  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
configuration 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. Specifically,
the distribution of the number of interaction neighbors a unit (amino acid)
has, the scaling of the structure's 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.
