Enzymatic Reaction in a Batch Reactor

k

1

in

3

dm

(mol.hr)

2000

k

2

in

-1

hr

300000

k

3

in

-1

hr

10000

Consider the following reaction scheme:

S+E

k

2

⇆

k

1

E.S

k

3

→

E+P

, where

S

is the substrate,

E

is an enzyme catalyst,

E.S

is the enzyme-substrate complex, which decomposes to give the product

P

, and

E

is the enzyme. This enzymatic reaction takes place in a batch reactor and the governing differential-algebraic system of equations is:

d

c

S

dt

=-

k

1

c

S

c

E

+

k

2

c

E.S

d

c

E.S

dt

=

k

1

c

S

c

E

-

k

2

c

E.S

-

k

3

c

E.S

d

c

P

dt

=

k

3

c

E.S

c

E0

=

c

E

+

c

E.S

=0.001mol/

3

dm

.

The initial conditions are:

c

S0

=1mol/

3

dm

,

c

E0

=0.001mol/

3

dm

,

c

P0

=0

, and

c

E.S0

=0.

These equations can be solved using the Mathematica built-in function, NDSolve. This approach is the rigorous one.

Another method, called the quasi-steady-state assumption, considers that

d

c

E.S

dt

=0

. The resulting governing equations are:

d

c

S

dt

=-

d

c

P

dt

=

-

k

3

c

E0

c

S

K+

c

S

, where

K=

k

2

+

k

3

k

1

.

This model is referred to as Michaelis-Menten kinetics. An analytical solution is possible for this model and is given by:

k

3

c

E0

t=

c

S0

-

c

S

+Kln

c

S0

c

S

.

The reaction rate constants

k

1

,

k

2

, and

k

3

are expressed in

3

dm

/(mol.hr),

-1

hr

, and

-1

hr

, respectively.

This Demonstration shows the substrate concentration, [S], (red curve) and the product concentration, [P], (blue curve) versus time obtained using the exact approach. The bold dots correspond to the quasi-steady-state approach. Agreement between both methods is obtained and justifies the utilization of the pseudo-steady-state hypothesis, which is also called the quasi-steady-state approach.