ASyntheticGenetic EdgeDetectionProgram
Jeffrey J. Tabor,1 Howard M. Salis,1 Zachary Booth Simpson,2,3 Aaron A. Chevalier,2,3 Anselm Levskaya,1
Edward M. Marcotte,2,3,4 Christopher A. Voigt,1,* and Andrew D. Ellington2,3,4
1Department of Pharmaceutical Chemistry, School of Pharmacy, University of California San Francisco, San Francisco, CA 94158, USA
2Center for Systems and Synthetic Biology
3Institute for Cellular and Molecular Biology
4Department of Chemistry and Biochemistry
University of Texas, Austin, TX 78712, USA
*Correspondence: cavoigt@picasso.ucsf.edu
DOI 10.1016/j.cell.2009.04.048
SUMMARY
Edge detection is a signal processing algorithm
common in artificial intelligence and image recogni-
tion programs. We have constructed a genetically
encoded edge detection algorithm that programs an
isogenic community of E. coli to sense an image of
light, communicate to identify the light-dark edges,
and visually present the result of the computation.
The algorithm is implemented using multiple genetic
circuits. An engineered light sensor enables cells
to distinguish between light and dark regions. In the
dark, cells produce a diffusible chemical signal
that diffuses into light regions. Genetic logic gates
are used so that only cells that sense light and the
diffusible signal produce a positive output. A mathe-
matical model constructed from first principles and
parameterized with experimental measurements of
the component circuits predicts the performance
of the complete program. Quantitatively accurate
models will facilitate the engineering of more com-
plex biological behaviors and inform bottom-up
studies of natural genetic regulatory networks.
INTRODUCTION
Living cells can be programmed with genetic parts, such as
promoters, transcription factors and metabolic genes (Andria-
nantoandro et al., 2006; Benner and Sismour, 2005; Canton
et al., 2008; Endy, 2005; Haseltine andArnold, 2007). These parts
can be combined to construct genetic versions of electronic
circuits, including switches (Atkinson et al., 2003; Gardner et al.,
2000; Kramer and Fussenegger, 2005; Kramer et al., 2004), logic
(Anderson et al., 2007; Guet et al., 2002; Rackham and Chin,
2005), memory (Ajo-Franklin et al., 2007; Gardner et al., 2000;
Ham et al., 2006), pulse generators (Basu et al., 2004), and oscil-
lators (Atkinson et al., 2003; Elowitz and Leibler, 2000; Fung et al.,
2005; Stricker et al., 2008; Tigges et al., 2009). The current chal-
lenge is to assemblemultiple genetic circuits into larger programs
for the engineering of more sophisticated behaviors (Purnick and
Weiss, 2009).
The characterization of transfer functions or the quantitative
relationship between circuit input(s) and output(s) (Bintu et al.,
2005a; Tabor et al., 2009; Voigt, 2006; Weiss et al., 1999) will aid
the development of accurate mathematical models (Ajo-Franklin
et al., 2007; Guido et al., 2006) which will allow complex genetic
programs to be examined in silico prior to physical construction.
Predictive models for the design of genetic programs will drive
applications in biotechnology and aid bottom-up studies of
natural regulatory systems.
Edge detection is a well-studied computational problem used
to determine the boundaries of objects within an image (Suel
et al., 2000). This process reduces the information content in
a complex image and is used in applications ranging from retinal
preprocessing (Maturana and Frenk, 1963) to the analysis of
microarray data (Kim et al., 2001). For a digital black and white
image, a typical algorithm operates by scanning for a white pixel
and then comparing the intensity of that pixel to its eight neigh-
boring pixels. If any of the neighbors is black, the algorithm clas-
sifies those pixels as being part of an edge. The serial nature of
this search process results in a computation time that increases
linearly with the number of pixels in the image. We aimed to
implement a parallel edge detection algorithm wherein each
bacterium within a spatially distributed population functions as
an independent signal processor. In this design, each bacterium
(up to 109 individuals for a 90 mm Petri dish image) processes
a small amount of local information simultaneously, and the pop-
ulation cooperates to find the edges.
