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

Stoichiometry and Energetics of Metabolic Transformations
Stoichiometry of Cell Growth and Product Formation
General Stoichiometry of Cell Growth; Medium Composition and Yield Coefficients

During growth, Cells consume substrates that provide them with energy and the precursor molecules required for the synthesis of new cellular mass. From a general perspective, this process requires that the nutrient medium contains all the chemical elements necessary to form new Cell mass, and that the Free energy of the assimilated substrates exceeds the free energy of the resulting cells and metabolic products (Fig. 5.28)*. In other words, the free energy of the newly formed substances must be lower than that of the utilized substrates.

Additionally, nutrients must possess an appropriate molecular Structure determined by the Specificity of the cellular Enzymes involved in catabolic and biosynthetic pathways. However, when studying cell growth as a unified process (which is how we will approach it throughout most of this section), incorporating the minute details of metabolic reaction mechanisms may prove not only unnecessary but counterproductive. The fundamental concept of cell growth as a single, holistic phenomenon depicted in Fig. 5.28 is that it is subject to specific stoichiometric constraints, regardless of the precise mechanisms or metabolic pathways utilized by the cellular "system" to achieve the "reaction" goal of cell growth. Therefore, when evaluating the suitability of various nutrients, we will simply consider those substrates that are known to be appropriate for the growth of the studied cell strain.

* In this section, metabolic products refer to Organic compounds distinct from the waste products excreted by cells into the environment. Nutrients or substrates are defined as substances consumed by The Cell from the medium during growth or The formation of metabolic products.

Class="center">

FIG. 5.28. Cellular growth as a unified process. Here, the "system" denotes a specific quantity of cells. During growth, the "system" acts as a catalyst, converting substrates into new cells and metabolic products.

The simplest form of stoichiometric constraints imposed on the COMPOSITION OF THE nutrient medium can be summarized as follows: if we wish to produce a total cell mass (X) (where capital letters in parentheses denote the total mass of the component specified) and if the weight fraction of element i in the cell is w_i, then the minimum required total content of element i across all substrates must equal w_i(X). In general, such calculations are best performed on a dry cell weight basis. To perform these calculations for various elements, one must know their relative Abundance within the cell. Detailed data on the Elemental Composition of the bacterium E. coli, provided in Table 2.1, serve as a good example, while Table 5.10 lists the main elemental contents (C, N, O, and H) for a range of microorganisms. Although the elemental composition of cells varies somewhat depending on the microbial species and even the composition of the nutrient medium for the same strain, these data provide a reliable estimate of the elemental composition of the cell mass. The meaning and application of the empirical cell formulas given in this table will be discussed in the next section.

Table 5.10. Elemental composition of selected microorganisms. The symbol μ denotes the specific growth rate, defined as the mass of cells produced per unit mass of cells per unit time (see Chapter 7)^a

* Reprinted with permission from: Atkinson, B., Mavituna, F., Biochemical Engineering and Biotechnology Handbook, p. 120, Macmillan Publishers Ltd., Surrey, England, 1983.

In reality, determining the medium composition required for efficient cell growth is significantly more complex because the basic stoichiometric principle outlined above fails to account for: first, the incorporation of a portion of the substrate elements into metabolic products excreted into the medium; second, limitations imposed by reaction rates; and third, specific metabolic properties of a given cell strain caused by the inhibitory effects of certain nutrients or metabolic products. We will examine the first complication later in this chapter, and discuss the second one now. Next, we will turn to yield coefficients—simple parameters used to quantify the stoichiometry of cell growth.

Although our primary focus in this section is on stoichiometry, kinetic factors must also be taken into account when developing approaches to simplify stoichiometric expressions for cell growth and product formation. The number of nutrients in a medium is usually very large, making it practically impossible to account for every single compound in a stoichiometric or kinetic analysis. Therefore, the system is typically described in a simplified manner based on a limited number of key compounds or elements that limit the process. The stoichiometrically limiting compound can be identified using cell growth stoichiometry by determining which substrate is depleted first in the cell growth "reaction" illustrated in Fig. 5.28. Other types of limiting compounds can be identified by studying The Effect of medium composition on cell growth. For a given strain under specific conditions (Temperature, pH, presence of other nutrients), the growth-rate-limiting nutrient component can be found experimentally as follows. Suppose cells are growing in a defined medium under set conditions. The concentration of one medium component is abruptly increased in this system, and the resulting change in the Cell Growth Rate is measured. Often, the medium is formulated so that only a single component limits the cell growth rate, while relatively minor Changes in the concentrations of all other components (compared to their baseline values) have virtually no effect on the growth rate. In principle, however, situations where the cell growth rate is simultaneously determined by multiple substrates are also possible.

