Deterministic and Stochastic Analysis of Simple Genetic Networks Adiel Loinger Ofer Biham Azi Lipshtat Nathalie 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 n = Hill Coefficient = Repression strength

The Auto-repressor

The Master Equation Probability for the cell to contain N A free proteins and N r bound proteins P(N A,N r ) :

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, (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, (2006) Lipshtat, Loinger, Balaban and Biham, Phys. Rev. E 75, (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