RESULTS
A Genetic Program for Edge Detection
The genetic edge detection algorithm programs a lawn of
bacteria to identify the light-dark boundaries within a projected
image of light (Figures 1A and 1B). To accomplish this, each
bacterium in the population executes the following pseudocode
(Figure 1C): IF NOT light, produce signal. IF signal AND NOT
(NOT light), produce pigment.
The ‘‘produce signal’’ and ‘‘produce pigment’’ functions make
the cell generate a diffusible communication signal and a black
pigment, respectively. The conversion of this pseudocode into
a molecular genetic system is shown in Figure 1D.
When cells sense that they are in the dark, they produce the
diffusible signal (Figure 1B). Cells that sense light do not make
the signal, but are allowed to respond to it. Thus only those cells
that are in the light but proximal to dark areas activate the output
which results in the enzymatic production of a black pigment.
The biological edge detection algorithm requires: (1) a dark
1272 Cell 137, 1272–1281, June 26, 2009 ª2009 Elsevier Inc.

sensor (NOT light), (2) cell-cell communication, and (3) X AND
(NOT Y) genetic logic. Each of these components is constructed
as an independent genetic circuit and the behavior is character-
ized. This data is used to parameterize a mathematical model
which in turn is used to analyze the complete system.
Construction and Characterization of Genetic Circuits
In an effort to make photographic bacteria, we previously con-
structed a dark sensor (Levskaya et al., 2005) based on genetic
parts from the blue-green algae Synechocystis (Yeh et al.,
1997). The sensor consists of a chimeric two-component system
and a two gene metabolic pathway to make the chromophore
phycocyanobilin (PCB) (Gambetta and Lagarias, 2001). To rewire
the two-component system, the osmosensing domain of the
E. coli protein EnvZwas replacedwith the photoreceptor domain
of the Synechocystis phytochrome Cph1. This programmed
phosphotransfer from EnvZ to OmpR and subsequent trans-
cription from the PompC to be repressed as a function of red light.
The sensor therefore functions as a genetic circuit with NOT
light logical behavior. When the dark sensor is connected to the
production of b-galactosidase, a plate of bacteria can print an
image of light as a pattern of black pigment (Figure 3A).
The transfer function, which characterizes how the output of
a circuit varies with input at steady-state, has been shown to be
a useful tool for connecting genetic circuits (Anderson et al.,
2007; Bintu et al., 2005a; Voigt, 2006; Weiss et al., 1999; Yoko-
bayashi et al., 2002).Here, the transfer function of thedark sensor
is determined in response to light in the 650 nm band (Figure 2A).
The dark sensor generatesmaximal transcriptional output at light
intensities between 0.000 and 0.002W/m2, reaches minimal
Figure 1. Bacterial Edge Detection
(A) Light is projected through a mask onto a large community (lawn) of bacteria grown on a Petri dish. The lawn computes the edges, or boundaries between light
and dark regions, and visually presents the output.
(B) To find the edges, bacteria in the dark produce a communication signal (green circles) that diffuses across the dark/light boundary. Bacteria in the dark cannot
respond to the communication signal. Only bacteria that are exposed to light and receive the signal become positive for the expression of a visible reporter gene.
The sum of this activity over the entire two-dimensional population is equivalent to the edges of the input image.
(C) (Top) A NOT light gate (lightning box + adjacent triangle) drives a cell-cell communication circuit (green X) and an inverter (red Y + adjacent triangle). These two
signals combine as inputs for a downstream AND gate (semi-circle) which drives the final output (Z). Because signal is inverted at Y, the gate driving Z can also be
described as an X AND (NOT Y) gate, and it is referred to as such throughout this work. (Bottom) Z is produced in only one of four possible combinations of X and Y
(presence of X, absence of Y).
