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

The mechanisms by which a protein's 3D structure can be determined based on its 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 is using protein contact maps to better understand proteins' properties. In this study, we explore the emergent behaviour of contact maps for different geometrically constrained models and compare them to real-world protein systems. Specifically, we derive an analytical approximation for the distribution of amino acid distances, denoted as $P(s)$, using a mean-field approach based on a geometric constraint model. This approximation is then validated for amino acid distance distributions generated from a 2D and 3D version of the geometrically constrained random interaction model. For real protein data, we show how the analytical approximation can be used to fit amino acid distance distributions of protein chain lengths of $L\approx100$, $L\approx200$, and $L\approx300$ generated from two different methods of evaluating a PCM, a simple cutoff based method and a shadow map based method. We present evidence that geometric constraints are sufficient to model the amino acid distance distributions of protein chains in bulk and amino acid sequences only play a secondary role, regardless of the definition of the PCM.