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

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

When a small number of living Cells is placed in a solution containing all necessary nutrients at a defined Temperature and pH, the cells will grow. Depending on Cell Morphology, this growth process can occur in two ways. Unicellular organisms undergo division as they grow, so the increase in their biomass (the mass of living organisms) is accompanied by an increase in cell number. This chapter will focus primarily on this type of growth, introducing METABOLISM/2.html">THE CONCEPT OF population growth. Mold Fungi grow in an entirely different manner: as the Organism develops, it is primarily the length and number of mycelial hyphae that increase. Thus, in a growing mold culture, the size and concentration of the organisms increase, but not necessarily their total number.

Cellular growth is closely linked to two other processes: the uptake of specific compounds from the surrounding environment by The Cell, and The excretion of metabolic end products back into the medium. As we will see later, the rates of these processes vary over an extremely wide range during growth. While it is generally impossible to predict these rates a priori, extensive practical experience shows that among the many known processes involving cell cultures, only a limited number of typical pathways for substrate utilization and metabolic product formation occur with high frequency. By examining these typical pathways first, we will be better prepared to tackle new challenges in the application of growing cell cultures.

The necessary and sufficient accuracy (and consequently the complexity) of a mathematical description of a system's kinetic properties depends, in turn, on its complexity and the intended application of the mathematical expressions. Before analyzing specific processes and their corresponding kinetic equations, we must develop a clear understanding of The Diversity of reactions and processes occurring within a cell population. While our Structure/133.html">Discussion will center on cell growth, the underlying principles, concepts, and reasoning apply equally to Other Aspects of chemical processes in living cells.

Let us begin with the most complex scenario: a general analysis of a growing cell population. We will then explore which simplifications can be made in this analysis and under what conditions such simplifications are justified. Fig. 7.1 outlines some of the parameters, phenomena, and interactions that govern The kinetics of cell population growth. First and foremost, we must recognize that two interacting systems must always be considered: the biological phase, consisting of the cell population, and the surrounding phase, or medium. Cells absorb nutrients and convert substrates (derived from the medium) into metabolic products. They generate heat, while the temperature of the medium, in turn, dictates the temperature of the cells. The medium exerts mechanical effects on the cells through hydrostatic pressure or hydrodynamic forces, as well as by altering its own viscosity due to the accumulation of cell mass and metabolic byproducts.

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FIG. 7.1. Some of the key parameters, phenomena, and interactions that determine the kinetics of cell population growth.

Regarding the growth medium, one of its most critical characteristics is its multicomponent nature: first, it must supply all essential nutrients for cell growth, and second, it accumulates various metabolic end products as the cells proliferate. Chemical Reactions can also take place within the medium itself, leading to the modification of metabolic products, as seen, for example, in penicillin Hydrolysis. Cells frequently consume or produce substances that alter the medium's acidity; the interplay between cellular processes and acid-base equilibria determines the environmental pH, which in turn affects cellular activity and transport mechanisms. Throughout cellular reactions, parameters such as temperature, Ionic strength, pH, and the rheological Properties of the broth may change over time.

The culture medium is frequently a multiphase system consisting of a liquid phase with dispersed gas bubbles, or two immiscible liquid phases, and occasionally one gas and two liquid phases. Recent advances in bioreactor design include Methods for simultaneously conducting cellular reactions and Separation processes by introducing specialized agents into the medium that promote The formation of an additional liquid or solid phase. Finally, due to the large volume of the bioreactor, high viscosity, and non-Newtonian behavior of the broth, different conditions may arise locally within distinct Regions of the vessel. All of these environmental parameters and variables significantly influence the kinetics of cell growth.

As for the most important Features of the cell population itself, as we know from previous chapters, each individual cell is a complex multicomponent system that is far from homogeneous, even at the microscopic level. A multitude of chemical reactions take place simultaneously within every cell, governed by an intricate network of regulatory systems. Thanks to these systems, the cell can alter—and indeed does alter—the rates and even the pathways of its internal chemical reactions in response to environmental conditions and medium composition. Long-term cultivation of a cell population can lead to the accumulation of spontaneous Mutations; moreover, operational conditions may impose selective pressure, resulting in slow evolutionary shifts in the genetic makeup of the strain. On the other hand, a growing cell culture always exhibits substantial population heterogeneity—meaning that at any given moment, even within the smallest culture volume, individual cells vary in age (some newly divided, others mature, and some undergoing division) and consequently in biochemical activity. As we learned in the previous chapter on the Cell Cycle, cells of different ages often possess distinct metabolic Functions and activities.

Obviously, it is practically impossible to construct a kinetic model that accounts for every single parameter and factor listed in Fig. 7.1. Therefore, we will now examine a series of approximations that simplify this picture and yield mathematical expressions capable of describing cell population growth kinetics. First, regarding the medium, it is generally assumed that—with the exception of a single component—all constituents are present in such high concentrations that their depletion has virtually no impact on overall process rates. Thus, one medium component becomes the rate-limiting nutrient for cell growth, and when analyzing The Effect of medium composition on growth kinetics, we need only consider the concentration of this specific component. In some cases, it becomes necessary to account for other medium constituents, such as an accumulating inhibitor; otherwise, our kinetic description would deviate too far from reality. For other environmental parameters, it is often justifiable to assume that their fluctuations do not substantially affect Microbial growth kinetics over the course of a typical experiment or process, provided these variations remain within standard operational limits. Furthermore, bioreactor control and regulation systems can maintain various environmental parameters, such as pH, temperature, and dissolved oxygen concentration, at constant levels. Nevertheless, for an adequate description across certain kinetic regimes, it may sometimes be necessary to incorporate multicomponent effects and multiple environmental parameters into the model.

