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

Kinetics of substrate utilization, metabolic product formation, and biomass generation in cell cultures
Transient-state cell growth kinetics
Main phases of cell growth in batch reactors

During specific periods of Cell culture growth in batch reactors, or following perturbations in continuous-flow chemostats, cell populations enter a transient state whose kinetics can be highly complex. Describing various patterns of transient cell growth may require Different types of mathematical growth kinetics models. Here, we examine several Examples of transient cell growth kinetics, focusing primarily on the most general case of population growth in a batch Reactor. Other Aspects of transient cell growth kinetics will be discussed when we explore bioreactor dynamics in Chapter 9.2.

As illustrated in Fig. 7.13, the Number of viable Cells in a typical batch process changes over time. Following the lag phase (latency phase), during which the population size remains virtually unchanged, a period of rapid growth begins, with cell numbers increasing exponentially over time. This stage of cell culture growth in a batch reactor is often called the logarithmic (or log) phase; however, we prefer the more precise term exponential growth, which we will use hereafter. Obviously, cells cannot multiply indefinitely in a closed vessel, and the exponential growth phase is eventually followed by a stationary phase, at which point The Cell count reaches its maximum. Finally, the stationary phase gives way to the death phase, during which the number of cells begins to decline. This phase also frequently exhibits an exponential decrease in the viable population.

Class="center">

FIG. 7.13. Growth curve of a culture in a batch reactor.

Any of these phases can play a crucial role in microbiological processes. For instance, process development objectives might include minimizing the duration of the lag phase while maximizing the length of the exponential growth phase (along with the Cell Growth Rate during that phase); the latter objective is achieved in part by artificially delaying the onset of the stationary phase. Toward the end of the process, achieving maximum cell population density is often vital, which is only possible by accounting for all variables affecting the culture growth phases. In the following Structure/133.html">Discussion, we will examine each phase individually and introduce mathematical models for The most significant phenomena at appropriate points. This discussion will continue in Section 7.3.2, where we consider mathematical models for overall cell growth in a batch reactor.

The duration of the lag phase following inoculation into fresh medium depends both on changes in nutrient composition (if any) and on the age and biomass of the inoculum. Transferring cells from one medium to another involves a drastic change in conditions that can affect living cells in various ways. First, as we already know, enzyme activity control and regulation systems incorporate adaptive mechanisms; specifically, when confronted with a novel nutrient, a cell will begin to assimilate it only after synthesizing new Enzymes. Therefore, transferring an exponentially growing culture from a glucose medium to a lactose medium is always accompanied by a period of low Cell Division rates, during which enzymes and Cofactors involved in lactose METABOLISM are synthesized within the cells. (What would happen if a culture grown on lactose were transferred into a glucose-lactose medium?) Similarly, a lag phase may occur following changes in nutrient concentrations. If the new medium is richer in the growth-limiting nutrient, that nutrient will initially be consumed not for cell growth, but to increase the concentrations of metabolizing enzymes. Conversely, a decrease in nutrient concentration may not trigger a lag phase; in this case, exponential growth resumes immediately upon the concentration drop, albeit at a reduced cell growth rate.

The activation of many intracellular enzymes requires specific low-molecular-weight compounds (Vitamins, cofactors) or Metal Ions (activators) capable of penetrating cell membranes. Transferring a small volume of a cell culture or inoculum into a substantially larger volume of medium will cause the outward diffusion of these essential catalytic substances if they are absent from the fresh medium or if the medium differs sharply in Ionic strength from the inoculum. In such cases, the growth rate will drop in accordance with the lowered concentration of intracellular activators, which in turn induces a lag phase while new synthesis mechanisms for these activators are "switched on." If concentrations of essential activators (ions and vitamins that the cell cannot synthesize) decline during these processes, the overall level of cellular activity may decrease irreversibly.

