Discretization orders for protein side chains

Proteins are important molecules that are widely studied in biology. Since their three-dimensional conformations can give clues about their function, an optimal methodology for the identification of such conformations has been researched for many years. Experiments of Nuclear Magnetic Resonance (NMR) are able to estimate distances between some pairs of atoms forming the protein, and the problem of identifying the possible conformations satisfying the available distance constraints is known in the scientific literature as the Molecular Distance Geometry Problem (MDGP). When some particular assumptions are satisfied, MDGP instances can be discretized, and solved by employing an ad-hoc algorithm, named the interval Branch & Prune. When dealing with molecules such as proteins, whose chemical structure is known, a priori information can be exploited for generating atomic orderings that allow for the discretization. In previous publications, we presented a handcrafted order for the protein backbones. In this work, we propose 20 new orders for the 20 side chains that can be present in proteins. Computational experiments on artificial and real instances from NMR show the usefulness of the proposed orders.     

The discretizable molecular distance geometry problem

Given a simple weighted undirected graph G=(V,E,d) with d:E???₊, the Molecular Distance Geometry Problem (MDGP) consists in finding an embedding x:V???³ such that ??x _u ?x _v ??=d _uv for each {u,v}??E. We show that under a few assumptions usually satisfied in proteins, the MDGP can be formulated as a search in a discrete space. We call this MDGP subclass the Discretizable MDGP (DMDGP). We show that the DMDGP is NP-hard and we propose a solution algorithm called Branch-and-Prune (BP). The BP algorithm performs remarkably well in practice in terms of speed and solution accuracy, and can be easily modified to find all incongruent solutions to a given DMDGP instance. We show computational results on several artificial and real-life instances.     

A hybrid genetic algorithmic approach to the maximally diverse grouping problem

The maximally diverse grouping problem (MDGP) is a NP-complete problem. For such NP-complete problems, heuristics play a major role in searching for solutions. Most of the heuristics for MDGP focus on the equal group-size situation. In this paper, we develop a genetic algorithm (GA)-based hybrid heuristic to solve this problem considering not only the equal group-size situation but also the different group-size situation. The performance of the algorithm is compared with the established Lotfi–Cerveny–Weitz algorithm and the non-hybrid GA. Computational experience indicates that the proposed GA-based hybrid algorithm is a good tool for solving MDGP. Moreover, it can be easily modified to solve other equivalent problems.     

Solving the clique partitioning problem as a maximally diverse grouping problem

In this paper we show that the clique partitioning problem can be reformulated in an equivalent form as the maximally diverse grouping problem (MDGP). We then modify a skewed general variable neighborhood search (SGVNS) heuristic that was first developed to solve the MDGP. Similarly as with the MDGP, significant improvements over the state of the art are obtained when SGVNS is tested on large scale instances. This further confirms the usefulness of a combined approach of diversification afforded with skewed VNS and intensification afforded with the local search in general VNS.     

The interval Branch-and-Prune algorithm for the discretizable molecular distance geometry problem with inexact distances

The Distance Geometry Problem in three dimensions consists in finding an embedding in ${\mathbb{R}^3}$ of a given nonnegatively weighted simple undirected graph such that edge weights are equal to the corresponding Euclidean distances in the embedding. This is a continuous search problem that can be discretized under some assumptions on the minimum degree of the vertices. In this paper we discuss the case where we consider the full-atom representation of the protein backbone and some of the edge weights are subject to uncertainty within a given nonnegative interval. We show that a discretization is still possible and propose the iBP algorithm to solve the problem. The approach is validated by some computational experiments on a set of artificially generated instances.     

Tabu search with strategic oscillation for the maximally diverse grouping problem

We propose new heuristic procedures for the maximally diverse grouping problem (MDGP). This NP-hard problem consists of forming maximally diverse groups—of equal or different size—from a given set of elements. The most general formulation, which we address, allows for the size of each group to fall within specified limits. The MDGP has applications in academics, such as creating diverse teams of students, or in training settings where it may be desired to create groups that are as diverse as possible. Search mechanisms, based on the tabu search methodology, are developed for the MDGP, including a strategic oscillation that enables search paths to cross a feasibility boundary. We evaluate construction and improvement mechanisms to configure a solution procedure that is then compared to state-of-the-art solvers for the MDGP. Extensive computational experiments with medium and large instances show the advantages of a solution method that includes strategic oscillation.     

Low-Dimensional Embedding with Extra Information

A frequently arising problem in computational geometry is when a physical structure, such as an ad-hoc wireless sensor network or a protein backbone, can measure local information about its geometry (e.g., distances, angles, and/or orientations), and the goal is to reconstruct the global geometry from this partial information. More precisely, we are given a graph, the approximate lengths of the edges, and possibly extra information, and our goal is to assign two-dimensional coordinates to the vertices such that the (multiplicative or additive) error on the resulting distances and other information is within a constant factor of the best possible. We obtain the first pseudo-quasipolynomial-time algorithm for this problem given a complete graph of Euclidean distances with additive error and no extra information. For general graphs, the analogous problem is NP-hard even with exact distances. Thus, for general graphs, we consider natural types of extra information that make the problem more tractable, including approximate angles between edges, the order type of vertices, a model of coordinate noise, or knowledge about the range of distance measurements. Our pseudo-quasipolynomial-time algorithm for no extra information can also be viewed as a polynomial-time algorithm given an "extremum oracle" as extra information. We give several approximation algorithms and contrasting hardness results for these scenarios.     

