Combinatorial Algorithms for Protein Folding in Lattice Models: A Survey of Mathematical Results

Abstract

We present a comprehensive survey of combinatorial algorithms and theorems about lattice protein folding models obtained in the almost 15 years since the publication in 1995 of the first protein folding approximation algorithm with mathematically guaranteed error bounds [60]. The results presented here are mainly about the HP-protein folding model introduced by Ken Dill in 1985 [37]. The main topics of this survey include: approximation algorithms for linear-chain and side-chain lattice models, as well as off-lattice models, NP-completeness theorems about a variety of protein folding models, contact map structure of self-avoiding walks and HP-folds, combinatorics and algorithmics for side-chain models, bi-sphere packing and the Kepler conjecture, and the protein sidechain self-assembly conjecture. As an appealing bridge between the hybrid of continuous mathematics and discrete mathematics, a cornerstone of the mathematical difficulty of the protein folding problem, we show how work on 2D self-avoiding walks contact-map decomposition [56] can build upon the exact RNA contacts counting formula by Mike Waterman and collaborators [96] leading to renewed hope for analytical closed-form approximations for statistical mechanics of protein folding in lattice models. We also include in this paper a few new results, research directions within reach of rigorous results, and a set of open problems that merit future exploration. * Dedicated to Michael Waterman on the occasion of his 67th birthday.

Boxelization

“…This complexity also appears in related fields such as protein folding [Berger and Leighton 1998;Istrail and Lam 2009]. Some very nice mathematical results for linkages, planes, and polyhedra are summarized by Demaine and O'Rouke [2007].…”

Exact methods for lattice protein models

“…They enable performance-guaranteed approximations close to the optimum for both backbone-only [36,37] and sidechain models [38,39]. For a detailed overview of approximating methods, refer to reviews by Hart and Newman [40] or Istrail and Lam [41].…”

A new off-lattice HP model with side-chains for protein folding problem

“…[1,2,3] There is less energy when more hydrophobic (H) amino-acids (the hydrophobic type of amino acid depends on the middle nucleotide of the codon [4]) are in contact in the core of the folded 3D structure and more polar (P) amino-acids are in contact with water. As we know the amino-acid sequence and the hydrophobicity of every amino-acid, we can predict the 3D structure -this method is called HP folding [5]. The closest to our model type of HP model is the offlattice one, following the approach know as HP folding.…”