(D) Conversion of the edge detection algorithm into a molecular genetic system. (Left) The light-sensitive protein Cph8 is a chimeric sensor kinase bearing the
photoreceptor domain of the Synechocystis phytochrome Cph1 and the kinase domain of E.coli EnvZ (Levskaya et al., 2005). Cph8 requires the covalently asso-
ciated chromophore phycocyanobilin (PCB, blue pentagons) which is produced from heme by the products of the two constitutively expressed genes ho1 and
pcyA (Gambetta and Lagarias, 2001). In the presence of red light, the kinase activity of Cph8 is inhibited, precluding the transfer of a phosphoryl group (light green
circle) to the response regulator OmpR (orange dumbbell) and subsequent transcription from the ompC promoter (PompC). The dark sensor therefore functions as
a NOT light transcriptional logic gate. (Center) luxI and cI are expressed polycistronically from the NOT light gate. LuxI is a biosynthetic enzyme from V.fischeri that
produces the cell-cell communication signal 3-oxohexanoyl-homoserine lactone (AHL). CI is the transcriptional repressor protein from phage l. AHL binds to the
constitutively expressed transcription factor LuxR to activate expression from the Plux-l promoter while CI dominantly represses it. Plux-l therefore functions as an
X AND (NOT Y) transcriptional logic gate. (Right) The output of Plux-l is lacZ, the product of which (b-galactosidase) cleaves a substrate in the media to produce
black pigment (Experimental Procedures). The edge detection algorithm is encoded as 10,020 basepairs of DNA, carried on three plasmid backbones (Exper-
imental Procedures).
Cell 137, 1272–1281, June 26, 2009 ª2009 Elsevier Inc. 1273

output at 0.04W/m2 and is repressed rapidly and continuously
asa functionof light (Figure 2A). The transfer functionhas the form
flight =
K
K + Lðbmax ÿ bminÞ+ bmin (1)
where bmax = 298 and bmin = 125 are the maximum andminimum
output values (in Miller units), L is the intensity of light (W/m2) and
the fit parameter is K = 0.0017 W/m2 (R2 = 0.75) (Experimental
Procedures).
The edge detection algorithm also requires that neighboring
bacteria communicate. It has previously been shown that E.coli
can be programmed to communicate using the quorum sensing
system from V. fischeri (Anderson et al., 2006; Balagadde et al.,
2008; Basu et al., 2005; Weiss and Knight, 2001; You et al.,
2004). We placed this communication system under the control
of the dark sensor (Figure 3B). In this circuit, dark activates tran-
scription of luxI, the product of which catalyzes the formation of
the membrane diffusible compound 3-oxohexanoyl-homoserine
lactone (AHL) (Engebrecht and Silverman, 1984). AHL binds to
the constitutively expressed transcription factor LuxR to acti-
vate expression of b-galactosidase. This produces a pattern of
b-galactosidase expression similar to the photographic bacteria,
but with an additional blurring component due to AHL diffusion
across the dark/light boundary (Figure 3B).
In addition to communication, the edge detection algorithm
requires that b-galactosidase be expressed only where AHL
AND light, or equivalently NOT (NOT light), are present. This
requires genetic circuits that encode theNOT andAND logic func-
tions to be combined with the NOT light circuit. Genetic logic can
be constructed by rewiring regulatory interactions (Anderson
et al., 2007; Cox et al., 2007; Guet et al., 2002; Mayo et al.,
2006;Weiss et al., 1999). TheNOT function can be achieved using
a genetic inverter, which has previously been shown to invert the
activity of an input promoter (Yokobayashi et al., 2002). We con-
structed an inverter using the cI gene fromphage l, the product of
which forms a dimeric transcriptional repressor that turns OFF the
output promoter when the input promoter is ON. By inserting the
inverter between the dark sensor input and b-galactosidase
output, a negative bacterial photograph can be generated where
black pigment is produced only in the light (Figure 3C).
