Design Principles for Constructing Gene Regulatory Networks with
Specific Properties : A Modular Approach
www.sys-bio.org
Ravishankar R. Vallabhajosyula (rrao@kgi.edu), Vijay Chickarmane (vchickar@kgi.edu) and Herbert M. Sauro (hsauro@kgi.edu)
Keck Graduate Institute, 535 Watson Drive, Claremont, CA, 91711.
Modeling Gene Regulatory Networks as Interacting Modules
Motivation
An interesting question in Systems Biology is whether a Gene
Regulatory Network be broken down into interacting, functional
modules. We seek to answer this question by taking inspiration
from Electrical Engineering where complex networks are
designed to perform specific tasks. These complex networks
are made up of smaller units, such as filters, mixers, amplifiers
etc. Are there such analogous units in Gene Regulatory
Networks? Our goal is to draw up such a list, design networks
which perform specific tasks and then finally compare them
with real networks to study their similarity or differences. This
then will allow us to probe the design principles of Gene
Regulatory Networks.
Complex Networks: Using the above modules we designed
two networks which are meant to perform more complex tasks.
These networks are :
Results
Conclusions
We have designed three modules. Each of these modules
consists of genes which produce transcription factors. The
genes themselves are regulated by some of these transcription
factors. It is the nature of the regulation, and the network
connectivity which gives rise to interesting dynamics.
We have presented a modular approach to designing gene
regulatory networks. The basic modules consisted of band
pass filters and phase invertors. using these functional units
we designed more complex circuits such as a network which
exhibits two peaks in its frequency response, and an oscillatory
circuit that can adaptively tune its frequency to an external
oscillating signal.
The following two modules are defined in terms of the frequency
response between the input transcription factor and the output
protein concentration.
J0
S
b
Band-Pass Filter
In the network to the left, a transcrption
factor S activates a gene which
produces protein b. The protein b is
subject to decay. This simple network
has a low pass frequency response
between the input S and output b.
J1
w1
Frequency Tracking Filter: The network consists of a
control system for a biological oscillator whose frequency
is made to track the frequency of an external
oscillatory signal.
Low pass filter (LPF):
A gene which is regulated by a transcription factor and
produces protein x.
Band pass filter (BPF):
Two genes, each of which is regulated by a common
transcription factor, is based on the use of a Feed Forward
Loop [1].
In real biological networks the stochasticity due to chemical
reactions is often very important in determining the phenotype.
We plan to address the issue of how noise affects the functioning
of these networks.
Phase Inverter:
We consider two proteins, x, y which can form a complex.
The complex then dissociates, such that y gets degraded,
however x is conserved. Hence if the gene which produces
x is regulated by a transcription factor which is increased
in time, the immediate effect on y, is to reduce it, since
it gets degraded by x. This leads to a phase inversion
between x and y.
Low-Pass Filter
Double Peak Response: This network has a frequency
response which has two peaks at different frequencies.
Given the rapid advance of synthetic biology [2], these design
principles we hope will be useful in designing novel experiments.
These experiments will in turn further our understanding of
the working of the basic fundamental blocks of gene regulatory
networks.
Of the two transcription factors regulating the second
gene, one is an activator, and the other an inhibitor. (By
reducing the delay in the above circuit, it is possible to
increase the dynamic range of the transfer function, and
thereby obtain a high pass filter.)
Simple modules that can be built using biochemical networks
The corner frequency of a low-pass
filter is the value of the frequency
where the amplitude response begins
to be attenuated at higher frequencies.
This is given by the value of the
degradation parameter.
Reactions
System Equation
J0
S
b
N0
Steady State Value
We consider a two gene interaction network
using a feed forward loop [1]. We assume
that the transcrption factor, S activates a
second gene, producing b which inhibits
expression of the second gene (leading to
protein x).
J1
w1
x
J2
Phase Inverter Module
S however also activates the second gene.
This combination of activation-inhibition
by the same transcription factor, leads to
a band pass filter characteristic, between
the input S and output x. A band pass
filter has the unique property of being able
to eliminate frequencies above and below
a specified range.
w2
J3
Steady State Values
System Equations
Reactions
N1
k0
x
k1
w1
w2
k4
N2
y
k2
w3
k3
Reactions
System Equations
Steady State Values
An example of phase shifted variables
x and y from the phase inverter module
Amplitude
Amplitude
x
References
1. Mangan, S. and Alon, U. (2003) Structure and function of
the feed-forward loop network motif. PNAS, 100, No.21,
11980-11985.
2. Weiss, R. Basu, S., Hooshangi, S., Kalmbach, A., Karig, D.,
Mehreja, R. and Netravali, I. (2003) Genetic circuit building
blocks for cellular computation, communications and signal
processing. Natural Computing, 2, 47-84.
Phase
Phase
y
Acknowledgements
Time
We would like to thank DOE GTL program and DARPA Biocomp for support.
