Predicting Synthetic Gene Networks
Diego di Bernardo , Lucia Marucci , Filippo Menolascina ,
and Velia Siciliano
Synthetic biology aims at designing and building new biological functions in living organisms. The
complexity of cellular regulation (regulatory, metabolic, and signaling interactions, and their coordinated
action) can be tackled via the development of quantitative mathematical models. These models are useful
to test biological hypotheses and observations, and to predict the possible behaviors of a synthetic net-
work. Indeed, synthetic biology uses such models to design synthetic networks, prior to their construction
in the cell, to perform specifi c tasks, or to change a biological process in a desired way. The synthetic net-
work is built by assembling biological “parts” taken from different systems; therefore it is fundamental
to identify, isolate, and test regulatory motifs which occur frequently in biological pathways. In this
chapter, we describe how to model and predict the behavior of synthetic networks in two difference cases:
(1) a synthetic network composed of fi ve genes regulating each other through a variety of regulatory inter-
actions in the yeast Saccharomyces cerevisiae (2) a synthetic transcriptional positive feedback loop stably
integrated in Human Embryonic Kidney 293 cells (HEK293).
Key words: Synthetic biology , Mathematical modeling , Positive feedback loop , S. cerevisiae , HEK
293 , Microfl uidics
The synthetic network described herein has been built for
benchmarking modeling and reverse-engineering approaches
( 1, 3 ) . IRMA was designed to be isolated from the cellular envi-
ronment (Cuccato et al., Heredity 102:527–532, 2009), and to
respond to galactose or glucose, which respectively “switch” the
network on and off by triggering transcription of its genes. This
network (Fig. 1 ) is very articulated in its interconnections, which
include regulator chains, single input motifs, and multiple feedback
Network in Yeast
Wilfried Weber and Martin Fussenegger (eds.), Synthetic Gene Networks: Methods and Protocols,
Methods in Molecular Biology, vol. 813, DOI 10.1007/978-1-61779-412-4_4, © Springer Science+Business Media, LLC 2012
58 D. di Bernardo et al.
loops generated by the combination of transcriptional activa-
tors and repressors. We selected well-characterized promoter/
TF-encoding-genes pairs. We chose nonessential and nonredun-
dant TF-genes that can be knocked out without affecting yeast
viability. The following genes have been chosen for IRMA: as
activators and repressors encoding genes: SWI5, ASH1, CBF1,
GAL4 , and GAL80 ; as promoter genes: HO , ASH1 , MET16 ,
GAL10 (Fig. 1 ).
This synthetic network is shown in Fig. 2 . We took advantage of
the inducible Tet regulatory system: the expression of Tetracycline-
controlled transactivator tTA is self-controlled by a CMV-TET
promoter responsive to thetTA itself unless Tetracycline, or its
analogous Doxycycline, is added to the medium in which cells are
grown ( 4 ) . To follow the protein dynamics of the positive feedback
loop, a destabilized yellow variant of the enhanced green fl uores-
cent protein (d2EYFP) (Clontech), with a half-life of approxi-
mately 2 h, was expressed together with the tTA transactivator
from the same mRNA, via an Intra Ribosomal Entry Sequence
(IRES) in between of the transactivator tTA and the d2EYFP
(Fig. 2 ). In order to stably express in HEK293 cells the inducible
feedback loop and to better characterize its dynamics overtime, we
used a lentiviral vector ( 5, 6 ) , based on the multisite Gateway
technology provided by Invitrogen.
When deriving a model from experimental data, three major
approaches can be used: white-box, black-box, and gray-box. In
white-box modeling, the model and parameter values are entirely
derived from fi rst principles, while in black-box, the model is
completely derived from input–output data. The third alternative,
the so-called gray-box approach ( 7 ) , combines the two above
Loop in Mammalian
1.3. Derivation of
for Synthetic Network
Fig. 1. Diagram of the synthetic network in yeast. Solid lines model transcriptional interactions while dashed lines are
meant to represent protein-protein interactions.
594 Predicting Synthetic Gene Networks
approaches. Specifi cally, fi rst principles are used to partially derive
the model structure, while parameters in the model are estimated
from experimental data. The approach described in this chapter,
both for the yeast and the mammalian synthetic network, is a gray-
box one. In this case, modeling entails the following main steps to
be executed iteratively: (1) derivation of the model equations and
(2) estimation of the model parameters from experimental data
and/or literature. Step (1) requires introducing simplifying hypoth-
esis and choosing a proper formal framework. Among the different
mathematical formalisms, those based on differential equations are
commonly used to describe the average behavior of a population of
cells ( 8 ) . The Differential Equations modeling approach is based
on the following biological assumptions: the quantifi ed concentra-
tions are homogeneous in space and they are continuous quantities
in time. These assumptions hold true for processes evolving on
long time scales in which the number of molecules of the species in
the reaction volume is suffi ciently large. Step (2) is required to
estimate unknown model parameters from the available experimental
data. Experimental data are strongly affected by noise. Hence,
classical optimization methods, based on gradient descent from an
arbitrary initial guess of the parameters’ value, can be unfeasible.
The above considerations suggest looking at stochastic optimization
algorithms, such as Genetic Algorithms (GA) ( 9 ) , which provide a
fl exible approach to nonlinear optimization. Their application has
been proved to yield good results in the parametrization of syn-
thetic networks ( 10, 11 ) .
When the network is quite complex in terms of the number of
the unknown parameters, there is the need of going through itera-
tion between experiments and modeling, to gather more experi-
mental data if needed. We show an example of such iteration, in
the case of the yeast network.
Fig. 2. Design of the positive feedback loop in mammalian cells. The promoter CMV-TET
consist of seven direct repeats of a 42-bp sequence containing the tet operator sequences
( tetO ), located just upstream of the minimal CMV promoter (P min CMV ). The tetracycline-
controlled transactivator tTA derives from the addition of the VP16 activation domain to
the transcriptional repressor TetR. The d2EYFP is the destabilized yellow-green variant of
enhanced green fl uorescent protein with a half-life of approximately 2 h.