The full logic function AHL ANDNOT (NOT light) is implemented
at the two-input promoter Plux-l, which is activated by AHL-bound
LuxR but dominantly repressed by CI. By adding a constitutively
expressed copy of the luxR gene to the inverter, the two-dimen-
sional transfer function of this promoter can be determined in
batch culture experiments by exogenously varying AHL and light
whilemeasuring b-galactosidaseactivity as the output. Transcrip-
tion fromPlux-l increases proportional to the concentration of AHL
between 2 nM and 200 nM. At a given AHL concentration, tran-
scription is repressed approximately 4-fold by maximal (dark) CI
levels as compared to those in saturating light (Figure 2B, left).
The experimental data is used to fit a two-dimensional transfer
function (Figure 2B, right) that uses the Shea-Ackers formalism
(Shea and Ackers, 1985) to model transcription factor binding to
Plux-l as a function of AHL (u1) and CI (u2) concentrations,
flogicðu1; u2Þ=
ðc0 + c1fLuxÞ
1+ c0 + c1fLux + c2fnCI + c1c2fLuxfnCI
(2)
where fLux is the concentration of LuxR dimers bound to AHL
(Urbanowski et al., 2004) and fCI is the concentration of dimeric
CI (Koblan and Ackers, 1991). The parameters c0 to c2 are deter-
mined by fitting the output of flogic to the transcription measure-
ments (c0 = 0.04, c1 = 0.05, c2 = 0.011, R2 = 0.81) (Experimental
Procedures) and n is 1.5. Taken together, the data in Figure 2
demonstrate that the dark sensor and the X AND (NOT Y) logic
circuit function as needed for use in the edge detection algo-
rithm. Moreover the transfer functions of the two circuits are
Figure 2. Transfer Functions of the Dark Sensor and X AND (NOT Y) Logic Gate
(A) The transfer function of the dark sensor is determined in batch culture (Experimental Procedures and Figure S4) and fit to a sigmoidal function (Equation 1). The
error bars indicate ±1 standard deviation. (B) (Left) The transfer function of the X AND (NOT Y) logic gate as determined in batch culture experiments. AHL (X) was
added exogenously to the growth media while CI (Y) levels were controlled by varying the intensity of light (Experimental Procedures and Figure S5). The data
shown are single replicates of 5 assays taken over 5 separate days where the concentration of CI was altered in each assay. (Right) Mathematical Model. The
output of flogic (Equation 2) as a function of AHL andCI. b-galactosidase output levels (Z) for the experiment and themodel are normalized by dividing by the output
value in the absence of CI with maximum exogenous AHL (Experimental Procedures).
1274 Cell 137, 1272–1281, June 26, 2009 ª2009 Elsevier Inc.

properly matched; transcription from the X AND (NOT Y) gate
can be controlled by AHL and CI over the output ranges gener-
ated by the dark sensor.
Assembling Circuits into the Full Program
Figure 3 shows the stepwise assembly of the edge detection
algorithm from the component genetic circuits. When a lawn of
bacteria programmed with the edge detection algorithm is
exposed to an image of light, the community prints the dark-light
edges (Figure 3D), with an average edge width of 6.0 ± 1.8 mm
(n = 3) (Figure 5A). Figure 4 demonstrates that the bacterial lawns
can accurately solve the edges of a circle, a square, and the
silhouette of a man.
