Feed-forward Networks
Feed-forward Circuits
Activate
Repress
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Feed-forward Circuits
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Feed-forward Circuits
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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
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AND GATE
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Feed-forward Circuits
Coherent Type I Genetic Network
Noise Rejection
Circuit
Narrow Pulse
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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;
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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
Coherent Type I Genetic Network
Original
C1
AND GATE
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Feed-forward Circuits
Coherent Type I Genetic Network
No Delay
P1
P1
P3
Time
P3
Delay
Time
P1 in this model is a fixed species which is controlled
by the modeler. In the previous example, P1 was synthesized
from a gene and degraded and control was by a separate input
that modified the expression rate of P1.
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Feed-forward Circuits
Coherent Type I Genetic Network
p = defn cell
$G2 -> P2; Vmax2*P1^4/(Km1 + P1^4);
P2 -> $w; k1*P2;
$G3 -> P3; Vmax3*P1^4*P2^4/(Km1 + P1^4*P2^4);
P3 -> $w; k1*P3;
end;
p.Vmax2 = 1;
p.Vmax3 = 1;
p.Km1 = 0.5;
p.k1 = 0.1;
p.P1 = 0;
p.P2 = 0;
p.P3 = 0;
p.ss.eval;
println p.sv;
// Pulse width
// Set to 1 for no effect
// Set to 4 for full effect
h = 1;
p.P1
m1 =
p.P1
m2 =
p.P1
m3 =
= 0.3;
p.sim.eval (0, 10, 100, [<p.Time>, <p.P1>, <p.P3>]);
= 0.7; // Input stimulus
p.sim.eval (10, 10 + h, 100, [<p.Time>, <p.P1>, <p.P3>]);
= 0.3;
p.sim.eval (10 + h, 40, 100, [<p.Time>, <p.P1>, <p.P3>]);
m = augr (m1, m2);
m = augr (m, m3);
graph (m);
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Feed-forward Circuits
Coherent Type I Genetic Network
Question: What behavior would
you expect if the feed-forward
network is governed by an OR
gate?
OR GATE
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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
P3
I
P3 comes down even though P1 is still high !
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Incoherent Type I Genetic Network
Pulse Generator
P1, P3
P3
P1
Pulses are not
symmetric because
the rise and fall
times are not the
same.
TIME
P3 comes down even though P1 is still high !
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Incoherent Type I Genetic Network
Digital Pulse Generator
Pulses are symmetric
because the rise and
fall times are the same.
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Incoherent Type I Genetic Network
Pulse Generator
One potential
problem, if the base
line for P3 is not at
zero, the off transition
will result in an
inverted pulse. Avoid
this by arranging the
base line of P3 to be at
zero.
TIME
Inverted Pulse
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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
Steady State Concentration Detector
I1
Circuit is off at low concentration, off at high concentrations
but comes on intermediate concentrations. Width of the peak
can be controlled by the cooperativity transcription binding.
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Incoherent Type I genetic Network
Concentration Detector
Take the pulse generator model and use this
code to control it:
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// 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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Incoherent Type I genetic Network
Response Accelerator
Making this stronger
makes the initial rise
go faster.
Then, bring the
overshot down to the
desired steady state
with the repression
feed-forward.
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An Introduction to Systems
Biology: Design Principles of
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Biological Circuits.
Autoreguation – Negative Feedback
Response Accelerator
Strong Feedback
+ strong input
promoter
P
Weak Feedback
Input, I
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Summary
1. Persistence detector. Does not
respond to transient signals.
1. Pulse generator
2. Concentration detector.
2. Rise time of P3 is delayed but P3
fall time is not.
3. Response time accelerator.
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Sequence Control – Temporal Programs
Single-input Module (SIM)
The different thresholds can be
achieved by assigning different Kms
to each expression rate law.
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An Introduction to Systems
Biology: Design Principles of
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Biological Circuits.
Sequence Control – Temporal Programs
Parallel Concentration Detecting Feed-Forward Networks
The kinetics can be arranged so that
each successive feed-forward loop
peaks at a later time.
……
P3 rises first, followed by P5.
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Temporal Order Control of
Bacterial Flagellar Assembly
Driven by a proton gradient.
Runs at approximately
6,000 to 17,000 rpm. With the
filament attaching rotation is
slower at 200 to 1000 rpm
Can rotate in both directions.
Approximately 50 genes
involved in assembly of the
motor and control circuits.
http://www.youtube.com/watch?v=0N09BIEzDlI
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Temporal Order Control of Flagellar
Assembly
An Introduction to Systems
Biology: Design Principles of
Biological Circuits.
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Temporal Order Control of Flagellar
Assembly
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Temporal Order Control of Metabolic
Pathways - Arginine
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Temporal Order Control of
Metabolic Pathways
Arginine
Early
Late
Red means more expression
of that particular gene.
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Temporal Order Control of
Metabolic Pathways
Methionine
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Temporal Order Control of
Metabolic Pathways
Methionine
Increasing a pathway’s capacity by
sequential ordering of expression is
probably only employed when the
pathway is empty.
For pathways already in operation, eg
pathways like glycolysis, increasing the
capacity is achieved by simultaneous
increases. This is done to avoid wild
swings in existing metabolite pools.
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Amplifiers
Output, P
Input, I
Amplifiers
Amplifiers
The Effect of Negative Feedback
No Feedback
Output, P
Input, I
Amplifiers
The Effect of Negative Feedback
Negative Feedback
stretches the response
and reduces the gain,
but what else?
No Feedback
Output, P
Input, I
With Feedback
Output, P
Input, I
Simple Analysis of Feedback
yi
A
k
yo
Simple Analysis of Feedback
yi
A
k
Solve for yo:
yo
Simple Analysis of Feedback
yi
A
k
Solve for yo:
yo
Simple Analysis of Feedback
At high amplifier gain (A k > 1):
In other words, the output is completely independent
of the amplifier and is linearly dependent on the
feedback.
Simple Analysis of Feedback
Basic properties of a feedback amplifier:
1. Robust to variation in amplifier characteristics.
2. Linearization of the amplifier response.
3. Reduced gain
The addition of negative feedback to a gene
circuit will reduce the level of noise (intrinsic
noise) that originates from the gene circuit itself.