Modeling

Here, we would like to think 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 cut the antimicrobial peptide out and it diffuses and outflows in the culture rapidly
3. We can purify antimicrobial peptides with His-tag affinity column

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

<img src="" class="figure" alt="This is a logistic curve" width="350px",height="auto">

where a is a 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 as follow.

<img src="" class="figure" alt="This is differential equations" width="290px",height="auto">

<img src="" class="figure" alt="This is a differential equations" width="190px", height="auto">

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 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 α, β and N, A into X, Y)

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). the right side in both differential equations as follow.

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.

 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.

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