60 D. di Bernardo et al.
Once a model of a synthetic network has been derived, it is possible
to refi ne it by means of several alternative strategies. The primary
goal of model refi nement is the improvement of model predictions
as regards the network dynamics, i.e., how gene and protein expres-
sion change in time following a perturbation; in order to accomplish
this task, the experimenter may need to stimulate its synthetic net-
work with highly dynamical signals. This is usually done to elicit
nonlinear modes or other peculiar characteristics of the network
under development. Most of the synthetic networks documented so
far ( 12 ) use chemical compounds as inducers and thus the previous
requirement often translates in the need to quickly change the con-
centration of these compounds in the media where cells are grown.
On the contrary, effective strategies for data acquisition are needed
to measure changes in the concentration of the species of interest in
live cells. Here, we propose an integrated strategy intended to solve
both of these issues at once via “microfl uidics” devices and time-
lapse fl uorescence microscopy. Microfl uidics involves the manipula-
tion of very small fl uid volumes, enabling creation and control of
nanoliter-volume reactors, thus mimicking cellular microenviron-
ments. Microfl uidics devices can be conveniently used to fi nely con-
trol the concentration of compounds in the extracellular environment
during time lapse microscopy experiments. Data acquired from long
term stimulation of cells carrying fl uorescent tags and tracked by the
microscopy imaging can be conveniently used to improve the quality
of the mathematical model of the circuit of interest.
In the following sections, we illustrate how this platform can be
adapted in experimental contexts involving both simpler and higher
eukaryotic systems, namely mammalian cell lines and S. cerevisiae .
1. S. cerevisiae strains YM4271 background ( MATa ura3-52 his3-
D200 ade2-101 lys2-801 leu2-3 trp1-901 gal4-D542 gal80-D538
ade5::hisG ) ( 3 ) .
2. YEP medium: 10 g/L Bacto yeast extract, 20 g/L Bacto
3. YEPD medium: YEP containing 2% glucose.
4. YEPGR medium: YEP containing 2% galactose and 2% raffi nose.
5. SC medium: 6.7 g/L yeast nitrogen base without amino acids,
1.35 g/L amino acid powder mix.
6. 2 U/ul DNAse I (Roche).
7. Rneasy MiniElute Cleaneup Kit (Quiagen).
8. SuperScript III First-Strand Synthesis System (Invitrogen).
9. Platinum SYBR Green qPCR SuperMix-UDG with ROX
1.4. Refi ning
by Means Of
2.1. Yeast Culture,
614 Predicting Synthetic Gene Networks
10. 7000 ABI Real-Time PCR machine.
11. Applied Biosystems SDS software version 1.2.3 to perform
1. Cole-Parmer nano syringe pump (Cole-Parmer).
2. USB valve control system.
3. Pneumadyne 8-Valve Manifold.
4. Tygon Microbore Tubing I.D.: 0. 25” O.D.: 0.125” (Swagelok).
5. Tygon Microbore Tubing I.D.: 0.020” O.D.: 0.060” (VWR).
6. Double distilled H 2 O.
7. Sulforhodamine B (Sigma-Aldrich).
8. 60-mL syringe (BD).
9. 10-mL syringes (BD).
10. 22G sterile needles (BD).
11. 2 mL cryovials (BMA).
12. Microfl uidic device.
13. Inverted fl uorescence microscope with temperature and CO 2
conditioning and appropriate fl uorescence fi lters.
1. 293FT cells maintained at 37°C in a 5% CO 2 -humidifi ed
incubator, and cultured in Dulbecco’s Modifi ed Eagle’s
Medium DMEM (GIBCO BRL) supplemented with 10%
heat-inactivated fetal bovine serum (FBS) (Invitrogen), 1%
L -glutammine, 1% MEM Nonessential Amino Acids, 1% MEM
Sodium pyruvate, and 1% antibiotic/antimycotic solution
2. Hek 293 cells maintained at 37°C in a 5% CO 2 -humidifi ed
incubator, and cultured in Dulbecco’s Modifi ed Eagle’s
Medium DMEM (GIBCO BRL) supplemented with 10%
heat-inactivated fetal bovine serum (FBS) (Invitrogen), 1%
L -glutamine, and 1% antibiotic/antimycotic solution
3. Doxycicline (Clontech) dissolved in tissue-culture water to a
fi nal concentration of 10 m g/mL, stored in aliquots at −20°C,
and then added to tissue culture dishes as required.
4. Polybrene (Invitrogen) dissolved in tissue-culture water to a
fi nal concentration of 6 mg/mL, stored in aliquots at −20°C,
and then added to tissue culture dishes to a fi nal concentration
of 6 m g/mL.
5. Blasticidin (Sigma) dissolved in tissue-culture water to a fi nal
concentration of 100 mg/mL stored in aliquots at −20°C, and
then added to tissue culture dishes to a fi nal concentration
of 3 m g/mL.
2.2. Microfl uidics and
2.3. Mammalian Cells
Culture and Lentiviral
62 D. di Bernardo et al.
1. Taq Phusion (Fynnzymes) is used to a fi nal concentration of
0.02 U/ m L; the HF buffer is added to a fi nal concentration
of 1×, primers and dNTPs are supplied to a fi nal concentra-
tion of 0.5 m M each and 200 m M each, respectively.
2. Taq DNA polymerase (Invitrogen) is used to a fi nal concentra-
tion of 1 U/ m L; the PCR buffer is added to a fi nal concen-
tration of 1× MgCl 2 1.5 mM, primers, and dNTPs are supplied
to a fi nal concentration of 0.5 m M each and 200 m M each
3. 5 U/ m L of NheI and EcoRV restriction enzymes (Roche).
4. 2 U/ m L of T4 DNA Ligase, and T4 DNA Ligase buffer to a
fi nal concentration of 1×.
