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Deterministic and Stochastic Analysis of Simple Genetic Networks

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Deterministic and Stochastic Analysis of Simple Genetic Networks

Adiel Loinger

MS.c Thesis of

under the supervision of

Ofer Biham

- Introduction
- The Auto-repressor
- The Toggle Switch
- The general switch
- The BRD and PPI switches
- The exclusive switch

- The Repressillator

- Mice and men share 99% of their DNA
- Are we really so similar to mice?

- 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

Bone Cells

Muscle Cells

- 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

genetic network

- 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

- The E. coli transcription network
- Taken from: Shen-Orr et al. Nature Genetics 31:64-68(2002)

- 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

n = Hill Coefficient

= Repression strength

Problems -

- Wrong dynamics
- Very crude formulation (unsuitable for more complex situations)

Rate equations – Michaelis-Menten form

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

P(NA,Nr) :

Probability for the cell to contain NAfree proteins and Nr bound proteins

The Master Equation

- 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

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

- Exists in the lambda phage
- Also synthetically constructed by collins, cantor and gardner (nature 2000)

- Rate equations
- Master equation – too long …

- 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

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)

- 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

- 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…

The BRD Switch - Bound Repressor Degradation

The PPI Switch –

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

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

BRD

PPI

- 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

- 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)

A genetic oscillator synthetically built by Elowitz and Leibler

Nature 403 (2000)

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

Rate equation results

- MC simulations results

- 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