Team:Lethbridge Canada/math

From 2013hs.igem.org

(Difference between revisions)
 
(36 intermediate revisions not shown)
Line 9: Line 9:
<link rel="stylesheet" href="https://2013hs.igem.org/Team:Lethbridge_Canada/bootstrap?action=raw&ctype=text/css" />
<link rel="stylesheet" href="https://2013hs.igem.org/Team:Lethbridge_Canada/bootstrap?action=raw&ctype=text/css" />
<link rel="stylesheet" href="https://2013hs.igem.org/Team:Lethbridge_Canada/concept?action=raw&ctype=text/css" type="text/css" media="screen" />
<link rel="stylesheet" href="https://2013hs.igem.org/Team:Lethbridge_Canada/concept?action=raw&ctype=text/css" type="text/css" media="screen" />
-
 
</head>
</head>
<body>
<body>
Line 30: Line 29:
<div class="cleafix>">
<div class="cleafix>">
<div id="navigation">
<div id="navigation">
-
<li class="navigation_project decrease_opacity"><a href="#" class="strict">Project</a>
+
<li class="navigation_project"><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/project" class="strict">Project</a>
<ul>
<ul>
<li><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/project">Description</a></li>
<li><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/project">Description</a></li>
 +
                                                <li><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/project#video_oxy">Visual Modeling</a></li>
<li><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/math">Math Model</a></li>
<li><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/math">Math Model</a></li>
 +
<li><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/results">Results</a></li>
 +
<li><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/wikifreeze">Wikifreeze</a></li>
</ul>
</ul>
</li>
</li>
<li class="navigation_parts decrease_opacity"><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/parts" class="strict">Parts</a></li>
<li class="navigation_parts decrease_opacity"><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/parts" class="strict">Parts</a></li>
-
<li class="navigation_notebook decrease_opacity"><a href="#" class="strict">Notebook</a>
+
<li class="navigation_notebook decrease_opacity"><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/protocols" class="strict">Notebook</a>
<ul>
<ul>
<li><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/protocols#protocol_header">Protocols</a></li>
<li><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/protocols#protocol_header">Protocols</a></li>
Line 43: Line 45:
<li><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/notebook_april">Notebook: April</a></li>
<li><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/notebook_april">Notebook: April</a></li>
<li><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/notebook_may">Notebook: May</a></li>
<li><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/notebook_may">Notebook: May</a></li>
 +
<li><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/notebook_june">Notebook: June</a></li>
</ul>
</ul>
</li>
</li>
<li class="navigation_safety decrease_opacity"><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/safety" class="strict">Safety</a></li>
<li class="navigation_safety decrease_opacity"><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/safety" class="strict">Safety</a></li>
-
<li class="navigation_outreach decrease_opacity"><a href="#" class="strict">Outreach</a>
+
<li class="navigation_outreach decrease_opacity"><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/outreach" class="strict">Outreach</a>
<ul>
<ul>
<li><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/outreach">Overview</a></li>
<li><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/outreach">Overview</a></li>
Line 52: Line 55:
<li><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/interviews">Interviews</a></li>
<li><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/interviews">Interviews</a></li>
<li><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/videos">Videos</a></li>
<li><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/videos">Videos</a></li>
 +
<li><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/surveys">Parent Surveys</a></li>
 +
<li><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/novel_study">Novel Study</a></li>
</ul>
</ul>
</li>
</li>
-
<li class="navigation_team decrease_opacity"><a href="#" class="strict">Team</a>
+
<li class="navigation_team decrease_opacity"><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/team" class="strict">Team</a>
<ul>
<ul>
<li><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/team#the_team">The Team</a></li>
<li><a href="https://2013hs.igem.org/Team:Lethbridge_Canada/team#the_team">The Team</a></li>
Line 68: Line 73:
<div id="mainContent">
<div id="mainContent">
-
<div id="math_contents">
+
<img class="math_banner_image" src="https://static.igem.org/mediawiki/2013hs/d/dc/Lethbridge_hs_igem_2013_banner_mathmodel.png">
 +
 
