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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation Synchronisation of Biological Clock Signals Capturing Coupled Repressilators from a Control Systems Perspective Thomas Hinze Mathias Schumann Stefan

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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Synchronisation of Biological Clock Signals

Capturing Coupled Repressilators from a Control Systems Perspective Thomas Hinze Mathias Schumann Stefan Schuster

thomas.hinze@uni-jena.de

Friedrich Schiller University Jena Department Bioinformatics at School of Biology/Pharmacy Modelling Oscillatory Information Processing Group 4th International Conference on Bio-Inspired Systems and Signal Processing (BIOSIGNALS2011)

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Human Daily Rhythm: Trigger and Control System

www.wikipedia.org Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Biological Clocks

Significance

  • Nanoscaled oscillatory reaction systems
  • High precision and self-sustainability
  • Robust and reliable control systems for

manifold processes

  • Adaptability to specific environmental conditions

(e.g. cycles of light/darkness)

  • Infradian (period > 1 day), circadian (≈ 1 day), and

ultradian (< 1 day) rhythms

  • Several independent evolutionary origins
  • Prototypes for fine-grained clock synchronisation
  • Medicine, agriculture, bionics, material sciences, biology

= ⇒ Keeping environmental time within living organisms

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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SLIDE 4

Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Biological Clocks

Significance

  • Nanoscaled oscillatory reaction systems
  • High precision and self-sustainability
  • Robust and reliable control systems for

manifold processes

  • Adaptability to specific environmental conditions

(e.g. cycles of light/darkness)

  • Infradian (period > 1 day), circadian (≈ 1 day), and

ultradian (< 1 day) rhythms

  • Several independent evolutionary origins
  • Prototypes for fine-grained clock synchronisation
  • Medicine, agriculture, bionics, material sciences, biology

= ⇒ Keeping environmental time within living organisms

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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SLIDE 5

Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Biological Clocks

Significance

  • Nanoscaled oscillatory reaction systems
  • High precision and self-sustainability
  • Robust and reliable control systems for

manifold processes

  • Adaptability to specific environmental conditions

(e.g. cycles of light/darkness)

  • Infradian (period > 1 day), circadian (≈ 1 day), and

ultradian (< 1 day) rhythms

  • Several independent evolutionary origins
  • Prototypes for fine-grained clock synchronisation
  • Medicine, agriculture, bionics, material sciences, biology

= ⇒ Keeping environmental time within living organisms

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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SLIDE 6

Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Biological Clocks

Significance

  • Nanoscaled oscillatory reaction systems
  • High precision and self-sustainability
  • Robust and reliable control systems for

manifold processes

  • Adaptability to specific environmental conditions

(e.g. cycles of light/darkness)

  • Infradian (period > 1 day), circadian (≈ 1 day), and

ultradian (< 1 day) rhythms

  • Several independent evolutionary origins
  • Prototypes for fine-grained clock synchronisation
  • Medicine, agriculture, bionics, material sciences, biology

= ⇒ Keeping environmental time within living organisms

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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SLIDE 7

Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Biological Clocks

Significance

  • Nanoscaled oscillatory reaction systems
  • High precision and self-sustainability
  • Robust and reliable control systems for

manifold processes

  • Adaptability to specific environmental conditions

(e.g. cycles of light/darkness)

  • Infradian (period > 1 day), circadian (≈ 1 day), and

ultradian (< 1 day) rhythms

  • Several independent evolutionary origins
  • Prototypes for fine-grained clock synchronisation
  • Medicine, agriculture, bionics, material sciences, biology

= ⇒ Keeping environmental time within living organisms

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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SLIDE 8

Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Biological Clocks

Significance

  • Nanoscaled oscillatory reaction systems
  • High precision and self-sustainability
  • Robust and reliable control systems for

manifold processes

  • Adaptability to specific environmental conditions

(e.g. cycles of light/darkness)

  • Infradian (period > 1 day), circadian (≈ 1 day), and

ultradian (< 1 day) rhythms

  • Several independent evolutionary origins
  • Prototypes for fine-grained clock synchronisation
  • Medicine, agriculture, bionics, material sciences, biology

= ⇒ Keeping environmental time within living organisms

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
slide-9
SLIDE 9

Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Biological Clocks

Significance

  • Nanoscaled oscillatory reaction systems
  • High precision and self-sustainability
  • Robust and reliable control systems for

manifold processes

  • Adaptability to specific environmental conditions

(e.g. cycles of light/darkness)

