1. Circadian Rhythms 안용열 ( 물리학과 )

2. 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?

3. ‘ 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)

4. 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

5. 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, … )

6. 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

7. 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?

8. 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?

9. 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

10. Mechanism in a diagram Positive elementNegative element

11. 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

12. Basic concepts ODE(ordinary differential equation) Ex) pendulum

13. Basic concepts Phase space Trajectory

14. Geometric paradigm of dynamics Classical method –Find analytical solution –Approximations (linearization) With trajectory in phase space,  Find “ Geometry ” of phase space

15. Geometry of dynamics

16. Fixed point and stability analysis Fixed point : a point where Give a small disturbance, then watch linear terms –Stable, unstable, saddle

17. Limit cycle  “ clock ” Isolated closed trajectory Only in nonlinear system(linear systems won ’ t be isolated) Stable limit cycle Linear system

18. Slaving principle(pseudo-steady state) For “ fast ” variable and “ slow ” variable Fast variable is a “ slave ” of slow variable  reduction of number of variables

19. 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

20. 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)

21. Stochastic resonance

22. “ The clock ” Gene A Gene R A A 1 A A 1 50 0.01 A 50 R 5 C + 2 10 0.5 500 50 100 1 0.2 1

23. The clock ’ s state Expressed genes mRNAs R A A C R R C

24. 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

25. Analysis of “ the clock ” Fixed point Null cline

26. Stochastic resonance in “ the clock ” No noise With noise

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

28. Synchronization of nonlinear oscillators Huygens - pendulum clock

29. Sync in nonlinear oscillators Winfree model Modified general model(Kuramoto)

30. 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

31. 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

32. Organization of Circadian Clock

33. 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)

34. References About nonlinear science and mathematical tools –A.T.Winfree, “ The Geometry of Biological Time ” (1990)  2 nd 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)

35. 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)

36. 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)

37. 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)