For about US$25 (1995 dollars), your local college bookstore will probably sell you a tinkertoy-like set of model atoms. Using the included connector rods, you can connect the atoms to create cool hydrocarbons and many other molecules! With enough sets, I suppose you could model proteins, even your own DNA.

I have some excellent drinking straw/ plastic hub sets of this. Made by a company in Oxford and sold in dillons and other bookstores for no obvious reason. Several sets can, indeed be used to make DNA or short peptides. Since the scale of the sets I have is about 1cm to the Ångstrom even a few bases of DNA are about a meter high. Alpha helicices are about the length of a kitchen roll and satsfyingly stable with hydrogen bonds in.

Sadly, proteins are next to impossible. Even the smallest are around 80-100 amino acids which is 800-1000 atoms minimum and would be the size of a large beach ball. Additionally, the secondary structure is held together by non-covalent forces that have no plastic analog. Those who still make physical (as opposed to computational) models of proteins use(d) metal rods as scaffold to attach the structure to.

I think that watson and crick used some sort of meccano set to first model DNA - and they also were among the first to work out alpha helix structure. Long live the physical molecular model!

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