5. LR Clonase II plus enzyme mix (Invitrogen).
1. Personal computer equipped with MATLAB Simulink
2. Image processing algorithm implemented in the control
To construct the IRMA containing strain, sequential PCR-based
genomic integrations were made sequentially. All the integrations
were confi rmed by PCR.
1. The 2× HA-hphMX4 cassette was amplifi ed by PCR and
inserted in front of the stop codon of ASH1 gene in YM4271
strain resulting in P278 strain.
2. To generate P280 strain MET16 promoter was amplifi ed from
W303 and cloned in YIplac128 between Hind III and Sac I.
3. GAL4 ORF was cloned between Sac I and Nde I, thus resulting
in p MET16pGAL4 .
4. The MET16pGAL4-LEU2 cassette was integrated in SHE2
5. CBF1 ORF was amplifi ed from W303 and cloned among Bam
HI and Pac I of p FA6a-GFP (S65T)-kanMX6 .
6. The CBF1-GFP-kanMX6 cassette was integrated downstream
of the HO promoter of P280 strain, obtaining P324.
7. ASH1 promoter was cloned in Pst I and Bam HI of YIplac211,
and then GAL80-3xFLAG was inserted between Bam HI and
8. The ASH1pGAL80-3XFLAG-URA3 was integrated in SWI5
locus, thus yielding P326.
of Gene Circuit
3.1. Construction of
S. cerevisiae Strains
634 Predicting Synthetic Gene Networks
9. ACE2 gene was deleted in the strain P326 by integrating
natMX4cassette from pAG25.
10. GAL10pSWI5AAA-MYC9-KlTRP1 was integrated in CBF1
locus resulting in IRMA containing strain.
Further details can be found in ( 3 ) .
In order to analyze the dynamic behavior of the network, we analyzed
expression profi les of network genes by quantitative real-time
RT-PCR following two different perturbation experiments; in the
fi rst we shifted cells from glucose (YEPD medium) to galactose
(YEPGR medium) (“switch-on” experiments) and from galactose
to glucose (“switch-off” experiments). In the second, we overex-
pressed each of the fi ve network genes in cells that were grown
either in glucose or galactose ( 3 ) .
1. Prepare the total RNA.2. Treat 1 m g of RNA with 2.5 U of
2. Clean up with RNeasy MiniElute Cleaneup Kit (Quiagen).
3. Reverse-transcribe the RNA cleaned using SuperScript III
First-Strand Synthesis System.
4. Set up quantitative real-time PCR reactions in duplicates using
Platinum SYBR Green qPCR SuperMix-UDG with ROX.
5. Amplify the cDNA thus obtained using a 7000 ABI Real-Time
We constructed the synthetic positive feedback loop into a lentivi-
ral vector system to allow integration of the circuit in mammalian
cells. To this end, we used the ViraPower Promoterless Lentiviral
Gateway Expression System (Invitrogen), which takes advantage
of the site-specifi c recombination properties of bacteriophage
lambda, making the transfer of single DNA sequences faster than
the usual cloning strategies.
1. Cloning of a destabilized yellow-green variant of enhanced
green fl uorescent protein (d2EYFP) in the p MAtTA-IRES-
EGFP vector: the d2EYFP was amplifi ed from pd2EYFP-1
(Clontech) by PCR using the High-Fidelity DNA Polymerase
Taq Phusion, with a forward primer containing a NheI recognition
sequence and a reverse primer containing an EcoRV recogni-
tion sequence. The PCR product and p MAtTA-IRES-EGFP
were then digested with NheI-EcoRV restriction enzymes and
the d2EYFP ligated in place of EGFP (ratio ng of vector/ng of
d2EYFP = 1/3–1/5).
2. Generating the pENTR vectors: in order to produce a lentiviral
vector by using the gateway system we fi rst generated the pENTR
vectors containing the genes and the promoters of interest fl anked
by specifi c recombination sites. The p MAtTA-IRES-d2EYFP
and Testing of the
in Hek293 Cells
64D. di Bernardo et al.
was linearized with the AseI restriction enzyme and recombined
with the pDONR221 (invitrogen) following the manufacturer
instruction. In this way we generated p ENTRtTA-IRES-
d2EYFP vector with specifi c recombination sites. The CMV-
TET promoter was amplifi ed from pTRE2 (clontech) by PCR.
The PCR was performed with the Taq polymerase provided by
Invitrogen that adds a single deoxyadenosine (A) to the 3 ¢ ends
of PCR products. This allows PCR inserts to ligate effi ciently
with the pENTR5 ¢ -TOPO vector which is supplied linearized
with single 3 ¢ -deoxythymidine (T) overhangs, obtaining the
pENTR5 ¢ -TOPO- CMV-TET with specifi c recombination sites.
Finally, we performed a recombination reaction between the
pENTR tTA-IRES-d2EYFP ,
and the p Lenti/R4R2/V5-DEST using the LR clonase enzyme
according to manufacturer instructions. The lentivirus was
then produced in 293FTcells as described in the instructions
provided by Invitrogen.
3. Cell culture for lentiviral transduction: To transduce cells with
the virus produced, 500000 HEK293 cells were plated and
incubated overnight. The day of transduction the medium was
removed and 1 mL of the virus was added to the cells together
with polybrene to a fi nal concentration of 6 m g/mL. After an
overnight incubation the medium containing the virus was
removed and replaced with complete culture medium containing
the blasticidin to a fi nal concentration of 3 mg/mL to select
for stably transduced cells.
4. To test the dynamics of the autoregolatory loop, we performed
two sets of time-series experiments in which stably integrated
HEK293 cells were imaged using time-lapse microscopy and
fl uorescence of d2EYFP was quantifi ed. For both the experi-
mental designs in the fi rst time point, cells were treated with
Doxycycline to “switch off” the network, since Doxycycline
prevents the tTA transactivator to bind the CMV-TET respon-
sive promoter. In the fi rst set of experiments, the dynamics
were followed for 37 h at 37°C, while in the second set the
temperature was reduced to 32 °C to limit cell motility and
thus facilitate image analysis ( 13 ) .
pENTR5 ¢ -TOPO- CMV-TET ,
The network of interest it the one showed in Fig. 1 . Details about
the network construction are reported in Subheadings 2.1 and 3.1 .
