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<section class="bg-light content" style="background-size: 100%;margin-top: 0em;padding-top: 1.1em;" class="height-90 parallax">
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          <h1 class="mx-auto my-0 text-center">Mathematical Modelling</h1>
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<h3 class="mx-auto my-0 text-center">Introduction</h3>
	<p>Mathematical Modelling was incorporated into Phase 1 of our project as a way to better describe our system. It is significantly important since it is an efficient way to optimize our system and devices for Phase 2 which will be focused on experimentally testing and characterizing our prototypes and parts, respectively. To build the mathematical models based on Ordinary Differential Equations (ODEs), SimBiology, a MATLAB feature, was used. This MATLAB feature provides tools to model, simulate and analyze biological systems and is highly beneficial for providing accurate quantitative results of our system. Mathematical models were built for both genetic circuits, Mercury and RDX, in order to simulate the expected behavior of each device. </p>
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<h3 class="mx-auto my-0 text-center">Cell Signaling Kinetics</h3>
<p>To model our system and the reactions that take place within our devices, we have used the Law of Mass Action kinetics to determine how the concentrations of the species change with respect to time. For the first part of our model in Phase 1, we have assumed that reactions are irreversible. For Phase 2, after sufficient experimental data is obtained, reactions will be assumed to be reversible due to the fact that most biological reactions are deemed reversible and enough data will be collected to be able to build these reversible models.</p>
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<h6 class="mx-auto my-0 text-center">We consider a first order irreversible reaction as the following:</h6>
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<h6 class="mx-auto my-0 text-center">From the above reaction, we get the following differential equation which describes the change in concentration of the species B with respect to time.</h6>
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<h6 class="mx-auto my-0 text-center">For a reversible reaction system, we consider a first order reversible reaction as the following:</h6>
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<h6 class="mx-auto my-0 text-center">From the above reaction, we get the following differential equation which describes the change in concentration of the species B with respect to time.</h6>
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<h3 class="mx-auto my-0 text-center">Translation Rate</h3>
<p>Using the assumptions the 
<a href="https://2016.igem.org/Team:Sydney_Australia/ InternalCellular">2016 Sydney Team</a> (https://2016.igem.org/Team:Sydney_Australia/ InternalCellular) found in a literary search, we used the translation rate of 17.1 amino acids/seconds. This rate uses the assumption that the factor affecting it is the speed of the ribosome. The translation factor is also affected by the amount of available mRNA transcripts for the ribosome to bind to. The values used as Translation Factors are:</p>

<center><img src="https://static.igem.org/mediawiki/2020/9/98/T--RUM-UPRM--TABLA1INGE.png" style="width: 80%"> </center>
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<center><img src="https://static.igem.org/mediawiki/2020/0/07/T--RUM-UPRM--TABLA2INGE.png" style="width: 80%"> </center>
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<center><img src="https://static.igem.org/mediawiki/2020/c/cc/T--RUM-UPRM--TABLA3INGE.png" style="width: 80%"> </center>
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<center><img src="https://static.igem.org/mediawiki/2020/d/d3/T--RUM-UPRM--TABLA4INGE.png" style="width: 80%"> </center>
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<center><img src="https://static.igem.org/mediawiki/2020/5/53/T--RUM-UPRM--TABLA5INGE.png" style="width: 80%"> </center>
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<center><img src="https://static.igem.org/mediawiki/2020/7/74/T--RUM-UPRM--TABLA6INGE.png" style="width: 80%"> </center>
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<h3 class="mx-auto my-0 text-center">Mathematical Models for the Mercury and RDX Genetic Circuits</h3>
<p>For the Mercury and RDX Genetic Circuits, a mathematical model was developed for each one of their devices, for Mercury: Device #1: Detection and Absorption of Mercury, Device #2: Bioremediation of Mercury and Device #3: Killswitch of Mercury Parts, and for RDX: Device #1: Detection of RDX, Device #2: Biodegradation of RDX and Device #3: Killswitch of RDX Parts. For each device, we have provided the ODE diagram representation in SimBiology and Concentration (mM) versus time (hr) plots to describe the system’s behavior. An overall analysis for all devices shows similar behaviors of the concentration as a function of time, as can be observed in the graphs for all devices, the time is directly proportional to the concentration of translated proteins, as the time increases, the concentration of translated proteins increases. Further parameters can be considered for this system once experimental data is available during Phase 2 of the project in order to obtain more accurate results of the system that is being proposed and studied.</p>
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<h6 class="mx-auto my-0 text-center"><b>Mercury Device #1: Detection and Absorption of Mercury</b></h6>
<h6 class="mx-auto my-0 text-center"><b>Mercury Device #2: Bioremediation of Mercury</b></h6>
<h6 class="mx-auto my-0 text-center"><b>Mercury Device #3: Killswitch of Mercury Parts</b></h6>
<h6 class="mx-auto my-0 text-center"><b>RDX Device #1: Detection of RDX</b></h6>
<h6 class="mx-auto my-0 text-center"><b>RDX Device #2: Biodegradation of RDX</b></h6>
<h6 class="mx-auto my-0 text-center"><b>RDX Device #3: Killswitch of RDX Parts</b></h6>
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<h3 class="mx-auto my-0 text-center">Phase 2 Plans</h3>
<p>The mathematical model presented above is highly beneficial to describe the Mercury and RDX prototype that is being proposed; it provides a quantitative insight into our system and its behavior, making it possible to optimize it through experimental work during Phase 2 of the project. This second phase will consist of recollecting experimental data in order to calculate the different kinetic constants to relate each component of the circuit and provide more accurate results through constant alteration and improvement of this model.</p>
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