
Nonlinear systems 107
dM
F
(t)
dt
= k
p
P O
F
(t − τ
m
)e
−µτ
m
h
1 − b
1 − e
T (t)/c
i
(3.49b)
− k
ρ
ρ
M
F
(t) − M
F
(t − τ
ρ
)e
−µτ
ρ
− (k
d
D + µ) M
F
(t) (3.49c)
dE(t)
dt
=
1
2
k
ρ
ρM
F
(t − τ
e
)e
−µτ
e
− (γ + µ) E(t) (3.49d)
dT (t)
dt
= K
K
n
H
i
K
n
H
i
+ T
n
H
(t)
E(t) − g
T (t)
T (t) + K
g
+ d
T
ext
e + T
ext
[1 + T (t)/f]
− µT (t)
(3.49e)
In Eq. 3.49, R is total repressor concentration, O is to tal operon concentra-
tion, P is mRNA polymerase (mRNAP) concentration, O
F
(t) is free operon
concentration, M
F
(t) is free mRNA concentration, E(t) is total enzyme con-
centration, T (t) is tryptophan concentration, K
r
is the repression equilibrium
constant, K
t
is the the rate equilibrium constant between the total repre ...