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 44

Since the product P is produced by the third reaction and it is not consumed, the
dynamics of the product P concentration can be modeled as in Eq. ( 2.21 ).

d [ P ]

= k 3 [ ES n ] . (2.21)

dt

Suppose that the complex ES n reaches a steady state faster than the product P . Then,
Eq. ( 2.20 ) becomes

d [ ES n ]

dt

n

= k 1 [ E ][ S ] − ( k 2 + k 3 ) [ ES n ]= 0 . (2.22)

Solving Eq. ( 2.22 )for [ E ] gives

]= (

k 2 + k 3 ) [ ES n ]

[ E . (2.23)

n

k 1 [ S ]

Assuming that the total amount of enzyme is conserved throughout the reaction, we
have

E 0 =[ E ]+[ ES n ] , (2.24)

where E 0 stands for the total enzyme concentration. Substituting Eq. ( 2.23 )into
Eq. ( 2.24 ) and then solving it for [ ES n ] , we obtain

n

E 0 [ S ]

[ ES n ]=

n .

k 2 + k 3

+[ S ]

k 1

Then the dynamics of the concentration of P given by Eq. ( 2.21 ) becomes

n

d [ P ] V max [ S ]

=

n ,

(2.25)

dt K m +[ S ]

where n V max and K m are positive constants. The parameters V max and K m are as in

Eq. ( 2.15 ), and n is a Hill coefficient . This equation is known as the Hill equation.
Unlike the Michaelis-Menten equation, the Hill equation has three adjustable param-
eters. The right-hand side of the equation f

,

n

V max [ S ]

( [ S ] ) = is a strictly increasing

n

K m [ S ]

function of [ S ] and satisfies the following two conditions: (i) lim f ( [ S ] ) = V max ,

[ S ]→∞

K 1 / n

V max

and (ii) f

2 . Biologically, this reaction can never take place at a
rate larger than V max .Thevalue K 1 / m is the substrate concentration necessary for this
reaction to occur at a rate equal to half of themaximum rate, V max . Since V max =

( [ S ]= ) =

m

k 3 E 0 ,
increasing the initial enzyme concentration linearly increases the reaction rate. When
n 1, the Hill function behaves like a step function. For example, for large n values,

f ( [ S ] ) ≈ 0 when [ S ] is small. On the other hand, f ( [ S ] ) = V max when [ S ] is big

(see Figure 2.3 ).
When the initial substrate concentration is conserved throughout the course of the
reaction and the initial enzyme concentration is much smaller than that of the substrate