Deterministic and Stochastic Analysis of Simple Genetic Networks

Adiel Loinger

Ofer BihamAzi LipshtatNathalie Q. Balaban

Introduction

The E. coli transcription network

Taken from: Shen-Orr et al. Nature Genetics 31:64-68(2002)

Outline

The Auto-repressor

The Toggle Switch

The Repressilator

The Auto-repressor

Protein A acts as a repressor to its own gene

It can bind to the promoter of its own gene and suppress the transcription

Rate equations – Michaelis-Menten form

Rate equations – Extended Set

The Auto-repressor

0 1

0 1

[ ](1 [ ]) [ ] [ ](1 [ ]) [ ]

[ ][ ](1 [ ]) [ ]

d Ag r d A A r r

dtd r

A r rdt

[ ][ ]

1 [ ]nd A g

d Adt k A

0 1/k n = Hill Coefficient

= Repression strength

The Auto-repressor

The Auto-repressor

The Master Equation

,0( , ) [ ( 1, ) ( , )]rA r N A r A r

dP N N g P N N P N N

dt

[( 1) ( 1, ) ( , )]A A r A A rd N P N N N P N N

0 ,1 ,0[ ( 1) ( 1, 1) ( , )]r rN A A r N A A rN P N N N P N N

1 ,0 ,1[ ( 1, 1) ( , )]r rN A r N A rP N N P N N

Probability for the cell to contain NA free proteins and Nr bound proteins

P(NA,Nr) :

The Genetic Switch

A mutual repression circuit. Two proteins A and B negatively

regulate each other’s synthesis

The Genetic Switch

Exists in the lambda phage Also synthetically constructed by

collins, cantor and gardner (nature 2000)

The Genetic Switch

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

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

Gardner et al., Nature, 403, 339 (2000)

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

Warren an ten Wolde, PRL 92, 128101 (2004)

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

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

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…

Bistable Switches

The BRD Switch - Bound Repressor Degradation

The PPI Switch –

Protein-Protein Interaction. A and B proteins form a complex that is inactive

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

The Exclusive Switch

But 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

The Exclusive Switch

The Repressillator

A genetic oscillator synthetically built by Elowitz and Leibler

Nature 403 (2000)

It consist of three proteins repressing each other in a cyclic way

The Repressillator

Monte Carlo Simulations

Rate equations

The Repressillator

Monte Carlo Simulations

(50 plasmids)

Rate equations

(50 plasmids)

Loinger and Biham, preprint (2007)

Summary

We have studied several modules of genetic networks using deterministic and stochastic methods

Stochastic analysis is required because rate equations give results that may be qualitatively wrong

Current work is aimed at extending the results to large networks, and including post-transcriptional regulation and other effects