Microfluidic device fabrication
A three-layer device was designed in AutoCAD that consisted of interaction channels, growth chambers, and main channels. The microfluidic master was pattered in three stages of photolithography using a micropattern generator (Heidelberg Instruments μPG 101). Unlike centrifuges, the spin coater protocol used for microfabrication is specified by an rpm. For the first layer, the silicon wafer was baked for 10 min at 200 °C and spin-coated at 4000 r.p.m. using SU-8 2000.5 (MicroChem) to generate 0.5 μm height. This layer was exposed to the interaction channels at 58 mW with a 47% dwell time using a 4 mm writehead, followed by a post-exposure bake for 30 min at 95 °C. The second layer was spin-coated at 3000 r.p.m. using a 26 : 1 mixture of SU-8 2000.5 to SU-8 3005, to produce 1.5 μm height. After aligning to the first layer, the wafer was exposed to the second patterning layer (growth chambers). Following an additional post-exposure bake, a third layer of SU-8 3025 photoresist was spin-coated at 3000 r.p.m. to generate 25 μm height. The wafer was exposed to the final layer consisting of the main channels, resistors, and inlets. Following a final post-exposure bake, the features were developed using SU-8 developer (MicroChem). The master was treated overnight with vapor phase (tridecafluoro-1,1,2,2-tetrahydrooctyl) trichlorosilane (Gelest) at room temperature. To fabricate each device, a 7 : 1 mixture of polydimethylsiloxane (Sylgard 184) to curing agent (Sylgard 184) was used to coat the master. After curing overnight at 100 °C, inlet and outlet holes were punched using a biopsy corer (WellTech). The surfaces were exposed to air plasma (Harrick Plasma PCD-32G) for 23 s to ionize the surface of the device to bond to the glass coverslips (ThermoFisher). Finally, the surfaces were bonded and baked for 1 hr at 100 °C to seal the device channels. For each experiment, the microfluidic device was flushed with 0.5% Tween 20 (Sigma-Aldrich) to prevent cells from adhering to the device. To load the cells into the growth chambers, a vacuum pressure of 330 mm Hg was applied.
Dye gradient experiment
The chemical gradients in the interaction channels were analyzed by administering 10 μM fluorescein (Sigma-Aldrich) and water at a flow rate of 200 μL/h into individual main channels. Paired growth chambers (n = 3) connected by each interaction channel length were continuously imaged using a 600 ms exposure time. Fluorescence and phase-contrast Images were collected using a Ti-E Eclipse inverted microscope (Nikon) using the GFP filter (Chroma) 470 nm/40 nm (ex), 525/50 nm (em). To analyze the images, the fluorescence of each growth chamber and 1 μm increments along the length of each interaction channel at steady state were determined.
Sender–receiver quorum-sensing experiments
Sender and receiver plasmids (Supplementary Fig. 2) were constructed using standard Gibson assembly protocols using primers synthesized by Integrated DNA Technologies and verified by Sanger Sequencing (Functional Biosciences). The sender (A6c_LuxI_GFP48) and receiver (E2c_LuxR_RFP or pJH9-35) plasmids were transformed into E. coli strains BW2778361 and MG1655Z162 (Table 2), respectively. An initial set of cultures were inoculated into LB media (Lennox, Sigma-Aldrich) containing 25 μg/mL chloramphenicol (Sigma-Aldrich) and cultured overnight at 37 °C with shaking. After ∼16 h, 1 μL of the cultures were diluted into 3 mL LB media containing 25 μg/mL chloramphenicol and incubated at 37 °C with shaking to early stationary phase (OD600 0.7–1.1). Next, we measured the OD600 of these cultures and centrifuged 1 mL at 3500 × g. The supernatant was removed and the pellet was resuspended to a final OD600 of ∼20. Cells were loaded into the device such that each growth chamber had two to three cells at the beginning of the experiment. In each experiment, the device was connected to three syringes (5 mL) containing LB media supplemented with 25 μg/mL