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Fast protein folding in the hydrophobichydrophilic model within threeeights of optimal
 1ZDD 34 1045 4.0 2.703 3.12 1Q2N 0.66 0.61 1VII 36 14280 7.4 3.047 12.59 1UNC 0.74 0.70 1EOM 37 36000 3.4 3.093 17.41 1I5H 0.47 0.49 1EDO 46 36000 7.2 3.656 11.54 1NBL 0.55 0.56 2IGD 61 174960 11.5 7.469 8.01 1MVK 0.79 0.74 1YPA 64 420840 9.4 6.687 0.34 2
, 1996
"... We present performanceguaranteed approximation algorithms for the protein folding problem in the hydrophobichydrophilic model (Dill, 1985). Our algorithms are the first approximation algorithms in the literature with guaranteed performance for this model (Dill, 1994). The hydrophobichydrophilic m ..."
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Cited by 68 (4 self)
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We present performanceguaranteed approximation algorithms for the protein folding problem in the hydrophobichydrophilic model (Dill, 1985). Our algorithms are the first approximation algorithms in the literature with guaranteed performance for this model (Dill, 1994). The hydrophobichydrophilic model abstracts the dominant force of protein folding: the hydrophobic interaction. The protein is modeled as a chain of amino acids of length n that are of two types; H (hydrophobic, i.e., nonpolar) and P (hydrophilic, i.e., polar). Although this model is a simplification of more complex protein folding models, the protein folding structure prediction problem is notoriously difficult for this model. Our algorithms have conformation that has linear (3n) or quadratic time and achieve a threedimensional protein a guaranteed free energy no worse than threeeighths of optimal. This result answers the open problem of Ngo et al. (1994) about the possible existence of an efficient approximation algorithm with guaranteed performance for protein structure prediction in any wellstudied model of protein folding. By achieving speed and nearoptimality simultaneously, our algorithms rigorously capture salient features of the recently proposed framework of protein folding by Sali et al. (1994). Equally important, the final conformations of our algorithms have significant secondary structure (antiparallel sheets, ^sheets, compact hydrophobic core). Furthermore, hypothetical folding pathways can be described for our algorithms that fit within the framework of diffusioncollision protein folding proposed by Karplus and Weaver (1979). Computational limitations of algorithms that compute the optimal conformation have restricted their applicability to short sequences (length < 90). Because our algorithms trade computational accuracy for speed, they can construct nearoptimal conformations in linear time for sequences of any size. 1.
Lattice and OffLattice Side Chain Models of Protein Folding: Linear Time Structure Prediction Better Than 86% of Optimal (Extended Abstract)
 J. Comput. Biol
, 1997
"... ) William E. Hart Sorin Istrail y Abstract This paper considers the protein structure prediction problem for lattice and offlattice protein folding models that explicitly represent side chains. Lattice models of proteins have proven extremely useful tools for reasoning about protein folding i ..."
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Cited by 26 (2 self)
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) William E. Hart Sorin Istrail y Abstract This paper considers the protein structure prediction problem for lattice and offlattice protein folding models that explicitly represent side chains. Lattice models of proteins have proven extremely useful tools for reasoning about protein folding in unrestricted continuous space through analogy. This paper provides the first illustration of how rigorous algorithmic analyses of lattice models can lead to rigorous algorithmic analyses of offlattice models. We consider two side chain models: a lattice model that generalizes the HP model (Dill 85) to explicitly represent side chains on the cubic lattice, and a new offlattice model, the HP Tangent Spheres Side Chain model (HPTSSC), that generalizes this model further by representing the backbone and side chains of proteins with tangent spheres. We describe algorithms with mathematically guaranteed error bounds for both of these models. In particular, we describe a linear time performanc...
S.: Robust proofs of NPhardness for protein folding: General lattices and energy potentials
 Journal of Computational Biology
, 1997
"... This paper addresses the robustness of intractability arguments for simplified models of protein folding that use lattices to discretize the space of conformations that a protein can assume. We present two generalized NPhardness results. The first concerns the intractability of protein folding inde ..."
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Cited by 21 (4 self)
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This paper addresses the robustness of intractability arguments for simplified models of protein folding that use lattices to discretize the space of conformations that a protein can assume. We present two generalized NPhardness results. The first concerns the intractability of protein folding independent of the lattice used to define the discrete proteinfolding model. We consider a previously studied model and prove that for any reasonable lattice the proteinstructure prediction problem is NPhard. The second hardness result concerns the intractability of protein folding for a class of energy formulas that contains a broad range of mean force potentials whose form is similar to commonly used pair potentials (e.g., the LennardJones potential). We prove that proteinstructure prediction is NPhard for any energy formula in this class. These are the first robust intractability results that identify sources of computational complexity of proteinstructure prediction particular problem formulations. Key words: protein folding; intractability, robustness, lattice models. that transcend
On the Complexity of String Folding
 Discrete Applied Mathematics
, 1996
"... A fold of a finite string S over a given alphabet is an embedding of S in some fixed infinite grid, such as the square or cubic mesh. The score of a fold is the number of pairs of matching string symbols which are embedded at adjacent grid vertices. Folds of strings in two and threedimensional m ..."
