Difference between revisions of "Team:Freiburg/Testpage"

 
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/*========= END: style for navigation bar ==========*/
 
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<h2> Modeling</h2>
  
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<div class="highlightBox">
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<h4>Note</h4>
  
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</div>
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<h2> Introduction</h2>
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<p>
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Nowadays, more and more information on biochemical networks is collected, and a lot is already known yet. Understanding biochemical networks is in general important for understanding signaling pathways within organisms, especially regarding medical health issues. Furthermore, it is needed for building new networks from scratch using synthetic biology.
 +
<br>
 +
However, the more components a network involves, the harder it gets to estimate how a complex network reacts to changes from both the inside and the outside. This is especially a problem, if entities change both in time and space. Experimentally studying all different components of a network for different possible conditions often is not only time-consumptive but also very expensive and therefore not viable.<br>
 +
Mathematical modeling of biological networks therefore is a powerful tool to support experimental research, predicting the behaviour of networks both concerning time and space. These predictions can then be validated experimentally.
  
 +
<p>
  
 +
Two crucial biological processes within the DiaCHIP are cell-free expression and the diffusion of proteins to the specific chemical surface. Cell-free expression limits the DiaCHIP regarding time, protein diffusion regarding space thereby determining the maximum protein spot density one can still resolve. Therefore, we decided to model both cell-free expression and protein diffusion to predict the behaviour of protein concentration in time and space.
  
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With a model like this, it is possible to find the minimum time needed to get a sufficiently high concentration of protein for antibody detection. This reduces the time needed for diagnosis using the DiaCHIP. Furthermore, not only protein, but also other molecules' concentrations can be simulated. Therefore, bottlenecks within the biochemical system as well as limiting substrates can be predicted. Specifically increasing the concentration of such limiting substrates as well as substances involved in bottleneck processes may increase not only the overall protein yield but also speed up synthesis to account for an even faster diagnosis.<br>
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Moreover, a cell-free expression model has huge potential for application on all systems dealing with protein expression in general. Although designed for cell-free systems, it represents the central dogma of biology and therefore mathematically describes a lot of processes involved in many biochemical networks.
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<h2>Overview</h2>
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<p>
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With our model we aim to simulate cell-free expression to predict the amount of protein synthesized during the course of time. We set up a system of ordinary differential equations (ODE System) based on the law of mass action, describing the kinetics of the single processes occurring during transcription and translation. Starting from a detailed description including <strong>XX</strong> different entities and XX different parameters, we defined additional assumptions to simplify our system to a total of ZZ entities and ZZ parameters. Kinetic constants and all other parameter values needed were researched in literature. If no values were found, appropriate values were suggested. As the final model can not be solved analytically, it was solved numerically using Mathematica. [REFERENCE]
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</p>
  
</script>
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<h2> Model Decription </h2>
  
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<h3> 1. Transcription</h3>
  
<script type="text/javascript">
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# ODEs: 30 (Simplified: 14) (Shared: cmRNA)<br>
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# Parameter: (Simplified: 9) (Shared: lDNA)<br>
  
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<h4> Detailed System </h4>
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[PICTURE] - DONE
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                  </a>
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                  <div class="thumbcaption">
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                      <p><strong>Figure 1: </p>
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                  </div>
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              </div>
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</div>
  
</script>
 
  
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[UNDERLYING ASSUMPTION] - DONE
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<br>
  