A linear assignment approach for the least-squares protein morphing problem

This work addresses the computation of free-energy differences between protein conformations by using morphing (i.e., transformation) of a source conformation into a target conformation. To enhance the morphing procedure, we employ permutations of atoms: we seek to find the permutation σ that minimizes the mean-square distance traveled by the atoms. Instead of performing this combinatorial search in the space of permutations, we show that the best permutation can be found by solving a linear assignment problem. We demonstrate that the use of such optimal permutations significantly improves the efficiency of the free-energy computation.     

Recent advances on the Discretizable Molecular Distance Geometry Problem

The Molecular Distance Geometry Problem (MDGP) consists in finding an embedding in R³${\mathbb{R}}^3$ of a nonnegatively weighted simple undirected graph with the property that the Euclidean distances between embedded adjacent vertices must be the same as the corresponding edge weights. The Discretizable Molecular Distance Geometry Problem (DMDGP) is a particular subset of the MDGP which can be solved using a discrete search occurring in continuous space; its main application is to find three-dimensional arrangements of proteins using Nuclear Magnetic Resonance (NMR) data. The model provided by the DMDGP, however, is too abstract to be directly applicable in proteomics. In the last five years our efforts have been directed towards adapting the DMDGP to be an ever closer model of the actual difficulties posed by the problem of determining protein structures from NMR data. This survey lists recent developments on DMDGP related research.     

Maximally diverse grouping: an iterated tabu search approach

The maximally diverse grouping problem (MDGP) consists of finding a partition of a set of elements into a given number of mutually disjoint groups, while respecting the requirements of group size constraints and diversity. In this paper, we propose an iterated tabu search (ITS) algorithm for solving this problem. We report computational results on three sets of benchmark MDGP instances of size up to 960 elements and provide comparisons of ITS to five state-of-the-art heuristic methods from the literature. The results demonstrate the superiority of the ITS algorithm over alternative approaches. The source code of the algorithm is available for free download via the internet.     

Double variable neighbourhood search with smoothing for the molecular distance geometry problem

We discuss the geometrical interpretation of a well-known smoothing operator applied to the Molecular Distance Geometry Problem (MDGP), and we then describe a heuristic approach based on Variable Neighbourhood Search on the smoothed and original problem. This algorithm often manages to find solutions having higher accuracy than other methods. This is important as small differences in the objective function value may point to completely different 3D molecular structures.     

Global Optimization on Funneling Landscapes

Molecular conformation problems arising in computational chemistry require the global minimization of a non-convex potential energy function representing the interactions of, for example, the component atoms in a molecular system. Typically the number of local minima on the potential energy surface grows exponentially with system size, and often becomes enormous even for relatively modestly sized systems. Thus the simple multistart strategy of randomly sampling local minima becomes impractical. However, for many molecular conformation potential energy surfaces the local minima can be organized by a simple adjacency relation into a single or at most a small number of funnels. A distinguished local minimum lies at the bottom of each funnel and a monotonically descending sequence of adjacent local minima connects every local minimum in the funnel with the funnel bottom. Thus the global minimum can be found among the comparatively small number of funnel bottoms, and a multistart strategy based on sampling funnel bottoms becomes viable. In this paper we present such an algorithm of the basin-hopping type and apply it to the Lennard–Jones cluster problem, an intensely studied molecular conformation problem which has become a benchmark for global optimization algorithms. Results of numerical experiments are presented which confirm both the multifunneling character of the Lennard–Jones potential surface as well as the efficiency of the algorithm. The algorithm has found all of the current putative global minima in the literature up to 110 atoms, as well as discovered a new global minimum for the 98-atom cluster of a novel geometrical class.     

Backbone colorings for graphs: Tree and path backbones

We introduce and study backbone colorings, a variation on classical vertex colorings: Given a graph G = (V,E) and a spanning subgraph H of G (the backbone of G), a backbone coloring for G and H is a proper vertex coloring V → {1,2,…} of G in which the colors assigned to adjacent vertices in H differ by at least two. We study the cases where the backbone is either a spanning tree or a spanning path. We show that for tree backbones of G the number of colors needed for a backbone coloring of G can roughly differ by a multiplicative factor of at most 2 from the chromatic number χ(G); for path backbones this factor is roughly . We show that the computational complexity of the problem “Given a graph G, a spanning tree T of G, and an integer ?, is there a backbone coloring for G and T with at most ? colors?” jumps from polynomial to NP‐complete between ? = 4 (easy for all spanning trees) and ? = 5 (difficult even for spanning paths). We finish the paper by discussing some open problems. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 137–152, 2007     

An improved hybrid global optimization method for protein tertiary structure prediction

First principles approaches to the protein structure prediction problem must search through an enormous conformational space to identify low-energy, near-native structures. In this paper, we describe the formulation of the tertiary structure prediction problem as a nonlinear constrained minimization problem, where the goal is to minimize the energy of a protein conformation subject to constraints on torsion angles and interatomic distances. The core of the proposed algorithm is a hybrid global optimization method that combines the benefits of the αBB deterministic global optimization approach with conformational space annealing. These global optimization techniques employ a local minimization strategy that combines torsion angle dynamics and rotamer optimization to identify and improve the selection of initial conformations and then applies a sequential quadratic programming approach to further minimize the energy of the protein conformations subject to constraints. The proposed algorithm demonstrates the ability to identify both lower energy protein structures, as well as larger ensembles of low-energy conformations.     

