Systems Biology (Methods in Molecular Biology)

X 1!

k 4

D

These reactions lead to the following ordinary differential

equations:

dX 1

dt

¼k 1 Ak 2 BX½Šþ 1 k 3 ½ŠX 12 ½ŠX 2 k 4 ½ŠðX 1 16 Þ

dX 2

dt

¼k 2 BX½Š 1 k 3 ½ŠX 12 ½ŠðX 2 17 Þ

where the rate constantsk 1 tok 4 and speciesA,Bare real and

positive. Note thatX 1 andX 2 are reactant species in dimensionless

form and, generally,

P^2

i¼ 1

dXi

dt^6 ¼0 meaning that the law of mass

conservation is not observed for the Brusselator as it considers a

non-equilibrium and nonlinear dissipative system. It can be shown

that the Brusselator can reach equilibrium state under certain con-

ditions, whenX 1 ¼A,X 2 ¼(B/A) andB<1+A^2 with all rate

constants set to 1. Figure7 shows stable dynamics of speciesX 1 and

X 2 to different values of parameters, demonstrating limit-cycle and

damped oscillations.

There have also been other works, subsequently, extending the

Brusselator model, for example the Oregonator developed by Field

and Noyes [26] which considers a third autocatalytic species, to

reflect more realistic chemical dynamics of the B-Z reactions. Fun-

damentally, the other works are also based on non-equilibrium and

nonlinear conditions, to model self-organizing chemical systems

displaying emergent responses that do not show “sum of the

parts” or linear dynamics.

3.3 Goodwin Model The Brusselator was developed for chemical systems. For studying
biological rhythms considering the regulation of genes and

(^8) ab
7 6 5 4 3 2 1 0
1.8
1.6
1.4
1.2
0.6
0.8
1
0 5 10 15
Time (-) Time (-)
20 25 30 0 5 10 15 20 25 30
Fig. 7Brusselator dynamics. (a) Limit cycle oscillations withX 10 ¼X 20 ¼1,A¼1,B¼4,k 1 ¼k 2 ¼k 3 ¼1,
(b) Damped oscillations withX 10 ¼X 20 ¼1,A¼1,B¼1.5,k 1 ¼k 2 ¼k 3 ¼1.x-axis represents time andy-
axis represents concentration in arbitrary units.X 1 isredwhileX 2 isblue
184 Kumar Selvarajoo