Chapter 9 – Emerging Biophysics Techniques 415
b Assuming that dimer formation and the binding of the dimer to a promotor is rapid,
and that E is always relatively low with the enzyme saturated, indicate what shapes
the separate graphs for the rate of production of X and of its degradation as a function
of C take in steady state. From the shape of these graphs, what type of function might
this specific biological circuit be used for in a synthetic device?
Answers
a
A reaction-limited regime is one in which the time scale that chemical reactions
occur is much larger than the diffusional time required for mixing of the chemical
reactants. Assuming a reaction-limited regime allows us to define simple chem
ical reaction equations and associated rate equations, which have no spatial
dependence. For production of X we can say that
(9.5)
C
k PX
k P
production =
+
3
2
5
For the degradation of X, there are two components, one due to the enzyme that
can be modeled using the Michaelis–Menten equation (see Chapter 8) and the
other due to spontaneous degradation of X:
(9.6)
C
k E
C
k
k
k
C
k C
degradation =
+
(
)
+
+
−
6
4
6
4
7
/
Thus,
(9.7)
C
C
C
k PX
k P
k E
C
k
k
k
C
procduction
degradation
=
+
=
+
−
+
(
)
+
−
3
2
5
6
4
6
4
/
+ k C
7
However, if the enzyme is saturated, then k4 ≫ (k−4 + k6) and the degradation rate
is approximately constant:
(9.8)
C
k PX
k P
k E
k C
=
+
−
−
3
2
5
6
7
b
The degradation rate versus C under these conditions is a straight line of gradient
−k7 and intercept −k6E. To determine the shape of the production curve, we need
to first determine the steady-state concentration of the dimer in terms of C. From
conservation we can say that
(9.9)
P
P
P X
total =
+
⋅
2
At steady state of the dimer binding to the promoter,
(9.10)
k P X
k PX
P X
k
k
P
P X
X
P X
P
X
k
total
total
2
2
2
2
2
2
2
2
2
2
2
0
1
⋅
−
=
∴⋅
=
−
⋅
(
)
∴⋅
=
−
−
−
−
−
−
+
∴⋅
−
⋅
=
+
2
2
2
2
2
2
2
2
/
/
k
X
P P
P X
P
k
k
k
k
X
total
total
/