Corynebacterium glutamicum - JUWEL - Forschungszentrum Jülich

2. Theory
For batch fermentations, the equations can be simplified since all fluxes are 0 7 . For
chemostat fermentations at steady state, all time derivatives are 0 which simplifies the
balance to yield:
− r = �
f
Ff (Cf − Css)
V
(2.6)
which can readily be solved using the steady state concentration Css.
For fed-batch fermentations, such general simplifications cannot be made. The concentrations
can however be simulated over time when the initial concentrations C(0) and
the fluxes are known and kinetic equations for the conversion rates as a function of the
concentrations are given:
2.3.2. Kinetics
� t
C(t) =C(0) +
t=0
˙Cdt (2.7)
Now the challenge is to find kinetic descriptions for the conversion rates r as a function
of the concentrations for all relevant substances: biomass, substrate(s) and product(s).
One common type of kinetic equations is the Monod kinetics, which is based on the
Michaelis-Menten type enzyme kinetics (Michaelis and Menten, 1913):
r = vmaxCSCC
Km + CS
(2.8)
where CS is the concentration of substrate, CC the catalyst concentration and Km
the Michaelis-Menten constant, which is a binding constant, equal to the concentration
needed to achieve an enzyme specific conversion rate 8 of half the maximal specific conversion
rate vmax. For more detailed explanation of the theoretical background of this
equation and similar enzyme kinetic models, see for instance Biselli (1992); Roels (1983).
For biomass growth on one limiting substrate, the Monod kinetics (Monod, 1942) can
be written analogously as:
CSCX
rx = µmax
KS + CS
rs = − rx
Yxs
(2.9)
(2.10)
7 except for the fluxes for for instance pH control and the flux of samples out of the reactor. The first
of which is usually of little importance as long as sufficiently high concentrations of the correction
fluids can be used. The last point is usually only of importance in research where many samples are
taken and small biorectors are being used. However, since this does not change the concentrations
but changes all amounts in an equal way, and since no substrates are added into the system, this can
also be neglected.
8 This is the conversion rate per unit enzyme, for instance in mmolsubstratemmol −1
enzymes −1
12