ΔX ¼ IF Q ! Q 0 THEN MU À R
ð
ÞÃX ELSE 0:
ð8:1Þ
The change in the nutrient concentration outside the cell, ΔN, depends on the
nutrient passing through the cell wall,
ΔN ¼ IF N > 0 THEN R Ã Q Ã X À V Ã X ELSE 0:
ð8:2Þ
R * Q * X is the return of the nutrients to the external environment, depending on
the internal nutrient concentration Q. The nutrient balance equation for the internal
concentration of the nutrient is
ΔQ ¼ IF Q ! Q 0 THEN V À MU Ã Q ELSE 0:
ð8:3Þ
Equation (8.3) is a form of the Monod equation used to predict the change in the
internal concentration.
The rates V and MU are calculated from a now-standard Michaelis-Menten
formula. The general form for this equation is derived from the enzyme-substrate
equation discussed in the previous chapter [2]:
V ¼ VM Ã N= KN þ N
ð
Þ
ð 8:4Þ
MU ¼ MU BAR Ã Q À Q 0
ð
Þ = KQ þ Q À Q 0
ð
Þ
ð
Þ
ð 8:5Þ
with VM the maximum rate of nutrient uptake per unit biomass, KN the half
saturation constant for nutrient uptake, KQ the half saturation constant for growth,
and MU BAR the maximum biomass growth rate.
Our model is shown in Fig. 8.2. Recognize, that this model differs from the
specification of the reaction rate in the previous chapter in that the reaction rate here
is not based on the product of the concentrations of the compounds. Also, note well
Internal
Concentration Q
Cell Biomass X with
Growth Rate MU
MU*X
R*X
R*Q*X
V*X
Fig. 8.1
76
8 Two-Stage Nutrient Uptake