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Edge detection is a signal processing algorithm common in artificial intelligence and image recognition 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 mathematical 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 complex biological behaviors and inform bottom-up studies of natural genetic regulatory networks.
Living cells can be programmed with genetic parts, such as promoters, transcription factors and metabolic genes (Andrianantoandro et al., 2006; Benner and Sismour, 2005; Canton et al., 2008; Endy, 2005; Haseltine and Arnold, 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 oscillators (Atkinson et al., 2003; Elowitz and Leibler, 2000; Fung et al., 2005; Stricker et al., 2008; Tigges et al., 2009). The current challenge is to assemble multiple genetic circuits into larger programs for the engineering of more sophisticated behaviors (P. Purnick and R. Weiss, in preparation).
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 neighboring pixels. If any of the neighbors is black, the algorithm classifies 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 90mm Petri dish image) processes a small amount of local information simultaneously, and the population cooperates to find the edges.
The genetic edge detection algorithm programs a lawn of bacteria to identify the light-dark boundaries within a projected image of light (Figure 1a,b). To accomplish this, each bacterium in the population executes the following pseudocode (Figure 1c):
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 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 characterized. This data is used to parameterize a mathematical model which in turn is used to analyze the complete system.
In an effort to make photographic bacteria, we previously constructed 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 EnvZ was replaced with the photoreceptor domain of the Synechocystis phytochrome Cph1. This programmed phosphotransfer from EnvZ to OmpR and subsequent transcription 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 β-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; Yokobayashi et al., 2002). Here, the transfer function of the dark sensor is determined in response to light in the 650nm band (Figure 2a). The dark sensor generates maximal transcriptional output at light intensities between 0.000 and 0.002W/m2, reaches minimal output at ~0.04W/m2 and is repressed rapidly and continuously as a function of light (Figure 2a). The transfer function has the form
where βmax = 298 and βmin = 125 are the maximum and minimum 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 transcription 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 activate expression of β-galactosidase. This produces a pattern of β-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 β-galactosidase be expressed only where AHL AND light, or equivalently NOT (NOT light), are present. This requires genetic circuits that encode the NOT and AND logic functions 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). The NOT 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 constructed an inverter using the cI gene from phage λ, 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 β-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 AND (NOT (NOT light))) is implemented at the two-input promoter Plux-λ, 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-dimensional transfer function of this promoter can be determined in batch culture experiments by exogenously varying AHL and light while measuring β-galactosidase activity as the output. Transcription from Plux-λ increases proportional to the concentration of AHL between 2nM and 200nM. At a given AHL concentration, transcription 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-λ as a function of AHL (u1) and CI (u2) concentrations,
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 determined by fitting the output of flogic to the transcription measurements (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 algorithm. Moreover the transfer functions of the two circuits are properly matched; transcription from the X AND (NOT Y) gate can be controlled by AHL and CI over the output ranges generated by the dark sensor.
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 ± 1.8mm (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.
A model of the complete edge detector system is constructed based on the individually measured dark sensor and logic transfer functions flight and flogic (Experimental Procedures). The model quantifies the dynamics of light-dependent production of AHL and CI, AHL diffusion, production of the β-galactosidase reporter and degradation of all products. Assuming that AHL diffusion is the slowest process, the system is described by the equations,
where u1 is the AHL concentration on the plate (nM), u2 is the concentration of CI dimers (nM), and u3 is the concentration of β-galactosidase in Miller Units. The diffusivity and half-life of AHL are obtained from previously published values (D = 1.67x10-7 cm2/sec, κ = 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 (κ1 = 0.03 nM/hr). The maximum CI and β-galactosidase concentrations are determined by fitting the experimental data to the individual transfer functions (κ3 = 0.8 nM/Miller, κ4 = 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 β-galactosidase on a plate of bacteria expressing the edge detector and each of the sub-circuits for each of the light patterns shown in Figures Figures33 and and4.4. The quantitative accuracy of the model is evaluated in Figure 5. Figure 5a shows a one-dimensional analysis 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 consequently the β-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).
