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<h1 style="margin-bottom: 0px; text-align:left">Model&thinsp;I</h1>
<h2 style="color:#ffffff; text-align:left">Selection system</h2>
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Model&thinsp;I describes the <i>in vivo</i> <a href="https://2016.igem.org/Team:Bielefeld-CeBiTec/Project/Selection/Bacterial_Two-Hybrid_System">bacterial two hybrid selection system</a>:
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	<img src="https://static.igem.org/mediawiki/2016/0/08/Bielefeld_CeBiTec_2016_10_16_Mod_Frosch.png
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	<b>Figure 1</b>: Selection pathway. 
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	Plasmid 1 and plasmid 2 each produce one protein of the bacterial two hybrid system, respectively. When both proteins come together and bind on the DNA the reporter gene is transcribed, here beta-lactamase. The beta-lactamase then diffuses from the cytoplasm into the periplasm, where it encounters the antibiotic ampicillin and deactivates it. For detailed description see <a href="https://2016.igem.org/Team:Bielefeld-CeBiTec/Project/Selection/Bacterial_Two-Hybrid_System">bacterial two hybrid system</a>.
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 The selection pathway can be illustrated by the following reaction sequence:
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	<img src="https://static.igem.org/mediawiki/2016/4/44/Bielefeld_CeBiTec_2016_10_16_Mod_chemical.png" alt="reaction sequence" width="600px">
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	<b>Figure 2</b>: Reaction pathway of the selection system.
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The 0 stands for degeneration of proteins or complexes and the + for the interaction of two proteins or complexes.
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 The reaction sequence starts with the expression of the Evobody-RpoZ translational fusion construct [(a)] on the one hand and the target protein-DNA binding domain translational fusion construct [(b)] on the other hand. 
 Both resulting fusion proteins are degraded over time [(c)]. 
 In the figure the degeneration is marked with a 0. 
 Furthermore, together with the DNA binding site for the DNA binding domain (DNABD) both fusion proteins form the bacterial two hybrid complex [(d)]. 
 When the complex is built the RNA polymerase gets recruited and transcribes the beta-lactamase. 
 Just like the fusion proteins the beta-lactamase will be degraded after a while [(e)]. 
 When the beta-lactamase and ampicillin meet in the periplasm the ampicillin will be deactivated and the cell survives.
<br><br>
Based on the reaction sequence the chemical reaction equations can be written down:

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(I) <i>k<sub>exprP1</sub></i> stands for the expression rate of plasmid 1, which holds the Evobody-RpoZ sequence [<i>s</i><sup>-1</sup>].<br>
k<sub>degEvo</sub></i> stands for the degeneration rate of the Evobody-RpoZ protein [<i>s</i><sup>-1</sup>].
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(II) <i>k<sub>exprP2</sub></i> stands for the expression rate of plasmid 2, which holds the target-DNA binding domain (DNABD) protein [<i>s</i><sup>-1</sup>].
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<i>k<sub>degTarget</sub></i> stands for the degeneration of the target-DNABD protein [<i>s</i><sup>-1</sup>].
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(III) <i>k<sub>ass</sub></i> stands for the association rate of the bacterial two hybrid complex [(<i>s*M</i>)<sup>-1</sup>].
<i>k<sub>diss</sub></i> stands for the disassociation rate of the bacterial two hybrid complex [<i>s</i><sup>-1</sup>].
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(IV) <i>k<sub>exprbla</sub></i> stands for the expression rate of beta-lactamase (bla) [<i>s</i><sup>-1</sup>].
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(V) <i>k<sub>transpbla</sub></i> stands for the transport rate of beta-lactamase from cytoplasm to periplasm [<i>s</i><sup>-1</sup>].
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(VI) <i>k<sub>degbla</sub></i> stands for the degeneration rate of the beta-lactamase [<i>s</i><sup>-1</sup>].
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(VII) <i>k<sub>transpamp</sub></i> stands for the transport rate of ampicillin from medium to cytoplasm and back [<i>s</i><sup>-1</sup>].
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(VIII) <i>k<sub>diff</sub></i> stands for the diffusion rate of beta-lactamase and ampicillin [(<i>s*M</i>)<sup>-1</sup>].
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<div class="container text_header"><h3>Differential equations</h3></div>
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Next the language of chemistry has to be translated into the language of mathematics. A system of ordinary differential equations is generated. As this system has to be solved numerically, we have chosen a notation the software package MATLAB&reg; understands:
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	<img src="https://static.igem.org/mediawiki/2016/4/43/Bielefeld_CeBiTec_2016_10_16_Mod_DGLs.png
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 	stand<sub>expr</sub> stands for standard expression [M].
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[Evobody] or Evobody stands for Evobody-RpoZ protein [M].
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[Target] or Target stands for target-DNABD protein [M].
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[DNA] stands for DNA [M].
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[amp<sub>Peri</sub>] stands for ampicillin in periplasm [M].
