethanol and CO₂ production (clearly state all assumptions). Knowing The amount of sugar introduced into the reaction mixture (contained in the grape juice and crystalline sucrose), the dry cell mass at the end of the process, and the results of the previous exercise (item c), formulate a carbon mass balance equation; state the assumptions made in the equation.

e) Human Sensory Organs are considered to be very sensitive measuring devices. Try your experimental product by Smell and Taste!

7.2. Temperature dependence of Cell Growth Rate. Johnson, Eyring, and Polissar (The Kinetic Basis of Molecular Biology, John Wiley and Sons, N.Y., 1954) proposed the following equation for the temperature dependence of the specific growth rate μ of E. coli in the range from 18 to 43 °C:

where a = 0.3612e^24.04; ∆H₁ = 15 kcal/mol; ∆H₂ = 150 kcal/mol; ∆S = 476.46 cal/(g-mol∙K).

a) Plot this function as ln μ versus 1/T.

b) Show that this equation can be expressed as the product of two Functions, the shape of which is suggested by the graph in Exercise 7.2a. How can the physical meaning of these two functions and the parameter values given above be explained?

c) What assumptions associated with irreversible inactivation are made in this interpretation of the temperature dependence of μ?

7.3. Single- and multi-substrate microbiological process kinetics. A culture is grown on a simple medium containing 0.3% (w/v) glucose. At time t = 0, the culture (as an inoculum) is diluted with a large volume of identical sterile medium. The growth process is monitored by measuring The change in optical density (OD) at 420 nm over time; the obtained data are listed in Column 1 of Table 7U3.1. Column 2 of the same table shows the OD changes of the same culture grown in a complex medium and diluted with a nutrient solution containing 0.15% glucose and 0.15% lactose (w/v). Assume that OD (at 420 nm) is proportional to The Cell mass concentration, with an OD of 0.175 corresponding to a concentration of 0.1 mg of Cells (on a dry weight basis) per 1 mL. Calculate the maximum specific growth rate μ_max, the lag-phase duration t_lag, and the overall yield coefficients Y (expressed in grams of cells per gram of substrate), assuming the substrate is always completely utilized. Explain the shape of the growth curves for each case.

Table 7U3.1^a

^a From: Biochemical Reasoning. Kerridge D., Tipton K. (eds.), prob. 39, W. B. Benjamin, Inc., Menlo Park, Calif., 1972.

^b The optical density measurement was preceded by diluting the sample with an equal volume of cell-free medium.

7.4. Kinetics of cell inactivation and death; chlorination. Several equations have been proposed to describe The rate of cell inactivation or death. One of the earliest published equations postulated an exponential decrease over time in the viable spore count of the anthrax bacillus in the presence of 5% phenol [Chick H., Investigation of the Laws of Disinfection, J. Hyg., 8, 698 (1908)]. The corresponding inactivation equation, dN/dt = -kN, is frequently referred to as Chick's law. Three other mathematical models have also been applied:

In the last three equations, the apparent first-order rate constant varies with time.

a) Integrate each of the given equations; demonstrate what shape the corresponding graphical plots will take and how the parameters of each model can be determined from these graphs.

b) Table 7U4.1 presents data reflecting the dependence of the degree of inactivation of an E. coli population on time and chlorine concentration. Do these data conform to any of the equations given above?

Table 7U4.1. Dependence of the inactivation rate of E. coli cells on chlorine concentration and exposure time (pH 8.5; 2–5 °C); the percentage of cells retaining viability is indicated^a

^a From: Fair G. M., Geyer J. C., Okun D. A., Water and Wastewater Engineering, pp. 31–39, John Wiley and Sons, Inc., New York, 1968.

c) For a given disinfectant, the time required to kill or inactivate a specific fraction of a population is expressed by the equation (concentration)^a × (time) = const. Upon inactivation by chlorine (in the form of HOCl), the value of a is 0.86 for E. coli, 3 for Adenoviruses, 1 for poliovirus, and 2 for Coxsackie virus A. Explain why such Variability in The values of a might (or might not) have been expected.

