Synchronization of Noisy Circadian Oscillators Francis J. Doyle III - - PowerPoint PPT Presentation

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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

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SLIDE 1

Francis J. Doyle III

Department of Chemical Engineering Biomolecular Science and Engineering Program Institute for Collaborative Biotechnologies UC, Santa Barbara

Synchronization of Noisy Circadian Oscillators

Workshop on Uncertain Dynamical Systems, Udine, August 25, 2011

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SLIDE 2

Circadian Clock: Coordinator of Life Processes

The master pacemaker resides within the SCN synchronizing peripheral tissue clocks Malfunction of circadian clocks contributes to the development of behavioral and metabolic disorders

Locomotor activity Sleep/wake Circadian food intake pattern Metabolism Hormones Blood pressure

Duez et al. J Appl Physiol (2004) Duez et al. Arterioscler Thromb Vac Biol (2010)

Behavioral and physiological processes are organized along an approximate 24-hour (circadian) cycle Central and peripheral clocks synchronize physiology to environmental cycles of light and nutrients

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SLIDE 3

Multi-scale Networks in Circadian Rhythms

[adapted from Herzog, 2007]

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SLIDE 4
  • 1. Nodes in Network are Stochastic

[Herzog et al., 2004]

in vivo explants isolated

Period Cycle-to-cycle variation

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SLIDE 5
  • 2. Nodes are Qualitatively Different from Network

[Liu et al., 2007]

Dissociated Cry2 -/- SCN Neurons Dissociated Cry1 -/- SCN Neurons

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SLIDE 6
  • 3. Nonstationary Noise Characteristics

[Meeker et al., 2010] [PER2:LUC bioluminesence data, Herzog lab]

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SLIDE 7
  • Genetic control:
  • Per1, Per2, and Rev-erbα

activated by BMAL1 and activated gene repressed by PER:CRY

  • Clk and Bmal1 activated by

RORc and repressed by REV-ERBα

  • Cry1, Cry2, and Rorc have

both regulatory elements

  • Per1 and Per2 may possess

an additional (unmodeled) regulatory element

Model Structure – Promoter Domain

[Mirsky et al., PNAS, 2009]

E-BOX Per1, Per2, Rev-erbα

CLK BMAL1 PER CRY

RORE Clk, Bmal1 E-BOX RORE Cry1, Cry2, Rorc

CLK BMAL1 PER CRY RORc RORc REV-ERBα REV-ERBα

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SLIDE 8

Model Structure – Feedback Loops

[Mirsky et al., PNAS, 2009]

  • Expressed as

21 ODEs

  • 8 genes
  • Mass-action

kinetics (except transcription, which is Michaelis- Menten)

per1 cry2 cry1 per2 per1 mRNA per2 mRNA cry1 mRNA cry2 mRNA PER1 PER2 CRY1 CRY2 PER 2:CRY1 PER 2:CRY2 PER 1:CRY1 PER 1:CRY2 bmal1 bmal1 mRNA rev-erba mRNA rev- erba REV-ERBA BMAL1 rorC rorC mRNA RORc clk clk mRNA CLK:BMAL1 per1 cry2 cry1 per2 per1 mRNA per2 mRNA cry1 mRNA cry2 mRNA PER1 PER2 CRY1 CRY2 PER 2:CRY1 PER 2:CRY2 PER 1:CRY1 PER 1:CRY2 bmal 1 bmal1 mRNA rev-erba mRNA rev- erba REV-ERBA BMAL1 rorC rorC mRNA RORc clk clk mRNA CLK:BMAL1 per1 cry2 cry1 per2 per1 mRNA per2 mRNA cry1 mRNA cry2 mRNA PER1 PER2 CRY1 CRY2 PER 2:CRY1 PER 2:CRY2 PER 1:CRY1 PER 1:CRY2 bmal 1 bmal1 mRNA rev-erba mRNA rev- erba REV-ERBA BMAL1 rorC rorC mRNA RORc clk clk mRNA CLK:BMAL1 Per1 Cry2 Cry1 Per2 per1 mRNA Per2 mRNA Cry1 mRNA Cry2 mRNA PER1 PER2 CRY1 CRY2 PER 2:CRY1 PER 2:CRY2 PER 1:CRY1 PER 1:CRY2 Bmal1 Bmal1 mRNA Rev-erba mRNA Rev-erbα REV-ERBα BMAL1 Rorc rorC mRNA RORc Clk Clk mRNA CLK CLK:BMAL1

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SLIDE 9

CLOCK ¡ BMAL1 ¡ BMAL1 ¡ PE-­‑BOX ¡ PE-­‑BOX ¡ Per1 ¡& ¡2 ¡ Cry1 ¡& ¡2 ¡ PER1 ¡&2 ¡ CRY1 ¡& ¡2 ¡ VIP ¡ CRY1 ¡& ¡2 ¡ PER1 ¡&2 ¡

Model Structure – Intercellular Coupling

[To et al., Biophys J, 2007]

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SLIDE 10

Model Analysis

 Test hypotheses  In silico clinical trials  Refine design of experiment  Probe for understanding  Determine best control input

Simulation and Analysis Model Experimentation

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SLIDE 11
  • 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

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SLIDE 12
  • 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

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SLIDE 13
  • 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

Wavelet Tools

[Harang et al., FOSBE, 2009; Harang et al., JBR, 2011]

