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

Kinetics of substrate utilization, metabolic product formation, and biomass generation in cell cultures
Kinetics of balanced growth
Monod equation for cell growth kinetics

The relationship shown in Fig. 7.4 is characteristic only of cases where the specific Cell Growth Rate is independent of x and s. [Here, s refers to the mass concentration of the substrate that limits cell growth.] The system becomes determinate if cell growth rate is limited by a certain nutrient.

Before discussing the details of how growth rate depends on nutrient supply, let us examine the general theoretical and practical approaches to formulating cell culture media. Depending on their composition, media are divided into two types. Synthetic media are defined as those with a precisely known chemical composition. As shown in Table 7.1, such media are prepared from an inorganic salt solution supplemented with the required carbon, nitrogen, and Energy Sources, as well as all essential Vitamins. Inorganic salts not only provide ions necessary for normal Cell Functioning but also act as a buffer, mitigating large pH fluctuations during population growth. Complex media contain Materials of undetermined composition. For instance, medium 4 in Table 7.1 is classified as complex because the exact Chemical composition of the Yeast extract is unknown. Other complex media include meat broth, Blood broth, corn steep liquor, and wastewater.

Class="center">Table 7.1. Examples of Synthetic and Complex Media^a

^a Stanier R. Y., Doudoroff M., Adelberg E. A., The Microbial World, 3rd ed., p. 79. Prentice-Hall, Inc., Englewood Cliffs, N. J., 1970.

^b Synthetic.

^c Complex.

^g If the medium is autoclaved, glucose should be sterilized separately and added aseptically. When heated in the presence of other substances—especially phosphates—sugars undergo partial decomposition, yielding compounds that are highly toxic to A number of microorganisms.

The primary requirement for a growth medium is to ensure a high rate of cell growth and/or product synthesis. However, it does not follow from this (as one might purely intuitively assume) that all nutrients should be supplied in large excess. First, an excessive nutrient concentration can inhibit cell growth or even lead to cell death. Second, if Cells grow too rapidly, accumulating metabolic end products may disrupt normal cellular biochemical processes. Therefore, overall cell growth is typically limited by restricting The amount of a single nutrient in the medium.

FIG. 7.5. Specific growth rate of E. coli as a function of growth-limiting nutrient concentration in a glucose medium (a) and a Tryptophan medium (for a tryptophan-requiring mutant) (b). (Reprinted by permission of the publisher from: Stanier R. Y., Doudoroff M., Adelberg E. A., The Microbial World, 3rd ed., p. 315, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1970.)

If the concentration of one essential medium component is varied while the concentrations of all other components remain constant, the dependence of cell growth rate on the concentration of that essential nutrient typically exhibits a hyperbolic curve, as shown in Fig. 7.5. The mathematical expression reflecting the functional dependence of the specific cell growth rate, μ, on the concentration of an essential nutrient was proposed by Monod in 1942. The Monod equation has the same form as the Langmuir adsorption isotherm (1918) and the conventional single-substrate enzyme-catalyzed reaction rate equation proposed by Henri (1902) and Michaelis and Menten (1913), establishing that

In this equation, μ_max is the maximum growth rate achieved at s ≫ K_s and constant concentrations of all other essential nutrients. K_s is the concentration of the growth-limiting nutrient at which the specific growth rate is half of its maximum value. As a first approximation, K_s can be considered the intermediate zone between the range of low concentrations where μ depends linearly on s and the range of high concentrations where μ becomes independent of s. As shown in Fig. 7.5, the K_s values for E. coli strains growing in glucose- and tryptophan-limited media are 0.22 × 10^-4 M and 1.1 ng/mL, respectively.

Given our familiarity with cell biochemistry, it should be obvious that the Monod equation is likely an oversimplification. However, as is often the case in other areas of technology, a relatively simple equation can satisfactorily describe certain relationships even when the physical meaning of the model parameters is unknown or perhaps even nonexistent. Nevertheless, in some specific situations, the Monod equation can be assigned a definite physical meaning. One of the most illustrative situations of this type occurs when cell growth rate is limited by The rate of membrane transport mediated by permeases (recall Section 5.7).

The appeal of the Monod equation (7.10) lies in its simplicity; however, applying this simple expression requires great caution. First, the value of K_s is often quite small, so s is usually much larger than K_s, and the expression s/(K_s + s) can be regarded as adequate for calculating deviations of μ from μ_max as the concentration s decreases. This relationship also implies that the specific growth rate μ is not zero at any non-zero concentration of the growth-rate-limiting component. In general, this condition cannot be considered proven for s ≪ K_s.

