Team:ITB Indonesia/Modeling/Difussion

From 2013.igem.org

(Difference between revisions)
Line 7: Line 7:
<h2 class="title">Difussion</h2>
<h2 class="title">Difussion</h2>
<div class="entry clearfix">
<div class="entry clearfix">
-
<p>First thing to do is calculate how much aflatoxin would  enter the cell every time. Aflatoxin diffused into cell through simple  diffusion mechanism, it means that aflatoxin difusion is drived by  concentration gradient.<br />
+
<p>First thing to do is calculate how much aflatoxin would  enter the cell every time. Aflatoxin diffused into cell through simple  diffusion mechanism, it means that aflatoxin difusion is drived by  concentration gradient.</p>
-
  Assumption :</p>
+
<div style="border:solid #F60">To simplify our model, we  assume that aflatoxin homogenely diffused into cell </div>
-
<ul>
+
<p><strong>0. General equation  of diffusion through membrane</strong><br />
-
  <li>Aflatoxin homogenely diffused into cell</li>
+
  Diffusion is frequently modelled if the system needs to  transport some molecule through cell membrane. Mathematical model for  diffusional phenomena through membrane is :<br />
-
</ul>
+
  <img src="file:///C|/Users/User/AppData/Roaming/Adobe/Dreamweaver CS5/en_US/OfficeImageTemp/clip_image002_0001.png" alt="" width="214" height="35" /><br />
-
<p><img src="difussion_clip_image002.png" alt="" width="214" height="35" /><br />
+
   Where n represents the number of molecule involved in  diffusion (So, dn/dt can be stated as &ldquo;molecule flux through cell membrane&rdquo;). Parameters of the equation :</p>
-
   Parameters of the equation :</p>
+
<table border="1" cellspacing="0" cellpadding="0">
<table border="1" cellspacing="0" cellpadding="0">
   <tr>
   <tr>
-
     <td width="73" valign="top"><br />
+
     <td width="73" align="center" valign="middle"><strong>Variable</strong></td>
-
      <strong>Variable</strong></td>
+
     <td width="302" align="center" valign="middle"><p><strong>Definition</strong></p></td>
-
     <td width="302" valign="top"><p><strong>Definition</strong></p></td>
+
     <td width="113" align="center" valign="middle"><p><strong>Value</strong></p></td>
-
     <td width="113" valign="top"><p><strong>Value</strong></p></td>
+
     <td width="127" align="center" valign="middle"><p><strong>Source</strong></p></td>
-
     <td width="127" valign="top"><p><strong>Source</strong></p></td>
+
   </tr>
   </tr>
   <tr>
   <tr>
-
     <td width="73" valign="top"><p>P</p></td>
+
     <td width="73" align="center" valign="middle"><p>P</p></td>
-
     <td width="302" valign="top"><p>Permeability aflatoxin-membrane E. coli</p></td>
+
     <td width="302" align="center" valign="middle"><p>Permeability aflatoxin-membrane E. coli</p></td>
-
     <td width="113" valign="top"><p>1,01 x 10-4 cm/s</p></td>
+
     <td width="113" align="center" valign="middle"><p>1,01 x 10-4 cm/s</p></td>
-
     <td width="127" valign="top"><p>Calculated (See Sect 1)</p></td>
+
     <td width="127" align="center" valign="middle"><p>Calculated (See Sect 1)</p></td>
   </tr>
   </tr>
   <tr>
   <tr>
-
     <td width="73" valign="top"><p>A</p></td>
+
     <td width="73" align="center" valign="middle"><p>A</p></td>
-
     <td width="302" valign="top"><p>E. coli membrane cell area</p></td>
+
     <td width="302" align="center" valign="middle"><p>E. coli membrane cell area</p></td>
-
     <td width="113" valign="top"><p>Changing with time</p></td>
+
     <td width="113" align="center" valign="middle"><p>Changing with time</p></td>
-
     <td width="127" valign="top"><p>Calculated (see Sect 2)</p></td>
+
