**4.5 Product Formation Kinetics**

Product formation kinetics fall into one of the following three types.

- Growth Associated Product Formation
- Non-Growth Associated Product Formation
- Mixed Mode Product Formation

**Figure 4-5a **Growth Associated Product Formation

**Figure 4-5b **Non**-**Growth Associated Product
Formation

**Figure 4-5c **Mixed Mode Product Formation

Typical time-profiles of these three cases are illustrated above. In Type I shown in Fig 4-5a, product is formed simultaneously with growth of cells. That is product concentration increases with cell concentration. The metabolic quotient for P can be expressed as a function of µ,

It is clear from the above, the proportionality constant, a is the yield
coefficient, YP/X. Anaerobic fermentation of sugars
by *Saccharomyces cerevisae* is an example of Type I. Illustrated below
are actual data for this bioprocess.

**Figure 4-5 **Ethanol Fermentation data for yeast illustrates
Type I product formation kinetics. Note that formation of alcohol is proportional
to cell concentration.

In Type II, product formation is unrelated to growth rate, but is a function of cell concentration. This is expressed as

Antibody formation by hybridoma, and some antibiotic fermentation exhibit this type of behavior.

In the third category, product formation is a combination of growth rate and cell concentration. That is,

Many biochemical processes fall into this category. Note that if b is
zero and a is Y_{P/X}, this case reduces to Type I. If a = 0, it
reduces to non-growth associated case. Therefore let us consider this more
general case for further analysis.

In a batch reactor, product accumulation can be obtained by carrying out mass balance on the product.

Rate of Product Formation = Accumulation of Product

For constant V,

If we consider exponential phase only, X = X_{0} Exp(µ_{m}t).
That is, substituting in the above gives

Integrating from t = 0, P = P_{0} we get

The above expression can be used to calculate the amount of product concentration at the end of a growth cycle.

Example 4-1

McCallion reported growth of Thermoanaerobacter ethanolicus under controlled pH of 7.0. Using appropriate graphs calculate YX/G and YLA/G, where G and LA refer to glucose and lactate. Is lactic acid formation growth associated ? Can you estimate an approximate value for q

_{glucose}?

Time [h]

Glucose [g/L]

Lactate (LA) [g/L]

Cell (X) [g/L]

0

19.50

0.45

0.01

13

16.88

3.88

0.41

14

14.85

4.94

0.54

16

13.11

6.98

0.92

18

10.40

8.98

0.99

19

8.91

10.30

1.05

20

7.75

10.83

1.15

22

5.18

12.57

1.30

24

3.64

14.58

1.35

37

0.25

16.03

0.69

SolutionFirst plot cell concentration versus time. The slope in the exponential growth phase is approximately 0.24 h

^{-1}. Notice that the culture growth slows down shortly after 15 h. One could also analyze the information numerically. Such an approach unfortunately will lack a good overview of the phases of growth.Now plot LA vs X and S vs X.

Plot above shows that the exponential behavior deviates at t = 16 h. A straight line drawn during the exponential phase gives growth rate.

Except for one point, all others lie nearly on a straight line. The data at 0.7 g/L of cell corresponds to the declining phase of growth, which may be ignored for current analysis. The line has a negative slope because glucose decreases as cell concentration increases.

Yields from the above graphs: