Deterministic and Stochastic Analysis of Simple Genetic Networks

Deterministic and Stochastic Analysis of Simple Genetic NetworksAdiel Loinger

Ofer BihamAzi LipshtatNathalie Q. Balaban

IntroductionThe E. coli transcription network

Taken from: Shen-Orr et al. Nature Genetics 31:64-68(2002)

OutlineThe Auto-repressor

The Toggle Switch

The Repressilator

The Auto-repressorProtein A acts as a repressor to its own geneIt can bind to the promoter of its own gene and suppress the transcription

The Auto-repressorRate equations Michaelis-Menten form

Rate equations Extended Set

n = Hill Coefficient= Repression strength

The Auto-repressor

The Auto-repressorThe Master Equation

Probability for the cell to contain NA free proteins and Nr bound proteins P(NA,Nr) :

The Genetic SwitchA mutual repression circuit.Two proteins A and B negatively regulate each others synthesis

The Genetic SwitchExists in the lambda phageAlso synthetically constructed by collins, cantor and gardner (nature 2000)

The Genetic SwitchPrevious 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)

The SwitchStochastic analysis using master equation and Monte Carlo simulations reveals the reason:For weak repression we get coexistence of A and B proteinsFor strong repression we get three possible states:A dominationB dominationSimultaneous repression (dead-lock)None of these state is really stable

The SwitchIn 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 proteinsBut there exist other options

Bistable SwitchesThe BRD Switch - Bound Repressor DegradationThe PPI Switch Protein-Protein Interaction. A and B proteins form a complex that is inactive

The Exclusive SwitchAn overlap exists between the promoters of A and B and they cannot be occupied simultaneouslyThe rate equations still have a single steady state solution

The Exclusive SwitchBut stochastic analysis reveals that the system is truly a switchThe probability distribution is composed of two peaksThe separation between these peaks determines the quality of the switch

Lipshtat, Loinger, Balaban and Biham, Phys. Rev. Lett. 96, 188101 (2006)Lipshtat, Loinger, Balaban and Biham, Phys. Rev. E 75, 021904 (2007)

k=1k=50

The Exclusive Switch

The RepressillatorA genetic oscillator synthetically built by Elowitz and Leibler Nature 403 (2000)It consist of three proteins repressing each other in a cyclic way

The RepressillatorMonte Carlo Simulations

Rate equations

The RepressillatorMonte Carlo Simulations (50 plasmids)

Rate equations (50 plasmids)

Loinger and Biham, preprint (2007)

SummaryWe have studied several modules of genetic networks using deterministic and stochastic methodsStochastic analysis is required because rate equations give results that may be qualitatively wrongCurrent work is aimed at extending the results to large networks, and including post-transcriptional regulation and other effects