  • Infradian (period > 1 day), circadian (≈ 1 day), and

ultradian (< 1 day) rhythms

  • Several independent evolutionary origins
  • Prototypes for fine-grained clock synchronisation
  • Medicine, agriculture, bionics, material sciences, biology

= ⇒ Keeping environmental time within living organisms

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
slide-10
SLIDE 10

Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Biological Clocks

Significance

  • Nanoscaled oscillatory reaction systems
  • High precision and self-sustainability
  • Robust and reliable control systems for

manifold processes

  • Adaptability to specific environmental conditions

(e.g. cycles of light/darkness)

  • Infradian (period > 1 day), circadian (≈ 1 day), and

ultradian (< 1 day) rhythms

  • Several independent evolutionary origins
  • Prototypes for fine-grained clock synchronisation
  • Medicine, agriculture, bionics, material sciences, biology

= ⇒ Keeping environmental time within living organisms

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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SLIDE 11

Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Biological Clocks

Significance

  • Nanoscaled oscillatory reaction systems
  • High precision and self-sustainability
  • Robust and reliable control systems for

manifold processes

  • Adaptability to specific environmental conditions

(e.g. cycles of light/darkness)

  • Infradian (period > 1 day), circadian (≈ 1 day), and

ultradian (< 1 day) rhythms

  • Several independent evolutionary origins
  • Prototypes for fine-grained clock synchronisation
  • Medicine, agriculture, bionics, material sciences, biology

= ⇒ Keeping environmental time within living organisms

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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SLIDE 12

Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Chronobiology science of biological rhythms and clock systems

βιοζ life λογοζ science rhythm ρυθµοζ χρονοζ time

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Circadian Clock

  • Undamped biochemical oscillation
  • Period approx. 24 hours persisting under constant

environmental conditions (e.g. permanent darkness DD or permanent light LL)

  • Entrainment – adaptation to external stimuli

(e.g. light-dark cycles induced by sunlight)

  • Temperature compensation within a physiological range
  • Reaction systems with at least one feedback loop

p e r t u r b a t i

  • n

concentration substrate time

= ⇒ Biological counterpart of frequency control system

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Circadian Clock

  • Undamped biochemical oscillation
  • Period approx. 24 hours persisting under constant

environmental conditions (e.g. permanent darkness DD or permanent light LL)

  • Entrainment – adaptation to external stimuli

(e.g. light-dark cycles induced by sunlight)

  • Temperature compensation within a physiological range
  • Reaction systems with at least one feedback loop

p e r t u r b a t i

  • n

concentration substrate time

= ⇒ Biological counterpart of frequency control system

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Cyanobacterium Synechococcus elongatus

“Simplest and earliest cells known to exhibit circadian phenomena” www.genome.jgi−psf.org www.wikipedia.org 1µm

  • Prokaryotic autotrophic picoplankton in tropical seas
  • Assumed to be on earth for more than 3.5 billion years
  • Clock: Phosphorylation cycle without gene expression

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

  • 1. Motivation

Chronobiology and Circadian Rhythms

  • 2. Definitions

Specifications for Synchronisation of Oscillatory Signals

  • 3. Repressilator

Gene Regulatory Network with Oscillatory Behaviour

  • 4. Internal Synchronisation

Simulation Studies using Coupled Repressilators

  • 5. External Synchronisation

Frequency Control Systems with Phase-Locked Loop

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Entrainment vs. Synchronisation

Entrainment

  • Oscillating signal (frequency, phase, and amplitude)

dynamically adapts to (varying) external stimulus. External stimulus itself not influenced. Synchronisation

  • External: Entrainment to external

stimulus (e.g. light-dark cycle induced by sunlight) + adaptation to signal shape

  • f external stimulus
  • Internal: oscillating signals

mutually adapt, converge to a common signal = ⇒ Entrainment can be seen as special case of synchronisation

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Properties of Synchronous Oscillations (I)

Undamped oscillations

  • Modelled oscillation results from solution of ordinary

differential equations (ODEs) describing dynamical behaviour of the biochemical clok system

  • Eigenvalues of Jacobian matrix (real parts < 0) mostly

indicate undamped oscillations

  • Limit cycles (represented by orbital courses) as method of

choice for numerical data

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Properties of Synchronous Oscillations (II)

