Positive Feedback and Bistability
1-23-09
Systems and Synthetic Biology
BIOE 498A
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 = ?
dS1/dt = ?
S1
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
v1
and
v2
v2
12
v1
10
8
6
4
2
0
0
1
2
3
4
S1
5
6
v1
Positive
Feedback
v2
16
Perturbations around a stable point
14
v1
and
v2
v2
δ S1
12
v1
10
8
6
4
2
0
0
1
2
3
4
S1
5
6
v1
Positive
Feedback
v2
16
Perturbations around a stable point
14
v2
δ S1
12
v1
and
v2
v2 > v1
v1
10
8
6
4
2
0
0
1
2
3
4
S1
5
6
v1
Positive
Feedback
v2
16
Perturbations around a stable point
14
v1
and
v2
v2
δ S1
12
v2 > v1
v1
10
8
Therefore: dS1/dt is
negative
6
4
2
0
0
1
2
3
4
S1
5
6
v1
Positive
Feedback
v2
16
Perturbations around a unstable point
14
v1
and
v2
v2
12
v1
10
8
δ S1
6
4
2
0
0
1
2
3
4
S1
5
6
v1
Positive
Feedback
v2
16
Perturbations around a unstable point
14
v1
and
v2
v2
12
v1
10
8
δ S1
6
v1 > v2
4
2
0
0
1
2
3
4
S1
5
6
v1
Positive
Feedback
v2
16
Perturbations around a unstable point
14
v1
and
v2
v2
12
v1
10
8
δ S1
Therefore: dS1/dt is
positive
6
v1 > v2
4
2
0
0
1
2
3
4
S1
5
6
v1
Positive
Feedback
v2
16
v2
14
v1
and
v2
12
v1
10
8
6
4
2
0
0
1
2
3
4
S1
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
PPlane
http://math.rice.edu/~dfield/dfpp.html
Stable points
Joshua L. Cherry and Frederick R. Adler. How to make a Biological Switch. J. Theor. Biol. (2000) 203, 117-133
Bifurcation Diagram
Bifurcation point
Stable
Steady state
value of A
Stable
Unstable
Stable
h
Bistability with Hysteresis
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).
More Hysteresis and 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).
Network structures involving toggle switches
Can you guess where this is useful?