For each species in the network, i.e., each mRNA (capital letters)
and correspondent protein concentration (small letters), we wrote
one equation, which expresses its change in time as the result of
production and degradation:
Modeling the Yeast
3.4.1. Step (i). Derivation
of the Model from First
65 4 Predicting Synthetic Gene Networks
([ 5],[ 1]; ,,,)[ 1],
[ 1][ 1],
(; ,)[ 4],
[ 4][ 4],
([4 ];,)[ 5],
v HSwiAshk k h h d CBF
CBF d Cbf
v H Cbf k hd GAL
v H Galk hd SWI
[ 5][ 5],
([ 5]; , )
([ 5];,)[ 1],
[ 1][ 1].
SWI d Swi
v HSwi d GAL
v HSwi k h d ASH
The fi rst two terms on the right-hand side of the mRNA equations
represent the production, where a are the basal transcription rates;
v are the maximal transcription rates modulated by the Hill
( ; , )
y k h
( ; , )
z k h
( , ; , , , )
y z k h k h
( ; , )
y k h H
( ; , ).
z k h
These are used to model transcriptional activation, or repression;
y and z represent transcription factor levels, h are the Hill coeffi -
cients (pure numbers that refer to the cooperativity of the activa-
tion binding reaction) and k are the Michaelis-Menten constants,
equal to the amount of transcription factor needed to reach half
maximal activation (or repression). For protein equations, the pro-
duction rates are b , i.e., the maximal translation rates. Degradations
of mRNAs and proteins are represented by d , i.e., the degradation
constants. The amount of free Gal4 depends on the interactions
of the galactose pathway with the network genes. For the units of
measurement, please refer to Table 1 . Summing up, when writing
the above model, we made the following assumptions: (A1) the
66D. di Bernardo et al.
transcriptional activity of each promoter is leaky ( a ); (A2)
the degradation kinetics of both mRNAs and proteins are
first-order; (A3) the protein production terms are proportional to
the corresponding mRNA concentrations; (A4) the transcriptional
activation–repression of each promoter by a transcription factor can
be modeled as a Hill function ( 2 ) .
Note that the concentration of Gal4free is the amount of Gal4
protein that is not involved in the formation of the protein-protein
complex with Gal80 and hence activates the GAL10 promoter
driving SWI5 expression.
Parameters of the mathematical model the yeast synthetic
model Refi ned model Exp. id.
k 1 (a.u.)
k 2 (a.u.)
k 3 (a.u.)
k 4 (a.u.)
1.884 1 1
30 0.035 0.035
0.229 0.037 0.037
0.216 0.09 Glu 0.09 Glu
0.01 Gal 0.01 Gal
k 5 (a.u.)
k 6 (a.u.)
a 1 (a.u. min. −1 )
a 2 (a.u. min. −1 )
a 3 (a.u. min. −1 )
a 4 (a.u. min. −1 )
a 5 (a.u. min. −1 )
v 1 (a.u. min. −1 )
v 2 (a.u. min. −1 )
v 3 (a.u. min. −1 )
0.16 1.884 1.884
0.160 1.884 1.884
0 0 –
1.10 × 10 −4 1.49 × 10 −4 –
3.2 × 10 −4 3 × 10 −3 –
0 7.4 × 10 −4 –
7.37 × 10 −5 6.1 × 10 −4 –
0.065 0.04 –
0.002 8.82 × 10 −4 –
0.025 0.002 Glu
v 3 Glu /v 3 Gal 9
v 4 (a.u. min. −1 )
v 5 (a.u. min. −1 )
d 1 (min −1 )
d 2 (min −1 )
d 3 (min −1 )
d 4 (min −1 )
0.007 0.014 –
0.002 0.018 –
0.033 0.022 –
0.042 0.047 –
0.047 0.421 –
0.141 0.098 –
674 Predicting Synthetic Gene Networks
In order to identify model’s parameters and validate the model, we
collected mRNAs expression levels during a time course experi-
ment, by shifting cells from glucose (YEPD medium) to galactose
(YEPGR medium) (“switch-on” experiment) as described in ( 3 )
and in Subheadings 2.1 and 3.2 . Data are shown in Fig. 3a .
We included as the fi rst point of the time-series the expression
level of the network genes after growing cells overnight in glucose,
just before shifting them from glucose to galactose ( 3 ) . The second
point, taken after 10 min, is measured just after the shift has
occurred. The averaged gene expression profi les (Fig. 3a ) show that
the standard washing steps, needed to shift cells from glucose
medium to the fresh new galactose-containing medium, induce a
transient increase in mRNA levels of GAL4 and GAL80 (Fig. 3a ,
gray bars). This effect is not dependent on galactose addition, but
uniquely on the washing steps ( 3 ) , and it is probably due to the
transient deprivation of carbon source during washing, which atten-
uates the degradation levels of GAL4 and GAL80 mRNAs ( 14 ) .
Moreover, the activation of CBF 1 appears to be delayed with
respect to the other Swi5 targets, respectively GAL80 and ASH1
(Fig. 3a ). Such delay is physically due to the sequential recruitment
of chromatin modifying complexes to the HO promoter, which
follows binding of Swi5 ( 15, 16 ) .
We then performed four additional experiments, shifting cells
from galactose to glucose, thus “switching off” gene expression in
the network, as described ( 3 ) and in Subheadings 2.1 and 3.2 .
model Refi ned model Exp. id.
d 5 (min −1 )
b 1 (min −1 )
b 2 (min −1 )
0.018 0.050 –
1 1 1
1 1 1
1 1 1
1 4 4
1 1 1
1 1 1
1 4 4
0.223 0.201 –
0.285 0.167 –
10 −4 Glu
t (min) 100 100 –
68D. di Bernardo et al.
