Team:Freiburg Bioware/Modeling/Virus Infection

From 2010.igem.org

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<h1>Model for Virus Infection</h1>
<h1>Model for Virus Infection</h1>
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<p style="text-align:justify;">
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<p style="text-align: justify;">
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As in the previous model for the virus production we established a ODE model based on the law of mass action. The following paragraph explains the reaction scheme and our model assumptions. In the subsequent paragraphs the system of differential equations is specified and the implementation in MathWorks® MATLAB is discussed.<br>
+
As in the previous model for the virus production we established a ODE
 +
model based on the law of mass action. The following paragraph explains
 +
the reaction scheme and our model assumptions. In the subsequent
 +
paragraphs the system of differential equations is specified and the
 +
implementation in MathWorks® MATLAB is discussed.<br>
The last section deals with our modeling results.
The last section deals with our modeling results.
<br>
<br>
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</p>
</p>
<h2>Reaction Scheme</h2>
<h2>Reaction Scheme</h2>
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<p style="text-align:justify;">
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<p style="text-align: justify;">
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To simplify the mathematical description of the reaction scheme we divide the cell into four compartments: the <b>extracellular matrix</b> (all quantities with the index <i>ext</i>), the space in <b>endosomes</b> (<i>end</i>), the <b>cytoplasm</b> (<i>cyt</i>) and the <b>nucleus</b> (<i>nuc</i>).<br>  
+
To simplify the mathematical description of the reaction scheme we
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A target cell is transduced by <b>viral particles</b> (<i>V</i>) in the extracellular matrix. Depending on their <b>degree of modification</b> (<i>m</i>) and thus their specificity they can bind to <b>receptors</b> (<i>R</i>) on the cell surface. Once a receptor has formed a <b>complex with a virus particle</b> (<i>VR</i>) receptor dimerization (<i>R<sub>2</sub></i>) occurs and the whole <b>complex</b> (<i>VR<sub>2</sub></i>) is invaginated and endosomes are formed. The virus is released from the endosome to the cytoplasm and is transported to the nucleus where uncoating of the capsid is initiated and the <b>single stranded DNA</b> (<i>ssDNA</i>) is released.<br>
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divide the cell into four compartments: the <b>extracellular matrix</b>
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Finally viral mRNA is processed and transported into the cytoplasm and the <b>enzyme for therapeutic approach</b> (<i>E</i>) is produced.  
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(all quantities with the index <i>ext</i>), the space in <b>endosomes</b>
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<br><br>
+
(<i>end</i>), the <b>cytoplasm</b> (<i>cyt</i>) and the <b>nucleus</b>
 +
(<i>nuc</i>).<br>
 +
A target cell is transduced by <b>viral particles</b> (<i>V</i>) in
 +
the extracellular matrix. Depending on their <b>degree of modification</b>
 +
(<i>m</i>) and thus their specificity they can bind to <b>receptors</b>
 +
(<i>R</i>) on the cell surface. Once a receptor has formed a <b>complex
 +
with a virus particle</b> (<i>VR</i>) receptor dimerization (<i>R<sub>2</sub></i>)
 +
occurs and the whole <b>complex</b> (<i>VR<sub>2</sub></i>) is
 +
invaginated and endosomes are formed. The virus is released from the
 +
endosome to the cytoplasm and is transported to the nucleus where
 +
uncoating of the capsid is initiated and the <b>single stranded DNA</b>
 +
(<i>ssDNA</i>) is released.<br>
 +
Finally viral mRNA is processed and transported into the cytoplasm and
 +
the <b>enzyme for therapeutic approach</b> (<i>E</i>) is produced. <br>
 +
<br>
</p>
</p>
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<table>
<table>
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<tbody>
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<tr>
<td>
<td>
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<center>
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<center><img
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<img width="380" src="https://static.igem.org/mediawiki/2010/e/e1/Freiburg10_VirusInfectionScheme01.png" alt="Reaction scheme for the virus production" />
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src="https://static.igem.org/mediawiki/2010/e/e1/Freiburg10_VirusInfectionScheme01.png"
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alt="Reaction scheme for the virus production" width="380">
</center>
</center>
</td>
</td>
<td>
<td>
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<center>
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<center><img
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<img width="359" height="335" src="https://static.igem.org/mediawiki/2010/9/90/Freiburg10_VirusInfection01.png" alt="Reaction scheme for the virus production" />
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src="https://static.igem.org/mediawiki/2010/9/90/Freiburg10_VirusInfection01.png"
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alt="Reaction scheme for the virus production" height="335" width="359">
</center>
</center>
</td>
</td>
 +
</tr>
 +
</tbody>
</table>
</table>
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<br><br>
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<br>
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<br>
<h2>Reduced Reaction Scheme</h2>
<h2>Reduced Reaction Scheme</h2>
-
<p style="text-align:justify;">
+
<p style="text-align: justify;">
-
The modeling approach only neglects linear transport processes and describes the receptor binding and dimerization in terms of the <b>law of mass action</b> (<i>LMA</i>).
+
The modeling approach only neglects linear transport processes and
-
<br><br>
+
describes the receptor binding and dimerization in terms of the <b>law
 +
of mass action</b> (<i>LMA</i>).
 +
<br>
 +
<br>
</p>
</p>
<center>
<center>
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<img width="359" height="227" src="https://static.igem.org/mediawiki/2010/8/83/Freiburg10_VirusInfection02.png" alt="Reaction scheme for the virus production" />
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<img
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</center>
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src="https://static.igem.org/mediawiki/2010/8/83/Freiburg10_VirusInfection02.png"
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<br><br>
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alt="Reaction scheme for the virus production" height="227" width="359"></center>
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<br>
 +
<br>
<h2>Differential Equations</h2>
<h2>Differential Equations</h2>
-
<p style="text-align:justify;">
+
<p style="text-align: justify;">
-
The ODE model consists of 7 equations containing 9 rate constants. To the equation for the extracellular virus concentration a degradation term is added corresponding to the immune response of the target system. The temporal behavior of <i>ssDNA</i> is completed by a linear degradation term.<br>
+
The ODE model consists of 7 equations containing 9 rate constants. To
 +
the equation for the extracellular virus concentration a degradation
 +
term is added corresponding to the immune response of the target
 +
system. The temporal behavior of <i>ssDNA</i> is completed by a linear
 +
degradation term.<br>
</p>
</p>
-
 