     <td width="127" align="center" valign="middle"><p>Calculated (see Sect 2)</p></td>
   </tr>
   </tr>
   <tr>
   <tr>
-
     <td width="73" valign="top"><p><img src="difussion_clip_image004.png" alt="" width="18" height="19" /></p></td>
+
     <td width="73" align="center" valign="middle"><p><img src="file:///C|/Users/User/AppData/Roaming/Adobe/Dreamweaver CS5/en_US/OfficeImageTemp/clip_image004_0001.png" alt="" width="18" height="19" /></p></td>
-
     <td width="302" valign="top"><p>Concentration gradient between inner and outer side of the    cell</p></td>
+
     <td width="302" align="center" valign="middle"><p>Concentration gradient between inner and outer side of the    cell</p></td>
-
     <td width="113" valign="top"><p>-</p></td>
+
     <td width="113" align="center" valign="middle"><p>-</p></td>
-
     <td width="127" valign="top"><p>Depends on case</p></td>
+
     <td width="127" align="center" valign="middle"><p>Depends on case</p></td>
   </tr>
   </tr>
</table>
</table>
-
<h3><strong>I. Permeability  aflatoxin-membrane</strong></h3>
+
<p><strong>I. Permeability  aflatoxin-membrane</strong><br />
-
<p>
+
   Permeability value between solute and solvent is very  specific for each case, and finding an analogous case to our system is really  difficult. We try to tinker some diffusional equation to find permeability  between aflatoxin and membrane with really few data.<br />
-
   Permeability value between solute and solvent can be described through this equation :<br />
+
  Generally, permeability can be described through this equation :<br />
-
<img src="difussion_clip_image006.png" alt="" width="59" height="34" /></p>
+
<img src="file:///C|/Users/User/AppData/Roaming/Adobe/Dreamweaver CS5/en_US/OfficeImageTemp/clip_image006_0002.png" alt="" width="59" height="34" /></p>
<table border="1" cellspacing="0" cellpadding="0">
<table border="1" cellspacing="0" cellpadding="0">
   <tr>
   <tr>
-
     <td width="73" valign="top"><br />
+
     <td width="73" align="center" valign="middle"><strong>Variable</strong></td>
-
      <strong>Variable</strong></td>
+
     <td width="302" align="center" valign="middle"><p><strong>Definition</strong></p></td>
-
     <td width="302" valign="top"><p><strong>Definition</strong></p></td>
+
     <td width="113" align="center" valign="middle"><p><strong>Value</strong></p></td>
-
     <td width="113" valign="top"><p><strong>Value</strong></p></td>
+
     <td width="127" align="center" valign="middle"><p><strong>Source</strong></p></td>
-
     <td width="127" valign="top"><p><strong>Source</strong></p></td>
+
   </tr>
   </tr>
   <tr>
   <tr>
-
     <td width="73" valign="top"><p>D</p></td>
+
     <td width="73" align="center" valign="middle"><p>D</p></td>
-
     <td width="302" valign="top"><p>Diffusivity constant of aflatoxin-membrane E. coli</p></td>
+
     <td width="302" align="center" valign="middle"><p>Diffusivity constant of aflatoxin-membrane E. coli</p></td>
-
     <td width="113" valign="top"><p>2,05 x 10-9 cm2/s</p></td>
+
     <td width="113" align="center" valign="middle"><p>2,05 x 10-9 cm2/s</p></td>
-
     <td width="127" valign="top"><p>Calculated (see below)</p></td>
+
     <td width="127" align="center" valign="middle"><p>Calculated (see below)</p></td>
   </tr>
   </tr>
   <tr>
   <tr>
-
     <td width="73" valign="top"><p>K</p></td>
+
     <td width="73" align="center" valign="middle"><p>K</p></td>
-
     <td width="302" valign="top"><p>Partition constant of aflatoxin-membrane E. coli</p></td>
+
     <td width="302" align="center" valign="middle"><p>Partition constant of aflatoxin-membrane E. coli</p></td>
-
