Systems Biology (Methods in Molecular Biology)

proteins, Brian C. Goodwin developed, in 1965, a highly simplified

regulatory network with a negative feedback mechanism to display

oscillatory behaviors [27]. His coupled oscillator model simulated

synchronous locking and sub-harmonic resonance arising from the

interaction of the oscillators. By varying the coupling constants, the

Goodwin model was able to show a wide range of oscillatory

frequencies.

The single-oscillator Goodwin model is illustrated by

dX 1

dt

¼

k 1

k 2 þk 3 ½ŠX 2

k 4 ð 18 Þ

dX 2

dt

¼k 5 ½ŠX 1 k 6 ð 19 Þ

whereX 1 represents a mRNA whileX 2 is the protein coded byX 1.

k 1 andk 5 represent the formation rate constants whilek 4 andk 6

represent the decay or depletion rate constants ofX 1 andX 2 ,

respectively.k 2 andk 3 control the negative regulation ofX 1 inde-

pendently and dependently byX 2 , respectively. Figure8a shows the

suppression ofX 1 onX 2 leads to periodic oscillatory dynamics.

To consider multiple crosstalk regulation between mRNAs and

proteins, Goodwin proposed a coupled oscillator

dX 11

dt

¼

k 11

k 21 þk 31 ½ŠþX 21 k 41 ½ŠX 22

k 51 ð 20 Þ

dX 12

dt

¼

k 12

k 22 þk 32 ½ŠþX 21 k 42 ½ŠX 22

k 52 ð 21 Þ

dX 21

dt

¼k 1 ½ŠX 11 k 2 ð 22 Þ

dX 22

dt

¼k 3 ½ŠX 12 k 4 ð 23 Þ

As seen from Fig.8b–d, the Goodwin’s coupled oscillator can

be used to produce several complex oscillatory patterns according

to the parameter values chosen, compared to the one in Fig.8a.

Therefore, the model has been investigated on numerous occasions

to understand emergent biological oscillatory and self-organizing

responses [28–30]. However, unlike the Brusselator, the Goodwin

models could not produce limit-cycle oscillation (a closed trajec-

tory observed in phase-space plots, where at least one other trajec-

tory spirals into it as time approaches infinity). Nevertheless, there

have been subsequent modifications to this model, for example by

Griffith in 1968 that overcame the limitation by making the repres-

sor term a sigmoidal Hill coefficient larger than 8 [31].

3.4 Synthetic
Biological Oscillator

To understand oscillatory behaviors in actual living systems, Elo-

witz and Leibler developed an artificial biological clock, called the

repressilator, in the bacteriaEscherichia coli[32]. Starting from a

Complex Biological Responses Using Simple Models 185