<strong>Figure 5:</strong> Effect of varying d<sub>MS</sub> and duration on time courses. Time courses with different number of pulses demonstrate the range in the distinguishing capability for a changing d<sub>MS</sub>. As d<sub>MS</sub> increases, the ability to differentiate pulsing and more sustained inputs increase.
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<strong>Figure 6:</strong> Effect of varying d<sub>MS</sub> and duration on time courses. Time courses with different number of pulses demonstrate the range in the distinguishing capability for a changing d<sub>MS</sub>. As d<sub>MS</sub> increases, the ability to differentiate pulsing and more sustained inputs increase.
<strong>Figure 6:</strong> Varying induced degradation rate affects IFFL's decoding performance. <strong> (Left) </strong> when d<sub>MS</sub>=6.2x, the highest vector difference is 1.7, and lowest value is -1.2. <strong> (Right) </strong> when d<sub>MS</sub> is 26.2x, the highest red value is 2.8, the lowest value is -0.94. We could see that IFFL's performance improved in general. The column corresponds to on2=4 has much higher value compared with the same column when d<sub>MS</sub> = 6.2x. In addition, the column for On2=1 when d<sub>MS</sub> = 26.2x also has much higher value as the saturation of blue areas decrease. However, for some cases where IFFL is just moderately better at decoding input structure than Naive, IFFL's performance becomes worse, observing that values of right-side columns decrease.
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<strong>Figure 7:</strong> Varying induced degradation rate affects IFFL's decoding performance. <strong> (Left) </strong> when d<sub>MS</sub>=6.2x, the highest vector difference is 1.7, and lowest value is -1.2. <strong> (Right) </strong> when d<sub>MS</sub> is 26.2x, the highest red value is 2.8, the lowest value is -0.94. We could see that IFFL's performance improved in general. The column corresponds to on2=4 has much higher value compared with the same column when d<sub>MS</sub> = 6.2x. In addition, the column for On2=1 when d<sub>MS</sub> = 26.2x also has much higher value as the saturation of blue areas decrease. However, for some cases where IFFL is just moderately better at decoding input structure than Naive, IFFL's performance becomes worse, observing that values of right-side columns decrease.
<strong>Figure 7:</strong> Variance as direct translation to noise. Variance of IFFL with respect to all combination of off and on time is exponentially lower than that of Naive.
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<strong>Figure 8:</strong> Variance as direct translation to noise. Variance of IFFL with respect to all combination of off and on time is exponentially lower than that of Naive.
<strong>Figure 8:</strong> Contour graph shows that at off-on ratio of 1:2, variability in concentration of expression is highest with respect to parameter value range. This variance is still much lower than that of naive model.
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<strong>Figure 9:</strong> Contour graph shows that at off-on ratio of 1:2, variability in concentration of expression is highest with respect to parameter value range. This variance is still much lower than that of naive model.
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Revision as of 23:50, 17 October 2018
Math Modeling
Abstract
In order to model IFFL's characteristics and its ability to encode and decode time-domain inputs relative to its alternative, Naive scheme, we developed system of Ordinary Differential Equations (ODE) to gain insight on parameters and their interactions leading to the behaviors of the output expressions. Then, we began implementing methods to assess IFFL's decoding ability under noise as means to develop a framework for acquiring relevant biological information in the time-domain. This step includes numerical computations of all possible concentration expression under different temporal structures of input that could be translated into vector analysis (heat map) to observing the behavior of variance under fixed temporal structures with respect to regions of parameter values as direct implication to noise response (illustrated using mesh and contour graphs). Overall, these methods support IFFL's ability to decode time-domain information much more accurately than its counterpart as well as having superior fidelity to noise response. With these findings, we hope that future iGEM team could develop IFFL systems that would be able to react to its input in an almost lossless manner.