     <td width="113" valign="top"><p>0,64</p></td>
+
     <td width="113" align="center" valign="middle"><p>0,64</p></td>
-
     <td width="127" valign="top"><p>[9]</p></td>
+
     <td width="127" align="center" valign="middle"><p>[9]</p></td>
   </tr>
   </tr>
   <tr>
   <tr>
-
     <td width="73" valign="top"><p>d</p></td>
+
     <td width="73" align="center" valign="middle"><p>d</p></td>
-
     <td width="302" valign="top"><p>E. coli membrane thickness</p></td>
+
     <td width="302" align="center" valign="middle"><p>E. coli membrane thickness</p></td>
-
     <td width="113" valign="top"><p>13 nm</p></td>
+
     <td width="113" align="center" valign="middle"><p>13 nm</p></td>
-
     <td width="127" valign="top"><p>[2]</p></td>
+
     <td width="127" align="center" valign="middle"><p>[2]</p></td>
   </tr>
   </tr>
</table>
</table>
<p>To determine the value of diffusivity constant, we use  Stoke-Einstein equation<br />
<p>To determine the value of diffusivity constant, we use  Stoke-Einstein equation<br />
-
   <img src="difussion_clip_image008.png" alt="" width="62" height="38" /><br />
+
   <img src="file:///C|/Users/User/AppData/Roaming/Adobe/Dreamweaver CS5/en_US/OfficeImageTemp/clip_image008_0000.png" alt="" width="62" height="38" /><br />
-
   where the radii value of solute (r) can be determined with  the help of molecular weight<br />
+
   where the radii value of solute (r) can be determined with  the help of molecular weight data<br />
-
   <img src="difussion_clip_image010.png" alt="" width="109" height="42" /> <br />
+
   <img src="file:///C|/Users/User/AppData/Roaming/Adobe/Dreamweaver CS5/en_US/OfficeImageTemp/clip_image010_0000.png" alt="" width="109" height="42" /> <br />
-
   So, to simplify our equation, we try to find the correlation between diffusivity constant and solute molecular weight. It can be done through dividing two sets of case (diffusion of protein with well-known molecular weight and diffusion of aflatoxin) and the result is<br />
+
   So, to simplify our equation, we try to find the correlation between diffusivity constant and solute molecular weight. It can be done through dividing two sets of case (diffusion of protein with well-known molecular weight and diffusion of aflatoxin) and the result is<br />
-
  <img src="difussion_clip_image012.png" alt="" width="88" height="38" /></p>
+
<img src="file:///C|/Users/User/AppData/Roaming/Adobe/Dreamweaver CS5/en_US/OfficeImageTemp/clip_image012_0000.png" alt="" width="88" height="38" /></p>
<table border="1" cellspacing="0" cellpadding="0">
<table border="1" cellspacing="0" cellpadding="0">
   <tr>
   <tr>
-
     <td width="73" valign="top"><br />
+
     <td width="73" align="center" valign="middle"><strong>Variable</strong></td>
-
      <strong>Variable</strong></td>
+
     <td width="302" align="center" valign="middle"><p><strong>Definition</strong></p></td>
-
     <td width="302" valign="top"><p><strong>Definition</strong></p></td>
+
     <td width="113" align="center" valign="middle"><p><strong>Value</strong></p></td>
-
     <td width="113" valign="top"><p><strong>Value</strong></p></td>
+
     <td width="127" align="center" valign="middle"><p><strong>Source</strong></p></td>
-
     <td width="127" valign="top"><p><strong>Source</strong></p></td>
+
   </tr>
   </tr>
   <tr>
   <tr>
-
     <td width="73" valign="top"><p>ρ</p></td>
+
     <td width="73" align="center" valign="middle"><p>ρ</p></td>
-
     <td width="302" valign="top"><p>Aflatoxin density</p></td>
+
     <td width="302" align="center" valign="middle"><p>Aflatoxin density</p></td>
-
     <td width="113" valign="top"><p>1,64 g/cm3</p></td>
