Team:HokkaidoU Japan/Modeling

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Microbusters

• Project Description
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• Ag43 Secretion System
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• His-tag Inactivation System
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• Modeling

Modeling

Here, we would like to consider 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 cuts the antimicrobial peptides out and they diffuses and outflows in the culture rapidly
3. 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.

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