Systems Biology (Methods in Molecular Biology)

Under the hypothesis of homogeneity of the solution where

reactions occur, the mass action law asserts that the time evolution

of the concentrations of the chemical substances is described by the

system of ordinary differential equations:

dAj

dt

¼

Xn

i¼ 1

ri μijνij



Aν 1 i^1 Aνmim ð 2 Þ

wherej¼1,...,m, and the symbols represent both the chemical

substance and its concentration;riis the rate constant. The rate

equations2 are derived under the following assumptions.

1. Chemical reactions. When they occur, are due to elastic colli-
   sions between the reactants.


2. Homogeneity of the reacting substances in the solution.


3. Thermal equilibrium of the solution.

At the atomic and molecular scale, chemical reactions between

molecules can occur only if molecules collide or approach each

other to small distances where bounding forces become meaning-

ful. These chemical bounding forces are of electrical or quantum

origin, and at distances larger than the mean free path they become

less important when compared with the kinetics associated with the

molecular motion. As chemical reactions only occur if the chemical

substances involved collide, the vector fields associated with the

right-hand side of Eq.2 are in general quadratic, representing

binary collisions. Higher order polynomial vector fields are possible

but, at the microscopic level, they are associated to triple or higher

order collisions, a situations that occurs with a very low probability.

The equation2 can also be written in the matrix form,

dxj

dt

¼ΓωðÞA ð 3 Þ

whereΓis thenmmatrix andAT¼(A 1 ,...,Am)

ωðAÞ¼

r 1 ðμ 1 jν 1 jÞAν 1 i^1 ... Aνm^1 m

...: ...: ...

rnðμnjνnjÞAν 1 n^1 ... Aνmnm

0

@

1

A ð 4 Þ

in generaln 6 ¼m. Associated with the differential equations one has

the conservation laws:

d

dt

ðÞ¼Aνk 0 ð 5 Þ

The input of genetic regulatory networks contains the list of

transcriptional activators and repressors of the network. If, in gen-

eral, the genes are catalytic substances presented in any genetically

controlled biological process, the usual threshold concept in

72 Rodolfo Guzzi et al.