7.5. Kinetics of cell growth and metabolite formation. Konak proposed a generalized logistic equation [An Equation for Batch Bacterial Growth, Biotech. Bioeng., 17, 271 (1975)]:

where N_∞ is the population mass in the stationary phase; a and b are constants; and N is the cell mass.

a) Show that the maximum growth rate is achieved at N/N_∞ = a / (a + b) and that its value can be determined from the equation

b) Demonstrate how you could determine each of the model parameters using the given equations and sketches of the corresponding plots.

c) Show that by substituting P for N only in the left-hand side of the equation and selecting appropriate values for a and b, one can obtain an expression describing the rate of cell product formation according to growth-associated, non-growth-associated, or mixed mechanisms corresponding to the Luedeking–Piret model reviewed in this chapter.

7.6. Cellular metabolic pathways and the kinetics of metabolite formation. a) Explain why it is often more difficult to quantify the kinetics of product formation than the kinetics of substrate utilization or biomass production.

b) Review the literature describing the biochemistry, Methods, and technology of the microbiological production of a specific antibiotic, vitamin, or amino acid. Write A brief Overview describing the stages of microbial synthesis, indicating (where data are available) the rate-limiting and equilibrium steps, as well as the presence of inhibition, activation, enzyme induction, or repression. Conclude by proposing (or discussing literature-reported) kinetics for the formation of this metabolite based on The sequence of biosynthetic reactions.

7.7. The Thiele modulus of a cell. It has been shown that the Thiele moduli of many cells and cellular Organelles (membranes, Mitochondria, Ribosomes, etc.) are equal to or slightly less than unity [Weisz P. V., Science, 179, 433 (1973)]. Explain this phenomenon from a quantitative perspective, assuming that cells can produce Enzymes over a very wide concentration range. For simplicity, assume that only a single enzyme is present in the system. In your reasoning, account for the volumetric rate, overall growth rate, and energy and other costs associated with enzyme synthesis. Note that this value of the Thiele modulus implies that the kinetics of cellular processes determined in the absence of external mass-transfer resistance (Chap. 4) will reflect the true kinetics of these processes, as was assumed throughout Chapter 7.

7.8. Dual-inhibition kinetics. It has been shown that the growth of Candida utilis on sodium acetate is inhibited by the substrate (acetate) and pH [Jackson J. V., Edwards V. H., Biotech. Bioeng., 17, 943 (1975)].

a) Express the growth rate of C. utilis in terms of the appropriate constants, hydrogen ion concentration, and total Substrate Concentration, given the following assumptions:

1. The dependence of the substrate inhibition degree on the total substrate concentration is analogous to noncompetitive substrate inhibition of enzymes and can be described by the Andrews equation [equation (7.32)].
2. The maximum growth rate μ_max is a similar function of the hydrogen ion concentration.
3. The substrate exhibits inhibitory activity only in its protonated form, i.e., the equilibrium HS (active inhibitor) ↔ S⁻ (inactive inhibitor) + H⁺ exists.

b) Show how the plot of 1/μ versus 1/s (in Lineweaver–Burk coordinates) will change as a function of pH. How can the parameters of this model be determined?

Similarly, pH affects A number of processes used in wastewater Treatment. An example of the kinetics of an anaerobic microbiological process is given in Section 14.4.6.

7.9. Monod kinetics in a CSTR. Consider an organism whose growth kinetics obey the Monod equation with μ_max = 0.5 h⁻¹ and K_s = 2 g/L.

a) At what dilution rate D will the maximum overall rate of cell mass production be achieved, assuming that the culture is at steady state, the Contents of the continuous stirred-tank Reactor are perfectly mixed, cells do not die within the culture, and S₀ = 50 g/L, Y = 1 (g cell mass / g substrate)?

b) How many equal-capacity reactors connected in series will be required to reduce the substrate concentration to 1 g/L at the value of D found in the previous exercise?

