Computational design of synthetic regulatory networks from a genetic library to characterize the designability of dynamical behaviors

Institute of Systems and Synthetic Biology (ISSB), Genopole - Université d'Évry Val d'Essonne - CNRS UPS3201, 91030 Évry Cedex, France.
Nucleic Acids Research (Impact Factor: 9.11). 08/2011; 39(20):e138. DOI: 10.1093/nar/gkr616
Source: PubMed
The engineering of synthetic gene networks has mostly relied on the assembly of few characterized regulatory elements using
rational design principles. It is of outmost importance to analyze the scalability and limits of such a design workflow. To
analyze the design capabilities of libraries of regulatory elements, we have developed the first automated design approach
that combines such elements to search the genotype space associated to a given phenotypic behavior. Herein, we calculated
the designability of dynamical functions obtained from circuits assembled with a given genetic library. By designing circuits
working as amplitude filters, pulse counters and oscillators, we could infer new mechanisms for such behaviors. We also highlighted
the hierarchical design and the optimization of the interface between devices. We dissected the functional diversity of a
constrained library and we found that even such libraries can provide a rich variety of behaviors. We also found that intrinsic
noise slightly reduces the designability of digital circuits, but it increases the designability of oscillators. Finally,
we analyzed the robust design as a strategy to counteract the evolvability and noise in gene expression of the engineered
circuits within a cellular background, obtaining mechanisms for robustness through non-linear negative feedback loops.


Available from: Javier Carrera
Computational design of synthetic regulatory
networks from a genetic library to characterize
the designability of dynamical behaviors
Guillermo Rodrigo
, Javier Carrera
and Alfonso Jaramillo
Institute of Systems and Synthetic Biology (ISSB), Genopole - Universite
vry Val d’Essonne - CNRS
UPS3201, 91030 E
vry Cedex, France,
Instituto de Biologı
a Molecular y Celular de Plantas, CSIC Universidad
cnica de Valencia, 46022 Valencia, Spain and
Instituto ITACA, Universidad Polite
cnica de Valencia,
46022 Valencia, Spain
Received March 25, 2011; Revised May 31, 2011; Accepted July 14, 2011
The engineering of synthetic gene networks has
mostly relied on the assembly of few characterized
regulatory elements using rational design principles.
It is of outmost importance to analyze the scalability
and limits of such a design workflow. To analyze the
design capabilities of libraries of regulatory
elements, we have developed the first automated
design approach that combines such elements to
search the genotype space associated to a given
phenotypic behavior. Herein, we calculated the
designability of dynamical functions obtained from
circuits assembled with a given genetic library.
By designing circuits working as amplitude filters,
pulse counters and oscillators, we could infer new
mechanisms for such behaviors. We also high-
lighted the hierarchical design and the optimization
of the interface between devices. We dissected the
functional diversity of a constrained library and we
found that even such libraries can provide a rich
variety of behaviors. We also found that intrinsic
noise slightly reduces the designability of digital
circuits, but it increases the designability of oscilla-
tors. Finally, we analyzed the robust design as
a strategy to counteract the evolvability and noise
in gene expression of the engineered circuits within
a cellular background, obtaining mechanisms for ro-
bustness through non-linear negative feedback
Over the past decade, we have witnessed the expansion of
synthetic biology (1), where the attempts for cell
reprogramming to perform new tasks have fructified in
the engineering of several synthetic regulatory circuits
(2–20). Usually, the design of synthetic circuits has been
inspired on the use of mathematical models (21,22) and
empirical engineering rules inferred from natural examples
(23,24), although requiring in many cases a genetic
fine-tuning to achieve the desired behavior (25). It is
expected that the widespread use of libraries of previously
well-characterized genetic regulatory elements (26–29),
together with the ability of engineering combinatorially
those elements (30), will allow avoiding trial-and-error
procedures, which are not efficient for optimizing and
implementing complex systems. Those designed circuits
may be later fine-tuned with directed evolution techniques,
although there is no general methodology for the de novo
network engineering. In fact, this bottom-up approach is
commonly used in other areas of engineering where a set
of off-the-shelf parts with precise specifications of their
operating points can be used to engineer sophisticated
systems, and has been already successful to engineer
novel biological circuits (12,19).
Large efforts in generating not only genetic diversity,
especially libraries of promoters (19,31–36), but also
post-transcriptional regulatory elements (6,14,37–39) and
synthetic transcription factors (40,41), encourage to use a
combinatorial approach to design artificial circuits (see
Supplementary Data for further details). In addition, the
quantitative characterization of these regulatory elements
allows inferring simple phenomenological mathematical
models, which could be used to construct the model of a
system that assembles different elements. In that way,
several synthetic biology-oriented design tools have been
developed to make available a library of mathematical
models created from that genetic diversity, together
with an interface to create gene circuits by wiring
elements (42–47). Notably, such a genetic diversity is
translated into a functional diversity when assembling
circuits, and these circuits could be readily compiled into
nucleic acid sequences. However, the design is reduced to
*To whom correspondence should be addressed. Tel: +33 169474444; Fax: +33 169474437; Email:
Published online 24 August 2011 Nucleic Acids Research, 2011, Vol. 39, No. 20 e138
ß The Author(s) 2011. Published by Oxford University Press.
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Page 1
examine one-by-one all possible combinations (e.g.
simulating the dynamical behavior), resulting in a
tedious design process. Thereby, the evolutionary algo-
rithms and optimization techniques (48–52) allow us to
automate this process to find the desired circuits and
finally depict the functional diversity of a library of
regulatory elements. Our novel approach allows
assembling models of regulatory elements from a library
and couples this with an automated design strategy.
