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Protein denaturation is the loss of a protein's native spatial organization, often with loss of biological function. Changes in temperature, pH or surroundings can disrupt protein secondary structure, protein tertiary structure or protein quaternary structure. Protein denaturation ordinarily preserves the peptide bonds and protein primary structure; it is different from cutting the chain. Recovery by protein folding is possible in some cases, but removing the denaturing conditions does not guarantee that every protein recovers. See Clark, Douglas and Choi: denaturation, environmental conditions and loss of function.

In the diagram, unfolding preserves all eleven peptide bonds. Cleaving the connection between positions 6 and 7 leaves two six-residue fragments: each contains five peptide bonds, so there are $5+5=10$ in total. Simply cooling or otherwise restoring suitable conditions cannot recover the original chain by protein folding alone; the missing covalent bond would have to be remade. Conversely, protein aggregation can impede recovery without breaking any peptide bond. These distinctions explain why loss of function does not by itself reveal whether a protein unfolded, associated with other chains, or suffered chemical damage.

Protein aggregation is the association of proteins into clusters, often involving exposed regions of partly folded or unfolded polypeptide chains. It can prevent recovery of normal function while leaving the protein primary structures intact. It is distinct from both unfolding one chain and the functional assembly described by protein quaternary structure. See Cooper: partly folded chains, aggregation and chaperone protection.

The protein primary structure specifies the ordered amino acid residues of each polypeptide chain, joined through peptide bonds. A chain is conventionally written from its N-terminus to its C-terminus. In the broader chemical sense, a complete protein primary structure also specifies covalent bond crosslinks between chain positions, including protein disulfide bonds; a bare sequence string leaves that information unstated. This describes chemical connectivity rather than the spatial arrangement of protein tertiary structure or protein quaternary structure. See IUPAC Gold Book: primary structure, sequence and chain crosslinking and IUPAC-IUB, 3AA-16: writing residue sequences from the amino end. Some conventions reserve protein primary structure for sequence alone and call the complete chemical connectivity the covalent structure; the distinction should be stated when comparing descriptions. See IUPAC-IUB polypeptide-conformation recommendations, section 6.1: the narrower sequence-only convention.

In this toy example M denotes methionine, A alanine and G glycine. Both polypeptide chains have six amino acid residues, five peptide bonds, an M at the N-terminus and a G at the C-terminus. Their amino acid composition is identical, yet position 2 is A in MAGAMG and M in MMAAGG. There are $6!/(2!2!2!)=90$ distinct orders with these counts, a multinomial coefficient. Even fixing those two end residues leaves $4!/(1!2!1!)=12$ orders. The counts constrain the answer without determining it.

For multiple polypeptide chains, record each sequence separately and specify any protein disulfide bonds between cysteine positions. Mature insulin, for example, has an A chain of 21 amino acid residues and a B chain of 30 amino acid residues. Their combined 51 residues give $(21-1)+(30-1)=49$ peptide bonds, rather than the 50 of one unbroken 51-residue chain. See Cooper, The Cell: insulin's two chains and cysteine crosslinks, figure 2.16. A protein disulfide bond joins side groups; it does not concatenate the C-terminus of one chain with the N-terminus of another. The three insulin protein disulfide bonds connect A6 to A11 within the A chain, and A7 to B7 and A20 to B19 between chains. Positions are numbered from each chain's N-terminus. See Replacement of the interchain disulfide bridge-forming amino acids A7 and B7 by glutamate impairs the structure and activity of insulin: native disulfide positions.

Why keep this information? Protein primary structure lets us distinguish changes of chemical identity from changes of shape, and compare corresponding positions between proteins. Protein folding can bring distant amino acid residues together at a protein binding site while preserving their sequence. The attractive idea is that a readable linear record can constrain a complicated three-dimensional object; it is not, by itself, a guarantee of a unique fold in every environment. See protein folding and the ribonuclease A refolding experiment. Merely bending an intact chain preserves its protein primary structure; hydrolysis of a peptide bond, or changing which positions a protein disulfide bond connects, changes its chemical connectivity.

The protein quaternary structure is the arrangement of multiple polypeptide chains in a functional assembly. A single-chain protein has protein tertiary structure without requiring protein quaternary structure. For an exact bookkeeping example, two intact model chains of twelve amino acid residues each contain $2\times 12=24$ residues and $2\times 11=22$ peptide bonds. Assembly alone adds no peptide bond joining their ends. Stable functional assembly should also be distinguished from protein aggregation. See Alberts et al.: proteins as assemblies of polypeptide subunits.

Protein structure describes both the chain's chemical organization and its spatial arrangement. Four levels answer different questions: protein primary structure gives the order; protein secondary structure gives recurring local patterns; protein tertiary structure gives one chain's overall arrangement; protein quaternary structure gives the assembly of multiple chains. These are levels of description, not four mandatory stages that every protein must pass through. See Clark, Douglas and Choi: four levels of protein structure.