+
     <td width="113" align="center" valign="middle"><p>1,64 g/cm3</p></td>
-
     <td width="127" valign="top"><p>[7]</p></td>
+
     <td width="127" align="center" valign="middle"><p>[7]</p></td>
   </tr>
   </tr>
   <tr>
   <tr>
-
     <td width="73" valign="top"><p>MW</p></td>
+
     <td width="73" align="center" valign="middle"><p>MW</p></td>
-
     <td width="302" valign="top"><p>Aflatoxin molecular weight</p></td>
+
     <td width="302" align="center" valign="middle"><p>Aflatoxin molecular weight</p></td>
-
     <td width="113" valign="top"><p>312,3</p></td>
+
     <td width="113" align="center" valign="middle"><p>312,3</p></td>
-
     <td width="127" valign="top"><p>[8]</p></td>
+
     <td width="127" align="center" valign="middle"><p>[8]</p></td>
   </tr>
   </tr>
</table>
</table>
-
<h3><strong>II. Aflatoxin  membrane cell area</strong></h3>
+
<div>
-
<p>
+
  <p style="border:solid #F60"><u><strong>How we did it in Simbiology?</strong></u><br />
-
   To simulate the effect of cell growth to diffusion phenomena, we modify the diffusion equation. Membrane cell area will be increased along with cell number, and it affected by cell growth.<br />
+
    <img src="file:///C|/Users/User/AppData/Roaming/Adobe/Dreamweaver CS5/en_US/OfficeImageTemp/clip_image014.jpg" alt="" width="219" height="85" /><br />
-
  We simplify this problem by assuming cell growth is like growing sphere. When the cell number doubled, it can be stated that the membrane cell area and cell volume doubled too.<br />
+
    We create a reversible  reaction with customized reaction rate (&lsquo;Unknown&rsquo; kinetic law), where k_permin  as forward reaction rate and k_permout as backward reaction rate. k_permin and  k_permout is stated by repeatedAssignment rule :<br />
-
  <img src="difussion_clip_image014.jpg" alt="" width="424" height="145" /><br />
+
    <strong>k_permin = P*Acell*((container.mol_out/container) -  (insideCell.mol_in/insideCell))</strong><br />
-
  Cell membrane area can be evaluated every time by this  equation :<br />
+
    <strong>k_permout = P*Acell*(-(container.mol_out/container) +  (insideCell.mol_in/insideCell))</strong><br />
-
   <img src="difussion_clip_image016.png" alt="" width="59" height="19" /> <br />
+
  Notice that k_permin and  k_permout is the same as dn/dt in membrane diffusion general equation</p>
-
   where the value of A0 represents membrane cell area of one E. coli cell and n is cell number at certain time. The value of n can be determined with cell growth kinetic :<br />
+
</div>
-
   <img src="difussion_clip_image018.png" alt="" width="127" height="19" /> <br />
+
<p><strong>II. Aflatoxin  membrane cell area</strong><br />
 +
  When we discussed with team, there is still one problem in  this model :</p>
 +
<p align="center"><br />
 +
  <em>&ldquo;Our biosensor is a  live device and can keep replicating even in the middle of analysis process.  How it will affect this model?&rdquo;</em></p>
 +
<p><br />
 +
   Good thinking! To simulate the effect of cell growth to  diffusion phenomena, we modify the diffusion equation. Membrane cell area will  be increased along with cell number, and it affected by cell growth.</p>
 +
<div style="border:solid #F60">We simplify this problem by assuming cell growth is like growing sphere. When the cell number doubled, it can be stated that the membrane cell area and cell volume doubled too.<br />
 +
<img src="file:///C|/Users/User/AppData/Roaming/Adobe/Dreamweaver CS5/en_US/OfficeImageTemp/clip_image016.jpg" alt="" width="424" height="145" /></div>
 +
<p>Cell membrane area can be evaluated every time by this  equation :<br />
