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.
Having defined what the model was supposed to answer, and what parts of the process were going to be included, the next step was to design a system of differential equations that would account for the various pieces of the model. The first draft of this system is shown below, along with a brief description of what each equation represents, and what the various terms in each equation account for.
dP/dt eqn img hereThis equation represented the pheromone concentration, (matA or mata) available to interact with the receptors of the cell. The equation was designed to account for the pheromones secreted by the yeast cells, the binding of the pheromone to the receptor, and the possible unbinding of the pheromone from the receptor. The secretion of the pheromone from one yeast cell was assumed to be independent of the changes exhibited by the other yeast cells. It was also assumed that this pheromone was equally distributed around the whole environment of the cells.
dB/dt eqn img here dU/dt eqn img here These two equations represented the two possible states of the pheromone receptor, (Ste2 or Ste3) bound to pheromone or unbound. While these proteins like most proteins will decay overtime and must be reproduced, due to the relatively small time scale that these proteins significantly contribute to the behavior of the system, these terms were assumed to be negligible. Furthermore, this model also assumes that all unbound receptors are capable of being bound and that all bound receptors are capable of signal transduction. dS/dt eqn img here dA/dt eqn img hereThese two equations represented a hypothetical signal that directly stimulated the production of the team's fluorescent hetero-transcription factor. The S equation signified the series of protein kinase interactions, caused by the production of bound pheromone receptor and a decay of the signal as the proteins are turned off. The A equation represented the total sum of the factors contributing to the synthesis of the construct, including the signal from the S equation, the auto-regulation of the construct, and the loss of the signal as the construct is synthesized.
dX/dt eqn img hereFinally this equation represented the concentration of the synthetic hetero-transcription factor that is produced by the signal, and is lost over time due to decay.
After creating this simple model, the team began doing research on the kinetic rate constants and binding affinities for the various parts of the model. During this research, the team came across a published model from Kofahl and Klipp's paper, Modeling the dynamics of the yeast pheromone pathway.
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