Deterministic and Stochastic Analysis of Simple Genetic Networks

Adiel LoingerMS.c Thesis of

under the supervision of

Ofer Biham

Outline

Introduction The Auto-repressor The Toggle Switch

The general switch The BRD and PPI switches The exclusive switch

The Repressillator

Introduction

Mice and men share 99% of their DNA

Are we really so similar to mice?

Introduction

The origin of the biological diversity is not only due the genetic code, but also due differences in the expression of genes in different cells and individuals Muscle Cells Bone Cells

Introduction

The DNA contains the genetic code It is transcribed to the mRNA The mRNA is translated to a protein The proteins perform various

biochemical tasks in the cell

The Basics of Protein Synthesis

Introduction

The transcription process is initiated by the binding of the RNA polymerase to the promoter

If the promoter site is occupied by a protein, transcription is suppressed

This is called repression

Transcriptional Regulation

In this way we get a network of interactions between proteins

genetic network

Introduction

The E. coli transcription network

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

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

Problems - Wrong dynamics Very crude formulation (unsuitable

for more complex situations)

Rate equations – Michaelis-Menten form

Rate equations – Extended setBetter than Michaelis-MentenBut still has flaws: A mean field approach Does not account for fluctuations

caused by the discrete nature of proteins and small copy number of binding sites

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 Auto-repressor

The master and rate equations differ in steady state

The differences are far more profound in more complex systems

For example in the case of … Lipshtat, Perets, Balaban and Biham,

Gene 347, 265 (2005)

The Toggle Switch

The General Switch

A mutual repression circuit. Two proteins A and B negatively

regulate each other’s synthesis

The General Switch

Exists in the lambda phage Also synthetically constructed by

collins, cantor and gardner (nature 2000)

The General Switch

Rate equations

Master equation – too long …

[ ][ ]

1 [ ]

[ ][ ]

1 [ ]

n

n

d A gd A

dt k B

d B gd B

dt k A

0 1

0 1

0 1

0 1

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

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

[ ][ ](1 [ ]) [ ]

[ ][ ](1 [ ]) [ ]

B A A

A B B

AA A

BB B

d Ag r d A A r r

dtd B

g r d B B r rdtd r

A r rdtd r

B r rdt

The General Switch

A well known result - The rate equations have a single steady state solution for Hill coefficient n=1:

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

1 4 / 1[ ] [ ]

2

kg dA B

k

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 specie 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

Bistable Switches

The BRD and PPI switches exhibit bistability at the level of rate equations.

[ ]( )[ ]

1 [ ] 1 [ ]

[ ]( )[ ]

1 [ ] 1 [ ]

rn

rn

d kd A gd A

dt k B k A

d kd B gd B

dt k A k B

[ ]( [ ])[ ]

1 [ ]

[ ]( [ ])[ ]

1 [ ]

n

n

d A gd B A

dt k B

d B gd A B

dt k A

BRD PPI

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)

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

Rate equation results MC simulations results

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 other networks, such as oscillators and post-transcriptional regulation