 +
<div id="description_contents">
<h1>Contents:</h1>
<h1>Contents:</h1>
<ul>
<ul>
Line 91: Line 98:
<li><a href="#references">References</a></li>
<li><a href="#references">References</a></li>
</ul>
</ul>
-
</div>
+
</div><br></br>
<div id="math_entries">
<div id="math_entries">
Line 97: Line 104:
<div id="the_goal">
<div id="the_goal">
<h1>The Goal:</h1>
<h1>The Goal:</h1>
-
<p>By mathematically modeling the maximum rate at which DH5a can produce proteins, the correct ratio of enzyme to protein can be determined. One group is cells are built to maximally express the gene of the oxytocin-neurophysin1 protein. With respect to protein levels, the amount of enzyme required to completely cut each oxytocin-neurophysin1 can be determined. The aim of this is to optimize the expression process to have exactly the amount of enzyme required and not produce an excess or too little. It is important to note that the model determines the theoretical maximum of protein production, not the experimental value.</p>
+
<p>By mathematically modeling the maximum rate at which DH5&alpha; can produce proteins, the correct ratio of enzyme to protein can be determined. One group of cells is built to maximally express the gene of the oxytocin-neurophysin I protein. With respect to protein levels, the amount of enzyme required to completely cut each oxytocin-neurophysin I can be determined. The aim of this is to optimize the expression process to have exactly the amount of enzyme required and not produce an excess or too little. It is important to note that the model determines the theoretical maximum of protein production, not the experimental value.</p>
<p><a href="#header" class="math_back_to_top">Back to top</a></p>
<p><a href="#header" class="math_back_to_top">Back to top</a></p>
-
</div>
+
</div><br></br>
<!-- \-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/ -->
<!-- \-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/ -->
Line 105: Line 112:
<div id="bacterial_growth">
<div id="bacterial_growth">
<h1>Bacterial Growth:</h1>
<h1>Bacterial Growth:</h1>
-
<p>The purpose of including bacterial growth is so the amount of proteins produced per individual cell can be multiplied by the number of DH5a cells present to calculate the total number of proteins produced in that time interval of 1 minute. In the bacterial growth expression, the two variables are represented as such:</p>
+
<p>The purpose of including bacterial growth is so the amount of proteins produced per individual cell can be multiplied by the number of DH5&alpha; cells present to calculate the total number of proteins produced in that time interval of 1 minute. In the bacterial growth expression, the two variables are represented as such:</p>
<div class="image_container">
<div class="image_container">
<img src="https://static.igem.org/mediawiki/2013hs/0/0a/Leth_hs_2013_growth.png" id="img_growth" class="math_img" />
<img src="https://static.igem.org/mediawiki/2013hs/0/0a/Leth_hs_2013_growth.png" id="img_growth" class="math_img" />
<ul>
<ul>
-
<li><i>b<sub>i</sub></i> = initial number of DH5a bacteria present</li>
+
<li><i>b<sub>i</sub></i> = initial number of DH5&alpha; bacteria present</li>
<li><i>t</i> = time (manipulated variable)</li>
<li><i>t</i> = time (manipulated variable)</li>
</ul>
</ul>
Line 116: Line 123:
<p>We assumed a doubling time of 30 minutes.</p>
<p>We assumed a doubling time of 30 minutes.</p>
<p><a href="#header" class="math_back_to_top">Back to top</a></p>
<p><a href="#header" class="math_back_to_top">Back to top</a></p>
-
</div>
+
</div><br></br>
<!-- \-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/ -->
<!-- \-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/ -->
Line 125: Line 132:
<div id="transcription">
<div id="transcription">
<h2>Transcription:</h2>
<h2>Transcription:</h2>
-
<p>The expression below calculates the amount of mRNA that is produced by an individual cell in the time period of 1 minute. Credit is given to the 2009 Beijing iGEM team for the average speed of transcription in E. coli. </p>
