I am working with an Arabidopsis mutant with an F-box protein knocked out. It has been shown that F-box proteins targets must first be phosphorylated (Skowrya et al., 1997). I have heard of phosphorylation sites, but I can't find out whether every protein has them. Can any protein be phosphorylated?
Phosphorylation can occur on specific amino acids only, what you have called phosporylation sites. These amino acids are Ser, Tyr, Asp, Thr and His. In theory any of these amino acids may be phosphorylated, but in reality it may not actually occur for a number of reasons. Some of these are because of the change in overall charge of the protein which can influence the 3D conformation, or the amino acids are not accessible to specific kinases, etc. If you ask for the purposes of doing a Western blot, then the antibody specification sheet should indicate whether a phosphorylated form exists and there should be a reference to the literature describing this modification.
One important thing is missing in the other answers: not only phosphorylation will happen only at selected aminoacids, but it will not happen at all of those.
So, not all of the Ser/Thr/Tyr of a protein can be phosphorylated because they could be structurally unaccessible to protein kinase and because they need to be in a specific motif in order to be phosphorylated.
Phosphorylation requires exposed serine, threonine, tyrosine, or histidine residues (in eukaryotes). This is because the transfer of phosphate groups to proteins is mediated by a class of proteins called kinases. Kinases can have broad or specific activity.
This review ought to have most of the answers to your questions :
I like Nico's response best +1. I did find an interesting review on phosphoarginine and phospholysine - the list of possible phosphorylation sites grows. Not only is it the spatial context of the amino acid to be (or not) phosphorylated, as Nico writes and Leonardo implies, but also the temporal. Are the target protein and protein kinase co-expressed? If you're looking at specific tissues (root, flower, leaf) and the kinase in question is only produced in seedlings, maybe that potential phosphorylation won't really occur.
For one of the most comprehensive databases of protein post-translational modification (including phosphorylation, methylation, acetylation, ubiquitination, etc.), check out PhosphoSite. You can find links to sequences, diseases, motifs, publications, antibodies, mass spec experiments, structures, you name it.