In this work, we tackle for the first time fundamental
questions that naturally emerge from that approach. What
functional circuits can we engineer with a given library of
regulatory elements? What is the diversity of possible
behaviors and what is the designability (defined as the
fraction of assembled circuits that follow a given
behavior) of each one? Is one behavior easier to design
than others? Certainly, these features depend on the
employed library. We also wonder what is the sensitivity
of the results to the regulatory elements; in other words,
how many functional circuits involve a given regulatory
element? In addition, we look at the robustness of a circuit
by locally perturbing its parameters and evaluating the
resulting fitness. At fixed network topology, we further
analyze the whole parameter space that provides the
targeted functionality, which accounts for the robustness
of all operative points and asymptotically tends to a value
that we call asymptotic robustness. Indeed, this property
accounts for the ability to design such a circuit given the
limitation of the number of genetic elements, and it could
be important to analyze the natural occurrence of certain
genetic architectures. All in all, to solve these questions,
we developed a computational framework to assemble,
simulate and design circuits, and that allowed us to
explore the functional diversity that came from the
assembled circuits with certain behavior (Figure 1).
The design of circuits was accomplished by a selection
step according to a dynamical behavior-based fitness
function that can also account for robustness. Initially,
we applied the methodology to design several functional
circuits with unlimited genetic diversity (given by the par-
ameter space) and study their asymptotic robustness.
Then, we designed complex circuits by plugging functional
modules. Subsequently, we dissected the whole dynamical
spectrum of a limited library of regulatory elements and
analyzed the properties of the resulting circuits. We also
analyzed the dependence of these results on the constitu-
ent library and how they could change when the
stochasticity of the cell is taken into account. Later, we
showed the application of our methodology to design a
robust circuit. Finally, we discussed the reliability and
implementability of the designed circuits.
Mathematical model for gene regulatory circuits
For modeling genetic circuits, we used a coupled system of
differential equations. We considered three different types
of species: mRNA (it can also be non-coding), proteins
(mainly transcription factors) and small molecules that
interact with proteins to activate or inhibit their regulatory
ability (e.g. isopropyl b-
IPTG—inhibits the activity of LacI). Likewise, the
production of the i
mRNA (x
) from a regulated
promoter follows
¼ Cfðy
, ð1Þ
where the term f(y
) is the transcription rate (y
and u
represent the concentrations of the j-th protein and its
regulating chemical, respectively), C the gene copy
number, the mRNA degradation coefficient and m the
growth rate of the cell (dilution term). C = 1 is assumed to
be constant in this work. For the computational design of
a circuit, we did not impose variability on m but
we assumed a constant value (e.g. m = 0.02 min
). For
simplicity, we assumed that all genes in an operon (i.e.
controlled by the same promoter) have the same mRNA
expression. The term f(y
) accounts for protein–DNA
and protein–molecule interactions (22), and for constitu-
tive promoters it is constant (see Supplementary Data for
further details). Importantly, our approach is independent
of the choice of this function, thus giving a big degree of
freedom to the kinetic characterization from experimental
data. Afterwards, the production of i-th protein (y
given by
¼ gðx
, ð2Þ
where the term g(x
) is the translation rate and the
protein degradation coefficient. The term g(x
accounts for post-transcriptional regulatory mechanisms,
such as riboswitches, allowing a further genetic element,
such as a trans-RNA, to control translation (6). In case
of no post-transcriptional elements, the translation
Figure 1. Scheme of the design platform adopted by harnessing a
library of models of composable regulatory elements. We explore the
functional networks that can be engineered either by exhaustive
combinatorial assembly or by heuristic optimization.
e138 Nucleic Acids Research, 2011, Vol. 39, No. 20 PAGE 2 OF 12
Page 2
rate is proportional to the mRNA concentration. In
addition, in this work we only considered first-order
degradation kinetics. The description of the construction
of the stochastic model is detailed in the Supplementary
Library of models of regulatory elements
To construct a library, each genetic regulatory element
was modeled by transfer functions that related the
output to the input values. These functions can be fitted
from experimental data. As DNA fragments, mathemat-
ical models can be assembled in a standard way to
simulate the behavior of circuits. Here, we only allowed
joining promoter and genes, or genes and genes (i.e. two
consecutive promoters was not allowed; such a construc-
tion should be specified as a whole part). One useful
format to store a mathematical model (molecular
species, kinetic parameters and DNA sequence) is
systems biology markup language (SBML) (53). Hence,
we had a single SBML file for the model of each biological
part; similarly as crystallographic data is stored in protein
data bank (PDB) format. The models for promoter parts
only account for the transcription rate, whereas for gene
parts the model accounts for the translation rate and the
degradation and dilution rates of mRNA and proteins.
We selected Hill-function models because their over-
whelming use in current characterization of transcription
regulation works. In the future, when more advanced
models may be used to fit characterization data,
they could be readily used with our computational
design procedure. A range of variation can be specified
for some kinetic parameters; likewise, the corresponding
value will be susceptible to be changed during the design
Exhaustive versus heuristic design
Multiple circuits can be constructed by harnessing the
available regulatory elements. To computationally
explore the functional diversity that offers such a library
and the designability of certain behaviors, two different
strategies can be adopted, and our approach provides an
automated implementation of them. On the one hand,
following the exhaustive design strategy, all possible
circuits, up to a maximal number of elements, are
constructed and simulated. Having the large collection
of dynamics, a post-processing step is applied to find
those circuits that behave according to the design specifi-
cations. This approach allows obtaining the whole
functional diversity and designability. On the other
hand, a heuristic design strategy provides a probabilistic
sampling frame of the functional diversity. It allows itera-
tively assembling models of existing elements and
evaluating the performance of the resulting circuit accord-
ing to a dynamical behavior-based fitness function (54).
