284
Chapter 7
in determining the cAMP level in the cell at different glucose concentrations. This illustrates a
general problem when a genetically structured model is combined with overall models for cell
function: Certain mechanisms may be described in great detail - in this case the gene expression,
whereas other processes are described by completely
empirical expressions.
Hereby the
performance of the overall model is largely determined by the performance of the empirical
expressions in the model, and it may be adequate to apply a simpler model for the gene expression.
The real strength of the genetically structured models is, however, not its linkage to the overall
growth model, but rather the possibility offered to analyze the influence of specific model
parameters on the process. Thus, using the above model the importance of the different equilibrium
constants, which are related to the binding affinities e.g. of the repressor to the operator, can be
studied in detail. This can be done by comparison with experimental data for the mRNA level,
preferably at conditions where the overall cell activity is the same in all experiments.
Example 7.7 A model for diauxic growth
Based on Eqs. (7.52) and (7.58), Harder and Roels (1982) developed a structured model for diauxic growth.
It describes the synthesis of mRNA encoding for the three enzymes necessary for lactose metabolism and
also for translation of the mRNA into proteins (which are collected in one compartment called XE). The
residual biomass, including building blocks for mRNA and enzyme synthesis, is pooled into one
compartment
X
which constitutes almost all of the cell mass, i.e.,X *l.
Synthesis of mRNA is described by
^ +
0 ; v,
—k ]f{ji)Q \Q 2
(1)
where/(P) is a linear function of the specific growth rate. The function/(P) is used to describe the way the
activity of the cell (e.g., expression of genes) increases with the specific growth rate. The expression is
completely empirical, but one could combine the Harder and Roels model with one of the two-compartment
models of Section 7.4.1 and replace y(P) in (1) with the concentration of the active compartment. In Eq. (1)
the functions
Q]
and
Q2
of Eqs. (7.52) and (7.58) both appear as factors. The fraction of repressor-free
operators and the fraction of activated promoters must both be high to obtain a rapid mRNA synthesis.
The rate of synthesis of enzymes necessary for lactose metabolism is assumed to be first order in the mRNA
concentration, i.e.,
X + X £
0 , v2
= k2X m
(2)
The half-life of mRNA is short due to rapid degradation by an assumed first-order process:
~
X mRNA + X = 0
, V3 =
k^X mRNA
(3)
Similarly, degradation of the lactose-metabolizing enzymes is included as one first-order process:
- X E+X = 0
; v4
=kAX £
(4)
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