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[NOMENCLATURE]
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<div class="accordion" style="margin-top:100px">
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  <div class="accordion-section">
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    <a class="accordion-section-title" href="#accordion-nomenclature_transc">Nomenclature (Transcription)</a>
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#bubble_sidebar {
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[ODE SYSTEM] - DONE
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<div class="accordion" style="margin-top:100px">
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<h4> Simplified System </h4>
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[UNDERLYING ASSUMPTION] - DONE
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[NOMENCLATURE]
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<br>
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[ODE SYSTEM] - DONE
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<div class="accordion" style="margin-top:100px">
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    <a class="accordion-section-title" href="#accordion-2">ODE System Transcription (Simplified)</a>
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<div class="image_box center">
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<h1 style="text-align:right">Cell-Free Expression of Immobilized DNA</h1>
+
<br>
<p>
+
<h3> 2. Translation</h3>
An important advantage of the DiaCHIP is the possibility to ship and store information encoded by DNA. From a DNA template array protein arrays can be produced on demand. In order to obtain this template, DNA is fixed on a silicone slide forming one side of our microfluidic chamber. Making use of a cell-free expression system, the DNA can then be transcribed and translated into the respective proteins resulting in the final protein array. The coding sequence of the proteins is genetically fused to a tag allowing their binding to a specific surface on the opposite side of the chamber.
+
</p>
+
  
<h2>Successful binding of DNA to the Silicone Slide</h2>
+
# ODEs: 61 (Simplified: 52) (Shared: cmRNA)<br>
 +
# Parameter: (Simplified: 32) (Shared: lDNA)<br>  
  
<div class="image_box left">
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<h4> Detailed System </h4>
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[PICTURE] - DONE
                      <a href="https://static.igem.org/mediawiki/2015/2/2b/Freiburg_PDMS_with_PDITC.png" class="lightbox_trigger">
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<div class="image_box center">
                      <img src="https://static.igem.org/mediawiki/2015/2/2b/Freiburg_PDMS_with_PDITC.png" width="250px">
+
  <div class="thumb2 trien" style="width:900px">
                        <div class="thumbcaption">
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                <div class="thumbinner">
                      </a>
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                    <a href="https://static.igem.org/mediawiki/2015/9/9a/Freiburg_tl_overview-wiki.jpg" class="lightbox_trigger">
                          <p><strong>Figure 1: Scheme of the APTES/PDITC surface.</strong> The PDMS surface is layered with PDITC. This enables the binding of the amino-labeled DNA.</p>
+
                    <img src="https://static.igem.org/mediawiki/2015/e/e6/Freiburg_tl_overview-wiki_preview.jpg" width="900px">
                      </div>
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                  </a>
                </div>
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                  <div class="thumbcaption">
</div>
+
                      <p><strong>Figure 1: </p>
 +
                  </div>
 +
              </div>
 
</div>
 
</div>
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 +
[UNDERLYING ASSUMPTION] - DONE
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<br>
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 +
[NOMENCLATURE]
 +
<div class="accordion" style="margin-top:100px">
 +
  <div class="accordion-section">
 +
    <a class="accordion-section-title" href="#accordion-nomenclature_translat">Nomenclature (Translation)</a>
 +
    <div id="accordion-nomenclature_translat" class="accordion-section-content" style="display:none; padding:15px;">
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<div class="image_box center">
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[ODE SYSTEM] - DONE
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<div class="accordion" style="margin-top:100px">
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    <a class="accordion-section-title" href="#accordion-3">ODE System Translation (Detailed)</a>
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<h4> Simplified System </h4>
 +
 +
[UNDERLYING ASSUMPTION] - DONE
 +
<br>
 +
 +
[NOMENCLATURE]
 +
 +
[ODE SYSTEM] - DONE
 +
<div class="accordion" style="margin-top:100px">
 +
  <div class="accordion-section">
 +
    <a class="accordion-section-title" href="#accordion-4">ODE System Translation (Simplified)</a>
 +
    <div id="accordion-4" class="accordion-section-content" style="display:none; padding:15px;">
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<div class="image_box center">
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<div class="image_box center">
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<div class="image_box center">
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<div class="image_box center">
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</div>   
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<div class="image_box center">
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    <div class="thumb2 trien" style="width:100%;">
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<h2>3. Diffusion</h2>
 +
 +
<h3>3.1 Introduction and Motivation</h3>
  