7.10. Specifics of CSTR design and analysis under cell growth inhibition.

a) Plot the dependence of x and s on D

where S₀ = 10 g/L, x₀ = 0, K_s = 1 g/L, K_i = 0.01 g/L, i = 0.05 g/L, μ_max = 0.5 h⁻¹, Y_x/s = 0.1 g cells / g substrate. On the same graph, plot the dependencies of x and s at i = 0.

b) Suppose a CSTR was designed to operate in the absence of inhibitors. Find the equations determining the ratios [s/s_i] and [xD/(xD)_i], i.e., the ratios of the model-predicted substrate concentration and biomass productivity to the corresponding observed values. How does the presence of the inhibitor affect the reactor behavior with respect to (xD)_max and washout?

c) Repeat the solution to exercises 7.10a and 7.10b assuming that i depends on x (i = x/10).

7.11. Yield Coefficient. When Modeling cell growth kinetics with endogenous METABOLISM or maintenance energy taken into account, the corresponding equation (assuming an excess of substrate) can be written as follows:

where Y' is the true growth yield coefficient (grams of cells produced per gram of substrate consumed for cell growth); k_e is the grams of substrate consumed for maintenance energy per gram of cells; Y is the apparent yield coefficient (grams of substrate consumed per gram of cell mass produced). The dependence of the yield coefficient on the dilution rate (in double reciprocal coordinates) is shown in Fig. 7U11.1.

FIG. 7U11.1. Dependence of the yield coefficient on the dilution rate in double reciprocal coordinates. [Reprinted from: Kirsh, Sykes, Prog. Ind. Microbiol., 9, 155 (1971); data adapted from: Tempest D. W., Herbert D., Phipps P. J., in Microbial Physiology and Continuous Culture, Powell E. O. et al. (eds.), HMSO, London, 1967.]

FIG. 7U12.1. Pathway and stoichiometry of citric acid Biosynthesis (AcCoA — acetyl-CoA; CIT — citrate; G6P — glucose-6-phosphate; GLU — glutamate; ICT — isocitrate; MAL — malate; OAA — oxaloacetate; OGT — a-ketoglutarate; PYR — Pyruvate; SUC — succinate; GOX — glyoxylate). [Reprinted with permission from: Aiba S., Matsuoka M., Identification of Metabolic Model: Citrate Production from Glucose by Candida lipolytica, Biotech. Bioeng., 21, 1373 (1979).]

a) Show that for a chemostat, x and s are related by

b) What equation should relate k_e, Y', and D in this case?

c) Do the growth data for A. aerogenes on a glycerol medium shown in Fig. 7U11.1 fit this model? Justify your answer quantitatively.

7.12. Stoichiometry and Rates of Metabolic Conversions. A simplified scheme for citric acid biosynthesis in Candida lipolytica is given in Fig. 7U12.1. In the scheme, symbols denote the key stoichiometric and kinetic parameters of the main reactions in the TCA and glyoxylate cycles. Here, v_i represents the specific carbon flux rates through Various metabolic pathways; ψ_car, ψ_pro, and a_i are stoichiometric coefficients; μ is the specific growth rate; v is the specific glucose uptake rate; Q_i is the formation rate of specific metabolic products. [Aiba S., Matsuoka M., Biotech. Bioeng., 21, 1373 (1979).]

a) Assuming that all metabolites are in a quasi-steady state, derive expressions relating the intrinsic rates of elementary reactions to the observed overall rates of glucose utilization, CO₂ production, citrate, and isocitrate formation.

b) Write the overall carbon mass balance equation. Does this equation depend on the expressions derived in Exercise 7.12a?

c) To assess The Importance of individual Metabolic Pathways in citrate biosynthesis, two models were proposed: one lacking The Glyoxylate cycle (v₇ = 0) and the other with a blocked TCA cycle (v₈ = 0). Using the equations derived in the two previous exercises and the experimental data from Table 1 of the paper by Aiba and Matsuoka, calculate the relative intracellular metabolic fluxes for these two models at D = 0.122 and 0.0769 h^-1. Based on your calculations, which of the two models appears to be more plausible?