+
<p>The expression below calculates the amount of mRNA that is produced by an individual cell in the time period of 1 minute. Credits given to the 2009 Beijing iGEM team for the average speed of transcription in <i>E. coli</i>. </p>
<div class="image_container">
<div class="image_container">
<img src="https://static.igem.org/mediawiki/2013hs/7/75/Leth_hs_2013_transcription.png" id="img_transcription" class="math_img" />
<img src="https://static.igem.org/mediawiki/2013hs/7/75/Leth_hs_2013_transcription.png" id="img_transcription" class="math_img" />
<ul>
<ul>
-
<li>4200nt/min = the maximum rate at which DNA is transcribed in to mRNA in E. coli</li>
+
<li>4200nt/min = the maximum rate at which DNA is transcribed in to mRNA in <i>E. coli</i></li>
<li><i>l<sub>tgene</sub></i> = gene length in (from RBS to the double terminator)</li>
<li><i>l<sub>tgene</sub></i> = gene length in (from RBS to the double terminator)</li>
</ul>
</ul>
Line 137: Line 144:
<p>The resultant number is the number of mRNA strands produced per minute in one cell.</p>
<p>The resultant number is the number of mRNA strands produced per minute in one cell.</p>
<p><a href="#header" class="math_back_to_top">Back to top</a></p>
<p><a href="#header" class="math_back_to_top">Back to top</a></p>
-
</div>
+
</div><br></br>
<div id="mrna_decay">
<div id="mrna_decay">
Line 159: Line 166:
<p>Because the exponent of the function is approximately or equal to 0, the entire expression for decay is equal to 1. The numerator of the exponent is essentially 0, so nearly any denominator would give the entire exponent a value of 0 when simplified. Although this logic nullifies the purpose of this term, it is important to note that mRNA decay is existent.</p>
<p>Because the exponent of the function is approximately or equal to 0, the entire expression for decay is equal to 1. The numerator of the exponent is essentially 0, so nearly any denominator would give the entire exponent a value of 0 when simplified. Although this logic nullifies the purpose of this term, it is important to note that mRNA decay is existent.</p>
<p><a href="#header" class="math_back_to_top">Back to top</a></p>
<p><a href="#header" class="math_back_to_top">Back to top</a></p>
-
</div>
+
</div><br></br>
<div id="transcription_mrna_decay">
<div id="transcription_mrna_decay">
Line 171: Line 178:
</div>
</div>
-
</div>
+
</div><br></br>
<!-- \-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/ -->
<!-- \-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/ -->
Line 184: Line 191:
<p><a href="#header" class="math_back_to_top">Back to top</a></p>
<p><a href="#header" class="math_back_to_top">Back to top</a></p>
-
</div>
+
</div><br></br>
<!-- \-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/ -->
<!-- \-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/ -->
Line 195: Line 202:
<img src="https://static.igem.org/mediawiki/2013hs/c/c7/Leth_hs_2013_translation.png" id="img_translation" class="math_img" />
<img src="https://static.igem.org/mediawiki/2013hs/c/c7/Leth_hs_2013_translation.png" id="img_translation" class="math_img" />
<ul>
<ul>
-
<li>2400Aa/min = the rate at which DNA is transcribed in to mRNA in E. coli</li>
+
<li>2400Aa/min = the rate at which DNA is transcribed in to mRNA in <i>E. coli</i></li>
<li><i>l<sub>protein</sub></i> = length of the protein to be expressed (Aa)</li>
<li><i>l<sub>protein</sub></i> = length of the protein to be expressed (Aa)</li>
</ul>
</ul>
</div>
</div>
-
<p>Credit is given to the 2009 Beijing iGEM team for the average speed of translation in E. coli.</p>
+
<p>Credits given to the 2009 Beijing iGEM team for the average speed of translation in <i>E. coli</i>.</p>
<p><a href="#header" class="math_back_to_top">Back to top</a></p>
<p><a href="#header" class="math_back_to_top">Back to top</a></p>
 +
<br></br>
<div id="derive_translation">
<div id="derive_translation">
Line 227: Line 235:
<p><a href="#header" class="math_back_to_top">Back to top</a></p>
<p><a href="#header" class="math_back_to_top">Back to top</a></p>
