A topological perspective into the sequence and conformational space of proteins

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. It is customarily thought that computations of this magnitude will require 'million years' with the present day computers. 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. * 1. Introduction The prediction of tertiary structures of proteins from the constituent aminoacid sequences constitutes a major challenging problem in science, on account of its complexity and importance 1 .Despite considerable efforts in unraveling the mysterious connection between the native structures of proteins and their most stable sequences, the mechanism underlying the energy landscape is not yet clear 2 .However, valuable insights have hitherto been gathered from minimalistic lattice models of proteins and simple fundamental rules seem to govern the native structures. The highly popular Hydrophobic-Polar (HP) model 3 for a chain of N aminoacid residues (or nodes) has provided information regarding the designable protein sequences vis a vis conformations for two and three dimensional lattices of moderate chain lengths. The lattice models provide a basis for comprehending the ground state of the native structures of proteins if systematically and exhaustively analyzed. The exact enumeration has been recognized as an NP hard problem 4 which restricts the conformational analysis to chains of small lengths. Further, the tertiary structures of proteins is dictated solely by the HP sequence 5 and a recent experimental evidence highlights the importance of the sequence alignment using a protein-like polymer sequence for the coil to globule transition 6. It is therefore imperative to carry out an exact sequence-conformation mapping from a new perspective for fairly long aminoacid chain lengths. The applicability of spin glass models to coil to globule transitions has also been demonstrated 7. In the case of protein sequences, the energies and degeneracies of 2 N states need to be …