Coupling, entrainment, and synchronization of biological oscillators Didier Gonze Unité de Chronobiologie Théorique Service de Chimie Physique - CP 231 Université Libre de Bruxelles Belgium
Introduction Christiaan Huygens (1629-1695) It is quite worths noting that when we suspended two clocks so constructed from two hooks imbedded in the same wooden beam, the motions of each pendulum in opposite swings were so much in agreement that they never receded the least bit from each other and the sound of each was always heard simultaneously. Further, if this agreement was disturbed by some interference, it reestablished itself in a short time. For a long time I was amazed at this unexpected result, but after a careful examination finally found that the cause of this is due to the motion of the beam, even though this is hardly perceptible.
Introduction S.H. Strogatz: "In the animal world, groups of Southeast Asian fireflies(*) provide a spectacular example of synchronization. Along the tidal rivers of Malaysia, Thailand and New Guinea, thousands of fireflies congregate in trees at night and flash on and off in unison. When they first arrive, their flickerings are uncoordinated. But as the night goes on, they build up the rhythm until eventually whole treefuls pulsate in silent concert" (*) "firefly" = luciole
Introduction "Talking to Strangers" (from "The Trials of Life", D. Attenborough, 1992). http://www.youtube.com/watch?v=obo_pkstyzc
Introduction Youtube: synchronization of metronomes (by Tom & Irvine) http://www.youtube.com/watch?v=yysnky4whym
Introduction Synchronization of Cellular Clocks in the Suprachiasmatic Nucleus Yamaguchi et al (2003) Science 302:1408-1412. http://www.sciencemag.org/cgi/content/full/sci;302/5649/1408/dc1
Overview Response of an oscillator to an external perturbation Harmonic vs limit-cycle oscillators Phase response curves, isochrones Periodic forcing of an oscillator Entrainment Arnold tongues, phase locking Synchronization of mutually coupled oscillators Local vs global coupling In-phase vs out-of-phase synchronization From 2 to N oscillators
Limit-cycle oscillators An oscillator is a system that execute a periodic behavior. A swinging pendulum returns the same point in space (x) at regular intervals; furthermore its velocity (v) also rises and falls with regularity. The amplitude of the oscillations depends on the initial perturbation; i.e. the height from which it is released (θ). pendulum θ x=0 v=v max x=x max v=0 spring Biological oscillators, in contrast, tend to have not only a characteristic period, but also a characteristic amplitude. In the phase space their trajectories correspond to a limit cycle. If a perturbation is exerted on such a system, they will automatically come back to their normal behavior, i.e. to their limit cycle. They indeed incorporate a dissipative mechanism to damp oscillators that grow too large and a soure of energy to pump up those that become too small. circadian clock Strogatz, Sci. Am. dec 1993, pp.68-74
Limit-cycle oscillators Lotka-Volterra system A model for circadian clock The Lotka-Volterra model behaves like an harmonic oscillator: its period and amplitude depend on the initial conditions This model for circadian clock generates limitcycle oscillations, characterized by a fixed period and amplitude. If a perturbation is applied, the system returns to its limit cycle.
Perturbation and phase shift A model for circadian clock Unperturbed oscillations Perturbed oscillations
Perturbation and phase shift A model for circadian clock Following the perturbation, the oscillations come back to the stable limit cycle, but they are now phase shifted with respect to the unperturbed oscillations: the maximum of the oscillations does not occur at the same time in each case. Thus, a transient perturbation induced a "permanent" phase shift in the oscillations. Such a phase shift can be expressed as a fraction of the period. It is usually comprised between -period/2 and period/2. A negative value indicates a phase advance, whereas a positive value indicates a phase delay. It can also be expressed as an angle (and comprised between - π and +π).
Phase response curve Model for circadian clock The sign and the amplitude of the phase depend on the amplitude of the perturbation and on the time at which the perturbation is exerted. The curve that shows the phase shift as a function of the time (phase) of the perturbation is called the phase response curve (PRC). phase advance phase delay Here we arbitrarily defined phase 0 as the minimum of mrna variable
Phase response curve: Circadian clocks PER-TIM model for circadian rhythms in Drosophila In Drosophila, light is known to activate the degradation of TIM protein. It was shown that a transient light pulse can phase shift the circadian oscillations in Drosophila. Leloup J-C & Goldbeter A (1998) A model for circadian rhythm in Drosophila incorporating the formation of a complex between the PER and TIM protein. J. Biol. Rhythms. 13: 70-87.
