286
Chapter 7
the enzyme of interest. A more general expression for synthesis of mRNA encoding a given protein
than that used in Example 7.7 is:
r mRNA ~ ^ 'm rh r X g ~ k m Q \ 0 -2 Q l X g
(7 • 6 0 )
where
kn
is the overall transcription rate constant, Tl* is the overall transcription efficiency, and
Xs
is
the copy number of the gene to be transcribed (could be given as number of genes per g DW). The
overall transcription efficiency is given as the product of
Qu Q2,
and
Q2.
The factors
Qx
and
Q
2
represent, respectively, the fraction of repressor free operators and the fraction of activated
promoters, i.e., those that may bind RNA polymerase. These factors are not necessarily identical
with those derived for the lac-operon above, e.g., if the control mechanism involves an anti-inducer
Q
2
is not given by Eq. (7.58) (see Problem 7.3). The factor
Q
2
is the fraction of promoters that form
complexes with the RNA polymerase, i.e., it is a function of the cellular content of RNA
polymerases.
The overall transcription rate constant
is a function of the environmental conditions, and Lee and
Bailey (1984c) specified it as a function of the specific growth rate (see Note 7.6). With mRNA
being very unstable, it is necessary to include degradation of mRNA in the model. This is normally
done as a first-order process as illustrated in Example 7.7:
^’mRNA,dcg
^m,deg
X mRNA
(7.61)
Translation of the mRNA to form the desired protein is generally described by Eq. (7.62).
rp = k p& mRNA
(7.62)
where Ap is the overall translation rate constant (see Note 7.6) and £ is the translation efficiency (this
is often set to unity). Similar to the degradation of mRNA, a turnover of protein is often included as
a first-order process:
= k p,tegX
?
(7-63)
The above model for protein synthesis is generally applicable, and the parameter values have been
identified for many different systems (see Note 7.6). The model is, however, often simplified in
order to keep its complexity at a reasonably low level.
Note 7.6. Mechanistic parameters in the protein synthesis model
Lee and Bailey (1984c) used the above model for an analysis of the influence of the specific growth rate on
the productivity of recombinant
E. coli.
Because of the mechanistic nature of the model, each of the
parameters has a physical meaning, and here we wall illustrate how Lee and Bailey calculated the model
parameters.
The overall transcription rate constant and the overall translation rate constant are given by:
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