De novo protein structure prediction from amino acid sequence is one of the most challenging problems in computational biology. As one of the extensively explored mathematical models for protein folding, Hydrophobic-Polar (HP) model enables thorough investigation of protein structure formation and evolution. Although HP model discretizes the conformational space and simplifies the folding energy function, it has been proven to be an NP-complete problem. In this paper, we propose a novel protein folding framework FoldingZero, self-folding a de novo protein 2D HP structure from scratch based on deep reinforcement learning. FoldingZero features the coupled approach of a two-head (policy and value heads) deep convolutional neural network (HPNet) and a regularized Upper Confidence Bounds for Trees (R-UCT). It is trained solely by a reinforcement learning algorithm, which improves HPNet and R-UCT iteratively through iterative policy optimization. Without any supervision and domain knowledge, FoldingZero not only achieves comparable results, but also learns the latent folding knowledge to stabilize the structure. Without exponential computation, FoldingZero shows promising potential to be adopted for real-world protein properties prediction.
Approximation algorithms offer rigorous mathematical tools and fold a protein structure within polynomial time. However, it may lead to a weak approximation ratio, resulting in a structure far from the optimal solution. Hart and Istrail proposed an approximation algorithm with ratio 3 8 of the optimal score for the 3D cubic lattice structure @cite_16 . An improved approximation algorithm with 2 5 performance guarantees was further developed by the same authors @cite_1 . For the 2D square lattice, an approximation algorithm @cite_7 can achieve the approximation ratio of 1 3.
@cite_16 @cite_1 @cite_7
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We consider the problem of protein folding in the HP model on the two-dimensional square lattice. This problem is combinatorially equivalent to folding a string of 0's and 1's so that the string forms a self-avoiding walk on the lattice and the number of adjacent pairs of 1's is maximized. We present a linear-time 1 3-approximation algorithm for this problem, improving on the previous best approximation factor of 1 4. The approximation guarantee of this algorithm is based on an upper bound presented by Hart and Istrail [6] and used in all previous papers that address this problem. We show that this upper bound cannot be used to obtain an approximation factor better than 1 2. Abstract In this paper we study the Steiner tree problem in graphs for the case when vertices as well as edges have weights associated with them. A greedy approximation algorithm based on “spider decompositions” was developed by Klein and Ravi for this problem. This algorithm provides a worst case approximation ratio of 2 ln k , where k is the number of terminals. However, the best known lower bound on the approximation ratio is (1− o (1)) ln k , assuming that NP ⊈ DTIME [ n O (log log n ) ], by a reduction from set cover. We show that for the unweighted case we can obtain an approximation factor of ln k . For the weighted case we develop a new decomposition theorem and generalize the notion of “spiders” to “branch-spiders” that are used to design a new algorithm with a worst case approximation factor of 1.5 ln k . We then generalize the method to yield an approximation factor of (1.35+ e ) ln k , for any constant e >0. These algorithms, although polynomial, are not very practical due to their high running time, since we need to repeatedly find many minimum weight matchings in each iteration. We also develop a simple greedy algorithm that is practical and has a worst case approximation factor of 1.6103 ln k . The techniques developed for this algorithm imply a method of approximating node weighted network design problems defined by 0–1 proper functions as well. These new ideas also lead to improved approximation guarantees for the problem of finding a minimum node weighted connected dominating set. The previous best approximation guarantee for this problem was 3 ln n by Guha and Khuller. By a direct application of the methods developed in this paper we are able to develop an algorithm with an approximation factor of (1.35+ e ) ln n for any fixed e >0. ABSTRACT This paper considers the protein energy minimization problem for lattice and off-lattice protein folding models that explicitly represent side chains. Lattice models of proteins have proven 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, 1985) 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 performance guaranteed approximation algorithm