We present performance-guaranteed approximation algorithms for the protein folding problem in the hydrophobic-hydrophilic model (Dill, 1985). Our algorithms are the first approximation algorithms in the literature with guaranteed performance for this model (Dill, 1994). The hydrophobic-hydrophilic 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 three-dimensional protein a guaranteed free energy no worse than three-eighths 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 well-studied model of protein folding. By achieving speed and near-optimality 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 diffusion-collision 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 near-optimal conformations in linear time for sequences of any size. 1.

Three open problems on folding/unfolding are discussed: (1) Can every convex polyhedron be cut along edges and unfolded at to a single nonoverlapping piece? (2) Given gluing instructions for a polygon, construct the unique 3D convex polyhedron to which itfolds. (3) Can every planar polygonal chain be straightened?

ured in standard deviation units) than has been previously demonstrated by more sophisticated methods. The arrangement of hydrophobic and polar residues alone as evaluated by our novel scoring scheme, is unexpectedly effective at recognizing native folds in general. It is surprising that a simple binary pattern of hydrophobic and polar residues apparently selects a given unique fold topology. 7 1995 Academic Press Limited *Corresponding author Keywords: protein folding; hydrophobic interaction; fold recognition; contact potential; threading Introduction Since the classic work by Kauzmann (1959), it has been hypothesized that hydrophobic interactions play a major role in organizing and stabilizing the architecture of proteins. This phenomenon, loosely defined, is related to the relative insolubility of non-polar substances in water (Tanford, 1980). It has long been observed that residues with hydrophobic side-chains tend to segregate into the interior of a globular protein, thus con

) William E. Hart Sorin Istrail y Abstract This paper considers the protein structure prediction problem for lattice and off-lattice 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 off-lattice 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 off-lattice model, the HP Tangent Spheres Side Chain model (HP-TSSC), 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...

The protein structure prediction problem is one of the most (if not the most) important problem in computational biology. This problem consists of nding the conformation of a protein with minimal energy. Because of the complexity of this problem, simplied models like Dill's HP-lattice model [15, 16] have become a major tool for investigating general properties of protein folding. Even for this simplied model, the structure prediction problem has been shown to be NP-complete [5, 7]. We describe a constraint formulation of the HP-model structure prediction problem, and present the basic constraints and search strategy. Of course, the simple formulation would not lead to an ecient algorithm. We therefore describe redundant constraints to prune the search tree. Furthermore, we need bounding function for the energy of an HP-protein. We introduce a new lower bound based on partial knowledge about the nal conformation (namely the distribution of H-monomers to layers). 1 Intr...

We introduce a new sampling algorithm, the equi-energy sampler, for efficient statistical sampling and estimation. Complementary to the widely used temperature-domain methods, the equi-energy sampler, utilizing the temperature–energy duality, targets the energy directly. The focus on the energy function not only facilitates efficient sampling, but also provides a powerful means for statistical estimation, for example, the calculation of the density of states and microcanonical averages in statistical mechanics. The equi-energy sampler is applied to a variety of problems, including exponential regression in statistics, motif sampling in computational biology and protein folding in biophysics.

We analyze the sequence to structure map for all uniquely folding sequences of short HP (hydrophobic-polar) model proteins on a square lattice (Lau and Dill 1989; Dill et al. 1995) to investigate aspects we consider relevant for evolution. Ranking structures by their frequencies we find few very frequent and many rare structures. The distribution can be empirically described by a generalized Zipf's law. All structures are relatively compact, yet the most compact ones are rare. Most sequences falling to the same structure belong to "neutral nets". These graphs in sequence space are connected by point mutations and centered around prototype sequences which tolerate the largest number (up to 55%) of neutral mutations. Profiles have been derived from these homologous sequences. Frequent structures conserve hydrophobic cores only while rare ones are sensitive to surface mutations as well. Shape space covering, i.e. the ability to transform any structure into most others with few point muta...

Abstract—We present an immune algorithm (IA) inspired by the clonal selection principle, which has been designed for the protein structure prediction problem (PSP). The proposed IA employs two special mutation operators, hypermutation and hypermacromutation to allow effective searching, and an aging mechanism which is a new immune inspired operator that is devised to enforce diversity in the population during evolution. When cast as an optimization problem, the PSP can be seen as discovering a protein conformation with minimal energy. The proposed IA was tested on well-known PSP lattice models, the HP model in two-dimensional and three-dimensional square lattices’, and the functional model protein, which is a more realistic biological model. Our experimental results demonstrate that the proposed IA is very competitive with the existing state-of-art algorithms for the PSP on lattice models. Index Terms—Aging operator, clonal selection algorithms, functional model proteins, hypermacromutation operator, hypermutation operator, immune algorithms (IAs), protein structure prediction problem, two-dimensional HP model, three-dimensional HP model. I.

Inverse protein folding concerns the identification of an amino acid sequence that folds to a given structure. Sequence design problems attempt to avoid the apparant difficulty of inverse protein folding by defining an energy that can be minimized to find protein-like sequences. We evaluate the practical relevance of two sequence design problems by analyzing their computational complexity. We show that the canonical method of sequence design is intractable, and describe approximation algorithms for this problem. We also describe an efficient algorithm that exactly solves the grand canonical method. Our analysis shows how sequence design problems can fail to reduce the difficulty of the inverse protein folding problem, and highlights the need to analyze these problems to evaluate their practical relevance. 1 Introduction The goal of the inverse protein folding problem (IPF) is to design a polymer sequence that folds to a given target conformation. Three criteria have been proposed for...

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.