Team:HokkaidoU Japan/Modeling

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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 following differential equation

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 as follow.

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 parameters α and β values arbitrarily and find these graph below(Figure.1).

Figure1. 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

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.

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