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The mechanisms by which a protein's 3D structure can be determined based onits amino acid sequence have long been one of the key mysteries of biophysics.Often simplistic models, such as those derived from geometric constraints,capture bulk real-world 3D protein-protein properties well. One approach isusing protein contact maps to better understand proteins' properties. Here, weinvestigate the emergent behaviour of contact maps for different geometricallyconstrained models and real-world protein systems. We derive an analyticalapproximation for the distribution of model amino acid distances, $s$, by meansof a mean-field approach. This approximation is then validated for simulationsusing a 2D and 3D random interaction model, as well as from contact maps ofreal-world protein data. Using data from the RCSB Protein Data Bank (PDB) andAlphaFold~2 database, the analytical approximation is fitted to protein chainlengths of $L\approx100$, $L\approx200$, and $L\approx300$. While a universalscaling behaviour for protein chains of different lengths could not be deduced,we present evidence that the amino acid distance distributions can beattributed to geometric constraints of protein chains in bulk and amino acidsequences only play a secondary role.

What geometrically constrained folding models can tell us about real-world protein contact maps