Feed-forward Networks
Systems and Synthetic Biology: 498A
Copyright © 2008: Sauro
Feed-forward Networks
1. Estimating the frequency of each isomorphic
subgraph in the target network.
2. Generating a suitable random graph to
test the significance of the frequency data.
3. Compare the target network with the
random graph.
Occurrences of the feed-forward loop
motifs as generated by the software
MAVisto [1]. The displayed network is part
of yeast data supplied with the MAVisto
software. The software is very straight
forward to use and will identify a wide
variety of motifs. Other similar tools include
FANMOD and the original tool mFinder.
F. Schreiber and H. Schwobbermeyer. MAVisto: a tool for
the exploration of network motifs. Bioinformatics,
21(17):3572–3574, 2005.
Copyright © 2008: Sauro
Feed-forward Circuits
The sign of an
interaction
can be determined
either from basic
biochemistry studies
or by looking at
microarray
expression profiles.
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
Dynamic Properties
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First Translate Non-stoichiometric
Network into a Stoichiometric Network
C1
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First Translate Non-stoichiometric
Network into a Stoichiometric Network
C1
?
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Feed-forward Circuits
Dynamic Properties
What does this actually mean?
AND GATE?
Input A
Input B
AND
OR
XOR
1
1
1
1
0
1
0
0
1
1
0
1
0
1
1
0
0
0
0
0
OR GATE?
Or something else?
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Feed-forward Circuits
Coherent Type I Genetic Network
Noise Rejection
Circuit
Narrow Pulse
C1
Wide Pulse
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An Introduction to the Kinetics of
Gene Expression
.…to a different slides show
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Feed-forward Circuits
Coherent Type I Genetic Network: AND Gate
C1
AND GATE
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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
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. NOTE THE DELAY.
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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
(Control, Simple Control System)
Delay
cAMP is made when
glucose runs out
No Delay
Nature Reviews Genetics 8, 450-461 (June 2007) | doi:10.1038/nrg2102
Network motifs: theory and experimental approaches Uri Alon
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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
Coherent Type I Genetic Network
Question: What behavior would
you expect if the feed-forward
network is governed by an OR
gate?
1. No delay on activation.
2. Delay on deactivation.
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:
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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Incoherent Type I genetic Network
Response Accelerator
Making this stronger
makes the initial rise
go faster.
Then, bring the
overshoot down to
the desired steady
state with the
repression feedforward.
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An Introduction to Systems
Biology: Design Principles of
29
Biological Circuits.
Summary
C1
1. Persistence detector. Does not
respond to transient signals.
I1
1. Pulse generator
2. Concentration detector.
AND: Delay on start, no delay on
deactivate.
3. Response time accelerator.
OR: No delay on start, delay on
deactivate.
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Other Motifs
1. Single-input Module (SIM)
2. Auto-regulation
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Sequence Control – Temporal Programs
Single-input Module (SIM)
E1
E2
E3
Input: X
The simplest approach is to have
different thresholds can be achieved
by assigning a different K and Vmax
to each expression rate law, easily
generated through evolutionary
selection.
An Introduction to Systems
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Biology: Design Principles of
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Biological Circuits.
Sequence Control – Temporal Programs
More Complex Arrangement
Parallel Concentration Detecting Feed-Forward Networks –
Generating Pulse Trains
The kinetics can be arranged so that
each successive feed-forward loop
peaks at a later time.
……
P3 rises first, followed by P5.
This allows pulse trains to be
generated.
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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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Auto-regulation – Negative Feedback
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Negative Feedback - Homeostasis
V1
V1, V2
P
Negative Feedback - Homeostasis
V1
V2
V1, V2
Steady State!
P
Negative Feedback - Homeostasis
V2
V1, V2
V2
V1
P
P is very sensitive to changes in V2 (k2)
Negative Feedback - Homeostasis
V2
V1
V1, V2
V2
P
P is less sensitive to changes in V2 (k2)
Negative Feedback - Homeostasis
V2 = 0.3
V1
V2 = 0.2
V1, V2
V2 = 0.1
S1
P is much less sensitive to changes in V2 (k2)
Auto-regulation – Negative
Feedback Response Accelerator
Strong Feedback
+ strong input
promoter
P
Weak Feedback
Input, I
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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.