Geometric constraints in protein folding
The intricate three-dimensional geometries of protein tertiary structures underlie protein function and emerge through a folding process from one-dimensional chains of amino acids. The exact spatial sequence and con1guration of amino acids, the biochemical environment and the temporal sequence of distinct interactions yield a complex folding process that cannot yet be easily tracked for all proteins. To gain qualitative insights into the fundamental mechanisms behind the folding dynamics and generic features of the folded structure, we propose a simple model of structure formation that takes into account only fundamental geometric constraints and otherwise assumes randomly paired connections. We find that despite its simplicity, the model results in a network ensemble consistent with key overall features of the ensemble of Protein Residue Networks we obtained from more than 1000 biological protein geometries as available through the Protein Data Base. Speci1cally, the distribution of the number of interaction neighbors a unit (amino acid) has, the scaling of the structures spatial extent with chain length, the eigenvalue spectrum and the scaling of the smallest relaxation time with chain length are all consistent between model and real proteins. These results indicate that geometric constraints alone may already account for a number of generic features of protein tertiary structures.\n\nAuthor summaryHow proteins fold constitutes one of the most persistent, broad, and exciting open research questions at the intersection of biology, chemistry, and physics. Which mechanisms induce a one-dimensional sequence of amino acids to form into a complex three-dimensional (3D) structure? Proteins in their active 3D structure impact most of the basic processes inside cells, including gene regulation, cell metabolism, and the creation of protein structures themselves. Yet, a general rule about which conditions lead to which speci1c 3D protein structures remains unknown to date.\n\nHere, we demonstrate how a simple model that takes only fundamental geometric constraints into account and otherwise assumes randomly paired connections, naturally generates an ensemble of folded structures that exhibits many of its coarse scale features consistent with those of protein residue networks resulting from tertiary structures of biological proteins. Speci1cally, we tested a set of more than 1000 biological proteins and model structures and extracted a range of ensemble properties, including the spatial extension with chain size, the distribution of the number of interacting neighbors in the folded structure, the spectrum of Laplacian eigenvalues, and the distribution of the dominant non-trivial eigenvalue. We found that all of those properties are consistent between the ensemble of biological protein residue networks and the networks emerging in a self-organized way from the simple model.\n\nThese results indicate that coarse ensemble properties of 3D protein structures are already induced by geometric constraints alone such that only finer scales of the folded structures of individual proteins are speci1cally controlled by the details of their amino acid sequences. Such simple models provide a new angle of analyzing protein structures at the coarse scale of ensembles and may help understand core mechanisms underlying the complex folding process.