For that, we used Monte Carlo Simulated Annealing
(MCSA) as optimization scheme (55). A movement in
the fitness landscape consists in a replacement, addition
or deletion of a given regulatory element. To evaluate the
fitness function, we first calculated the average distance
(metric function) for all genes i considered as outputs,
for a given target behavior k, between the current circuit
dynamics (y
) and the target one (z
) according to
ðtÞÞ logðz
, ð3Þ
where T is the final time (e.g. the time to reach the steady
state). The metric is in logarithmic scale to properly
balance the species concentrations, since they can vary
in several orders of magnitude in biological systems. The
function (t) is a weighting factor to only evaluate the
circuit dynamics in a specified temporal domain ((t):
[0,T] ! [0,1]). Subsequently, the fitness function that
aggregates all targets we used reads
, ð4Þ
is a normalization constant to adjust the fitness
value to the metric function (e.g.
= 3), and g
gives the
scalar weight in logarithmic scale for optimizing target k
indicates that target k has 10 times more
priority than l). If
for one k then we assumed
= 0. Importantly, this fitness function (c belongs to
the interval [0,1]) penalizes those circuits that do not
satisfy simultaneously all targets. Being D the fitness
update after a movement, this is accepted with probability
max{1, exp(D /T
)}, where T
is the MCSA
temperature. T
is continuously adjusted during the
optimization process following an exponential cooling
Circuit design and asymptotic robustness
Initially, we constructed a library of artificial regulatory
elements, including all types of logic combinatorial pro-
moters of two entries. Additionally, the kinetic parameters
characterizing those elements were specified as a range of
variation. This feature allows that the genetic sequence of
many biological parts could be easily modified to create
a new part with diminished binding affinity or stability by
a single mutation. Otherwise, it is much more difficult to
find a suitable mutation that would increase the binding
or stability. Therefore, by allowing this range in the
parameter space, we would enlarge the search space
while still maintaining the linking with the genotype,
because the parts from an optimal solution could be
readily engineered to follow a model agreeing with the
designed parameters. The nominal values were taken
from several experimental studies (11,12,27,36). Thus,
the genetic diversity was almost unlimited, being the
design space defined by topological and parameter modi-
fications of the circuit. To explore this space we applied
the heuristic design strategy to find the optimal assemblies
of elements and parameterizations that gave functional
circuits. First, we repeatedly applied the optimization
method to design all possible circuits relying on a
feed-forward loop (FFL) structure for one-stripe pattern
formation (56–59). We found six different architectures
AGE 3 OF 12 Nucleic Acids Research, 2011, Vol. 39, No. 20 e138
Page 3
for working as an amplitude filter (Figure 2), where five of
them corresponded to incoherent FFLs (all except 2A),
and in architectures 2B and 2E repression dominated
over activation. Moreover, in Supplementary Figure S1,
we show the circuits corresponding to inverse amplitude
filters. Certainly, the two regulatory branches with
opposite sign are responsible for such a behavior, and
the combinatorial promoter of the downstream
promoter is central to get a variety of functionally analo-
gous circuits. Interestingly, some of those architectures
have been found involved in developmental processes
(60,61). In Supplementary Figure S2, we illustrate a
possible implementation of a FFL-based circuit respond-
ing to intermediate concentrations of IPTG, together
with its characteristic transfer function. Notably, we
did not exhaustively construct all possible FFL circuits
from the library for their scoring. Instead, we prob-
abilistically sampled the fitness landscape and we
always found a solution corresponding to one of the
six FFL structures presented. Moreover, this approach
can be applied to design functional circuits without
accounting for the designability of the desired behavior,
and then study the intrinsic properties of the circuit
irrespective to the library, such as its asymptotic
We then investigated the asymptotic robustness of those
FFL circuits, which functioned as amplitude filters with
a fold-change (F) of at least one order of magnitude at the
detection point (F 10). By constraining the sign of the
regulations (fixed topology), we obtained a parameter
space of 210
different combinations for each
topology shown in Figure 2. Accordingly, the highest
asymptotic robustness was reached by the architecture
2E with the 21.29%, followed by the architectures 2A
with the 17.17% and 2F with the 17.66%, indicating
that those circuits have a one-stripe pattern-prone
structure. Interestingly, this could be because the input
gene (X) has a non-monochromatic regulatory mode
(i.e. both activator and repressor) in these topologies.
On the contrary, the architecture 2C was highly sensitive
to parameter variations with asymptotic robustness of
only the 2.54%. The architectures 2B and 2D with the
8.60% and 7.69%, respectively, were in between.
However, despite of its low asymptotic robustness, the
structural core 2C is broadly found in many natural
systems. For instance, in the Drosophila patterning circuit-
ry, gene hb represses both genes kni and Kr and kni
also represses Kr (60). In addition, the core 2B is the
FFL motif most abundant within the regulatory map of
bacteria and yeast (62). That the genes involved in the
structures 2B and 2C have a monochromatic regulatory
mode could explain the increasing presence of these
circuits. Moreover, from a synthetic perspective, pro-
moters type NOR and IMPLIES could be engineered by
placing contiguously the corresponding operators in the
promoter region (7,12,36).
