Difference between revisions of "Team:HokkaidoU Japan/Modeling"

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<P>Here, we would like to think the following system as a mathmatical model;
+
<P>Here, we would like to consider the following system as a mathmatical model;
 
  <ol>
 
  <ol>
 
  <li><i>E. coli</i> can produce Ag43 whose &alpha;-domain is replaced with His-tag recombinant antimicrobial peptides</li>
 
  <li><i>E. coli</i> can produce Ag43 whose &alpha;-domain is replaced with His-tag recombinant antimicrobial peptides</li>
<li>The antimicrobial peptide constitutively can be secreted through Ag43 system. There is a large amount of AspN in liquid culture and AspN cut the antimicrobial peptide out and it diffuses and outflows in the culture rapidly</li>
+
<li>The antimicrobial peptide constitutively can be secreted through Ag43 system. There is a large amount of AspN in liquid culture and AspN cuts the antimicrobial peptides out and they diffuses and outflows in the culture rapidly</li>
 
  <li> We can purify antimicrobial peptides with His-tag affinity column </li>
 
  <li> We can purify antimicrobial peptides with His-tag affinity column </li>
 
</ol>
 
</ol>
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<P>where <i>a</i> is a rate of maximum population growth and K is carrying capacity and defining b=a/K then gives the differential equation.
+
<P>where <i>a</i> is a rate of maximum population growth and K is a carrying capacity and defining b=a/K  
Next, we add the term of toxicity of the antimicrobial peptide to this equation and we describe amount of antimicrobial peptides in the second differential equation as follow.</p>
+
 
 +
Next, we add the term of toxicity of the antimicrobial peptide to this equation and we describe the amount of antimicrobial peptides in the second differential equation as follow.</p>
 
<div align="center">
 
<div align="center">
 
  <img src="https://static.igem.org/mediawiki/2015/e/e2/HokkaidoU_modeling_formula2.png" class="figure" alt="This is differential equations" width="290px",height="auto"><br>
 
  <img src="https://static.igem.org/mediawiki/2015/e/e2/HokkaidoU_modeling_formula2.png" class="figure" alt="This is differential equations" width="290px",height="auto"><br>
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</div>
 
</div>
 
   
 
   
  <P>where c is rate of toxicity of the antimicrobial peptide, e is rate of expression of the antimicrobial peptide f is rate of outflow of the antimicrobial peptide
+
  <P>where c is a rate of toxicity of the antimicrobial peptide, e is a rate of expression of the antimicrobial peptide f is a rate of outflow of the antimicrobial peptide
We can take 1 for 3 constants (a, b, c) of the right side in the first formula using the flexibilities of the scale (In scale transformation, e, f will change into &alpha;, &beta; and N, A into X, Y) <br>
+
We can take 3 constants (a, b, c) of the right side for 1 in the first formula using the flexibilities of the scale (In scale transformation, e, f will change into &alpha;, &beta; and N, A into x, y) <br>
  
 
<div align="center">
 
<div align="center">
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</div><br>
 
</div><br>
  
Here, at arbitrary parameters &alpha; and &beta; , find these phase diagram below(Fig. 1).</p>
+
Here, at arbitrary parameters &alpha; and &beta; , find these phase diagram below (Fig. 1).</p>
 
   
 
   
 
<div align="center">
 
<div align="center">
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<P>Define minute displacement as (&delta;x, &delta;y). the right side in both differential equations as follow.</p>
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<P>Define minute displacement as (&delta;x, &delta;y) and the right side in both differential equations as follow.</p>
  
 
<div align="center">
 
<div align="center">
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</div>
 
</div>
  
<p>Determine eigenvalues of Jacobian matrix in this time and if two eigenvalues are negative, we can find the fixed point stable, if positive we can find the fixed point instable.</p>
+
<p>Determine eigenvalues of Jacobian matrix in this time and if two eigenvalues both are negative, we can find the fixed point stable, but if one side or both is positive we can find the fixed point instable.</p>
 
   The result of calculation is <br>
 
   The result of calculation is <br>
 
    
 
    
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<img src="https://static.igem.org/mediawiki/2015/3/3f/Fomula7_hokkaidoUmodelling.png" style="" class="figure" alt="This is a graph " width="190px", height="auto">
 
<img src="https://static.igem.org/mediawiki/2015/3/3f/Fomula7_hokkaidoUmodelling.png" style="" class="figure" alt="This is a graph " width="190px", height="auto">
 
</div>  
 
</div>  
  <p>Therefore, we illustrated that amount of AMP and population of bacteria will be constant at last regardless of parameter &alpha; and &beta; value.</p>
+
  <p>Therefore, we illustrated that the amount of AMP and the population of bacteria will be constant at last regardless of parameter &alpha; and &beta; value. Thus we can gain AMPs constantly.</p>
  
  

Revision as of 01:09, 19 September 2015

main2

Microbusters

Modeling

Here, we would like to consider the following system as a mathmatical model;

  1. E. coli can produce Ag43 whose α-domain is replaced with His-tag recombinant antimicrobial peptides
  2. The antimicrobial peptide constitutively can be secreted through Ag43 system. There is a large amount of AspN in liquid culture and AspN cuts the antimicrobial peptides out and they diffuses and outflows in the culture rapidly
  3. We can purify antimicrobial peptides with His-tag affinity column


We want to make sure if we obtain antimicrobial peptides through these system constantly or not.

First, we want to describe the number of host cells growing without toxicity of the peptide as the differential equation. The logistic equation is a model of population growth first published by Pierre Verhulst. The logistic model is described by the following differential equation

This is a logistic curve

where a is a rate of maximum population growth and K is a carrying capacity and defining b=a/K Next, we add the term of toxicity of the antimicrobial peptide to this equation and we describe the amount of antimicrobial peptides in the second differential equation as follow.

This is differential equations
This is a differential equations

where c is a rate of toxicity of the antimicrobial peptide, e is a rate of expression of the antimicrobial peptide f is a rate of outflow of the antimicrobial peptide We can take 3 constants (a, b, c) of the right side for 1 in the first formula using the flexibilities of the scale (In scale transformation, e, f will change into α, β and N, A into x, y)

This is differential equations

Here, at arbitrary parameters α and β , find these phase diagram below (Fig. 1).

This is a graph

Fig. 1 phase diagram at arbitrary parameters

We can expect that the amount of antimicrobial peptides and population of bacteria will be constant at last regardless of parameter α and β value. So, we would like to make sure the fixed points of these differential equations is stable or not. Let each of differential equations equal to zero, and solve them then we can get the fixed points of these equations

This is ///

Define minute displacement as (δx, δy) and the right side in both differential equations as follow.

This is ///
This is ///

Determine eigenvalues of Jacobian matrix in this time and if two eigenvalues both are negative, we can find the fixed point stable, but if one side or both is positive we can find the fixed point instable.

The result of calculation is
This is ///
This is a graph

Therefore, we illustrated that the amount of AMP and the population of bacteria will be constant at last regardless of parameter α and β value. Thus we can gain AMPs constantly.

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