Phase response curve: Circadian clocks PER-TIM model for circadian rhythms in Drosophila Limit-cycle oscillations in constant conditions (constant darkness, DD) pulse duration v dt(ll) v dt(dd) pulse amplitude (strength) In the model, light is "included" in the TIM degradation rate, v dt. Leloup J-C & Goldbeter A (1998) J. Biol. Rhythms. 13: 70-87.
Phase response curve: Circadian clocks PER-TIM model for circadian rhythms in Drosophila Phase shifting induced by a light pulse: depending on the time (phase) at which the light pulse is given, the oscillations are advance, delayed, or no phase shifted compared the the unperturbed oscillations. Phase shift Leloup J-C & Goldbeter A (1998) J. Biol. Rhythms. 13: 70-87.
Phase response curve: Circadian clocks PER-TIM model for circadian rhythms in Drosophila Comparison with experimental data Wild type Short period mutant Experimental data (Hall JC & Rosbash M (1987) Genes and biological rhythms. Trends Genet. 3, 185-91) Theoretical phase response curve (Leloup J-C & Goldbeter A (1998) J. Biol. Rhythms. 13: 70-87)
Phase response curve: Circadian clocks PER-TIM model for circadian rhythms in Drosophila Influence of the light pulse strength and duration on the PRC Leloup J-C & Goldbeter A (1998) J. Biol. Rhythms. 13: 70-87.
Phase response curve PRC: type 1 vs type 0 Figure from D. Gonze, PhD thesis (2001) It is usual to distinguish between two types of phase response curves: In type 1 phase response curve, the maximum phase shift observed is much smaller than half of the period of the oscillations (cf. cases A and B illustrated here). This situation is thus encountered when the effect of the perturbation is rather small. In contrast, in type 0 phase response curve, large shifts are obtained and the phase response curves is discontinuous (cf. case C shown here). Such a situation occurs when the system is always reset to the same phase, regardless of the phase of the perturbation; for example when the perturbation brings the system to a steady state.
Phase response curve PRC: type 1 vs type 0 An alternative way to represent phase response curves is to plot the new phase as a function of the previous phase (phase of the perturbation). In this case the CRP is refered to as phase resetting curve. In this representation, type 1 PRC have a slope of about 1, whereas type 0 PRC have a slope of about 0. Type 0 PRC are sometimes said strong resetting. Figure from scholarpedia (phase response curve): http://www.scholarpedia.org/article/phase_response_curve
Phase response curve
Phase response curve
Phase response curve http://as.vanderbilt.edu/johnsonlab/prcatlas/
Phase response curve Actogram Phase response curve Oscillator design Phase response curves: Allow links between the core (genetic) clock and the overt (physiological) rhythm Give insights on the oscillatory mechanism (PRC0 <-?-> Hopf bifurcation/relaxation oscillator) Give information on entrainment properties [see later]... See also: Granada A, Hennig RM, Ronacher B, Kramer A, Herzel H. (2009) Phase response curves elucidating the dynamics of coupled oscillators. Methods Enzymol. 2009;454:1-27
Periodic perturbation So far we have seen how an oscillator responds to a transient perturbation. Now: How does an oscillator behave in presence of a periodic perturbation????
Entrainment Periodic forcing An oscillator that is under the control of periodic external signal is said periodically forced. Such a forcing can lead to entrainment. Entrainment implies that the period of the forced oscillator is exactly the one of the external signal. This also means that the phase of the oscillations are locked (i.e. the maximum of a variable occurs always at the same phase with respect to the external forcing). The oscillations are said phase-locked. Think about circadian rhythms: any living organism is subject to the periodic variation of light and temperature induced by the day & night cycle.
Entrainment: Circadian clocks Entrainment of circadian oscillations in Drosophila Circadian oscillations (characterized by a free running period of about 24h) are entrained by the external light-dark cycle and thus gets a period of exactly 24h. Limit cycle oscillations in DD Entrained oscillations in LD In constant darkness (DD), the period of the limit cycle oscillations is 23.8h. The light-dark cycle (LD cycle) is modeled by a periodic, square-wave, variation of parameter v dt. Under appropriate LD12:12 conditions, entrainment occurs: the oscillations get a period of exactly 24.0h and are phase locked (the maximum of per mrna appears every day at the middle of the night).