for the HP side chain model that... We present an improved semidefinite programming based approximation algorithm for the MAX CUT problem in graphs of maximum degree at most 3. The approximation ratio of the new algorithm is at least 0.9326. This improves, and also somewhat simplifies, a result of Feige, Karpinski and Langberg. We also observe that results of Hopkins and Staton and of Bondy and Locke yield a simple combinatorial 4 5-approximation algorithm for the problem. Finally, we present a combinatorial 22 27-approximation algorithm for the MAX CUT problem for regular cubic graphs. Protein structure prediction remains one of the significant challenges in computational biology. We have previously shown that our kinetostatic compliance method can overcome some of the key difficulties faced by other de novo structural prediction methods, such as the very small time steps required by the molecular dynamics approaches, or the very large number of samples required by the sampling based techniques. In this paper we extend the previous free energy formulation by adding the solvent effects, which contribute predominantly to the folding phenomena. We show that the addition of the solvation effects, which complement the existing Coulombic and van der Waals interactions, lead to a physically effective energy function. Furthermore, we achieve significant computational speed-up by employing efficient algorithms and data structures that effectively reduce the time complexity from O(n2) to O(n), n being the number of atoms. Our simulations are consistent with the general behavior observed in protein folding, and show that the hydrophobic atoms tend to pack inside the core of the molecule in an aqueous solvent, while a vacuum environment produces no such effect.Copyright © 2013 by ASME In-vitro methods for protein structure determination are time-consuming, cost-intensive, and failure-prone. Because of these expenses, alternative computer-based predictive methods have emerged. Predicting a protein’s 3-D structure from only its amino acid sequence—also known as ab initio protein structure prediction (PSP)—is computationally demanding because the search space is astronomically large and energy models are extremely complex. Some successes have been achieved in predictive methods but these are limited to small sized proteins (around 100 amino acids); thus, developing efficient algorithms, reducing the search space, and designing effective search guidance heuristics are necessary to study large sized proteins. An on-lattice model can be a better ground for rapidly developing and measuring the performance of a new algorithm, and hence we consider this model for larger proteins (>150 amino acids) to enhance the genetic algorithms (GAs) framework. In this paper, we formulate PSP as a combinatorial optimization problem that uses 3-D face-centered-cubic lattice coordinates to reduce the search space and hydrophobic-polar energy model to guide the search. The whole optimization process is controlled by an enhanced GA framework with four enhanced features: 1) an exhaustive generation approach to diversify the search; 2) a novel hydrophobic core-directed macro-mutation operator to intensify the search; 3) a per-generation duplication elimination strategy to prevent early convergence; and 4) a random-walk technique to recover from stagnation. On a set of standard benchmark proteins, our algorithm significantly outperforms state-of-the-art algorithms. We also experimentally show that our algorithm is robust enough to produce very similar results regardless of different parameter settings. An important question in theoretical computer science is to determine the best possible running time for solving a problem at hand. For geometric optimization problems, we often understand their complexity on a rough scale, but not very well on a finer scale. One such example is the two-dimensional knapsack problem for squares. There is a polynomial time (1 + ϵ)-approximation algorithm for it (i.e., a PTAS) but the running time of this algorithm is triple exponential in 1 ϵ, i.e., Ω(n221 ϵ). A double or triple exponential dependence on 1 ϵ is inherent in how this and several other algorithms for other geometric problems work. In this paper, we present an EPTAS for knapsack for squares, i.e., a (1+ϵ)-approximation algorithm with a running time of Oϵ(1)·nO(1). In particular, the exponent of n in the running time does not depend on ϵ at all! Since there can be no FPTAS for the problem (unless P = NP) this is the best kind of approximation scheme we can hope for. To achieve this improvement, we introduce two new key ideas: We present a fast method to guess the Ω(221 ϵ) relatively large squares of