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| − | <div class = "h1">Model Building Process</div> | + | <div class = "h1">Hypoxia induced fluorescence</div> |
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| | <div class="pagediv"> | | <div class="pagediv"> |
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| | <div class="para"> | | <div class="para"> |
| − | <div class="h3">Hypoxia induced fluorescence </div>
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| − | In addition to the induction of iRFP fluorescence by nitric oxide, we also tested the induction of iRFP fluorescence with a hypoxia promoter. We expected iRFP fluorescence to increase with increased hypoxic conditions (less oxygen) when using NarK promoter and fdhf promoters, both characterized as hypoxia-inducible. | + | In addition to the sinduction of iRFP fluorescence by nitric oxide, we also tested the induction of iRFP fluorescence with a hypoxia promoter. We expected iRFP fluorescence to increase with increased hypoxic conditions (less oxygen) when using NarK promoter and fdhf promoters, both characterized as hypoxia-inducible. |
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| | Transcription of the fdhf promoter is regulated by an RNA polymerase with sigma factor 54 whose binding is dictated by presence of an additional activator complex consisting of FhlA and formate. Only when the FhlA-formate complex is present will the sigma-54 polymerase initiate transcription. This process is induced by formate, but is also heavily repressed by presence of oxygen, giving it characterization as a hypoxia sensor. | | Transcription of the fdhf promoter is regulated by an RNA polymerase with sigma factor 54 whose binding is dictated by presence of an additional activator complex consisting of FhlA and formate. Only when the FhlA-formate complex is present will the sigma-54 polymerase initiate transcription. This process is induced by formate, but is also heavily repressed by presence of oxygen, giving it characterization as a hypoxia sensor. |
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| − | <img src="https://static.igem.org/mediawiki/2016/3/32/Promoter_Strengh_vs_Time_paper.png" style="display: block; margin: auto; width: 80%">
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| − | <br><br>
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| − | The first step of our model is to describe the rate of change of enzymes based on promoter strength. Here we assumed that the enzyme production rate is directly proportional to strength of the promoter. Therefore, we were able to use a mass-action kinetics equation of promoters to describe the enzyme concentration. The equation is shown below:
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| − | <img src="https://static.igem.org/mediawiki/2016/2/28/Protomter_Equations_new.png" style="display: block; margin: auto; width: 25%">
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| − | In this equation, Ai is the concentration of enzyme i, ki is the production rate of each enzyme i, kd is the degradation rate of all enzymes, and t is time. By solving this equation, we derived the equation of enzyme concentration against time.
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| − | <br><br>
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| − | <img src="https://static.igem.org/mediawiki/2016/5/54/Promoter_ODE_new.png" style="display: block; margin: auto; width: 25%">
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| − | <br><br>
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| − | Since we assumed that the promoter strength is proportional to the promoter concentration, we would use the equation to fit our data using least squares method (Fig. 1).
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| − | <br><br>
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| − | <img src="https://static.igem.org/mediawiki/2016/thumb/3/3a/Fitted_Lines_of_Promoter_Strength_vs_Time.png/800px-Fitted_Lines_of_Promoter_Strength_vs_Time.png" style="display: block; margin: auto; width: 80%">
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| − | <b>Figure 1.</b> Linear regressions fitted to normalized fluorescence vs time. The circles represent data from Jones et al., 2013. The solid lines are our regression lines. The colors indicate with which promoters the circles and lines correspond.
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| − | <br><br>
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| − | In general, the regression lines are able to capture the change of strength of each enzyme over time. In this way, the parameters are determined. The table below lists the parameter values.
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| − | <img src="https://static.igem.org/mediawiki/2016/7/7e/Promoter_Strength_Fit_Parameters.png" style="display: block; margin: auto; width: 60%">
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| − | <b>Table 1.</b> Parameters realted to promoter strength and degradation of molecules. In the table, ki (i = 1,2,3,4,5) are the production rate coefficients of promoter I (i = 1,2,3,4,5), and kd is the degradation rate coefficient of all promoters.
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| − | <br><br>
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| − | <div class='h3'>2. Modeling the Steady-state Violacein Yield</div>
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| − | After we finished the regression model of each promoter, we created a second model to describe the violacein biosynthetic pathway. The pathway (Fig. 2) involves five enzyme-catalyzed reactions and one non-enzymatic reaction (Lee et al, 2013).