Entrainment Question: Are forced oscillators always entrained by a periodic external signal? No! This depends on the signal strength, on the type of oscillator, on the period of the oscillators, on the nature of the coupling, etc... Question: If the system is not entrained, how does it behave? What are the alternative dynamical behaviors?
Entrainment: Arnold tongues The behavior depends on the signal strength (A) and on the relative period of the oscillator (ω 0 ) and of the forcing signal (ω). A entrainment quasi-periodic behavior (QP) quasi-periodic behavior (QP) ω 0 ω Entrainment will occur when the period of the oscillator and of the external signal are relatively close, and for a sufficient amplitude of the signal. Outside the entrainment region, quasi-periodic behavior occurs.
Entrainment: Arnold tongues The behavior depends on the signal strength (A) and on the relative period of the oscillator (ω 0 ) and of the forcing signal (ω). A entrainment 1:2 entrainment 1:1 entrainment 2:1 QP QP QP QP ω 0 /2 ω 0 Sub-harmonic entrainment occurs when the period of the oscillator is multiple of the period of the external signal. Sub-harmonic entrainment means that the oscillations undergoes 2 oscillations/forcing period (case ω 2ω 0, noted 2:1), one oscillation/2 forcing periods (case ω 1/2ω 0, noted 1:2), etc. 2ω 0 ω
Entrainment: Arnold tongues The behavior depends on the signal strength (A) and on the relative period of the oscillator (ω 0 ) and of the forcing signal (ω). A entrainment 1:2 entrainment 2:3 entrainment 1:1 entrainment 3:2 entrainment 2:1 QP QP QP QP QP QP ω 0 /2 2ω 0 /3 ω 0 3ω 0 /2 2ω 0 ω For any rational ratio between the period of the oscillator and the period of the external signal, sub-harmonic entrainment may occur. These entrainment regions are called Arnold tongues (from the name of the russian mathematician Vladimir Arnold).
Entrainment: Arnold tongues The behavior depends on the signal strength (A) and on the relative period of the oscillator (ω 0 ) and of the forcing signal (ω). A CHAOS CHAOS CHAOS CHAOS 1:2 2:3 1:1 2:3 2:1 QP QP QP QP QP QP QP QP QP QP QP ω 0 /2 2ω 0 /3 ω 0 3ω 0 /2 2ω 0 ω At large amplitude of the periodic signal, chaos occurs.
Entrainment Phase response curve What can we learn about entrainment properties from Phase Response Curves? PRC as a gauge of entrainment: The magnitude of the phase-shifting exhibited by the clock is a gauge to the limit of entrainment. Indeed, one can expect that PRC with large phase-shift (e.g. PRC of type 0) can permit synchronization to LD cycle of a broader range as compared with low amplitude PRC. Moreover the shape of the PRC gives insights on the entrainment phase. Ref: Johnson, 1992 Ex: Geier et al, JBR 2005 PRC as a measure of reentrainment time (i.e. after a jet lag). Ref: Winfree, 1980? Ex: Leloup & Goldbeter, JTB 2013
Entrainment: Circadian clocks Entrainment and chaos in a model for circadian clocks In Neurospora (as well as in mammals), light enhances the transcription of the clock gene frq (genes per1-3 in mammals). Light is considered in the model as a periodic forcing of the parameter v s : v s takes its basal value during the dark phase and a higher value (v max ) in the light phase. Gonze D, Goldbeter A (2000) Entrainment versus chaos in a model for a circadian oscillator driven by light-dark cycles. J. Stat. Phys. 101: 649-663.
Entrainment: Circadian clocks Entrainment and chaos in a model for circadian clocks (A) No forcing: autonomous, limit cycle oscillations. (B) Average amplitude forcing: entrainment. (C),(D) High amplitude forcing: chaos. At low amplitude, the systems undergoes quasi-periodic oscillations (not shown): these oscillations look very regular, but they are progressively shifted whith respect to the light-dark forcing. Behavior observed as a function of the period and the amplitude of the forcing. Gonze, Goldbeter (2000) J. Stat. Phys. 101: 649-663.
