Oscillate is a novel system utilizing variable inputs to trigger variable expression. Using a combination of biological low-pass and high-pass filters, we developed a functional gene expression system which is able to respond to frequency-variable input signals.
We provide evidence that our genetic bandpass filter allows expression of GFP to occur only between a narrow range of temperature frequency, while discriminating against frequencies outside the band pass filters range.
Figure 1. System overview (a) Low pass filter with GFP (BBa_K2561003) (b) High pass filter with GFP (BBa_K2561005)(c) Low pass filter with hrpS for AND gate (BBa_K2561004)
(d) High pass filter with hrpR for AND gate (BBa_K2561006)
(e) AND gate reporter (BBa_K2561007)
We specifically designed our novel low-pass and high-pass filters to be modular, so that they could be tested and function individually without need for the full system to be intact. The effects of the individual tests run on each the low-pass and high pass filters are to be taken as oscillatory responses. Significant expression shows activation of the system and lack therein shows no response. However, expression should be cyclical and mimic the input signal.
We first confirmed the functionality of both the low pass and the high pass filter individually, by subjecting cultures with only the low pass or high pass inserted to oscillating temperatures between 37 °C and 22 °C for the period we were testing. Using a plate reader and taking samples of bacteria every 30 minutes, the ratio of fluorescence to bacterial cell density was plotted over time. Bacterial cell density was determined by measuring absorbance of the culture at 600 nm. Fluorescence was measured by exciting the GFP with 485 nm light and recording its emission at 510 nm.
Figure 2. The oscillatory input signal took the form of a square wave with varying period.
To account for the stability of the GFP, we report the change in the fluorescence/absorbance ratio between successive time points as the normalized fluorescence. As a result, this tends to lead to variations in the amplitude of the signal read; however, the frequency of the signal output is of importance. The plots of the high-pass and low-pass cultures resulted in an oscillating function with the low-pass and high-pass filters acting opposite each other, indicating that both parts were individually functional at frequencies within the band.
We initially tested with an oscillation period of 2 hours (138.8 μHz) to confirm the low and high pass were individually functional. This initial frequency was predicted by our deterministic model based on the approximate parameters we had determined from literature searches. Continued testing allows for the model to learn, and will further help fit parameters to values.
We continued subjecting our cultures with the individual parts in them to different frequency oscillations in temperature to find a parameter that produced the maximum fluorescent expression in both. We also tested for frequencies in which the ratio of fluorescence to bacterial cell density was inverted between the high pass and low pass cultures, so that the maxima of the high pass were aligned with the minima of the low pass and vice versa. This response is crucial for optimal bandpass functionality.
During our testing, we also tested for the upper bound (cut-off) frequencies of fluorescence for the low pass. By testing a wide range of periods, from 10 minute periods to 3 hour periods, we determined the bounds as the frequency at which the sinusoidal expression of GFP was no longer observed.
Figure 3. The high and low pass filters work opposite each other - when one peaks in expression, the others is at a local minimum.
First however, to verify that the RNA thermometer and σF were working properly, we measured fluorescence with a step-input. The results below indicate that the RNA thermometer melt temp is around 37°C, and that the sigma factor and RNA thermometer are working as intended.
Figure 4. Verification of low-pass components.
Analysis
We fit the data using Fourier analysis with the MATLAB Curve Fitting Toolbox. Using nonlinear least-squares, we determined the effectiveness of the filter by examining the angular frequency, ⍵, of the Fourier function. Given that ⍵ = 2πf, it was clear that simply fitting one term to the data was inadequate, although there was a clear correlation. Like an electronic band-pass, we would expect the frequency of the input oscillation to match the frequency of the output.
To remedy the fit, we added more terms to the Fourier series representing the output oscillations. Adding terms accounts for noise in the data by accounting for the harmonics - integer multiples of the frequency - that are overlaid. By accounting for the harmonics (second and third), we demonstrate a progressively better fit and that our system effectively passes frequencies within the band.