<center>
<center>
-
<img width="655" height="314" src="https://static.igem.org/mediawiki/2010/8/85/Freiburg10_VirusInfection04.png" alt="Reaction scheme for the virus production" />
+
<img
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</center>
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src="https://static.igem.org/mediawiki/2010/8/85/Freiburg10_VirusInfection04.png"
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<br><br>
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alt="Reaction scheme for the virus production" height="314" width="655"></center>
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+
<br>
 +
<br>
<h3>Model Extensions</h3>
<h3>Model Extensions</h3>
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<p style="text-align:justify;">
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<p style="text-align: justify;">
-
The model is extended by a non-linear dependency of the internalization rate constant of the <b>degree of modification</b> (<i>m</i>) and the production efficiency of functional virus particles reduces the amount of available <i>ssDNA</i> in the nucleus.<br>
+
The model is extended by a non-linear dependency of the internalization
-
The assumed functional dependency is shown in the two figures below.<br>  
+
rate constant of the <b>degree of modification</b> (<i>m</i>) and the
 +
production efficiency of functional virus particles reduces the amount
 +
of available <i>ssDNA</i> in the nucleus.<br>
 +
The assumed functional dependency is shown in the two figures below.<br>
</p>
</p>
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<img width="400" src="https://static.igem.org/mediawiki/2010/5/54/Freiburg10_ModificationEfficiency.png" alt="" />
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<table style="text-align: left; width: 90%;" border="0" cellpadding="2"
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cellspacing="2">
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<tbody>
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<br><br>
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<tr>
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<td style="vertical-align: top; width: 468px;"><img
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src="https://static.igem.org/mediawiki/2010/5/54/Freiburg10_ModificationEfficiency.png"
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alt="" width="500"></td>
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<td style="vertical-align: top; width: 384px;"><br>
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</td>
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</tr>
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</tbody>
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</table>
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<br>
 +
<br>
<h2>Methods and Simulation</h2>
<h2>Methods and Simulation</h2>
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<p style="text-align:justify;">
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<p style="text-align: justify;">
-
The ODE model was implemented in MathWorks® MATLAB R2010b. Integration of the differential equations was achieved using the stiff integrator <i>ode15s</i> with automatic integration step size management.<br>
+
The ODE model was implemented in MathWorks® MATLAB R2010b. Integration
-
To calibrate the dynamics of the mathematical model to those of biological system we used time lapse data of fluorescence experiments as well as published values for the rate constants.
+
of the differential equations was achieved using the stiff integrator <i>ode15s</i>
-
<br><br>
+
with automatic integration step size management.<br>
-
The parameters used are given in the table below. Also you can download the MATLAB source code.
+
To calibrate the dynamics of the mathematical model to those of
 +
biological system we used time lapse data of fluorescence experiments
 +
as well as published values for the rate constants.
 +
<br>
 +
<br>
 +
The parameters used are given in the table below. Also you can download
 +
the MATLAB source code.
<br>
<br>
</p>
</p>
-
<img width="405" height="337" src="https://static.igem.org/mediawiki/2010/c/c7/Freiburg10_RateConstants02.png" alt="" />
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<img
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<br><br>
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src="https://static.igem.org/mediawiki/2010/c/c7/Freiburg10_RateConstants02.png"
-
 
