simple six coupled linear differential equations model approximat-
ing the gene regulatory network of repressing genes, they investi-
gated the key regulatory parameters that are required to produce
unstable steady-state, resulting in regular oscillations:
dXi
dt
¼Xiþ
k 1
1 þYjn
k 2 ð 24 Þ
dYi
dt
¼k 3 YjXi
ð 25 Þ
whereXi(i¼ lacl,tetR,cl) represent mRNAs (genes) andYj
(j¼λcl, Lacl, TetR) represent proteins concentrations. Figure9a
shows the oscillations in protein concentrations with time.
Subsequently, a plasmid encoding the repressilator and a
reporter protein were constructed and inserted into the bacteria.
50 a
cd
b
40
30
20
10
-10
-20
-30
80
40
60
20
-20
0
0
0
40
30
20
10
-10
0
20 40
Time (-) Time (-)
Time (-) Time (-)
60 80 100
0 200 400 600 800 1000 0 200 400 600 800 1000
0 200
200
150
100
50
-50
0
400
s
600 800 1000
Fig. 8Simple and Complex Goodwin dynamics. Simulations with single oscillator with parameters (a)X 0 ¼7,
Y 0 ¼10,k 1 ¼72,k 2 ¼36,k 3 ¼1,k 4 ¼2,k 5 ¼1,k 6 ¼0, and coupled oscillators (b)X 10 ¼7,Y 10 ¼10,
X 20 ¼7,Y 20 ¼10,k 11 ¼k 12 ¼360,k 21 ¼36,k 22 ¼43,k 31 ¼1.0,k 32 ¼0,k 41 ¼0.1,k 42 ¼1,k 51 ¼5,
k 52 ¼5,k 1 ¼0.5,k 2 ¼0,k 3 ¼0.6,k 4 ¼0, (c)k 31 ¼0.5,k 41 ¼0.2,k 51 ¼8,k 52 ¼8, (d)k 31 ¼0.1,k 32 ¼1.
Only parameters that are different from (b) are listed for (c)and(d).x-axis represents time andy-axis represents
concentration in arbitrary units.X 1 isredwhileX 2 isbluefor (a),X 1 isdark bluewhileX 2 islight blueforb–d
186 Kumar Selvarajoo