 
<p>
 
<p>
The core component of the DiaCHIP is the microfluidic chamber composed of a glass slide for protein immobilization and a PDMS (polydimethylsiloxane) flow cell. The silicone PDMS needs to be activated to enable the generation of the DNA template array by binding of the respective sequences. <a href="https://2015.igem.org/Team:Freiburg/Project/Surface_Chemistry"target="_blank">Oxygen plasma</a> is used to first activate the surface of the PDMS slide. This allows to cover it with the silane APTES and finally apply the <a href="https://2015.igem.org/Team:Freiburg/Project/Surface_Chemistry"target="_blank">crosslinker PDITC</a>. The DNA sequence to be immobilized requires an amino group to be covalently immobilized on the PDITC surface. The structure of this surface is schematically visualized in figure 1. The same surface chemistry can be used to immobilize proteins unspecifically.
+
In the final step of cellfree expression proteins being produced are diffusing inside the microfluidic chamber. We modeled an ideal case to provide a tool:
 
<br>
 
<br>
To obtain an expression cassette for GFP with such an amino group, the target sequence was amplified by PCR using an amino-labeled reverse primer. Additionally, the forward primer was labeled by the fluorescent dye Cy3 to enable visualization by fluorescence microscopy. Successful amplification of the target sequence was verified by agarose gel electrophoresis.
+
On the PDMS slide spots of bound DNA produce proteins with steadily decreasing production
</p>
+
rate. The product is distributed homogeneously on the spot and starts diffusing freely in the cell-free
 +
mix. Furthermore besides convection through gravitation any interaction is assumed to be negligibly small. The coated iRIf glass is expected to be an ideal sink; any proteins reaching the slide are bound and therefore do not contribute to diffusion anymore.
  
 +
<p>
  
<div class="image_box right">
+
What knowledge did we want to gain by modeling?
<div class="thumb2 trien" style="width:300px">
+
<br>
                  <div class="thumbinner">
+
- Time optimization: When is the most efficient time to stop the expression?
                      <a href="https://static.igem.org/mediawiki/2015/thumb/f/f5/Freiburg_spotting_microarrayscanner.png/735px-Freiburg_spotting_microarrayscanner.png" class="lightbox_trigger">
+
<br>
                      <img src="https://static.igem.org/mediawiki/2015/thumb/f/f5/Freiburg_spotting_microarrayscanner.png/735px-Freiburg_spotting_microarrayscanner.png" width="250px"> 
+
- Product optimization: How much of the totally produced proteins does bind to the surface?
                        <div class="thumbcaption">
+
<br>
                      </a>
+
- Spot distance optimization: How is the bound protein distributed on the glass slide?
                          <p><strong>Figure 2: Immobilization of DNA on a PDMS slide.</strong> A: Top view on the slide indicating the spotting pattern. B: Cy3 fluorescence indicating successful immobilization of amino-labeled DNA. As a negative control, Cy3- but not amino-labeled DNA was spotted.</p>
+
<br>
                      </div>
+
<br>
                </div>
+
In order to achieve this we constructed the following system.
</div>
+
</div>
+
  
 +
<br>
  
 +
<h3>3.2 Model System</h3>
 
<p>
 
<p>
Coupling of DNA to the PDMS slide was achieved using a DNA concentration of 25&nbsp;ng/µl. Either 1 or 3 µl were spotted directly onto the slide using a distinct pattern (figure 2A). To verify that immobilization occurs specifically for amino-labeled DNA, we used non-amino-labeled DNA as a negative control and spotted the same amount. The slide was subsequently incubated over night and the DNA solution was dried afterwards at 60°C. After washing the slide, immobilized DNA was visualized by fluorescence microscopy. The resulting fluorescence pattern clearly corresponds to the spotting pattern on the slide, thereby confirming that the immobilization of DNA was successful. Spots that were incubated with amino-labeled DNA show a distinct Cy3 fluorescence signal, whereas DNA that was not labeled with an amino group was not bound to the surface (figure 2B).  
+
Effective modeling depends on wisely chosen assumptions.
</p>
+
  
<h2 class="left">Cell-Free Expression of GFP From Spotted DNA</h2>
+
Assume the following setup:
 +
<br>
 +
(PIC)
 +
<br>
  
