Deterministic and Stochastic Analysis of Simple Genetic Networks Adiel Loinger MS.c Thesis of under...
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Transcript of Deterministic and Stochastic Analysis of Simple Genetic Networks Adiel Loinger MS.c Thesis of under...
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