164
J.-F. Cornet et al.
However, such assumptions (no accumulation in liquid phase, no transport
by liquid flow) may be questionable in the case of total CO2 at high pH values,
the partition coefficient Ki taking much greater values than for other dissolved
gases such as 02, N2, etc. This case of importance for the culture of photosynthetic micro-organisms will be discussed in more detail in Sect. 5.
2.1.3 Characterization of Growth and Product Formation by Several
Stoichiometric Equations
The drawback of a single stoichiometric equation model is that it implies
constant yield representation of the growth process, i.e. independent of external
conditions such as light energy supply or nutrient limitations. Also, the biomass
elemental formula is held constant.
A number of situations occur in which this kind of model fails for cultures of
photosynthetic micro-organisms where yields and biomass composition are
known to change as a function of the external conditions applied.
Models are therefore needed in which changes in the organism's composition
are considered. A class of potentially useful models results from a single extension of the previous global approach, in which amounts and properties of
biomass are specified by several variables. These models are generally termed
compartment models and combine a better description of the behaviour of the
cultures under different operating conditions with moderate complexity.
For such compartment models, the amount of biomass is not only specified
by the total biomass concentration Cx, but also by a biotic state vector X [1],
which contains the relative proportions of the components defining biomass.
Accordingly, the various chemical compounds can be treated independently,
each being synthesized according to a particular stoichiometric equation. The
time course of cell composition can be properly represented provided the rate of
synthesis for each part of the biotic vector is a function of the external conditions.
When using this approach for the growth of Spirulina platensis, the biotic
state vector has to be characterized by proteins, carbohydrates, lipids, nucleic
acids, sulfated glycogen contents and by a complex sulfated exopolysaccharide.
A further degree of refinement allows for distinctions within the protein pool
and the pigments (phycocyanins, chlorophylls), which is important when studying the effects of mineral limitations. The stoichiometries of synthesis for each of
these macromolecules are established as for a single stoichiometric equation
model: determination of the elemental formula from elemental analysis and/or
from their chemical formula and calculation of the stoichiometric coefficients.
The general formulation of the mathematical model associated with this
description can be easily deduced from the single stoichiometry model. Let
subscript XA stand for the active biomass, i.e. the fraction of biomass that
actually ensures growth and metabolite production, so excluding reserves, let
(J~) be the average molar specific rate of the j th reaction and v u the stoichiometric
J.-F. Cornet et al.
However, such assumptions (no accumulation in liquid phase, no transport
by liquid flow) may be questionable in the case of total CO2 at high pH values,
the partition coefficient Ki taking much greater values than for other dissolved
gases such as 02, N2, etc. This case of importance for the culture of photosynthetic micro-organisms will be discussed in more detail in Sect. 5.
2.1.3 Characterization of Growth and Product Formation by Several
Stoichiometric Equations
The drawback of a single stoichiometric equation model is that it implies
constant yield representation of the growth process, i.e. independent of external
conditions such as light energy supply or nutrient limitations. Also, the biomass
elemental formula is held constant.
A number of situations occur in which this kind of model fails for cultures of
photosynthetic micro-organisms where yields and biomass composition are
known to change as a function of the external conditions applied.
Models are therefore needed in which changes in the organism's composition
are considered. A class of potentially useful models results from a single extension of the previous global approach, in which amounts and properties of
biomass are specified by several variables. These models are generally termed
compartment models and combine a better description of the behaviour of the
cultures under different operating conditions with moderate complexity.
For such compartment models, the amount of biomass is not only specified
by the total biomass concentration Cx, but also by a biotic state vector X [1],
which contains the relative proportions of the components defining biomass.
Accordingly, the various chemical compounds can be treated independently,
each being synthesized according to a particular stoichiometric equation. The
time course of cell composition can be properly represented provided the rate of
synthesis for each part of the biotic vector is a function of the external conditions.
When using this approach for the growth of Spirulina platensis, the biotic
state vector has to be characterized by proteins, carbohydrates, lipids, nucleic
acids, sulfated glycogen contents and by a complex sulfated exopolysaccharide.
A further degree of refinement allows for distinctions within the protein pool
and the pigments (phycocyanins, chlorophylls), which is important when studying the effects of mineral limitations. The stoichiometries of synthesis for each of
these macromolecules are established as for a single stoichiometric equation
model: determination of the elemental formula from elemental analysis and/or
from their chemical formula and calculation of the stoichiometric coefficients.
The general formulation of the mathematical model associated with this
description can be easily deduced from the single stoichiometry model. Let
subscript XA stand for the active biomass, i.e. the fraction of biomass that
actually ensures growth and metabolite production, so excluding reserves, let
(J~) be the average molar specific rate of the j th reaction and v u the stoichiometric