Team:RHIT/Modeling

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<p>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.</p><br />
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<p>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.</p>
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<br />dP/dt eqn img here<br /><br />
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image 1 here
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<p>This 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.</p><br />
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<h4>Figure 1. Independent binding sites</h4>
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<br />dB/dt eqn img here<br />
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image 2 here
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<br />dU/dt eqn img here<br /><br />
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<h4>Figure 2. Cooperative binding</h4>
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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.</p><br />
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<p>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.</p>
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<br />dS/dt eqn img here<br />
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Signal/Response Diagrams here
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<br />dA/dt eqn img here<br /><br />
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<p> 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.</p>
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<p>These 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.</p><br />
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<br />dX/dt eqn img here<br /><br />
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<p>Finally 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.</p><br />
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<p>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, <i>Modeling the dynamics of the yeast pheromone pathway</i>.</p><br />
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Revision as of 14:44, 17 August 2012

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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:



  • Mating Factor Binds to Ste2/Ste3 homodimer (Ste2 for mat alpha / Ste3 for mat a)
  • Gpa1 subunit exchanges GDP for GTP
  • Gpa1 releases Ste4/G-gamma heterodimer
  • Ste4/Ste18 (through Ste4), transmits the signal to the following complexes, Ste5/Ste11, Ste20 protein kinase, and a Far1/Cdc24 complex
  • Ste4 binds to Cdc24, Far1 brings Cdc24 to Cdc42, Cdc24 facilitates the exchange of GDP for GTP, causing activated Cdc42 to bind to the complex between Ste20 and Bem1, resulting in the activation of Ste20 * Ste20 is in a low-activity state before being bound by Cdc42
  • Ste4 binds to Ste5, Ste5 acts as an adaptor to bring Ste4-beta and Ste11 toward Ste20
  • Ste20 phosphorylates Ste11 (aided by Ste50), causing it to activate
  • Ste11 then activates Ste7 by phosphorylating it
  • Ste7 in turn activates Fus3 and Kss1 by phosphorylating them *Ste5 also plays a role in all both of these events
  • Fus3 and Kss1 phosphorylate Ste12, Dig1 and Dig2, in their phosphorylated state Dig1 and Dig2 release activated Ste12 *Kss1 holds the Dig1/Dig2 complex together with Ste12
  • Activated Ste12 binds to DNA to facilitate transcription of pheromone response genes

    • 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.

    image 1 here

    Figure 1. Independent binding sites

    image 2 here

    Figure 2. Cooperative binding

    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.

    Signal/Response Diagrams here

    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.

    Stochastic text

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