The averaged time-series gene expression profi les (Fig. 3b ) were
used for validating model predictive performance. We refer to this
dataset as the “switch-off” dataset.
Finally, we collected another set of experiments to be used to fur-
ther validate the model’s predictive ability. We measured gene expres-
sion responses of the fi ve network genes following exogenous
overexpression of each of the fi ve genes under the control of a strong
constitutive promoter, as described in ( 3 ) and in Subheadings 2.1 and
3.2 . Such overexpression experiments were performed both in glucose
Fig. 3. Identifi cation and validation results of time-series, phenomenological model. Circles
represent average expression data for each of the IRMA genes at different time points.
Dashed lines represent standard errors. Solid lines represent in silico data. ( a ) Identifi cation
results of the phenomenological model on the average 5 h “switch-on” time-series. ( b )
Validation of the phenomenological model on the average 3 h “switch-off” dataset.
694 Predicting Synthetic Gene Networks
and in galactose. We refer to these two experimental datasets as the
“Galactose steady-state” and “Glucose steady-state” (Fig. 4a, c ).
At this stage, we had to properly refi ne the model both to be able
to capture the features highlighted by the gene expression profi les
and to reduce the number of parameters to be estimated. First of
all, we made the following extra modeling assumptions: (A5) a fi x
time delay, t , equal to 100 min, is added in the activation of the
HO promoter by Swi5; (A6) a transient decrease in the mRNA
degradation of GAL4 and GAL80 of value Db1 and Db2 is added for
an interval of 10 min to describe the effect of the washing steps.
Due to the lack of protein concentrations measurements, we
also assumed that the protein concentrations are monotonically
increasing functions of their corresponding mRNA concentrations
at any time (A7).
In order to defi ne the active amount of Gal4 (GAL4 free in the
above Equations), we needed to describe the effect of the galactose
pathway on the network dynamics. In the literature, very detailed
models of the galactose pathway have been presented ( 17 ) . Such
paradigms can be simplifi ed in a number of ways, but it often leads
to include in the model nonmeasurable complexes concentrations.
Thus, we decided to use a phenomenological approach, assuming
that (A8) the protein-protein interaction between Gal80 and Gal4
can be modeled as a direct inhibition of GAL80 on the promoter of
SWI5 , and that the strength of such inhibition depends on the
medium (strong inhibition in glucose, weak inhibition in galactose).
Actually we assumed that the GAL10 promoter is activated by
GAL4 and noncompetitively inhibited by GAL80. The resulting
phenomenological Delay Differential Equations (DDEs) model is:
SWI5(t - )
 ([ 4])
We use the symbol ^ to indicate medium-dependent quanti-
ties. Thus, we are assuming that the Michaelis-Menten coeffi cient
of the phenomenological description of the inhibition of GAL80 is
dependent on the medium.
3.5.1. Step (i). Derivation
of the Phenomenological
70D. di Bernardo et al.
Fig. 4. Experimental and simulated overexpression experiments ( a , c ). In vivo expression levels of IRMA genes after
overexpression of each gene (perturbed gene; indicated by the black dots on the bars) from the constitutive GPD promoter
( gray bars ) and after transformation of the empty vector ( white bars ). IRMA cells were transformed with each of the constructs
containing one of the fi ve genes or with the empty vector. At least, three difi erent colonies were grown in glucose ( b ) and
in galactose-rafi nose ( a ) up to the steady-state levels of gene expression. Quantitative PCR data are represented (average
data from different colonies) ( b , d ). In silico expressionlevels of IRMA genes obtained by simulating the overexpression of
each gene with the phenomenological model ( e , f ). In silico expression levels of IRMA genes obtained by simulating the
overexpression of each gene with the refi ned model.
714 Predicting Synthetic Gene Networks
For the sake of simplicity, we set all of the Hill coeffi cients to 1. For
the identifi cation of the remaining parameters, we used the “switch-
on” dataset, using as initial values the simulated steady-state mRNA
levels in glucose. Parameters’ identifi cation results, obtained by
using the Genetic Algorithm and hand refi nement ( see Note 6 ) are
shown in Fig. 3 a and the inferred parameters in Table 1 . The model
captures the delay in CBF1 activation and the small variations of
GAL4 and GAL80 .
In order to validate the model predictive performance, we used
the “Glucose steady-state” and “Galactose steady-state” over-
expression experiments, and compared them with their in silico
counterparts by simulating the overexpression of each of the fi ve
genes (Fig. 4b ), (D)). We further validated the predictive perfor-
mance against the “switch-off” time-series by simulating in silico
the “switch-off” experiment (i.e., setting the medium-dependent
parameters to their values in glucose and starting the simulation
from the steady-state equilibrium in galactose) (Fig. 3b ). The phenom-
enological DDEs model has good descriptive and predictive
performance. However, the 24 identifi ed parameter values are
likely to be different from their physical values. For example, model
parameters (Table 1 ) indicate that the inhibition of Ash1 on CBF1
is so weak that can be neglected, even if in the literature it has been
reported otherwise ( 16 ) .
At this point, we needed to clarify the biological properties of the
HO promoter by taking direct measurements of the promoters’
parameters ( 3 ) . We thus measured the transcriptional response of
the promoters of GAL10 , MET16 , ASH1 , and HO; the latter when
regulated by both Swi5 and Ash1 ( see Note 7 ). For each promoter,
we fi tted to data the equation at steady state of the gene whose
expression the promoter drives. Of note, it became apparent from
the new experimental data and the fi tting results that galactose not
only weakens the inhibition of Gal80 on the GAL1 0 promoter
(assumption (A8)), but also allows a faster activation of the GAL10
promoter. Moreover, in galactose such activation is possible for
values of GAL4 lower than in glucose.