 +
   <img src="file:///C|/Users/User/AppData/Roaming/Adobe/Dreamweaver CS5/en_US/OfficeImageTemp/clip_image018.png" alt="" width="59" height="19" /> <br />
 +
   where the value of A0 represents membrane cell area of one E. coli cell and n is cell number at certain time. The value of n can be determined with cell growth kinetic :<br />
 +
   <img src="file:///C|/Users/User/AppData/Roaming/Adobe/Dreamweaver CS5/en_US/OfficeImageTemp/clip_image020.png" alt="" width="124" height="19" /> <br />
   So, the equation&rsquo;s final form to evaluate cell membrane area  every time become :<br />
   So, the equation&rsquo;s final form to evaluate cell membrane area  every time become :<br />
-
   <img src="difussion_clip_image020.png" alt="" width="150" height="19" /> <br />
+
   <img src="file:///C|/Users/User/AppData/Roaming/Adobe/Dreamweaver CS5/en_US/OfficeImageTemp/clip_image022.png" alt="" width="147" height="19" /> <br />
-
   With the same principle, we can evaluate cell volume every time :<br />
+
   With the same principle, we can evaluate cell volume every time :<br />
-
   <img src="difussion_clip_image022.png" alt="" width="145" height="19" /><br />
+
   <img src="file:///C|/Users/User/AppData/Roaming/Adobe/Dreamweaver CS5/en_US/OfficeImageTemp/clip_image024.png" alt="" width="142" height="19" /><br />
-
To gather the value of A0 dan V0, we use the data that bacteria has area to volume ratio 3:1 [4]. Cell density and wet cell mass of E. coli can be known from literature</p>
+
  To gather the value of A0 dan V0, we use the data that  bacteria has area to volume ratio 3:1 [4]. Cell density and wet cell mass of E. coli can be known from literature</p>
<table border="1" cellspacing="0" cellpadding="0">
<table border="1" cellspacing="0" cellpadding="0">
   <tr>
   <tr>
-
     <td width="205" valign="top"><br />
+
     <td width="205" align="center" valign="middle"><strong>Variable</strong></td>
-
      <strong>Variable</strong></td>
+
     <td width="205" align="center" valign="middle"><p align="center"><strong>Value</strong></p></td>
-
     <td width="205" valign="top"><p align="center"><strong>Value</strong></p></td>
+
     <td width="205" align="center" valign="middle"><p align="center"><strong>Source</strong></p></td>
-
     <td width="205" valign="top"><p align="center"><strong>Source</strong></p></td>
+
   </tr>
   </tr>
   <tr>
   <tr>
-
     <td width="205" valign="top"><p align="center">Cell density</p></td>
+
     <td width="205" align="center" valign="middle"><p align="center">Cell density</p></td>
-
     <td width="205" valign="top"><p align="center">1,105 g/ml</p></td>
+
     <td width="205" align="center" valign="middle"><p align="center">1,105 g/ml</p></td>
-
     <td width="205" valign="top"><p align="center">[5]</p></td>
+
     <td width="205" align="center" valign="middle"><p align="center">[5]</p></td>
   </tr>
   </tr>
   <tr>
   <tr>
-
     <td width="205" valign="top"><p align="center">Wet cell mass</p></td>
+
     <td width="205" align="center" valign="middle"><p align="center">Wet cell mass</p></td>
-
     <td width="205" valign="top"><p align="center">10-12 g</p></td>
+
     <td width="205" align="center" valign="middle"><p align="center">10-12 g</p></td>
-
     <td width="205" valign="top"><p align="center">[6]</p></td>
+
     <td width="205" align="center" valign="middle"><p align="center">[6]</p></td>
   </tr>
   </tr>
</table>
</table>
Line 135: Line 141:
   <table border="1" cellspacing="0" cellpadding="0">
   <table border="1" cellspacing="0" cellpadding="0">
     <tr>
     <tr>
-
       <td width="205" valign="top"><br />