Entrainment: Circadian clocks Entrainment and chaos in a model for circadian clocks Chaotic attractor obtained when the amplitude of the forcing signal (LD cycle) is large. This behavior is associated with a large variability of the peak-to-peak interval (there is no entrainment) and of the phase of the oscillations (there is no phase-locking). Gonze, Goldbeter (2000) J. Stat. Phys. 101: 649-663.
Entrainment: Circadian clocks Loss of entrainment of circadian clocks can be linked with physiological dysfunctions. Non-24 h SPS (QP+Jump) Quasi-periodic behavior obtained with the mammalian circadian clock model. Here, QP is associated with jumps in the phase. Source: J.-C. Leloup Uchiyama et al. (1996) Sleep 19:637-40.
Entrainment: Circadian clocks Reentrainment of circadian clocks after a jet lag can also be studied with the model. Phase Advance (short night) Phase Delay (long day) Source: J.-C. Leloup
Entrainment: NFkB
Entrainment: NFkB
Entrainment: NFkB
Entrainment of synthetic clocks (exp.)
Entrainment of synthetic clocks (exp.)
Entrainment Entrainment can also occur between two oscillators, when one oscillator (slave oscillator) is under the control of the other one (master oscillator or pacemaker). x x z y z y master oscillator slave oscillator
Entrainment Entrainment can also occur between two oscillators, when one oscillator (slave oscillator) is under the control of the other one (master oscillator or pacemaker). Examples: Many tissues undergo circadian oscillations, but most of them are peripheral clocks which are under the control of (and entrained by) the central circadian pacemaker located in the suprachiasmatic nuclei (SCN) of the brain. In some cells, the cell cycle is entrained by the circadian clock. The molecular mechanism of the cell cycle in under the control of the circadian clock and, consequently cells divide with a period of 24h. The segmentation clock controlling the developement appears to involve a network of coupled oscillators.
Entrainment: circadian clock - cell cycle Coupling between the circadian clock and the cell cycle Wee1
Entrainment: circadian clock - cell cycle Coupling between the circadian clock and the cell cycle Circadian clock Cell cycle Cyclin Cdc25 M + M k sw Wee1 Z + Z Wee1 + M=active cdc2 kinase Z=active cyclin protease Source: Claude Gérard
Entrainment: circadian clock - cell cycle
Entrainment: circadian clock - cell cycle Coupling between the circadian clock and the cell cycle
Entrainment: circadian clock - cell cycle Coupling between the circadian clock and the cell cycle
Entrainment: circadian clock - cell cycle Coupling between the circadian clock and the cell cycle
Entrainment: Segmentation clock Coupled oscillators in the segmentation clock
Entrainment: Segmentation clock Coupled oscillators in the segmentation clock FGF oscillator Wnt oscillator Notch oscillator Goldbeter & Pourquié (2008) Modeling the segmentation clock as a network of coupled oscillations in the Notch, Wnt and FGF signaling pathways J. Theor. Biol 252: 574-585
Entrainment: Segmentation clock Coupled oscillators in the segmentation clock FGF osc Wnt osc Notch osc FGF osc Wnt osc Notch osc When the 3 oscillators are uncoupled, they oscillate independently, with their free running period. Goldbeter & Pourquié (2008) J. Theor. Biol 252: 574-585 When the 3 oscillators are coupled, they all oscillate with the same period, equals to the one of the FGF oscillator. Wnt and Notch oscillators are entrained. Note that the amplitude of the entrained oscillators are also affected.
Mutual coupling and synchronization
Mutual coupling and synchronization When two oscillators are mutually coupled... Source: Youtube: synchronization of metronomes (by Tom & Irvine): http://www.youtube.com/watch?v=yysnky4whym http://www.youtube.com/watch?v=w1tmzascr-i Voir aussi: Ligeti: poème symphonique pour 100 metronomes http://www.youtube.com/watch?v=x8v-udhcdyg
Mutual coupling How to couple two oscillators? Direct coupling Global coupling A variable of one oscillator exerts an influence on the second oscillator and vice versa. A variable is common to the two oscillators. x x x x z y y z z y z
Mutual coupling How to couple N oscillators? full coupling local coupling random coupling global coupling In the case of a local coupling, the topology of the network might be important. y y =? y = common variable belonging to each oscillator. y = common variable controlled by each oscillator. y = quorum sensing.
