Bistable and Oscillatory Systems

Bistable Systems Systems which display two stable steady states with a third unstable state are usually termed bistable systems.

Bistable Systems Time S1 High State Low State

Bifurcation Plots Parameter: k1 Steady State: S1 Unstable branch Stable branch

Bistable Systems Oscill8 Emery Conrad

Bistable Systems S1 k1 v2 v1 v2 Perturbations around a stable point

Bistable Systems S1 k1 v2 v1 v2 Perturbations around a stable point  S1

Bistable Systems S1 k1 v2 v1 v2 v2 > v1 Perturbations around a stable point  S1

Bistable Systems S1 k1 v2 v1 v2 v2 > v1 Therefore: dS1/dt is negative Perturbations around a stable point  S1

Bistable Systems S1 k1 v2 v1 v2 Perturbations around a unstable point  S1

Bistable Systems S1 k1 v2 v1 v2 v1 > v2 Perturbations around a unstable point  S1

Bistable Systems S1 k1 v2 v1 v2 Therefore: dS1/dt is positive Perturbations around a unstable point v1 > v2  S1

Bistable Systems S1 k1 v2 v1 v2 Therefore: dS1/dt is positive Perturbations around a unstable point v1 > v2  S1

Irreversible Bistable System S R Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell John J Tysony, Katherine C Chenz and Bela Novak Current Opinion in Cel Biology, vol 15, (2003)

Natural Examples Lac Operon Lac Genes Lactose (Internal) Lactose (External) Allolactose (Internal) Permease Membrane Positive Feedback Loop

Natural Examples Lac Operon Multistability in the lactose utilization network of Escherichia coli, Ertugrul M. Ozbudak et al, Nature, vol 247, (2004), Green fluorescence measures the level of expression in the Lac operon.

Natural Examples Sin (Sporlation Inhibition) Operon Sporulation is an expensive and dramatic response to stress. The sin operon is central to the timing and early dynamics of sporulation.

Natural Examples Sin Operon sinI sinR SinI SinR Poor conditions leads to the stimulation of SinI expression. SinI represses constitutive production of SinR Inactivation of SinR leads to activation of the sporulation pathway. The Bacillus subtilis sin Operon: An Evolvable Network Motif Christopher A. Voigt, Denise M. Wolf and Adam Arkin Genetics, vol 169, (2004)

Natural Examples Sin Operon sinI sinR SinR Poor conditions leads to the stimulation of SinI expression. SinI represses constitutive production of SinR Inactivation of SinR leads to activation of the sporulation pathway. The Bacillus subtilis sin Operon: An Evolvable Network Motif Christopher A. Voigt, Denise M. Wolf and Adam Arkin Genetics, vol 169, (2004)

Natural Examples Sin Operon sinI sinR SinR Sporulate (Output) Poor conditions leads to the stimulation of SinI expression. SinI represses constitutive production of SinR Inactivation of SinR leads to activation of the sporulation pathway. The Bacillus subtilis sin Operon: An Evolvable Network Motif Christopher A. Voigt, Denise M. Wolf and Adam Arkin Genetics, vol 169, (2004)

Natural Examples Sin Operon sinI sinR SinI SinR Environmental Signal, Poor Conditions Sporulate (Output) Poor conditions leads to the stimulation of SinI expression. SinI represses constitutive production of SinR Inactivation of SinR leads to activation of the sporulation pathway. The Bacillus subtilis sin Operon: An Evolvable Network Motif Christopher A. Voigt, Denise M. Wolf and Adam Arkin Genetics, vol 169, (2004)

Natural Examples Sin Operon sinI sinR SinI SinR Environmental Signal, Poor Conditions Sporulate (Output) The Bacillus subtilis sin Operon: An Evolvable Network Motif Christopher A. Voigt, Denise M. Wolf and Adam Arkin Genetics, vol 169, (2004)

Natural Examples Sin Operon sinI sinR SinI SinR Environmental Signal, Poor Conditions Sporulate (Output) The Bacillus subtilis sin Operon: An Evolvable Network Motif Christopher A. Voigt, Denise M. Wolf and Adam Arkin Genetics, vol 169, (2004)

Natural Examples Arkin: DARPA Meeting, 2002

Synthetic Bistable Systems Prediction and measurement of an autoregulatory genetic module Farren J. Isaacs, Jeff Hasty, Charles R. Cantor, and J. J. Collins, PNAS, 100:7714 (2003)

Synthetic Bistable Systems Nature 403, (2000) Construction of a genetic toggle switch in Escherichia coli Timothy S. Gardner, Charles R. Cantor and James J. Collins

Modifying a Bistable System

p = defn cell $Xo -> So; k0*Xo; So -> S1; k1*So + Vmax*So*S1^n/(15 + S1^n); S1 -> $X1; k2*S1; end;

Relaxation Oscillator

So S1

Relaxation Oscillator A relaxation oscillator has two parts, a threshold device, for example a bistable system, and a negative feedback loop.

Relaxation Oscillator

Synthetic Relaxation Oscillator Cell, Vol. 113, 597–607, May 30, 2003, Development of Genetic Circuitry Exhibiting Toggle Switch or Oscillatory Behavior in Escherichia coli, Mariette R. Atkinson Michael A. Savageau Jesse T. Myers and Alexander J. Ninfa

Negative Feedback

V1, V2 V1 V2 = 0.3 V2 = 0.2 V2 = 0.1 S1

Negative Feedback Phase Shift Oscillator

Ring Oscillator

Ring Oscillator Repressilator

Bistable Systems CELL SIGNALLING DYNAMICS IN TIME AND SPACE Boris N. Kholodenko, Nat Rev Mol Cell Biol. Vol 7(3), (2006)