Optical super-resolution microscopy enables the observations of individual bio-molecules. The arrangement and dynamic behaviour of such molecules is studied to get insights into cellular processes which in turn lead to various application such as treatments for cancer diseases. STFC's Central Laser Facility provides (among other) public access to super-resolution microscope techniques via research grants. The access includes sample preparation, imaging facilities and data analysis support. Data analysis includes single molecule tracking algorithms that produce molecule traces or tracks from time series of molecule observations. While current algorithms are gradually getting away from "connecting the dots" and using probabilistic methods, they often fail to quantify the uncertainties in the results. We have developed a method that samples a probability distribution of tracking solutions using the Metropolis-Hastings algorithm. Such a method can produce likely alternative solutions together with uncertainties in the results. While the method works well for smaller data sets, it is still inefficient for the amount of data that is commonly collected with microscopes. Given the observations of the molecules, tracking solutions are discrete, which gives the proposal distribution of the sampler a peculiar form. In order for the sampler to work efficiently, the proposal density needs to be well designed. We will discuss the properties of tracking solutions and the problems of the proposal function design from the point of view of discrete mathematics, specifically in terms of graphs. Can mathematical theory help to design a efficient proposal function?