Fig. 7.2 summarizes the core principles of the various approximations and descriptive approaches that can be useful in the mathematical Analysis of the cellular phase. According to this framework, developed by Fredrickson and Tsuchiya, approaches to analyzing microbiological systems are classified based on the number of components used to describe the cells, and whether the cells are treated as a heterogeneous population of distinct entities (which they truly are) or as a population of averaged cells (in which case they differ fundamentally very little from any solute component). Multicomponent cell models are termed structured, whereas single-component models are unstructured. Describing a cell population while accounting for its heterogeneity is known as a segregated approach, whereas an unsegregated description considers only the average properties of the cells. As illustrated in Fig. 7.2, the real-world situation corresponds to a structured, segregated system. If cell heterogeneity does not significantly affect the kinetics of the processes under study, one can adopt the "average cell approximation," thereby simplifying the description from segregated to unsegregated. In the so-called "balanced growth" state, all synthetic activities of the cells are coordinated in such a way that population proliferation does not alter the average cell composition. Under these conditions, it is permissible to use models that ignore the multicomponent nature of cells. As we will see below, the analysis and description of cell population growth typically rely on the most idealized scenario: an unsegregated, unstructured model.

FIG. 7.2. Various approaches to describing cell population growth kinetics.

On the other hand, we will encounter numerous situations where it is more advantageous to treat the biological phase as a more complex system. For instance, under non-steady-state conditions (which are particularly typical of batch microbiological processes), balanced growth conditions hold true even in the first approximation only for a relatively short period of time. It is well established that during a batch process, both the cellular COMPOSITION OF THE population and the rates and types of reactions occurring within it can vary over extremely wide ranges, in which case more detailed models become useful. Furthermore, structured models make it possible to incorporate the known features of the cellular biochemical reaction network directly into mathematical expressions. Similarly, incorporating the features of the cell cycle into cell growth kinetics without major difficulties (the segregated approach) helps to enhance the value and broaden the applicability range of the model. The most crucial elements of segregated models are the kinetics and regulatory features of single-cell growth.

Living cells are extremely small systems, and therefore The amount of any chemical component within them is strictly limited. Indeed, the values listed in Table 5.6 are very small compared to the 10²³ order of magnitude typical of a chemist's molecule count. A limiting case is DNA, of which a slowly growing bacterium contains only a single molecule; clearly, in this case, the very concept of "DNA concentration within the cell" becomes highly ambiguous. The same can be said of trace ions, Organelles, and many other cellular components. Although in some of the models discussed below we will describe intracellular events as continuous, it should be recognized that such a description is merely a convenient approximation of a typical averaged cell within a cell population (see below). Reactions and mass transfer processes involving a limited number of molecules must be treated as stochastic events. To describe such events, stochastic models of cell populations have been developed, although they do not offer significant advantages over simpler deterministic models. The predictability of cell population behavior can be illustrated, for example, by evaluating the accuracy of a deterministic description for a very small population, such as a typical inoculum (usually about 10⁵ cells). Let the distribution of individual cell cycle durations be described by a normal distribution:

Let us assume and the standard deviation σ to be 0.5. We can then state that, with a confidence coefficient of 0.95, the cell cycle duration of an individual cell is

Next, let us consider a suspension containing a sufficiently large number of cells, for example, m cells. Suppose each of the m cells grows independently of the others, i.e., the cell suspension contains m independent samples. Then the 95% confidence limit for the population doubling time will be where

Consequently, as m increases, the uncertainty in the population doubling time decreases rapidly. Assuming, for example, as before, σ = 0.5 and it is easy to find that the 95% confidence limits for t for a population of m cells are

In essence, this straightforward calculation reaffirms the well-known principle that, as in any stochastic process with a relatively large number of events (such as chemical reactions or phenomena associated with fluid transport), variations in population characteristics can be predicted with a high degree of accuracy, even when the standard deviations of individual cell characteristics are large. Thus, the growth of an inoculum leading to an increase in concentration from 10⁴ to 10⁸ cells per mL obviously provides sufficient averaging across all growth stages, making it possible to determine the population doubling time with high accuracy. Similarly, one can reasonably discuss The rate of DNA Synthesis in a typical cell of the population, even though in each individual cell only one or two DNA molecules are synthesized at a rate that at any given moment may differ sharply from the average value.

It is impossible to discuss kinetics without data on the Reactor design used to measure process rates and evaluate their kinetics. In the next section, we will briefly review the respective material balances for Two Types of ideal bioreactors. Then we will proceed to analyze the kinetics of cell growth, starting from the situation schematically depicted in the upper left corner of Fig. 7.2, i.e., from the simplest models of cell growth kinetics, substrate utilization, and metabolite production. In subsequent sections of this chapter, we will examine other situations also illustrated in Fig. 7.2 and attempt to assess The Importance of various conceptual and mathematical approaches to analyzing cell population growth kinetics. In general, developing a kinetic model of a cell population is a sort of art that requires, first, keeping in mind the ultimate goal for which the model is intended; second, a carefully considered and well-substantiated choice of the primary variables and parameters affecting the most important processes; and, third, a certain conceptual and mathematical flexibility necessary to translate the qualitative CHARACTERISTICS OF THE system into the language of practical mathematical expressions.