-
</div>
+
</div><br></br>
<!-- \-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/ -->
<!-- \-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/ -->
Line 240: Line 248:
<p><a href="#header" class="math_back_to_top">Back to top</a></p>
<p><a href="#header" class="math_back_to_top">Back to top</a></p>
-
</div>
+
</div><br></br>
<!-- \-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/ -->
<!-- \-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/ -->
Line 246: Line 254:
<div id="equation">
<div id="equation">
<h1>Assembling the Equation:</h1>
<h1>Assembling the Equation:</h1>
-
<p>Bacteria don’t double forever. They will reach a maximum population for their environment size. For this reason, to accurately model protein production we need to make a piece-wise equation. We let <i>j</i> equal the amount of time that it takes for DH5a to reach their maximum population size. Before time j we use the expression for start time <i>i</i>  up to time <i>j</i>. Because after time <i>j</i>, DH5a no longer double, we replace the time variable with <i>j</i>  so the graph from this point will be a horizontal line parallel to the time-axis. Time <i>k</i>  is the end time of the protein production period. The area under the curve from <i>i</i>  to <i>k</i>  is the total amount of protein expressed (<i>n<sub>p</sub></i>).</p>
+
<p>Bacteria don’t double forever. They will reach a maximum population for their environment size. For this reason, to accurately model protein production we need to make a piece-wise function. We let <i>j</i> equal the amount of time that it takes for DH5&alpha; to reach their maximum population size. Before time <i>j</i> we use the expression for start time <i>i</i>  up to time <i>j</i>. Because after time <i>j</i>, DH5&alpha; no longer double, we replace the time variable with <i>j</i>  so the graph from this point will be a horizontal line parallel to the time-axis. Time <i>k</i>  is the end time of the protein production period. The area under the curve from <i>i</i>  to <i>k</i>  is the total amount of protein expressed (<i>n<sub>p</sub></i>).</p>
<div class="image_container">
<div class="image_container">
<img src="https://static.igem.org/mediawiki/2013hs/e/e4/Leth_hs_2013_complete.png" id="img_complete" class="math_img" />
<img src="https://static.igem.org/mediawiki/2013hs/e/e4/Leth_hs_2013_complete.png" id="img_complete" class="math_img" />
</div>
</div>
 +
 +
<p>When plotted, the function creates a graph that looks as follows:</p>
 +
 +
<div class="image_container">
 +
<img id="img_graph" src="https://static.igem.org/mediawiki/2013hs/4/4b/Leth_hs_2013_graph.png" id="img_graph" class="math_img" />
 +
</div>
 +
 +
<p>The graph illustrates that at time <i>i</i> the protein output is zero and increases until time <i>j</i>. This uses the first part of the piece-wise function. From time <i>j</i> to time <i>k</i> (end time) the graph is constant and deals with the second part of the piece-wise function.</p>
<p><a href="#header" class="math_back_to_top">Back to top</a></p>
<p><a href="#header" class="math_back_to_top">Back to top</a></p>
-
</div>
+
</div><br></br>
<!-- \-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/ -->
<!-- \-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/-\-/ -->
Line 268: Line 284:
<tr>
<tr>
<td><i>b<sub>i</sub></i></td>
<td><i>b<sub>i</sub></i></td>
-
<td>Initial number of E. coli</td>
+
<td>Initial number of <i>E. coli</i></td>
</tr>
</tr>
Line 278: Line 294:
<tr>
<tr>
<td><i>j</i></td>
<td><i>j</i></td>
-
<td>Time it takes for E. coli to reach its maximum population size</td>
+
<td>Time it takes for <i>E. coli</i> to reach its maximum population size</td>
</tr>
</tr>
Line 342: Line 358:
<tr>
<tr>
-
<th>NEC1</th>
+
<th>NEC I</th>
<th>Value</th>
<th>Value</th>
</tr>
</tr>
Line 355: Line 371:
<td>2289bp</td>
<td>2289bp</td>
</tr>
</tr>
-
<table>
+
</table>
<p><a href="#header" class="math_back_to_top">Back to top</a></p>
<p><a href="#header" class="math_back_to_top">Back to top</a></p>

