Systems Biology (Methods in Molecular Biology)

EþA$

k 1

k 1

EA!

kcat

EþB

In this case, the rate ofBformation can be shown to be (see

ref.8 for detailed derivation):

dB

dt

¼

Vmax½ŠA

½ŠþA KM

ð 5 Þ

whereVmax¼kcat[E]tandKM¼kcatþk 1 k^1.

The reaction rate increases with increasing [A], approaching an

asymptotic atVmax, when all enzymes (limiting factor) are bound to

A(Fig.2B). [E]tis the total enzyme concentration andkcatis the

maximum number of enzymatic reactions catalyzed per second.

Subsequent work on this basic principle led to the extension of

the kinetics to represent more complex scenarios, such as multi-

substrate ping-pong and ternary-complex mechanisms [8].

There are certain classes of enzymes with multiple active sites

that alter the reaction kinetics in complex ways. For example, the

reaction activity for lower substrate concentration is inefficient

while at higher concentration the activity is highly efficient, result-

ing in S-shape reaction rates (Fig.2C). This usually occurs through

substrate concentration-dependent conformation changes that vary

the enzyme affinity.

The commonly used allosteric reactions adopt the Hill equa-

tion, which is a modified form of the Michaelis-Menten kinetics:

dB

dt

¼

Vmax½ŠAn

½ŠAnþKI

ð 6 Þ

wherenis the Hill coefficient that describes the cooperativity, and

KIis a constant that is different toKM. Note that negative or

positive cooperativity is represented whenn<1orn>1, respec-

tively. Whenn¼1, the Hill equation becomes Michaelis-Menten

kinetics.

Overall, there are various forms of enzyme kinetics, depending

on the types of intermediates or co-factor affecting the overall

reactions. There are entire books just dealing with different types

of enzyme kinetics, and the details are beyond the scope of this

chapter.

2.3 From Reactions
to Networks

As the development of computing power progressed significantly in

the 1960s, there have been numerous efforts to model complete

biological network modules, such as energy metabolic pathways

and immune signaling cascades. Here, we consider a series of

chemical reactions that form pathways and networks.

Consider a closed system withnspecies,X¼(X 1 ,X 2 ,...,Xn),

that are connected through chemical reactions. Given a perturba-

tion to one of the species in the system, the resultant changes in the

176 Kumar Selvarajoo