  The precise sequence of aminoacids plays a central role in the tertiary
structure of proteins and their functional properties. The Hydrophobic-Polar
lattice models have provided valuable insights regarding the energy landscape.
We demonstrate here the isomorphism between the protein sequences and
designable structures for two and three dimensional lattice proteins of very
long aminoacid chains using exact enumerations and intuitive considerations.We
emphasize that the topological arrangement of the aminoacid residues alone is
adequate to deduce the designable and non-designable sequences without explicit
recourse to energetics and degeneracies. The results indicate the computational
feasibility of realistic lattice models for proteins in two and three
dimensions and imply that the fundamental principle underlying the designing of
structures is the connectivity of the hydrophobic and polar residues.