Subsequently, we used the optimization method to
design a circuit able to count. This is an interesting
example that could already unveil many of the issues we
meet in more complex networks. Cells may take advantage
of this sort of circuits to regulate fundamental processes,
such us telomere length control (63), where a machinery to
count molecules or events is required. Counters have
different stable states and rely on memory-like architec-
tures that allow retaining the initial state, unless a perturb-
ation switches the system (3,14,18,64). Generally, the
underlying mechanism of biological counters consists
in overcome certain threshold after a specific number of
consecutive pulse-like events. Herein, we attempted the
design of a two-pulse counter, where we imposed that
the system had to reach three different states. We
applied the optimization method to design all possible
two-gene circuits. We found that, within a delimited
time domain, all possible circuits were functional and
reached three states. However, those circuits based on
an activator–repressor core had a meta-stable state,
which falls into the basin of attraction of one of the two
stable states after certain time (Supplementary Figure S3).
Figure 2. Schemes of several FFL-based gene circuits optimized to operate as amplitude filters (also called band detectors). The mathematical model
for each circuit is provided in SBML format in the Supplementary File
e138 Nucleic Acids Research, 2011, Vol. 39, No. 20 PAGE 4 OF 12
Page 4
In Figure 3, we show the phase diagrams for the circuits
based on a monochromatic regulatory core and one
self-activation (see in Supplementary Figure S4 the
corresponding phase diagrams for the circuits based on
an activator-repressor core). The addition of another
self-activation on the buffer gene allows having a symmet-
ric multi-stable device (64). We then computed the asymp-
totic robustness of these two circuits, by exploring
exhaustively all possible parameterizations (for that we
discretized the parameter space into 2 10
combinations). The double repression core allowed
tristability in the 0.2422% of the cases, whereas the
double activation core in the 0.1787% (relatively low in
both cases).
Next, we attempted the design of a two-pulse counter
relying on just two states. As design specifications, we
imposed pulses of 10 min within an interval of 50 min
with amplitude of 100-fold. The designed circuit with a
plausible implementation counting pulses of IPTG is
shown in Supplementary Figure S5. However, the func-
tioning of this system (i.e. number of pulses it is able
to count) depends on the pulse length and interval.
In addition, we attempted the automated design of a
tunable genetic timer. These devices consist of memories
that change the state of operation according to an external
signal and the time to accomplish this transition (time to
reach the steady state) can be modulated by another signal
(19). The designed circuit with a plausible implementation
is shown in Supplementary Figure S6, together with its
characteristic transfer function. This circuit consisted in
a coherent FFL coupled to a memory-like mechanism
based on a self-activation, and it existed a threshold
in the IPTG concentration from which the circuit
responded to different levels of it.
Hierarchical design and modularity
Once a functional genetic device is obtained, either from
computational or rational design methods, and experi-
mentally validated, it can be integrated in the library as
a new element to be used in the construction of more
complex systems. As a first approach, we included in
our library of regulatory elements a circuit previously
optimized to operate as a tristable. Remarkably, the
incorporation into the design procedure of black-box
modules enhances the optimization of the impedance
matching, where the output of a device serves directly as
the input of a downstream one, and could considerably
enlarge the functional diversity of the library. Hence,
following such a design approach, we were able to
obtain complex functions with modular systems. In our
particular case, we designed a system coupling a tristable,
an amplitude filter, and a frequency-tunable oscillator
(Figure 4). Initially, this tristable gave a low concentra-
tion. After a pulse of 20 min with amplitude of 1000-fold
in the inducer, the device switched its state to reach the
intermediate concentration level, and subsequently the
oscillator changed its frequency and the amplitude filter,
which operates as a detector of the intermediate state,
reached its ON state. After a second pulse, the device
switched to its high concentration point, inducing a new
change in the frequency of the oscillator and giving the
detector back to its OFF state. Interestingly, the
frequency-tunable oscillator evolved to couple two differ-
ent regulatory mechanisms, and the external signal
switched from one mechanism to another, then changing
the frequency of the oscillations. In addition, with the
consideration of delayed reactions (e.g. due to translation
and multimerization) we could obtain complex oscilla-
tions, which can drive to a route toward chaos (65).
In fact, this mechanism has been previously applied to
design genetic oscillators with a minimal number of
elements (16). In Supplementary Figure S7, we show
a genetic clock designed by optimization coupled to a
tristable, which modulates its frequency and the shape of
its oscillations according to the states of the switch-like
However, one important issue in such an approach is
the possibility of the loss of function of a device when
plugging it to a downstream module. This effect, usually
called retroactivity (66), emerges when a transcription
factor plays two different roles in both modules, and
is indeed a consequence of the limited protein amount.
This result may have significant consequences on the
dynamics of the system, even when the stochasticity of
the cell is taken into account (67). Here, our modeling
neglects this effect by assuming that the concentration of
free protein is always much higher than the protein bound
to DNA (22); also as an imposition to ensure modularity
in the design and to be able to combine different elements
from the library. Although for many systems this
approach is valid (9), it could be found some examples
where such a model is not too accurate. To solve this
problem in practice, one strategy would be to impose as
a design constraint that the output gene had no regulatory
effects on the circuit. Likewise, this output could be used
as the input in further downstream modules with increased
guarantees of a proper functioning. Thereby, in our
system of Figure 4, gene U could be split into two genes,
one for working within the tristable device and another
for setting the amplitude filter and the oscillator, although
still it would exist a coupling between these two devices
due to a common input.
Figure 3. Schemes of two two-gene circuits designed to reach
tristability, showing the corresponding phase diagrams. Filled and
open circles represent stable and unstable states respectively. The math-
ematical model for each circuit is provided in SBML format in the
Supplementary File
PAGE 5 OF 12 Nucleic Acids Research, 2011, Vol. 39, No. 20 e138
Page 5
Figure 4. (A) Scheme of a complex regulatory system comprising a frequency-tunable oscillator and a state detector, designed by using the tristable
device as an element of the library. Moreover, we show the transfer functions of the different devices that form the system. (B) Dynamics of the
output genes of the complex system. Pulses in the input (I) of 20 min and 1000-fold of amplitude were applied at t = 1000 and 2000 min.
e138 Nucleic Acids Research, 2011, Vol. 39, No. 20 PAGE 6 OF 12
Page 6
Functional diversity and designability
We further studied the designability of a given dynamical
behavior. For that, we constructed a library of SBML
models of well-characterized regulatory elements previ-
ously implemented in vivo. Likewise, the corresponding
kinetic parameters were fitted from experimental data
and kept fixed. Using this library, we constructed by
in silico assembly all possible architectures up to three
genes, giving 501 952 different circuits (see the different
configurations in Figure 5A). We systematically imposed
-cI as output gene in all circuits. Thereby, we computed
the dynamics of all circuits to perform an analysis of the
behaviors that could be obtained with such a library. For
this work, we considered a library of 36 elements,
involving 5 genes and 31 synthetic promoters. As genes,
we contemplated the classical repressors LacI, TetR and
-cI, and the activators AraC and LuxR. Moreover, we
built a library with three constitutive promoters with dif-
ferent transcription rates, 16 single promoters involving 4
lacO, 4 tetO, 2 araO, 2 luxO (36), 2
O (11) and 2
(5) and 12 combinatorial promoters involving 4 lacO–
tetO, 3 araO–lacO, 2 araO–tetO (36), 1
O–lacO (12),
1 luxO–
O (7) and 1 luxO–lacO (68). The models also
accounted for the external molecules (IPTG, anhydrote-
tracycline –aTc–, L(+)-arabinose and acyl homoserine
lactone—AHL—) that modify the regulatory ability of
the transcription factors and represent the inputs of
the circuits. These SBML models are provided as
Supplementary Data and they contain the corresponding
kinetic parameters. Then, for each external inducer we
considered three different states (low, intermediate and
high), giving 81 environmental conditions for all combin-
ations, and four more conditions in which the inducers
had a pulse-like dynamics.
By compiling all numerical results (details in
Supplementary Data), we were able to dissect the dynam-
ical spectrum of the library (i.e. its functional diversity),
which included circuits operating as oscillators, amplitude
filters, memories and different logic gates (Figure 5B).
As expected, the majority of the circuits functioned as
logic gates, and because the external signals (IPTG, aTc,
arabinose and AHL) always activated transcription, the
set of NAND and NOR gates was highly reduced. In
addition, 1% of the assembled circuits was able to
exhibit oscillations. Furthermore, we found amplitude
filters in the 0.016% of the cases, and memories in the
0.436%. Certainly, this spectrum depends on the value
of F specified to differentiate between two concentration
levels (F gives their ratio). Herein, we imposed F 10,
although we also performed a screening to see the effect
of different values of F. Not surprisingly, as higher is
F, the number of functional circuits decreases
(Supplementary Table S1). In addition, we studied the
effect of the initial condition on the output gene finding
that the results were almost independent of this. Certainly,
the initial condition only affects in memory-like circuits,
but this effect was captured by imposing pulse-like
dynamics on the input. Interestingly, the repertoire of
designed circuits was essentially based on minimal cores
that provided the required functional mechanism
(Supplementary Figure S8). These cores illustrate the
design principles in which the dynamical spectrum is
based on. However, the use of a limited library and a
partial set of input conditions, while allowing an exhaust-
ive exploration, prevent obtaining a comprehensive
analysis of the design principles. For instance, as we
have shown above, a double activation core gives a
memory-like mechanism but it was not found in the
repertoire of circuits. In addition, all amplitude filters
were based on the FFL architecture 2C, although
further circuits, not necessarily FFLs, can be employed
to read morphogen gradients (59). We did not obtain
further topologies because the monochromatic regulatory
mode and the lack of cooperation between transcription
Scalability and sensitivity of designability
Furthermore, we investigated the dependence of the
designability of a function on the existing elements of
the library by calculating the degree of sensitivity of
each regulator over the resulting dynamical spectrum
(Figure 6). Accordingly, LacI appeared to be the most
important regulator, indeed for this particular case of
study, since it participated in the majority of the
Figure 5. (A) Graphical representation of the exhaustive design
strategy. Starting from a library of composable genetic regulatory
elements (mathematical models provided in SBML format in the
Supplementary File, we constructed all possible circuits up
to three genes for simulation. (B) Dynamical spectrum of the library by
exhaustive exploration (functional diversity). We represent the percent-
age of circuits that behave as oscillators, amplitude filters, memories
and logic gates (designability). To differentiate between two states of a
circuit, we imposed at least one order of magnitude in concentration.
PAGE 7 OF 12 Nucleic Acids Research, 2011, Vol. 39, No. 20 e138
Page 7
functional circuits. In the specific case of the amplitude
filters, since their mechanism relied on two different re-
pressions (Supplementary Figure S8), LacI and TetR
participated in all circuits. Certainly, the addition of
more regulatory elements in the library would enlarge
the designability of the different behaviors, and the iden-
tification of the regulatory cores in Supplementary Figure
S8 would lead to rationally decide on the elements of more
interest. In addition, we studied whether the designability
could be estimated by sampling a small subset of
assembled circuits instead of an exhaustive exploration.
This would provide further support to the heuristic explor-
ation by means of optimization methods. Interestingly, we
found similar results for the dynamical spectrum of the
library when analyzing the dynamics of about 1000
circuits (corresponding to the 0.2% of the total circuits)
(Supplementary Figure S9). This suggests that even a
small fraction of assembled circuits is representative of
the whole population of circuits. By exploiting this fact,
we could analyze the functional diversity and designability
of several libraries of models at a minimal computational
cost or we could study how to enrich the library with
new regulatory elements.
We further studied the designability of the different
behaviors when considering the stochasticity inherent to
the cellular processes (we focused on intrinsic noise, details
in Supplementary Data) (69). Since the stochastic simula-
tion entails a higher computational cost, we considered
a subset of circuits as before to perform this study, and
because it is expected this will not strongly affect the
results. For each condition of inputs, we considered
the average value and standard deviation of the output
(computed using the time dynamics after a transient
period). In general, we found similar results as in the
deterministic regime (Supplementary Figure S10). We
could explain this by the fact that in most cases gene
expression is high enough, which minimizes the effect of
intrinsic noise, although in some cases a particular circuit
topology could also help in such a noise reduction (59).
However, we found an increase of almost a doubling in
the number of oscillators. By examining the circuits, we
realized that circuits based on an activation–repression
mechanism with fast damped oscillations in the determin-
istic regime and that were identified as stable circuits were
then selected as noise-induced oscillators. For the other be-
haviors, the designability results in the stochastic regime
were slightly lower. The maximal reduction in
designability was of 20% in the case of YES/NOT
gates. In the circuits that were selected according to the
deterministic solution but not to the stochastic one, there
is an increase of noise in protein expression that prevents
identifying different states of operation.
Afterwards, we wondered whether a unique circuit
could exhibit different behaviors. Interestingly, we
observed special circuits that displayed multifunctionality
according to different input conditions (e.g. oscillators
working as amplitude filters, memories or logic gates).
Supplementary Table S2 shows the number of circuits
with two functions and the corresponding statistical
significance. For instance, the 0.3% of the total set of
circuits functioning as oscillators and memories held the
two functions by properly setting the environmental
factors (statistical significance assessed by bootstrapping).
In addition, we calculated the number of circuits with
multifunctionality (Supplementary Table S3) showing a
tendency log-normal in the distribution (slope=1.463,
= 0.982, P < 0.1). This sort of circuits is appealing for
cellular regulation and organization because the circuitry
rewiring instrumental to change the function is accom-
plished by an on-the-fly reprogramming sentence (70).
As well as a single gene can attain several functions
[e.g. a protein with different enzymatic properties (71)],
a multifunctional circuit can be exploited by the cell to
exert a conditional control of different responses.
Robust design for a cell environment
One of the main problems in synthetic biology is the
reliability of the engineered circuits when they are
deployed in a continuous cellular background. Certainly,
if there is not a selective pressure to maintain a circuit and
it results in a cost for the cell, the genetic parts that form it
will accumulate mutations (involving point mutations, in-
sertions and deletions) and a quantity of them will entail
the loss of function of the circuit (27,72). Thereby,
we implemented a strategy based on robust design to
counteract this undesirable effect by which the resulting
circuit has the ability of maintaining the same behavior
under perturbations in the kinetic parameters of the model
(i.e. a sort of mutational robustness) (54)]. This also deals
with the eventual parameter uncertainty of the inferred
models. To this end, we expanded the metric function to
account for robustness, scalarizing the corresponding
multi-objective optimization problem, given by
¼ð1 Þ
, ð5Þ
Figure 6. Sensitivity analysis of the dynamical spectrum. We release
one regulatory element of the library (in particular, one gene) to
analyze its contribution to the dynamical spectrum (we represent the
remaining number of functional circuits relative to the total).
e138 Nucleic Acids Research, 2011, Vol. 39, No. 20 PAGE 8 OF 12
Page 8
where is the degree of robustness specified (i.e. for ¼ 0
no robustness is imposed), and
is the average of the
metric function over a large number of perturbations. As a
case of study, and since the promoter region is usually a
preferential site to accumulate mutations (72), we just
focused on perturbations in the binding affinities that
randomly change the parameter values up to 2-fold
(changes in all parameters simultaneously). In addition,
to accelerate the convergence future work could just
evaluate that function in a more reduced neighborhood
with a first-order approximation (73).
Accordingly, we attempted the design of a robust amp-
litude filter satisfying the (input, output) specifications of
(Figure 7A). The architecture
of the resulting circuit involves a FFL and for its imple-
mentation we would need a promoter type NAND, which
could be engineered by using artificial operator sites that
would be recognized by heterodimers from chimeric
proteins (74). We then used this circuit as starting point
(keeping fixed the topology) to further apply the optimiza-
tion procedure with ¼ 0. As we could observe, the new
circuit reaches a higher fitness at ¼ 0, which indicates
that it exists a cost for robustness (Figure 7B). While being
this cost low, this allows the circuit to maintain an
operative mode even under genetic perturbations.
In addition to the evolvability of living systems, gene
circuits are subjected to stochastic processes that may
prevent the expected behavior. The use of stochastic
models could allow designing more robust circuits, or at
least circuits that maintain their deterministic behavior.
One way to explore the effect of noise in the design of
circuits would be to in silico evolve a circuit to be robust
against noise and compare it with a circuit not evolved
with such a selective pressure. We applied our optimiza-
tion method to design circuits that maintained a targeted
dynamics in presence of noise (both intrinsic and extrin-
sic), obtaining different circuits with self-repressions and
low translation rates (Supplementary Figure S11) (69).
On the contrary, this approach could be applied to
design circuits displaying high noise levels (75), since
they may provide to the cell certain advantage under
unpredictable environmental conditions (20).
In this work, we have tackled the problem of the
designability of a given gene dynamics provided a
library of composable regulatory elements, considering
that the functional circuits come from combining different
elements of the library. This measure of designability
quantifies the entropy of a given dynamical behavior.
For that, we have developed a computational method-
ology that allows exploring the diversity of behaviors
that can be obtained by assembling circuits by means of
two different design strategies: one based on heuristic op-
timization and other based on exhaustive simulation of
circuits. We have taken advantage of current characteriza-
tions of regulatory elements into libraries of mathematical
models (26–29), allowing to rapidly select the regulatory
element of interest for our circuit. Although the emergence
of unexpected behaviors is always an issue in synthetic
biology, it is anticipated that the use of standardized
parts allows reducing the endless tweaking process when
engineering a synthetic gene circuit (12,19). Using a proper
mathematical formulation, we were able to generate a
large collection of genetic circuits by assembling those
regulatory elements, and identify the functional subset ac-
cording to certain specifications. Initially, we constructed
an artificial library of models to design circuits by opti-
mization toward a configuration satisfying the specifica-
tions. We designed filters and counters of gene expression,
which allowed us to find new regulatory mechanisms
able to provide such behaviors. Sometimes the behavior
requires a very precise genotype, making improvable to
get many cells with such behavior in a heterogeneous
population. To investigate this, we have defined the
concept of asymptotic robustness, which provides a
measure of the maximum genotypic heterogeneity for
a given of phenotypic behavior. In the long term, it is
expected that synthetic biology projects will provide
many examples of standardized circuits with a given
dynamics, which could be incorporated into the available
libraries. Then, one could extend our analysis to such
Figure 7. (A) Scheme of a genetic circuit optimized to operate as a
robust amplitude filter. U and Z are the input and output, respectively.
(B) Fitness function versus the robustness weight for the robust circuit
shown above (red) and the same circuit optimized with ¼ 0 (blue),
illustrating the cost of robustness.
PAGE 9 OF 12 Nucleic Acids Research, 2011, Vol. 39, No. 20 e138
Page 9
cases. One issue here would involve the interfacing of such
modules. To analyze this, we exploited one of our
designed circuits as a single element of the library to
obtain a complex system involving such a functional
unit, illustrating a hierarchical design approach and
allowing the design of plug-and-play devices with
optimal impedance matching.
Given a library of regulatory elements, it is possible to
construct many circuits with various dynamical behaviors.
But some behaviors occur more often than others. To
quantitatively analyze this, we computed the designability
of a set of useful behaviors. There, we constructed a more
reduced library of regulatory elements to assemble all
possible circuits up to three genes and process their
dynamics. Remarkably, the library involved promoters
that had been previously characterized and even used for
engineering various synthetic circuits in the bacterium
Escherichia coli. Interestingly, we found that a limited
library could encode a large number of behaviors.
Certainly, our computational method allowed construct-
ing and simulating the dynamics of this large set of circuits
and assisted to dissect the spectrum of dynamical behav-
iors and study their designability. We found that a same
genotype could have several functions depending on the
external signals. Nevertheless, as the size of the circuits
and the number of elements of the library increases, the
exhaustive design strategy becomes unpractical, thus
requiring heuristic methods. Since noise is an important
factor that affects the dynamics of a circuit, we also
included it in our analysis. The consideration of intrinsic
noise slightly reduces the designability of digital circuits,
but it increases the designability of oscillators. This is
understandable from the fact that digital devices are
steady-state based circuits, where noise could only spoil
the behavior. On the other hand, oscillatory circuits are
dynamical systems, where the noise could contribute to
enhance the behavior. In addition, we expect that our
results would be maintained when the library is
enhanced by incorporating more accurate experimental
measurements of the transcription regulation elements.
That the new models could be more elaborated and
accurate, they would not much change the fitness land-
scape and thus the designability of behaviors.
Herein, as opposite to other optimization-based design
approaches (48–52) where it is difficult or even impossible
to compile into a reliable DNA sequence the designed
circuit, our automated approach scrutinizes in a combina-
torial way the available genetic diversity to optimize
a circuit with the desired function, and finally output
a DNA sequence that encodes this designed circuit.
However, the evolvability of genetic circuits in a continu-
ous cellular background is a delicate issue. Importantly,
the robust design approach entails that the expected
behavior can be also observed throughout the cells of a
population. Indeed, natural systems are robust to coun-
teract mutations that, in most cases, entail unpredictable
changes in the function of these elements (76). Moreover,
as higher is the load of the circuit for the cell (i.e. quantity
of resources required for function expression), faster will
be the loss of function of the circuit because deleterious
mutations will be selected (27,72). Hence, we tackled the
design of robust circuits by accounting for parameter
sensitivity into the objective function. We illustrated that
such a circuit pays a cost by decreasing its fitness in order
to gain robustness, which will lead to a higher evolution-
ary stability. Alternatively, or even in combination, we
could design circuits that could, while preserving its
central function, induce a benefit to the cell, by accounting
for the interface between the circuit and the cellular
interactome (77), and then assess the selection and
reliability of the engineered circuit.
Interestingly, one possible extension to our work would
be the development of a more complex, hierarchically
distributed design platform (26). Herein, more diverse,
characterized regulatory elements would be considered,
involving transcriptional, riboregulatory, metabolic and
signaling elements. These different regulatory elements
would be combined to yield complex functional genetic
circuits, involving different regulatory mechanisms.
