Probably your best bet is to embed other SWT widgets in the StyledText. Each widget will perform a certain calculation in a background thread and update its visual representation as the results come in. Note that you can do the calculations in the background but the rendering must happen in the SWT thread. So you can't do many complex things during rendering. If things become too slow, use cache images (create a couple of off-screen images where you can render the results and then simply draw those images).
Extend these widgets from Canvas
as this is the widget intended for custom rendering. It will also allow you to react to different events (i.e. display additional information when the user hovers over an enzyme cut).
Be careful with the enzyme cuts, though: They vary in height. I suggest to give this widget a bit more space by default (even when it's not used) so the text doesn't jump a lot while the widget calculates and adds cuts.