<div class="image_box left">
+
<p>
<div class="thumb2 trien" style="width:300px">
+
Illustrated above is a crosssection of the system around one spot (A1). Due to assumption A2 the number of geometrical degrees of freedom can be reduced to 2.  
                  <div class="thumbinner">
+
                      <a href="lightbox_trigger" href="https://static.igem.org/mediawiki/2015/thumb/9/99/Freiburg_20150911_DNA_on_PDMS_7.0_oven30min_expr30min.jpg/794px-Freiburg_20150911_DNA_on_PDMS_7.0_oven30min_expr30min.jpg" class="lightbox_trigger">
+
                      <img src="https://static.igem.org/mediawiki/2015/thumb/9/99/Freiburg_20150911_DNA_on_PDMS_7.0_oven30min_expr30min.jpg/794px-Freiburg_20150911_DNA_on_PDMS_7.0_oven30min_expr30min.jpg" width="250px"> 
+
                        <div class="thumbcaption">
+
                      </a>
+
                          <p><strong>Figure 3: Cell-free expressed GFP confirmed by fluorescence microscopy.</strong></p>
+
                      </div>
+
                </div>
+
</div>
+
</div>
+
  
 +
<br>
 +
<br>
 +
(PIC: parameters/mats)
 +
<br>
 +
<br>
 +
 +
Including movement due to gravitation the physical process is described by the diffusion convection equation:
 +
 +
<br>
 +
<br>
 +
(Formel: Diffgleichung)
 +
<br>
 +
<br>
 
<p>
 
<p>
To confirm that DNA was not only bound to the PDMS slide but is also suited for cell-free expression, we flushed the microfluidic chamber described above with our cell-free  expression mix. After incubation for two hours at room temperature the expressed GFP was detected using a standard fluorescence microscope (figure 3).  
+
It is an inhomogeneous parabolic partial differential equation (PDE). The diffusion constant &kappa; depends on the media and materials involved in the system, the inhomogenity describes sources of the system. The velocity vector v is assumed to be the sedimentation velocity (A4) and therefore a constant. The initial and boundary conditions are the following:
  
More details on the vector design and cloning strategies to generate the needed DNA can be found <a href="https://2015.igem.org/Team:Freiburg/Methods/Cloning">here</a>.  
+
<br>
 +
<br>
 +
(Formel: ICs and BCs)
 +
<br>
 +
<br>
 +
<p>
 +
The problem can be solved numerically by application of a "finite differences method". The complete space M is split up into squares. Each square inhibits a concentration value. In an iterative method the diffusion between neighbouring squares is calculated in small time steps. If the step size is chosen small enough and the grid fine enough the diffusion can be simulated. The algorithm is the following:
 +
 
 +
<br>
 +
<br>
 +
(PIC: Algorithmus)
 +
<br>
 +
<br>
 +
<p>
 +
In a simulation using python (v3.4.2, together with numpy and scipy) depending on time step size, square width, diffusion constant and spot width we could determine binding behaviour on the slide as function of time:
 +
 
 +
<br>
 +
<br>
 +
(Eingangsparameter, Graphik oder GIF...)
 +
<br>
 +
<br>
 
</p>
 
</p>
  
</div>
+
<h3>3.3 Numerical approach</h3>
 +
Our approach used the Crank-Nicholson-method: A functions change in one time step is determined by the average of momentary and following time step of the right hand side of the PDE:
  
