This pathway has two inputs, but only has two states: the target gene expressed or not. If “Pulse on” input signal is higher than “pulse off”, it will lead to the expression of gene of interest. Otherwise, it will not lead to expression.
In order to expand our scope of the filter, we also design the eukaryotic pathway which has the function of corresponding filter/switch. Meanwhile, we also need to understand its working condition clearly.
Modeling
In this pathway, the time of spreading of the substance or each combination/response reaction, compared to the whole time scales (protein), is negligible, so we adopted simplified differential equation to simulate the pathway this time.

Parameter details
more details 
Fomular
more details
Parameter  Description 

trc pulseoff  rate of pulseoff transcription 
trc pulseon  rate of pulseon transcription 
copynum  copy number of plasmid 
trc 1  rate of ABF2 transcription 
trl pp2cs  rate of pp2cs translation 
trl snRk  rate of snRk translation 
trl ABF2  rate of ABF2 translation 
trl Gene  rate of Gene_of_interset translation 
deg mPP2Cs  rate of mPP2C degradation 
deg mSnRk2.2  rate of mSnRk2.2 degradation 
deg mABF2  rate of mABF2 degradation 
deg mGene  rate of mGene_of_interset degradation 
deg pp2cs  rate of pp2cs degradation 
deg snRk  rate of snRk degradation 
deg ABF2  rate of ABF2 degradation 
deg Gene  rate of Gene_of_interset degradation 
Vmax mSnRk2.2  maximum transcription rate of promoter RD29A 
Vmax mgene  maximum transcription rate of promoter RD29A 
KmABF2  apparent association constant for ABF2 binding with RD29A 
Vmax ABF2  maxinum rate of Phosphorylation 
Vmax PP2C  maxinum rate of Dephosphorylation 
KmSnRK  MichaelisMenten constant of Phosphorylation 
KmPP2C  MichaelisMenten constant of Dephosphorylation 
Analysis
We have only on/off states of the eukaryotic system, so the complexity of this model will be lower, but it probably means that the system can have a better stability and robustness, for that reason, we will focus on the sensitivity analysis of parameters, meanwhile, validate the designed functions work out or not.

Test of switch function
Before we begin to do the test, we need to find out whether our design has reached the expected goal. So we changed the signal intensity ratio of pulse on and pulse off and do the statistics of the steadystate concentration of protein under different ratio.
It is clear that there exists a distinct boundary between opened state and closed state. It can be regarded as the turning point of the system when the ratio is about 1.

Test of filter function
We adopted a simple sine function as input to verify the filtering performance.* (The input signal in the figure is to show the result more clearly so that we have moved the graph. The actual strength is 0.1sin（t）)
We can see that the output signal, which does not contain obvious waveform, is still stable. And the ABF2 phosphorylation/dephosphorization of the system has played the role as the "capacity" as we analyzed before, so as to improve the filtering effect of the system.

Parameter sensitivity analysis/robustness evaluation
The main approximate treatment in this circuit should be substrate disaggregation coefficient correction, when SnRK2.2 / PP2Cs concentration is low and supersaturated by the substrate, meanwhile, part of the substrate cannot be catalytic in time. So we mainly analyzed the velocity of substrate producing rate for testing the sensitivity of the parameters, which is named trc1.
Similarly, we keep pathways in the condition of open and choose to change the ratio of signal strength to find out the final expression level and the response time for analyzing.
The expression level of the final production varies with the changing of trc1, but it still maintains itself in a high expression level, which has a minor influence on our switching or the filtering function.
Even though there exist some twists in the responding time, all the floating range is still in 10%. And our circuit shows a great stability after we enlarged the numerical value of trc1.
Summary
Our eukaryotic pathway can be worked out as a filter and a swich in a steady way. And the analyze of sensitivity of parameters also shows that we should use the promoter, which has much bigger velocity of transcription, to help maintain a great robustness.