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| − | <img src= "https://static.igem.org/mediawiki/2016/thumb/2/2f/Violacein_Biosynthetic_Pathway.png/737px-Violacein_Biosynthetic_Pathway.png" style="display: block; margin: auto; width: 80%">
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| − | <b>Figure 2.</b> Violacein synthetic pathway. The purple arrows highlight the five enzymatic and one non-enzymatic steps of violacein production from two molecules of tryptophan. The five enzymes are indicated by bolding (VioA, VioB, etc.).
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| − | The model was developed as three major parts. A pseudocode of this model is provided here.
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| − | <br>
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| − | <b>Define ODE System</b>
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| − | <ol>
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| − | <li>Calculate the production and degradation rate of each molecule in the pathway from the concentration of reagents and parameters.</li>
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| − | <li>Obtain the rate of change of each molecule based on the production and degradation rates.</li>
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| − | </ol>
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| − | <b>Solve the System of Nonlinear Equations at Steady State</b>
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| − | <ol>
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| − | <li>Solve the system of nonlinear equations at steady state starting at an initial guess X0.</li>
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| − | <li>Use the result as a new initial guess; repeat the numerical method to solve the system of equations again.</li>
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| − | <li>Calculate the relative error of each chemical in the new result.</li>
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| − | <li>If the maximum error is smaller than 0.0001%, output violacein concentration at steady state as the final result.</li>
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| − | </ol>
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| − | <b>Optimize Parameters to Fit Experimental Data</b>
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| − | <ol>
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| − | <li>Set the initial guess of the parameters.</li>
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| − | <li>Load the data from literature, which include the choice of promoter for each gene and the corresponding violacein yield determined experimentally.</li>
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| − | <li>For each promoter selection scenario, pass the promoter types and the temporary parameters to the steady-state model.</li>
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| − | <li>Obtain the violacein yield predicted by the steady-state model for each promoter selection scenario.</li>
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| − | <li>Compute the residual sum of squares (RSS) of between the predicted violacein yields and the violacein yields given by experiment.</li>
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| − | <li>Determine the optimal parameters by minimizing the RSS (least square method).</li>
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| − | </ol>
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| − | Using the principles of mass action kinetics, we derived the system of ODE equations in the model. The equations involves 17 parameters (Table 2). Five parameters (kA, kB, kC, kD and kE) are related to the production rates of the five enzymes, which depend only the strength of the promoter type. Another parameter, kd, is the degradation coefficient of all molecules due to the growth of E.coli. The value of this parameter is fixed and shown in Table1. In addition to these known parameters, the equations include 11 undetermined parameters related to the reaction rates at specific steps in the violacein synthetic pathway. As described in the pseudocode, we used least square regression to determine the optimal values of these parameters.
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| − | Each one of the11 differential equations describes the rate of change of specific molecule in the system. The equations consider the production, consumption, and degradation rates of the molecules. Degradation of molecules is described by first order decay. Therefore, the rate of degradation of a molecule depends on a degradation constant and the degradation coefficient. The degradation coefficient is identical for all molecules since it only depends on E.coli growth rate.
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| − | <div class="h3"> Differential Equations in the Model</div>
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| − | <img src="https://static.igem.org/mediawiki/2016/c/cf/Enzyme_Production_Rate.png" style="display: block; margin: auto; width: 80%">
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| − | <img src="https://static.igem.org/mediawiki/2016/thumb/7/7f/Chemical_Production_Rate_1.png/1200px-Chemical_Production_Rate_1.png" style="display: block; margin: auto; width: 100%">
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| − | <img src="https://static.igem.org/mediawiki/2016/thumb/f/fd/Chemical_Production_Rate_2.png/1199px-Chemical_Production_Rate_2.png" style="display: block; margin: auto; width: 100%">
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| | + | </div> |
| | <div class="fixed_flyer" id = "sec5" style="position:relative;z-index:5"> | | <div class="fixed_flyer" id = "sec5" style="position:relative;z-index:5"> |
| | <div class = "h1">Results</div> | | <div class = "h1">Results</div> |
