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)

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)

Stochastic resonance

“The clock”

Gene A Gene R

AA

1 AA

1

50 0.01

A50

R 5

C

+

2

100.5

500 50

50 100

10.2

1

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

Analysis of “the clock”

Fixed point

Null cline

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)