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Positive Feedback and Bistability. BIOE 423: 2013. 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. v1 = ? v2 = ? dS/dt = ?. v2.

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Presentation Transcript
stable state
Stable state

Transient state

Stable state

Simulation of biochemical network

Stable

steady state

multiple stable states
Multiple stable states

Different starting points lead to different steady states

positive feedback
Positive Feedback

v1 = ?

v2 = ?

dS/dt = ?

v2

v1

positive feedback1

Positive Feedback

p = defn cell

$Xo -> S1; 0.5 + Vmax*S1^n/(15 + S1^n);

S1 -> $X1; k1*S1;

end;

p.Xo = 1;

p.X1 = 0;

p.S1 = 1;

p.n = 4;

p.Vmax = 10;

p.k1 = 2;

5

positive feedback2

Positive Feedback

High State

S1

Low State

Time

6

positive feedback3

Positive Feedback

v1

v2

Perturbations around a stable point

v2

S1

v1

k1

positive feedback4

Positive Feedback

v1

v2

Perturbations around a stable point

v2

S1

 S1

v1

k1

positive feedback5

Positive Feedback

v1

v2

Perturbations around a stable point

v2

S1

v2 > v1

 S1

v1

k1

positive feedback6

Positive Feedback

v1

v2

Perturbations around a stable point

v2

S1

v2 > v1

 S1

v1

Therefore: dS1/dt is negative

k1

positive feedback7

Positive Feedback

v1

v2

Perturbations around a unstable point

v2

S1

v1

 S1

k1

positive feedback8

Positive Feedback

v1

v2

Perturbations around a unstable point

v2

S1

v1

 S1

v1 > v2

k1

positive feedback9

Positive Feedback

v1

v2

Perturbations around a unstable point

v2

S1

v1

Therefore: dS1/dt is positive

 S1

v1 > v2

k1

positive feedback10

Positive Feedback

v1

v2

Perturbations around a unstable point

v2

S1

v1

Therefore: dS1/dt is positive

 S1

v1 > v2

k1

where in nature do we find multiple steady states
Where in nature do we find multiple steady states?

Eukaryotic cell differentiation

Bacterial differentiation and adaptation

www.phri.org/research/res_pidubnau.asp

http://weirdscience.ca/2007/

bistability of the lac operon
Bistability of the lac operon

Where is the positive feedback?

genetic toggle switch
Genetic Toggle Switch

dA/dt = ?

dB/dt = ?

Where is the positive feedback?

Synthetic toggle switch has been built using lacI and tetR 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
Flip-Flop (Latch)

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.

flip flop
Flip-Flop

0

0

NOR

1

0

0

0

NOR

0

1

Making NOR gates is ‘relatively’ easy and requires only two operator sites

downstream of the RNA polymerase binding site (promoter).

flip flop1
Flip-Flop

0

0

NOR

1

0

0

0

NOR

0

1

flip flop2
Flip-Flop

0

0

NOR

1

0

0

0

NOR

0

1

0

0

NOR

1

1

0

0

NOR

0

1

flip flop3
Flip-Flop

0

0

NOR

1

0

0

0

NOR

0

1

1

0

NOR

1

1

0

0

NOR

0

1

flip flop4
Flip-Flop

0

0

NOR

1

0

0

0

NOR

0

1

1

0

NOR

0

1

0

0

NOR

0

1

flip flop5
Flip-Flop

0

0

NOR

1

0

0

0

NOR

0

1

1

0

NOR

0

1

0

0

NOR

0

0

flip flop6
Flip-Flop

0

0

NOR

1

0

0

0

NOR

0

1

1

0

NOR

0

1

0

1

NOR

0

0

flip flop7
Flip-Flop

0

0

NOR

1

0

0

0

NOR

0

1

1

1

NOR

0

1

0

1

NOR

0

0

flip flop8
Flip-Flop

0

0

NOR

1

0

0

0

NOR

0

1

0

1

1

1

NOR

NOR

0

0

0

1

0

0

1

1

NOR

NOR

0

0

0

0

flip flop9
Flip-Flop

0

0

NOR

1

0

0

Toggle A to reset P1

Toggle B to set P1

0

NOR

0

1

0

0

0

0

NOR

NOR

1

1

0

0

0

1

0

0

NOR

NOR

0

1

1

1

bifurcation diagram
Bifurcation Diagram

Stable

Steady state value of A

Stable

Unstable

Stable

h

bistability with hysteresis
Bistability with Hysteresis

Stable state

State Variable

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
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).