chloramphenicol, 0.1% Tween 20 (Sigma-Aldrich), and 62.5 ng/mL anhydrotetracycline hydrochloride (Cayman Chemicals), as well as a fourth syringe (5 mL) containing the same media supplemented with 0.1% arabinose (Sigma-Aldrich). During the microscopy experiment, the microfluidic device was incubated at 37 °C in a custom-designed temperature incubation chamber. The main channels were flushed at a rate of 300 μL/h to wash away excess cells from the growth chamber. The flow rate of the inlet containing arabinose (I22, Fig. 1) and the corresponding inlet on the opposite side (I11) were set to 10 μL/h to prevent cell growth and clogging within the inlet and resistor and to reduce pressure differences across the device. Fluorescence and phase-contrast images were collected using a Ti-E Eclipse inverted microscope (Nikon) every 7 min at 21 different positions. Fluorescence was imaged using the following filters (Chroma): GFP: 470 nm/40 nm (ex), 525/50 nm (em) or RFP: 560 nm/40 nm (ex), 630/70 nm (em). The device was incubated for a period of time to allow cells to grow and divide. After the growth chambers had filled with cells, the media was switched to test conditions described in Table 1. For Experiment 1 (Table 1), the arabinose inlet (I22, Fig. 1) and the corresponding inlet on the opposite side (I11, Fig. 1) were switched to 200 μL/h and the flow rate through the remaining inlets (I12, I21) were set to 0 μL/h. The forced oscillation experiments (Experiments 3 and 4, Table 1) used 10 mL syringes to extend the duration of the experiment. Flow rates of the 0.1% arabinose (inlet I22) and 0% arabinose (inlet I21) were alternated out of phase between 200 and 0 μL/h for a period of time. One of the receiver inlets (I11) flowed continuously at a rate of 200 μL/h and the other inlet (I12) was set to 0 μL/h for the duration of the experiment.
Dual-feedback oscillator experiments
The E. coli strain CY027 was transformed separately with plasmids pC220 and pC239 or pC236 and pC239 to construct the activator and repressor43, respectively using a standard chemical transformation protocol (Table 2). Overnight cultures were inoculated into LB media (Lennox) containing 50 μg/mL kanamycin and 100 μg/mL spectinomycin and incubated overnight at 37 °C with shaking. After ∼16 h, 1 μL of the overnight cultures were diluted into 3 mL LB media and incubated at 37 °C with shaking to early stationary phase (OD600 0.7–1.1).
Cells were loaded into the device following the procedure specified above. Following cell loading, the microfluidic chip was placed in the custom-designed temperature incubation chamber at 37 °C. All four inlets were connected to syringes (10 mL) containing LB media with kanamycin (50 μg/mL), spectinomycin (100 μg/mL), and 0.1% Tween 20. Syringes connected to inlets I22 and I11 also contained 1 mM isopropyl β-d-1-thiogalactopyranoside (IPTG) (Sigma).
The cells were initially grown in the device at 37 °C with inlets I12 and I21 flowing at 200 μL/h, and inlets I22 and I11 flowing at 10 μL/h to prevent cell growth and clogging. Phase-contrast and fluorescence images were collected every 7 min at 21 different positions. Once the growth chambers were filled with cells (Table 1), the inlets (I12 and I21) containing the pre-culture media were set to 0 μL/h and the inlets (I11 and I22) containing the test media were set to 200 μL/h.
Amino acid auxotroph experiments
E. coli strains ΔmetA63 and ΔpheA63 (Table 2) were transformed with plasmids A6c_GFP64 and A6c_RFP64, respectively, using a standard chemical transformation protocol. The plasmids harbored an IPTG-inducible fluorescent reporter. An initial set of cultures were inoculated into LB media (Lennox) containing chloramphenicol (25 μg/mL) and incubated overnight at 37 °C with shaking. After ∼16 h, 1 μL of the overnight cultures were diluted into 3 mL of LB containing 25 μg/mL chloramphenicol and 1 mM IPTG (Sigma-Aldrich) and incubated at 37 °C with shaking until early stationary phase (OD600 0.7–1.1).
