Enzyme Inhibition with Time Delay
Enzyme Inhibition with Time Delay
Consider the following consecutive enzyme reactions:
I
s
k
1
→
Y
1
z
→
Y
2
k
2
→
where is a substrate supply maintained at constant value, and are reaction rate constants, and molecules of inhibit the reaction step with . Assume the inhibitor molecules must travel to the position of the regulatory enzyme by diffusion or active transport causing a time delay ; the equations describing the system are
I
s
k
1
k
2
n
Y
2
Y
1
z
→
Y
2
z=
k
1
1+
n
α((t-τ))
y
2
τ
d
y
1
dt
k
1
I
s
y
1
d
y
2
dt
y
1
k
2
y
2
z=
k
1
1+
n
α((t-τ))
y
2
y
1
k
1
I
s
z
y
2
k
1
I
s
k
2
where and denote concentrations, is time, and isa positive constant. The system has an equilibrium when = and =.This Demonstration investigates solutions of the system starting at a small perturbation away from the equilibrium.
y
1
y
2
t
α
y
1
k
1
I
s
z
y
2
k
1
I
s
k
2
ϵ