Phase-locked Loops for Chemical Control of Oscillation Frequency A - - PowerPoint PPT Presentation

▶
phase locked loops for chemical control of oscillation
SMART_READER_LITE
LIVE PREVIEW

Phase-locked Loops for Chemical Control of Oscillation Frequency A - - PowerPoint PPT Presentation

Dec 31, 2023 209 likes •798 views

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback Simul. Studies Generalisation Prospectives Phase-locked Loops for Chemical Control of Oscillation Frequency A prototype of biological clocks and their entrainment by

slide-1
SLIDE 1

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Phase-locked Loops for Chemical Control of Oscillation Frequency

A prototype of biological clocks and their entrainment by light? Thomas Hinze1,2 Benedict Schau1 Christian Bodenstein1

1Friedrich Schiller University Jena

Department of Bioinformatics at School of Biology and Pharmacy Modelling Oscillatory Information Processing Group

2Saxon University of Cooperative Education, Dresden

{thomas.hinze,christian.bodenstein}@uni-jena.de

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-2
SLIDE 2

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Human Daily Rhythm: Trigger and Control System

www.wikipedia.org Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-3
SLIDE 3

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Chronobiology science of biological rhythms and clock systems

βιοζ life λογοζ science rhythm ρυθµοζ χρονοζ time

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-4
SLIDE 4

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Circadian Clock

  • Sustained biochemical oscillation (endogenous rhythm)
  • Free-running period close to but typically not exactly 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

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-5
SLIDE 5

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Circadian Clock

  • Sustained biochemical oscillation (endogenous rhythm)
  • Free-running period close to but typically not exactly 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

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-6
SLIDE 6

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Circadian Clock

  • Sustained biochemical oscillation (endogenous rhythm)
  • Free-running period close to but typically not exactly 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

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-7
SLIDE 7

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Circadian Clock

  • Sustained biochemical oscillation (endogenous rhythm)
  • Free-running period close to but typically not exactly 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

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-8
SLIDE 8

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Circadian Clock

  • Sustained biochemical oscillation (endogenous rhythm)
  • Free-running period close to but typically not exactly 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

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-9
SLIDE 9

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Circadian Clock

  • Sustained biochemical oscillation (endogenous rhythm)
  • Free-running period close to but typically not exactly 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

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-10
SLIDE 10

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

General Scheme of a Simple Control Loop

u(t) = C(D(w(t),y(t)))

controller actuator plant sensor

x(t) = P(v(t)) y(t) = F(x(t)) v(t) = A(u(t)) stimulus external system output v(t) u(t) y(t) w(t) x(t)

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-11
SLIDE 11

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Frequency Control using Phase-locked Loop

coupled

  • ne or several

elementary 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

signal error (reference) stimuli external Adapted from T. Hinze, M. Schumann, C. Bodenstein, I. Heiland, S. Schuster. Biochemical Frequency Control by Synchronisation of Coupled Repressilators: An In-silico Study of Modules for Circadian Clock Systems. Computational Intelligence and Neuroscience 2011:262189, 2011 Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-12
SLIDE 12

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Combine Reaction Network Modules

affects frequency signal tuning signal

  • utput

signal error (reference) stimuli external network candidate 1 network candidate 1

  • scillator

candidate 1

  • T. Hinze, C. Bodenstein, I. Heiland, S. Schuster. Capturing Biological Frequency Control of Circadian Clocks by

Reaction System Modularization. ISSN 0926-4981, ERCIM News 85:27-29, 2011 Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-13
SLIDE 13

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Mass-action Reaction Kinetics at a Glance

Modeling Temporal Behaviour of Chemical Reaction Networks

Assumption: number of effective reactant collisions Z proportional to reactant concentrations (Guldberg 1867) A + B

ˆ k

− → C . . . . ZC ∼ [A] and ZC ∼ [B], so ZC ∼ [A] · [B] Production rate generating C: vprod([C]) = ˆ k · [A] · [B] Consumption rate of C: . . . . . .vcons([C]) = 0

d [C] d t

= vprod([C]) − vcons([C])

d [C] d t

= ˆ k · [A] · [B] Initial conditions: [C](0), [A](0), [B](0) to be set

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-14
SLIDE 14

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Mass-action Reaction Kinetics at a Glance