The cells were loaded into the device following the procedure outlined above. Following cell loading, the microfluidic chip was placed in the custom-designed temperature incubation chamber at 37 °C. The media always contained 1× MOPS Buffer (Teknova), 1× ACGU mix (Teknova), chloramphenicol, 0.1% Tween 20, 1.32 mM potassium phosphate dibasic (Teknova), and 0.2% glucose (Teknova), whereas the amino acid composition varied across experiments (Table 1). The amino acid solutions consisted of either EZ Amino Acids (AA, Teknova) or a modified amino acid solution (AA*) (Table 1). The AA* solution consisted of 0.4 mM l-asparagine (VWR), 0.01 mM calcium pantothenate (VWR), 0.2 mM l-histidine (VWR), 10 mM l-serine (VWR), 0.8 mM l-alanine (Fisher Scientific), 0.4 mM l-lysine (Fisher Scientific), 0.1 mM l-tryptophan (Fisher Scientific), 0.4 mM l-aspartic acid (Dot Scientific), 0.1 mM l-cysteine (Dot Scientific), 0.8 mM l-glycine (Dot Scientific), 0.4 mM l-isoleucine (Dot Scientific), 0.8 mM l-leucine (Dot Scientific), 0.01 mM para-amino benzoic acid (Dot Scientific), 0.4 mM l-proline (Dot Scientific), 0.4 mM l-threonine (Dot Scientific), 0.6 mM l-valine (Dot Scientific), 5.2 mM l-arginine (Sigma), 0.01 mM di-hydroxy benzoic acid (Sigma), 0.6 mM l-glutamic acid (Sigma), 0.01 mM para-hydroxy benzoic acid (Sigma), 0.01 mM thiamine (Sigma), 0.2 mM l-tyrosine (Sigma), and 0.6 mM l-glutamine (Acros Organics). In minimal media supplemented with AA*, varying concentrations of methionine (Dot Scientific) and/or phenylalanine (Dot Scientific) were added. The AA amino acid solution consisted of all components in AA* plus 0.2 mM methionine and 0.4 mM phenylalanine.
The cells were grown for a period of time at 37 °C with 1 mM IPTG prior to the media switch as described in Table 1 to fill the growth chambers. Phase-contrast and fluorescence images were collected every 10 min at 21 different positions. After the growth chambers were filled with cells, the inlets (I12 and I21) containing the pre-culture media were set to 0 μL/h and the inlets (I11 and I22) containing the test media were set to 200 μL/h.
Amino-acid measurements
The ΔmetA and ΔpheA strains were inoculated into LB (Lennox) containing chloramphenicol (25 μg/mL) and grown overnight at 37 °C with shaking. After 16 h, 10 μL of the cultures were transferred into 3 mL of fresh LB containing chloramphenicol (25 μg/mL) and incubated at 37 °C with shaking until early stationary phase (OD600 0.7–1.1). Immediately following, the cultures were centrifuged at 3500 × g for 5 min, supernatant was removed, and the cells were inoculated into MOPS EZ Rich Defined Medium lacking M and F at an initial OD600 of 0.05. For the ΔmetA strain, 0, 2, 5, 10, or 200 μM M was added to the media. For ΔpheA strain, 0, 4, 10, 20, or 400 μM F was added to the media. The cultures were incubated at 37 °C with shaking for 3 h. After recording the OD600 of each culture, the cells were centrifuged at 3500 × g for 10 min, the supernatant was filtered by a 0.2 μm filter (GE Healthcare) and the concentrations of M or F were measured with a fluorometric assay kit (BioVision) or by liquid chromatography-mass spectrometry (LC-MS), respectively. Concentrations of M in the filtered conditioned media of ΔpheA cultures were measured with a fluorometric methionine assay kit (BioVision) with a 0.5 μM limit of detection. Raw fluorescence measurements were converted to methionine concentrations using a standard curve.
The analysis of F concentrations in the filtered conditioned media of ΔmetA was performed on a Shimadzu LC-MS2020. All solvents and reagents used for analysis were HPLC grade or higher quality. Methanol and formic acid were sourced from Fisher Scientific and Acros Organics, respectively. Water was prepared in house with a Millipore Milli-Q water purification system. Separations were performed at 40 °C on a Discovery BIO wide pore C5-5 column (15 cm × 2.1 mm × 5 µm) from Millipore-Sigma with a paired Supelguard (2 cm × 4 mm × 5 µm) guard column. The running buffer was a binary gradient of water with 0.1% v/v formic acid (Buffer A) and methanol (Buffer B) according to the following protocol: 4 min at 5% B, a linear gradient from 5% to 20% for 4 min, a linear gradient from 20% B to 95% B for 2 min, 2 min at 95% B, a linear gradient from 95% B to 5% B for 2 min, equilibration at 5% B for 6 min. The total flow rate was 0.2 ml min-1. Under these conditions, methionine and phenylalanine eluted at 3.8 and 5.8 min, respectively. The ion source was operated in electrospray ionization mode with a cone voltage of 4.5 kV, the interface was held at 400 °C, and the desolvation line at 250 °C. The dry nitrogen was supplied to the nebulizer at 1.5 L/min and drying gas at 15 L/min. The mass spectrometer was run in selective ion monitoring mode for monitoring m/z 150 for methionine and m/z 166 for phenylalanine with a scan time of 1 s. Standards were prepared for each run by adding known concentrations of methionine and phenylalanine to fresh media. The standard curve was run before and after the sample batch and each sample was run twice for technical replicates.