7.13. Dependence of maximum productivity and washout on dilution rate. a) Using equations (7.16) and (7.18), show that the ratio D_m0/D_w0 (where m₀ is the maximum productivity and w₀ is washout) depends only on (s_f/K_s) and is given by the expression

b) Show that in the limiting cases, the preceding equation can be simplified as follows:

c) Discuss the operational Stability of the reactor and its controllability in the limiting cases described in part b.

d) Derive expressions for Y_P/Х = f(D) and the corresponding derivative df/dD applicable at the maximum productivity (pD)_mах.

7.14. Kinetics of inhibition. If Microbial growth is inhibited by a volatile metabolic product (e.g., ethanol), the cell growth rate can be enhanced by continuously evacuating the headspace above the culture broth.

a) Show that for a batch process described by equation (7.33), the time dependence of the substrate concentration for the overall reaction S→0.3 P+cell mass is expressed by the equation

where x is the biomass concentration (g/L). Assume the yield coefficient Y_x/s is constant.

b) By integrating the above equation, find the expression determining the ratio x(t)/x_i(t), where x and x_i are the biomass concentrations in the absence and presence of the inhibitor, respectively.

c) Determine the ratio x(t)/x_i(t) (see part b) at x₀ = 10^-6 g/mL, Y_X/S = 0.1 g cells/g substrate, K_s = 0.22 g/L, μ_mах = 0.408 h^-1, K_I = 16 g/L for two substrate concentrations: s₀ = 5.0 g/L and s₀ = 70 g/L (assume that the Molecular Weight of S is three times that of P).

7.15. Lag phase in a batch process. a) It has been shown that during the growth of Aerobacter aerogenes in a batch process using an ammonium sulfate medium, the duration of the lag phase (t_lаg) is approximately proportional to the inoculum volume and inversely proportional to the cell concentration in the inoculum [3]. The model proposed by Dean and Hinshelwood [3] explains this by stating that the lag phase ends when the concentration of a specific substance A inside the cell reaches a threshold value c'. These researchers also assumed that during the lag phase, the concentration c of substance A varies with time according to:

с = аV + а'n₀t + а''t       (1)

where V is the inoculum volume; a is the concentration of substance A per unit volume of the inoculum; n₀ is the number of cells per unit volume of culture (n₀ can be considered constant since the cell growth rate during the lag phase is negligible); a' is the average rate of increase in the concentration of substance A (due to its synthesis by other cells) per unit time per cell; and a" is the increase in the concentration of substance A due to intracellular synthesis. Using equation (1), determine t_lag and discuss your result in relation to the experimentally observed dependence. Analyze the assumptions made in deriving equation (1).

b) It is well known that the duration of the lag phase is affected by the physiological state of the inoculum population. Suppose the inoculum contains n₀ viable cells and n_d dead cells, and that the growth of viable cells begins immediately upon inoculation at a constant specific rate μ. Determine the dependence of μ_арр (the observed apparent specific growth rate estimated from the total cell concentration) on time. Based on your findings, discuss The Effect of inoculum viability and cell density on the apparent duration of the lag phase.

7.16. Microbial population growth. A new microorganism has been discovered in which each parent cell divides to form three daughter cells. Using the experimental data on population growth given below, calculate the average time between two consecutive cell divisions.

7.17. Kinetics of biotransformations. A solid surface-bound monolayer of cells catalyzes The conversion of substrate S into product P, with negligible cell growth. Assuming that the rate of substrate utilization per unit surface area is described by the equation

where

since P inhibits the reaction. In the above equations, s and p denote the concentrations of S and P at the surface, respectively.

The concentrations of S and P in the bulk solution far from the surface are s₀ and p₀, respectively. The Mass transfer coefficients for S and P are h_s and h_p, respectively.

Express the rate of P formation per unit surface area in terms of the observable concentrations in the solution, s₀ and p₀, given that Y_P/S moles of P are formed per mole of S consumed.