Modeling Temporal Behaviour of Chemical Reaction Networks

Assumption: number of effective reactant collisions Z proportional to reactant concentrations (Guldberg 1867) A + B

ˆ k

− → C . . . . ZC ∼ [A] and ZC ∼ [B], so ZC ∼ [A] · [B] Production rate generating C: vprod([C]) = ˆ k · [A] · [B] Consumption rate of C: . . . . . .vcons([C]) = 0

d [C] d t

= vprod([C]) − vcons([C])

d [C] d t

= ˆ k · [A] · [B] Initial conditions: [C](0), [A](0), [B](0) to be set

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-15
SLIDE 15

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Mass-action Reaction Kinetics at a Glance

Modeling Temporal Behaviour of Chemical Reaction Networks

Assumption: number of effective reactant collisions Z proportional to reactant concentrations (Guldberg 1867) A + B

ˆ k

− → C . . . . ZC ∼ [A] and ZC ∼ [B], so ZC ∼ [A] · [B] Production rate generating C: vprod([C]) = ˆ k · [A] · [B] Consumption rate of C: . . . . . .vcons([C]) = 0

d [C] d t

= vprod([C]) − vcons([C])

d [C] d t

= ˆ k · [A] · [B] Initial conditions: [C](0), [A](0), [B](0) to be set

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-16
SLIDE 16

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Mass-action Reaction Kinetics at a Glance

Modeling Temporal Behaviour of Chemical Reaction Networks

Assumption: number of effective reactant collisions Z proportional to reactant concentrations (Guldberg 1867) A + B

ˆ k

− → C . . . . ZC ∼ [A] and ZC ∼ [B], so ZC ∼ [A] · [B] Production rate generating C: vprod([C]) = ˆ k · [A] · [B] Consumption rate of C: . . . . . .vcons([C]) = 0

d [C] d t

= vprod([C]) − vcons([C])

d [C] d t

= ˆ k · [A] · [B] Initial conditions: [C](0), [A](0), [B](0) to be set

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-17
SLIDE 17

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Mass-action Reaction Kinetics at a Glance

Modeling Temporal Behaviour of Chemical Reaction Networks

Assumption: number of effective reactant collisions Z proportional to reactant concentrations (Guldberg 1867) A + B

ˆ k

− → C . . . . ZC ∼ [A] and ZC ∼ [B], so ZC ∼ [A] · [B] Production rate generating C: vprod([C]) = ˆ k · [A] · [B] Consumption rate of C: . . . . . .vcons([C]) = 0

d [C] d t

= vprod([C]) − vcons([C])

d [C] d t

= ˆ k · [A] · [B] Initial conditions: [C](0), [A](0), [B](0) to be set

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-18
SLIDE 18

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Mass-action Reaction Kinetics at a Glance

Modeling Temporal Behaviour of Chemical Reaction Networks

Assumption: number of effective reactant collisions Z proportional to reactant concentrations (Guldberg 1867) A + B

ˆ k

− → C . . . . ZC ∼ [A] and ZC ∼ [B], so ZC ∼ [A] · [B] Production rate generating C: vprod([C]) = ˆ k · [A] · [B] Consumption rate of C: . . . . . .vcons([C]) = 0

d [C] d t

= vprod([C]) − vcons([C])

d [C] d t

= ˆ k · [A] · [B] Initial conditions: [C](0), [A](0), [B](0) to be set

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-19
SLIDE 19

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Mass-action Kinetics: General ODE Model

Chemical reaction system a1,1S1 + a2,1S2 + . . . + an,1Sn

ˆ k1

− → b1,1S1 + b2,1S2 + . . . + bn,1Sn a1,2S1 + a2,2S2 + . . . + an,2Sn

ˆ k2

− → b1,2S1 + b2,2S2 + . . . + bn,2Sn . . . a1,hS1 + a2,hS2 + . . . + an,hSn

ˆ kh

− → b1,hS1 + b2,hS2 + . . . + bn,hSn, results in ordinary differential equations (ODEs) d [Si] d t =

h

  • ν=1
  • ˆ

kν · (bi,ν − ai,ν) ·

n

  • l=1

[Sl]al,ν

  • with

i = 1, . . . , n.

