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| − | <h1 class="h-sectionT">MODEL OF RFP PRODUCTION</h1> | + | <h1 class="h-sectionT">MATHEMATICS SECTION</h1> |
| | + | <p>According to our bench project we divided the Math Modelling in two: (1) RFP production taking in account the regulation of MerR gene, which turns RFP expression possible only in the presence of Mercury. This model allow us to understand the force of the new promoters we designed by the speed of RFP’s production; (2) A model to predict the production of the complex Omp-A-Phytochelatin and bioaccumulation of mercury.</p> |
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| − | <h3 style="color:#b54df5;margin-left: 2em">1. Modelling</h3> | + | <div class="imgCentralizada" > |
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| − | <p>We want to characterize the promoter’s velocity of expression due to presence of mercury, so we will attach an RFP gene to it to produce.</p> | + | < a href="https://static.igem.org/mediawiki/2016/ 2/ 22/ UFAM_UEA_MATH_%281%29. pdf" target=" blank"><img style="width: 15em; heigth: 15em" src="https://static.igem.org/mediawiki/2016/ b/ ba/ UFAM_UEA_MATH_% 283%29.png"/></ a> |
| − | <p>First we will model the production of RFP due to Hg<sup>2+</sup> at stationary time phase with this configuration. Then we add the initial phase consideration</p>
| + | <a href="https://static.igem.org/mediawiki/2016/ b/ b4/ UFAM_UEA_MATH_% 282%29. pdf" target=" blank"><img style="width: 13em; heigth: 13em" src="https://static.igem.org/mediawiki/2016/5/ 5b/ UFAM_UEA_MATH_% 284%29.png"/></a> |
| − | | + | </div> |
| − | <p>1.1 Stationary time phase. </p>
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| − | <p>The GMO we have is:</p>
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| − | <div class="imgCentralizada"><img src="https://static.igem.org/mediawiki/2016/thumb/1/1b/UFAM_UEA_Team_math_%281%29.png/799px-UFAM_UEA_Team_math_%281%29.png" />
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| − | <p>Where MerR is a repressor and releases when H2+ is presented in the interior of the cell, we will note this amount as Hgin and the exterior as Hgout. And Promoter nP is the promoter which velocity we want to characterize. The activation of the gene in the pressence of Hgin is represented like:</p>
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| − | <div class=" imgCentralizada"><img src= "https://static.igem.org/mediawiki/2016/8/8f/UFAM_UEA_Team_math_%282%29.png" /> | + | |
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| − | <p>where G is the inactive form of gene repressed by merR, and X is its active form. Through the law of mass action we derive the diferential equation</p>
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| − | <p>(1)<img src="https://static.igem.org/mediawiki/2016/6/6b/UFAM_UEA_Team_math_%283%29.png" width=250 height=60/> <p>
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| − | <p>At the equillibrium state, ie. <img src="
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| − | https://static.igem.org/mediawiki/2016/4/47/UFAM_UEA_Team_math_%284%29.png" width=80 height=40/>, we have that the proportion of genes in the activated gene is</p>
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| − | <p>(2)<img src="
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| − | https://static.igem.org/mediawiki/2016/1/17/UFAM_UEA_Team_math_%285%29.png" width=250 height=60/></p>
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| − | <p>Where <img src="
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| − | https://static.igem.org/mediawiki/2016/3/30/UFAM_UEA_Team_math_%286%29.png" width=80 height=40/></p> | + | |
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| − | <p>So this is the average production rate of a typical gene, so the average mRNA production will be</p> | + | |
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| − | <p>(3)<img src="
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| − | https://static.igem.org/mediawiki/2016/5/5e/UFAM_UEA_Team_math_%287%29.png" width=350 height=80/></p> | + | |
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| − | <p>Where α is the production of mRNA due to the stochastic nature of the binding of merR for repressing the RNA Polymerase and δ1 is the degradation factor of mRNA. The reaction for production of RFP from mRNA is</p>
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| − | <div class=" imgCentralizada"><img src= "https://static.igem.org/mediawiki/2016/4/43/UFAM_UEA_Team_math_%288%29.png" /> | + | |
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| − | <p>so we derive the diferential equation</p>
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| − | <p>(4)<img src="
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| − | https://static.igem.org/mediawiki/2016/7/73/UFAM_UEA_Team_math_%289%29.png" width=350 height=80/></p>
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| − | <p>where δ2 is the degradation constant of [RFP]. The variation of exterior mercury is </p>
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| − | <div class="imgCentralizada"><img src="https://static.igem.org/mediawiki/2016/3/38/UFAM_UEA_Team_math_%2810%29.png" />
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| − | <p>from here we derive the differential equation</p>
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| − | <p>(5)<img src="
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| − | https://static.igem.org/mediawiki/2016/1/1e/UFAM_UEA_Team_math_%2811%29.png" width=350 height=80/></p>
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| − | <p>As the interior mercury is used by the inactive form of the gene, the variation of Hgin<sup>2+</sup> is</p>
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| − | <p>(6)<img src="
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| − | https://static.igem.org/mediawiki/2016/thumb/7/7f/UFAM_UEA_Team_math_%2812%29.png/800px-UFAM_UEA_Team_math_%2812%29.png" width=350 height=80/></p>
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| − | <p>To solve this system we will assume that the permeability of the cell memebrane to mercury is instantaneus then we have that <img src="
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| − | https://static.igem.org/mediawiki/2016/7/76/UFAM_UEA_Team_math_%2813%29.png" width=80 height=40/> then <img src="
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| − | https://static.igem.org/mediawiki/2016/c/c5/UFAM_UEA_Team_math_%2814%29.png" width=150 height=60/> so we have that the variation of the interior mercury is</p>
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| − | <p>(7)<img src="
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| − | https://static.igem.org/mediawiki/2016/5/53/UFAM_UEA_Team_math_%2815%29.png" width=350 height=80/></p> | + | |
