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For example, I have a paper that says: "the atomic RMS distribution about the mean coordinate positions is 0.41 +/- 0.04 A for the backbone atoms and 0.82 +/- 0.04 A for all atoms"
And I have no idea what any of this means, I found several papers with this kind of information but none of them explain why this is relevant in order to understand the structure of a protein.
The "RMS distribution about the mean coordinate positions" could be restated as "certainty of the structure". In other words, the main backbone that defines the large-scale shape in the structure is expected to be accurate to within about 0.41 angstrom, while the atoms more on the periphery (or otherwise not forming the backbone of the structure) are less accurately solved. How important this is will depend on what you want to do with the structure. If you want to get an idea for what it looks like then the accuracy is less important, if you want to do docking then the more accurate the atoms around any sort of pocket/interaction surface are the better.