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SLIDE 14

Period Variability Analysis

[Harang et al., JBR, 2011] Data from E. Herzog Lab

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SLIDE 15

Experimental & Modeling Comparison

[Harang et al., JBR, 20101]

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SLIDE 16
  • 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

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SLIDE 17

Phase Response Curves

Maximal phase advance Maximal phase delay Dead zone Maximal phase advance Maximal phase delay Dead zone

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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)

  • Cumulative Parametric Phase Sensitivity
  • sustained perturbation
  • related to classical sensitivity
  • solve adjoint linear variational equation
  • state “disturbance”
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SLIDE 19

General Kernel for PRC

  • The slope of the phase sensitivity

predicts the response to an infinitesimal pulse

[Taylor et al., IEEE TAC, 2008]

  • requires differentiation and adjoint
  • general input signal
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Stochastic Sensitivity Analysis

  • Deterministic vs. Stochastic
  • What is sensitivity? behavior = density function
  • Sensitivity measure:

[Costanza & Seinfeld, J. Chem. Phys., 1981]

t1 f(y(p)) t y y t y

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SLIDE 21
  • Classic Sensitivity
  • Alternative measures of “perturbations”
  • Kolmogorov-Smirnov Distance
  • Kullback-Leibler Divergence

Sensitivity Analysis for Stochastic Systems

[Mirsky et al., IET Sys. Biol., 2011]

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SLIDE 22

The Stochastic Clock is Resistant to Noise

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SLIDE 23

Less Robust Ordering in Cell Cycle

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SLIDE 24
  • 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

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SLIDE 25

CLOCK ¡ BMAL1 ¡ BMAL1 ¡ PE-­‑BOX ¡ PE-­‑BOX ¡ Per1 ¡& ¡2 ¡ Cry1 ¡& ¡2 ¡ PER1 ¡&2 ¡ CRY1 ¡& ¡2 ¡ VIP ¡ CRY1 ¡& ¡2 ¡ PER1 ¡&2 ¡

Cellular ¡Network ¡Model ¡-­‑ ¡Interac@ons ¡

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SLIDE 26

Global + Local Cues = Robust Performance

[To et al., Biophys. J.; 2007; Wang and Doyle, Automatica, 2011] Global + Local Cues

S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡

GPS provides reference

Sensors

S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡

Sensors

Range improvement via beamforming

S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡ S ¡

Sensors

S ¡ S ¡ S ¡ S ¡ S ¡

Information dissemination and distributed storage

Timing and Phase Synchronization in Wireless Networks

2007

Global Local

Global:

  • 1. Satellites
  • 2. GPS
  • 3. Command centers

Local:

  • 1. Field commands
  • 2. Personal

communications

  • 3. Cellular, WIFI, WIMAX

Master (global)- slave (local) architecture

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SLIDE 27
  • 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 [Wang & Doyle III, Automatica, 2011]

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SLIDE 28
  • 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 Frequency Noise [Wang & Doyle, Automatica, 2011]

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SLIDE 29
  • System Dynamics (ai,j = aj,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

Sync Rate of Oscillator Networks

Linear Phase Model with Delays [Wang & Doyle, CDC, 2010]

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  • Positive Coupling
  • Negative Coupling

g = 1 g = 2 g = 3 g = 4 g = 5 =0.01 0.7000 0.5000 0.3500 0.3000 0.2500 =0.50 1.8500 1.4500 1.1500 1.1000 0.7500

Global cue

ai,j = 1 ai,j = 2 ai,j = 3 ai,j = 4 ai,j = 5 =0.01 0.9500 0.8500 0.9000 0.9500 0.9500 =0.50 1.2000 1.4000 2.8500 4.3000 8.5000

Local cue Global cue

g = 1 g = 2 g = 3 g = 4 g = 5 = 2 13.150 4.9000 4.2500 2.4500 2.3000

30

Local cue

ai,j = 1 ai,j = 2 ai,j = 3 ai,j = 4 ai,j = 5 = 2 2.3000 4.2500 6.2500 8.3500 10.7000

Sync Rate of Oscillator Networks

Linear Phase Model with Delays [Wang & Doyle, CDC, 2010]

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  • System Dynamics
  • Theorem: A stronger global cue leads to a faster synch rate,
  • a stronger local cue brings a faster synch rate if phase differences (deviation

from the global cue) are in (-π/2, π/2)

  • a stronger local cue can reduce or increase synch rate if phase differences

(deviation from the global cue) are in (-π, -π/2)U(π/2, π)

Sync Rate of Oscillator Networks

Nonlinear Phase Model (Kuramoto) [Wang & Doyle, ACC, 2011]

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Summary

 Insights into robust yet fragile

architectures in gene regulatory network

 Role of intercellular coupling in

achieving robust synchronization

 Robust timekeeping performance

in the presence of stochastic noise

Simulation and Analysis Model Experimentation

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SLIDE 33

Acknowledgments

  • Neda Bagheri [Northwestern]
  • Dr. Peter Chang
  • Dr. Rudi Gunawan [ETH]
  • Kirsten Meeker
  • Henry Mirsky [U. Michigan]
  • Stephanie Taylor [Colby College]
  • T.-L. To [MIT]
  • Dr. Yongqiang Wang
  • Felipe Nuñez
  • Peter St. John
  • Rich Harang
  • Professor Linda Petzold

Experimental Collaborators:

  • E. Herzog (WashU), S. Kay (UCSD)