The length of the lag phase is heavily influenced by the growth phase of the inoculum, which in turn is obtained by cultivating cells in a small batch reactor. The volume of the inoculum is also a critical variable; as mentioned earlier, it affects the extent to which substances such as vitamins and activators diffuse into the medium. For example, transferring a population of young cells into a medium rich in intermediates (such as Amino Acids) proceeds without a lag phase; conversely, when transferring the same inoculum into a medium containing ammonium sulfate, the cells will become depleted of vital intermediates due to outward transport into the solution. If the culture is in the exponential growth phase at the time of transfer, the inoculum medium should already contain significant concentrations of metabolic intermediates, thereby minimizing the diluting effect of the transfer. Transferring an older culture (in or near the stationary phase) to an ammonium sulfate medium will result in a significantly longer lag phase.

FIG. 7.14. When the medium contains equal initial concentrations of glucose and xylose, diauxic, two-phase growth of an E. coli culture is observed in a batch reactor. (Reprinted with permission from Stanier, R. Y., Doudoroff, M., Adelberg, E. A., The Microbial World, 3rd ed., p. 308, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1970.)

When a medium contains multiple carbon sources, a series of sequential lag phases can sometimes be observed (Fig. 7.14). This phenomenon, known as diauxie (diphasic growth), is caused by shifts in metabolic pathways during growth. In diauxism, cells preferentially consume one carbon source first; once that nutrient is exhausted, they must redirect their activity from growth to the "retooling" required to assimilate the new carbon source. This sequential nutrient utilization is likely rooted in The phenomenon of catabolite repression discussed in Section 6.1.4.

Attempts to minimize culture growth time and overall process duration have typically focused on reducing the length of the lag phase characteristic of microbial batch processes. Based on the material we have already covered and other data (described in the literature cited at the end of the chapter), the following General Principles emerge:

1. The inoculum culture should exhibit maximum metabolic activity and be in the exponential growth phase at the time of inoculation.

2. The medium used to grow the inoculum should be chemically similar in composition to the large-scale microbiological process medium.

3. To prevent diffusion-driven losses of essential intermediates and activators, it is advisable to use a larger inoculum volume (up to 5–10% of the fresh medium volume).

By the end of the lag phase, the microbial population has adapted to the new conditions. The cells can now multiply rapidly, and the cell mass (or viable cell count) doubles at regular time intervals. The increase in cell number during this period can be described by the equation

where

x = x₀ at t = t_1as      (7.356)

Thus, The rate of increase of x is proportional to x. Integrating equation (7.35), we obtain

From this, it follows that the time required for the population to double is

As with steady-state cell growth in chemostats, characterizing a population during exponential growth in a batch process requires only a single parameter, μ (or t_d). As a first approximation, batch culture growth during this phase can be considered balanced. Consequently, studying cell culture growth in a batch reactor yields valuable data on balanced growth kinetics, provided the investigation is restricted to the exponential growth phase.

Exponential growth ends when a critical process variable (such as the concentration of a nutrient or a toxic byproduct) reaches a level that no longer Supports the maximum specific growth rate. The depletion of a growth-limiting nutrient can trigger a sharp drop in the cell growth rate, since the overall rate of nutrient utilization peaks rapidly during the exponential phase. To derive an approximate mathematical expression for this phenomenon, let us assume that prior to the onset of the stationary phase, the consumption rate of nutrient A is proportional to the viable cell mass concentration:

Let us assume further that exponential growth continues until a stationary phase is established, and that exponential growth starts at time zero. Then

where x₀ is the mass concentration of living cells at the onset of exponential growth.

If at time zero the concentration of A is a₀, it follows from equations (7.38) and (7.39) that A will be completely utilized when

Here x_s is the mass concentration of the population at the moment when substance A is depleted and the population transitions into the stationary phase. Consequently, x_s represents the maximum biomass concentration in a batch process (cf. Fig. 7.13). By transforming equation (7.40), one can obtain an equation expressing the maximum biomass concentration at the time of nutrient depletion:

The linear dependence of x_s on the initial nutrient concentration a₀ has been observed experimentally in many cases (x₀ is often so small that it can be neglected). An example of such cell population behavior is shown in Fig. 7.15. At the same time, the plot of n_s (the cell number density in the stationary phase) for the bacterium A. aerogenes versus lactose concentration deviates noticeably from the dependence predicted by equation (7.41). Obviously, the onset of the stationary phase and the maximum biomass concentration can also be influenced by other factors.

