274
Chapter 7
and the yield coefficient will therefore also increase for increasing substrate concentration.
The present two-compartment model, which solves some of the conceptual problems with the Williams
model, has also been extended for description of recombinant
E. coli
fermentations (Nielsen
et a l
, 1991c;
Strudsholm
e ta l
, 1992),_____________________________________________________________________
Many other simple structured models are presented in the literature (see Harder and Roels (1982)
and Nielsen and Villadsen (1992) for reviews). Most of these are similar in structure to the models
described above, but the same ideas may be formulated very differently by different authors. Thus
Powell (1967) introduced a class of structured models which he called "bottleneck models." They
are based on an assumption of one cellular element being the bottleneck for cellular growth. These
models are formally identical with the two-compartment models where the active compartment is
the bottleneck for growth, and Powell also infers that the bottleneck may be the PSS. Other models
are based on an extension of unstructured models, where some or all of the rate constants are
described as functions of the environmental conditions, e.g.,
parameter, and / is a function o f the substrates, which determines the "target value" o f the
parameter
Sweere
et al.
(1988) used this concept to improve on the capability of the Sonnleitner
and Kappeli model (see Example 7.3) to describe dynamic growth conditions, and the revised
model fits experimental data for batch fermentation better than the original model. However, the
approach is empirical, and the parameters in Eq. (7.37) cannot be estimated from steady-state
experiments alone (the adaptation time
has to be estimated from a transient experiment).
7.4.2 Cybernetic Models
Most academic fermentation studies are made with only a single limiting substrate, but in industrial
processes several different components of a complex substrate may become rate-controlling in
various parts of the fermentation. Modeling the parallel uptake of substrate components that serve
different purposes in the microorganism, e.g., an energy source like glucose and a nitrogen source
like ammonia, can be done in a rather simple fashion as illustrated with Eq. (7.18)-(7.20). Modeling
the sequential uptake of different substrates that serve the same purpose in the microorganism, e.g.,
glucose and lactose, is, however, much more difficult. Sequential uptake of substrates in batch
fermentations normally results in different exponential growth phases separated by lag phases
where synthesis of enzymes necessary for metabolism of the next substrate is carried out. This is
referred to as
diauxic growth
with two substrates (triauxic growth with three substrates). The
sequential uptake of substrates is a consequence of complex regulatory structures in the cell. Thus,
there is typically glucose repression (or more precise carbon catabolite repression) on the utilization
(7.37)
where
is a characteristic time for adaptation to new environmental conditions, £,ma*
is a
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