Circadian Rhythms
안용열
(물리학과)
Index
• Intro - What is the circadian rhythm?
• Mechanism in reality
• How can we understand it?
 Nonlinear dynamics
–
–
–
–
Limit cycle
Linearization and stability
Stochastic resonance
Coupled nonlinear oscillators
• Summary - What have we learned?
‘Circadian’ rhythm?
• ‘circa’ means ‘round about’
• ‘dies’ means ‘a day’
 ‘About-a-day-period behavioral
rhythm’
• Sleep-wake cycle, Insect eclosion, …
• Circadian rhythm vs. cell cycle?(ref)
Is 24 hours a long time?
• If we think that a day is long time…
 A trap!-Two short period oscillator model
 long period is extremely sensitive to
changes in the short period.
• ‘because long periods are inconvenient in
the laboratory’ (Winfree)
 aging, female endocrine cycle,
replacement of membrane phospholipids
What we know about
circadian rhythms I
• Scale
– In temporal scale  About 24 hours(ref)
– In spatial scale  From a single cell to
complex multicelluar organisms in
synchrony
– In the kingdom of life  from bacteria to
mammals (synechococcus, neurospora,
drosophila, mouse, human,…)
What we know about
circadian rhythms II
• Reliability
– Period conservation under temperature
variation (temperature compensation)
– Immunity to many kinds of chemical
perturbation
– Sensitivity to visible light of an
appropriate color
– Slow entrainment to outside environment
Dunlap’s viewpoint about
circadian clock research
• Mechanism - how does the clock
work?
• Input – how does outer world entrain
the clock?
• Output – how does the clock control
the entire organism?
Viewpoint of this
presentation(mech-specific)
• First, How can we make a 24-hours
clock in a single cell?
• We get a clock, then how do cells in
a tissue synchronize with each other?
• We get tissues in synchrony, then
how do tissues synchronize all over
the body?
Discovered Mechanism in a cell
• Positive element vs. negative element
– Positive element enhance both
– Negative element inhibit positive element
– Negative element has ‘slower’ dynamics
• This mechanism is fundamental in the
neuron interaction model(ref)
– Simplest example which has a limit cycle
Mechanism in a diagram
Positive element
Negative element
How can we understand it?
• Nonlinear dynamics!
• Why nonlinear?
– Nonlinear systems are ubiquitous
• Zoology Metaphor
– Linear systems can be broken down into
parts (superposition principle. 2+2=4)
nonlinear  emergence, holism,
stability…
– Noise tolerance
Basic concepts
• ODE(ordinary differential equation)
Ex) pendulum
Basic concepts
• Phase space
Trajectory
Geometric paradigm of
dynamics
• Classical method
– Find analytical solution
– Approximations (linearization)
• With trajectory in phase space,
 Find “Geometry” of phase space
Geometry of dynamics
Fixed point and stability analysis
• Fixed point : a point where
• Give a small disturbance, then watch
linear terms
– Stable, unstable, saddle
Limit cycle  “clock”
• Isolated closed trajectory
• Only in nonlinear system(linear
systems won’t be isolated)
Linear system
Stable limit cycle
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
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
noise-induced
dynamics(Stochastic resonance)
• Noise  what is to be removed
• Noise  what is important in dynamics
• Noise “enhance” signal (stochastic
resonance, coherent resonance)
– Climate change (Phys.Rev.Lett., 88,038501)
– Sensory system(PRL, 88,218101)
• Noise can do “work”
– Molecular ratchet, Parrondo’s paradox(ref)
Stochastic resonance
“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
The clock’s state
80
0.8
Expressed
genes
0.6
mRNAs
60
40
0.4
A
20
0.2
30
40
50
1500
30
60
C
2000
1000
R
C
R
40
50
60
2000
1500
A
1000
500
500
30
40
50
60
250
500
750
1000
1250
1500
1750
R
Analysis of “the clock”
• “The Clock” has so many variable.