7.18. Mathematical models of cellular product formation and maintenance metabolism. It has been repeatedly suggested that the Luedeking–Piret model, which relates cellular growth to metabolic product formation [Equation (7.93)]:

r_fp = aμх + βх

and the maintenance energy model developed by Pirt [Proc. Royal Soc., Series B, 163, 224–231 (1965)]:

where Y is the observed yield coefficient based on substrate; Y' is the theoretical yield coefficient (neglecting maintenance metabolism); and m is the maintenance coefficient (g substrate/g cells per hour), are equivalent. Do you agree with this assumption? Justify your answer.

7.19. Williams’ structured model of cell growth. Following the reasoning of Williams, let us divide the biomass into two parts. The first part includes intermediates, enzymes, and other components involved in the Formation of the structural and genetic material of the cell; this is equivalent to the synthetic fraction of biomass in Williams’ model. The second part comprises the structural and genetic biomass material. Assume that the fractions of the First and Second PARTS OF THE biomass are f₁ and f₂ of the total biomass, respectively, and that the sum of f₁ and f₂ equals unity.

Let us write the model equations in their intrinsic form:

where

a is the amount of the synthetic biomass fraction formed per unit mass of the growth rate-limiting substrate utilized;

β is the amount of structural (genetic) biomass formed per unit mass of synthetic biomass utilized.

a) Consider balanced growth, in which f₁ = f₂ = 0; this state can be achieved if s remains constant for a sufficiently long time. How does p depend on s under balanced growth conditions?

b) Show that the Monod model will approximately describe the state of balanced growth if K₂ ≫ 1 and k₂ ≫ ak₁K₂. What (approximately) will be the values of f₁ and f₂ under these conditions?

c) Let x₁ = cf₁ and x₂ = cf₂, where c = x₁ + x₂. Write the differential equations for x₁, x₂, and s that describe the chemostat system. Also, write the differential equation for the population density n; assume that y [the amount of structural (genetic) material in the cell] is constant.

d) Plot lg c versus t, lg n versus t, and m (average cell size m = c/n) versus t for batch cell growth starting from two different initial values: f₁ : f₂ = 0,40 and f₁ = 0,05. In both cases, assume that c₀ = 2∙10^-3 g/L and s₀ = 5,0 g/L. Use the following constants for the plots: a = ß = 0,50; k₁ = 6,0 h^-1; k₂ = 3,0 h^-1; K₁ = 0,2 g/L; K₂ = 0,25 g/L; cell mass 1,1∙10^-13 g.

e) Using the parameters listed in Section 7.19, d, plot μ as a function of s for balanced growth. On the same graph, plot μ as a function of s According to the Monod model. Choose the constants μ_m and K_s in the Monod model such that p asymptotically approaches a constant value at high s, and the derivative of μ with respect to s at s = 0 is the same for both the Monod model and the general model.

f) Plot c, s, and m as Functions of the dilution rate for a steady-state chemostat at s_j = 5,0 g/L.

g) How will c₁, s, and m change upon a "shift-up" or "shift-down" of chemostat operating conditions from D = 0,1 h^-1 to D = 0,3 h^-1, and from D = 0,3 h^-1 to D = 0,1 h^-1? Assume that the shift is preceded by a steady state of balanced growth.

h) Discuss your calculated results in comparison with the actual behavior of bacterial populations.

References

The journals listed below publish research on the kinetics of cell population growth, alongside many Other Aspects of biochemical engineering and applied biology: Biotechnology and Bioengineering, Enzyme and Microbial Technology, Biotechnology Letters, Journal of Fermentation Technology, Journal of Chemical Technology and Biotechnology, Journal of Applied Chemistry and Biotechnology, Process Biochemistry, Trends in Biotechnology, Agricultural and Biological Chemistry, Applied Microbiology, Applied Biochemistry and Microbiology, and others. The multi-volume series Advances in Biochemical Engineering and Biotechnology (formerly Advances in Biochemical Engineering) publishes exceptional review articles that frequently discuss future Perspectives in biochemical engineering. Review and original research papers are also featured in Annual Reports on Fermentation Processes, biochemical engineering publications from the Annals of the New York Academy of Sciences, and annuals such as Progress in Industrial Microbiology and Society for General Microbiology Symposium.