+
       <td width="205" align="center" valign="middle"><strong>Variable</strong></td>
-
        <strong>Variable</strong></td>
+
       <td width="205" align="center" valign="middle"><p align="center"><strong>Value</strong></p></td>
-
       <td width="205" valign="top"><p align="center"><strong>Value</strong></p></td>
+
     </tr>
     </tr>
     <tr>
     <tr>
-
       <td width="205" valign="top"><p align="center">A0</p></td>
+
       <td width="205" align="center" valign="middle"><p align="center">A0</p></td>
-
       <td width="205" valign="top"><p align="center">2,715 x 10-18 m3</p></td>
+
       <td width="205" align="center" valign="middle"><p align="center">2,715 x 10-18   m3</p></td>
     </tr>
     </tr>
     <tr>
     <tr>
-
       <td width="205" valign="top"><p align="center">V0</p></td>
+
       <td width="205" align="center" valign="middle"><p align="center">V0</p></td>
-
       <td width="205" valign="top"><p align="center">9,05 x 10-19 m3</p></td>
+
       <td width="205" align="center" valign="middle"><p align="center">9,05 x 10-19   m3</p></td>
     </tr>
     </tr>
   </table>
   </table>
-
</div>
+
</div>
 +
<div>
 +
  <p style="border:solid #F60"><u><strong>How we did it in  Simbiology?</strong></u><br />
 +
    Uncheck the &ldquo;Constant  Value&rdquo; in parameter A (cell area) and insideCell (cummulative cell volume).  Then we apply this rule (because we use &ldquo;rate&rdquo; as rule type, the equation must  be written in the form of first derivative or dx/dt) :<br />
 +
  <img width="594" height="119" src="file:///C|/Users/User/AppData/Roaming/Adobe/Dreamweaver CS5/en_US/OfficeImageTemp/clip_image025.png" alt="Text Box: A= A_0.n_o.exp⁡(μ_net.t)  à Acell = Aonecell*n0*myu*exp(myu*time)  V=V_0.n_o.exp⁡(μ_net.t)  à insideCell = Vonecell*n0*myu*exp(myu*time)" /></p>
 +
</div>
 +
<div>
 +
  <p style="border:solid #F60">Another perspective from  our model instructor, Mochamad Apri, said that involving cell growth into  diffusion equation will be valid <strong>if our system&rsquo;s analysis time is far more  greater than cell growth rate</strong>. If our system&rsquo;s analysis time is below cell  growth rate, it will be wise to assume that cell growth is negligible for  diffusion phenomena.</p>
 +
</div>
 +
<p><strong>III. Average  aflatoxin concentration in each of our cell</strong><br />
 +
  On previous section, we simulate diffusion phenomena by  assuming all cell is gathered into one big cell with volume = n*Vcell and  membrane area = n*Acell. How we calculate aflatoxin concentration in each of  our cell? The answer is simple, just divide aflatoxin molecule in &ldquo;big cell&rdquo;  with cell number at recent time. But how can we do this simple division in  Simbiology? Simbiology is designed to handle reaction equation, and we need  another way to do this simple equation.<br />
 +
  After long time searching, the solution is really easy. We  just copy the same reaction rate from diffusion phenomena to this  &ldquo;distribution&rdquo; reaction (actually there is no such reaction in real world, it  just helped us find the average aflatoxin concentration) and divide it by  cellNum (parameter for cell number).<br />
 +
  <img src="file:///C|/Users/User/AppData/Roaming/Adobe/Dreamweaver CS5/en_US/OfficeImageTemp/clip_image026.png" alt="" width="371" height="119" /><br />
 +
  diffusion reaction  rate                   : k_permin –  k_permout<br />
 +
  distrib reaction rate        : (k_permin – k_permout)/cellNum</p>
 +
We hope this will help other team who face the same  problem with us.
</div>
</div>
</div>
</div>