| − | <p>And assume that mRNA is at Quasi-Steady state, ie. <img src="
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| − | https://static.igem.org/mediawiki/2016/5/52/UFAM_UEA_Team_math_%2816%29.png" width=80 height=40/>, then we have
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| − | that the amount of mRNA is
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| − | </p>
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| − | <p>(8)<img src="https://static.igem.org/mediawiki/2016/4/4d/UFAM_UEA_Team_math_%2817%29.png" width=350 height=80/></p>
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| − | <p>Then we have that the velocity of production of RFP is</p>
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| − | <p>(9)<img src="https://static.igem.org/mediawiki/2016/0/0f/UFAM_UEA_Team_math_%2818%29.png" width=350 height=80/></p>
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| − | <p>So we will also assume that the degradation of RFP is very slow, δ2 = 0, and also the stochastic production of mRNA is very slow, α = 0. Then finaly we have that the production of RFP is</p>
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| − | <p>(10)<img src="
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| − | https://static.igem.org/mediawiki/2016/c/c7/UFAM_UEA_Team_math_%2819%29.png" width=350 height=80/></p>
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| − | <p>Where <img src="
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| − | https://static.igem.org/mediawiki/2016/e/e2/UFAM_UEA_Team_math_%2820%29.png" width=120 height=60/></p>
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| − | <p>1.2. Initial phase. Since bacteria grow exponentially, it is often useful to plot the logarithm of the relative population size [Y = ln(N/N0)] against time .So lets use the Gompertz equation to model this. The three phases of the growth curve can be described by three parameters: the maximum specific growth rate, µm is defined as the tangent in the inflection point; the lag time, λ, is defined as the x-axis intercept of this tangent; and the asymptote [A = ln(N/N0)] is the maximal value reached. Here we are not considering the death rate.<sup>[1]</sup></p>
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| − | <p>(11)<img src="
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| − | https://static.igem.org/mediawiki/2016/f/fb/UFAM_UEA_Team_math_%2821%29.png" width=300 height=60/></p>
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| − | <p>Lets add that the amount of bacteria changes to our model. So we have more bacteria. As Vmáx depends on the amount of bacteria, so we can propose that a single bacteria has a velocity <img src="
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| − | https://static.igem.org/mediawiki/2016/9/90/UFAM_UEA_Team_math_%2822%29.png" width=120 height=60/> production of RFP per unit of time and then</p>
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| − | <p>(12)<img src=" | + | |
| − | https://static.igem.org/mediawiki/2016/e/ef/UFAM_UEA_Team_math_%2823%29.png" width=300 height=60/></p>
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| − | <p>So our model now is</p>
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| − | <p>(12)<img src="
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| − | https://static.igem.org/mediawiki/2016/1/10/UFAM_UEA_Team_math_%2824%29.png" width=300 height=60/></p>
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| − | <p>Here we are assuming that when a bacter appears it alredy has all the properties necessary to work, so that delay is inside the growing time.</p>
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| − | <h3 style="color:#b54df5;margin-left: 2em">2. For Phytochelatin modelling:</h3>
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| − | <p>As the Phitoquelatin is expressed linked with Omp-A we can treat it like a single complex C. So we have the reaction scheme:</p>
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| − | <div class="imgCentralizada"><img src="https://static.igem.org/mediawiki/2016/3/38/UFAM_UEA_Team_math_%2810%29.png" />
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| − | <div class="imgCentralizada"><img src="https://static.igem.org/mediawiki/2016/a/a6/UFAM_UEA_Team_math_%2826%29.png.png" />
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| − | | + | |
| − | <p>Where M is the free mercury and C∗ is the mercury linked with the complex C, kC is the rate of production of the complex and δC is the rate of degradation of the complex. As we put the bactery with the mercury after some hours we can assume that the cell are full populated so we have that [CT] = [C] + [C∗] then we only need to know how vary [C∗] because [CT] is constant.</p>
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| − | <p>(1)<img src="
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| − | https://static.igem.org/mediawiki/2016/1/11/UFAM_UEA_Team_math_%2827%29.png" width=300 height=60/></p>
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| − | <p>Also as the total concentration of mercury does not change we have that M0 = [M]+[C*] where [M] is the concentration of free mercury, [C*] is the concentration of the mercury linked with Phitoquelatin that is linked with Omp-A per cell.</p>
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| − | <p>(2)<img src="
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| − | https://static.igem.org/mediawiki/2016/4/4c/UFAM_UEA_Team_math_%2828%29.png" width=300 height=60/></p>
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| − | <p>We can assume that the rate of splitting of the complex C∗ is 0, because Mercury and Phitoquelatin have very hight affinity.</p>
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| − | <p>(3)<img src="
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| − | https://static.igem.org/mediawiki/2016/6/60/UFAM_UEA_Team_math_%2829%29.png" width=300 height=60/></p>
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| − | </br></br>
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| − | <p>References:<p>
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| − | <p><i>[1] MH Zwietering, Il Jongenburger, FM Rombouts, and K Van’t Riet. Modeling of the bacterial growth curve. Applied and environmental microbiology, 56(6):1875–1881, 1990</1></p>
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| | </div> | | </div> |
MATHEMATICS SECTION
According to our bench project we divided the Math Modelling in two: (1) RFP production taking in account the regulation of MerR gene, which turns RFP expression possible only in the presence of Mercury. This model allow us to understand the force of the new promoters we designed by the speed of RFP’s production; (2) A model to predict the production of the complex Omp-A-Phytochelatin and bioaccumulation of mercury.
