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Generally, if a biochemical reaction contains several substances and they form a network, complex results will display. Look at the following simple model:
reactions
ratio constant
A1 --> 2A1 ---------------------------(k1)
A1+ A2 --> 2A2 ---------------------(k2)
A2 --> 0 -----------------------------(k3)
From the model we get the following equations:
d [A1]/dt = -k2 [A1][ A2] + k1 [A1]
Figure 1. Limit Cycle
d [A2]/dt = k2[A1][ A2]
– k3 [A2]
If k1=5.0, k2=1.0, k3=5.0 and the initial densities of A1 and A2 are 1.0 and 1.0, the above equation will lead limit cycle as shown in Figure 1.
Limit cycle is a relative simple phenomenon, the following reaction may lead chaos:
Willamowski-Rossler network: [Aguda,B.D. & Clarker,B.L.,J.Chem.Phys.1988,89:7428-7434]
reactions
ratio constant ratio constant in
opposite direction
T1 + X --> 2X k1 k1_
X + Y --> 2Y k2 k2_
T5 + Y --> T2 k3 k3_
X + Z --> T3 k4 k4_
T4 + Z --> 2Z k5 k5_
Table
1: The parameters and values:
|
parameter |
value |
parameter |
value |
|
k1 |
30.0 |
k1_ |
0.25 |
|
k2 |
1.0 |
k2_ |
10-4 |
|
k3 |
10.0 |
k3_ |
10-3 |
|
k4 |
1.0 |
k4_ |
10-3 |
|
k5 |
16.5 |
k5_ |
0.5 |
The densities of the three reactants (X,Y,Z) are 1.0,1.0,1.0.
Figure
2: Fluctuating of the reactants’ densities.
Figure 3: Chaos
The models shown above is simple compared with the real network in human body, so we can imagine the complexity in reality.