Circadian Rhythms

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Circadian Rhythms

• ???
• (????)

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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?

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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)

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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

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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,)

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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

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Dunlaps 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?

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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?

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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

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Mechanism in a diagram
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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. 224) nonlinear ?
  emergence, holism, stability
• Noise tolerance

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Basic concepts

• ODE(ordinary differential equation)
• Ex) pendulum

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Basic concepts

• Phase space

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Geometric paradigm of dynamics

• Classical method
• Find analytical solution
• Approximations (linearization)
• With trajectory in phase space,
• ? Find Geometry of phase space

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Geometry of dynamics
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Fixed point and stability analysis

• Fixed point a point where
• Give a small disturbance, then watch linear terms
  
• Stable, unstable, saddle

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Limit cycle ? clock

• Isolated closed trajectory
• Only in nonlinear system(linear systems wont be
  isolated)

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Slaving principle(pseudo-steady state)

• For fast variable and slow variable
• Fast variable is a slave of slow variable
• ? reduction of number of variables

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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

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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, Parrondos paradox(ref)

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Stochastic resonance
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The clock
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The clocks state
C
R
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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

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Analysis of the clock
Null cline
Fixed point
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Stochastic resonance in the clock
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Synchronization of the clocks

• Clock ? Limit cycle or oscillator
• Interacting clocks ? coupled oscillators

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Synchronization of nonlinear oscillators
Huygens - pendulum clock
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Sync in nonlinear oscillators

• Winfree model
• Modified general model(Kuramoto)

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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

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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 SCNs period determination

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Organization of Circadian Clock
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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)

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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)

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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)

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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)

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References

• Stochastic resonance
• L.Gammaitoni et al., Rev. Mod. Phys. 70, 223
  (1998)
• Molecular ratchet Parrondos 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)