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[amp<sub>Medium</sub>] stands for ampicillin in medium [M].
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[bla<sub>Cyt</sub>] stands for beta-lactamase in cytoplasm [M].
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[bla<sub>Peri</sub>] stands for beta-lactamase in periplasm [M].
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[Evobody-Target] stands for Evobody-Target protein complex [M].
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[Target-DNA] stands for Target-DNA complex [M].
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[Evobody-Target-DNA] stands for Evobody-Target-DNA complex [M].
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<i>k<sub>dissEvoTarget</sub></i> stands for disassociation rate of the Evobody-Target complex [<i>s</i><sup>-1</sup>].
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<i>k<sub>assEvoTarget</sub></i> stand for association rate of the Evobody-Target complex [(<i>s*M</i>)<sup>-1</sup>].
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<i>k<sub>dissTargetDNA</sub></i> stand for disassociation rate of the Evobody-Target complex [<i>s</i><sup>-1</sup>].
<br>
<i>k<sub>assTargetDNA</sub></i> stand for association rate of the Evobody-Target complex [(<i>s*M</i>)<sup>-1</sup>].
<br>
<i>k<sub>degTarget</sub></i> stands for degeneration rate of the target protein [<i>s</i><sup>-1</sup>].
<br>
<i>k<sub>degEvo</sub></i> stands for degeneration rate of the Evobodies [<i>s</i><sup>-1</sup>].
<br>
<i>k<sub>exprblaETD</sub></i> stands for the expression rate of beta-lactamase with the bacterial two hybrid complex [<i>s</i><sup>-1</sup>].
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<div class="container text_header"><h3>Diffusion</h3></div>
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The deactivation reaction of ampicillin through beta-lactamase is a diffusion-controlled reaction. An equation that considers the rate at which the reactants diffuse is needed (Atkins and De Paula 2010):
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	<img src="https://static.igem.org/mediawiki/2016/6/60/Bielefeld_CeBiTec_2016_10_16_Mod_diff1.jpg
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	<img src="https://static.igem.org/mediawiki/2016/1/13/Bielefeld_CeBiTec_2016_10_16_Mod_diffusion_normal.png
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	<b>Figure 3</b>: Diffusion of two proteins without consideration of an active centrum.
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This equation describes the general diffusion of two molecules (figure 3). 
But in our case the enzyme beta-lactamase only reacts with ampicillin, if its active centrum is in close proximity to ampicillin (figure 4 B). 
When beta-lactamase and ampicillin meet correctly, ampicillin is deactivated as shown in figure 4, section B to D. Section E indicates a wrong encounter of beta-lactamase and ampicillin.
That&apos;s why we decided to modify the equation according to (Bisswanger 2008):
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	<img src="https://static.igem.org/mediawiki/2016/5/52/Bielefeld_CeBiTec_2016_10_16_Mod_diff2.png
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	<img src="https://static.igem.org/mediawiki/2016/0/05/-Bielefeld_CeBiTec_2016_10_16_Mod_diffusion.png
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	<b>Figure 4</b>: Diffusion of two proteins with consideration of an active centrum.
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 <i>R<sub>min</sub></i> is defined as the distance in which the two reactant molecules, here ampicillin and beta-lactamase, react. This is the case, when they are in very close proximity. D is the sum of the diffusion coefficients of the two reactants in the solution. In our case the solution is the periplasm. The term diffusion describes the movement of a substance in a solution; the bigger the diffusion coefficient, the larger the diffusion velocity. 
The angle <i>&alpha;</i> restricts the reaction area of the enzyme to its active center (see opening in figure 4). So a reaction between beta-lactamase and ampicillin only takes place, when ampicillin meets the active center of beta-lactamase. 
<br>
Finally, the equation for the diffusion rate <i>k<sub>diff</sub></i> must be multiplied with the Avogadro constant <i>N<sub>A</sub></i> to gain the unit [(<i>s*M</i>)<sup>-1</sup>].
<br><br>
<i>R<sub>min</sub></i> was computated under the assumption that beta-lactamase and ampicillin react when they are in close vicinity and that the molecules are nearly spherical. Therefore, the minimal radius of a sphere that could contain the given mass of protein was estimated using the formula from (Erickson 2009):
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	<img src="https://static.igem.org/mediawiki/2016/6/6a/Bielefeld_CeBiTec_2016_10_16_Mod_prot_size.jpg
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for V in nm<sup>3</sup>, M in Dalton and R<sub>min</sub> in nanometer.
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For beta-lactamase we obtained 2.0277 nm and for ampicillin 0.4649 nm.
<br>
To compute the diffusion coefficients we used the program WinHydroPro v1.00 PUB  [Ortega, 2011]. We received 2.852*10<sup>-9</sup> for beta-lactamase and 1.237*10<sup>-8</sup> for ampicillin.
The angle <i>&alpha;</i> was computed with the open-source program PyMOL (PyMOl 2013). <i>&alpha;</i> was determined to be 25&deg;.