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-20
SLIDE 20

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Mass-action vs. Saturation Kinetics

Kinetics Activation (rate law) Repression (rate law) Mass-action (no saturation) v = k · [S] − Michaelis-Menten (saturation) v = K ·

[S] T+[S]

v = K ·

  • 1 −

[S] T+[S]

  • Higher-Order Hill

(saturation) v = K ·

[S]n T+[S]n

v = K ·

  • 1 −

[S]n T+[S]n

  • Michaelis Menten: Typical enzyme kinetics
  • Higher-order Hill (n ≥ 2): Typically for gene expression

using sigmoidal transfer function

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-21
SLIDE 21

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Plant: Controllable Core Oscillator

coupled

  • ne or several

elementary 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

signal error (reference) stimuli external Adapted from T. Hinze, M. Schumann, C. Bodenstein, I. Heiland, S. Schuster. Biochemical Frequency Control by Synchronisation of Coupled Repressilators: An In-silico Study of Modules for Circadian Clock Systems. Computational Intelligence and Neuroscience 2011:262189, 2011 Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-22
SLIDE 22

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Controllable Goodwin-type Core Oscillator

a, A, translation transportation transcriptional inhibition Z X k 5 k 4,K 4 k 6,K 6 Y k 3 K1 k 2,K 2

time (days) substrate concentration X Y Z k6 (nM/h) period length after transient phase (h) k4 (nM/h) period length after transient phase (h) k2 (nM/h) period length after transient phase (h)

  • T. Hinze, C. Bodenstein, B. Schau, I. Heiland, S. Schuster. Chemical Analog Computers for Clock Frequency

Control Based on P Modules. Proceedings of the Twelfth International Conference on Membrane Computing, to appear within series Lecture Notes in Computer Science, Springer Verlag, 2011 Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-23
SLIDE 23

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Core Oscillator: Dynamical Behaviour

˙ [X] = a A + K1[Z]2 − k2[X] K2 + [X] ˙ [Y] = k3[X] − k5[Y] − k4[Y] K4 + [Y] ˙ [Z] = k5[Y] − k6[Z] K6 + [Z]

  • B. Schau. Reverse-Engineering circadianer Oszillationssysteme als Frequenzregelkreise mit
  • Nachlaufsynchronisation. Diploma thesis, 2011

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-24
SLIDE 24

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Affecting Frequency by Degradation Rate of Z

k6

p e r i

  • d

l e n g t h ( h )

  • Velocity parameter k6 of Z degradation notably influences
  • scillation frequency
  • Period control coefficients assigned to each reaction

quantify influence on frequency

  • T. Hinze, C. Bodenstein, B. Schau, I. Heiland, S. Schuster. Chemical Analog Computers for Clock Frequency

Control Based on P Modules. Proceedings of the Twelfth International Conference on Membrane Computing, to appear within series Lecture Notes in Computer Science, Springer Verlag, 2011 Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-25
SLIDE 25

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Controller: Signal Comparator

coupled

  • ne or several

elementary 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

signal error (reference) stimuli external Adapted from T. Hinze, M. Schumann, C. Bodenstein, I. Heiland, S. Schuster. Biochemical Frequency Control by Synchronisation of Coupled Repressilators: An In-silico Study of Modules for Circadian Clock Systems. Computational Intelligence and Neuroscience 2011:262189, 2011 Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-26
SLIDE 26

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Signal Comparator: Multiplication Unit

1 2 3 4 5 6 7 10 20 30 40 50 60 70 80 90 100 substrate concentration time X1 X2 Y

X1 k 2 k 1 X2 Y

˙ [X1] = ˙ [X2] = ˙ [Y] = k1[X1][X2] − k2[Y] ODE solution for asymptotic steady state in case of k1 = k2: [Y](∞) = lim

t→∞

  • 1 − e−k1t

· ([X1](t) · [X2](t)) = [X1](0) · [X2](0) Input-output mapping: [Y] = [X1] · [X2]

  • T. Hinze, C. Bodenstein, B. Schau, I. Heiland, S. Schuster. Chemical Analog Computers for Clock Frequency

