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, µm 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 KS is called Monod constant or simply the substrate
saturation constant. The significance of KS is, when substrate
concentration is numerically equal to KS, 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, CDO, 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 KS and KDO are cell specific. KS is typically in the order of 10 mg/L for glucose and KDO is less than 1 mg/L for oxygen in the case of bacteria and yeast. KDO 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 Ks
here is 0.5 g L-1. Value of KDO is 0.1
mg L-1 Note that when CDO =
0.1 mg L-1, µ is half of its maximum at values of S >>
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).