Positive Feedback and Bistability Stable state Transient state

Positive Feedback and Bistability

Stable state Transient state Stable state Simulation of biochemical network Stable steady state

Multiple stable states Different starting points lead to different steady states

Positive Feedback v 1 = ? v 2 = ? v 1 d. S/dt = ?

Positive Feedback p = defn cell $Xo -> S 1; 0. 5 + Vmax*S 1^n/(15 + S 1^n); S 1 -> $X 1; k 1*S 1; end; p. Xo = 1; p. X 1 = 0; p. S 1 = 1; p. n = 4; p. Vmax = 10; p. k 1 = 2; 5

Positive Feedback High State S 1 Low State Time 6

v 1 Positive Feedback v 2 Perturbations around a stable point v 2 S 1 v 1 k 1

v 1 Positive Feedback v 2 Perturbations around a stable point v 2 S 1 k 1 v 1

v 1 Positive Feedback v 2 Perturbations around a stable point S 1 k 1 v 2 > v 1

v 1 Positive Feedback v 2 Perturbations around a stable point S 1 v 2 > v 1 Therefore: d. S 1/dt is negative k 1

v 1 Positive Feedback v 2 Perturbations around a unstable point v 2 S 1 v 1 S 1 k 1

v 1 Positive Feedback v 2 Perturbations around a unstable point v 2 S 1 v 1 > v 2 k 1

v 1 Positive Feedback v 2 Perturbations around a unstable point v 2 S 1 v 1 S 1 Therefore: d. S 1/dt is positive v 1 > v 2 k 1

Where in nature do we find multiple steady states? Eukaryotic cell differentiation http: //weirdscience. ca/2007/ Bacterial differentiation and adaptation www. phri. org/research/res_pidubnau. asp

Bistability of the lac operon Where is the positive feedback?

Genetic Toggle Switch d. A/dt = ? d. B/dt = ? Where is the positive feedback? Synthetic toggle switch has been built using lac. I and tet. R repressors. Gardner, T. S. Cantor, C. R. Collins, J. J. Construction of a genetic toggle switch in Escherichia coli. Nature (2000) 6767, pages 339 -342

Flip-Flop (Latch) A B 1 0 0 0 1 0 1 0 0 0 1 1 1 ? ? Flip-flops can be made either from NAND or NOR gates. In synthetic biology it is probably easier to construct OR like gates than AND gates. In addition an OR based flip-flop is quiescent when both inputs are low, meaning low protein levels. Latching occurs when one or other of the inputs is brought to a high state. 18

0 0 NOR Flip-Flop 1 0 A B NOR 1 1 0 0 1 0 NOR 0 1 0 Making NOR gates is ‘relatively’ easy and requires only two operator sites downstream of the RNA polymerase binding site (promoter). Copyright (c) 2010 19

Network structures involving toggle switches Developmental Switch

Bifurcation Diagram Stable Steady state value of A Stable Unstable Stable h

Bistability with Hysteresis State Variable Stable state Unstable state Stable state One of the parameters in the model Gianluca M. Guidi, and Albert Goldbeter. Bistability without Histeresis in Chemical Reaction Systems: A Theoretical Analysis of Irreversible Transitions between Multiple Steady States. Journal of Physical Chemistry (1997), 101 (49).

Bistability with Irreversibility Gianluca M. Guidi, and Albert Goldbeter. Bistability without Histeresis in Chemical Reaction Systems: A Theoretical Analysis of Irreversible Transitions between Multiple Steady States. Journal of Physical Chemistry (1997), 101 (49).