The robust ticking of a circadian clock David Zwicker, Jeroen van Zon,David Lubensky, Pim Altena,...

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 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


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


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


0 1 6K K K

Solve set of ODEs


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)


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)


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 not robust against variations growth rate!


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

The robust ticking of a circadian clock David Zwicker, Jeroen van Zon,David Lubensky, Pim Altena,...

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