If the growth rate of a cell population is related to the concentration of the growth-limiting nutrient by a mathematical expression resembling the Monod equation, it follows that specific relationships must exist between bioreactor operating conditions and the kinetic and stoichiometric parameters of The Cell population. To uncover these relationships, we can start with the mass balance equation for the growth-limiting substrate, coupled with the cell mass balance, since μ depends on s. In the substrate balance, we use the yield coefficient Y_x/s (see Section 5.10.1):

Then, the steady-state substrate mass balance equation can be written in the following form:

Substituting μ from equation (7.10) into this expression, we obtain

In this case, the cell mass balance equation will be as follows:

Equations (7.12) and (7.13) are often referred to as the Monod chemostat model.

If, as is usually the case, the feed nutrients are sterile (x_f = 0), the corresponding expressions for x and s can easily be found by rearranging these equations:

Equations (7.14) and (7.15) describe the steady-state dependence of x and s on the dilution rate (D = F/V). At very low flow rates for a given volume, D→0 and, consequently, s also approaches zero. Since almost all the substrate introduced into the Reactor is consumed by the cells, the cell mass concentration in the reactor effluent will be equal to s_fY_х/s.

As D is gradually increased, s also rises—initially in proportion to D, and then, as D approaches μ_mах, at an even faster rate. In the exact same manner, the cell mass concentration decreases, first linearly with D, and then more rapidly as D→μ_mах. As D approaches μ_mах, x eventually drops to zero; this means that the dilution rate D has just exceeded the maximum possible growth rate, and a steady-state solution to equation (7.14) is possible only when x = 0. The loss of all cells under steady-state conditions, known as washout, occurs when D exceeds D_mах; the latter parameter can be determined from equation (7.14) by Setting x = 0:

FIG. 7.6. Dependence of effluent Substrate Concentration s, cell mass concentration x, and cell production rate D_x on the continuous culture dilution rate D, calculated using the Monod model for a chemostat (μ_mах = 1 h^-1, K_s = 0.2 g/L, Y_х/s = 0.5, s_f = 10 g/L).

Figure 7.6 shows the plots of substrate and cell mass concentrations versus D, as described by equations (7.14) and (7.15) for the following set of parameter values:

Note that near the washout point, the reaction mixture becomes extremely sensitive to fluctuations in D; even a slight change in D leads to a relatively large variation in x and/or s.

This relationship must be kept in mind whenever the goal of a continuous microbial process is biomass production. The rate of cell production per unit volume of the reactor is given by Dx; the curve of Dx versus D (Fig. 7.6) exhibits a sharp maximum. The maximum cell production rate can be calculated by solving the differential equation

in which equation (7.14) is used to express x as a function of D. Solving equation (7.17) yields

If, as is often the case in practice, s_f ≫ K_s, then D_max.prod. approaches μ_mах and, consequently, lies close to the washout point. In such situations, illustrated in particular in Fig. 7.6, it is generally advisable to avoid aiming for the maximum biomass production rate in order to steer clear of the zone of highest sensitivity. In general, when tackling any optimization problem in search of relatively easily controllable parameters, It is important not to overlook the practical aspects of balancing sensitivity, controllability, and reliability.

When studying the patterns of end-product formation in continuous microbial processes, we introduce another yield coefficient, Y_р/х, defined as the ratio

Having examined the stoichiometry of metabolic product formation in Chapter 5, we already know that Y_р/х is constant in Type I microbial processes, but varies in others. Applying this coefficient allows us to write the steady-state material balance equation for a cellular metabolic product in the form

From this it follows that

Together with the equations given above that express the dependence of μ and x on process parameters, equation (7.21) makes it possible to calculate the product concentration in the reactor effluent. The reactor productivity with respect to the product is equal to pD and, for a constant Y_p/x, reaches a maximum when D has the value determined by equation (7.18). Therefore, when attempting to maximize process productivity, the sensitivity factor must also be taken into account. [At what value of D is pD maximized if Y_p/x = f(D)?]

FIG. 7.7. Experimentally observed growth parameters of a continuous culture (A. aerogenes) show qualitative agreement with the Monod model. (Reprinted from: Herbert D., A Theoretical Analysis of Continuous Culture Systems, in Society of Chemical Industries Monograph 12, p. 247, London, 1961.)

As an example, Fig. 7.7 presents the results of an experimental study on the growth of the bacterium Aerobacter aerogenes in a continuous flow reactor. The experimental data and the simple mathematical model described above show quite good qualitative agreement. At the same time, the observed cell mass production rate and substrate concentration remain approximately constant over a significantly broader range of conditions than predicted by the Monod equation calculations (Fig. 7.6). Furthermore, the Monod model fails to match experimental data at very low and very high dilution rates. In the next section, we will examine these two extreme regimes and try to understand why the Monod model is inapplicable in such cases.