Reaction-Diffusion Model
A model of the complete edge detector system is constructed
based on the individually measured dark sensor and logic trans-
fer functions flight and flogic (Experimental Procedures). Themodel
quantifies the dynamics of light-dependent production of AHL
andCI, AHL diffusion, production of the b-galactosidase reporter
and degradation of all products. Assuming that AHL diffusion is
the slowest process, the system is described by the equations,
vu1
vt =DV
2u1 + k1flight ÿ k2u1 (3)
u2 = k3flight (4)
u3 = k4flogicðu1; u2Þ (5)
where u1 is the AHL concentration on the plate (nM), u2 is the
concentration of CI dimers (nM), and u3 is the concentration
of b-galactosidase in Miller Units. The diffusivity and half-life
of AHL are obtained from previously published values (D =
1.67x10ÿ7 cm2/sec, k2 = 0.012 hr-1) (Basu et al., 2005; Flagan
et al., 2003). The production rate of AHL is a function of the
density of the bacteria on the plate and is obtained by fitting to
the edge profile (k1 = 0.03 nM/hr). The maximum CI and b-galac-
tosidase concentrations are determined by fitting the experi-
mental data to the individual transfer functions (k3 = 0.8 nM/
Miller, k4 = 289 Miller) (Experimental Procedures). Because the
system is an agarose plate, the reaction-diffusion model is
defined on polar coordinates with a no-flux boundary condition
on the outer border.
The model accurately describes the pattern of b-galactosi-
dase on a plate of bacteria expressing the edge detector and
each of the sub-circuits for each of the light patterns shown
in Figures 3 and 4. The quantitative accuracy of the model is
evaluated in Figure 5. Figure 5A shows a one-dimensional anal-
ysis of the circle pattern where the in silico and in vivo edge
intensity profiles are compared as a function of radial distance
from the center. For complex images, the edge intensity is
greater at acute angles and along convex arcs than flat edges.
In these areas there are more cells producing AHL per unit
area. This increases the local AHL concentration and conse-
quently the b-galactosidase output in adjacent illuminated
areas. The relationship between edge intensity and the angle
of line intersection is also accurately captured by the model
(Figure 5B).
Figure 3. Construction of the EdgeDetector
from Individual Genetic Circuits
Various circuits are constructed and the effect on
image processing is assayed (left) and compared
to the mathematical model (right). The details of
the genetic circuits and simulations are presented
in the Experimental Procedures and Figure S2.
(A) The bacterial photography circuit, where the
b-galactosidase output is expressed directly
under the control of the light-sensitive PompC1157
promoter. This produces a positive print of the
projected image.
(B) Cell-cell communication components are
added by placing luxI under the control of the
dark sensor and expressing luxR constitutively.
This produces a positive image with an additional
blurring component due to AHL diffusion.
(C) A genetic inverter is inserted between the dark
sensor input and the b-galactosidase output. This
produces an inverted (negative) print of the image,
where the light regions are printed as dark and
vice-versa.
(D) A population of cells programmed with the
complete edge detector system produces black
pigment only at the boundary between light and
dark regions. The model solutions are reported in
Miller units (color bars, right).
Cell 137, 1272–1281, June 26, 2009 ª2009 Elsevier Inc. 1275

communication greatly reduced the information-processing
requirement for each member of the population while simple
genetic logic allowed the proper integration of local signals for
the formation of the final pattern.
Edge Detection is used for the identification of objects in
a wide variety of in silico image processing applications (Suel
et al., 2000) and has also been shown to be a natural function
of the retina (Maturana and Frenk, 1963). In silico edge detection
algorithms address each pixel of an image in series, resulting in
a computation time that increases linearly with the number of
pixels. In the bacterial edge detector the computation is
massively parallel, resulting in a computation time that is inde-
pendent of image size. This strategy is also an example of
‘‘Amorphous Computing’’ (Abelson et al., 2000) whereby a
computation is performed as the emergent result of many
spatially distributed processors working together locally without
the need for global coordination. The applications of biological
amorphous computers are still largely unexplored but are
intriguing in light of the astounding feats of self-organization
and information processing seen in natural pattern forming and
neural network systems.
