In order to model the behavior of incoherent feedforward loop and compare to that of naïve system, we developed a heat sensitive model which illustrates the interaction between mf-Lon, m-Scarlet and C1. We then performed analysis for both IFFL and naïve system: by including noise in input, we used stochastic model to compare noise difference and variance of the output, m-Scarlet, regarding to fixed on and off time as well as parameter regions in these two systems. In addition, we made heat maps to investigate the relationship between parameters such as duration of off time and on time of the input and performance of IFFL in distinguishing different inputs compared to naïve system. Furthermore, regarding to induced degradation rate of mf-Lon to m-Scarlet, we compared different sets of heat maps (each set has its own degradation rate) to see the impact of changing induced degradation rate of mf-Lon to m-Scarlet on superiority of IFFL to naïve system. Both of them provide insights that will inform future teams how to appropriately use IFFL under certain circumstances.
Results
When Zhang C, Tsoi. R, Wu. F and You. L demonstrated that fundamental property of incoherent feedforward loop (IFFL) can be used to exhibit temporal adaptation, they further indicated this property serves as the foundation for distinguishing input signals with varying temporal input. To demonstrate this important ability, we built quantitative ordinary differential equations (ODE) model and used heat maps and mesh graphs to examine both vector difference and noise difference between IFFL and naïve. It is shown that under most(certain/ideal) circumstances, IFFL indeed outperforms naïve system in distinguishing different structure of temporal input. Thus IFFL is capable of interpreting information in dynamic input signal more accurately.
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 and m-Scarlet will have stair case output, which is identical as naïve system. That means IFFL loses its ability to distinguish different type of temporary input. 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 difference between two on time (for example, on1=4 and on2=9). When IFFL outperformed naïve system, the vector difference is positive (red areas) and vice versa (blue areas). (Left) when off time = 1, most areas are red in graph. So that means in this case IFFL works better in general. (Right) when off time = 10, most areas are white in graph, and some part are blue. Therefore in this case IFFL work almost as same as naïve, and sometime it works better and other time naïve works better.
Figure 2: 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.
Figure 3: 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 test is 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 temporary structure diminishes. Therefore below a certain point, as dR increase, the time input distinguishing ability of IFFL increase.
Figure 4: 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 drew the 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 of degradation rate leads to better performance of IFFL in general. For some parts, 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 parts, 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 increase.
Figure 5: Changing of Induced Degradation Rate affects IFFL performance. (Left) when dR=6.2, the highest vector difference is 1.7, and lowest value is -1.2. (Right) when we changed to dR = 26.2, the highest value is 2.8, the lowest valye is -0.94. We could see IFFL performance improves in general. The column corresponds to on2=4 has much higher value compared with that column when dR = 6.2. In addition, the column for On2=1 when dR = 26.2 also has much higher value, see the saturation of blue areas decrease. However, for some cases where IFFL is just moderately superior than Naïve, IFFL performances 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 6: 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 7: 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.
Overview
In our project, we developed our model based on Zhang's paper Processing Oscillatory Signals by Incoherent Feedforward Loops. The original paper model an IFFL where X is protease, R is reporter, k2 is the production rate of X, k1is the production rate of R, dx is endogenous degradation rate of X, dR0 is endogenous degradation rate of R, dR is the induction of the degradation of R by X, kx is the half activation threshold.
Figure 8: system of ODEs to define the IFFL network
The analytical solution for above model is:
Figure 9: Analytical Solution.
Therefore, based on original model, we developed a heat-induced IFFL model:
In this system, the promoter inhibitor protein CI is degraded by heat at high temperature. So when temperature is 30 ºC, CI begins to increase, thus inhibiting the transcription process of both mf-Lon and m-Scarlet; when temperature is 37 ºC, CI degrades to 0 immediately, thus enabling the transcription process of both mf-Lon and m-Scarlet, and then mf-Lon degrades m-Scarlet once it reaches half-activation threshold. We analysed difference between heat-inducible IFFL model and its naïve model with details, with main focuses on vector difference of temporal structure distinguishing abilities and noise between IFFL and Naïve system, refer to the results part.
Discussion
Reference
[1] Zhang C, Tsoi R, Wu F, You L (2016) Processing Oscillatory Signals by Incoherent Feedforward Loops. PLoS Comput Biol 12(9): e1005101.
