Single-molecule localisation microscopy: accounting for chance co-localisation between foci in bacterial cells

Using single-molecule fluorescence microscopes, individual biomolecules can be observed within live bacterial cells. Using differently coloured probes, physical associations between two different molecular species can be assessed through co-localisation measurements. However, bacterial cells are finite and small (~ 1 μm) relative to the resolution limit of optical microscopes (~ 0.25 μm). Furthermore, the images produced by optical microscopes are typically two-dimensional projections of three-dimensional objects. These limitations mean that a certain proportion of object pairs (molecules) will inevitably be assigned as being co-localised, even when they are distant at molecular distance scales (nm). What is this proportion? Here, we attack this problem, theoretically and computationally, by creating a model of the co-localisation expected purely due to chance. We thus consider a bacterial cell wherein objects are distributed at random and evaluate the co-localisation in a fashion that emulates an experimental analysis. We consider simplified geometries where we can most transparently investigate the effect of a finite size of the cell and the effect of probing a three-dimensional cell in only two dimensions. Coupling theory to simulations, we also study the co-localisation expected due to chance using parameters relevant to bacterial cells. Overall, we show that the co-localisation expected purely due to chance can be quite substantial and describe the parameters that it depends upon.