| Construct | Terms (in fit) | Period (T) (min) | Expected ⍵ (min-1) | Fit ⍵ (min-1) | R2 | 95% Confidence Interval (⍵) |
|---|---|---|---|---|---|---|
| High Pass | 1 | 120 | 0.0523 | 0.1245 | 0.6491 | (0.11, 0.13) |
| High Pass | 2 | 120 | 0.0523 | 0.06166 | 0.7472 | (0.05, 0.07) |
| Low Pass | 1 | 120 | 0.0523 | 0.07642 | 0.5065 | (0.05, 0.09) |
| Low Pass | 3 | 120 | 0.0523 | 0.04002 | 0.9164 | (0, 0.12) |
| Low Pass | 1 | 40 | 0.157 | 0.1590 | 0.4248 | (0.13, 0.18) |
| Low Pass | 2 | 40 | 0.157 | 0.1765 | 0.9280 | (0.17, 0.18) |
| High Pass | 1 | 180 | 0.035 | 0.0386 | 0.3790 | (0.03, 0.05) |
| High Pass | 3 | 180 | 0.035 | 0.0435 | 0.6255 | (0.034, 0.052) |
| Low Pass | 1 | 180 | 0.035 | 0.076 | 0.6348 | (0.06, 0.08) |
| Low Pass | 4 | 180 | 0.035 | 0.0187 | 0.9050 | (0.01, 0.03) |
| Low Pass | 1 | 10 | 0.628 | 0.4439 | 0.4241 | (0.37, 0.51) |
| Low Pass | 3 | 10 | 0.628 | 0.2447 | 0.4631† | (0.12, 0.36) |
†Adjusted R-square.
Figure 5a-5d. Low and high pass at T = 120 min. (a) Low pass with one Fourier term (b) Low pass with three Fourier terms (harmonics) (c) High pass with one Fourier term (d) High pass with two Fourier terms
Figure 6a-6b. Low pass at T = 40 min. (a) Low pass fit with one Fourier term (b) Low pass fit with two Fourier terms
Figure 7a-7d. Low and High pass at T = 180 min. (a) Low pass with one Fourier term (b) Low pass with three Fourier terms (c) High pass with one Fourier term (d) High pass with four Fourier terms
While adding terms to the Fourier series inherently improves the fit, it is important to note that the R^{2} is not the only important result of the analysis. From our conversation with Dr. Edwin Kan, an electrical engineer studying important biological signals, we realized that in addition to a strong coefficient of determination, we needed to ensure the signal frequency matched the input closely. If a curve we fit to the oscillatory output has a strong R^{2}, but is not close to the correct frequency, then the signal was likely attenuated and the correlation is a result of over-fitting.
Additionally, we can attribute some of the difference in the input and output frequency to lag time in the system. If constant, this would only create a phase-shift in the output frequency; however, even small variations will influence the output oscillatory frequency.
Based on the results of the analysis, we found that the low-pass filter cutoff frequency was fH = 1633 µHz. This frequency corresponds to a period of 10 minutes. Signals at higher frequencies than the cutoff will be attenuated by our system.
Band Pass
To test the low pass and high pass constructs in combination (the true band pass), we first had to subclone the constructs (BBa_K2561011 and BBa_K2561004) into vectors with different antibiotic resistance tags so that they could be transformed into the same cell and successful cotransformants be isolated. However, we ran into problems with incompatible origins of replication while using the pSB1A3 and pSB1k3.m1 vectors. We thus subcloned the low pass construct (BBa_K2561004) using XbaI and PstI into the pCOLADuet1 vector; once verified, we transformed the low pass and high pass constructs into BL21(DE3) cells. Successful clones were grown in terrific broth (TB) at either 30 or 37 degrees Celsius and diluted to a final OD600 of 0.2. Cells were grown at both temperature points to factor out basal expression before testing. Cells were placed in shaker incubators at either 30 degrees or 37 degrees and transferred between incubators at the necessary times; samples were collected at logical times in each oscillation to get a uniform curve.
Like the data for the individual low and high pass filters, we applied a Fourier analysis to the data and searched looked for the angular frequency to match the input. We determined the cutoff frequency and period at the high frequency end to be T = 20 min and f = 833 μHz (0.05 min—1). At the cutoff frequency, it is clear that a Fourier analysis yields a value for the output frequency not equal to the input frequency (based primarily on the second harmonic). While using the first harmonic yields a frequency within a wide confidence interval, it is a poor fit for the data. We also verified that the filter passes frequencies that fall within the bandwidth. Testing at a period of 120 min revealed that the first harmonic of the Fourier series produces a tight confidence interval for ω that contains the true frequency with good correlation.
We’d like to thank the Fromme lab for generously providing us with the pCOLADuet1 vector and the Baskin lab for allowing us to share their equipment for this part of the project.
| Construct | Terms (in fit) | Period (T) (min) | Expected ⍵ (min-1) | Fit ⍵ (min-1) | R2 | 95% Confidence Interval (⍵) |
|---|---|---|---|---|---|---|
| Band Pass | 1 | 20 | 0.314 | 0.328 | 0.1253† | (0.214, 0.442) |
| Band Pass | 2 | 20 | 0.314 | 0.218 | 0.7215† | (0.149, 0.286) |
| Band Pass | 1 | 120 | 0.0523 | 0.04712 | 0.9218 | (0.039, 0.055) |
†Adjusted R2
Figure 8. Band pass construct tested at 20 minute period (attenuated). Second harmonic fit.
Figure 9. Band pass construct tested at 120 minute period (passed). First harmonic fit.