Circadian Rhythms

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 ina 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. Positive element Negative element Mechanism in a diagram

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. Trajectory Basic concepts • Phase space

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

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

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

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” Null cline Fixed point

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

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) 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)

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)