In order to capture the behavior observed from the new experiments,
we considered two additional parameters in the model to be explic-
itly dependent on the medium. Thus, we refi ned the previous
model by changing the equation of SWI5 , which became
d [ 15],
3.5.2. Step (II).
Identifi cation of Model’s
Parameters and Model
and Reidentifi cation
of the Correspondent
Parameters (Step (II))
3.5.4. Step (i).
Refi ned Model
72 D. di Bernardo et al.
where again the symbol ^ indicates parameters dependent on the
medium. From the analysis of data, we found that the value assumed
by v 3 in galactose is 9 times bigger than the one in glucose.
Analogously, the value of k 4 is 9 times bigger in glucose than in
galactose ( see Table 1 ).
From the promoter dataset, we estimated 16 parameters, including
the medium-dependent ones (Table 1 ). From such data, we could
neither fi t degradation constants nor the washing effect parame-
ters. Thus, the remaining 17 parameters were evaluated again from
the “switch-on” experiment (Table 1 ). In simulations, the initial
values of mRNA concentrations were set to the steady state values
predicted by the model in glucose. The in silico “switch-on”
time-series is shown in Fig. 5a . Also in this case, we tested the
predictive ability of the model performing in silico the previously
described “Glucose steady-state” and “Galactose steady-state”
overexpression experiments (Fig. 4e, f ) and the “switch-off” time-
series (Fig. 5b ). Now, the identifi ed parameters confi rm that the
Ash1 inhibition of the HO promoter is indeed strong, as reported
in the literature ( 16 ) .
There are still discrepancies between the in vivo and in silico
initial values of CBF1 , SWI5 , and ASH1 in the “switch-off” data-
set, and in the predicted steady state of mRNA levels in galactose.
We attribute them to the unmodeled effect of protein dynamics,
which have been removed from the original model due to the lack
of experimental measurements. In particular, we noticed that the
Gal4 protein is stable ( 18 ) , and therefore even a small, or transient,
increase in its mRNA level is able to induce the GAL10 promoter,
regulating Swi5 in our network. Since we do not explicitly model
protein dynamics, a small increase in GAL4 mRNA cannot fully
activate GAL10 in the model and does not cause the increase in
SWI5 mRNA seen in vivo. The quality of the fi tting and the predic-
tions could be further improved by modeling the proteins levels of
all the genes in the network. However, in the actual version of the
network, it is not possible to measure protein levels with the excep-
tion of only one gene (Cbf1). Thus, the assumption of steady state
for protein dynamics is required, not only to simplify the model
but also mainly to not introduce the problem of overfi tting and
nonuniqueness of parameters for proteins.
The network of interest is the one showed in Fig. 2 . Details about
the network construction are reported in Subheadings 2.3 , 2.4 and
3.3 . Again, the formal framework we chose is based on Ordinary
Differential Equations, since we are measuring the average behavior
of a population of cells uniformly infected by the virus.
For each species, i.e., each mRNA (italic capital letters) and
correspondent protein concentration (roman small letters), we
wrote one equation, which expresses the change in concentration
3.5.5. Step (ii).
of the Refi ned Model
Feedback Loop in
3.6.1. Step (i). Derivation of
the Mathematical Model
734 Predicting Synthetic Gene Networks
of the species in a given time interval, as the result of a production
and a degradation. We assumed:
Hill functions to model the rate of gene transcription, includ-
ing basal activity to describe the leakiness of the CMV-TET
Linear degradation for all genes and proteins (A2)
Linear dynamics for the translation (A3)
Michaelis–Menten-like modeling of the effect of the inducer
Distinct dynamics for the unfolded (inactive) and folded
(active) forms of the reporter protein (d2EYFP) (A5)
Fig. 5. Identifi cation and validation results of time-series, phenomenological model. Circles
represent average expression data for each of the IRMA genes at different time points.
Dashed lines represent standard errors. Solid lines represent in silico data. ( a ) Identifi cation
results of the phenomenological model on the average 5 h “switch-on” time-series.
( b ) Validation of the refi ned model on the average 3 h “switch-off” dataset.
74 D. di Bernardo et al.
The last assumption was introduced to take into account time
delay for the maturation of d2EYFP protein. Thus, we derived two
differential equations as in ( 11 ) : one for the translation of mRNA
to the unfolded d2EYFP protein, and one for the folded protein
d2EYFP. Letting x 1 be the tTA/d2EYFP mRNA concentration, x 2
the tTA protein concentration, x 3 the unfolded d2EYFP protein
concentration, and x 4 the folded d2EYFP protein concentration,
the equations describing the network become:
( ) ,
v d x
v xd x
K xd x
= a + - a-
Note that, due to the presence of the IRES, the concentrations
of tTA protein and d2EYPF protein depend on the same variable
( x 1 ), that is the amount of tTA mRNA.
As in the case of the yeast synthetic, network, to identify model’s
parameters and validate the model predictive ability, we performed
two sets of “switch off” time-series experiments as described in
Subheading 3.3 . HEK293 cells expressing the positive feedback
loop, were imaged using time-lapse microscopy and fl uorescence
of d2EYFP was quantifi ed. At the beginning of the experiment
(fi rst time point) cells were treated with Doxycycline to “switch
off” the network, since Doxycycline prevents the ttA transactivator
to bind the CMV-TET responsive promoter. In the fi rst set of
experiments the dynamics were followed for 37 h at 37°C, while in
the second set the temperature was reduced to 32°C to limit cell
motility and thus facilitate image analysis.
We fi rst simulated the model using, when available, the parameters
values previously estimated in the literature ( see ref. 11 ) , and
reported in Table 2 . However, using such parameters, we were not
able to correctly reproduce the experimental “switch-off” dynamic
behavior of the postive feedback loop. In particular, using the
reference parameters the in-silico simulation shows faster dynamics
than the in vivo data. Thus, we used the experimental “switch off”
time series to identfy the model parameters using a Genetic
Algorithm optimization method, letting the parameters vary in a
neighborhood of the literature value. In this case, hand refi nement
of Mathematical Models
for Synthetic Network
75 4 Predicting Synthetic Gene Networks
( see Note 6 ) was not necessary, since the number of parameters to
estimate was lower than in the yeast network case, and the search
interval was smaller. Of note, during the parameter fi tting proce-
dure, we tried to minimize the changes to parameters previously
estimated in the literature.