A linear-time algorithm for solving the molecular distance geometry problem with exact inter-atomic distances

We describe a linear-time algorithm for solving the molecular distance geometry problem with exact distances between all pairs of atoms. This problem needs to be solved in every iteration of general distance geometry algorithms for protein modeling such as the EMBED algorithm by Crippen and Havel (Distance Geometry and Molecular Conformation, Wiley, 1988). However, previous approaches to the problem rely on decomposing an distance matrix or minimizing an error function and require O(n²) to O(³) floating point operations. The linear-time algorithm will provide a much more efficient approach to the problem, especially in large-scale applications. It exploits the problem structure and hence is able to identify infeasible data more easily as well.     

A least-squares approach for discretizable distance geometry problems with inexact distances

A branch-and-prune (BP) algorithm is presented for the discretizable distance geometry problem in $$\mathbb {R}^K$$ with inexact distances. The algorithm consists in a sequential buildup procedure where possible positions for each new point to be localized are computed by using distances to at least K previously placed reference points and solving a system of quadratic equations. Such a system is solved in a least-squares sense, by finding the best positive semidefinite rank K approximation for an induced Gram matrix. When only K references are available, a second candidate position is obtained by reflecting the least-squares solution through the hyperplane defined by the reference points. This leads to a search tree which is explored by BP, where infeasible branches are pruned on the basis of Schoenberg’s theorem. In order to study the influence of the noise level, numerical results on some instances with distances perturbed by a small additive noise are presented.     

Extensive facility location problems on networks with equity measures

This paper deals with the problem of locating path-shaped facilities of unrestricted length on networks. We consider as objective functions measures conceptually related to the variability of the distribution of the distances from the demand points to a facility. We study the following problems: locating a path which minimizes the range, that is, the difference between the maximum and the minimum distance from the vertices of the network to a facility, and locating a path which minimizes a convex combination of the maximum and the minimum distance from the vertices of the network to a facility, also known in decision theory as the Hurwicz criterion. We show that these problems are NP-hard on general networks. For the discrete versions of these problems on trees, we provide a linear time algorithm for each objective function, and we show how our analysis can be extended also to the continuous case.     

Local search based heuristics for global optimization: Atomic clusters and beyond

Finding good solutions to large scale, hard, global optimization problems, is a demanding task with many relevant applications. It is well known that, in order to tackle a difficult problem, an algorithm has to incorporate all of the available information on the problem domain. However, as we will show in this paper, some general purpose methods and the ideas on which they are built can provide guidance towards the efficient solution of difficult instances. Most of this paper will be devoted to heuristic techniques applied to the problem of finding a minimum energy configuration of a cluster of atoms in R³${\mathbb{R}}^3$. This is a very relevant problem with a large set of applications which has triggered considerable research efforts in the last decade. We will show how some relatively simple ideas can be used to generate fairly efficient methods which have been employed to discover many new cluster structures. In this paper we will introduce some of the main algorithmic ideas which have proven to be particularly successful in the field of global optimization applied to atomic cluster conformation problems. We will mainly discuss Basin Hopping methods, as well as their population–based variant, and some specific technique based on “direct moves”. These methods, although designed for the specific problem, have a much wider applicability. In fact, one of the aims of this paper is also that of suggesting that similar ideas can be employed in order to design innovative methods even for totally different global optimization problems, like, e.g., circle packing and space trajectory planning.     

Routing-efficient CDS construction in Disk-Containment Graphs

In wireless networks, Connected Dominating Sets (CDSs) are widely used as virtual backbones for communications. On one hand, reducing the backbone size will reduce the maintenance overhead. So how to minimize the CDS size is a critical issue. On the other hand, when evaluating the performance of a wireless network, the hop distance between two communication nodes, which reflect the energy consumption and response delay, is of great importance. Hence how to minimize the routing cost is also a key problem for constructing the network virtual backbone. In this paper, we study the problem of constructing applicable CDS in wireless networks in terms of size and routing cost. We formulate a wireless network as a Disk-Containment Graph (DCG), which is a generalization of the Unit-Disk Graph (UDG), and we develop an efficient algorithm to construct CDS in such kind of graphs. The algorithm contains two parts and is flexible to balance the performance between the two metrics. We also analyze the algorithm theoretically. It is shown that our algorithm has provable performance in minimizing the CDS size and reducing the communication distance for routing.