In the end we computed the reproduction rate in dependence of the ampicillin concentration in the periplasm, adapting the formula of Potsdam&apos;s:
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	<img src="https://static.igem.org/mediawiki/2016/3/39/Bielefeld_CeBiTec_2016_10_16_Mod_grrate.jpg
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 where <i>n</i> is the ampicillin concentration in periplasm. The time point 20 minutes was chosen, because the division rate of an <i>E. coli</i> cell is around 20 minutes.
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 <div class="container text_header"><h3>Experimental validation</h3></div>
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A theoretically implemented model must always be validated with experimentally measured data.
Therefore, we designed and conducted an amount of appropriate experiments.
Firstly, we performed a beta-lactamase PCA colorimetric assay, and, secondly, a cultivation experiment. The cultivation experiment was used to measure the beta-lactamase concentration with the Nitrocefin assay over time. The experiments were essential to adapt model I.

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<div class="container text_header"><h4>Beta-lactamase PCA colorimetric assay</h4></div>
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To quantify the amount of active beta-lactamase in the periplasm of an <i>E. coli</i> bacterium and thus validate our mathematical program in this aspect, we performed an [osmotic shock] and a <a href="https://2016.igem.org/Team:Bielefeld-CeBiTec/Experiments/Protocols#assays#Nitro_assay">beta-lactamase PCA colorimetric assay</a> (Andr&eacute; Galarneau et. al 2002). The osmotic shock leads to the disruption of the periplasm and spreading of membrane proteins. To yield the maximum amount of active beta-lactamase an osmotic shock is used that does not use detergents that are harmful for the activity of beta-lactamase (Jalalirad 2013). With the beta-lactamase PCA colorimetric assay we determined the amount of beta-lactamase through the reduction of Nitrocefin. When Nitrocefin is split by beta-lactamase it changes its color from yellow to red like in figure 5. To measure how the beta-lactamase production changes over time we cultivated <i>E. coli</i> with a beta-lactamase plasmid for 6 hours and took probes every hour. Afterwards we did a lysis and a Nitrocefin assay. The results can be seen here:
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	<img src="https://static.igem.org/mediawiki/2016/7/71/Bielefeld_CeBiTec_2016_10_16_Mod_nitrocefin.jpg
" alt="Nitrocefin assay" width="600px">
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	<b>Figure 5</b>: Nitrocefin assay.
	<br>
	From left to right: water control and time points t0 to t6 of a <i>E. coli</i> cultivation. 
During a cultivation period samples were taken from the culture every hour (t0 to t6). Cells in this sample were lysed, and measurements with a Nitrocefin assay were conducted.
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	<img src="https://static.igem.org/mediawiki/2016/f/fb/Bielefeld_CeBiTec_2016_10_16_Mod_nitrocefin_OD.png
" alt="Growth curve of a culture, that carries a beta-lactamase plasmid" width="600px">
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	<b>Figure 6</b>: Growth curve of a culture, which carries a beta-lactamase plasmid.
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In figure 5 it can be seen that the beta-lactamase concentration rises with proceeding cultivation. Also we measured the beta-lactamase concentration of the whole culture at every time point and computed with the use of the OD (figure 6) the beta-lactamase concentration for one cell. 1 OD is converted to 8&thinsp;*&thinsp;10<sup>8</sup> cells. One cell comprises around 1.27848&thinsp;*&thinsp;10<sup>-6</sup>&thinsp;M beta-lactamase in the periplasm. We adapted our system accordingly.
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<div class="container text_header"><h3>References</h3></div>
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			<ul>
<li>Atkins, Peter W.; Paula, Julio de (2010): Physical chemistry. 9. ed. Oxford u.a.: Oxford Univ. Press.</li>
                        <li>Bisswanger, Hans (2008): Enzyme kinetics. Principles and methods. 2., rev. and updated ed. Weinheim: Wiley-VCH. Online accessible at http://site.ebrary.com/lib/alltitles/docDetail.action?docID=10301967.</li>
<li>Erickson, Harold P. (2009): Size and shape of protein molecules at the nanometer level determined by sedimentation, gel filtration, and electron microscopy. In: Biological procedures online 11, S. 32–51. DOI: 10.1007/s12575-009-9008-x.</li>
<li>Galarneau, André; Primeau, Martin; Trudeau, Louis-Eric; Michnick, Stephen W. (2002): |[beta]|-Lactamase protein fragment complementation assays as in vivo and in vitro sensors of protein|[ndash]|protein interactions. In: Nature Biotechnology 20 (6), S. 619–622. DOI: 10.1038/nbt0602-619.</li>
<li>Jalalirad, Reza (2013): Selective and efficient extraction of recombinant proteins from the periplasm of Escherichia coli using low concentrations of chemicals. In: Journal of industrial microbiology & biotechnology 40 (10), S. 1117–1129. DOI: 10.1007/s10295-013-1307-1.</li>
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