EXAMPLE 1 : Building a Network with maximal responses at two different frequencies
Low Frequency
Module 1
Introduction
Characteristics of the Frequency Response
Amplitude in a linear scale
LF2
LF1
The figure on the left shows how the double peak frequency filter
network has been constructed. The input to the network is through
the boundary node S, which drives the two band-pass modules
LF1 and HF1. Output from these first stage modules then drives
the second stage modules LF2 and HF2.
Low frequency Path
b0
Our approach comprises of the following steps.
b2
w
w
The final readout is a gene which has two activations set up from
each module. This has the effect of summing the contributions
from the low and high frequency paths.
N
1. By convolving two band pass filters we can generate a response
which is narrow. This effect can be seen in the adjacent figure,
in the frequency responses of x and x2.
x0
N
2. We then design another similar network which exhibits a narrow
response, but which peaks at a higher frequency.
w
N
x2
The figures on the right display the frequency response of the
readout with respect to input S. As can be seen, the frequency
response has two peaks, one at 0.01 corresponding to the low
frequency path, and the other at 100 corresponding to the high
frequency path.
w
Out
S
Input
3. The response from the two filters are “added” by allowing the
end protein products to be transcription factors for a readout
gene. The result is two narrow peaks in the frequency domain.
(between readout gene and input)
Low Frequency
Module 2
Double peak frequency filter
Biological systems are often subject to external stimuli that in
some cases is a periodic function of time. In this example we
consider a biological network that can sense changes in the external
stimulus at multiple frequency bands.
It can be seen that using this method, the final frequency response
of the readout gene with respect to the input can be made as
narrow as desired by adding more filter stages. We have used
only two such stages to demonstrate the feasibility of this approach.
The phase inverter module is constructed by
using two species (gene products), one of which
is regulated by an external signal. In this
example, N1 is assumed to be a periodic input.
x and y form a complex, which when it
dissociates conserves x, whereas y is degraded.
Hence the net effect of the reaction on the
concentration of y is to pull it down when x
goes up, and to raise it when x goes down. This
leads to a phase inversion between x and y.
b1
b3
w
Amplitude in logarithmic scale and Phase
Amplitude
w
w
A narrow frequency response can be obtained by convolving
different filters. Here x and x2 are two band-pass filter modules.
The output from the first module is used as input for the second
module, leading to the high attenuation of x2. This is depicted
below.
Input
First
Band Pass
x
Second
Band Pass
x1
N
w
N
x3
w
Phase
High frequency Path
Output
HF2
HF1
High Frequency
Module 1
x2
High Frequency
Module 2
EXAMPLE 2 : An Artificially constructed Gene Network for tracking frequency changes in an external source
Time Evolution of the Biological oscillator (blue)
compared to the External signal (red).
Gene Network Model of the Frequency tracking circuit
Introduction
We construct a network that tracks an external signal varyng in time.
Such a circuit is widely used, for example, in electronic systems.
Specifically we consider a biological oscillator, which is made to track
an external signal which also oscillates, but at a frequency which varies
in time. We therefore require the frequency of the biological oscillator
to “track” the frequency of the external signal. The motivation for this
example could be circadian rhythms which adjust their frequencies
according to the external conditions.
The input external signal is assumed to be some form of ligand that can
diffuse into the cell and activate a transcription factor.
External
Oscillator
M
In the network figure displayed on the left, the third and fourth
complexes contain the information about the frequency difference
between the reference oscillatory frequency and the biological oscillator
frequency. This difference is then extracted as the readout of a gene
which uses one of the complexes as an activator and the other as an
inhibitor. This is then used as a feedback to alter the biological oscillator’s
frequency such that it tracks the frequency of the external signal.
M
1
First
LP
1
Int
Complex
3
Third
Complex
1
Internal
Oscillator
The panel of figures on the right shows the time series of one of the
proteins of the biological oscillator whose frequency can be seen to
slowly vary, in effect tracking the external signal frequency.
Fourth
Second
LP
Complex
Phase
Inverter
Module
M
2
Sub
Complex
Int
2
M
4
t = 200 to 400
This is also seen below where we superimpose the plot of the control
signal onto the biological oscillator, and the parameter which determines
the variation of the external signal frequency. After an initial transient
the control signal begins to follow the external oscillator frequency.
2
Similar to electronic circuits, we have “mixers”-represented as protein
complexes, band pass filters and phase inverters all of which have been
described earlier.
t = 400 to 600
Tracking Performance over time
A Schematic representation of the tracking circuit
0.18
0.16
LP
M
1
Int
1
0.14
M
1
LP
Phase
Inverter
2
Int
M
3
Proportional to Frequency
The basic idea is that using these modules the cellular network compares
its frequency with that of the external signal frequency and generates
an appropriate feedback which can correct for the difference.
t = 0 to 200
4
2
t = 600 to 800
External Signal
Biological Oscillator
0.12
0.1
0.08
t = 800 to 1000
0.06
Sub
0.04
M
2
Feedback
Mixer
0.02
Subtractor
0
200
400
600
800
1000
Time
1200
1400
1600
1800