Stochastic Simulations of Genetic Regulatory Networks:

The Genetic Toggle Switch

Adiel Loinger

Ofer BihamNathalie Q. BalabanAzi LipshtatYishai ShimoniBaruch BarzelGuy HetzroniDan Mendels

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Introduction

E. coli transcriptional regulation network

Data taken from RegulonDB

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Introduction

Transcriptional Repression:

genepromoter

When the promoter is vacant the gene is expressed (mRNAs and proteins are being synthesized)

genepromoter

When the promoter is occupied by a repressor, transcription is suppressed (mRNAs and proteins are not synthesized)

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• The Hill-coefficient h represents the number of repressors required to bind simultaneously in order to perform the regulation

• The repression strength is represented by a parameter k (the ratio between binding and unbinding rates).

Introduction

A

h = 1h = 2

AA

Some important parameters:

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Introduction

Deterministic and Stochastic Analysis

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The Genetic Switch

• A mutual repression circuit.• Two proteins A and B negatively regulate each other’s

synthesis• This architecture is also called the general switch

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The Genetic Switch

• Exists in the lambda phage• Also synthetically constructed on plasmids in E. coli by

Gardner, Cantor and Collins [Nature 403,339 (2000)]

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The Genetic Switch

• Previous studies using deterministic rate equations concluded that for Hill-coefficient h=1 there is a single steady state solution and no bistability.

• Conclusion - cooperative binding (Hill-coefficient h>1) is required for a switch

Gardiner, Cantor and Collins, Nature, 403, 339 (2000)

Cherry and Adler, J. Theor. Biol. 203, 117 (2000)

Warren and ten Wolde, Phys. Rev. Lett. 92, 128101 (2004)

Walczak et al., Biophys. J. 88, 828 (2005)

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The Switch

• Stochastic analysis using master equation and Monte Carlo simulations reveals the reason:

• For weak repression we get coexistence of A and B proteins

• For strong repression we get three possible states: A domination B domination Simultaneous repression (dead-lock)

• None of these state is really stable

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The Switch

• In order that the system will become a switch, the dead-lock situation (= the peak near the origin) must be eliminated.

• Cooperative binding does this – The minority protein type has hard time to recruit two proteins

• But there exist other options…

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The Exclusive Switch

An overlap exists between the promoters of A and B and they cannot be occupied simultaneously

The rate equations still have a

single steady state solution

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The Exclusive Switch

• However, stochastic analysis reveals that the system is truly a switch

• The probability distribution is composed of two peaks

• The separation between these peaks determines the quality of the switch

Lipshtat, Loinger, Balaban and Biham, Phys. Rev. Lett. 96, 188101 (2006)

Lipshtat, Loinger, Balaban and Biham, Phys. Rev. E 75, 021904 (2007)

k=1

k=50

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The Exclusive Switch

• Spontaneous transitions occur between the two states of the switch

• The stability of the switch is characterized by the mean time between transitions

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Plasmids

• The synthetic toggle switch was encoded on plasmids in E. coli.

• Plasmids are circular self replicating DNA molecules which include only few genes.

• The number of plasmids in a cell can be controlled.

How does the number of plasmids affect the switching time?

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The effect of plasmids copy number

• Warren and ten Wolde [PRL 92, 128101 (2004)] showed that for a single plasmid with h = 2, the exclusive switch is more stable than the general switch.

This does not hold for a high plasmid copy number

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The effect of plasmids copy number

Loinger and Biham, preprint

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Summary

• Stochastic analysis is required for studying genetic circuits with feedback

• Subtle features may play a major role• Additional research topics include:

Other types of modules (Repressilator, Mixed feedback loop, etc)

Other levels of regulation (Post-Transcriptional, Protein-Protein Interactions)

Analysis of complex networks