Synchronization of Noisy Circadian Oscillators Francis J. Doyle III Department of Chemical Engineering Biomolecular Science and Engineering Program Institute for Collaborative Biotechnologies UC, Santa Barbara Workshop on Uncertain Dynamical Systems, Udine, August 25, 2011
Circadian Clock: Coordinator of Life Processes Behavioral and physiological Locomotor activity processes are organized along an Sleep/wake approximate 24-hour (circadian) cycle Circadian food intake pattern Central and peripheral clocks synchronize physiology to Metabolism environmental cycles of light and Hormones Blood pressure nutrients Duez et al. J Appl Physiol (2004) The master pacemaker resides within the SCN synchronizing peripheral tissue clocks Malfunction of circadian clocks contributes to the development of behavioral and metabolic disorders Duez et al. Arterioscler Thromb Vac Biol (2010)
Multi-scale Networks in Circadian Rhythms [adapted from Herzog , 2007]
1. Nodes in Network are Stochastic Cycle-to-cycle Period variation in vivo explants isolated [Herzog et al ., 2004]
2. Nodes are Qualitatively Different from Network [Liu et al., 2007] Dissociated Cry1 -/- SCN Neurons Dissociated Cry2 -/- SCN Neurons
3. Nonstationary Noise Characteristics [Meeker et al., 2010] [PER2:LUC bioluminesence data, Herzog lab]
Model Structure – Promoter Domain [Mirsky et al., PNAS , 2009] • Genetic control: PER CRY • Per1, Per2, and Rev-erb α activated by BMAL1 and CLK BMAL1 E-BOX Per1, Per2, Rev-erb α activated gene repressed by PER:CRY • Clk and Bmal1 activated by REV-ERB α RORc and repressed by RORc RORE Clk, Bmal1 REV-ERB α • Cry1, Cry2, and Rorc have PER CRY both regulatory elements REV-ERB α RORc CLK BMAL1 • Per1 and Per2 may possess RORE E-BOX Cry1, Cry2, Rorc an additional (unmodeled) regulatory element
Model Structure – Feedback Loops [Mirsky et al., PNAS , 2009] per1 per2 cry1 cry2 per1 Per1 Per2 per2 cry1 Cry1 Cry2 cry2 per1 per2 cry1 cry2 • Expressed as 21 ODEs CLK:BMAL1 CLK:BMAL1 CLK:BMAL1 CLK:BMAL1 per1 mRNA per2 mRNA cry1 mRNA cry2 mRNA per1 mRNA per2 mRNA cry1 mRNA cry2 mRNA per1 mRNA Per2 mRNA Cry1 mRNA Cry2 mRNA per1 mRNA per2 mRNA cry1 mRNA cry2 mRNA • 8 genes BMAL1 PER1 PER2 CRY1 CRY2 REV-ERBA BMAL1 PER1 PER2 CRY1 CRY2 REV-ERBA CLK BMAL1 PER1 PER2 CRY1 CRY2 REV-ERB α BMAL1 PER1 PER2 CRY1 CRY2 REV-ERBA PER 2:CRY1 • Mass-action PER 2:CRY1 PER 2:CRY1 PER 2:CRY1 PER 2:CRY2 PER 2:CRY2 PER 2:CRY2 PER 2:CRY2 kinetics (except PER 1:CRY1 PER 1:CRY1 PER 1:CRY1 PER 1:CRY1 PER 1:CRY2 transcription, PER 1:CRY2 PER 1:CRY2 PER 1:CRY2 which is bmal1 rorC rev- Bmal1 bmal Rorc rorC rev- Rev-erb α bmal1 mRNA bmal rorC rev- rev-erba Michaelis- Rev-erba mRNA bmal1 mRNA Bmal1 mRNA erba rev-erba bmal1 mRNA rev-erba 1 erba 1 erba mRNA mRNA clk mRNA Menten) clk Clk clk mRNA clk Clk mRNA clk mRNA clk mRNA rorC mRNA rorC mRNA rorC mRNA rorC mRNA RORc RORc RORc RORc
Model Structure – Intercellular Coupling [To et al., Biophys J , 2007] P E-‑BOX ¡ Cry1 ¡& ¡2 ¡ CRY1 ¡& ¡2 ¡ CRY1 ¡& ¡2 ¡ CLOCK ¡ BMAL1 ¡ BMAL1 ¡ PER1 ¡&2 ¡ P E-‑BOX ¡ Per1 ¡& ¡2 ¡ PER1 ¡&2 ¡ VIP ¡
Model Analysis Model Test hypotheses In silico clinical trials Refine design of experiment Probe for understanding Simulation and Analysis Determine best control input Experimentation
Analysis Toolkit • Data Analysis • Wavelets • Synchronization indices • Mathematical Model Analysis • Sensitivity analysis for phase (VRC, PIPRC) • Sensitivity analysis for stochastic systems (KS, KLD measures) • Formal Control Theoretic Methods (stability, coupling) • Often requires abstractions
Analysis Toolkit • Data Analysis • Wavelets • Synchronization indices • Mathematical Model Analysis • Sensitivity analysis for phase (VRC, PIPRC) • Sensitivity analysis for stochastic systems (KS, KLD measures) • Formal Control Theoretic Methods (stability, coupling) • Often requires abstractions
Wavelet Tools [Harang et al., FOSBE , 2009; Harang et al., JBR , 2011] • Morlet wavelet • Simultaneous estimation of phase, frequency, amplitude • No strong parametric assumptions • Magnitude (scale, translation) ~ strength of freq at time point • Methodology • Generate continuous wavelet transform table (CWT) • Select CWT ridges (local maxima) frequency evolution • Extensions to stochastic populations of cells • Period distribution of ensemble • Period duration of cells
