Upload
collin-riley
View
223
Download
0
Embed Size (px)
Citation preview
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
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
Geometric paradigm of dynamics
• Classical method– Find analytical solution– Approximations (linearization)
• With trajectory in phase space, Find “Geometry” of phase space
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)
Stable limit cycle
Linear system
Slaving principle(pseudo-steady state)
• For “fast” variable and “slow” variable• Fast variable is a “slave” of slow
variable reduction of number of variables
-0.5 0.5 1
0.2
0.4
0.6
0.8
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)
The clock’s state
30 40 50 60
0.2
0.4
0.6
0.8
Expressedgenes
30 40 50 60
20
40
60
80
mRNAsR
A
30 40 50 60
500
1000
1500
2000
A
C
R
250 500 750 1000 1250 1500 1750
500
1000
1500
2000
R
C
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
Synchronization of “the clocks”
• Clock Limit cycle or oscillator• Interacting clocks coupled
oscillators
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
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)