Revision as of 02:30, 28 September 2013

Difussion

First thing to do is calculate how much aflatoxin would enter the cell every time. Aflatoxin diffused into cell through simple diffusion mechanism, it means that aflatoxin difusion is drived by concentration gradient.

To simplify our model, we assume that aflatoxin homogenely diffused into cell

0. General equation of diffusion through membrane
Diffusion is frequently modelled if the system needs to transport some molecule through cell membrane. Mathematical model for diffusional phenomena through membrane is :

Where n represents the number of molecule involved in diffusion (So, dn/dt can be stated as “molecule flux through cell membrane”). Parameters of the equation :

Variable

Definition

Value

Source

P

Permeability aflatoxin-membrane E. coli

1,01 x 10-4 cm/s

Calculated (See Sect 1)

A

E. coli membrane cell area

Changing with time

Calculated (see Sect 2)

Concentration gradient between inner and outer side of the cell

-

Depends on case

I. Permeability aflatoxin-membrane
Permeability value between solute and solvent is very specific for each case, and finding an analogous case to our system is really difficult. We try to tinker some diffusional equation to find permeability between aflatoxin and membrane with really few data.
Generally, permeability can be described through this equation :

Variable

Definition

Value

Source

D

Diffusivity constant of aflatoxin-membrane E. coli

2,05 x 10-9 cm2/s

Calculated (see below)

K

Partition constant of aflatoxin-membrane E. coli

0,64

[9]

d

E. coli membrane thickness

13 nm

[2]

To determine the value of diffusivity constant, we use Stoke-Einstein equation

where the radii value of solute (r) can be determined with the help of molecular weight data

So, to simplify our equation, we try to find the correlation between diffusivity constant and solute molecular weight. It can be done through dividing two sets of case (diffusion of protein with well-known molecular weight and diffusion of aflatoxin) and the result is

Variable

Definition

Value

Source

ρ

Aflatoxin density

1,64 g/cm3

[7]

MW

Aflatoxin molecular weight

312,3

[8]

How we did it in Simbiology?

We create a reversible reaction with customized reaction rate (‘Unknown’ kinetic law), where k_permin as forward reaction rate and k_permout as backward reaction rate. k_permin and k_permout is stated by repeatedAssignment rule :
k_permin = P*Acell*((container.mol_out/container) - (insideCell.mol_in/insideCell))
k_permout = P*Acell*(-(container.mol_out/container) + (insideCell.mol_in/insideCell))
Notice that k_permin and k_permout is the same as dn/dt in membrane diffusion general equation

II. Aflatoxin membrane cell area
When we discussed with team, there is still one problem in this model :


“Our biosensor is a live device and can keep replicating even in the middle of analysis process. How it will affect this model?”


Good thinking! To simulate the effect of cell growth to diffusion phenomena, we modify the diffusion equation. Membrane cell area will be increased along with cell number, and it affected by cell growth.

We simplify this problem by assuming cell growth is like growing sphere. When the cell number doubled, it can be stated that the membrane cell area and cell volume doubled too.

Cell membrane area can be evaluated every time by this equation :

where the value of A0 represents membrane cell area of one E. coli cell and n is cell number at certain time. The value of n can be determined with cell growth kinetic :

So, the equation’s final form to evaluate cell membrane area every time become :

With the same principle, we can evaluate cell volume every time :

To gather the value of A0 dan V0, we use the data that bacteria has area to volume ratio 3:1 [4]. Cell density and wet cell mass of E. coli can be known from literature

Variable

Value

Source

Cell density

1,105 g/ml

[5]

Wet cell mass

10-12 g

[6]

So the value of A0 and V0 is :

Variable

Value

A0

2,715 x 10-18 m3

V0

9,05 x 10-19 m3

How we did it in Simbiology?
Uncheck the “Constant Value” in parameter A (cell area) and insideCell (cummulative cell volume). Then we apply this rule (because we use “rate” as rule type, the equation must be written in the form of first derivative or dx/dt) :
Text Box: A= A_0.n_o.exp⁡(μ_net.t)  	à	Acell = Aonecell*n0*myu*exp(myu*time)  V=V_0.n_o.exp⁡(μ_net.t)  	à	insideCell = Vonecell*n0*myu*exp(myu*time)

Another perspective from our model instructor, Mochamad Apri, said that involving cell growth into diffusion equation will be valid if our system’s analysis time is far more greater than cell growth rate. If our system’s analysis time is below cell growth rate, it will be wise to assume that cell growth is negligible for diffusion phenomena.

III. Average aflatoxin concentration in each of our cell
On previous section, we simulate diffusion phenomena by assuming all cell is gathered into one big cell with volume = n*Vcell and membrane area = n*Acell. How we calculate aflatoxin concentration in each of our cell? The answer is simple, just divide aflatoxin molecule in “big cell” with cell number at recent time. But how can we do this simple division in Simbiology? Simbiology is designed to handle reaction equation, and we need another way to do this simple equation.
After long time searching, the solution is really easy. We just copy the same reaction rate from diffusion phenomena to this “distribution” reaction (actually there is no such reaction in real world, it just helped us find the average aflatoxin concentration) and divide it by cellNum (parameter for cell number).

diffusion reaction rate                   : k_permin – k_permout
distrib reaction rate        : (k_permin – k_permout)/cellNum

We hope this will help other team who face the same problem with us.