2. 5. Theoretical model
When S. cerevisiae is grown aerobically using glucose as
substrate, biomass and ethanol are produced and a diauxic pattern can be
observed. After depletion of glucose ethanol is consumed by the cells.
The process can be simulated by the following differential equations:
\(\frac{dX}{dt}=\ \mu_{G}X+\ \mu_{E}X\)
\(\frac{dG}{dt}=\ -\frac{\mu_{G}X}{Y_{X/G}}\)
\(\frac{dE}{dt}=\ \frac{\mu_{G}X}{Y_{E/G}}-\ \frac{\mu_{E}X}{Y_{X/E}}\)
G, E and X are the glucose, ethanol and the biomass concentrations,
respectively. \(\mu_{G}\) and \(\mu_{E}\) are the specific growth rates
on glucose and ethanol, respectively. \(Y_{X/G}\), \(Y_{E/G}\) and\(Y_{X/E}\) are the yield coefficients with respect to the conversion
from glucose to biomass, glucose to ethanol and ethanol to biomass,
respectively. The yield coefficients have been fixed to values which are
determined by pre-runs of the process (\(Y_{X/G}=0.175\ g_{X}/g_{G}\),\(Y_{E/G}=\ 0.473\ g_{E}/g_{G}\) and\(Y_{X/E}=0.598\ g_{X}/g_{E}\)).
The specific growth rates (\(\mu_{G0}\) and \(\mu_{E0}\)) which are the
kinetic parameters of the process model are determined during the
model-based calibration procedure.