The simulations of the model with the identifi ed parameters
(Table 2 ) are shown in Fig. 6 . The model is now able to recapitulate
the positive feedback loop dynamics in response to different inducer
concentrations and experimental settings ( see Subheading 3.3 ). We
estimated a different the degradation rate of the reporter protein
(d2EYFP) in the two experimental conditions. As mentioned above,
in the fi rst set of experiments (Dataset 1) the cells were kept at
37°C, while, in the second round of experiments (Dataset 2), we
used a lower temperature (32°C) to limit cell motility. Figure 6 (A)
and (C) show the response of the system using the same amount of
inducer (1 m g/mL), with cells at 37°C and 32°C, respectively. The
dynamics of the “switch off” are faster if the temperature is higher,
Parameters of the mathematical model of the positive feedback loop
in mammalian cells
Parameter Description Reference value Estimated value
K 1 (nM) Michaelis-Menten
a 1 (nM min −1 )
v 1 (nM min −1 )
Basal activity CMV-TET promoter 0.085 0.085
v 2 (min −1 )
d 1 (min −1 )
General translation rate 0.02 0.02
d 2 (min −1 ) Degradation rate
d 3 (min −1 ) Degradation rate
– 0.0020 (Dataset 1)
0.0014 (Dataset 2)
h 1 Hill coeffi cient of the
q (nM) Affi nity Doxycycline
K f (min −1 ) Folding rate d2EYFP 0.015 0.015
76D. di Bernardo et al.
Fig. 6. Dynamical behavior of the positive feedback loop in mammalian cells. In this fi gure model predictions of the dynamics
characterizing the circuit for varying concentrations of Doxycycline (1 m g/mL for ( a ) and ( c ), 100 ng/mL for ( b ) and 10 m g/mL
for ( d )) have been reported. The sample time is equal to 15 min. The cells were treated with the antibiotic at t = 0 (min).
Model predictions are reported in with thick line while experimental results are represented in blue . In ( a ), the cells were
kept at 37°C and observed up to 37 h. In ( b – d ) the cells were kept at 32°C and observed up to 61 h. In ( e ) we report the
comparison of the dynamics of the circuit obtained by varying the strength of the positive feedback loop. C line = model
simulation of the system including the positive feedback loop using the inferred parameter values (Table 2 ). B line = model
simulation of the system reducing the strength of the positive feedback loop. A line = model simulation of the system
removing the positive feedback loop.
774 Predicting Synthetic Gene Networks
as the cells metabolism is faster ( 13 ) . In the model, we captured this
behavior by changing the degradation rate of the reporter protein
(parameter d 3 in Table 2 ).
Another parameter we estimated, the maximal transcription
rate of the CMV-TET promoter (Table 2 ), was quite different form
the reported value in literature. The physical meaning is that the
strength of the positive-feedback loop is much stronger than previ-
ously estimated, at least in the cell-line we used in this experiment
(HEK 293). The presence of the positive feedback loop is the key
to understand the dynamics of the network, because it makes it
harder for the d2EYFP to be down regulated by Doxycycline.
Indeed, in Fig. 6 e we analyzed how the presence of the positive
feedback loop affects the switch-off dynamics: decreasing its
strength (green line) or removing it (black line), the network is
switched off much faster. The model was further refi ned as
described in ( 19 ) .
In order to carry out experiments with S. cerevisiae or mammalian
cell lines, appropriate microfl uidic device designs must be con-
ceived. The design of such devices should be carefully considered
on the basis of the morphological characteristics of the cells used:
10 m m in height channels, for example, may be suited for S. cerevi-
siae but may be not compatible with many mammalian cell types.
For this reason the experimenter is highly recommended to check
both morphological and physiological properties of the cells to be
employed before moving to the device design step. The setup of
experiments for S. cerevisiae and mammalian cell lines is quite simi-
lar and differs only in the preparation of cells and temperature/
CO 2 conditioning settings. Depending on the motility level of cells
you may be interested in lowering it: two effective ways to accom-
plish this is either (a) by lowering the temperature in the microin-
cubator at 32°C or (b) using drugs such as Cytochalasin D,
ladrunculin, Nocodazole, etc. Advantages and disadvantages of
both the types of approaches should be carefully evaluated before
choosing for one or the other since temperature-based strategy
may interfere with cells’ metabolism (thus adding uncertainty to
the quantifi cation results) while drugs-based strategies (mainly
acting on cytoskeleton and microtubules development) may be
characterized by unwanted side effects.
1. Day 0.
(a) Fill the 50-mL Falcon tube with 5 mL Synthetic Complete
(b) Add 500 m L Galactose 20% (w/v).
(c) Add yeast cells from the solid medium culture.
(d) Place the Falcon tube in the shaker at 30°C/150–200 rpm.
on Microfl uidics
3.7.1. Yeast culture
78D. di Bernardo et al.
2. Day 1.
(a) Measure O.D. 600 and dilute in 5 mL so as to obtain a fi nal
O.D. 600 of 0.5.
(b) Grow for 4–6 h.
This protocol assumes you designed and built a microfl uidic device
with n ports, w of which are inlets.
This section deals with the specifi c aspects of the computing system
behind the control law. The implementation presented uses
MATLAB Simulink but other software packages like National
Instruments LabView can be used as well. Here, we provide details
of both the steps needed to acquire and process data from the
1. Program your microscopy control scheme to acquire bright
fi eld as well as fl uorescent fi elds at regular time intervals (5 min
with exposure times of 200–400 ms have been found to work
fi ne for yeasts).