 +
<br>
 +
(For_07)
 +
<br>
 +
 +
In case of convection and diffusion first and second derivations have to be determined. In (finite) differential form these can be approximated by
 +
 +
<br>
 +
(For_06)
 +
<br>
 +
 +
Proecting those onto our given problem the diffusion-convection equation parts are given by:
 +
 +
<br>
 +
(For_10)
 +
<br>
 +
 +
where we presupposed &Delta;x=&Delta;z. For the sake of convenience constance had been bundled together
 +
 +
<br>
 +
(For_12)
 +
<br>
 +
 +
which lead to a large system of equations of the form:
 +
 +
<br>
 +
(For_12)
 +
<br>
 +
 +
What can be seen is the chosen geometrical approach: Diffusion and convection are done only with direct grid neighbours (two vertical and two horizontal ones). Logically this holds true only for non-border grid points. For the borders boundary conditions have to be considered: We assumed the left and right edges of the box to be open, the upper (PDMS side) and lower side to be closed ones. This can be achieved by the following conditions:
 +
 +
<br>
 +
(For_19)
 +
<br>
 +
 +
 +
 +
 +
<h3>3.4 Assumptions</h3>
 +
<p>
 +
A1: The spots are equally distributed and distance to the borders of the microfluidic chamber is at least one
 +
spot-to-spot distance.
 +
 +
<p>
 +
 +
A2: The spots are perfect circles and the binding site at the iRIf glass is homohegeneous -> Around one spot
 +
cylindrical symmetry is given.
 +
 +
<p>
 +
 +
A3: The area of produced proteins is as thick as the one of bound ones -> &chi; = &epsilon;
 +
 +
<p>
 +
 +
A4: Movement by virtue of gravitation is considered to already be in a stationary status (constant movement)
 +
 +
</br>
 +
</br>
 +
 +
</div>
 
</html>
 
</html>
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{{Freiburg/wiki_content_end}}
 
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Latest revision as of 21:40, 18 September 2015

""

Modeling

Note

Introduction

Nowadays, more and more information on biochemical networks is collected, and a lot is already known yet. Understanding biochemical networks is in general important for understanding signaling pathways within organisms, especially regarding medical health issues. Furthermore, it is needed for building new networks from scratch using synthetic biology.
However, the more components a network involves, the harder it gets to estimate how a complex network reacts to changes from both the inside and the outside. This is especially a problem, if entities change both in time and space. Experimentally studying all different components of a network for different possible conditions often is not only time-consumptive but also very expensive and therefore not viable.
Mathematical modeling of biological networks therefore is a powerful tool to support experimental research, predicting the behaviour of networks both concerning time and space. These predictions can then be validated experimentally.

Two crucial biological processes within the DiaCHIP are cell-free expression and the diffusion of proteins to the specific chemical surface. Cell-free expression limits the DiaCHIP regarding time, protein diffusion regarding space thereby determining the maximum protein spot density one can still resolve. Therefore, we decided to model both cell-free expression and protein diffusion to predict the behaviour of protein concentration in time and space.

With a model like this, it is possible to find the minimum time needed to get a sufficiently high concentration of protein for antibody detection. This reduces the time needed for diagnosis using the DiaCHIP. Furthermore, not only protein, but also other molecules' concentrations can be simulated. Therefore, bottlenecks within the biochemical system as well as limiting substrates can be predicted. Specifically increasing the concentration of such limiting substrates as well as substances involved in bottleneck processes may increase not only the overall protein yield but also speed up synthesis to account for an even faster diagnosis.
Moreover, a cell-free expression model has huge potential for application on all systems dealing with protein expression in general. Although designed for cell-free systems, it represents the central dogma of biology and therefore mathematically describes a lot of processes involved in many biochemical networks.

Overview

With our model we aim to simulate cell-free expression to predict the amount of protein synthesized during the course of time. We set up a system of ordinary differential equations (ODE System) based on the law of mass action, describing the kinetics of the single processes occurring during transcription and translation. Starting from a detailed description including XX different entities and XX different parameters, we defined additional assumptions to simplify our system to a total of ZZ entities and ZZ parameters. Kinetic constants and all other parameter values needed were researched in literature. If no values were found, appropriate values were suggested. As the final model can not be solved analytically, it was solved numerically using Mathematica. [REFERENCE]

Model Decription

1. Transcription

# ODEs: 30 (Simplified: 14) (Shared: cmRNA)
# Parameter: (Simplified: 9) (Shared: lDNA)

Detailed System

[PICTURE] - DONE

Figure 1:

[UNDERLYING ASSUMPTION] - DONE
[NOMENCLATURE] [ODE SYSTEM] - DONE

Simplified System

[UNDERLYING ASSUMPTION] - DONE
[NOMENCLATURE]
[ODE SYSTEM] - DONE

2. Translation

# ODEs: 61 (Simplified: 52) (Shared: cmRNA)
# Parameter: (Simplified: 32) (Shared: lDNA)

Detailed System

[PICTURE] - DONE

Figure 1:

[UNDERLYING ASSUMPTION] - DONE
[NOMENCLATURE] [ODE SYSTEM] - DONE

Simplified System

[UNDERLYING ASSUMPTION] - DONE
[NOMENCLATURE] [ODE SYSTEM] - DONE

3. Diffusion

3.1 Introduction and Motivation

In the final step of cellfree expression proteins being produced are diffusing inside the microfluidic chamber. We modeled an ideal case to provide a tool:
On the PDMS slide spots of bound DNA produce proteins with steadily decreasing production rate. The product is distributed homogeneously on the spot and starts diffusing freely in the cell-free mix. Furthermore besides convection through gravitation any interaction is assumed to be negligibly small. The coated iRIf glass is expected to be an ideal sink; any proteins reaching the slide are bound and therefore do not contribute to diffusion anymore.

What knowledge did we want to gain by modeling?
- Time optimization: When is the most efficient time to stop the expression?
- Product optimization: How much of the totally produced proteins does bind to the surface?
- Spot distance optimization: How is the bound protein distributed on the glass slide?

In order to achieve this we constructed the following system.

3.2 Model System

Effective modeling depends on wisely chosen assumptions. Assume the following setup:
(PIC)

Illustrated above is a crosssection of the system around one spot (A1). Due to assumption A2 the number of geometrical degrees of freedom can be reduced to 2.

(PIC: parameters/mats)

Including movement due to gravitation the physical process is described by the diffusion convection equation:

(Formel: Diffgleichung)

It is an inhomogeneous parabolic partial differential equation (PDE). The diffusion constant κ depends on the media and materials involved in the system, the inhomogenity describes sources of the system. The velocity vector v is assumed to be the sedimentation velocity (A4) and therefore a constant. The initial and boundary conditions are the following:

(Formel: ICs and BCs)

The problem can be solved numerically by application of a "finite differences method". The complete space M is split up into squares. Each square inhibits a concentration value. In an iterative method the diffusion between neighbouring squares is calculated in small time steps. If the step size is chosen small enough and the grid fine enough the diffusion can be simulated. The algorithm is the following:

(PIC: Algorithmus)

In a simulation using python (v3.4.2, together with numpy and scipy) depending on time step size, square width, diffusion constant and spot width we could determine binding behaviour on the slide as function of time:

(Eingangsparameter, Graphik oder GIF...)

3.3 Numerical approach

Our approach used the Crank-Nicholson-method: A functions change in one time step is determined by the average of momentary and following time step of the right hand side of the PDE:
(For_07)
In case of convection and diffusion first and second derivations have to be determined. In (finite) differential form these can be approximated by
(For_06)
Proecting those onto our given problem the diffusion-convection equation parts are given by:
(For_10)
where we presupposed Δx=Δz. For the sake of convenience constance had been bundled together
(For_12)
which lead to a large system of equations of the form:
(For_12)
What can be seen is the chosen geometrical approach: Diffusion and convection are done only with direct grid neighbours (two vertical and two horizontal ones). Logically this holds true only for non-border grid points. For the borders boundary conditions have to be considered: We assumed the left and right edges of the box to be open, the upper (PDMS side) and lower side to be closed ones. This can be achieved by the following conditions:
(For_19)

3.4 Assumptions

A1: The spots are equally distributed and distance to the borders of the microfluidic chamber is at least one spot-to-spot distance.

A2: The spots are perfect circles and the binding site at the iRIf glass is homohegeneous -> Around one spot cylindrical symmetry is given.

A3: The area of produced proteins is as thick as the one of bound ones -> χ = ε

A4: Movement by virtue of gravitation is considered to already be in a stationary status (constant movement)