Overview
Zhang et al. demonstrated that fundamental property of Incoherent Feedforward Loop(IFFL) can be used to exhibit temporal adaptation [1]. They further indicated this property could serve as the foundation for distinguishing input's temporal structure. To demonstrate this important ability, we built quantitative ordinary differential equations (ODE) model inspired by the generic model in Zhang's paper. Specifically, this model was meant to simulate the Temperature-Sensitive IFFL system developed by the wetlab team (Analysis of our model can be found here). Here, promoter inhibitor protein CI is quickly degraded at a specific temperature, which activates the production pathway for mf-Lon and m-Scarlet. The denominator seen under the production rate is intended to highlight leakiness in production even when there is a high concentration of CI, but this event is not observed during numerical testings due to low parameter values being used. These parameter values implemented for testing are derived and altered from MCMC and hard-coded values done in our chemically-induced model.
To assess Zhang's claim, we needed to come up with a system that serves as a counterpart to IFFL Decoder. The best way to measure the "distinguishability" performance was to take different types of inputs that encode information in the time domain and simulate these inputs being processed by the Decoder, which converts them into the amplitude domain. If inputs that are distinguishable in the time domain become outputs that are well-distinguishable in the amplitude domain, then we call that successful decoding. We did a computational experiment where we compared the use of our IFFL decoder against a “naive decoding circuit” (Above ODE) which consists of a gene that is activated by the input without any additional regulation. The naive decoding circuit represents the transfer of information from an endogenous biological signal into a synthetic circuit in the simplest possible way.
To compare the relative successful decoding of IFFL, we derived quantitative method of representing the difference between reporter's expression concentration decoded through different temporal inputs - call it on time or duration - with fixed off time as vectors for both systems. The vector represents the output of the IFFL system or the naive system with the same on time and off time. For example, in our case, one can imagine the vectors for IFFL and Naive systems as both 80-dimensional vectors in 80-dimensional space since we have 80 time points, then a determining value was derived by taking the vector difference between Naive and IFFL using the Euclidean Distance equation.
Figure. 1: Euclidean Distance equation for 2 n-dimensional vectors
Finally, these values were organized in a "heat map", where the shadings in red represented relatively how much better IFFL was at decoding the input signal at that two specific different on time, and vice versa for the blue shadings. Overall, this assessment can be summed up as IFFL's relative "distinguishability" strength of time-domain inputs from amplitude-domain output.
Figure. 2: First two graphs represent two different distinct inputs of different on times with fixed off time. Based on these inputs, expression concentrations are calculated for both IFFL and Naive, with their differences represented as a vector, and the final values being shown in the heat map below. The bottom right graph shows general behavior of the systems at two different on times: The second subplot showing expression by IFFL and the third by Naive. The last subplot shows the concentration differences between those two distinct inputs as a function of time. Higher difference in the last subplot indicates better decoding ability.
Figure. 3: Likewise, decoding ability of Naive is better than IFFL at these two particular on times, which is represented by blue shadings.
With IFFL's ability to detect small variations in temporal structure from the input, we would expect that the system would be susceptible to noise. To test this hypothesis, we performed the following numerical testing for both IFFL and Naive model and compare the variance as direct translation to noise response. Gaussian noise with average value 0 and standard deviation of 0.1 was applied to the concentration of CI:
Start with fixed value for off-on ratio, calculate all concentration expressions of m-Scarlet within a specific parameter region of m-Scarlet production rate. Variance is then taken for that specific off-on ratio, then on ratio is incremented, then off ratio is incremented as on ratio resets. Each variance for respective off-on ratio is the average variance of concentration expression as function for varying parameter value range of m-Scarlet production rate.
Figure. 4: Scheme of Variance Test as Direct Translation to Noise Response for both IFFL and Naive.
We believe that this is the appropriate method to assess the system's ability to respond to noise. Production rate of m-Scarlet was selected to be the changing parameter region since this is analogous to either changing values for degradation rate of mf-Lon or the induced degradation rate of m-Scarlet by the protease. The method not only covered output expressions in wide range of parameter regions, but it also guaranteed to illustrate output's behavior across all possible range of structural inputs.