 pick up two slowest variable : R, C
• Can the reduced system exhibit
‘clock’ – limit cycle – behavior?
 stability analysis of fixed point and
application of poincare-bendixon
theorem
Analysis of “the clock”
Null cline
Fixed point
Stochastic resonance
in “the clock”
No noise
With noise
Synchronization
of “the clocks”
• Clock  Limit cycle or oscillator
• Interacting clocks  coupled
oscillators
Synchronization of nonlinear
oscillators
Huygens
- pendulum clock
Sync in nonlinear oscillators
• Winfree model
• Modified general model(Kuramoto)
SCN – The master clock
•
•
•
•
In the hypothalamus of the brain
Recept light signal from retina
About 20000 neuron
Negative elements : Period(Per),
Cryptochrome(Cry)
• Positive elements: Clock, Bmal1
Synchronization in SCN
• SCN  coupled oscillators
• If f(-x) = -f(x), and if K s are all
symmetric,
• Then collective frequency is mean of all.
• Cell, 91,855 : hamster SCN’s period
determination
Organization of Circadian
Clock
What have we learned?
• Study PHYSICS!
– Abundant Nonlinearity in biology
– Nonlinear dynamics is important for
dynamical systems (ex. circadian clock)
– Noise effects are important in life
– Organisms actively use noise. (muscle,
circadian clock)
References
• About nonlinear science and mathematical tools
– A.T.Winfree, “The Geometry of Biological Time” (1990)
 2nd edition published in 2001
– S.H.Strogatz, “Nonlinear dynamics and chaos” (1994)
– J.D.Murray, “Mathematical Biology” (1993)
– H.R.Wilson, “Spikes, decisions, and actions” (1999)
• About coupled oscillators
– A.T.Winfree, “The geometry of biological time” (1990)
- S.H.Strogatz, “Sync” published in 2003
- S.H.Strogatz et al., “Coupled oscillators and biological
synchronization”, Scientific american vol 269, No. 6 (1993)
– S.H.Strogatz, From Kuramoto to Crawford, Physica D, 143, 1
(2000)
– C.L et al. and S.H.Strogatz, Cell, 91,855 (1997)
References
• About single cell level circadian rhythm
– J.C.Dunlap, “Molecular bases for Circadian Clocks”, Cell, vol 96, 271
(1999) (Review)
– N.Barkai and S.Leibler, Nature, 403, 268 (1999)
– J.M.G.Vilar et al., PNAS, 99, 5988 (2002)
– N.R.J.Glossop et al., Science, 286, 766 (1999) (mechanism of
drosophila clock genes)
– S.Panda et al., “Circadian rhythm from flies to human”, Nature,
417,329 (2002)
• Why circadian, circannual rhythms are not precisely one day
or one year?
– H.Daido, Phys. Rev. Lett. 87, 048101 (2001)
• The circadian oscillator can be synchronized by light without
input from eyes
– U.Schibler, Nature, 404, 25 (2000)
References
• About synchronization between tissues or
organisms
– U.Schibler, et al., “A web of circadian pacemaker”, Cell,
111,919 (2002)
– S.M.Reppert et al., “Coordination of circadian timing in
mammals”, Nature, 418,935 (2002)
– M.H.Hastings, nature, 417,391 (2002)
– K.Stokkan et al., Science, 291,490 (2001)
– J.D.Levine et al., Science, 298,2010 (2002)
• Cancer connection
– M.Rosbash et al., Nature, 420,373 (2002)
References
• Stochastic resonance
– L.Gammaitoni et al., Rev. Mod. Phys. 70, 223 (1998)
• Molecular ratchet & Parrondo’s paradox
–
–
–
–
R.D.Astumian et al., Phys.Rev.Lett.,72,1766 (1994)
G.P.Harmer et al., Nature, 402,864(1999)
J.M.R.Parrondo et al., Phys.Rev.Lett., 85, 5226 (2000)
R.Toral et al., cond-mat/0302324 (2003)