Types of synchronization Types of synchronization See animation: Scholarpedia: synchronization (by Pikovsky & Rosenblum) http://www.scholarpedia.org/article/synchronization
Synchronization: cell cycle Synchronization in antiphase in a minimal coupled model for the cell cycle coupling functions Romond et al (1999) Alternating oscillations and chaos in a model of two coupled biochemical oscillators driving successive phases of the cell cycle Annals NY acad. Sci. 30:180-193
Synchronization: cell cycle Synchronization in antiphase in a minimal coupled model for the cell cycle Under "appropriate" conditions, the model can generate anti-phase sycnhronization, with alternating oscillations of S-phase and M- phase cyclins (C 1 and C 2 ).
Synchronization: cell cycle The cell cycle as a multiple oscillators system Gérard C, Goldbeter A (2009) Temporal self-organization of the cyclin/cdk network driving the mammalian cell cycle. Proc Natl Acad Sci USA 106, 21643-21648. Gérard C, Goldbeter A (2011) From simple to complex patterns of oscillatory behavior in a model for the mammalian cell cycle containing multiple oscillatory circuits. Chaos 20, 045109
Synchronization from cell-cell coupling Various mechanims allow cells to communicate with each others Examples: (A) Delta-Notch signaling (B) FGF/RTK signaling (C) Glutamate, GABA, Dopamine, Norepinephrine (D) Dopamine, Vasopressin (hypothalamus), Melatonine (Pineal), Insulin (Pancreas) (Bottom, right) Cell-cell coupling by gap junctions (via connexins) (not shown) Quorum sensing (bacteria)
Synchronization of circadian clocks Synchronization of a population of circadian clocks in the SCN Single cell model X = clock gene mrna concentration Y = clock protein concentration Z = gene transcription repressor
Synchronization of circadian clocks Synchronization of a population of circadian clocks in the SCN Coupling through the mean field Circadian clock: Neurotransmitter: Average neurotransmitter concentration: Gonze et al (2005) Biophys. J. 89: 120-129 (mean field)
Synchronization of circadian clocks Synchronization of a population of circadian clocks in the SCN Coupling through the mean field: results Synchronization of 10000 cells (circadian oscillators) In absence of coupling, each oscillator oscillates with its own period (the periods are distributed around 24h). When the coupling strength is sufficient, the mean field (average neurotransmitter concentration) oscillates and entrains each individual oscillator. These oscillators are then synchronized, nearly inphase.
Synchronization of circadian clocks Synchronization of a population of circadian clocks in the SCN Coupling through the mean field: results Coupling of 2 cells (circadian oscillators) The behavior of this system can be explored for the case of 2 coupled oscillator. Here is shown the dynamical behavior obtained as a function of the coupling strength K and of the ration of the periods of the oscillators r. Several dynamical behavors can occurs in this system: quasiperiodicity (QP), stable steady state (SSS), synchronization or complex oscillations (PD and chaos)
Synchronization of metabolic oscillators Synchronization of metabolically driven insulin secretion Pedersen, Bertram, Sherman (2005) Intra- and inter-islet synchronization of metabolically driven insulin secretion. Biophys J. 89:107-19.
Synchronization of metabolic oscillators Synchronization of metabolically driven insulin secretion fast synchronization of insulin secretion slow synchronization of glycolysis Pedersen, Bertram, Sherman (2005) Intra- and inter-islet synchronization of metabolically driven insulin secretion. Biophys J. 89:107-19.
Synchronization of coupled repressilators
Synchronization of coupled repressilators Q increases (Q <-> diffusion rate <-> couping strength) R = order parameter (measure the level of synchronization) Effect of coupling strength, Q Effect of cell-cell variability, Δβ
Synchronization of coupled relaxation osc.
Synchronization of synthetic clocks (exp.)
Synchronization of synthetic clocks (exp.) https://www.youtube.com/watch?v=rbawc2gdyjq
References Books Pikovsky A, Rosenblum M, Kurths J (2001) Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge Univ Press. Winfree A (1980) The Geometry of Biological Time, Springer. Strogatz S (2003) Sync: The Emerging Science of Spontaneous Order, Penguin Press. Review papers Strogatz (1993) Coupled oscillators and biological synchronization, Sci. Am., dec. 1993. Glass (2001) Synchronization and rhythmic processes in physiology, Nature 410: 277-284. Rosenblum & Pikovsky (2003) Synchronization: from pendulum clocks to chaotic lasers and chemical oscillators, Contemp. Phys. 44: 401-416.