Oscillatory Circuits
Copyright © 2008: Sauro
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Simple Bistable Circuit
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Oscillator Circuits
Repressilator
Classic Feedback Oscillator
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Oscillator Circuits
Relaxation Oscillator
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Oscillator Circuits
Excitable, Noise Driven Oscillator
p545
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Oscillator Circuits
Excitable, Noise Driven Oscillator
dP1/dt = 0.004 + 0.08*P1^2/(0.2^2+P1^2) – P1/(1+P1+P2)
dP2/dt = 0.8/(1+(P1/0.222)^5) - P2/(1+P1+P2)
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Oscillator Circuits
1. Stable point
2. Saddle point
3. Unstable point
Excitable, Noise Driven Oscillator
Red Line: Nullcline for P1
Orange Line: Nullcline for P2
P2
2.
1.
3.
As we vary P1 how does
P2 change (Orange)
As we vary P2 how does
P1 change (Red)
P1
Software
Copyright (c)Tool:
2008 http://math.rice.edu/~dfield/dfpp.html
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Circuits
Behaviors
Designs
1. Memory less switches
2. Bistable Switches/Toggle
3. Oscillators
4. Noise Filters
5. Robustness
6. Response Acceleration
7. Logical Decisions
8. Signal Amplifiers
9. Pulse Generators
10. Event Sequencing
11. …….
1.
2.
3.
4.
5.
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Negative Feedback
Positive Feedback
Negative Feedforward
Positive Feedforward
Cooperativity
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Logic Circuits
AND/NAND
A B
A
B
OR/NOR
A
B
AND
NAND
1
1
1
0
0
1
0
1
1
0
0
1
0
0
0
1
A B
OR
NOR
1
1
1
0
0
1
1
0
1
0
1
0
0
0
0
1
NOT
A
XOR/NXOR
A
B
A B
A
B
0
1
1
0
XOR
NXOR
1
1
0
1
0
1
1
0
1
0
1
0
0
0
0
1
All gates can be derived from the NAND Gate.
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OR/NOR/XOR/NXOR/NOT can be Made
from Combinations NAND Gates
NOT
OR
A
A
B
XOR
OR
AND
NAND
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Flip-Flop (Latch)
A
B
1
0
1
0
0
0
1
0
0
1
0
1
0
0
0
1
1
1
0
0
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.
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0 0
NOR
Flip-Flop
1
0
A B
NOR
1
1
0
0
1
0
1
0
0
0
0
1
0
NOR
0 1
0
Promoter
Operators O1, O2
Gene
downstream
Making NOR gates is ‘relatively’ easy and requires only two operator sites
downstream of the RNA polymerase binding site (promoter).
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0 0
NOR
Flip-Flop
1
0
A B
NOR
1
1
0
0
1
0
1
0
0
0
0
1
0
NOR
0 1
0
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0 0
NOR
Flip-Flop
1
0
A B
NOR
1
1
0
0
1
0
1
0
0
0
0
1
0
NOR
0 1
0 0
NOR
0
1
1
0
NOR
0 1
0
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0 0
NOR
Flip-Flop
1
0
A B
NOR
1
1
0
0
1
0
1
0
0
0
0
1
0
NOR
0 1
1 0
NOR
0
1
1
0
NOR
0 1
0
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0 0
NOR
Flip-Flop
1
0
A B
NOR
1
1
0
0
1
0
1
0
0
0
0
1
0
NOR
0 1
1 0
NOR
0
0
1
0
NOR
0 1
0
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0 0
NOR
Flip-Flop
1
0
A B
NOR
1
1
0
0
1
0
1
0
0
0
0
1
0
NOR
0 1
1 0
NOR
0
0
1
0
NOR
0 0
0
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0 0
NOR
Flip-Flop
1
0
A B
NOR
1
1
0
0
1
0
1
0
0
0
0
1
0
NOR
0 1
1 0
NOR
0
0
1
0
NOR
0 0
1
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0 0
NOR
Flip-Flop
1
0
A B
NOR
1
1
0
0
1
0
1
0
0
0
0
1
0
NOR
0 1
1 1
NOR
0
0
1
0
NOR
0 0
1
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0 0
NOR
Flip-Flop
A B
1
0
NOR
1
1
0
0
1
0
1
0
0
0
0
1
0
NOR
0 1
1 1
NOR
0
0 1
NOR
0
1
0
0
0
NOR
0 0
1
NOR
0 0
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0
1
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0 0
NOR
Flip-Flop
A B
1
0
0
NOR
0 1
0 0
NOR
1
1
0
0
1
0
1
0
0
0
0
1
Toggle A to reset P1
Toggle B to set P1
0
0 0
NOR
1
0
0
1
0
NOR
1 1
NOR
0
NOR
0 1
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1
0
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XOR Gate
OR
AND
NAND
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XOR Gate
Operators O1, O2
Promoter
0,1 = 0
Gene
0,1 = 0
0,0 = 1
1,1 = 1
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Half Adder
A
B
S
C
1
1
0
1
0
1
1
0
1
0
1
0
0
0
0
0
0+0=0
0+1=1
1+0=1
1 + 1 = 0 carry 1
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Half Adder
A
B
S
C
1
1
0
1
0
1
1
0
1
0
1
0
0
0
0
0
0+0=0
0+1=1
1+0=1
1 + 1 = 0 carry 1
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Feed-forward Circuits
Activate
Repress
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Feed-forward Circuits
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Feed-forward Circuits
C1
I1
Relative abundance of different FFL types in Yeast and E. coli. Data taken from
Mangan et al. 2003.