In addition to new elements, inherent effects such as the
variation of cell growth rate due to different culture
media, the delay in the biochemical reactions and the
parameter uncertainty of the models are important ques-
tions that would be explored. Furthermore, the design of
circuits could be combined with tools for the design of
synthetic DNA sequences. This would exploit the inter-
actions between nucleic acids and the reengineering of
natural proteins. Promoters with targeted transcription
rates or multiple operators (35,36), small RNAs with
targeted secondary structures (6), or chimeric proteins
acting as new transcription factors (74) are examples of
what we could design computationally. These all elements
would be modeled by transfer functions and these would
be stored in a library (virtual or real). Importantly, it
could be also specified a degree of evolvability, by which
the value of the kinetic parameters characterizing that
element would be susceptible to be changed after specific
mutations in its sequence. Finally, the cellular chassis in
which the circuit is going to be deployed could be also
introduced as a generalized element by modeling the
host elements that require the circuit for its expression
(78). This would allow to provide a prediction of the
response of the engineered cell under the conditions for
which the circuit was designed, and consequently improve
the design process.
Supplementary Data are available at NAR Online.
The authors thank M. Suarez and J. Forment for helping
with programming and computational resources.
Funding for open access charge: Generalitat Valenciana
(BFPI-2007-160); HPC-Europa programme (RII3-CT-
2003-506079); Spanish Ministry of Education and
Science (TIN-2006-12860); Structural Funds ERDF;
e138 Nucleic Acids Research, 2011, Vol. 39, No. 20 P
AGE 10 OF 12
Page 10
FP6-NEST 043340 (BioModularH2); FP7-ICT-043338
Genopole/UEVE (A3405); Fondation pour la Recherche
Conflict of interest statement. None declared.
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    • "However, not all parts interact with each other, and hence not all designs are biologically plausible [Densmore et al. 2010]. Without considering such constraints, the possible solution space for biological systems would grow exponentially [Rodrigo et al. 2011]. Information about interactions between SVPs can be used to identify suitable SVPs, restricting the number of solutions to biologically plausible designs. "
    [Show abstract] [Hide abstract] ABSTRACT: Modelling and computational simulation are crucial for the large-scale engineering of biological circuits since they allow the system under design to be simulated prior to implementation in vivo. To support automated, model-driven design it is desirable that in silico models are modular, composable and use standard formats. The synthetic biology design process typically involves the composition of genetic circuits from individual parts. At the most basic level, these parts are representations of genetic features such as promoters, ribosome binding sites (RBSs), and coding sequences (CDSs). However, it is also desirable to model the biological molecules and behaviour that arise when these parts are combined in vivo. Modular models of parts can be composed and their associated systems simulated, facilitating the process of model-centred design. The availability of databases of modular models is essential to support software tools used in the model-driven design process. In this article, we present an approach to support the development of composable, modular models for synthetic biology, termed Standard Virtual Parts. We then describe a programmatically accessible and publicly available database of these models to allow their use by computational design tools.
    Full-text · Article · Dec 2014 · ACM Journal on Emerging Technologies in Computing Systems
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    • "The system parameters in the dynamic model of genetic circuits can be identified from these measurement data using system identification methods [22]. In [38,39], a class of robust genetic circuits has been constructed by selecting the applicable promoter-RBS component from a promoter-RBS library. Therefore, to realize our proposed genetic logic circuits including the genetic waveform-shaping circuit and genetic counter, one first establishes the measurement device which exhibits fluorescence concentrations of a series of a repressor or activator gene with different promoter- RBS components and TFs via fluorescence measurement [41] and then rebuild a promoter-RBS library with information of system parameters in terms of our mathematical model describing the behaviors of genetic logic gates by using the system identification methods [30,35]. "
    [Show abstract] [Hide abstract] ABSTRACT: Background Rhythmic clock widely occurs in biological systems which controls several aspects of cell physiology. For the different cell types, it is supplied with various rhythmic frequencies. How to synthesize a specific clock signal is a preliminary but a necessary step to further development of a biological computer in the future. Results This paper presents a genetic sequential logic circuit with a clock pulse generator based on a synthesized genetic oscillator, which generates a consecutive clock signal whose frequency is an inverse integer multiple to that of the genetic oscillator. An analogous electronic waveform-shaping circuit is constructed by a series of genetic buffers to shape logic high/low levels of an oscillation input in a basic sinusoidal cycle and generate a pulse-width-modulated (PWM) output with various duty cycles. By controlling the threshold level of the genetic buffer, a genetic clock pulse signal with its frequency consistent to the genetic oscillator is synthesized. A synchronous genetic counter circuit based on the topology of the digital sequential logic circuit is triggered by the clock pulse to synthesize the clock signal with an inverse multiple frequency to the genetic oscillator. The function acts like a frequency divider in electronic circuits which plays a key role in the sequential logic circuit with specific operational frequency. Conclusions A cascaded genetic logic circuit generating clock pulse signals is proposed. Based on analogous implement of digital sequential logic circuits, genetic sequential logic circuits can be constructed by the proposed approach to generate various clock signals from an oscillation signal.
    Full-text · Article · May 2014 · BMC Systems Biology
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    • "Their approach is applicable only to combinatorial circuits. Rodrigo and Jaramillo [9] evaluate the network configurations obtained by metaheuristic search algorithms with behaviour based fitness functions [10]. This is very similar to an approach of Huynh et al. [8] and does not solve the problems of the evaluators used by them. "
    [Show abstract] [Hide abstract] ABSTRACT: We present several measures that can be used in de novo computational design of biological systems with information processing capabilities. Their main purpose is to objectively evaluate the behavior and identify the biological information processing structures with the best dynamical properties. They can be used to define constraints that allow one to simplify the design of more complex biological systems. These measures can be applied to existent computational design approaches in synthetic biology, i.e., rational and automatic design approaches. We demonstrate their use on a) the computational models of several basic information processing structures implemented with gene regulatory networks and b) on a modular design of a synchronous toggle switch.
    Full-text · Article · Mar 2014 · IEEE/ACM Transactions on Computational Biology and Bioinformatics
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