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Cited by 17 (0 self)
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A fold of a finite string S over a given alphabet is an embedding of S in some fixed infinite grid, such as the square or cubic mesh. The score of a fold is the number of pairs of matching string symbols which are embedded at adjacent grid vertices. Folds of strings in two and threedimensional meshes are considered, and the corresponding problems of optimizing the score or achieving a given target score are shown to be NPhard. 1 Introduction The motivation for the stringfolding problems considered here lies in computational biology. Prediction of the threedimensional structure of a protein from its known linear sequence of amino acids is an important practical open problem, which seems to be extremely challenging. The way in which a protein folds determines many of its biological and chemical properties. A natural approach is to look for a spatial configuration achieving a minimum free energy level. The energy is determined by such factors as the number of chemical bonds esta...
Opportunities for Combinatorial Optimization In Computational Biology
, 2003
"... This is a survey designed for mathematical programming people who do not know molecular biology and want to learn the kinds of combinatorial optimization problems that arise. After a brief introduction to the biology, we present optimization models pertaining to sequencing, evolutionary explanations ..."
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Cited by 14 (0 self)
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This is a survey designed for mathematical programming people who do not know molecular biology and want to learn the kinds of combinatorial optimization problems that arise. After a brief introduction to the biology, we present optimization models pertaining to sequencing, evolutionary explanations, structure prediction and recognition. Additional biology is given in the context of the problems, including some motivation for disease diagnosis and drug discovery. Open problems are cited with an extensive bibliography, and we o er a guide to getting started in this exciting frontier.
Protein Folding, Spin Glass and Computational Complexity
 In Proceedings of the 3rd DIMACS Workshop on DNA Based Computers, held at the University of Pennsylvania, June 23 – 25
, 1997
"... . A reduction from "Ground State of Spin Glass" in statistical mechanics to a minimumenergy model of protein folding is made, which shows that the latter is NPcomplete (high complexity) . The reduction approximates true folding of a protein. The method also enables to show that even if the backbone ..."
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Cited by 9 (0 self)
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. A reduction from "Ground State of Spin Glass" in statistical mechanics to a minimumenergy model of protein folding is made, which shows that the latter is NPcomplete (high complexity) . The reduction approximates true folding of a protein. The method also enables to show that even if the backbone of the protein is fixed, the folding of the sidechains is NPcomplete. In a separate second part, the possibility of synthesizing proteins to solve arbitrary instances of the spin glass problem is speculated upon. 1. Introduction The motivation for this work is the speculation of exploiting nature's capability of protein folding to solve computationally intractable problems. One way of investigating this idea is to encode known NPcomplete problems in terms of protein folding. The main content of this paper is to do this for the spin glass problem. We construct a protein that achieves the encoding, i.e., the folded protein provides a solution to spin glass. More precisely, albeit incident...
Spatial Codes and the Hardness of String Folding Problems
 Proceedings of the ACMSIAM Symposium on Discrete Algorithms
, 1998
"... We present a general technique for proving NPhardness (under randomized polynomial time reductions) of string folding problems over a finite alphabet. All previous such intractability results have required an unbounded alphabet size. These problems correspond to the protein folding problem in varia ..."
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Cited by 7 (0 self)
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We present a general technique for proving NPhardness (under randomized polynomial time reductions) of string folding problems over a finite alphabet. All previous such intractability results have required an unbounded alphabet size. These problems correspond to the protein folding problem in variants of the hydrophobichydrophilic (or HP) model with a fixed number of monomer types. Our proof also establishes the MAX SNPhardness of these problems (again under randomized polynomial time reductions). This means that obtaining even an approximate solution to the protein folding problem, to within some fixed constant factor, is NPhard. Our technique involves replacing the symbols of an unbounded alphabet by codewords over a fixed alphabet, and has two novel aspects. The first is the essential use of the approximation hardness of the source problem in the reduction, even for the proof of NPhardness. The second is the concept of spatial codes, a variant of classical errorcorrecting cod...
Combinatorial Algorithms for Protein Folding in Lattice Models: A Survey of Mathematical Results
, 2009
"... “... a very nice step forward in the computerology of proteins. ” Ken Dill 1995[1] 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 ..."
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Cited by 3 (0 self)
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“... a very nice step forward in the computerology of proteins. ” Ken Dill 1995[1] 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 HPprotein folding model introduced by Ken Dill in 1985 [37]. The main topics of this survey include: approximation algorithms for linearchain and sidechain lattice models, as well as offlattice models, NPcompleteness theorems about a variety of protein folding models, contact map structure of selfavoiding walks and HPfolds, combinatorics and algorithmics of sidechain models, bisphere packing and the Kepler conjecture, and the protein sidechain selfassembly 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 selfavoiding walks contactmap decomposition [56] can build upon the exact RNA contacts counting
Prediction of Protein Structures Using Simple Exact Models
, 1996
"... this paper so as not to obfuscate the meaning with minor details. ..."
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Cited by 1 (0 self)
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this paper so as not to obfuscate the meaning with minor details.
Opportunities for Combinatorial Optimization
"... This is a survey designed for mathematical programming people who do not know molecular biology and want to learn the kinds of combinatorial optimization problems that arise. After a brief introduction to the biology, we present optimization models pertaining to sequencing, evolutionary explanations ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
This is a survey designed for mathematical programming people who do not know molecular biology and want to learn the kinds of combinatorial optimization problems that arise. After a brief introduction to the biology, we present optimization models pertaining to sequencing, evolutionary explanations, structure prediction and recognition. Additional biology is given in the context of the problems, including some motivation for disease diagnosis and drug discovery. Open problems are cited with an extensive bibliography, and we oer a guide to getting started in this exciting frontier.