Systems Biology (Methods in Molecular Biology)

2 Modeling Biochemical Reactions

2.1 Mass Action
Kinetics

The earliest research in biochemical reactions, around the

mid-1800s, dealt with chemical equilibrium. That is, chemical

systems where perturbation from a stable steady-state subsequently

returns to its original state are said to be in equilibrium. For

example, heating oil at room temperature and, subsequently,

removing the heat results in the temperature of the oil decay to its

original reading. This happens in a “closed” system where external

conditions, such as pressure, energy, mass, do not enter or leave the

system that is in cooling.

To illustrate mathematically, consider a simple case involving

only the decay process, that is, the point onward when a heat source

is removed. The temperature (T) that changes over time follows the

deterministic Newton’s Law of Cooling:

dT

dt

¼kT½ŠðT 0 1 Þ

whereT 0 is the room temperature andkis the rate of cooling. The

minus sign indicates temperature drop in time. Equation1 is called

an ordinary differential equation, where the rate of change of

Tdepends only on its concentration and a rate constant.

Now consider a closed chemical reaction of molecular species

AintoB:

A!B

The arrow indicates the equilibrium state lies far to the right,

that is, the reverse reaction (B!A) proceeds only at an infinitesi-

mal extent and can be ignored. For every speciesBformed (con-

centration units in mol), anAspecies disappears:

dB½Š

dt

¼

dA½Š

dt

¼k 1 ½ŠðA 2 Þ

The constantk 1 is called the rate constant and it has the unit of

per second. Here, the masses of chemicals are conserved, and the

rate constant provides a direct measure of how fast the reaction

is. The higher thek 1 value, the faster the reaction. This type of

reaction is called a first-order reaction, as its rate only depends on

the first power of the reactant concentration (Fig.2A).

A second-order reaction occurs typically when two same species

react to form a chemical product. An example:

2 A!

k 2

A 2

The rate of such a reaction is proportional to the second power

of the concentration of reactant, for the reaction can occur only

when two molecules of the same species collide:

174 Kumar Selvarajoo