9: How many macrophages does it take to kill a virus?

Some organisms move around more or less at random, so we can use diffusion to “model” how these cells move. This works fine, as long as you accept the assumption / simplification that these organisms are mindless little robots that walk around changing directions randomly. Sometimes this is a reasonable concession to make in exchange for the simplicity and power of the diffusion model, other times it is not. It’s a choice that you, the person implementing the model, need to make.

Here is an example of using diffusion to model a cell moving (adapted from Essential Mathematical Biology by N. F. Britton):

The macrophage is a specialised cell that defends the body against foreign materials. Within the lungs, macrophages engulf inhaled viruses and digest them. Macrophages move in a way that is very similar to diffusion – they ooze along at a stately 3 micrometres per minute, but change directions more or less randomly about every 5 minutes.


Consider an alveolus (diameter = 300 micrometres) that contains a single macrophage and a single virus. Assume the macrophage and virus start at opposite ends of the cell and the macrophage moves in a straight line (i.e. not by diffusion) towards the virus. How long will it take for the macrophage to arrive?