Transcript Genetic Toggle Switch without Cooperative Binding

Deterministic and
Stochastic Analysis of
Simple Genetic Networks
MS.c Thesis of
Adiel Loinger
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 Basics of Protein Synthesis




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
Introduction
Transcriptional Regulation



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
In this way we get a network of
interactions between proteins
genetic
network
Introduction

The E. coli
transcription
network

Taken from: ShenOrr et al. Nature
Genetics 31:6468(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
The Auto-repressor

Rate equations – Michaelis-Menten form
d [ A]
g

 d [ A]
n
dt 1  k[ A]
n = Hill Coefficient
k   0 / 1 = Repression strength

Rate equations – Extended Set
d [ A]
 g (1  [r ])  d [ A]   0 [ A](1  [ r ])  1[ r ]
dt
d [r ]
  0 [ A](1  [r ])  1[r ]
dt
The Auto-repressor
Rate equations – Michaelis-Menten form
Problems  Wrong dynamics
 Very crude formulation (unsuitable
for more complex situations)
Rate equations – Extended set
Better than Michaelis-Menten
But 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
P(NA,Nr) : Probability for the cell to contain NA free
proteins and Nr bound proteins
The Master Equation
d
P( N A , N r )  g Nr ,0 [ P( N A  1, N r )  P( N A , N r )]
dt
d[( N A  1) P( N A  1, N r )  N A P( N A , N r )]
 0 [ Nr ,1 ( N A  1) P( N A  1, N r  1)   Nr ,0 N A P( N A , N r )]
1[ Nr ,0 P( N A  1, N r  1)   Nr ,1 P( N A , N r )]
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
d [ A]
g

 d [ A]
dt
1  k[ B]n
d [ B]
g

 d [ B]
dt
1  k[ A]n

Master equation –
too long …
d [ A]
 g (1  [rB ])  d [ A]   0 [ A](1  [rA ])  1[rA ]
dt
d [ B]
 g (1  [rA ])  d [ B]   0 [ B](1  [rB ])  1[rB ]
dt
d [rA ]
  0 [ A](1  [rA ])  1[rA ]
dt
d [rB ]
  0 [ B](1  [rB ])  1[rB ]
dt
The General Switch

A well known result - The rate equations have a single
steady state solution for Hill coefficient n=1:
1  4kg / d  1
2k
Conclusion - Cooperative binding (Hill coefficient n>1)
is required for a switch
[ A]  [ B] 

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


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



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 (deadlock)
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.
BRD
dr k
d [ A]
g


(
d

)[ A]
dt
1  k[ B]n
1  k[ A]
dr k
d [ B]
g


(
d

)[ B]
dt
1  k[ A]n
1  k[ B]
PPI
d [ A]
g

 (d   [ B])[ A]
dt
1  k[ B]n
d [ B]
g

 (d   [ A])[ B]
dt
1  k[ A]n
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
k=1
k=50
Lipshtat, Loinger, Balaban and Biham, Phys. Rev. Lett. 96,188101 (2006)
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