Applying Mechanistic Models in Bioprocess Development - Measurement, Monitoring, Modelling, and Control of Bioprocesses - page 142

Table 1 Stoichiometric matrix describing aerobic growth of budding yeast
Component i ? C 1

Glucose

C 2
Ethanol

C 3
Oxygen

C 4
Biomass

Symbols G E O X

mol l -1 gl -1 mol l -1 gl -1 mol l -1 gl -1 C-mol l -1 gl -1 Units

Process j ;
Biomass growth by glucose
oxidation (Eq. 1 )

Y X Oxid -1 -10 0 a Y OG b

Red

Biomass growth by glucose
reduction (Eq. 2 )

-1 -1 j Y EG 0 0 g Y XG

Biomass growth by ethanol
oxidation (Eq. 3 )

0 0 -1 -1 k Y OE l Y XE

C 6 H 6 O þ kO 2 þ l 0 : 15 NH 3 ½ ! lC 1 H 1 : 79 O 0 : 57 N 0 : 15 þ mCO 2 þ nH 2 O ð 3 Þ

For each pathway, a mass balance can be established for each atomic element
(e.g. C or N). To solve such elemental balances for carbon, hydrogen, and oxygen,
one stoichiometric coefficient for each pathway has to be assumed. Since the
biomass yield coefficients are often easily estimated from experimental data, they
are typically the ones that are assumed. Therefore, only the coefficients b, g and l,
or the corresponding mass yields Y X Oxid ,Y XG

Red , and Y XE will be considered as model
parameters; i.e., the other stoichiometric coefficients are fixed based on Eqs. 1 - 3 .
Furthermore, a process matrix can be used to describe the rates of consumption
and production of each of the model variables (glucose, ethanol, oxygen, and
biomass), as well as the fluxes in each pathway. Details on the use of this matrix
notation are provided by Sin and colleagues [ 14 ]. The interested reader can find
additional details on elemental mass and energy balances applied to fermentation
processes elsewhere [ 15 , 16 ].
In the case of the model used as an example here, the total glucose consumption
and ethanol consumption rates (when considered individually) are mathematically
described using Monod-type kinetics (Eqs. 4 - 6 ). The maximum uptake rates for
glucose, ethanol, and oxygen (r i,max ) are model parameters, and they are character-
istic of the S. cerevisiae strain being used. The same goes for the substrate saturation
constants: K G , K E ,andK O . The maximum oxygen uptake rate (r O,max ) corresponds to
the respiratory capacity, as it reflects the maximum rate for oxidation of glucose or
ethanol when any of these carbon sources is in excess. The ethanol uptake rate
includes a term accounting for glucose repression; i.e., ethanol consumption is only
observed for low concentrations of glucose. The strength of inhibition (i.e., how low
the glucose concentration should be before ethanol consumption is allowed) is
defined by the inhibition constant K i . The specific growth rate of biomass is defined as
the sum of the growth resulting from each pathway, and is estimated based on the
yield of biomass on the substrate and the corresponding uptake rate (Eq. 7 ).

G
G+K G

r Total
G

¼ r Oxid
G

þ r Red
G

¼ r G ; max ð 4 Þ