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Fig. 5.2
Feedback loop of FRQ protein synthesis in circadian oscillators
5.3
Protein Synthesis Control Using Differential
Geometry Methods
It will be shown that using a differential geometric approach and the computation
of Lie derivatives one can arrive into the linear canonical form for the dynamical
model of the circadian oscillator.
First, the nonlinear model of the circadian oscillator is written in the following
state-space form:
0
1
0
1
0
1
K
i
K
i
Cx
3
v
m
x
1
K
m
Cx
1
v
s
x
1
x
2
x
3
0
x
1
0
@
A
C
@
A
D
@
A
K
s
x
2
(5.5)
v
d
K
d
Cx
2
K
1
x
2
C K
2
x
3
K
1
x
2
K
2
x
3
Next, the state variable
z
1
D h
1
.x/ D x
1
is defined. Next, the variable
z
2
D
L
f
h
1
.x/ is computed. It holds that
@h
1
@h
1
@h
1
z
2
D L
f
h
1
.x/ D
@x
3
f
3
)
z
2
D L
f
h
1
.x/ D f
1
)L
f
h
1
.x/ D
v
s
@x
1
f
1
C
@x
2
f
2
C
(5.6)
K
i
x
1
K
m
Cx
1
K
i
Cx
3
v
m
Equivalently
@
z
2
@
z
2
@
z
2
z
3
D L
f
h
1
.x/ D L
f
z
2
)
z
3
D
@x
1
f
1
C
@x
2
f
2
C
@x
3
f
3
)
.K
m
Cx
1
/
2
h
v
s
K
m
Cx
1
i
v
s
K
i
nx
n
1
(5.7)
K
i
K
i
Cx
3
v
m
K
m
x
1
z
3
D
v
m
3
.K
i
Cx
3
/
2
.K
1
x
2
K
2
x
3
/
