Bistability in the Lactose Operon of Escherichia coli: A Comparison of Differential Equation and Boolean Network Models - Mathematical Concepts and Methods in Modern Biology - page 42

Equations ( 2.8 )-( 2.9 ) now describe the dynamics of the single enzyme single
substrate reaction in Eq. ( 2.7 ). However, with an additional assumption, these two
equations can further be simplified. In an enzymatic reaction, the enzyme-substrate
complex reaches a steady state much earlier than the product does. This allows us to
take d [ ES ]
dt

≈ 0. Therefore,

d [ ES ]

= k 1 [ E ][ S ]− ( k 2 + k 3 ) [ ES ]= 0 . (2.11)

dt

Solving Eq. ( 2.11 )for [ E ] yields

]= (

k 2 + k 3 ) [ ES ]

[ E . (2.12)

k 1 [ S ]

After plugging from Eq. ( 2.12 ) into the enzyme conversation equation given by

Eq. ( 2.10 ) and solving the resultant equation for

[ E ]

[ ES ] , we obtain [ ES ] in terms of

only [ S ] :

E 0 [ S ]

[ ES ]= . (2.13)

k 2 + k 3
k 1

+[ S ]

Substituting [ ES ] given by Eq. ( 2.13 ) into Eq. ( 2.9 ) yields

d [ P ] V max [ S ]

=

] ,

(2.14)

dt K m +[ S

where

k 2 + k 3

V max = k 3 E 0 , K m = (2.15)

k 1

are two positive parameters. This equation is called the Michaelis-Menten equation.
The right-hand side of this equation f

V max [ S ]

( [ S ] ) = is an increasing function of

K m +[ S ]

the substrate concentration with two properties: (i) lim f ( [ S ] ) = V max , and (ii)

[ S ]→∞

V max

2 . From a biological point of view, this reaction never occurs at a
rate greater than V max . K m is the substrate concentration at which the reaction rate is
equal to half of its maximum value V max . Furthermore, since V max =

f ( [ S ]= K m ) =

0, the
reaction rate is a linear increasing function of the initial enzyme concentration E 0 ,
which means that higher initial enzyme concentration makes the reaction go faster
and this relationship between the rate and E 0 is linear.
If the total substrate concentration is conserved throughout the course of the reac-
tion, we can write

k 3 E 0 >

S 0 =[ S ]+[ ES ]+[ P ] . (2.16)

Since the initial concentration of the enzyme is usually much smaller than the initial
substrate concentration, the concentration of ES is negligibly small compared to either
the substrate or the product concentration. Hence, ( 2.16 ) simplifies to

S 0 =[ S ]+[ P ] . (2.17)