Enzyme Inhibition with Time Delay

time

350.

delay

4.

k

1

1

α

0.0005

ϵ

0.1

Consider the following consecutive enzyme reactions:

I

s

k

1

→

Y

1

z

→

Y

2

k

2

=1/2

→

,

where

I

s

is a substrate supply maintained at constant value,

k

1

and

k

2

are reaction rate constants, and

n

molecules of

Y

2

inhibit the reaction step

Y

1

z

→

Y

2

with

z=

k

1

1+

n

α(

y

2

(t-τ))

. Assume the inhibitor molecules must travel to the position of the regulatory enzyme by diffusion or active transport causing a time delay

τ

; the equations describing the system are

d

y

1

dt

=

k

1

I

s

-z

y

1

,

d

y

2

dt

=z

y

1

-

k

2

y

2

,

z=

k

1

1+

n

α(

y

2

(t-τ))

,

y

1

(t≤0)=

k

1

I

s

z

,

y

2

(t≤0)=

k

1

I

s

k

2

,

where

y

1

and

y

2

denote concentrations,

t

is time, and

α

is

a positive constant. The system has an equilibrium when

y

1

=

k

1

I

s

z

and

y

2

=

k

1

I

s

k

2

.This Demonstration investigates solutions of the system starting at a small perturbation

ϵ

away from the equilibrium.