2. Set up your algorithm so that once images are acquired the
image processing routine is launched on them:
(a) Your image processing algorithm should use bright fi eld
image to locate cells.
(b) Once cells are located, a binary mask image is used to
compute a mean fl uorescence of cells.
(c) A mean intensity quantifi cation of the red fl uorescence fi eld
can be used to match the galactose/raffi nose time profi le
obtained experimentally with the desired one and this infor-
mation can be used to understand whether clogging or other
detrimental phenomena are taking place in the device.
1. To obtain a lentiviral vector by recombination of three plasmids,
we found that the optimal concentration of each plasmid is:
pENTR5 ¢ -TOPO promoter 10 ng, pENTR gene 10 ng, p Lenti/
R4R2/V5-DEST 120 ng, according to the fmoles required for
2. We suggest to dephosphorylate the p MAtTA-IRES-d2EYFP
linearized with the AseI restriction enzyme before performing
the recombination reaction in pDONR221 to avoid the
possibility of forming colonies in the negative control because
the plasmid is prone to becoming circular again. We, thus,
recommend to add 5 U of the Alkaline Phosphatase, Calf
Intestinal (CIP) (New England Biolabs) to the vector linearized
3.8. Setting Up the
794 Predicting Synthetic Gene Networks
and to leave the reaction at 55°C for 45 min, and then to purify
the fragment with the QIA quick PCR purifi cation kit
3. It is really necessary to control that all the concentrations of
the vectors you fi nd in the ViraPower Promotrless Lentiviral
Gateway Expression System are correct.
4. The Doxycycline and Blasticidine working solution can be
stored at 4°C up to 1 month.
5. As reported, when you are modeling a synthetic network, to
decide what can be simplifi ed, and what needs to be modeled
in more details, it is necessary to go through iterative refi ne-
ment steps both in the model and in the experimental dataset.
6. When the number of unknown parameters is high, as in yeast
network case, and the physical feasible range for them is large,
it is not easy to identify them. Even though stochastic identifi -
cation techniques provide a fl exible approach to nonlinear
optimization, it is good to proceed with hand refi nement.
Thus, we proceeded in this way:
(a) Multiple identifi cation of each parameter running GA
more than one time letting the parameter vary in a physical
feasible range. In this way, we obtained different values for
each parameter (even with the same setting, since the algo-
rithm is stochastic) that, however, remain quite close to
each other. Thus, this step allows to signifi cantly restrict-
ing the search range.
(b) Hand refi nement of the parameters in the narrow range
obtained in step (a).
7. Since each experiment is costly and time consuming, the best
option in the iterative analysis of a biological network is, at
each step, to only perform those experiments that the mathe-
matical modeling deems indispensable. In the case of the yeast
network, we could have performed the promoter strength
experiments from the beginning, since the Hill functions were
almost unchanged during the model refi nement, with the
exception of the GAL10 and HO promoters modeling.
However, the need of performing such experiments arose after
the identifi cation of Model C, since we did not trust the identi-
fi ed Hill functions parameters.
8. The quality of the microscopy section is of primary importance
in experiments involving S. cerevisiae expressing fl uorescent
reporters due to the thickness of the wall protecting yeasts
from the external environment. Yeast experiments of this type
are best carried out at magnifi cations like 40× and 60×. High
numerical aperture objectives should be preferred to make the
image processing easier. High-quality mercury lamps are
80 D. di Bernardo et al.
needed to obtain good fl uorescent fi eld images; CCD cameras
can lower the Signal to Noise Rate (SNR) associated to the
images when compared to conventional cameras ( 20 ) .
9. Attention should be paid to the air-bubbles washing steps:
PDMS is porous and the user can take advantage of this prop-
erty to get rid of the air-bubbles just by injecting ddH 2 O at
high pressure (just by pressing the plunger of the syringes used
to fl ush the device). Once there are no more bubbles in the
device, the experimenter should pay attention to the discon-
nection of the ddH 2 O syringes: the needles should be gently
disconnected and pressing the plunger for 10 s before unplugging
the related needle can help in generating the pressure needed
to prevent air from reentering the port.
10. All media, sugars and dyes to be used should be fi ltered with
0.22 m m fi lters so as to avoid the presence of small particles
that can accelerate the clogging of the device or, alternatively,
mislead the image segmentation algorithm. The preparation of
the media in a fume hood does provide a strong advantage in
11. When sugars such as galactose are used as inducers, supple-
menting with supporting metabolic sources should be consid-
ered as an option. Growth of yeast cells in media featuring
galactose can be quite slow: having cells slowly moving in a
fi eld can be a good advantage from the point of view of the
image processing algorithm, unfortunately the metabolic
implications of these conditions cannot always be predicted
with accuracy thus adding uncertainty to our experiments.
12. In the proposed protocol, we used a red fl uorescent dye,
namely, Sulforhodamine B, whose excitation and emission
spectra largely overlap with red fl uorescent protein tag (e.g.,
Cherry etc.). If the protein to be controlled is tagged with
such a fl uorescent protein it may help using a different dye
(e.g., Atto 655) to track the inducer compound concentration.
Nevertheless, it should be noted that choosing such option
may imply buying new fi lters (Atto 655, for example, emits in
the far red and requires a specifi c fi lter like the Cy5.5-A
(Semrock, PN FF685-Di01-25 × 36)).
13. The image processing algorithm is one of the key points of the
whole setup: several free software packages have been pro-
posed in the fi eld of cellular microscopy being CellProfi ler
( 21 ) , Cell-ID ( 22 ) , CellTracker ( 23 ) . Nevertheless, it should
be noted that achieving a good quality of segmentation often
may require a long session of parameter fi ne-tuning and, more
often, fl exibility of these packages is an added value and not a
core characteristic. Experienced users should consider devel-
oping their own code on the basis of their specifi c application
and experimental context: testing the dependence of the
814 Predicting Synthetic Gene Networks Download full-text
segmentation results on the experimental conditions may help
in optimizing the code and make it more robust to varying
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