the robust ticking of a circadian clock david zwicker, jeroen van zon,david lubensky, pim altena,...
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The robust ticking of a circadian clock
David Zwicker, Jeroen van Zon,David Lubensky, Pim Altena, Pieter Rein ten Wolde
Beijing, July 27, 2010
Synechococcus
Elongatus
Introduction: circadian rhythms
In general, circadian rhythms are:Free running ~24 hour oscillationsEntrained to light
Circadian rhythms are very robust
Circadian clocks are extremely stable higher organisms: cell-cell interactions clock cyanobacteria stable at single cell level
Mihalcescu, Hsing, Leibler, Nature (2004)
Correlation time: 166 days!
Key questions:
How does the clock work?
How can it be so stable?
Clock cyanobacterium S. elongatus ideal model system
Circadian rhythms: oscillations gene expression
Three genes crucial: kaiA, kaiB, kaiCkaiBC forms an operonExpression kaiBC oscillatesContinuous overexpression KaiC represses kaiBCTemporal overexpression KaiC resets phase
Transcription-translation cycle (TTC)!
kaiBC
C
CC
C
Golden, Johnson, Kondo, Science (1998)
Circadian rhythms: oscillations protein modification
Protein phosphorylation cycle (PPC)!
KaiC is hexamer with two phosphorylation sites per monomer
In the dark: no gene expression, yet oscillations of phosphorylation level!
Kondo lab, Science (2005A)
KaiC circadian oscillationsin the test tube
Kondo lab, Science (2005B)
Test tube with KaiA, KaiB, KaiC and ATP (and water): oscillations!
Is the PPC the principal pacemaker?
Oscillations of gene expression without oscillations of phosphorylation level
TTC exists without a PPC!
Is the TTC perhaps the pacemaker after all?
Kondo lab, Genes & Development (2008)
phosphorylation
gene expression
Key question:
Why does the system have a PPC and a TTC?
Overview
PPC in vitro
PPC in vivo
TTC in vivo
PPC + TTC
Overview
PPC in vitro
PPC in vivo
TTC in vivo
PPC + TTC
Overview models for PPC
1. Monomer shuffling:• Emberly & Wingreen (PRL, 2006); Yoda, Eguchi,Terrada,
Sasai (PLCB, 2007); Mori et al. (PB, 2007);
2. Differential affinity or sequestration• Van Zon, Lubensky, Altena, Ten Wolde (PNAS, 2007);
Clodong et al. (MSB, 2007); Rust et al. (Science 2007);
For overview PPC models, see Markson & O’Shea (FEBS Lett, 2009)
Roadmap to working PPC model
1. Individual KaiC hexamers phosphorylate and dephosphorylate in a cyclical manner
2. Action of KaiA and KaiB synchronizes KaiC phosphorylation cycles by differential affinity
Allosteric cycle in KaiC phosphorylation
ADP
ATP
The subunits of a KaiC hexamer can exist in two conformational states, active and inactive
All subunits switch conformation collectively (MWC model)
ATP/ADP binding to subunits stabilizes the active state
Subunits with ATP bound become phosphorylated
Phosphorylated subunits are preferably in the inactive state
unphosphorylated
phosphorylated
Allosteric cycle in KaiC phosphorylation
ADP
ATP
Fast ATP/ADP binding and unbinding:
Partition function hexamer in state :
Free energy of hexamer in state :
Allosteric cycle: thermodynamics
ADP
ATP
Free energy of hexamers:
Active
Inactive
Allosteric cycle: thermodynamics
ADP
ATP
Free energy system including ATP hydrolysis:
Allosteric cycle: flipping kinetics
ADP
ATP
Nucleotide binding
Conformational transition
Flipping rate depends exponentially on degree of phosphorylation!
p = 3
Allosteric cycle in KaiC phosphorylation
No macroscopic oscillations due to lack of synchronization between the cycle of individual KaiC hexamers!
Conformational transition
Synchronization by differential affinity: a toy model
KaiA stimulates phosphorylation of active KaiC
[KaiA] is smaller than [KaiC] KaiA binds with differential
affinity: it binds most strongly to less phosphorylated, active KaiC
Synchronization by differential affinity: a toy model
0 1 6K K K
Solve set of ODEs
Synchronization by differential affinity: a toy model
KaiA binds and stimulates the laggards!
Full model Kai system: KaiC + KaiA
KaiA + KaiC
0 4 8 12 16 20 240
0.2
0.4
0.6
0.8
1
time(hour)
average phosphorylation
Kageyama et al. Mol. Cell 2006
1
1
i i
i i
i i i
C C
C C
C A C A C A
KaiA stimulates
phosphorylation of active KaiC
Binding of KaiA stabilizes active KaiC
Full model Kai system: KaiC + KaiB
0 4 8 12 16 20 240
0.2
0.4
0.6
0.8
1
time(hour)
average phosphorylation
2
2 2 1
2i i
i i
C B B C
B C B C
KaiB does not stimulate phosphorylation
KaiB stabilizes inactive KaiC by binding it, restoring the cycle
Xu et al. EMBO J. (2003)
KaiC
KaiC+KaiB
Full model: KaiC + KaiA + KaiB
0 12 24 36 48 60 720
0.2
0.4
0.6
0.8
1
time(hour)
average phosphorylation
Nakajima et al. Science 2005
2 2 2
2 2 2 2 1
2i i
i i
B C A A B C
A B C A B C
KaiB-KaiC binds to and sequesters KaiA, leading to another form of differential affinity
Phase portrait
Changing KaiA Changing KaiB
Experiment: Kageyama et al. Mol. Cell 2006
Model
Conclusions deterministic PPC model
Oscillations over large range of KaiA and KaiB concentrations
Reproduces experiments on subsets of Kai proteins
Robustness against variations in parameters: temperature compensation (not shown)
Our model makes several testable predictions
J. S. van Zon, D. K. Lubensky, P. R. H. Altena, P. R. ten Wolde,PNAS 107, 7420 (2007)
http://www.arxiv.org/abs/q-bio.MN/0703009
PPC in vitro: robustness to noise
PPC is highly robust against noise!
PPC in vivo: PPC plus constitutive gene expression
PPC in vitro
PPC in vivo
TTC in vivo
PPC + TTC
PPC in vivo: PPC plus constitutive gene expression
PPC not robust against variations growth rate!
PPC in vivo: PPC plus constitutive gene expression
High growth rate: phosphorylation level new KaiC cannot catch up before degradation.
PPC plus TTC
PPC in vitro
PPC in vivo
TTC in vivo
PPC + TTC
PPC plus TTC
TTC+PPC highly intertwined
System differs from conventional coupled phase oscillators
PPC plus TTC: robustness
PPC plus TTC highly robust.
TTC-only model
PPC in vitro
PPC in vivo
TTC in vivo
PPC + TTC
Comparison performance
Only PPC+TTC robust over full range growth rates!
PPC - TTC: origin enhanced stability
High copy number PPC
Modification leads to delay
Sharp threshold crossings enhances robustness to noise
Conclusions and outlook
Both TTC plus PPC are needed for robustness against variations in growth rate
Mechanism follows from simple argument on comparison protein decay timescale with oscillation period
Also higher organisms employ protein modification
Test by putting system in E.coli?
http://www.arxiv.org/abs/q-bio.MN/1004.2821