1. Biochemical and Biological Engineering Science, Blakebrough N. (ed.), vol. 1, Academic Press, Inc., New York, 1967. This book provides useful reference material on bioreactor design. Chapter 6 (Luedeking R., Fermentation Process Kinetics) is particularly recommended.
2. Aiba S., Humphrey A., Millis N., Biochemical Engineering and Equipment. Moscow: Pishchevaya Promyshlennost, 1978. This advanced textbook on biochemical engineering covers batch process kinetics (Ch. 4), discusses continuous processes in detail (Ch. 5), and dedicates Chapter 9 to liquid sterilization issues.
3. Dean A. C. R., Hinshelwood C. N., Growth, Function and Regulation in Bacterial Cells, Oxford University Press, London, 1966. In addition to mathematical methods for analyzing metabolic transformations, this monograph examines extensive experimental data on other aspects of cell growth and its regulation. This book is highly recommended as supplementary reading; its only drawback is an unwarranted skepticism toward molecular biology.
4. Microbial Growth, Dawson P. S. S. (ed.), Dowden, Hutchinson and Ross, Inc., Stroudsburg, Pa., 1974. An excellent collection of classic papers (including Monod's seminal work) dedicated to The Study of microbial growth.
5. Roels J. A., Energetics and Kinetics in Biotechnology, Elsevier, Amsterdam, 1983. This monograph provides a thorough description of both kinetic models for MICROBIAL GROWTH AND product formation, as well as associated problems in chemical Thermodynamics.
6. Foundation of Biochemical Engineering: Kinetics and Thermodynamics in Biological Systems, ACS Symposium Series 207, Blanch H. W., Papoutsakis E. T., Stephanopoulos G. (eds.), American Chemical Society, Washington, D.C., 1983. A fairly comprehensive collection of review articles covering process kinetics ranging from enzymatic reactions to cell population growth.

It is very useful to consult the following review articles on the MATHEMATICAL MODELING OF cell growth:

7. Tsuchiya H. M., Fredrickson A. G., Aris R., Dynamics of Microbial Cell Populations; Adv. Chem. Eng., 6, 125 (1966).
8. Fredrickson A. G., Megee R. D., III, Tsuchiya H. M. Mathematical Models for Fermentation Processes, Adv. Appl. Microbiol., 23, 419 (1970).

The following studies examine the problems of microbial growth inhibition by metabolic products and substrates:

9. Aiba S., Skoda M., Nagatani M., Kinetics of Product Inhibition in Alcohol Fermentation, Biotech. Bioeng., 10, 845 (1968).
10. Aiba S., Skoda M., Reassessment of the Product Inhibition in Alcohol Fermentation, J. Ferment. Technol. Japan, 47, 790 (1969).
11. Andrews J. F., A Mathematical Model for the Continuous Culture of Microorganisms Utilizing Inhibitory Substrates, Biotech. Bioeng., 10, 707 (1968).

The following papers are devoted to the Growth of filamentous organisms and their metabolic product formation.

12. Metz B., Kossen N. W. F., The Growth of Molds in the Form of Pellets — A Literature Review, Biotech. Bioeng., 19, 781 (1977).
13. Megee R. D., Kinoshita S., Fredrickson A. G., Tsuchiya H. M., Differentiation and Product Formation in Molds, Biotech. Bioeng., 12, 771 (1970).
14. Matsumura M., Imanaka T., Yoshida T., Taguchi H., Modeling of Cephalosporin C Production and Its Application to Fed-batch Culture, J. Ferment. Technol. Japan, 59, 115 (1981).

