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).
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 distinguishing 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 vector for both systems. This vector was calculated using the Euclidean distance among all points between the two expressions, then a determining value was derived by taking the vector difference between Naive and IFFL. Finally, these values were organized as 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.
Result
I. Effect of On/Off time
The conclusion we reached is that IFFL works better than Naïve system when off time is short, or the ratio of on time to the period is large. When on time is fixed, if off time is short, or ratio of on time to the period is large, input is not sustained, then mf-Lon will be less likely to reach the threshold to activate the induced degradation to m-Scarlet, thus produces stair case output, which is identical to the behavior seen in Naïve system. Therefore, short off time scenario is desired for effective use of IFFL.
Figure 1: Vector difference between IFFL and Naïve system relative to different off time when processing signals. vector difference is defined by the Euclidean distance of concentration expression of IFFL to naïve at fixed off time with two different on time (for example, on1=4 and on2=9). When IFFL outperformed Naïve system (difference in concentration expressions as a function of time between two distinct on time are large), the vector difference is positive (red areas) and vice versa (blue areas). (Left) when off time = 1, most areas are red, meaning that IFFL is relatively better at distinguishing its input structures (shadings). (Right) when off time = 10, most areas are white in graph, and some are blue. Notice the symmetry along the diagonal.
Figure 2: For the case offtime=1, on1=8 and on2=2, the specific behaviors of IFFL and Naive system are shown on the right. Since there is a huge positive difference, i.e., IFFL outperformed Naive, the value = 2.9 is positive and high.
Figure 3: for the case offtime=10, on1=8 and on2=2, the specific behaviors of IFFL and Naive system are shown on the right. Since Naive outperformed IFFL in this case, the value = -1.1 is negative and high.
Figure 4: The combined graphs of when off time=1, 2, 3, 4, 8 and 10. The trend is illustrated: as off time becomes longer, the IFFL gradually lose its ability to distinguish input signals.
II. Effects of Induced Degradation Rate
Another parameter we tested was the induced-degradation rate of mf-Lon to m-Scarlet, i.e., dR. According to Zhang's paper, when dR is sufficiently low, the ability for IFFL to distinguish different temporal structure diminishes. Therefore, below a certain point, as dR increase, the time input distinguishing ability of IFFL increases.
Figure 5: Effects of varying dR and duration on time courses. Time courses with different number of pulses demonstrate the range in the counting capability for a changing dR. As dR increases, the ability to differentiate pulsing and more sustained inputs increase.
Similarly, we created heat maps similar to Figure 1 above but with different dR. We then compared these heat maps with fixed off-time to see the impact of dR on distinguishing ability of IFFL.
By observing the vector differences on heat maps, we reached the conclusion that increase in 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 of dR causes even better performance and difference may reach 3; for some regions, when IFFL is moderately superior to naïve system (for example, difference = 0.3), increase of dR actually makes their performance worse. In addition, for some parts that naive system outperformed IFFL, IFFL performance also increases as dR increases.
Figure 6: Changing of Induced Degradation Rate affects IFFL performance. (Left) when dR=6.2x, the highest vector difference is 1.7, and lowest value is -1.2. (Right) when we changed dR to 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 dR = 6.2x. In addition, the column for On2=1 when dR = 26.2x also has much higher value as the saturation of blue areas decrease. However, for some cases where IFFL is just moderately superior than Naïve, IFFL's performance become worse, observing that values of right-side columns decrease somehow.
III. Robustness to Noise
With IFFL's ability to detect small changes in temporal structure from the input, one would expect that the system would be susceptible to noise. We performed the following numerical testing for both IFFL and naive model and compare the variance as direct translation of output's noisiness. Here, off-on ratio refers to the time ratio when the heat is turned on (CI degrades completely) or off, analogous to Period being off + on and Duration being the on time. Gaussian noise with average value 0 and standard deviation of 0.1 was applied concentration of CI:
Start with fixed value for off-on ratio, calculate all concentration expression of m-Scarlet within a specific range 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 7: 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 8: 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.
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 and weakens 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.
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 closely related input 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.