Period Variability Analysis [Harang et al., JBR , 2011] Data from E. Herzog Lab
Experimental & Modeling Comparison [Harang et al., JBR , 20101]
Analysis Toolkit • Data Analysis • Wavelets • Synchronization indices • Mathematical Model Analysis • Sensitivity analysis for phase (VRC, PIPRC) • Sensitivity analysis for stochastic systems (KS, KLD measures) • Formal Control Theoretic Methods (stability, coupling) • Often requires abstractions
Phase Response Curves Maximal phase Maximal phase advance advance Dead zone Dead zone Maximal Maximal phase delay phase delay
Analytic PRC Measures • Use “Sensitivity” in the closed-loop sense – Output response to a disturbance in the controlled system • State Impulse Phase Response Curve (State IPRC) • solve adjoint linear variational equation • state “disturbance” • Cumulative Parametric Phase Sensitivity • sustained perturbation • related to classical sensitivity
General Kernel for PRC • The slope of the phase sensitivity predicts the response to an infinitesimal pulse • requires differentiation and adjoint • general input signal [Taylor et al., IEEE TAC , 2008]
Stochastic Sensitivity Analysis • Deterministic vs. Stochastic f(y(p)) y y y t t t 1 • What is sensitivity? behavior = density function [Costanza & Seinfeld, J. Chem. Phys., 1981] • Sensitivity measure:
Sensitivity Analysis for Stochastic Systems [Mirsky et al., IET Sys. Biol. , 2011] • Classic Sensitivity • Alternative measures of “perturbations” • Kolmogorov-Smirnov Distance • Kullback-Leibler Divergence
The Stochastic Clock is Resistant to Noise
Less Robust Ordering in Cell Cycle
Analysis Toolkit • Data Analysis • Wavelets • Synchronization indices • Mathematical Model Analysis • Sensitivity analysis for phase (VRC, PIPRC) • Sensitivity analysis for stochastic systems (KS, KLD measures) • Formal Control Theoretic Methods (stability, coupling) • Often requires abstractions
Cellular ¡Network ¡Model ¡-‑ ¡Interac@ons ¡ P E-‑BOX ¡ Cry1 ¡& ¡2 ¡ CRY1 ¡& ¡2 ¡ CRY1 ¡& ¡2 ¡ CLOCK ¡ BMAL1 ¡ BMAL1 ¡ PER1 ¡&2 ¡ P E-‑BOX ¡ Per1 ¡& ¡2 ¡ PER1 ¡&2 ¡ VIP ¡
Global + Local Cues = Robust Performance [To et al., Biophys. J.; 2007; Wang and Doyle, Automatica , 2011] Global + Local Cues Global: 1. Satellites 2. GPS 3. Command centers 2007 Local: Global 1. Field commands 2. Personal communications 3. Cellular, WIFI, WIMAX Local Master (global)- slave (local) architecture Information GPS provides dissemination reference and distributed Range improvement storage via beamforming S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ Sensors Sensors Sensors Timing and Phase Synchronization in Wireless Networks
Sync Rate of Oscillator Networks Linear Phase Model [Wang & Doyle III, Automatica , 2011] • System Dynamics • Theorem For an oscillator network with a global cue and an arbitrary local cue (not necessarily symmetric or balanced), the oscillators will always synchronize, and synch rate is only determined by the global cue wyqthu@gmail.com
Sync Rate of Oscillator Networks Linear Phase Model with Frequency Noise [Wang & Doyle, Automatica , 2011] • System Dynamics • Theorem For an oscillator network with a global cue and an arbitrary local cue (not necessarily symmetric or balanced), the oscillators will always synchronize in the stochastic sense, and synch rate is only determined by the global cue wyqthu@gmail.com
Sync Rate of Oscillator Networks Linear Phase Model with Delays [Wang & Doyle, CDC , 2010] • System Dynamics ( a i , j = a j , i ) • Theorem (Positive Coupling) If delays in different channels are identical, when with determined by synch rate is only determined by the global cue; when , synch rate increases with an increase in the global cue, and it decreases with an increase in the local cue If delays in different channels are diverse, synch rate increases with an increase in the global cue • Theorem (Negative Coupling) Synch rate increases with an increase in the global cue and it decreases with an increase in the local cue
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