9.4.3 Stoichiometry

The stoichiometry for cell growth is very complex and varies with microorganism/nutrient system and environmental conditions such as pH, temperature, and redox potential. This complexity is especially true when more than one nutrient contributes to cell growth, as is usually the case. We shall focus our discussion on a simplified version for cell growth, one that is limited by only one nutrient in the medium. In general, we have

In order to relate the substrate consumed, new cells formed, and product generated, we introduce the yield coefficients. The yield coefficient for cells and substrate is

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A representative value of Y_c/s might be 0.4 (g/g).

The reciprocal of Y_c/s, i.e., Y_s/c,

gives the ratio of –ΔC_s (the substrate that must be consumed to increase the cell concentration of ΔC_c) to the increase in cell concentration ΔC_c.

Product formation can take place during different phases of the cell growth cycle. When product formation only occurs during the exponential growth phase, the rate of product formation is

9-63

where

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The product of Y_p/c and μ—that is, (q_P = Y_p/c μ)—is often called the specific rate of product formation, q_P, (mass product/volume/time). When the product is formed during the stationary phase where no cell growth occurs, we can relate the rate of product formation to substrate consumption by

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The substrate in this case is usually a secondary nutrient, which we discuss in more detail later when the stationary phase is discussed.

The stoichiometric yield coefficient that relates the amount of product formed per mass of substrate consumed is

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In addition to consuming substrate to produce new cells, part of the substrate must be used just to maintain a cell’s daily activities. The corresponding maintenance utilization term is

A typical value is

The rate of substrate consumption for maintenance, r_sm, whether or not the cells are growing is

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When maintenance can be neglected, we can relate the concentration of new cells formed to the amount of substrate consumed by the equation

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This equation can be used for both batch and continuous flow reactors.

If it is possible to sort out the substrate (S) that is consumed in the presence of cells to form new cells (C) from the substrate that is consumed to form product (P), that is,

the yield coefficients can be written as

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9-69B

These yield coefficients will be discussed further in the substrate utilization section.

Substrate Utilization

We now come to the task of relating the rate of nutrient (i..e., substrate) consumption, –r_s, to the rates of cell growth, product generation, and cell maintenance. In general, we can write

In a number of cases extra attention must be paid to the substrate balance. If product is produced during the growth phase, it may not be possible to separate out the amount of substrate consumed for cell growth (i.e., produce more cells) from that consumed to produce the product. Under these circumstances, all the substrate consumed for growth and for product formation is lumped into a single stoichiometric yield coefficient, Y_s/c, and the rate of substrate disappearance is

9-70

The corresponding rate of product formation is

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The Stationary Phase

Because there is no growth during the stationary phase, it is clear that Equation (9-70) cannot be used to account for substrate consumption, nor can the rate of product formation be related to the growth rate [e.g., Equation (9-63)]. Many antibiotics, such as penicillin, are produced in the stationary phase. In this phase, the nutrient required for growth becomes virtually exhausted, and a different nutrient, called the secondary nutrient, is used for cell maintenance and to produce the desired product. Usually, the rate law for product formation during the stationary phase is similar in form to the Monod equation, that is,

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The net rate of secondary nutrient consumption, r_sn, during the stationary phase is

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Because the desired product can be produced when there is no cell growth, it is always best to relate the product concentration to the change in secondary nutrient concentration. For a batch system the concentration of product, C_p, formed after a time t in the stationary phase can be related to the secondary nutrient concentration, C_sn, at that time.

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We have considered two limiting situations for relating substrate consumption to cell growth and product formation; product formation only during the growth phase and product formation only during the stationary phase. An example where neither of these situations applies is fermentation using lactobacillus, where lactic acid is produced during both the logarithmic growth and stationary phase.

The specific rate of product formation is often given in terms of the Luedeking–Piret equation, which has two parameters α (growth) and β (nongrowth)

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with

r_p = q_pC_c

The assumption here in using the β-parameter is that the secondary nutrient is in excess.

Example 9-4. Estimate the Yield Coefficients

Analysis: We first used the data in Table E9-4.1 to calculate the yield coefficient Y_s/c, Y_c/s, Y_s/p, Y_p/s, and Y_p/c. Next, we used nonlinear regression to find the Monod rate law parameters μ_max and K_s.