Auxotroph community batch culture experiment
Separate culture tubes containing LB with chloramphenicol (25 µg/mL) were inoculated with ΔmetA or ΔpheA and incubated overnight at 37 °C with shaking. After 16 h, the cultures were diluted into 5 mL of EZ Rich Medium (Teknova) containing chloramphenicol (25 µg/mL) and 1 mM IPTG and lacking M and F at a final OD600 of 0.05. The initial ratio of ΔmetA to ΔpheA was 10 : 1, 1 : 1, or 1 : 10 (n = 3, for each starting ratio). The cultures were incubated at 37 °C with shaking for at least 24 h before transferring the community to fresh media using a 1 : 100 dilution. At this transfer time, the OD600 of each culture was measured and a 2 µL sample was spotted onto a glass slide for cell counting with microscopy (20× magnification) on a Nikon Eclipse Ti. Four images comprising four distinct fields of view were taken of each sample and each image was a composition of phase contrast, GFP and RFP channels. Subsequently, ImageJ was used to extract the number of ΔmetA cells from the GFP channel and ΔpheA cells from the RFP channel for each image.
Population-level image analysis
For Experiments 1, 3, 6, 7, and 11–14 (Table 1), individual growth chambers were segmented in DeepCell65. Five neural networks were trained on 21 randomly selected images and binary masks (made using FIJI image analysis software66), which specified the growth chamber positions. The trained model was used to analyze the remaining microscopy images. The results of the trained networks (two to five depending on segmentation accuracy) were averaged to improve segmentation accuracy.
For Experiments 2, 4, and 5 (Table 1), growth chambers were segmented using custom code in Python that aligned each growth chamber across all time points. A binary mask denoting the growth chambers was applied to all time points. The DeepCell and alignment methods generated nearly identical fluoresence time-series data.
In each analyzed image, custom code (Python) was used to label the binary mask with the growth chamber positions and total areas and compute the average fluorescence intensity of each growth chamber. Segmented regions less than or greater than 1000 and 3500 pixel area were eliminated from the data set. Specific criteria were used to eliminate outliers from the datasets including (1) infrequent pressure fluctuations leading to loss of cells from the growth chambers, (2) device bonding issues leading to collapsed interaction channels or cells that enter the interaction channels, (3) growth chambers with unoccupied regions, (4) abnormal cell growth that significantly altered the total number of cells in the growth chamber, or (5) cell growth in the main channels that may have generated different media diffusion rates into the growth chambers. In all physically separated experiments (Experiments 1–7 and 11–14, Table 1), the connected chamber was excluded from the data set if a growth chamber was identified as an outlier based on these criteria (Table 1).
Population-level fluorescence time-series analysis
Fluorescent time-series measurements for each growth chamber in Experiments 6, 7, and 11–14 (Table 1) were analyzed by bootstrapping. Using this method, the biological replicate curves for a given interaction channel length were randomly sampled 10,000 times with replacement. In Experiment 1 (Table 1), background fluorescence was subtracted from the data by subtracting the minimum RFP fluorescence intensity across all growth chambers for model fitting (Fig. 1c, d).
The P-values for all bootstrapped datasets were computed by bootstrap hypothesis testing. Here, the null hypothesis H0 assumes that a sample of size n with mean x*obs and a sample of size m with mean y*obs are derived from the same population. This test is performed as follows:
Calculate the sample mean difference, t*obs, as t*obs = x*obs − y*obs
Merge two samples into one set of n + m observations.
Draw a bootstrap sample of n + m observations with replacement from the merged set.
Calculate the mean of the first n observations, x* and compute the mean y* for the remaining m observations in the bootstrap sample. The test statistic t* is evaluated as t* = x* − y*.
Repeat steps 3 and 4 B times where B ≥ 1000.
Evaluate the P-value as: P-value = number of times where t* > t*obs divided by B.
Reject H0 if P-value ≥ α, where α = 0.05.