Latest revision as of 19:50, 20 July 2013



The Goal:

By mathematically modeling the maximum rate at which DH5α can produce proteins, the correct ratio of enzyme to protein can be determined. One group of cells is built to maximally express the gene of the oxytocin-neurophysin I protein. With respect to protein levels, the amount of enzyme required to completely cut each oxytocin-neurophysin I can be determined. The aim of this is to optimize the expression process to have exactly the amount of enzyme required and not produce an excess or too little. It is important to note that the model determines the theoretical maximum of protein production, not the experimental value.

Back to top



Bacterial Growth:

The purpose of including bacterial growth is so the amount of proteins produced per individual cell can be multiplied by the number of DH5α cells present to calculate the total number of proteins produced in that time interval of 1 minute. In the bacterial growth expression, the two variables are represented as such:

  • bi = initial number of DH5α bacteria present
  • t = time (manipulated variable)

We assumed a doubling time of 30 minutes.

Back to top



mRNA Production:

Transcription:

The expression below calculates the amount of mRNA that is produced by an individual cell in the time period of 1 minute. Credits given to the 2009 Beijing iGEM team for the average speed of transcription in E. coli.

  • 4200nt/min = the maximum rate at which DNA is transcribed in to mRNA in E. coli
  • ltgene = gene length in (from RBS to the double terminator)

The resultant number is the number of mRNA strands produced per minute in one cell.

Back to top



mRNA Decay:

The stability of mRNA varies, but it will always decay. Half-lives can rage from seconds to hours.

  • ts = stagnant time of mRNA between transcription and translation
  • th = half-life

The above expression will act as a coefficient for how much mRNA will remain after the amount of time that mRNA sits stagnant in the cells between the processes of transcription and translation. However, we have assumed that immediately or very shortly after transcription, the mRNA then undergoes translation. The following logic describes this situation:

Because the exponent of the function is approximately or equal to 0, the entire expression for decay is equal to 1. The numerator of the exponent is essentially 0, so nearly any denominator would give the entire exponent a value of 0 when simplified. Although this logic nullifies the purpose of this term, it is important to note that mRNA decay is existent.

Back to top



Transcription and mRNA Decay Working Together:

When combining the previous two expressions, we can determine the amount of mRNA produced by one cell per minute.

Back to top



Total mRNA:

Expanding on the previous expression, we get the amount of total amount of mRNA produced by the total number of cells existing at any given time. This is achieved by multiplying the amount of mRNA produced by a single cell by the amount of cells that currently exist.

Back to top



Translation:

The following expression models the maximum amount of proteins produced per minute.

  • 2400Aa/min = the rate at which DNA is transcribed in to mRNA in E. coli
  • lprotein = length of the protein to be expressed (Aa)

Credits given to the 2009 Beijing iGEM team for the average speed of translation in E. coli.

Back to top



Deriving the Translational Constant:

By dividing the expression for translation by the expression for transcription, we can determine the rate at which one strand of mRNA can be translated in to one protein.

Because there are three nucleotides (also called a codon) of mRNA for every one amino acid, the gene length (nt) over the protein length (Aa) will always be equal to 3.

By simplifying this expression, we get the following term that is the rate of translation from mRNA.

We used the RBS BBa_B0034 which has an efficiency index of 1.0.

Back to top



Protein Production:

By taking the expression from Total mRNA section and incorporating the Translational Constant section we can determine the total protein production. By multiplying the amount of mRNA produced by the rate at which mRNA is translated in to protein, we can calculate the amount of protein produced. We do this by finding the area under the curve with respect to time.

Back to top



Assembling the Equation:

Bacteria don’t double forever. They will reach a maximum population for their environment size. For this reason, to accurately model protein production we need to make a piece-wise function. We let j equal the amount of time that it takes for DH5α to reach their maximum population size. Before time j we use the expression for start time i up to time j. Because after time j, DH5α no longer double, we replace the time variable with j so the graph from this point will be a horizontal line parallel to the time-axis. Time k is the end time of the protein production period. The area under the curve from i to k is the total amount of protein expressed (np).

When plotted, the function creates a graph that looks as follows:

The graph illustrates that at time i the protein output is zero and increases until time j. This uses the first part of the piece-wise function. From time j to time k (end time) the graph is constant and deals with the second part of the piece-wise function.

Back to top



Values

Variable Description
bi Initial number of E. coli
i Start time (t = 0)
j Time it takes for E. coli to reach its maximum population size
k End time
lgene Length of only gene to be expressed (nt)
ltgene Length of gene from RBS to double terminator (nt)
lprotein Length of protein (Aa)
np Number of proteins produced
t Elapsed time
th Half-life of mRNA
ts Time mRNA sits in the cell between transcription and translation
RBS RBS efficiency (B0034 = 1.0)
Preprooxyphysin Value
lgene 403bp
ltgene 418bp
NEC I Value
lgene 2277bp
ltgene 2289bp

Back to top