Making a sensor involves more than just the creation of the basic biological circuit, as there could be many factors that might improve the performance of the system. An example of these could be the strength of the RBS, the spacing between the promoter and the reporter gene, or the copy number of the plasmid used to house the circuit. While we couldn't nearly address all of these points in our project, we did want to look into the optimization of our system. To do this, we turned to mathematical modelling to create a visual representation of our system. We worked off of a basic framework published by Koutinas, M., et al.,[1,2] which was modelling the expression of the Ps promoter with XylR induction. The Ps promoter is the natural promoter found with the xylR gene, and the XylR protein can interact with both it and the Pu promoter we used in our project. Due to similarities between the Ps and Pu promoters, we assumed the deactivation rate of the two components were alike. For remaining values, we simply replaced the values of the Ps promoter with known values of the Pu promoter, keeping constants reported for XylR the same. In the end, we created five equations to represent the action of our system. Our model shows our biobrick producing the protein XylR, binding with xylene and the relationship to the output of our respective proteins. Each of our formulas are constructed as a rate in which a concentration or output is given of a specific substance in respect to time- otherwise known as a derivative. We will now expand upon the equations we modified and used. The first equation above represents the transcription of DNA into the XylR RNA transcript. RNA is the representation of the transcript, with P representing the number of plasmids inside a cell. J23100TC represents the activity of our promoter controlling XylR. ta is the transcription rate of the xylR into RNA, based on the transcription rate of E. Coli and the size of XylR, and KRNAdeg represents the degradation rate in the E. Coli. The second and third equations represent the inactive and active forms of XylR- or the concentration of XylR that has not bonded to xylene (XylRi) and the concentration of XylR that has bonded to xylene (XylRa). RNA is the substitution of our first equation. tr is the translation rate of our RNA into XylR. rXylR is the oligomerization constant of XylR, rR,XylR is the dissociation constant of active XylR, and xyl in the total concentration of xylene. αXylRi accounts for degradation of XylR. The fourth equation represents the concentration of xylene present. XylRi and XylRa refer to the concentrations of the inactive and active forms of XylR. rXylR is the oligomerization constant of XylR, rR,XylR is the dissociation constant of inactive XylR and xyl is the total concentration of xylene. αXylRi accounts for the degradation of XylRa, which would release the xylene it held. We multiplied by seconds per hour to get a domain more realistic to our needs. The fifth equation represents the output of the pu promoter. PuTC represents the activity of the Pu promoter or the output of protein. αpu is the deactivation rate of the Pu promoter. KXylRa is the activation coefficients of the Pu due to binding and nps,a is the hill coefficient for the interaction. β0 and βPS are the basal and maximal expression, respectively, of the Pu promoter. In order to create an output, we used the program Scilab. By inputting these equations into code we were able to have the program calculate our estimates and graph the results. In the end our output was around 5.5 mPoPS. This is a logical output and tells us our biobrick will successfully over express the protein. We set the output of our model in PoPS due to our system's flexibility when it comes to our reporter protein. This allows us to quickly add in an extra equation to turn our current output into fluorescence, LacZ output, or pigment production without having to worry about editing other parts of our model. Below is a copy of our code that we had used in our program. The resulting graph is the output of our Pu promoter. While we have yet to apply our modelling framework to the optimization of our sensor, we have created the foundation of a model that can be easily tweaked to test the effect of different variables on sensor performance. Through this we aim to do a sensitivity analysis of the various factors involved in our model in the future so that we can predict what to change for version 2 of our sensor. From there we can return to our model, feeding new data into it to create many future versions of our system, improving performance each time