Wool and silk come from animals, and both are proteins: long chains of amino acids joined by peptide bonds, the same -CONH- amide link that holds nylon's chains together.
Almost every peptide bond along a protein chain carries an N–H group and a C=O group, and these make hydrogen bonds with each other: the δ+ hydrogen of an N–H is drawn to a lone pair on the δ− oxygen of a C=O somewhere else along the chain, or on another chain. Silk and wool use these hydrogen bonds in two different ways.
Silk: flat sheets
Silkworms, the caterpillars of the silk moth, spin silk to build their cocoons. Nearly half of the amino acids in silk's main protein, fibroin, are glycine, and nearly a third are alanine. Their side groups are small, a hydrogen atom and a methyl group, so the chains lie straight and close, with every N–H hydrogen-bonded to a C=O on the chain next door. The chains make flat sheets, and the sheets stack closely on top of each other.
Silk fibres are smooth and fine, and each has a rounded triangular cross-section that bends and reflects light like a prism, which gives silk its shine. A gummy second protein, sericin, coats the raw silk and is washed off in hot, soapy water before the silk is woven.
Wool: coiled chains
Wool's protein is keratin, the same family of proteins as your hair and nails. Each keratin chain coils into a helix, like a spring, with 3.6 amino acids in every turn. Hydrogen bonds hold the coil in place: each C=O holds the N–H of the amino acid four places further along the same chain. Pairs of coils twist round each other, and bundles of them run along the fibre.
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Keratin is rich in cysteine. Where two cysteines on neighbouring chains meet, each loses the hydrogen from its -SH group and their sulfur atoms join with a covalent bond, an S–S bridge, which ties the chains together. Pull a wool fibre and its coils stretch out; let go, and the bridges and the coils pull it back. That springiness is why a wool jumper keeps its form and shakes out its creases.