Structured models of cell growth can be found in references [5–8], as well as in the following articles:

15. Fredrickson A. G., Formulation of Structured Growth Models, Biotech. Bioeng.. 18,1481 (1976).
16. Williams F. M., A Model of Cell Growth Dynamics, J. Theoret. Biol., 15, 190 (1967).
17. Harder A., Roels J. A., Application of Simple Structured Models in Bioengineering, in Advances in Biochemical Engineering, vol. 21, Fiechter A. (ed.), p. 55,Springer-Verlag,Berlin, 1982.
18. Bijkerk A. H. E., Hall R. J., A Mechanistic Model of the Aerobic Growth ofSaccharomycescerevisiae, Biotech. Bioeng., 19, 267 (1977).
19. PammentN. В., Hall R. Barford J. Р., Mathematical Modeling of Lag Phases in Microbial Growth, Biotech. Bioeng., 30, 349 (1978).
20. Shuler M. L.,DomachM. AL, Mathematical Models of the Growth of Individual Cells, in Foundations of Biochemical Engineering, Blanch H. W., Papoutsakis E. T., Stephanopoulos G. (eds.), p. 101, American Chemical Society, Washington, D.C., 1983.
21. Ramkrishna D., A Cybernetic Perspective of Microbial Growth, in Foundations of Biochemical Engineering, Blanch H.W.,Papoutsakis E. T., Stephanopoulos G. (eds.), p. 161, American Chemical Society, Washington, D.C., 1983.
22. Kompala D. S., Ramkrishna D., Tsao G. T., Cybernetic Modeling of Microbial Growth on Multiple Substrates, Biotech. Bioeng., 26, 1272 (1984).

The Kinetics of metabolic product formation are discussed in the following papers:

23. GadenЕ. L., Chem. Ind. Rev., 154 (1955), J. Biochem. Microbiol. Technol. Eng., 1 413 (1959).
24. Deindoerfer F. H., Adv. Appl. Microbiol., 2, 321 (1960).
25. OllisD. F., A Simple Batch Fermentation Model: Theme and Variations, Ann. N.Y. Acad. Sei. USA, 413, 144 (1983).
26. Pazoutova S., Votruba J., Rehdcek Z., A Mathematical Model of Growth and Alkaloid Production in the Submerged Culture of Claviseps purpurea, Biotech. Bioeng., 23, 2837 (1981).
27. Lee S. B., Bailey J. E., Analysis of Growth Rate Effect of Productivity of Recombinant EscherichiacoliPopulations, Biotech. Bioeng., 26, 66 (1984).
28. Lee S. B., Bailey J. E., Genetically Structured Models for lac Promoter — Operator Function in the EscherichiacoliChromosome and in Multicopy Plasmids: lac Operator Function, Biotech. Bioeng., 26, 1372 (1984).

Segregated kinetic models are described in the papers listed below.

29. Ramkrishna D., Statistical Models of Cell Populations, Adv. in Biochem. Eng, 11, 1 (1979).
30. Nishimura Y., Bailey J. E., On the Dynamics of Cooper — Helmstetter — Donachie Procaryote Populations, Math. Biosci., 51, 505 (1980).
31. Hjortso M. A., Bailey J. E., Steady-State Growth of Budding Yeast Populations in Well-Mixed Continuous Flow Microbial Reactors, Math. Biosci., 60, 235 (1982).
32. Shu P., Mathematical Models for the Product Accumulation in Microbial Processes, J. Biochem. Microbiol.Technol.Eng., 3, 95 (1961).

An excellent Introduction to the problem of sterilization is provided in the article:

33. Blakebrough N., Preservation of Biological Materials Especially by Heat Treatment, in Biochemical and Biological Engineering Science, vol. 2, Blakebrough N. (ed.), Academic Press, Inc., New York, 1968.

Factors affecting microbial resistance to sterilization are summarized in Chapters 20 and 21 of Frobisher's monograph (reference [2] in Chapter 1), while the issue of

phage elimination is examined in the article:

34. Hango etal, Phage Contamination and Control, in Microbial Production of Amino Acids, Kodansha Ltd., Tokyo, and John Wiley and Sons, Inc., New York, 1973.