**4.2 What Does µ Depend on?**

Specific growth rate (µ) depends on a number of factors such as
growth medium composition, temperature, pH and others. Experimental studies
have shown that one cannot increase growth rate beyond a certain maximum
value, µ_{m} due to inherent metabolic reaction rate limitations.
In general, when substrate, S is limiting growth, Monod (1949) reported
that growth rate variations can be expressed as

where K_{S} is called Monod constant or simply the substrate
saturation constant. The significance of K_{S} is, when substrate
concentration is numerically equal to K_{S}, growth rate is exactly
half of maximum growth rate. See Figure 4-1.

Figure 4-1 Monod Kinetics. Dependence of Growth Rate on Limiting
Substrate. Specific growth rate reaches a maximum value of 0.5 h^{-1}.
Value of KS here is 0.5 g L^{-1}. Note that
when S = 0.5 g L^{-1}, µ is half of its maximum.

The form of Eq(4-5) can be used to describe dependence of µ on
more than one limiting nutrient. In many practical applications availability
of oxygen for respiration often limits growth. When both substrate, S,
and dissolved oxygen concentration, C_{DO}, are both limiting growth,
specific growth rate can be mathematically described as

Figure 4-2 illustrates the behavior of maximum growth rate when two
substrates are limiting. The parameters K_{S} and K_{DO}
are cell specific. K_{S} is typically in the order of 10 mg/L for
glucose and K_{DO} is less than 1 mg/L for oxygen in the case of
bacteria and yeast. K_{DO} has been reported to be higher for mammalian
and insect cells.

Figure 4-2 Monod Kinetics when two substrates are limiting. Specific
Growth Rate reaches a maximum value of 0.5 h^{-1}. Value of K_{s}
here is 0.5 g L^{-1}. Value of KDO is 0.1
mg L^{-1} Note that when C_{DO }=
0.1 mg L^{-1}, µ is half of its maximum at values of S >>
Ks.

Let us now consider growth under conditions of only substrate limitations in a batch bioreactor. Incorporating the substrate limited condition, bioreactor material balance equation, Eq(4-2), can be modified and we may write,

In order to integrate the above, one of the variables, S, needs to be replaced in terms of X. The yield relationship, Eq(2-3), can be integrated as

which simplifies to

where subscript, 0 refers to initial concentration. Substituting for S from Eq(4-8) in Eq(4-7) and integrating gives,

For analyzing batch systems, use the above to calculate cell concentration and then calculate substrate concentration using Eq(4-8).