Individual genetic circuits that mimic basic electronic functions have previously been constructed (Anderson et al., 2007; Atkinson et al., 2003; Basu et al., 2004; Elowitz and Leibler, 2000; Gardner et al., 2000; Guet et al., 2002; Ham et al., 2006; Yokobayashi et al., 2002). The next step is to understand how to combine these functions to create more complex genetic programs (P. Purnick and R. Weiss, in preparation). This requires well-characterized parts and circuits that perform reliably when linked to others, without unforeseen higher-order effects (Kim and Tidor, 2003). Here, we have demonstrated that a number of well-characterized genetic circuits can be reliably combined to create a larger program. Different circuit combinations produce expected behaviors that can be predicted by a mathematical model parameterized with data from the characterization of the individual circuits. This will not be true for all circuit combinations, and understanding the origins of higher-order effects and how they can be accounted for in the design process is an outstanding challenge (Arkin and Fletcher, 2006). As the programs become larger, it will also become increasingly important to include information on how the engineered circuits impose burdens upon the host cells, including transcriptional, translational, metabolic, and energy resources (Canton et al., 2008; Tabor et al., 2008). Detailed studies of the interactions between synthetic circuits and host systems (Guido et al., 2007) may lead to new conclusions regarding regulatory and metabolic cross-talk and the ability of cells to tolerate or adapt to increased genetic load.
Logic and cell-cell communication form the core of regulatory networks that drive fundamental biological processes such as pattern formation (Kondo and Asai, 1995; Meinhardt and Gierer, 2000; Sick et al., 2006) and development (Freeman, 2000; Materna and Davidson, 2007). The combination of these two functions allows each cell in a population to respond appropriately to local signals without the need for information regarding its position within the global system. Here, we have programmed a population of bacteria to form a pattern corresponding to the edges within a projected image of light. As in natural systems, 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 independent 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 (Kobayashi et al., 2004) and two (Brenner et al., 2007) cell density-dependent 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, two genetically distinct populations of bacteria (AHL senders and receivers) were manually 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 tractability makes them amenable to rigorous mathematical analysis, which can be used to generate 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 Elowitz, 2005). The connection between primary DNA sequence and phenotype then closes the design cycle, expediting the engineering 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 sophistication of natural systems. This, in turn, will enable rigorous bottom-up testing of structure-function relationships in natural genetic systems.
The strain for all experiments in this study is E.coli JW3367 (E.coli K12 W3110, envZ-lacZ- NCBI-GI: 89110606) from which the Kanamycin resistance marker is removed (termed JW3367c). Transformations are plated on LB agar supplemented with 50μg/mL Kanamycin, 34μg/mL Chloramphenicol and 50μg/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 300μL 60% glycerol (sterile) to 700μL actively growing culture (log phase) and freezing at −80°C.
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) (Supplemental Information). 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 downstream of J13023 in its host plasmid via a suffix operation(Knight, 2002) using XbaI and PstI.
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 seamless 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 β-galactosidase directly, it produces smoother, higher contrast images than the shorter ompC promoter BBa_R0082.
Miller Assays are conducted in 700μL total volume with the Yeast ß-Galactosidase Assay Kit (Pierce, Cat# 75768) in sterile, clear 1.7mL microcentrifuge tubes at 28°C 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.
A “Light Camera”, 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 82V, 300W Philips FocusLine quartz bulb is used as the light source. The broad wavelength light is filtered through a 650nm bandpass filter (Edmund Optics catalogue #43-189), stenciled through a 34×24mm 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 0.15W/m2 in the 620-680nm band as determined by a EPP2000C Concave Grating spectrometer (Stellarnet, Oldsmar, FL). Dark areas of the images typically have 0.0000 to 0.0003W/m2 light over the same range. Bleedthrough light outside this band is negligible.
The output of the edge detector and other circuits are assayed using plates that are developed in the light camera. Detailed instructions for these assays are provided in the Supplemental Information. The agar slabs containing the bacteria are grown and exposed to light for 14 hours (Edge Detector, Inverter, Communication). The projector is then turned off and the slabs are allowed to ‘develop’ at 36°C in the dark for 10 hours. For photography the slabs are exposed to the image for 45 hours. The slabs are placed at 4°C overnight to stop bacterial growth and gene expression, before being photographed.
Photographs of the bacterial plates are taken with a Canon SD900 digital camera. All four sample plates shown in Figure 3 of the main text were taken in a single photograph and the brightness, levels and contrasts were optimized identically. This preserves the relative signal intensity of each of the plates.