The growth-rate-limiting component and the stoichiometrically limiting component are not necessarily the same substance. In other words, under certain conditions, cell growth rate may be limited by one compound, whereas batch growth may cease due to the exhaustion of a completely different compound. Yet, this possibility is frequently ignored in various mathematical expressions used for the quantitative description of Microbial growth, which can lead to notable difficulties in bioreactor operations and erroneous Conclusions when interpreting their performance. Bearing these potential complications in mind, we will nevertheless assume that cell growth is limited by a single nutrient. For a specific system and defined medium, the validity of the assumption that the growth-rate-limiting and stoichiometrically limiting components are identical can always be verified experimentally.

It is frequently observed that the total amount of cell mass produced during cell growth is directly proportional to the mass of the substrate consumed by the cells (typically a carbon-containing precursor, energy source, or oxygen) (Fig. 5.29). This proportional relationship is termed the yield coefficient:

It should be noted that the dimensions of the yield coefficient depend on how the quantities of cell mass and substrate are expressed (e.g., in units of mass or moles); this fact must always be taken into account when using literature values of Y. When multiple different substrates S_i are consumed simultaneously from the medium, several corresponding cell growth yield coefficients Y_x/si can be defined. Table 5.11 lists the yield coefficients for the growth of various microorganisms in media with different carbon sources; here, figures in the same row correspond to different Methods of measuring The amount of consumed substrate (in grams, moles, or grams of carbon per mole of substrate).

FIG. 5.29. Relationship between total biomass produced and the amount of substrate consumed during the aerobic growth of Aerobacter aerogenes Bacteria (a) and anaerobic growth of Propionibacterium pentosaceum bacteria (b) on various carbon sources. [Reprinted with permission from: a) Hadjipetrou, L. P., Gerrits, J. P., Tenlings, F. A. G., Stouthamer, A. H., Relation between Energy Production and Growth of Aerobacter aerogenes, J. Gen. Microbiol., 36, 139 (1964); b) Bauchop, T., Elsden, S. R., Growth of Microorganisms in Relation to their Energy Supply, J. Gen. Microbiol., 23, 457 (1960).]

If the yield coefficient remains approximately constant for a given cell culture in a given nutrient medium, equation (5.46) allows us to determine The change in cell mass concentration from the change in Substrate Concentration, and vice versa. In other words, one of these variables can be expressed in terms of the other, allowing one variable to be omitted during bioreactor design calculations. This stoichiometric relationship can also be useful for monitoring the operation of biological reactors; in this case, the substrate concentration can simultaneously serve as a measure of cell mass concentration, provided that both corresponding initial values are known. This is extremely convenient because substrate concentrations are typically much easier to measure than biomass concentrations. In practice, it is most convenient to measure the concentrations of substances exiting the aerobic bioreactor off-gas, such as CO₂. In this case, stoichiometric relationships are also used as a basis for correlating various biotechnological process variables. However, before examining such relationships, it is necessary to discuss the Variability of the yield coefficient and METABOLISM/2.html">THE CONCEPT OF maintenance metabolism.

There is no guarantee that an empirically determined yield coefficient, which represents an apparent stoichiometric ratio, remains constant for a given Organism in a given medium. Indeed, experiments have shown that the yield coefficient frequently varies with changes in the cell growth rate. To understand the reasons for these variations, it is sometimes helpful to divide substrate utilization into three parts: conversion of substrate into cell mass (assimilation), generation of energy for cell synthesis, and provision of energy for maintenance. Maintenance energy here refers to the Energy Expenditure required for cell survival or maintenance in a (potentially) viable state, independent of cell growth and new Cell Formation. This energy is consumed in The Active Transport of ions and other substances across cell membranes and in the turnover of degrading cellular constituents.

Table 5.11. Yield coefficients for the aerobic growth of selected microorganisms on various carbon sources. Here, the symbol Y_X/O2 denotes the coefficient expressed in grams of cell mass per gram of O₂ assimilated^a

^a Reprinted with permission from: Nagai, S., Mass and Energy Balances for Microbial Growth Kinetics, in Advances in Biochemical Engineering, vol. 11, Ghose, T. K., Fiechter, A., Blakebrough, N. (eds.), Springer-Verlag, New York, p. 53, 1979.

In the case of chemoheterotrophs, the same substrate serves as both the energy and carbon source, so total substrate utilization can be expressed by the following equation:

Dividing this equation by (∆X), we obtain

The true economic yield coefficient of cell growth, $\Delta X / (\Delta S)_{\text{assimilation}}$, remains relatively constant and represents a stoichiometrically well-defined value. However, this cannot be said of the overall yield coefficient $Y_{x/s}$, which is due to the dependence of the various $(\Delta S)$ terms on the right-hand side of equation (5.48) on growth conditions. Thus, in a rapidly growing cell population, the bulk of the substrate is utilized for assimilation and growth, whereas in a stationary or dormant cell population, the substrate is often consumed exclusively for maintenance (in which case $Y_{X/S} = 0$).

In the next section, we will examine other, more rigorous stoichiometric expressions for cell growth that account for carbon, hydrogen, nitrogen, and oxygen mass balance equations. In addition, we will explore a more detailed approach to Modeling cell growth, wherein growth is still treated as a unitary process while simultaneously incorporating the pivotal role of NADH and ATP in growth metabolism.