Team:HokkaidoU Japan/Modeling

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Modeling

Modeling

If the amount of antimicrobial peptides (here referred to as A) a bacteria cell produces 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 mathematical 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)

where r is rate of maximum population growth and K is carrying capacity. Dividing both sides by K and defining b=N/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)

where c is rate of toxicity of the antimicrobial peptide, d is rate of expression of the antimicrobial peptide e is rate of decomposition of the antimicrobial peptide

Here, we took 1 for 3 constants (a, b, c) of the right side in the first formula using the flexibilities of the scale. Though here we let parameter e value be 1 arbitarily, this value does not affect qualitatively as the formula shows. We find the graph below by regarding this formula as a function of parameter d.

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