Surprisingly little. Maybe things like binding pockets and broad classes of movement (hinges, etc.) can be inferred (sometimes). Otherwise, it's the other way around: the shape of the backbone is determined by the detailed interactions. Just think about alpha helices and beta sheets -- they're stabilized by hydrogen bonding patterns along the protein backbone, but the propensity of a particular residue to form a helix or a sheet is determined by interactions amongst the side-chain atoms in a particular context. The atomic interactions are the story, and the "fold" is really just the outline.

Also, of course, it's not just the atoms in the protein itself: add in water, ions, small molecules, other proteins, etc., and you have a real mess. A small molecule (say, a drug), can displace a water molecule, which has a cost, which causes something else to move a bit to compensate, stretching a bond, and so on and so forth, sometimes resulting in dramatic differences in the bound conformation of a protein. It's not uncommon -- it's essentially the norm. And it's not like there's just one conformation. The whole thing is really like a wiggly, jiggly, metastable mass, and the introduction of a slight perturbation is enough to send the blob spiraling into a new metastable state.

When you start to ask questions about how a protein moves over time, it becomes a much more complicated problem than predicting the overall conformation. Which is itself quite hard.