Cells and Worms - 1. The Theory

If you scatter 100 worms on a patch of soil 1 meter by 1 meter how many worms will fall on top of another worm? This might seem like a really pointless question, but it is surprisingly relevant to biological research using microscopes. It's also a surprisingly hard question to answer because worms are very wriggly! However, even this dry, theoretical, research problem provides the tools for making fun illustrations...

My work involves a lot of automated image analysis; taking a picture from a microscope and automatically analysing it to extract scientific data. To make sure an automated analysis is reliable you have to think about all the likely problems that might turn up, and with cells and microscopes a common problem is when two cells are lying on top of each other. The problems this causes are easy to imagine; if there are two cells with one nucleus lying on top of each other then it might look like one cell with two nuclei.

For some types of cells it is quite easy to work out how likely two are to touch or lie partly on top of each other when they are scattered randomly over a microscope slide. An example of an easy case is where all cells are circular and the same size; the approximate calculation is quite simple. Unfortunately the cells I work on are more worm-like in shape, about 17 microns long and 2 wide... if you scatter these cells over a slide how many will end up touching?

To work out the answer simulation is vital; the maths is just too complicated to do it analytically. A simulation of worm-like shapes proved to be quite simple:

1. Pick a random starting point, direction and curvature.
2. Start drawing a curved line from that point.
3. Occasionally re-randomise the curvature.
4. Stop once you have reached the length of the cell.
5. Draw the profile of the cell shape along that curve.

Following these simple rules and tweaking the parameters (e.g. the minimum and maximum curvature, frequency of randomising curvature, etc.) gives a simple algorithm for drawing a worm-like shape. With a bit of tweaking it could draw cells that look like trypanosomes. Using this drawing tool it was possible to measure the chance of a cell touching or lying on top of another cell already on the microscope slide. Just repeat the drawing process thousands of times and detect whether the newly drawn cell intersects with any previously drawn ones. Problem solved.

This process gave me the answer I needed, but it also provided a tool for drawing trypanosome-like shapes. Better than that, it was easy to adapt it to make sure no two cells overlapped and they fitted neatly together over the image... And just like that a dry, theoretical, research problem turned into a beautiful image:

This was also easy to adapt to other worm-like shapes, like earthworms:

Software used:

ImageJ