a suitable near-optimal packing instead of using brute-force enumeration. Secondly, we introduce an indirect guessing framework to define sizes of cells for the remaining squares. In the previous PTAS each of these steps needs a running time of Ω(n221 ϵ) and we improve both to Oϵ(1) · nO(1). We complete our result by giving an algorithm for two-dimensional knapsack for rectangles under (1 + ϵ)-resource augmentation. In this setting, we also improve the best known running time of Ω(n1 ϵ1 ϵ) to Oϵ(1) · nO(1) and compute even a solution with optimal profit, in contrast to the best previously known polynomial time algorithm for this setting that computes only an approximation. We believe that our new techniques have the potential to be useful for other settings as well. The search for efficient and predictive methods to describe the protein folding process at the all-atom level remains an important grand-computational challenge. The development of multi-teraflop architectures, such as the IBM BlueGene used in this study, has been motivated in part by the large computational requirements of such studies. Here we report the predictive all-atom folding of the forty-amino acid HIV accessory protein using an evolutionary stochastic optimization technique. We implemented the optimization method as a master-client model on an IBM BlueGene, where the algorithm scales near perfectly from 64 to 4096 processors in virtual processor mode. Starting from a completely extended conformation, we optimize a population of 64 conformations of the protein in our all-atom free-energy model PFF01. Using 2048 processors the algorithm predictively folds the protein to a near-native conformation with an RMS deviation of 3.43 A in <24 h. © 2007 Wiley Periodicals, Inc. J Comput Chem, 2007 Prediction of the three-dimensional structure of a protein from its amino acid sequence can be considered as a global optimization problem. In this paper, the Chaotic Artificial Bee Colony (CABC) algorithm was introduced and applied to 3D protein structure prediction. Based on the 3D off-lattice AB model, the CABC algorithm combines global search and local search of the Artificial Bee Colony (ABC) algorithm with the chaotic search algorithm to avoid the problem of premature convergence and easily trapping the local optimum solution. The experiments carried out with the popular Fibonacci sequences demonstrate that the proposed algorithm provides an effective and high-performance method for protein structure prediction. Lattice protein folding models are a cornerstone of computational biophysics. Although these models are a coarse grained representation, they provide useful insight into the energy landscape of natural proteins. Finding low-energy threedimensional structures is an intractable problem even in the simplest model, the Hydrophobic-Polar (HP) model. Description of protein-like properties are more accurately described by generalized models, such as the one proposed by Miyazawa and Jernigan (MJ), which explicitly take into account the unique interactions among all 20 amino acids. There is theoretical and experimental evidence of the advantage of solving classical optimization problems using quantum annealing over its classical analogue (simulated annealing). In this report, we present a benchmark implementation of quantum annealing for lattice protein folding problems (six different experiments up to 81 superconducting quantum bits). This first implementation of a biophysical problem paves the way towards studying optimization problems in biophysics and statistical mechanics using quantum devices. We prove a structure theorem for the feasible solutions of the Arora-Rao-Vazirani SDP relaxation on low threshold rank graphs and on small-set expanders. We show that if G is a graph of bounded threshold rank or a small-set expander, then an optimal solution of the Arora-Rao-Vazirani relaxation (or of any stronger version of it) can be almost entirely covered by a small number of balls of bounded radius. Then, we show that, if k is the number of balls, a solution of this form can be rounded with an approximation factor of O(sqrt log k ) in the case of the Arora-Rao-Vazirani relaxation, and with a constant-factor approximation in the case of the k-th round of the Sherali-Adams hierarchy starting at the Arora-Rao-Vazirani relaxation. The structure theorem and the rounding scheme combine to prove the following result, where G=(V,E) is a graph of expansion (G), is the k-th smallest eigenvalue of the normalized Laplacian of G, and (G) = disjoint S_1,...,S_k 1 > log^ 2.5 k phi(G) or k (G) >> log k sqrt log n loglog n (G), then the Arora-Rao-Vazirani relaxation can be rounded in polynomial time with an