282
Chapter 7
Note 7.5
Derivation of Eq. (7.52)
With the assumptions specified for the derivation of Eq. (7.52), we have
[ X ,\
« w + [ x , « s j
(1)
[ x . \ * lx „ ] + [ x „ x r]
(2)
f o j , « [ S J
(3)
Using Eqs. (2) and (7.46) the fraction of repressor-free operators is
o
. J &
l . ____ !____
a
f r . l
i
After multiplication by 1 +
in both the nominator and the denominator, we find
O = ____________ l +
____________
'
l + * ,[S te ]” + *2 ( [ * , ] + * . № „ ] ■
(5)
Using Eqs. (7.45), (1), and (3), we obtain the expression in Eq. (7.52) for the fraction of repressor-free
operators._________________________________________________________________________________
Small molecules that influence the transcription of genes are called effectors, and in the lac-operon
the effector (lactose) is an
inducer.
In other opérons there may, however, be a negative type of
control, and here the effector is called an
anti-inducer.
With an inducer the binding affinity to the
operator of the free repressor is much larger than that of the inducer-repressor complex, i.e.,
K2 »
FU,
whereas with anti-inducer it is the other way round. For an anti-inducer, the fraction of
repressor-free operators can be found from an expression similar to Eq. (7.52) (see Harder and
Roels (1982) and Problem 7.3).
The other control mechanism in the lac-operon is the so-called carbon catabolite repression, which
ensures that no enzymes necessary for lactose metabolism are synthesized as long as a preferred
substrate is available, e.g., glucose. The mechanisms behind carbon catabolite repression are not
completely known, but it is known that the binding of the RNA polymerase to the promotor is
strongly enhanced when a complex of cAMP and a protein called CAP (catabolite activator protein)
is bound to the promotor. When the concentration of glucose (or another energy source) in the
extracellular medium is high, the intracellular cAMP level is found to be low. Consequently, the
level of the cAMP-CAP complex is low at high glucose concentrations. The site of the binding of
the cAMP-CAP complex has been located in several opérons that are under carbon catabolite
repression, and binding of the complex to DNA has been found to promote helix destabilization
downstream. This in turn facilitates the binding of the RNA polymerase and hereby stimulates gene
expression. The carbon catabolite repression can be described by the following equilibria:
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