FIG. 7.15. Dependence of the maximum biomass concentration in a batch process on the initial concentration of the growth-limiting nutrient: a — Pseudomonas sp. in a fructose medium; b — A. aerogenes in a lactose medium; c — A. aerogenes in an ammonium tartrate medium. (Reprinted with permission from the publisher from: a) Stanier R. Y., Doudoroff M., Adelberg E. A., The Microbial World, 3rd ed., p. 313, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1970; b, c) Dean A. C. R., Hinshelwood C., Growth, Function and Regulation in Bacterial Cells, p. 72, Oxford University Press, London, 1956.)

If a toxin accumulates in the culture that inhibits the cell growth rate (compared to growth in the exponential phase), an equation of the following type may prove useful in describing The behavior of such a system (c_t is the toxin concentration):

In the special case of a linear dependence of the growth rate on c_t, this equation takes the form:

Here b is a constant. It is logical to assume that the rate of toxin production depends only on x and is proportional to x:

Therefore, (if c_t = 0 at t = 0), and the cell growth equation is transformed as follows:

The magnitude of the instantaneous effective specific growth rate μ_eff is equal to

This value decreases with time at an increasing rate, such that

Cell growth ceases when

According to equation (7.42), cell growth ceases only when c_t reaches a value equal to 1/b. Dilution of the given medium containing the toxin, or the Introduction of a non-nutritive substance that binds this toxin, should be accompanied by the resumption of cell growth and, consequently, an increase in the maximum biomass concentration x_s in the stationary phase. If growth is restrained by nutrient depletion, dilution with a nutrient-free solution does not affect x_s. These relationships can serve as criteria for a preliminary Assessment of the causes behind the slowdown and cessation of cell growth. More precise criteria are harder to formulate because, as we will see later, population growth under conditions of primary nutrient depletion slows down somewhat before the nutrient is fully utilized, whereas the cell growth rate in the presence of a toxin can become immeasurably small long before dx/dt reaches zero.

FIG. 7.16. Relationship between the initial nutrient concentration and the maximum biomass concentration in batch culture. At high nutrient concentrations, when they do not determine the maximum biomass concentration, the accumulation of toxic substances may become the concentration-limiting factor.

Fig. 7.16 illustrates the theoretically predicted dependence of the maximum biomass concentration on the initial concentration of a given nutrient. A gradual decrease in nutrient concentration eventually leads to a linear dependence of the maximum biomass concentration on the initial nutrient concentration. In this case, the absence of the exponential growth phase is obviously caused by nutrient limitation. Conversely, increasing the initial nutrient concentration may eventually cause x_s to become independent of a₀; this is presumably explained by the accumulation of toxic substances or The Influence of another cell-growth-limiting nutrient.

When studying Selection/30.html">The population as a whole, we must not lose sight of The Fate of individual cells. In general, populations are always heterogeneous, and the population growth curve in a batch process merely reflects a certain averaged parameter of an extremely complex system. During the exponential growth phase, for example, some cells divide to produce new, young cells, while others at the very same time grow and mature. Since cells of different ages typically vary in size and chemical composition, we can treat Cells of the same age as a distinct "substance". From this perspective, the growth of a single microorganism species results in a population containing a multitude of different "substances".

Population heterogeneity becomes particularly pronounced in the stationary and death phases. Thus, in the stationary phase, some cells divide while others die. Dead cells frequently undergo lysis (breakdown and dissolution); As a result, CARBOHYDRATES, amino acids, and other cellular components are released into the medium and serve as nutrients for the living members of the population. This cannibalism helps sustain the population mass during the stationary phase. Eventually, however, nutrient depletion and toxin accumulation prevent the population from sustaining itself any further, and the death phase ensues.

