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 45

n=2
n=8
n=40

0

[S]

FIGURE 2.3
Representative curves from the Hill equation given by Eq. ( 2.25 ) for various values of the
Hill coefficient n 2 when both K m and V max are constant. The curves are all sigmoidal

and they behave like a step function when n is big (solid curve).

concentration, Eq. ( 2.25 ) can be written in terms of the substrate consumption rate.
That is, when

S 0 =[ S ]+[ P ] , (2.26)

where S 0 is the initial substrate concentration, Eq. ( 2.25 ) becomes

n

d [ S ] V max [ S ]

=−

n .

dt K m +[ S ]

Notice that the Hill function reduces down to a Michaelis-Menten function when
n 1. Both the Hill function and the Michaelis-Menten function are commonly used

in modeling of biochemical reaction networks.
Exercise 2.2. The Hill equation is an approximation for multi-molecule binding
and it assumes simultaneous binding of n -molecules of a substrate S to the enzyme E
above. Now suppose that two molecules of the substrate S are undergoing a reaction
with an enzyme in an ordered manner as follows:

=

S k 1 S k 3

k 5
→ E + k 2 ES + k 4 ES 2 P + E . (2.27)

Derive a rate equation under the steady state assumption and compare it with the
Hill equation given in Eq. ( 2.25 ). When do these two equations become roughly the
same?