We argue that protein loops can be described by topological domain-wall solitons that interpolate between ground states which are the ␣ helices and ␤ strands. We present an energy function that realizes loops as soliton solutions to its equation of motion, and apply these solitons to model a number of biologically active proteins including 1VII, 2RB8, and 3EBX ͑Protein Data Bank codes͒. In all the examples that we have considered we are able to numerically construct soliton solutions that reproduce secondary structural motifs such as ␣-helix-loop-␣-helix and ␤-sheet-loop-␤-sheet with an overall root-mean-square-distance accuracy of around 1.0 Å or less for the central ␣-carbons, i.e., close to the limits of current experimental accuracy. DOI: 10.1103/PhysRevE.82.011916 PACS number͑s͒: 87.15.Cc, 05.45.Yv Solitons are ubiquitous and widely studied objects that can be materialized in a variety of practical and theoretical scenarios ͓1,2͔. For example solitons can be deployed for data transmission in transoceanic cables, for conducting electricity in organic polymers ͓1͔, and they may also transport chemical energy in proteins ͓3͔. Solitons explain the Meissner effect in superconductivity and dislocations in liquid crystals ͓1͔. They also model hadronic particles, cosmic strings, and magnetic monopoles in high energy physics ͓2͔ and so on. The first soliton to be identified is the Wave of Translation that was observed by John Scott Russell in the Union Canal of Scotland. This wave can be accurately described by an exact soliton solution of the Korteweg-de Vries ͑KdV͒ equation ͓1͔. At least in principle it can also be constructed in an atomary level simulation where one accounts for each and every water molecule in the Canal, together with all of their mutual interactions. However, in such a Gedanken simulation it would probably become a real challenge to unravel the collective excitations that combine into the Wave of Translation without any guidance from the known soliton solution of the KdV equation: Solitons can not be constructed simply by adding up small perturbations around some ground state. Instead, a ͑topological͒ soliton emerges when non-linear interactions combine elementary constituents into a localized collective excitation that is stable against small perturbations and cannot decay, unwrap or disentangle ͓1,2͔.In this Communication we argue that topological solitons describe proteins in their native folded state ͓4,6͔. We characterize a folded protein by the Cartesian coordinates r i of its N central ␣ carbons, with i =1, ... ,N. For many biologically active proteins these coordinates can be downloaded from protein data bank ͑PDB͒ ͓7͔. Alternatively, the protein can be described in terms of its bond and torsion angles that can be computed from the PDB data. For this we introduce the tangent vector t i and the binormal vector b iTogether with the normal vector n i = b i ϫ t i we then have three vectors that are subject to the discrete Frenet equation ͓8͔.Here, T 2 and T 3 are two of the standard gen...

We introduce a novel generalization of the discrete nonlinear Schrödinger equation. It supports solitons that we utilize to model chiral polymers in the collapsed phase and, in particular, proteins in their native state. As an example we consider the villin headpiece HP35, an archetypal protein for testing both experimental and theoretical approaches to protein folding. We use its backbone as a template to explicitly construct a two-soliton configuration. Each of the two solitons describe well over 7.000 supersecondary structures of folded proteins in the Protein Data Bank with sub-angstrom accuracy suggesting that these solitons are common in nature.