Relatively few studies have been devoted to the death phase, perhaps because many industrial microbiological batch processes terminate long before this phase is reached. It is generally accepted that population death follows an exponential law:

In this equation, t represents the time elapsed since the onset of the death phase. Equation (7.48) is based on the assumption that the same fraction of living cells dies at any given time.

One explanation for the exponential decrease in population size during the death phase is that cell death in culture is a random event, and the demise of a given cell is determined solely by the probability of this event. The drawback of this explanation is that it ignores the population's history. Dean and Hinshelwood suggested that in stable and dying populations, not only do living cells subsist on dead ones, but competing PARTS OF THE cell's metabolic machinery also struggle for scarce intermediate compounds at each other's expense [3]. If this assumption is accepted, it can be shown that the cell death rate must follow an exponential law. Other models of population decline will be discussed in Section 7.7.

To better understand the patterns of cell population growth in a batch process and to lay the groundwork for deriving the corresponding mathematical expressions, it is advisable to examine a series of experimental results concerning changes in population mass and chemical composition during batch processes. It should be emphasized once again that these results pertain not to an individual cell, but to the aggregate of numerous cells comprising the population at a given moment.

Dean, Hinshelwood, and other researchers have dedicated numerous studies to Changes in the composition of *A. aerogenes* cultures. In particular, they demonstrated that in a glucose-containing medium, the lag phase is reduced to a minimum if the inoculum is taken from a population in the stationary phase that followed glucose depletion. The obtained data are shown in Fig. 7.17. Note that during the Cytology/cytology/16.html">Early stages of the growth cycle, the curves for the average cell mass and the RNA/DNA ratio exhibit sharp maxima. Although The amount of DNA per unit of cell mass and The ratio of protein mass to RNA mass remain relatively constant, these data, combined with other information, make it easy to reconstruct the picture of heightened metabolic activity aimed at utilizing increased amounts of nutrients in the new medium and synthesizing large quantities of diffusing intermediate metabolites.

Fig. 7.18 presents data illustrating another interesting phenomenon, which we will revisit when examining the formation rates of cellular end products. The average RNA concentration in the cell increases in proportion to the population growth rate. It should be emphasized that this relationship is observed only when changes in the growth rate are driven by variations in the COMPOSITION OF THE nutrient medium. If the cell growth rate is altered, for example, by lowering or raising the Temperature, the RNA concentration during the exponential growth phase appears to remain virtually constant.

In Conclusion of this section, we will examine in somewhat greater detail The properties of cell populations and The Nature of enzyme activity changes throughout the Cell Cycle. During the exponential growth phase of *A. aerogenes*, the concentrations of two enzymes with Hydrogenase activity change very little, whereas the specific activity of asparagine deaminase varies over a quite wide range. The initial drop in this enzyme's activity is attributed to inoculum dilution, while the activity increase at the end of the cell cycle is presumably due to a decrease in the pH of the medium at this stage of the process.

FIG. 7.17. Changes in cell characteristics during the growth of *A. aerogenes* in a batch process. (Reprinted by permission from: Dean, A. C. R., Hinshelwood, C., Growth, Function and Regulation in Bacterial Cells, pp. 87–89, Oxford University Press, London, 1966.)

This fact once again demonstrates that a batch cell culture should be viewed as a complex unified system consisting of a cell population and a liquid medium of variable composition. In a sense, population growth in a batch process can be considered a function of the initial characteristics of both the cells and the medium; therefore, any population parameters after the process has begun depend on each of these two phases. Thus, in the general case, it is somewhat unjustified to consider the Properties of the cell population alone in a batch growth process, since these properties are determined by the interaction between the medium and the population.

FIG. 7.18. Dependence of RNA content on the growth rate of *A. aerogenes* for various carbon sources. (Reprinted from: Dean, A. C. R., Hinshelwood, C., Growth, Function and Regulation in Bacterial Cells, p. 92, Oxford University Press, London, 1966.)