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  • Noise  what is to be removed 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

10 1 “ The clock ” A C 2 1 + R 50 0.2

5 50 Gene A A 1 50 A 500 0.01

Gene R A 1 100 A 50 0.5

The clock ’ s state 0.8

0.6

0.4

0.2

2000 1500 1000 500 A 30 C Expressed genes 40 R 50 30 40 50 60 60 C 2000 1500 1000 500 80 60 40 A 20 R 30 40 mRNAs 50 60 250 500 750 1000 1250 1500 R 1750

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

Fixed point Analysis of “ the clock ” Null cline

Stochastic resonance in “ the clock ” No noise With noise

Synchronization of “ the clocks ” • Clock  Limit cycle or oscillator • Interacting clocks  oscillators coupled

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

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 2 nd edition published in 2001 – S.H.Strogatz,

“

Nonlinear dynamics and chaos

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– J.D.Murray,

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Mathematical Biology

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(1993) – H.R.Wilson, “ Spikes, decisions, and actions ” (1990) (1994) (1999) • About coupled oscillators – A.T.Winfree,

“

The geometry of biological time

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- S.H.Strogatz,

“

Sync

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published in 2003 (1990) - 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

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, 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)