Control Based on P Modules. Proceedings of the Twelfth International Conference on Membrane Computing, to appear within series Lecture Notes in Computer Science, Springer Verlag, 2011 Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-27
SLIDE 27

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Comparing Phases: Mathematical Background

Output of core oscillator ω = 2π/τ: y(t) = y(t + τ) = A0 +

∞

  • n=1

An cos(nωt + ϕn) Input of external reference signal ω′ = 2π/τ ′: z(t) = z(t + τ ′) = A′

0 + ∞

  • n=1

A′

n sin(nω′t + ϕ′ n)

For simplicity we assume that all higher harmonics are removed by a filter.

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-28
SLIDE 28

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Comparing Phases by Multiplication

Multiplication module: ˙ x = k(z(t)y(t) − x) lim

k→∞ x(t) = z(t)y(t)

Output of multiplication: z(t)y(t) = A′

0A0 + A′ 0A1 cos(ωt + ϕ1) + A0A′ 1 sin(ω′t + ϕ′ 1)

+ A′

1A1

2

  • sin((ω′ − ω)t + ϕ′

1 − ϕ1) + sin((ω′ + ω)t + ϕ′ 1 + ϕ1)

  • Low frequency term (ω′ ≈ ω) carries the phase-difference

information: φ′ − φ.

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-29
SLIDE 29

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Actuator: Global Feedback with Low-pass Filter

coupled

  • ne or several

elementary 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

signal error (reference) stimuli external Adapted from T. Hinze, M. Schumann, C. Bodenstein, I. Heiland, S. Schuster. Biochemical Frequency Control by Synchronisation of Coupled Repressilators: An In-silico Study of Modules for Circadian Clock Systems. Computational Intelligence and Neuroscience 2011:262189, 2011 Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-30
SLIDE 30

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Low-pass Filter as Global Feedback

  • Desensibilise global feedback by

signal smoothing, damping, and delay

  • Eliminate high-frequency oscillations by a low-pass filter

Simple linear reaction cascade forms a low-pass filter.

Samoilov et al. J Phys Chem 106, 2002

  • B. Schau. Reverse-Engineering circadianer Oszillationssysteme als Frequenzregelkreise mit
  • Nachlaufsynchronisation. Diploma thesis, 2011

Adjust kinetic parameters to obtain desired filtering.

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-31
SLIDE 31

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Low-pass Filter as Global Feedback

  • Desensibilise global feedback by

signal smoothing, damping, and delay

  • Eliminate high-frequency oscillations by a low-pass filter

Simple linear reaction cascade forms a low-pass filter.

Samoilov et al. J Phys Chem 106, 2002

  • B. Schau. Reverse-Engineering circadianer Oszillationssysteme als Frequenzregelkreise mit
  • Nachlaufsynchronisation. Diploma thesis, 2011

Adjust kinetic parameters to obtain desired filtering.

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-32
SLIDE 32

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Low-pass Filter as Global Feedback

  • Desensibilise global feedback by

signal smoothing, damping, and delay

  • Eliminate high-frequency oscillations by a low-pass filter

Simple linear reaction cascade forms a low-pass filter.

Samoilov et al. J Phys Chem 106, 2002

  • B. Schau. Reverse-Engineering circadianer Oszillationssysteme als Frequenzregelkreise mit
  • Nachlaufsynchronisation. Diploma thesis, 2011

Adjust kinetic parameters to obtain desired filtering.

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-33
SLIDE 33

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Low-pass Filter

X2 X3 k n Y k n−1 k 3 X1 k 2 k 1 Xn−1 X k n+1

time (days) substrate concentration [Y] time (days) substrate concentration [X]

˙ [X1] = k1[X] − k2[X1] ˙ [X2] = k2[X1] − k3[X2] . . . ˙ [X n−1] = kn−1[Xn−2] − kn[Xn−1] ˙ [Y] = kn[Xn−1] − kn+1[Y]

  • T. Hinze, C. Bodenstein, B. Schau, I. Heiland, S. Schuster. Chemical Analog Computers for Clock Frequency

