Difference between revisions of "Team:HokkaidoU Japan/Modeling"
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| − | <P>Here, we would like to | + | <P>Here, we would like to consider the following system as a mathmatical model; |
<ol> | <ol> | ||
| − | <li>E. coli can produce Ag43 | + | <li><i>E. coli</i> can produce Ag43 whose α-domain is replaced with His-tag recombinant antimicrobial peptides</li> |
| − | <li>The antimicrobial peptide constitutively can be secreted through Ag43 system. There | + | <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> | ||
| − | <br><br> We want to make sure we obtain antimicrobial peptides through these system constantly or not. </P> | + | <br><br> We want to make sure if we obtain antimicrobial peptides through these system constantly or not. </P> |
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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 </P> | 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 </P> | ||
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| + | <div align="center"><img src="https://static.igem.org/mediawiki/2015/e/e5/HokkaidoU_modeling_formula1.png" class="figure" alt="This is a logistic curve" width="350px",height="auto"></div> | ||
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| + | <P>where <i>a</i> is a rate of maximum population growth and K is a carrying capacity and defining b=a/K | ||
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| + | 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"> | ||
| + | <img src="https://static.igem.org/mediawiki/2015/4/4e/Hokkaidom2.png" class="figure" alt="This is differential equations" width="287px",height="auto"><br> | ||
| + | </div> | ||
| + | |||
| + | <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 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) <br> | ||
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| − | |||
<div align="center"> | <div align="center"> | ||
| − | + | <img src="https://static.igem.org/mediawiki/2015/8/88/Hokkaidom3.png" class="figure" alt="This is differential equations" width="335px",height="auto"> | |
| − | + | </div><br> | |
| − | + | Here, at arbitrary parameters α and β , find these phase diagram below (Fig. 1).</p> | |
| − | + | ||
| − | + | ||
| − | Here, | + | |
<div align="center"> | <div align="center"> | ||
| − | <img src="https://static.igem.org/mediawiki/2015/a/a6/Graph1hokkaidoUmodel.png" style="width:auto; height:auto;" class="figure" alt="This is a graph "></div> | + | <img src="https://static.igem.org/mediawiki/2015/a/a6/Graph1hokkaidoUmodel.png" style="width:auto; height:auto;" |
| + | class="figure" alt="This is a graph "> | ||
| + | </div> | ||
| + | <p class="caption">Fig. 1 phase diagram at arbitrary parameters</p> | ||
| − | + | <p>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. | |
| − | + | ||
| − | <p>We can expect that the amount of antimicrobial peptides and population of bacteria will be constant at last regardless of parameter | + | |
Let each of differential equations equal to zero, and solve them then we can get the fixed points of these equations </p> | Let each of differential equations equal to zero, and solve them then we can get the fixed points of these equations </p> | ||
<div align="center"> | <div align="center"> | ||
| − | <img src="https://static.igem.org/mediawiki/2015/ | + | <img src="https://static.igem.org/mediawiki/2015/4/4f/Hokkaidom4.png" style="width:563px; height:auto;" class="figure" alt="This is /// "> |
| + | </div> | ||
| − | + | <P>Define minute displacement as (δx, δy) and the right side in both differential equations as follow.</p> | |
| − | <P>Define | + | |
<div align="center"> | <div align="center"> | ||
| − | <img src="https://static.igem.org/mediawiki/2015/ | + | <img src="https://static.igem.org/mediawiki/2015/2/29/Hokkaidom1.png" style="" class="figure" alt="This is /// " width="307px", height="auto"><br> |
| − | <img src="https://static.igem.org/mediawiki/2015/ | + | <img src="https://static.igem.org/mediawiki/2015/1/1f/Hokkaidom5.png" style="" class="figure" alt="This is /// " width="637px",height="auto"> |
| + | </div> | ||
| − | + | <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> | |
| − | <p>Determine | + | |
The result of calculation is <br> | The result of calculation is <br> | ||
<div align="center"> | <div align="center"> | ||
| − | <img src="https://static.igem.org/mediawiki/2015/ | + | <img src="https://static.igem.org/mediawiki/2015/7/75/Hokkaidom6.png" style="" class="figure" alt="This is /// " width="550px", height="auto"><br> |
| − | <img src="https://static.igem.org/mediawiki/2015/3/3f/Fomula7_hokkaidoUmodelling.png" style="" class="figure" alt="This is a graph " width=" | + | <img src="https://static.igem.org/mediawiki/2015/3/3f/Fomula7_hokkaidoUmodelling.png" style="" class="figure" alt="This is a graph " width="175px", height="auto"> |
| − | + | </div> | |
| − | <p>Therefore, we illustrated that amount of AMP and population of bacteria will be constant at last regardless of parameter | + | <p>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.</p> |
Latest revision as of 02:44, 19 September 2015
Modeling
Here, we would like to consider the following system as a mathmatical model;
- E. coli can produce Ag43 whose α-domain is replaced with His-tag recombinant antimicrobial peptides
- 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
- 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
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.
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)
Here, at arbitrary parameters α and β , find these phase diagram below (Fig. 1).
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
Define minute displacement as (δx, δy) and the right side in both differential equations as follow.
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
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.
