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increases there is a cooperative activation of the rate, which
is characterized by the Hill number p. When the concen-
tration of D exceeds the intermediate value needed for half-
maximal activation, the activation constant K
D
, the increase
in rate diminishes as it approaches the maximum velocity
given by
g
MMax
. As the concentration of D approaches and
then exceeds the inhibition constant K
I
, there is a cooper-
ative repression of the rate, which is characterized by the
Hill number n. The loss of mRNA, M, is first-order with rate
constant
d
M
.
The rate of translation is first-order in the concentration
of M with rate constant
g
C
. At very low concentrations of R
there is a basal rate of loss of C given by
d
C
. As the
concentration of R increases there is a cooperative activa-
tion of the rate, which is characterized by the Hill number a.
When the concentration of R exceeds the intermediate
value needed for half-maximal activation, the activation
constant K
R
, the increase in rate diminishes as it approaches
the maximum velocity given by
d
CMax
. The reversible
dimerization of CI is characterized by the second-order rate
constant
g
D
and the first-order rate constant
b
D
.
(A)
(B)
FIGURE 15.7
(A) Kinetic model of the cI gene circuit for phage lambda.
Symbols: NA, nucleotide precursors; mRNA, transcript of cI gene; AA,
activated amino acid precursors; CI, monomeric form of the CI protein;
CI
e
CI, dimeric form of the CI protein; RecA*, activated form of the
RecA protein. (B) Simplified symbolic version of the kinetic model in (A).
Symbols: N, nucleotide precursors; M, transcript of cI gene; A, activated
amino acid precursors; C Monomeric form of the CI protein; D Dimeric
form of the CI protein; R, activated form of the RecA protein.
features of these models include first-order loss of CI
mRNA, first-order translation of the CI mRNA, dimeriza-
tion of CI monomers, and cooperative activation of pRM
transcription by low concentrations of CI dimers and
repression at high concentrations. These well-established
features of the cI gene circuit are included in a number of
detailed kinetic
[59
66]
models
that incorporate the recent experimental results of Dodd
et al.
[67]
.
The following is a kinetic model that includes these
conventional features of the system. It differs from existing
models by the assumption of a phenomenological Hill-like
function for the rate of pRM transcription and the potential
for cooperative RecA* catalysis of CI monomer degrada-
tion under substrate-limited conditions. The first assump-
tion appears to represent adequately the experimental data
of Dodd et al.
[67]
, as shown below. The second assumption
addresses the detailed kinetics of CI degradation, which is
unknown because it has been difficult to study for technical
reasons (J.W. Little, personal communication). Neverthe-
less, fitting experimental data with this model yields an
estimate of the capacity for regulation of CI degradation
that is consistent with the limited experimental data avail-
able
[68]
, as discussed below.
e
63]
and stochastic
[64
e
Estimation of Parameter Values
Dodd et al.
[67]
produced two sets of data critical for
understanding the mechanism that controls cI gene tran-
scription in lambda lysogens. In the first case, the pRM
promoter is associated with wild-type operator sequences
that allow binding of CI dimers at low concentration to the
operator sites OR1
OL2 to activate tran-
scription, and that allow binding of CI dimers at high
concentrations to the operator sites OR3 and OL3 to repress
transcription. In the second case, the pRM promoter is
associated with a disabled operator site OL3 that fails to
bind CI dimers, thereby eliminating repression at high
concentrations. We fit our model to these data and obtained
the values shown in
Table 15.4
(for details see
[69]
).
OR2 and OL1
e
e
Recast Equations
Equations
(4)
to (6) can be recast
[12,70]
exactly into
a generalized mass action (GMA) system of equations
within the power-law formalism by introducing two new
variables
x
2
¼
"
g
M
K
D
þ
#
D
p
g
MMax
þ
g
M
K
n
D
n
dM
dt
¼
I
D
p
1
D
n
d
M
M (4)
K
D
þ
k
n
I
þ
d
C
K
R
þ
d
CMax
R
a
K
R
þ
C
dC
dt
¼
2
g
D
C
2
g
C
M
þ
2
b
D
D
R
a
K
D
þ
x
1
þ
x
p
þ
n
1
K
n
I
K
R
þ
x
R
:
and
x
3
¼
(5)
dM
dt
¼
g
M
K
D
x
1
g
MMax
D
p
x
1
g
M
K
n
D
p
þ
n
x
1
2
þ
þ
dD
dt
¼
g
D
C
2
2
2
I
b
D
D
d
D
D
(6)
d
M
M
(7)
The behavior described by these equations, which represent
both mass-action and rational-function mechanisms, can be
described as follows.
At very low concentrations of D there is a basal rate of
transcription given by
g
M
. As the concentration of D
dC
dt
¼ g
C
M
þ
2
g
D
C
2
d
C
K
R
Cx
1
2
b
D
D
3
d
CMax
R
a
Cx
1
(8)
3