Control Based on P Modules. Proceedings of the Twelfth International Conference on Membrane Computing, to appear within series Lecture Notes in Computer Science, Springer Verlag, 2011 Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-34
SLIDE 34

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Low-pass Filter: Bode Plot as Characteristic Curve

frequency in 1/s cutoff frequency slope magnitude in dB

Magnitude dB = 10 · lg

  • amplitude of output signal

amplitude of input signal

  • Signals affected by smoothing delay throughout cascade
  • Oscillation waveform harmonisation into sinusoidal shape
  • Global filter parameters:

passband damping, cutoff frequency, slope

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-35
SLIDE 35

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Model of a Chemical Frequency Control Based on PLL

low−pass filter core oscillator

characteristic curve Bode plot

comparator

[Y] = [E][Z]

signal

core

  • scillator

filter low−pass plant actuator controller [F](t) [Z](t) external stimulus [E](t)

a, A, transfer function scaled tuning signal [D] = [U]+[F]*ae ae = Y k 3 X k 5 k 4,K 4 k 6,K 6 X2 X3 l 5 E m1 K1 k 8 k 8 k 7 k 7 k 7 U F l 4 l 3 X1 l 2 l 1 X4 Z D F Z X Y k 2,K 2

magnitude (dB) frequency (1/s) k6 (nM/h) period length after transient phase (h) length of 0.61 days) cutoff frequency: 0.000019 1/s (corresponds to period

  • T. Hinze, C. Bodenstein, B. Schau, I. Heiland, S. Schuster. Chemical Analog Computers for Clock Frequency

Control Based on P Modules. Proceedings of the Twelfth International Conference on Membrane Computing, to appear within series Lecture Notes in Computer Science, Springer Verlag, 2011 Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-36
SLIDE 36

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Period Lengths subject to Constant External Stimulus

constant external stimulus [E] period length (h) of signal [Z]

  • T. Hinze, C. Bodenstein, B. Schau, I. Heiland, S. Schuster. Chemical Analog Computers for Clock Frequency

Control Based on P Modules. Proceedings of the Twelfth International Conference on Membrane Computing, to appear within series Lecture Notes in Computer Science, Springer Verlag, 2011 Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-37
SLIDE 37

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Time to Entrainment to Different Period Lengths

period length (h) of external stimulus [E] time to entrainment (days) of signal [Z]

Natural period of core oscillator: 24.2h

  • T. Hinze, C. Bodenstein, B. Schau, I. Heiland, S. Schuster. Chemical Analog Computers for Clock Frequency

Control Based on P Modules. Proceedings of the Twelfth International Conference on Membrane Computing, to appear within series Lecture Notes in Computer Science, Springer Verlag, 2011, accepted Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-38
SLIDE 38

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Time to Entrainment to Different Initial Phase Shifts

initial phase shift (°) between external stimulus [E] and core oscillator output [Z] time to entrainment (days) of signal [Z] external stimulus' period length 24h 26' 24h worst case best case

Entrainment reached within convergence interval 1min

  • T. Hinze, C. Bodenstein, B. Schau, I. Heiland, S. Schuster. Chemical Analog Computers for Clock Frequency

Control Based on P Modules. Proceedings of the Twelfth International Conference on Membrane Computing, to appear within series Lecture Notes in Computer Science, Springer Verlag, 2011 Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-39
SLIDE 39

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Best Case and Worst Case Entrainment

time (days) period length (h) of signal [Z] external stimulus' period length and initial phase shift 24h and 325° 24h and 245°

Entrainment reached within convergence interval 1min

  • T. Hinze, C. Bodenstein, B. Schau, I. Heiland, S. Schuster. Chemical Analog Computers for Clock Frequency

Control Based on P Modules. Proceedings of the Twelfth International Conference on Membrane Computing, to appear within series Lecture Notes in Computer Science, Springer Verlag, 2011 Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-40
SLIDE 40

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Perturbed Core Oscillator

Unperturbed core oscillator at constant external signal A′

0:

dX dt = F(X) with limit cycle solution X0(t) = X0(t + τ). Perturbed core oscillator: dX dt = F(X) + ε sin (. . . ) kl ∂F ∂kl (X). Since ε is small the amplitude of the limit cycle is not affected and we can reduce the model to the phase dynamics!

