Deterministic and Stochastic Analysis of Simple Genetic Networks

1. Deterministic and Stochastic Analysis of Simple Genetic Networks Adiel Loinger Ofer Biham Azi Lipshtat Nathalie Q. Balaban

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

3. Outline • The Auto-repressor • The Toggle Switch • The Repressilator

4. 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

5. The Auto-repressor • Rate equations – Michaelis-Menten form • Rate equations – Extended Set n = Hill Coefficient = Repression strength

6. The Auto-repressor

7. The Auto-repressor P(NA,Nr) : Probability for the cell to contain NAfree proteins and Nr bound proteins The Master Equation

8. The Genetic Switch • A mutual repression circuit. • Two proteins A and B negatively regulate each other’s synthesis

9. The Genetic Switch • Exists in the lambda phage • Also synthetically constructed by collins, cantor and gardner (nature 2000)

10. 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, 128101 (2004) Walczak et al., Biophys. J. 88, 828 (2005)

11. 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

12. 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…

13. The BRD Switch - Bound Repressor Degradation The PPI Switch – Protein-Protein Interaction. A and B proteins form a complex that is inactive Bistable Switches

14. 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

15. 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) Lipshtat, Loinger, Balaban and Biham, Phys. Rev. E 75, 021904 (2007)

16. The Exclusive Switch

17. 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

18. The Repressillator • Monte Carlo Simulations • Rate equations

19. The Repressillator • Monte Carlo Simulations (50 plasmids) • Rate equations (50 plasmids) Loinger and Biham, preprint (2007)

20. 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