Positive Feedback and Bistability
Systems and Synthetic Biology
BIOE 424
Stable state
Transient state
Stable state
Stable
steady state
0.5
1.0
1.0
1.5
[s]
[s]
2.0
1.5
2.5
2.0
3.0
Simulation of biochemical network
0
2
4
6
t
8
10
0
10
20
t
30
40
6
8
[s]
10
12
Multiple stable states
Different starting
points lead to
different steady
states
0
10
20
30
40
50
30
40
50
6
8
[s]
10
12
t
0
10
20
t
Positive Feedback
v1 = ?
v2
v2 = ?
dS/dt = ?
v1
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 Feedback
High State
S1
Low State
Time
6
v1
Positive
Feedback
v2
16
Perturbations around a stable point
14
S1
v2
12
v1
10
8
6
4
2
0
0
1
2
3
4
k1
5
6
v1
Positive
Feedback
v2
16
Perturbations around a stable point
14
S1
12
v2
 S1
v1
10
8
6
4
2
0
0
1
2
3
4
k1
5
6
v1
Positive
Feedback
v2
16
Perturbations around a stable point
14
S1
12
v2
v2 > v1
 S1
v1
10
8
6
4
2
0
0
1
2
3
4
k1
5
6
v1
Positive
Feedback
v2
16
Perturbations around a stable point
14
S1
12
v2
v2 > v1
 S1
v1
10
8
Therefore: dS1/dt is
negative
6
4
2
0
0
1
2
3
4
k1
5
6
v1
Positive
Feedback
v2
16
Perturbations around a unstable point
14
S1
v2
12
v1
10
8
6
 S1
4
2
0
0
1
2
3
4
k1
5
6
v1
Positive
Feedback
v2
16
Perturbations around a unstable point
14
S1
v2
12
v1
10
8
6
 S1
v1 > v2
4
2
0
0
1
2
3
4
k1
5
6
v1
Positive
Feedback
v2
16
Perturbations around a unstable point
14
S1
v2
12
v1
10
8
6
Therefore: dS1/dt is
positive
 S1
v1 > v2
4
2
0
0
1
2
3
4
k1
5
6
v1
Positive
Feedback
v2
16
Perturbations around a unstable point
14
S1
v2
12
v1
10
8
6
Therefore: dS1/dt is
positive
 S1
v1 > v2
4
2
0
0
1
2
3
4
k1
5
6
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
Where is the positive feedback?
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)
A
B
1
0
1
0
0
0
1
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
1
0
0
0
0
1
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).
Copyright (c) 2010
19
0 0
NOR
Flip-Flop
1
0
A B
NOR
1
1
0
0
1
0
1
0
0
0
0
1
0
0
NOR
0 1
Copyright (c) 2010
20
0 0
NOR
Flip-Flop
1
0
A B
NOR
1
1
0
0
1
0
1
0
0
0
0
1
0
0
NOR
0 1
0 0
NOR
1
1
0
0
NOR
0 1
Copyright (c) 2010
21
0 0
NOR
Flip-Flop
1
0
A B
NOR
1
1
0
0
1
0
1
0
0
0
0
1
0
0
NOR
0 1
1 0
NOR
1
1
0
0
NOR
0 1
Copyright (c) 2010
22
0 0
NOR
Flip-Flop
1
0
A B
NOR
1
1
0
0
1
0
1
0
0
0
0
1
0
0
NOR
0 1
1 0
NOR
0
1
0
0
NOR
0 1
Copyright (c) 2010
23
0 0
NOR
Flip-Flop
1
0
A B
NOR
1
1
0
0
1
0
1
0
0
0
0
1
0
0
NOR
0 1
1 0
NOR
0
1
0
0
NOR
0 0
Copyright (c) 2010
24
0 0
NOR
Flip-Flop
1
0
A B
NOR
1
1
0
0
1
0
1
0
0
0
0
1
0
0
NOR
0 1
1 0
NOR
0
1
0
1
NOR
0 0
Copyright (c) 2010
25
0 0
NOR
Flip-Flop
1
0
A B
NOR
1
1
0
0
1
0
1
0
0
0
0
1
0
0
NOR
0 1
1 1
NOR
0
1
0
1
NOR
0 0
Copyright (c) 2010
26
0 0
NOR
Flip-Flop
A B
1
0
NOR
1
1
0
0
1
0
1
0
0
0
0
1
0
0
NOR
0 1
1 1
NOR
0 1
NOR
0
1
0
0
0
1
1
NOR
0 0
0
Copyright (c) 2010
NOR
0 0
27
0 0
NOR
Flip-Flop
A B
1
0
0
NOR
0 1
0 0
NOR
1
0
0
1
0
1
0
0
1
0
1
0
0
0
0
1
1
0
0
NOR
1 1
1
Toggle A to reset P1
Toggle B to set P1
0
0 0
NOR
NOR
Copyright (c) 2010
NOR
0 1
28
Network structures involving toggle switches
Developmental Switch
Bifurcation Diagram
Stable
Steady state
value of A
Stable
Unstable
Stable
h
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
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).