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-41
SLIDE 41

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Amplitude and Phase

Granada & Herzel. PLoS ONE 4(9): e7057, 2009

We can assign each point on the limit cycle X0 a specific phase value φ.

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-42
SLIDE 42

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Phase Reduction (Kuramoto 1984)

Demir et al. IEEE Transactions on circuits and systems 47(5):655-674, 2000

Oscillator phase dynamics:

dφ dt = ω + ε PRCl(φ) sin

  • φ′ − φ + ϕlpf
  • .

PRCl is the 2π-periodic phase response curve of kl.

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-43
SLIDE 43

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Phase Response Curve

http://en.wikipedia.org/wiki/Phase_response_curve Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-44
SLIDE 44

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Phase Difference

Phase difference ψ between oscillator and external signal: ψ = φ − φ′ dψ dt = ω − ω′ − ε PRCl(φ′ + ψ) sin

  • ψ − ϕlpf
  • ψ is a slowly changing variable compared to φ′ = ω′t, therefore

we may average the perturbation over one external cycle and consider ψ on the slow time scale: 1 τ ′ τ ′ PRCl(φ′(t) + ψ) dt = −Cτ

l ,

where Cτ

l = kl/τ ∂τ ∂kl is the period control coefficient.

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-45
SLIDE 45

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Phase Difference

Phase difference equation:

dψ dt = ω − ω′ ε + Cτ

l sin

  • ψ − ϕlpf
  • Phase-locking corresponds to (stable) steady-state solutions ψ0
  • f this equation:

φ(t) = φ′(t) + ψ0. Phase locking exists in a region enclosed by: ε± = ∓

  • ω − ω′ 1

Cτ

l

, the so called Arnold tongue.

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-46
SLIDE 46

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Arnold Tongue

  • B. Schau. Reverse-Engineering circadianer Oszillationssysteme als Frequenzregelkreise mit
  • Nachlaufsynchronisation. Diploma thesis, 2011

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-47
SLIDE 47

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Phase Lag

The phase lag can be easily determined from the derived

  • equation. For example consider ω = ω′ and Cτ

l < 0, the stable

solution then is: ψ0 = ϕlpf. That means the phase lag is completely determined by the low-pass filter.

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-48
SLIDE 48

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Conclusions

  • Chemical frequency control can utilise PLL
  • Prototypic modelling example for

entrainment of circadian clockworks

  • Chemical processing units in minimalistic manner
  • Variety of chemical implementations
  • Modularisation in (bio)chemical reaction systems

Some open questions

  • Identification of in-vivo counterparts
  • Replacement of individual processing units

(like different core oscillators)

  • Balancing advantages and limitations of the PLL approach
  • Inclusion of temperature entrainment (by Arrhenius terms)
  • Alternative concepts of frequency control

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-49
SLIDE 49

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Conclusions

  • Chemical frequency control can utilise PLL
  • Prototypic modelling example for

entrainment of circadian clockworks

  • Chemical processing units in minimalistic manner
  • Variety of chemical implementations
  • Modularisation in (bio)chemical reaction systems

Some open questions

  • Identification of in-vivo counterparts
  • Replacement of individual processing units

(like different core oscillators)

  • Balancing advantages and limitations of the PLL approach
  • Inclusion of temperature entrainment (by Arrhenius terms)
  • Alternative concepts of frequency control

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-50
SLIDE 50

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Conclusions

  • Chemical frequency control can utilise PLL
  • Prototypic modelling example for

entrainment of circadian clockworks

  • Chemical processing units in minimalistic manner
  • Variety of chemical implementations
  • Modularisation in (bio)chemical reaction systems

Some open questions

  • Identification of in-vivo counterparts
  • Replacement of individual processing units

(like different core oscillators)

  • Balancing advantages and limitations of the PLL approach
  • Inclusion of temperature entrainment (by Arrhenius terms)
  • Alternative concepts of frequency control

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-51
SLIDE 51

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Conclusions

  • Chemical frequency control can utilise PLL
  • Prototypic modelling example for

entrainment of circadian clockworks

  • Chemical processing units in minimalistic manner
  • Variety of chemical implementations
  • Modularisation in (bio)chemical reaction systems