Asymptotic or total adaption

  • Harmonisation of oscillating substrate concentration
  • after finite time tsync within
  • arbitrarily selectable ε-neighbourhood

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Properties of Synchronous Oscillations (III)

Monofrequential oscillation after tsync

  • Fast Fourier Transformation / Fourier analysis

(discrete data processing and comparison of peaks)

  • Laplace transform and subsequent algebraic processing

(preferably for sinusoidal signals)

  • Numerical exploration (e.g. sampling)

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Internal Clock Synchronsiation: Technical Protocols

Each node in a bidirectionally coupled computer network

  • Comprises a specific clock (potential deviations to others)
  • Can communicate with all other nodes by

sending/receiving local time stamps

  • Requests time stamps from others (mutually exchange)
  • Successively adjusts its local clock

(Lamport, Christian, Berkeley algorithms)

Berkeley algorithm. A. S. Tanenbaum and M. van Steen, Distributed Systems Principles and Paradigms, 2001 Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

External Clock Synchronsiation: Technical Protocols

Each node in unidirectionally coupled computer network

  • Comprises a specific clock (potential deviations to others)
  • Localised within hierarchial network structure
  • Retrieves time stamps exclusively from upper layers

(unidirectional signal transduction)

  • Successively adjusts its local clock by propagating time

stamps from clock(s) in root position

Network Time Protocol (NTP). de.wikipedia.org/wiki/Network_Time_Protocol Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

  • 1. Motivation

Chronobiology and Circadian Rhythms

  • 2. Definitions

Specifications for Synchronisation of Oscillatory Signals

  • 3. Repressilator

Gene Regulatory Network with Oscillatory Behaviour

  • 4. Internal Synchronisation

Simulation Studies using Coupled Repressilators

  • 5. External Synchronisation

Frequency Control Systems with Phase-Locked Loop

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Repressilator Prototype

In-vitro Oscillating Gene Regulatory Network

Eulowitz et al., Nature 403:335-338, 2000 Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Repressilator Model: Network Topology

Based on M.B. Elowitz, S. Leibler. A synthetic oscillatory network of transcriptional regulators. Nature 403:335-338, 2000 Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

ODEs Formalising Repressilator’s Dynamic Behaviour

d LacI_Protein d t = k_tl · LacI_mRNA − k_p · LacI_Protein d TetR_Protein d t = k_tl · TetR_mRNA − k_p · TetR_Protein d cI_Protein d t = k_tl · cI_mRNA − k_p · cI_Protein d LacI_mRNA d t = a0_tr + a_tr · KMn KMn + cI_Protein − k_tl · LacI_mRNA − k_r · LacI_mRNA d TetR_mRNA d t = a0_tr + a_tr · KMn KMn + LacI_Protein − k_tl · TetR_mRNA − k_r · TetR_mRNA d cI_mRNA d t = a0_tr + a_tr · KMn KMn + TetR_Protein − k_tl · cI_mRNA − k_r · cI_mRNA Reaction rates and parameter setting: k_tl = 6.93, k_p = 0.069, k_r = 0.347, a0_tr = 0.03, a_tr = 29.97, KM = 40, n = 3 resulted from parameter fitting based on available experimental data (Garcia-Ojalvo et al.). System implies sustained limit-cycle oscillations after transient phase.

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Dynamical Behaviour of the Repressilator (TetR)

Initialisation at limit cycle avoids transient phase = ⇒ Eliminates its influence on synchronisation time

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Period Control by Velocity of Protein Degradation

Variable degradation rates k_p = ln(2)/x (frequence parameter x = 3, . . . , 15) of proteins sufficient for clock advance or delay. Frequence control: prerequisite for synchronisability.

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Repressilator’s Transfer Function

Correlation between velocity of protein degradation and period. Identification of minimal period delimiting sustained oscillations.

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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SLIDE 30

Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

  • 1. Motivation

Chronobiology and Circadian Rhythms

  • 2. Definitions

Specifications for Synchronisation of Oscillatory Signals

  • 3. Repressilator

Gene Regulatory Network with Oscillatory Behaviour

  • 4. Internal Synchronisation

Simulation Studies using Coupled Repressilators

  • 5. External Synchronisation

Frequency Control Systems with Phase-Locked Loop

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Coupled Repressilators for Internal Synchronisation

Bidirectional diffusion of TetR proteins between either repressilators enable internal synchronisation. Diffusion parameter diff as additional rate constant (linear kinetics)

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Typical Synchronisation Run

time steps TetR abundance

Typical synchronisation run of two TetR-coupled repressilators, coupling strength diff = 0.04, initial phase shift 182◦.