Several other efforts have leveraged cell-cell communication
to program coordinated multicellular behaviors. These include a
genetically-encoded turbidostat (You et al., 2004), one (Kobaya-
shi et al., 2004) and two (Brenner et al., 2007) cell density-depen-
dent transcription regulators, a transcriptional pulse generator
(Basu et al., 2004), synthetic ecosystems (Balagadde et al.,
2008; Weber et al., 2007) and a pattern forming system (Basu
et al., 2005). In the latter case, twogenetically distinct populations
of bacteria (AHL senders and receivers) weremanually overlayed
in different configurations in order to generate different patterns.
By contrast, the edge detector is implemented within an isogenic
cell population that forms patterns in response to an external
input with no requirements for cell placement.
Synthetic systems such as these could be used as early in vivo
models for studying the ‘design principles’ that govern natural
processes. Their simplicity and tractabilitymakes themamenable
to rigorousmathematical analysis,whichcanbeused togenerate
rapidly testable hypotheses regarding the contribution of specific
parameters to overall function. Because regulatory motifs recur
ubiquitously in biology, the synthetic systems can then serve as
working models for their natural counterparts (Sprinzak and Elo-
witz, 2005). The connection between primary DNA sequence and
phenotype then closes the design cycle, expediting the engi-
neering of novel biological behaviors.
The construction of very large fragments of DNA (Cello et al.,
2002; Chan et al., 2005; Endy, 2008; Gibson et al., 2008a; Gibson
et al., 2008b; Smith et al., 2003; Tumpey et al., 2005) is no longer
a limitation in the engineering of biological systems. Predicting
the behavior of complex genetic programs de novo is now the
limiting step in the programming of cellular behavior. Thorough
characterization of the performance of simple genetic parts and
their resulting circuits will allow the development of predictive
mathematical tools which will be required to program cells and
cellular communities for functions which approach the sophisti-
cation of natural systems. This, in turn, will enable rigorous
bottom-up testing of structure-function relationships in natural
genetic systems.
EXPERIMENTAL PROCEDURES
Strain and Media
The strain for all experiments in this study is E. coli JW3367 (E. coli K12W3110,
envZ-lacZ- NCBI-GI: 89110606) from which the Kanamycin resistance marker
is removed (termed JW3367c). Transformations are plated on LB agar supple-
mented with 50mg/mL Kanamycin, 34mg/mL Chloramphenicol and 50mg/mL
Ampicillin as necessary. The strains are maintained in LB + 0.1M HEPES
pH = 8.0 supplemented with the antibiotics. Glycerol stocks of the strains
are maintained by adding 300 mL 60% glycerol (sterile) to 700mL actively
growing culture (log phase) and freezing at ÿ80C.
Edge Detection Plasmids
E. coli JW3367c is transformed with the light sensing plasmids pPLPCB (p15a
KanR) (Gambetta and Lagarias, 2001), pCph8 (ColE1, CmR) (Levskaya et al.,
2005) and a third plasmid carrying the circuit. All of the circuit plasmids are
based on the pSB4A3 BioBrick vector backbone (Shetty et al., 2008), which
contains the pSC101* origin of replication and AmpR. The pSC101* origin is
carried at 2-3 copies per cell (Lutz and Bujard, 1997). The edge detector
plasmid, pEDL3, is constructed from a series of DNA parts many of which
are Biobricks (Knight, 2002) (see the Supplemental Data available with this
article online). Other functional DNA elements used in the construction of the
edge detector are the weak ribosome binding site RBS3(Weiss, 2001) and
the ORF of the lacZ gene. The lacZ ORF is amplified from the plasmid
pEXPlacZ (Invitrogen) using primers that encode the Biobricks prefix and suffix
sequences, which carry the restriction sites EcoRI, XbaI (forward) and SpeI
and PstI (reverse) respectively. This allows the lacZ gene to be cloned down-
stream of J13023 in its host plasmid via a suffix operation(Knight, 2002) using
XbaI and PstI.