Result
I. Effect of On/Off time
IFFL is always better at decoding in the domain range where the on time for the input is long enough to trigger the transitional behavior of the reporter from stair-case to pulse-like steady state (see Figure. 2, IFFL subplot). This is due to the induced degradation rate of m-Scarlet by mf-Lon as the protease passes its half-activation threshold, which is triggered by sustained on time. When it comes to observing vector difference between two similarly closed on times (but not long enough to observe the transition behavior in reporter), IFFL and Naive performed similarly at distinguishing these temporal structures. In general, IFFL allows a wider range of distinguishable time-domain inputs relative to the Naive system.
Figure 5: The combined graphs of when off time=1, 2, 3, 4, 8 and 10. The trend is illustrated: as off time becomes longer, IFFL gradually lose its ability to distinguish input signals due to shorter on time relative to total period. Notice the heat map's symmetry along the diagonal.
II. Effects of Induced Degradation Rate
We also observed the heat map with varying induced-degradation rate of m-Scarlet by mf-Lon, dMS. According to Zhang's paper, when dMS is sufficiently low, the ability for IFFL to distinguish different temporal structure diminishes. Therefore, as dMS increases, the time input distinguishing ability of IFFL increases.
Figure 6: Effect of varying dMS and duration on time courses. Time courses with different number of pulses demonstrate the range in the distinguishing capability for a changing dMS. As dMS increases, the ability to differentiate pulsing and more sustained inputs increase.
By observing values on heat maps, we reached the conclusion that increase in induced degradation rate leads to better performance of IFFL in general. For some regions, when IFFL is extraordinarily superior to Naïve system (for example, when difference = 2.5), increase in dMS yields even better decoding performance; for some regions, when IFFL is moderately better at decoding input than the Naive system, increase in dMS lowers IFFL's ability to decode input structure.
Figure 7: Varying induced degradation rate affects IFFL's decoding performance. (Left) when dMS=6.2x, the highest vector difference is 1.7, and lowest value is -1.2. (Right) when dMS is 26.2x, the highest red value is 2.8, the lowest value is -0.94. We could see that IFFL's performance improved in general. The column corresponds to on2=4 has much higher value compared with the same column when dMS = 6.2x. In addition, the column for On2=1 when dMS = 26.2x also has much higher value as the saturation of blue areas decrease. However, for some cases where IFFL is just moderately better at decoding input structure than Naive, IFFL's performance becomes worse, observing that values of right-side columns decrease.
III. Robustness to Noise
The demonstration of noise damping property of IFFL can be explained
through proportionality effect of mf-Lon. Even with noisy inputs, mf-Lon is
produced accordingly to weaken the effect of stochastic expressions seen in
m-Scarlet. In the case of naive’s high variance, result shows very consistent
noise response through all combination of off and on ratios. Regardless,
IFFL is exponentially more robust to noise than naive.
Figure 8: Variance as direct translation to noise. Variance of IFFL with respect to all combination of off and on time is exponentially lower than that of Naive.
Figure 9: Contour graph shows that at off-on ratio of 1:2, variability in concentration of expression is highest with respect to parameter value range. This variance is still much lower than that of naive model.
Discussion
The goal of this model was to demonstrate the effectiveness of IFFL at processing information in time-domain inputs relative to the performance of its alternative, Naive system. Through this lens, aspects such as input distinguishing abilities and sensitivity to noise were examined through numerical computation and analysis, and the results indicate succinct distinction in IFFL's performance.
When structural inputs are widely different, it is obvious to detect such changes from the output expression in any arbitrary circuits. However, the ability to identify a wide range of temporal strucutures in terms of on and off ratios is the appealing characteristic of IFFL. Moreover, computational result revealing the motif's high fidelity in responding to noisy input ensures the functionality of the circuit compared to its alternative. With these findings, we hope future iGEM teams could built upon this foundation to go further on the insight of how natural systems encode time-domain information, and based on this knowledge, implement future synbio projects useful in this track.
Reference
[1] Zhang C, Tsoi R, Wu F, You L (2016) Processing Oscillatory Signals by Incoherent Feedforward Loops. PLoS Comput Biol 12(9): e1005101.