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Time to Synchronisation for Various Initial Phase Shifts

50 100 150 200 250 300 350 2000 4000 6000 8000 10000 12000 14000

diff = 0.01 diff = 0.04 diff = 0.07 diff = 0.1 diff = 0.13

180°

phase shift φ time to synchronize

initial phase shift time to synchronisation

diff = 0.01 diff = 0.04 diff = 0.07 diff = 0.10 diff = 0.13

Time to synchronisation subject to various initial phase shifts. Parameter diff= 0.01, . . . , 0.13 denotes coupling strength from weak to strong coupling. Initial antiphase rhythmicity (phase shift 180◦) between both repressilators causes the highest effort to synchronise both

  • scillatory signals by

mutual forcing.

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Time to Synchronisation for Various Initial Frequencies

Weak diffusion, diff =0.01, frequency parameter x ratio: 9.475 / 9.5

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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SLIDE 35

Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Time to Synchronisation for Various Initial Frequencies

Weak diffusion, diff =0.01, frequency parameter x ratio: 9.4 / 9.5

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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SLIDE 36

Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Time to Synchronisation for Various Initial Frequencies

Weak diffusion, diff =0.01, frequency parameter x ratio: 9.3 / 9.5

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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SLIDE 37

Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Time to Synchronisation for Various Initial Frequencies

Weak diffusion, diff =0.01, frequency parameter x ratio: 9.2 / 9.5

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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SLIDE 38

Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Time to Synchronisation for Various Initial Frequencies

Weak diffusion, diff =0.01, frequency parameter x ratio: 9.1 / 9.5

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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SLIDE 39

Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Time to Synchronisation for Various Initial Frequencies

Weak diffusion, diff =0.01, frequency parameter x ratio: 9.0 / 9.5

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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SLIDE 40

Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Frequency Synchronisation Window

Ratios of initial frequencies subject to synchronous frequency considering variety of coupling strengths diff = 0.01, . . . , 0.13: variant of an Arnold tongue

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
slide-41
SLIDE 41

Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

  • 1. Motivation

Chronobiology and Circadian Rhythms

  • 2. Definitions

Specifications for Synchronisation of Oscillatory Signals

  • 3. Repressilator

Gene Regulatory Network with Oscillatory Behaviour

  • 4. Internal Synchronisation

Simulation Studies using Coupled Repressilators

  • 5. External Synchronisation

Frequency Control Systems with Phase-Locked Loop

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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SLIDE 42

Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Frequency Control System with Phase-Locked Loop

coupled

  • ne or several

core oscillator(s) local feedback(s) global feedback path damping and delay) (loop filter for affects frequency signal tuning signal comparator frequency deviation) (phase difference or signal

  • utput

(reference) stimuli external error signal

Coupled repressilators as core oscillator of frequency control system able to manage external synchronisation to external stimuli (reference

  • scillation)

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Conclusions and Take Home Message

  • Repressilator as promising biochemical in-vitro model

system to explore synchronisation of circadian oscillations

  • Inherent oscillation similar but not equal to sinusoidal

course (hence not “symmetric”)

  • Repressilator coupling by diffusion of TetR protein enables

internal synchronisation.

  • Arbitrary initial phase shifts (also antiphasic behaviour)

become harmonised while adaptation to different initial frequencies spans a synchronisation window.

  • Coupled repressilators can be considered as part of a

frequency control system based on phase-locked loop (PLL) utilising external synchronisation.

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster
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Motivation Definitions Repressilator Internal Synchronisation External Synchronisation

Special Thanks go to . . .

German Federal Ministry of Education and Research, project 0315260A within Research Initiative in Systems Biology

... you for your attention. Questions?

Department Bioinformatics, FSU Jena

Stefan Schuster

Department Bioinformatics, FSU Jena

Mathias Schumann ... the funding organization ... my coworkers

Synchronisation of Biological Clock Signals

  • T. Hinze, M. Schumann, S. Schuster