Planning Process
Differential Model
Stochastic Model

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 cooperativity 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. Circuit diagram.

Figure 2. Independent binding sites.

Figure 3. Perfect cooperativity.

During the analysis of these systems, the value of S became an area of interest. In order to determine the nature of this function, the biological system from which the S is derived was studied. In order to study this system, a paper written by Kofahl and Klipp’s titled Modeling the dynamics of the yeast pheromone pathway was analyzed. There were two methods employed to analyze this model: the deterministic model and a stochastic model. The deterministic model detailed in the paper was recreated and the level of activated Ste12 was measured as a function of the amount of mating factor. The stochastic model was created by the NTNU team. Using Mohan, Shahrezaei, et. al.’s paper The scaffold protein Ste5 directly controls a switch-like mating decision in yeast, the Norwegian team made a direct correlation between the activation of Fus3 and the amount of mating factor, and measured activated Ste12 activity as a function of Fus3 activation. The results of these models indicated that for a given amount of mating factor, a constant level of Ste12 is produced. The diagrams of these models are shown below:

Figure 4. Steady state response diagrams.

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 represents the amount of protein. The first illustration depicts a system where once the signal passes a particular threshold the circuit is turned on and will continue to produce protein. The second depicts a system where there is a particular threshold of mating pheromone required to bring about a stable level of protein; if the signal falls below this level, the protein production is transient and returns to zero. The third depicts a system in which there is no discontinuity in protein production, and signal is required for all protein production.

Figure 5. Bifurcation diagrams.

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 values of the parameters; specifically, the decay terms must be less than the production terms of the system. The results of the model make good biological and intuitive 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.

For an in depth look at the work and analysis that went into developing the mathematical model for this system download this pdf.

The team had a collaborative effort with the iGEM team from Norwegian University of Science and Technology (NTNU). Their contribution to the project was the development of a stochastic model of the yeast pheromone response pathway presented in the Kofahl and Klipp’s paper. In particular, the NTNU team was looking at the system’s Ste12 activation response from varying amounts of initial mating factor. Using a paper from Nature, the team discovered that there was a directly proportional relationship between the amount of activated Ste12 and the amount of free Fus3. This allowed for a substantial simplification of the system. The result of their stochastic simulations for 250 micromolar mating factor is shown below:

This plot shows that a given amount of mating factor will result in a fairly steady amount of activated Ste12. Taking the average of the various values produced by the simulation allowed for the production of a dose-response curve of the system. This curve is shown below:

This result is significant because it supports the differential model’s conclusion that for a particular amount of mating factor, a relatively constant level of Ste12 activation is observed. This provided the team with more evidence for the assumption that the original input signal can be treated as a gradually decaying constant.

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