Photography, Inverter, and Communication Circuit Plasmids
The plasmids the carry the photography, inverter and cell-cell communication
circuits are pJT108, pJT106 and pJT105, respectively. The plasmids pJT105
and pJT106 are constructed by deleting single genes from pEDL3 using seam-
less inverse PCR and ligation with the Phusion Site Directed Mutagenesis Kit
(Finnzymes, Woburn, MA) according to manufacturer’s instructions. Plasmid
pJT108 is constructed by amplifying the PompC1157 genomic region of E. coli
RU1012(Utsumi et al., 1989) with overhanging homology regions to pJT103
and seamlessly replacing R0082 via the MEGAWHOP method (Miyazaki,
2003). The PompC1157 promoter (pJT108) is used for the bacterial photographs
because when read out by b-galactosidase directly, it produces smoother,
higher contrast images than the shorter ompC promoter BBa_R0082.
Miller Assays
Miller Assays are conducted in 700 mL total volume with the Yeast b-Galacto-
sidase Assay Kit (Pierce, Cat# 75768) in sterile, clear 1.7 mL microcentrifuge
tubes at 28C in ambient light according to the manufacturer’s instructions.
The reactions are quenched after visible yellow color develops and the
OD420 measurements are taken in VWR disposable cuvettes (VWR Cat#
97000-586) on a Cary 50 Bio spectrophotometer. The equation to calculate
Miller Units is (1000*OD420)/(t*V*OD600), where t is the reaction time of the
assay in minutes, and V is the volume of cell culture added to the reaction.
The Light Camera
A ‘‘Light Camera’’ (Incubator-Projector), which enables the projection of an
image onto a plate of growing bacteria, is constructed as described before
(Levskaya et al., 2005). A Kodak Ektagraphic III AMT projector equipped with
an 82 V, 300 W Philips FocusLine quartz bulb is used as the light source. The
broad wavelength light is filtered through a 650 nm bandpass filter (Edmund
Optics catalog #43–189), stenciled through a 34x24 mm slide printed with
a black and white image at 2032 dpi (mask), and focused through a lens.
The images projected onto the slabs have power characteristics of 0.08 to
0.15 W/m2 in the 620–680 nm band as determined by a EPP2000C Concave
Grating spectrometer (Stellarnet, Oldsmar, FL). Dark areas of the images typi-
cally have 0.0000 to 0.0003W/m2 light over the same range. Bleedthrough light
outside this band is negligible.
Cell 137, 1272–1281, June 26, 2009 ª2009 Elsevier Inc. 1277

The constants c0 to c2 reflect the apparent in vivo Gibbs free energies of
binding for each state and are determined by minimizing the differences
between the output of flogic and the two-input transfer function over the 25
different conditions of 3OC6HSL concentration and light intensity. The best
fit values are c0 = 0.04, c1 = 0.05 and c2 = 0.011 (R2 = 0.81).
Reaction-Diffusion Model
Given a light mask, the reaction-diffusion model calculates the time- and posi-
tion-dependent expression level of the b-galactosidase (b-gal) output gene.
The model consists of (1) a partial differential equation describing 3OC6HSL
production, degradation, and diffusion and (2) two algebraic equations
describing the steady-state concentrations of CI and b-gal in response to
3OC6HSL and light. In dimensionless form, these equations are
vu1
vt =
1
r
vu1
vr +
v2u1
vr2 +
1
r2
v2u1
vq2
+ k1flight ÿ k2u1 (8)
u2 = k3flight (9)
u3 = k4flogicðu1; u2Þ (10)
Where u1, u2, and u3 represent the concentrations of 3OC6HSL, CI, and b-gal
at a position on the plate whose polar coordinates are given by (r, q). The flight
and flogic functions quantify the transcription rates of the light-dependent
ompC promoter and the CI-repressed, LuxR::3OC6HSL-activated lux-l
promoter, respectively.
