Kirby Bauer Antibiotic Testing

1. Related to the two dimensional diffusion program from the FRAP problem from homework 3, and also a number of problems from this assignment, is Kirby-Bauer antibiotic testing. Make a theoretical model of bacterial density on a two dimensional plate, that describes the variation of bacterial density with time, when a wafer the shape of a circular disk is placed in medium. You don't have to implement this as code or solve the model analytically, but write down a precise mathematical model with parameters, such as the diffusion coefficient of the antibiotic and at least one parameter describing susceptibility to the antibiotic. The model should be time dependent. You can further assume that there is only one wafer and that the plate is infinite in size.

You should describe the following parts of the model:

(a) The growth of bacteria as a function of time. You can assume that when the bacteria reproduce, they stay at the same position. (Extending this to allow a change in position will be part of a later problem.)

(b) The diffusion of antibiotic from the wafer. You can model the wafer as a substance with the same diffusion coefficient as the surrounding agar plate. You can assume that the amount of antibiotic that gets absorbed by cells is negligible.

(c) The interaction between the antibiotic and the bacteria. This will depend on the kind of antibiotic. You can distinguish two extremes, a bacteriostatic agent, as opposed to one that is bactericidal.

• Qualitatively, describe how would you expect the bacterial density to vary with time?

2. If you want to go further, you can turn this into a two week long project by doing the following additions:

• Implement the model numerically to obtain the bacterial concentration as a function of time. You can start the bacteria off as being at a uniform concentration, and the antibiotic concentration as being uniform within a radius and zero outside of its.
• Study what happens if the initial concentration of antibiotic is not circularly symmetric, but say, starts as a square shaped wafer, or some other shape?
• What happens if the initial concentration of bacteria is not uniform. For example, if you add random noise to the concentration, or a sinusoidal fluctuation of a given wavelength?
• Now modify your model to include antibiotic the fact that antibiotic gets absorbed by bacteria, so the concentration of antibiotic will not be conserved.
• Instead of antibiotic, consider the case of a predatory bacteria or a virus. When the virus infects a bacterium, it will make a certain number of copies of itself. One is therefore turning the concentration of one component (bacteria) into another component (viruses). You can still assume that the virus particles diffuse while the bacteria does not.
• How do your results compare with studies of similar problems in the literature?

You can answer a lot of related questions and make changes to the modeling as well.