When inoculating a fresh medium, the cells encounter an environmental shock, which results in a lag phase. The length of this phase depends upon the type of organism, the age and size of the inoculum, any changes in nutrient composition, pH and temperature. When presented with a new nutrient the cell adapts itself to its new environment and normally produces the required enzyme.

Essentially all nutrients can limit the fermentation rate by being present in concentrations that are either too low or too high. At low concentrations, the growth rate is roughly proportional to concentration, but as the concentration increases, the growth rate rises rapidly to a maximum value, which is maintained until the nutrient concentration reaches an inhibitory level, at which point the growth rate begins to fall again. The same type of hyperbolic curve will be obtained for all essential nutrients as the rate-limiting nutrient. The effects of different nutrients on growth rate can best be compared in terms of the concentrations that support a half-maximal rate of growth, this is the saturation constant (/Q.[13] For carbon and energy sources this concentration is usually on the order of 10"5 to 10"6 M, which corresponds for glucose to a concentration between 20 and 200 mg/L. In general, Ks for respiratory enzymes, those associated with sugar metabolism, is lower than Ks for the hydrolytic enzymes, those associated with primary substrate attack.

The Monod equation is frequently used to describe the stimulation of growth by the concentration of nutrients as given by:

where, = specific growth rate, h'1

Ks = saturation constant, gL"1 at |imax/2

S = substrate concentration, gl"1

The saturation constant Ks for Saccharomyces cerevisiae on glucose is 25 mg/L, for Escherichia on lactose: 20 mg/L, and for Pseudomonas on methanol: 0.7 mg/L. Here, nmax is the maximum growth rate achievable when S ยป Ks and the concentration of all other essential nutrients is unchanged. The saturation constant Ks is the approximate division between the lower concentration range where p. is essentially linearly related to S and the higher rate where (j. becomes independent of

The effect of excessive nutrient or product concentrations on growth is often expressed empirically as:

where, = inhibition, constant, gl"1

I = concentration of inhibitor, gl"1

Equations 1 and 2 can be combined to illustrate the characteristics common to many substrates:

The kinetic models of Eqs. 1, 2 and 3 are illustrated in dimensionless form in Fig. 2. It can be seen that the adding of large amounts of substrate to provide high concentrations of product(s) and to overcome the rate-limiting effects of Eq. 1 can result in concentrations that are so high that the fermentation is limited by the effects of Eq. 2. The ideal operating range would be 1 < S/Ks < 2 where the growth rate is near its maximum and is relatively insensitive to substrate concentration.[4J The concentration ranges which enhance or inhibit fermentation activity vary with each microorganism, chemical species, and growth conditions.

S/K for (A); S/K for (C) with K. = 1000 K : I/K. for (B).

a o 1S1

S/K for (A); S/K for (C) with K. = 1000 K : I/K. for (B).

a o 1S1

Figure 2. (A) Monod Growth Model; (B) model for growth inhibition; and (C) model for substrate activation and inhibition of growth.

Product formation is related to the substrate consumption as follows: Eq.(4) AP = Yp/S

where, AP = product concentration - initial product concentration ingl-1.

AS = substrate concentration - initial substrate concentration in gl'1 Yp/s = product yield, g-P.g-S"1

This equation is especially useful when the substrate is a precursor for the product. Many other models are available. When calculating a material balance for medium formulation, the choice of models depends upon the data that are available.

Ethanol-from-biomass fermentation is an example of product inhibition, while high fermentable carbohydrate concentrations also can become inhibitive. Aiba and Shoda developed a mathematical model for the anaerobic fermentation, where nine constants are required to describe the model.'133

dS 1 iff 1 dP trv

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