approximation ratio O(sqrt log k ). Stronger approximation guarantees are achievable in time exponential in k via relaxations in the Lasserre hierarchy. Guruswami and Sinop [GS13] and Arora, Ge and Sinop [AGS13] prove that 1+eps approximation is achievable in time 2^ O(k) poly(n) if either > (G) poly(eps), or if SSE_ n k > sqrt log k log n (G) poly(eps), where SSE_s is the minimal expansion of sets of size at most s. Background The protein folding problem is a fundamental problems in computational molecular biology and biochemical physics. Various optimisation methods have been applied to formulations of the ab-initio folding problem that are based on reduced models of protein structure, including Monte Carlo methods, Evolutionary Algorithms, Tabu Search and hybrid approaches. In our work, we have introduced an ant colony optimisation (ACO) algorithm to address the non-deterministic polynomial-time hard (NP-hard) combinatorial problem of predicting a protein's conformation from its amino acid sequence under a widely studied, conceptually simple model – the 2-dimensional (2D) and 3-dimensional (3D) hydrophobic-polar (HP) model. The best approximation algorithm for Max Cut in graphs of maximum degree 3 uses semidefinite programming, has approximation ratio 0.9326, and its running time is @Q(n^3^.^5logn); but the best combinatorial algorithms have approximation ratio 4 5 only, achieved in O(n^2) time [J.A. Bondy, S.C. Locke, J. Graph Theory 10 (1986) 477-504; E. Halperin, et al, J. Algorithms 53 (2004) 169-185]. Here we present an improved combinatorial approximation, which is a 5 6-approximation algorithm that runs in O(n^2) time, perhaps improvable even to O(n). Our main tool is a new type of vertex decomposition for graphs of maximum degree 3. Abstract Protein structure prediction is a fundamental issue in the field of computational molecular biology. In this paper, the AB off-lattice model is adopted to transform the original protein structure prediction scheme into a numerical optimization problem. We present a balance-evolution artificial bee colony (BE-ABC) algorithm to address the problem, with the aim of finding the structure for a given protein sequence with the minimal free-energy value. This is achieved through the use of convergence information during the optimization process to adaptively manipulate the search intensity. Besides that, an overall degradation procedure is introduced as part of the BE-ABC algorithm to prevent premature convergence. Comprehensive simulation experiments based on the well-known artificial Fibonacci sequence set and several real sequences from the database of Protein Data Bank have been carried out to compare the performance of BE-ABC against other algorithms. Our numerical results show that the BE-ABC algorithm is able to outperform many state-of-the-art approaches and can be effectively employed for protein structure optimization. A realistic computer simulation of protein folding requires a comprehensive account of interaction energetics, placing a substantial demand on processing power. This paper presents an improved computational framework for protein folding software package PROTOFOLD, that enables efficient computation of solvation free energy effects in addition to Coulombic and van der Waals interactions. Efficient data structures have been utilized to speed-up the sequential running times from O(n2) to O(n), n being the number of atoms. It turns out, however, that an accurate evaluation of molecular surface areas characterizing the solvation effects imposes a computational bottleneck to the entire simulation. Massive computational power offered by Graphics Processing Units (GPU) was exploited to develop a simple and efficient Single-Instruction Multiple-Thread (SIMT) algorithm for the latter step. The running times were monitored for different steps of the folding simulation for different molecular sizes. Significant performance improvements were observed, yielding promising results where numerous iterative runs are needed.