Firstly, is important to remember that protein structures are dynamic due to the torsion angles between the N-terminal and C-terminal bonds. There are different conformations to expose different sequences to the outside of the protein to react/catalyze. So there is no one perfect conformation for a protein in a biological system.
The best models we have are taken through x-ray crystallography of the crystalized protein structure. This static portrayal of the protein may be inaccurate, because the biological system will expose the protein to different hydrophobic/hydrophilic interactions than the affect of the protein to itself.
Anfinsen's experiments were able to definitively prove the a proteins structure is coded by its amino acid sequence. To answer your question, this sequence is primarily responsible for a protein's core structure formation.
Many deletions/insertions occur during specific transition states that the protein is in. Enzymes actually target specific conformations of a protein that best produces the P product. See image:
