Post on 23-Feb-2016
description
Positive Feedback and Bistability
BIOE 423: 2013
Stable state
0 2 4 6 8 10
1.0
1.5
2.0
t
[s]
Simulation of biochemical network
Stable steady state
Transient state Stable state
0 10 20 30 40
0.5
1.0
1.5
2.0
2.5
3.0
t
[s]
Multiple stable states
0 10 20 30 40 50
68
1012
t
[s]
0 10 20 30 40 50
68
1012
t
[s]
Different starting points lead to different steady states
Positive Feedback
v1 = ?
v2 = ?
dS/dt = ?v1
v2
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
Time
S1
High State
Low State
6
Positive Feedback
S1
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6
k1
v2
v1
v1 v2
Perturbations around a stable point
Positive Feedback
S1
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6
k1
v2
v1
v1 v2
Perturbations around a stable point
S1
Positive Feedback
S1
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6
k1
v2
v1
v1 v2
v2 > v1
Perturbations around a stable point
S1
Positive Feedback
S1
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6
k1
v2
v1
v1 v2
v2 > v1
Therefore: dS1/dt is negative
Perturbations around a stable point
S1
Positive Feedback
S1
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6
k1
v2
v1
v1 v2
Perturbations around a unstable point
S1
Positive Feedback
S1
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6
k1
v2
v1
v1 v2
v1 > v2
Perturbations around a unstable point
S1
Positive Feedback
S1
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6
k1
v2
v1
v1 v2
Therefore: dS1/dt is positive
Perturbations around a unstable point
v1 > v2 S1
Positive Feedback
S1
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6
k1
v2
v1
v1 v2
Therefore: dS1/dt is positive
Perturbations around a unstable point
v1 > v2 S1
Where in nature do we find multiple steady states?
http://weirdscience.ca/2007/ www.phri.org/research/res_pidubnau.asp
Eukaryotic cell differentiation Bacterial differentiation and adaptation
Bistability of the lac operon
Where is the positive feedback?
Genetic Toggle Switch
Where is the positive feedback?
dA/dt = ?
dB/dt = ?
Gardner, T. S. Cantor, C. R. Collins, J. J. Construction of a genetic toggle switch in Escherichia coli. Nature (2000) 6767, pages 339-342
Synthetic toggle switch has been built using lacI and tetR repressors.
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 constructOR like gates than AND gates.
In addition an OR based flip-flop is quiescent when both inputs are low, meaning low protein levels. Latching occurswhen one or other of the inputs is brought to a high state. 18
Flip-Flop0
0
1
0
0
NOR
NOR
A B NOR
1 1 0
0 1 0
1 0 0
0 0 1
1
0 0
Making NOR gates is ‘relatively’ easy and requires only two operator sitesdownstream of the RNA polymerase binding site (promoter).
19
Flip-Flop0
0
1
0
0
NOR
NOR
A B NOR
1 1 0
0 1 0
1 0 0
0 0 1
1
0 0
20
Flip-Flop0
0
1
0
0
NOR
NOR
A B NOR
1 1 0
0 1 0
1 0 0
0 0 1
1
0 0
1
0
1
0
0
NOR
NOR1
0 0
21
Flip-Flop0
0
1
0
0
NOR
NOR
A B NOR
1 1 0
0 1 0
1 0 0
0 0 1
1
0 0
1
0
1
0
0
NOR
NOR1
1 0
22
Flip-Flop0
0
1
0
0
NOR
NOR
A B NOR
1 1 0
0 1 0
1 0 0
0 0 1
1
0 0
1
0
0
0
0
NOR
NOR1
1 0
23
Flip-Flop0
0
1
0
0
NOR
NOR
A B NOR
1 1 0
0 1 0
1 0 0
0 0 1
1
0 0
1
0
0
0
0
NOR
NOR0
1 0
24
Flip-Flop0
0
1
0
0
NOR
NOR
A B NOR
1 1 0
0 1 0
1 0 0
0 0 1
1
0 0
1
1
0
0
0
NOR
NOR0
1 0
25
Flip-Flop0
0
1
0
0
NOR
NOR
A B NOR
1 1 0
0 1 0
1 0 0
0 0 1
1
0 0
1
1
0
0
0
NOR
NOR0
1 1
26
Flip-Flop0
0
1
0
0
NOR
NOR
A B NOR
1 1 0
0 1 0
1 0 0
0 0 1
1
0 0
1
1
0
0
0
NOR
NOR0
1 1
0
1
0
0
0
NOR
NOR0
0 1
27
Flip-Flop0
0
1
0
0
NOR
NOR
A B NOR
1 1 0
0 1 0
1 0 0
0 0 1
1
0 0
0
0
1
1
1
NOR
NOR1
0 0
0
0
1
0
0
NOR
NOR1
0 0
Toggle A to reset P1Toggle B to set P1
28
Network structures involving toggle switches
Developmental Switch
Bifurcation Diagram
h
Steady state value of A
Stable Unstable
Stable
Stable
Bistability with Hysteresis
One of the parameters in the model
Unstable state
Stable state
Stable state
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).
State Variable
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).