The background intensity is measured to be 75 (where 0 is pure white and 255 is pure black), which is the ‘leaky’ pixel intensity value taken from the illuminated region of plate of bacteria expressing the inverter circuit in Figure 3C. This background intensity value is a constant through the entire image analysis.
The radial edge intensity profiles in Figure 5A are extracted from the three circle images (Figure S3) (Matlab Image Processing Toolbox, Mathworks). These high resolution images are first converted to grayscale and subjected to a 5 point nearest neighbor smoothing to reduce digital artifacts. For each circle image, the radial pixel profile is calculated in the following way: starting from the center of each circle, the pixel intensities at each radial position (r = 0 to 2.8 cm) and at a constant θ coordinate are extracted. A set of radial pixel profiles are then extracted by performing this procedure while varying the θ coordinate. This set is then averaged together to create a mean radial pixel profile. This analysis is repeated for each circle image, creating a total of three independent mean radial pixel profiles. The background intensity is subtracted from each mean radial pixel profile. Then, each profile is divided by its radial pixel value at the r = 0 cm position, which is at the center of the circle, creating three independent fold-change edge profiles. The average and standard deviation of the fold-change edge profiles is computed.
The raw grayscale pixel intensities from the asymmetrical silhouette mask are extracted at selected angle intersections (ImageJ, 1.40g, Wayne Rasband, NIH) and background corrected. The background corrected intensities are then normalized by dividing by the maximum value in the data set (x = 51°, y = 1.0 in Figure 5b). The experimental values are compared to Miller Unit predictions from varying the angle of intersection in the reaction diffusion model as described below.
The average edge width and error presented in the main text is determined by repeating three separate 36mm circle pattern plates on three different days. The plates (Figure S3) are photographed and analyzed using ImageJ 1.40g (Wayne Rasband, NIH). Raw images are converted to grayscale and inverted. The horizontal pixel intensity profile across the plate is determined using rectangle probe traversing the center of the circle. The width of the edge is determined from the pixel intensity profile by drawing a straight line from the start of the edge (point of rapid signal intensity rise) to the point at which the signal intensity drops to the approximate average maximum signal intensity in the nearby internal illuminated area. Two edges (one from the left side of the circle and one from the right) are measured on each plate. The individual widths measured from the three plates in Figure S3 are 0.86cm, 0.76cm, 0.46cm, 0.62cm, 0.37cm, 0.53cm. The error values reported are the standard deviation.
The transfer function of the dark sensor is determined in batch culture Miller Assay experiments using strain JW3367c carrying the plasmids pJT103, pPLPCB and pCph8. pJT103 is comprised of the shortened PompC promoter (BBa_R0082) upstream of the strong RBS (BBa_B0034)(Elowitz and Leibler, 2000) and the lacZ ORF in the pSB4A3 backbone (Figure S4).
Overnight starter cultures are inoculated from −80C stocks in 3mL LB + Ampicillin, Kanamycin and Chloramphenicol and grown to OD600 ~4. Cultures are then diluted into 1mL LB + 0.1M HEPES pH = 6.6 to a final OD600 of 0.001 and added to a single, internal well of a sterile 24-well plate (Falcon, Part# 351147). The plate (with lid on) is then fixed onto a VWR incubating mini shaker (Cat# 12620-942) from which the plastic lid has been removed, placed inside of a dark incubator, illuminated with a defined amount of 650nm filtered light and shaken at 420rpm for 330-345 minutes at 36°C. The light is measured as previously(Levskaya et al., 2005) with the spectrometer probe placed at the position equivalent to the x,y center of the assay plate. The cultures, which have reached log phase (final OD600 between 0.6 and 1.2), are immediately collected under a safe green light and pipetted into black 1.7mL microcentrifuge tubes. Then, 100μL of each sample is immediately used to determine the OD600 while 350μL is used for the Miller Assay.
The quantity flight is the expression rate of the light-repressed BioBrick ompC promoter as a function of light input and has been experimentally determined using a β-galactosidase readout (Figure 2a). We fit the experimental data to the sigmoidal function,
where the fit parameter is K = 0.0017 W/m2 (R2 = 0.75), the maximum expression level βmax = 298 Miller Units, the minimum expression level βmin = 125 Miller Units, and the light intensity in units of W/m2 passing through the mask at position (r,θ) is Lr,θ .