Some open questions

  • Identification of in-vivo counterparts
  • Replacement of individual processing units

(like different core oscillators)

  • Balancing advantages and limitations of the PLL approach
  • Inclusion of temperature entrainment (by Arrhenius terms)
  • Alternative concepts of frequency control

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-52
SLIDE 52

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Conclusions

  • Chemical frequency control can utilise PLL
  • Prototypic modelling example for

entrainment of circadian clockworks

  • Chemical processing units in minimalistic manner
  • Variety of chemical implementations
  • Modularisation in (bio)chemical reaction systems

Some open questions

  • Identification of in-vivo counterparts
  • Replacement of individual processing units

(like different core oscillators)

  • Balancing advantages and limitations of the PLL approach
  • Inclusion of temperature entrainment (by Arrhenius terms)
  • Alternative concepts of frequency control

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-53
SLIDE 53

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Conclusions

  • Chemical frequency control can utilise PLL
  • Prototypic modelling example for

entrainment of circadian clockworks

  • Chemical processing units in minimalistic manner
  • Variety of chemical implementations
  • Modularisation in (bio)chemical reaction systems

Some open questions

  • Identification of in-vivo counterparts
  • Replacement of individual processing units

(like different core oscillators)

  • Balancing advantages and limitations of the PLL approach
  • Inclusion of temperature entrainment (by Arrhenius terms)
  • Alternative concepts of frequency control

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-54
SLIDE 54

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Conclusions

  • Chemical frequency control can utilise PLL
  • Prototypic modelling example for

entrainment of circadian clockworks

  • Chemical processing units in minimalistic manner
  • Variety of chemical implementations
  • Modularisation in (bio)chemical reaction systems

Some open questions

  • Identification of in-vivo counterparts
  • Replacement of individual processing units

(like different core oscillators)

  • Balancing advantages and limitations of the PLL approach
  • Inclusion of temperature entrainment (by Arrhenius terms)
  • Alternative concepts of frequency control

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-55
SLIDE 55

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Conclusions

  • Chemical frequency control can utilise PLL
  • Prototypic modelling example for

entrainment of circadian clockworks

  • Chemical processing units in minimalistic manner
  • Variety of chemical implementations
  • Modularisation in (bio)chemical reaction systems

Some open questions

  • Identification of in-vivo counterparts
  • Replacement of individual processing units

(like different core oscillators)

  • Balancing advantages and limitations of the PLL approach
  • Inclusion of temperature entrainment (by Arrhenius terms)
  • Alternative concepts of frequency control

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-56
SLIDE 56

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Conclusions

  • Chemical frequency control can utilise PLL
  • Prototypic modelling example for

entrainment of circadian clockworks

  • Chemical processing units in minimalistic manner
  • Variety of chemical implementations
  • Modularisation in (bio)chemical reaction systems

Some open questions

  • Identification of in-vivo counterparts
  • Replacement of individual processing units

(like different core oscillators)

  • Balancing advantages and limitations of the PLL approach
  • Inclusion of temperature entrainment (by Arrhenius terms)
  • Alternative concepts of frequency control

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-57
SLIDE 57

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Conclusions

  • Chemical frequency control can utilise PLL
  • Prototypic modelling example for

entrainment of circadian clockworks

  • Chemical processing units in minimalistic manner
  • Variety of chemical implementations
  • Modularisation in (bio)chemical reaction systems

Some open questions

  • Identification of in-vivo counterparts
  • Replacement of individual processing units

(like different core oscillators)

  • Balancing advantages and limitations of the PLL approach
  • Inclusion of temperature entrainment (by Arrhenius terms)
  • Alternative concepts of frequency control

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein
slide-58
SLIDE 58

Motivation PLL Scheme Core Oscillator Signal Comparator Global Feedback

  • Simul. Studies

Generalisation Prospectives

Acknowledgements

Department of Bioinformatics at School of Biology and Pharmacy Friedrich Schiller University Jena Jena Centre for Bioinformatics Research Initiative in Systems Biology European Molecular Computing Consortium

Phase-locked Loops for Chemical Control of Oscillation Frequency

  • T. Hinze, B. Schau, C. Bodenstein