© 2013 ASME We present the best known algorithms for approximating the minimum-size undirected @math -edge connected spanning subgraph. For simple graphs our approximation ratio is @math . The more precise version of this bound requires @math , and for all such @math it improves the long-standing performance ratio of Cheriyan and Thurimella [SIAM J. Comput., 30 (2000), pp. 528-560], @math . The improvement comes in two steps. First we show that for simple @math -edge connected graphs, any laminar family of degree @math sets is smaller than the general bound ( @math versus @math ). This immediately implies that iterated rounding improves the performance ratio of Cheriyan and Thurimella. The second step carefully chooses good edges for rounding. For multigraphs our approximation ratio is @math . This improves the previous ratio @math [H. N. Gabow, M. X. Goemans, E. Tardos, and D. P. Williamson, Networks, 53 (2009), pp. 345-357]. It is of interest since it is known that for some constant @math , an approximation ratio @math implies @math . Our approximation ratio extends to the minimum-size Steiner network problem, where @math denotes the average vertex demand. The algorithm exploits rounding properties of the first two linear programs in iterated rounding. Freedman, Kitaev, and Wang (2002), and later Aharonov, Jones, and Landau (2009), established a quantum algorithm to "additively" approximate the Jones polynomial V(L;t) at any principal root of unity t. The strength of this additive approximation depends exponentially on the bridge number of the link presentation. Freedman, Larsen, and Wang (2002) established that the approximation is universal for quantum computation at a non- lattice, principal root of unity. We show that any value-distinguishing approximation of the Jones polynomial at these non-lattice roots of unity is #P-hard. Given the power to decide whetherjV(L;t)j b for fixed constants 0 1, T(G; x; y) is #P-hard to approximate within a factor of c even for planar graphs G. Along the way, we clarify and generalize both Aaronson's theorem and the Solovay- Kitaev theorem. The effect of training a neural network secondary structure prediction algorithm with different types of multiple sequence alignment profiles derived from the same sequences, is shown to provide a range of accuracy from 70.5 to 76.4 . The best accuracy of 76.4 (standard deviation 8.4 ), is 3.1 (Q3) and 4.4 (SOV2) better than the PHD algorithm run on the same set of 406 sequence non-redundant proteins that were not used to train either method. Residues predicted by the new method with a confidence value of 5 or greater, have an average Q3 accuracy of 84 , and cover 68 of the residues. Relative solvent accessibility based on a two state model, for 25, 5, and 0 accessibility are predicted at 76.2, 79.8, and 86.6 accuracy respectively. The source of the improvements obtained from training with different representations of the same alignment data are described in detail. The new Jnet prediction method resulting from this study is available in the Jpred secondary structure prediction server, and as a stand-alone computer program from: http: barton.ebi.ac.uk . Proteins 2000;40:502–511. © 2000 Wiley-Liss, Inc. We present an approximation algorithm for the problem of finding a minimum-cost k-vertex connected spanning subgraph, assuming that the number of vertices is at least 6k 2 . The approximation guarantee is six times the kth harmonic number (which is O(log k)), and this is also an upper bound on the integrality ratio for a standard linear programming relaxation. We present a new efficient algorithm for the search version of the approximate Closest Vector Problem with Preprocessing (CVPP). Our algorithm achieves an approximation factor of O(n sqrt log n ), improving on the previous best of O(n^ 1.5 ) due to Lag arias, Lenstra, and Schnorr hkzbabai . We also show, somewhat surprisingly, that only O(n) vectors of preprocessing advice are sufficient to solve the problem (with the slightly worse approximation factor of O(n)). We remark that this still leaves a large gap with respect to the decisional version of CVPP, where the best known approximation factor is O(sqrt n log n ) due to Aharonov and Regev AharonovR04 . To achieve these results, we show a reduction to the same problem restricted to target points that are close to the lattice and a more efficient reduction to a harder problem, Bounded Distance Decoding with preprocessing (BDDP). Combining either reduction with the previous best-known algorithm for BDDP by Liu, Lyubashevsky, and Micciancio LiuLM06 gives our main result. In the setting of CVP without preprocessing, we also give a reduction from (1+eps)gamma approximate CVP to gamma approximate CVP where the target is at distance at most 1+1 eps times the minimum distance (the length of the shortest non-zero vector) which relies on the lattice sparsification techniques of Dadush and Kun DadushK13 . As our final and most technical contribution, we present a substantially more efficient variant of the LLM algorithm (both in terms of run-time and amount of preprocessing advice), and via an improved analysis, show that it can decode up to a distance proportional to the reciprocal of the smoothing parameter of the dual lattice MR04 . We show that this is never smaller than the LLM decoding radius, and that it can be up to an wide tilde Omega (sqrt n ) factor larger.