Modeling of Growth Kinetics
281
K
[ X .X rn S j
4
\ x rn s < A x A
(7.48)
Application of Eqs. (7.45)-(7.48) is based on the following assumptions (Harder and Roels, 1982):
A macroscopic description can be used to express the influence of the reacting species on
the kinetics, i.e., the concentrations of the different components are used. However,
microorganisms only contain a few (1-4) copies of one type of operator per cell, and the
number of repressor proteins per cell is also low (10-20). For such small entities the
meaning of concentrations and o f thermodynamic equilibrium is disputable, and it may be
more correct to apply a stochastic modeling approach.
As in Michaelis-Menten kinetics for enzymes (Section 6.1) all reactions in (7.44) are
assumed to be equilibrium reactions. This is reasonable since the relaxation times for the
equilibria are much smaller than for most other cellular reactions.
Balances for the repressor, operator, and inducer are
[x,l =
(7.49)
M
=[x.]+[x.xr]+[x.xrnsbc]
(7.50)
M
=
[S ,J + n [ X rn S ,J + n [X ,X rn S ,J
(7.51)
where the index
t
refers to the total concentration. In wild-type
E. coli
there are 10-20 times more
repressor molecules than there are operators, and in this case the last two terms in Eq. (7.49) can be
neglected. Furthermore, with the weak binding of the inducer-repressor complex to the operator,
[A
'0AT1«iS'be] can be neglected in Eq. (7.50). Finally, Eq. (7.51) can be simplified by assuming that the
intracellular concentration of inducer molecules is in sufficient excess over repressor molecules,
and consequently that
+
n[XJC^iS]lK
.]
« [5^]. With these simplifications the fraction of
repressor free operators2 is found (see Note 7.5) to be
0
_
[x0]
_
i + . r , [ s j ;
'
[x,\
i+ * ,[ s ,J ;+ A :2[x r],
(7.52)
Since the transcription of the three genes in the operon is likely to be determined by the fraction of
repressor-free operators eq. (7.52) is valuable for description of the synthesis of the enzymes
necessary for lactose metabolism in a structured model that aims at describing diauxic growth on
glucose and lactose. The inducer concentration Shc is likely to be correlated with the extracellular
lactose concentration, whereas the total content of repressor protein can be assumed to be constant.
J Since the total number of operators o f a given type in the cell is very small, it does not make much sense to talk about
die fraction of repressor-free operators. However, in a description of enzyme synthesis one may use Eq. (7.52) as an
expression for the probability that the operator is repressor flee.
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