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 oscillations Entrained 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, kaiC kaiBC forms an operon Expression kaiBC oscillates Continuous overexpression KaiC represses kaiBC Temporal overexpression KaiC resets phase Transcription-translation cycle (TTC)! kaiBC C C C 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 oscillations in 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 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 Kageyama et al. Mol. Cell 2006  KaiA stimulates phosphorylation of active KaiC  Binding of KaiA stabilizes active KaiC

Full model Kai system: KaiC + KaiB  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 Nakajima et al. Science 2005  KaiB-KaiC binds to and sequesters KaiA, leading to another form of differential affinity

Phase portrait Changing KaiAChanging 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)

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?