Circadian Rhythm:
analysis in detail
안용열
(물리학과)
Index
•
•
•
•
•
•
Brief Review
Formulation
Reduction of formula
Analysis
New insight from other field
Conclusion
Brief Review
• Circadian rhythm exists in a single cell
• Circadian rhythm exists in a synchrony
over whole body
• Very simple system generates robust
circadian rhythm
• Noise and nonlinear dynamics play
important roles in understanding it
Mechanism in a diagram
Positive element
Negative element
“The clock”
C
1
A
10
50
50
500
A
Gene A
2
1
50
+
0.2
1
R
0.5
5
50
0.01
A
A
Gene R
1
100
A
Formulation
A
Gene R
A
50
Gene A
A
1
1
100
A
G A' t
G R' t
A
R
G AA t
G RR t
A
R
GA t A t
GR t A t
Formulation
10
M A' t
50
AA
G AA t
GA t
MA
MA t
500
A
Gene A
1
50
0.5
A
0.01
M R' t
A
RR
G RR t
R
GR t
MR
MR t
50
A 1
A
Gene R 100
Remaining Formulations
• I think it’s waste of time
• Results
GA' t
A
G AA t
A
GA t A t ,
G R' t
R
G RR t
R
GR t A t ,
G AA ' t
A
GA t A t
A
G AA t ,
G RR ' t
R
GR t A t
R
G RR t ,
M A' t
AA
A' t
A
M R' t
G AA t
MA t
RR
R' t
R
c' t
CA
G RR t
MR t
t R t
A
GA t
A G AA
R
CA
t
MA
R
GR t
t R t
A
c t ,
MA t ,
G RR t
MR
A t
A GA
MR t ,
Ac
t
R
R t ,
t
R
GR t
C
R t
A
,
Reducing
• We can solve the ODE with computer
now.
• But let’s try to get further insight into
the essential elements
• Let’s use Slaving principle (Quasisteady state assumption)
Flashback: Slaving principle
(pseudo-steady state)
• For “fast” variable and “slow” variable
• Fast variable is a “slave” of slow variable
 reduction of number of variables
1
0.8
0.6
0.4
0.2
-0.5
0.5
1
What is the slowest variable?
C
1
A
10
50
50
500
A
Gene A
2
1
50
+
0.2
1
R
0.5
5
50
0.01
A
A
Gene R
1
100
A
Using Slaving principle
0
A G AA
A
G A A,
0
R G RR
R
G R A,
0
A GA
A
A
G AA,
0
R GR
A
R
G RR,
0
AA G AA
0
A
0
RR G RR
MA
A
A
R
c' t
CA
MA
G AA
R
R' t
GA
MR
M A,
R G RR
GR
MR
CA
R R t
A
A
GA
R
GR
C
R
M R,
R R t
A
c t
Ac
t
R
R t ,
A
,
Result
R
dR
R
dt
MR
R
R
RR
R
R
1
2
1
C
2
dC
dt
1
2
MA
C
AA
R
C
2
MA
A
1
A
AA
MA
C
R
MA
CR
A
CR
1
2
A
A
A
A
A
AA
A
AA
1
2
A
A
A
A
1
A
2
A
2
CR
MA
A
MA
1
MA
C
R
A
AA
C
A
R
4
A
2
A
A
A
4
A
4
A
MA
C
A
Very complex, but low dimensional
A
A
A
A
R
A
C
R
R
AC
R
A
A
A
MA
A
A
A
A
2
4
CR
A
CR
MA
A
MA
A
A
A
A
2
A
A
MA
CR
2
A
A
AA
AA
A
A
AA
A
A
A
R
AC
R
It works
• Simplified version also oscillates
• But oscillation level and period differs
Simplified
version
R: solid line
C: dashed line
Full version
Finding Fixed Point
• Finding nullclines
C
C
1500
1750
1500
1250
1250
1000
1000
750
750
500
500
250
R
250
50
100
150
200
250
300
R`=0 nullcline
R
50
100
150
200
250
300
C`=0 nullcline
1500
Fixed point
1000
500
50
100
150
200
250
300
Flashback: Poincare-Bendixson
theorem
• If an annulus region in 2d
– Has no stable fixed point
– Has only trajectories which are confined in it
 There exist limit cycles
Existence of Limit cycle
• Physics in biosystem  The trajectory
of the system is confined
• If there is a fixed point and if it is
unstable one,
• by Poincare-Bendixon theorem, a
Limit cycle exists
Fixed point analysis - Is it stable?
• In 1-D
dx/dt
f(x)
stable
unstable
x
Fixed point
Fixed point analysis
• In 2-D
u = x – x0, v = y - y0
For the system
dx/dt = f(x,y)
dy/dt = g(x,y)
Fixed point : (x0, y0)
( f(x0,y0) = 0, g(x0,y0) = 0 )
Substitute them to f, g
Series expansion &
Picking up only linear term
Is Fixed point stable?
• Result : the fixed point is unstable for
broad parameter range.  a limit cycle
exists
• In the case of stable fixed point  We
can’t say about limit cycle. A trajectory
can sink to the fixed point.
Stochastic resonance
in “the clock”
No noise
With noise
In other field: Neuron
• Wilson-Cowan oscillator (Neural network
model)
• Simplest model that possesses a limit
cycle(typical circuit in neuron network)
E
E
E
I
E
I
Wilson-Cowan oscillator
•
•
•
•
•
E(t), I(t) : spike rate
dE/dt = 1/5 (-E + S(1.6E – I + K))
dI/dt = 1/10(-I + S(1.5E))
When K!=0, limit cycle exists
This oscillator shows qualitatively
same behavior with circadian clock
Flashback: The clock
dE/dt = 1/5 (-E + S(1.6E – I + K))
dI/dt = 1/10(-I + S(1.5E))
1
10
A
500
A
Gene A
2
50
50
1
50
C
+
0.2
1
R
0.5
5
50
0.01
A
A
1
Gene R 100
A
Conclusion
• We’ve learned a way how to
investigate such dynamics problem
I
• The configuration( E
)
is sufficient for limit cycle behavior
• The oscillators with such structure are
ROBUST
References
• Previous presentation
• S.H.Strogatz, “Nonlinear dynamics and chaos”
(1994)
• H.R.Wilson, “Spikes, decisions, and actions”
(1999)