Difference between revisions of "Team:HokkaidoU Japan/Modeling"
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<P>If amount of antimicrobial peptides (here referred to as A) bacteria cell produce is increased, conversely the number of host cells (referred to as N) will be decreased because of toxicity of the peptide. We want to express this relation as a mathmatical model. | <P>If amount of antimicrobial peptides (here referred to as A) bacteria cell produce is increased, conversely the number of host cells (referred to as N) will be decreased because of toxicity of the peptide. We want to express this relation as a mathmatical model. | ||
| − | 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 differential equation ( | + | 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 differential equation (Figure.1)</P> |
| + | |||
<img src="https://static.igem.org/mediawiki/2015/e/e5/HokkaidoU_modeling_formula1.png" class="photo" alt="This is a logistic curve "> | <img src="https://static.igem.org/mediawiki/2015/e/e5/HokkaidoU_modeling_formula1.png" class="photo" alt="This is a logistic curve "> | ||
| + | <p class="caption">Figure.1 the logistic model</p> | ||
| − | <P>where a is rate of maximum population growth and K is carrying capacity and defining b=a/K then gives the differential equation | + | <P>where a is rate of maximum population growth and K is carrying capacity and defining b=a/K then gives the differential equation. |
| − | 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 ( | + | 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 (Figure.2).</p> |
| − | <img src="https://static.igem.org/mediawiki/2015/e/e2/HokkaidoU_modeling_formula2.png" class="photo" alt="This is differential equations"> | + | |
| − | < | + | <img src="https://static.igem.org/mediawiki/2015/e/e2/HokkaidoU_modeling_formula2.png" class="photo" alt="This is differential equations"><br> |
| − | + | <img src="https://static.igem.org/mediawiki/2015/1/15/HokkaidoU_modeling_formula3.png" class="photo" alt="This is a differential equations"> | |
| + | <p class="caption">Figure.2 differential equations of the antimicrobial peptide toxicity</p> | ||
| + | |||
<P>where c is rate of toxicity of the antimicrobial peptide, e is rate of expression of the antimicrobial peptide f is rate of decomposition of the antimicrobial peptide | <P>where c is rate of toxicity of the antimicrobial peptide, e is rate of expression of the antimicrobial peptide f is rate of decomposition of the antimicrobial peptide | ||
We took 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 α, β) | We took 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 α, β) | ||
| − | Here, we change parameter α and β value and find these graph below.</p> | + | Here, we change parameter α and β value and find these graph below(Figure.3).</p> |
| + | |||
<img src="https://static.igem.org/mediawiki/2015/a/a6/Graph1hokkaidoUmodel.png" class="photo" alt="This is a graph "> | <img src="https://static.igem.org/mediawiki/2015/a/a6/Graph1hokkaidoUmodel.png" class="photo" alt="This is a graph "> | ||
| + | <p class="caption">Figure3. Population of bacteria expressing the antimicrobial peptide at various value of α and β</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. | ||
| + | Let each of differential equations equal to zero, and solve them then we can get the fixed points of these equations (figure.4)</p> | ||
| − | |||
| − | |||
<img src="https://static.igem.org/mediawiki/2015/7/79/HokkaidoU_modeling_formula4.png" class="photo" alt="This is /// "> | <img src="https://static.igem.org/mediawiki/2015/7/79/HokkaidoU_modeling_formula4.png" class="photo" alt="This is /// "> | ||
| − | + | <p class="caption">Figure.4 </p> | |
| − | <P>Define minute intervals as (δx, δy) | + | |
| + | |||
| + | <P>Define N≡X,A≡Y and minute intervals as (δx, δy). the right side in both differential equations as follow.</p> | ||
<img src="https://static.igem.org/mediawiki/2015/a/ab/Fomula5.5_hokkaidoUmodelling.png" class="photo" alt="This is /// "> | <img src="https://static.igem.org/mediawiki/2015/a/ab/Fomula5.5_hokkaidoUmodelling.png" class="photo" alt="This is /// "> | ||
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<p>Determine the value of eigenvalues of each matrixes 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 the value of eigenvalues of each matrixes and if two eigenvalues are negative, we can find the fixed point stable, if positive we can find the fixed point instable.</p> | ||
The result of calculation is | The result of calculation is | ||
| + | |||
<img src="https://static.igem.org/mediawiki/2015/1/1c/HokkaidoU_modeling_formula6.png" class="photo" alt="This is a graph "> | <img src="https://static.igem.org/mediawiki/2015/1/1c/HokkaidoU_modeling_formula6.png" class="photo" alt="This is a graph "> | ||
<img src="https://static.igem.org/mediawiki/2015/3/3f/Fomula7_hokkaidoUmodelling.png" class="photo" alt="This is a graph "> | <img src="https://static.igem.org/mediawiki/2015/3/3f/Fomula7_hokkaidoUmodelling.png" class="photo" alt="This is a graph "> | ||
| + | |||
<p>Therefore, we illustrated that amount of AMP and population of bacteria will be constant at last regardless of parameter α and β value.</p> | <p>Therefore, we illustrated that amount of AMP and population of bacteria will be constant at last regardless of parameter α and β value.</p> | ||
| + | |||
Revision as of 14:48, 17 September 2015
Modeling
If amount of antimicrobial peptides (here referred to as A) bacteria cell produce is increased, conversely the number of host cells (referred to as N) will be decreased because of toxicity of the peptide. We want to express this relation as a mathmatical model. 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 differential equation (Figure.1)
Figure.1 the logistic model
where a is rate of maximum population growth and K is carrying capacity and defining b=a/K then gives the differential equation. 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 (Figure.2).
Figure.2 differential equations of the antimicrobial peptide toxicity
where c is rate of toxicity of the antimicrobial peptide, e is rate of expression of the antimicrobial peptide f is rate of decomposition of the antimicrobial peptide We took 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 α, β) Here, we change parameter α and β value and find these graph below(Figure.3).
Figure3. Population of bacteria expressing the antimicrobial peptide at various value of α and β
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 (figure.4)
Figure.4
Define N≡X,A≡Y and minute intervals as (δx, δy). the right side in both differential equations as follow.
Determine the value of eigenvalues of each matrixes and if two eigenvalues are negative, we can find the fixed point stable, if positive we can find the fixed point instable.
The result of calculation is
Therefore, we illustrated that amount of AMP and population of bacteria will be constant at last regardless of parameter α and β value.