In the forced oscillation experiments (Experiments 3 and 4, Table 1), a peak finding algorithm (Python) was applied to the time-series gene expression data at steady state with minimum inter-peak threshold of 21 min. The amplitude was computed by subtracting the minimum and maximum of each oscillation and dividing this value by two. To calculate the SNR, a moving mean computed over 20 time points was subtracted from the data. The power spectra for each replicate was calculated using Welch’s method (Python) with a Hamming window applied across the length of the time-series. The power spectra were filtered to exclude frequencies lower than the signal bandwidth. The signal was defined as the total power of the signal bandwidth. The noise was computed as the total power of frequencies larger than the signal bandwidth. The power spectra for all the replicates for a given interaction channel length were randomly sampled with replacement 10,000 times. For each iteration, the SNR ratio was computed by dividing the signal by the noise. Bootstrap hypothesis testing, as described above, was used to compute P-values.
For the distributed gene circuit oscillator experiment (Experiment 5, Table 1), the fluorescence intensity of each fluorescent reporter was normalized by subtracting the global minimum of the reporter across all replicates, dividing by the global maximum of fluorescence across all replicates, and applying a moving mean of 20 time points to the data. A peak finding algorithm (Python) was applied to detect peaks with a minimum inter-peak distance of 70 min and a minimum peak height of 0.015 by analyzing the data after the media switch. The number of peaks detected, the amplitude of expression at each peak, and the distance between subsequent peaks were computed for each replicate.
In the spatially separated auxotroph experiments (Experiments 6, 7, 11–14, Table 1), the fluorescence background for each reporter was subtracted from the data and then the time-series was normalized by dividing by the maximum value. The change in fluorescence per unit time (ΔF Δt−1) was computed by determining the slope of a line fit to a 10 time point moving window and then multiplying by negative one. The global maximum of ΔF Δt−1 corresponded to the maximum growth rate. The doubling time was calculated using times the following equation:
$${mathrm{{Doubling}}};{mathrm{{time}}} = frac{{{mathrm{ln}}left( 2 right)}}{{max left( {Delta {mathrm{F}}Delta {mathrm{t}}^{ – 1}} right)}}.$$
In Experiment 11 (Table 1), the ΔF Δt−1 curves displayed a biphasic trend. To characterize the growth rate at each peak, the ΔF Δt−1 time-series was analyzed between the time point of the media switch and the time point corresponding to 25% of the maximum fluorescence. The local maxima within this time window were identified using the findpeaks algorithm (Python). The bootstrapped ΔF Δt−1 time-series were aligned by the first peak and the doubling times at the global maximum were calculated as described above. For the second growth phase, the doubling time was calculated at the maximum ΔF Δt−1 for the period of time between the global maximum and the time point corresponding to 25% of the maximum fluorescence.
Single-cell image analysis
Single-cell metrics were obtained with a custom machine learning approach implemented in Python with the Keras API running on top of TensorFlow67. We used two convolutional neural networks with U-Net architecture. First, we performed segmentation of individual cells in each image and then tracked each of the segmented cell instances over time. The segmentation network takes as an input the phase contrast images of cells grown in MISTiC and for each image and yields a binary mask segmenting the cells from the background. Training data were obtained from a separate experiment imaging fluorescently labeled E. coli at 60× magnification with phase-contrast and fluorescence images collected every 10 min. We used the fluorescence images to generate binary segmentation masks of the cells, which then served as the ground truth for the phase contrast images used for network training. A total of 1066 images were curated this way. The network was trained for 100 epochs using a stochastic gradient descent optimizer and a pixelwise weighted loss function to enforce the learning of narrow borders between adjacent cells. To minimize overfitting of the network to the training data, random affine transformations and elastic deformations were applied in real-time during the training process.
Cell tracking was performed with a separate U-Net similar to a method reported previously68. The input for this network is a set of consecutive binary segmentation masks. For each cell in the current segmentation, the network predicts the cell in the previous segmentation image from which the current cell was derived. This backwards tracking approach eliminates the need for the network to learn occurrences of cells leaving the chamber and reduces the number of classes to two (the tracked cell and the background). Using segmentations from the mixed auxotroph experiment, we curated 2656 sets of training images with a custom script in MATLAB. Training occurred for 200 epochs using an Adam optimizer and a class-weighted categorical cross-entropy loss function. Similarly, data augmentation was performed to reduce overfitting of the data.