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alt="" height="337" width="405">
-
<a href="https://2010.igem.org/Wiki/images/8/87/Freiburg10_VirusInfectionCode.m">get the m-File (MATLAB source code)!</a><br>  
+
<br>
-
 
+
<br>
-
 
+
<a
-
<br><br>
+
href="https://2010.igem.org/Wiki/images/8/87/Freiburg10_VirusInfectionCode.m">get
 +
the m-File (MATLAB source code)!</a><br>
 +
<br>
 +
<br>
<h2>Results and Discussion</h2>
<h2>Results and Discussion</h2>
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<center>
<center>
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<img width=800 src="https://static.igem.org/mediawiki/2010/8/8c/Freiburg10_VirusInfectionPlot01.png" alt="" />
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<img
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</center>
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src="https://static.igem.org/mediawiki/2010/8/8c/Freiburg10_VirusInfectionPlot01.png"
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<br><br>
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alt="" width="800"></center>
-
 
+
<br>
 +
<br>
<center>
<center>
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<img width=800 src="https://static.igem.org/mediawiki/2010/9/9e/Freiburg10_VirusInfectionModPlot01.png" alt="" />
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<img
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</center>
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src="https://static.igem.org/mediawiki/2010/9/9e/Freiburg10_VirusInfectionModPlot01.png"
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<br><br>
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alt="" width="800"></center>
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</html>
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{{:Team:Freiburg_Bioware/Footer}}
{{:Team:Freiburg_Bioware/Footer}}
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</body>
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</html>

Revision as of 20:21, 24 October 2010

{{:Team:Freiburg_Bioware/Head}} {{:Team:Freiburg_Bioware/menu_home}}

Model for Virus Infection

As in the previous model for the virus production we established a ODE model based on the law of mass action. The following paragraph explains the reaction scheme and our model assumptions. In the subsequent paragraphs the system of differential equations is specified and the implementation in MathWorks® MATLAB is discussed.
The last section deals with our modeling results.

Reaction Scheme

To simplify the mathematical description of the reaction scheme we divide the cell into four compartments: the extracellular matrix (all quantities with the index ext), the space in endosomes (end), the cytoplasm (cyt) and the nucleus (nuc).
A target cell is transduced by viral particles (V) in the extracellular matrix. Depending on their degree of modification (m) and thus their specificity they can bind to receptors (R) on the cell surface. Once a receptor has formed a complex with a virus particle (VR) receptor dimerization (R2) occurs and the whole complex (VR2) is invaginated and endosomes are formed. The virus is released from the endosome to the cytoplasm and is transported to the nucleus where uncoating of the capsid is initiated and the single stranded DNA (ssDNA) is released.
Finally viral mRNA is processed and transported into the cytoplasm and the enzyme for therapeutic approach (E) is produced.

Reaction scheme for the virus production
Reaction scheme for the virus production


Reduced Reaction Scheme

The modeling approach only neglects linear transport processes and describes the receptor binding and dimerization in terms of the law of mass action (LMA).

Reaction scheme for the virus production


Differential Equations

The ODE model consists of 7 equations containing 9 rate constants. To the equation for the extracellular virus concentration a degradation term is added corresponding to the immune response of the target system. The temporal behavior of ssDNA is completed by a linear degradation term.

Reaction scheme for the virus production


Model Extensions

The model is extended by a non-linear dependency of the internalization rate constant of the degree of modification (m) and the production efficiency of functional virus particles reduces the amount of available ssDNA in the nucleus.
The assumed functional dependency is shown in the two figures below.


Methods and Simulation

The ODE model was implemented in MathWorks® MATLAB R2010b. Integration of the differential equations was achieved using the stiff integrator ode15s with automatic integration step size management.
To calibrate the dynamics of the mathematical model to those of biological system we used time lapse data of fluorescence experiments as well as published values for the rate constants.

The parameters used are given in the table below. Also you can download the MATLAB source code.



get the m-File (MATLAB source code)!


Results and Discussion





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