The constants k1 and k2 quantify the maximum production rate and the
degradation rate of 3OC6HSL, respectively. The production rate of 3OC6HSL
is estimated so that the maximum concentration on the plate is 2.5 nM while
the degradation rate of 3OC6HSL is slow; it has a half-life of about 2.5 days
at pH 6.6 (Flagan et al., 2003). The conversion factor between the ompC tran-
scription rate, characterized by flight, and CI concentration is k3 = 0.8 nM/Miller.
The constant k4 is the maximum b-gal concentration, which is 289 Miller units.
This value was determined in batch culture experiments as described above at
500 nM (maximum) exogenous AHL in the absence of any CI protein (plasmid
pJT105).
When solving these equations, the space and time coordinates are de-
dimensionalized so that r* = r / R and t* = tD/R2 where r is the radial position
from the center of the plate, R is the radius of the plate, t is time and D =
1.67x10ÿ7 cm2/sec is the diffusivity of 3OC6HSL (Basu et al., 2005). The
system is an agarose plate with radius R = 4.25 cm (3.55 mm operating
depth), homogeneously filled with stationary bacteria. Because the bacterial
photographs are crisp in our system we assume that there is no appreciable
bacterial movement in the agarose plates. There is a no-flux boundary
condition (Neumann type) at r* = 1 and a uniformly zero initial 3OC6HSL
concentration.
The differential equations in Equations (8–10) are solved using the finite
difference method. We substitute 2nd order central differences for all spatial
derivatives to create a sparse system of ordinary differential equations. The
ordinary differential equations are solved using theMatlab (Mathworks, Natick,
MA) ode23 s stiff numerical integrator with a final time of 24 hr (t* = 0.0027). A
sufficient number of radial and axial elements are used to accurately resolve
each light mask. The solution yields the dynamics of edge formation in
response to a given light mask.
Quantifying the Effect of Angle of Intersection on Edge Intensity
The effect of changing the angle of intersection between light and dark
boundaries on the edge intensity is examined, comparing the model predic-
tions to the experimentally observed behaviors. We create a series of unit
circle in silico masks where q degrees of the circle are in the light with 360-q
degrees in the dark and where q is varied from 50 to 345 degrees. For each
mask, the solution of the reaction-diffusionmodel is computed, which predicts
the maximum edge intensity. The maximum edge intensity is the b-galactosi-
dase concentration at the edge location. The model predictions compare
favorably with the experimentally observed edge intensities of the asymmet-
rical silhouette mask at the selected angle intersections (Figure 5B). The image
analysis procedure to obtain the experimental data is described above.
Calculating the Radial b-gal Profile
The radial edge intensity profile of the circle images are compared to the in
silico radial b-galactosidase profile from the model solution (Figure 5A). We
compute the in silico radial b-galactosidase profile by first inputting the circle
light mask into the model and determining the solution. Then, the b-galactosi-
dase concentration in terms of Miller units (u3) is outputted along the radial
coordinate (r = 0 to 1.8 cm) and divided by the value of u3 at r = 0 to obtain
the normalized intensity in Figure 5A.
SUPPLEMENTAL DATA
Supplemental Data include Supplemental Experimental Procedures, five
figures, and Supplemental References and can be found with this article online
at http://www.cell.com/supplemental/S0092-8674(09)00509-1.
ACKNOWLEDGMENTS
We thank E.A. Davidson, L.A. Lavery, M. Levy, K. McGary, and A. Scouras for
helpful discussions; A. Nishimura for the JW3367E. coli strain; and L.A. Lavery,
C. Conboy, and D. Endy for assistance with Biobrick construction. This work
was supported by the National Science Foundation (SynBERC), NSF-
BES0547637, NIH EY016546, NIH AI067699, NIH R01GM077040, Office of
Naval Research, and the Pew and Packard Foundations. J.J.T. is supported
by a Kirschstein National Research Service Award.
Received: January 30, 2009
Revised: March 9, 2009
Accepted: April 13, 2009
Published: June 25, 2009
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