The two-input transfer function of the signal integrating promoter Plux-λ (BBa_R0065) is determined in strain JW3367c carrying plasmids pJT104, pPLPCB and pCph8. pJT104 is the edge detection plasmid pEDL3 from which the luxI ORF (BBa_C0061) has been removed (Figure S5). This allows independent control of CI and AHL abundance via light and exogenous addition to the media, respectively. The readout of the Plux-λ (BBa_R0065) promoter is β-galactosidase in Miller Units.
The quorum signal 3OC6HSL (N-(b-Ketocaproyl)-L-homoserine lactone, Sigma-Aldrich# K3007) is added at different concentrations across different wells of the plate while a single light intensity is applied to the entire plate. The data in Figure 2b are the result of 25 data points taken as 5 sets of 5 points over 5 days, where the 5 data points for a given light intensity are collected on a single day. The cell cultures are in mid-log phase and typically between OD600 0.6 and 0.85 at the time of the assay.
The steady-state concentration of β-gal is determined by the transcription rate of the LuxR-activated, CI-repressed Plux-λ promoter, which is quantified by the flogic function. The Shea-Ackers formalism is used to enumerate the binding states of LuxR and CI bound to the promoter(Ackers et al., 1982; Bintu et al., 2005b). The steady-state concentration of β-galactosidase is proportional to the probability of RNA polymerase initiating transcription. The expression for flogic(u1,u2) is
where n = 1.5 and fLux is the concentration of LuxR dimer bound to AHL (, LuxRtot = 2000 nM)(Urbanowski et al., 2004), which is
and fCI is the concentration of dimeric CI ( (Koblan and Ackers, 1991)), which is
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).
Given a light mask, the reaction-diffusion model calculates the time- and position-dependent expression level of the β-galactosidase (β-gal) output gene. The model consists of (i) a partial differential equation describing 3OC6HSL production, degradation, and diffusion and (ii) two algebraic equations describing the steady-state concentrations of CI and β-gal in response to 3OC6HSL and light. In dimensionless form, these equations are
Where u1, u2, and u3 represent the concentrations of 3OC6HSL, CI, and β-gal at a position on the plate whose polar coordinates are given by (r, θ). The flight and flogic functions quantify the transcription rates of the light-dependent ompC promoter and the CI-repressed, LuxR::3OC6HSL-activated lux-λ promoter, respectively.
The constants κ1 and κ2 quantify the maximum production rate and the degradation rate of 3OC6HSL, respectively. The production rate of 3OC6HSL is estimated so that the maximum AHL 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 transcription rate, characterized by flight, and CI concentration is κ3 = 0.8 nM/Miller. The constant κ4 is the maximum β-gal concentration, which is 289 Miller units. This value was determined in batch culture experiments as described above at 500nM (maximum) exogenous AHL in the absence of any CI protein (plasmid pJT105).
When solving these equations, the space and time coordinates are dedimensionalized 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 = 1.67×10−7 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 the Matlab (Mathworks, Natick, MA) ode23s stiff numerical integrator with a final time of 24 hours (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.
The effect of changing the angle of intersection between light and dark boundaries on the edge intensity is examined, comparing the model predictions to the experimentally observed behaviors. We create a series of unit circle in silico masks where θ degrees of the circle are in the light with 360-θ degrees in the dark and where θ is varied from 50 to 345 degrees. For each mask, the solution of the reaction-diffusion model is computed, which predicts the maximum edge intensity. The maximum edge intensity is the β-galactosidase concentration at the edge location. The model predictions compare favorably with the experimentally observed edge intensities of the asymmetrical silhouette mask at the selected angle intersections (Figure 5B). The image analysis procedure to obtain the experimental data is described above.
The radial edge intensity profile of the circle images are compared to the in silico radial β-galactosidase profile from the model solution (Figure 5A). We compute the in silico radial β-galactosidase profile by first inputting the circle light mask into the model and determining the solution. Then, the β-galactosidase 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.
We thank E.A. Davidson, L.A. Lavery, M. Levy, K. McGary, and A. Scouras for helpful discussions, A. Nishimura for the JW3367 E.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.
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