In addition to the synthetic biology project, the team also pursued the creation of a mathematical model of the modified biological system. The first step of the modeling process was to gain an understanding of the various chemical and biological agents, how they interacted with each other, and ultimately how these interactions caused the physiological changes in the cell. This was done by studying the mechanism presented in Bradwell's paper, A walk-through of the yeast pheromone response pathway. Once the key components were identified, a simplified explanation of the process of interest was created. Below is the mechanism the team used for the basis of the model:
After laying out this process, the team began making evaluations as to which parts of the process would be important, and what kind of questions could be answered by the model. After careful deliberation and discussion, the team decided that the questions of interest to the project were to determine the sensitivity of the circuit and the time after exposure that fluorescence is detectable. With those questions in mind, the system was broken into four distinct portions: initiation, transduction, production, and regulation.
The initiation portion of the model pertained to the interactions between the pheromone and the receptor. Since one of the goals of the model was to address the sensitivity of the construct, this portion was mostly conserved and incorporated into the model. The transduction portion covered most of the unmodified kinase cascade covered in the mechanism. Because of the relatively short time period of these interactions in comparison to the approximated time frame of the circuit, and the well characterized and unchanged nature of this part of the pathway, these interactions were removed from the model. The third portion of the model covered the transcriptional and translational activities necessary for the production of the construct. The final module of the model was accounting for the regulatory functions contained in the construct.
For the team’s mathematical modeling section, two differential models of the system were created: a case with one or more independent binding sites and a case that exhibits cooperatively with various binding sites. Both of these systems of equations are shown below. The behavior of the steady-state solutions of the system of equations were analyzed using algebraic and graphical methods. This analysis provided insight into the possible results from this kind of system.
Figure 1. Independent binding sites
The figures below illustrate the three possible solutions predicted by the mathematical model. The X-axis is a measure of the external mating pheromone concentration, and the Y-axis depicts the amount of protein. The first illustration depicts a system where once any signal or protein is present the circuit is turned on and continues to produce more protein up to a cap. The second depicts a system where there a particular threshold of mating pheromone required to bring about a stable level of protein, below this level, the protein production is transient and returns to zero. The third depicts a system similar to the second where there is a threshold, where the protein is created up to cap, however once the signal drops below that threshold, it again returns to zero.
While it is not yet possible to verify which, if any, accurately represents the actual system, the created models predict possible scenarios that would allow for the success of the project. Furthermore, the model predicts that the success of this project is dependent solely on the parameters of the system, in particular the ratios between _____. The results of the model make good biological sense and intuitively make sense. In order for the project to be successful, either the first or second depictions must hold true. The analysis, derivation, future work, and all the work leading to these conclusions are listed below in the named sections.
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