Following segmentation and tracking, the raw output was processed with custom code in Python to reconstruct cell lineage and obtain single-cell metrics. The instantaneous growth rate of each cell was computed from the cross-sectional area recorded during the 100 min window (10 data points) immediately following that instant. Growth rate was computed by fitting a line to each 100 min window and then dividing the slope of the line by the average cell area during that time interval. For all analyses, a minimum tracking duration of 100 min was imposed to enforce consistent computation of growth rate. For all analyses involving growth rate, statistical outliers were identified using a modified z-score computed on the chamber averaged growth rates at each time point
$$M_{mathrm{{i}}} = frac{{0.6745left( {x_{mathrm{{i}}} – tilde x} right)}}{{M_{mathrm{{d}}}}},$$
where (tilde x) represents the median growth rate and Md denotes the median absolute deviation69. Statistical outliers were detected using a threshold of Mi > 3.5. Growth chambers with more than one time point registering as an outlier were excluded from the analysis (Table 1). Experimental outliers occurred primarily due to segmentation and tracking errors caused by loss of focus or empty chambers at specific positions. Outliers were considered separately for each strain.
Model fitting
Custom code (MATLAB) was used for computational modeling. An ordinary differential equation model was developed to study inter-strain communication via chemical signal diffusion (quorum-sensing). Detailed descriptions of the diffusion and gene expression models are in the Supplementary Methods. The general mathematical form of the equations describing the concentration of AHL or fluorescein in each discretized spatial region is
$$dot x_i = Dleft( {x_{i – 1} + x_{i + 1} – 2x_i} right) – gamma x_i,$$
where xi and xi+1 represent concentrations in adjacent regions of the device. The parameters D and (gamma) denote the diffusion and degradation rates of the diffusible molecule, respectively. For the gene expression model, the general mathematical form for modeling transcription is
$$dot B_{mathrm{{m}}} = propto _Bfrac{{A^{mathrm{{n}}}}}{{K^{mathrm{{n}}} + A^{mathrm{{n}}}}} – gamma B_{mathrm{{m}}},$$
where A and Bm represent a transcription factor and its regulated transcript, respectively. The parameters (propto _B,n,K), and (gamma) denote the maximum transcription rate, Hill coefficient, half-maximum concentration or binding affinity and mRNA degradation rate, respectively. The general mathematical form for representing time delays due to sequential protein assembly, fluorescent protein maturation or media switching is
$${it{y}}_{it{j}} = {it{a}}left( {{it{y}}_{{it{j}} – 1} – {it{y}}_{it{j}}} right){it{for}},{it{j}} = 1:{it{N}}_.$$
The species yN represents the time-delayed species y1 and the delay time is computed by (N cdot a^{ – 1}_.)
The model was simulated using ode23s (MATLAB). A model with a variable number of delay equations was fit to the data using a genetic algorithm. The algorithm identified a best estimate for the parameter values and an optimal model structure by adjusting the number of delay equations to minimize the L2-norm between the model and the data. First, 100 parameter sets were randomly sampled using an upper and lower bound for each parameter. For each parameter set, the model was simulated and the L2-norm between the model and the data was computed. The parameters were ranked from lowest to highest L2-norm. The first parameter set (lowest L2-norm) was averaged with parameter sets 2-10, generating 9 new parameter sets. These parameter sets were combined with 81 randomly sampled parameter sets using an upper and lower bound for each parameter. This procedure was repeated until the L2-norm did not change significantly with additional iterations. The best estimates for the parameters are listed in Supplementary Table 3.
The parameters of the amino-acid cross-feeding model (Supplementary Methods) were fit using a genetic algorithm. The genetic algorithm can be most efficient with high-order systems and many unknowns. One of the challenges with the genetic algorithm is there is no proof of convergence and the rate of convergence can be slow if the initial guesses on the parameters are far from the minimizing set and the bounds on the parameters are too broad. In order to overcome these challenges, careful consideration was taken to determine the lower and upper bounds for each parameter. Initially, bounds were determined based on biologically relevant and feasible values. In addition, experimental observations were used to infer necessary relationships between parameters. The bounds on the parameters were adjusted accordingly. After this, the genetic algorithm was executed until the error became invariant for a sequence of 10 generations. As the genetic algorithm is not optimal, it is possible to arrive at slightly different values if we were to run the genetic algorithm longer or reinitiate at new random initial conditions. However, the qualitative fits remain fairly close, as do the parameter values. Nevertheless, given experimental error, it is not in our benefit to achieve an optimal fit, since such a fit does not imply better prediction of quantitative values of parameters.
Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
Source: Ecology - nature.com