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Feed-forward Circuits
Coherent Type I Genetic Network
C1
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Feed-forward Circuits
Coherent Type I Genetic Network
Noise Rejection
Circuit
Narrow Pulse
C1
Wide Pulse
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Feed-forward Circuits
Coherent Type I Genetic Network
p = defn cell
$G1 -> S1; Vmax1*X^4/(Km1 + X^4);
S1 -> $w; k1*S1;
$G2 -> S2; Vmax2*S1^4/(Km1 + S1^4);
S2 -> $w; k1*S2;
$G3 -> S3; Vmax3*S1^4*S2^4/(Km1 + S1^4*S2^4);
S3 -> $w; k1*S3;
end;
C1
p.Vmax1 = 1;
p.Vmax2 = 1;
p.Vmax3 = 1;
p.Km1 = 0.5;
p.X = 0;
p.k1 = 0.1;
p.S1 = 0;
p.S2 = 0;
p.S3 = 0;
p.ss.eval;
println p.sv;
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Feed-forward Circuits
Coherent Type I Genetic Network
// Pulse width
// Set to 1 for no effect
// Set to 2.4 for full effect
h = 2.4;
C1
p.X = 0.3;
m1 = p.sim.eval (0, 10, 100, [<p.Time>, <p.X>, <p.S3>]);
p.X = 0.7; // Input stimulus
m2 = p.sim.eval (10, 10 + h, 100, [<p.Time>, <p.X>, <p.S3>]);
p.X = 0.3;
m3 = p.sim.eval (10 + h, 40, 100, [<p.Time>, <p.X>, <p.S3>]);
m = augr (m1, m2);
m = augr (m, m3);
graph (m);
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Feed-forward Circuits
Incoherent Type I Genetic Network
I1
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Synthetic Incoherent Type I Genetic Network
I1
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Incoherent Type I Genetic Network
Pulse Generator
I1
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Incoherent Type I Genetic Network
Pulse Generator
I1
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Incoherent Type I Genetic Network
Pulse Generator
p = defn cell
$G1 -> P2; t1*a1*P1/(1 + A1*P1);
P2 -> $w; gamma_1*P2;
$G3 -> P3; t2*b1*P1/(1 + b1*P1 + b2*P2 + b3*P1*P2^8);
P3 -> $w; gamma_2*P3;
end;
p.P2
p.P3
p.P1
p.G3
p.G1
I1
=
=
=
=
=
0;
0;
0.01;
0;
0;
p.t1 = 5;
p.a1 = 0.1;
p.t2 = 1;
p.b1 = 1;
p.b2 = 0.1;
p.b3 = 10;
p.gamma_1 = 0.1;
p.gamma_2 = 0.1;
// Time course response for a step pulse
p.P1
m1 =
p.P1
m2 =
= 0.0;
p.sim.eval (0, 10, 100, [<p.Time>, <p.P1>, <p.P3/1>]);
= 0.4; // Input stimulus
p.sim.eval (10, 50, 200, [<p.Time>, <p.P1>, <p.P3/1>]);
m = augr (m1, m2);
graph (m);
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Incoherent Type I genetic Network
Concentration Detector
I1
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Incoherent Type I genetic Network
Concentration Detector
I1
// Steady state response
n = 200;
m = matrix (n, 2);
for i = 1 to n do
begin
m[i,1] = p.P1;
m[i,2] = p.P3;
p.ss.eval;
p.P1 = p.P1 + 0.005;
end;
graph (m);
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