Discretizable Molecular Distance Geometry Problem Research Articles

A Branch-and-Bound Algorithm for the Molecular Ordered Covering Problem.

The Discretizable Molecular Distance Geometry Problem (DMDGP) plays a key role in the construction of three-dimensional molecular structures from interatomic distances acquired through nuclear magnetic resonance (NMR) spectroscopy, with the primary objective of validating a sequence of distance constraints related to NMR data. This article addresses the escalating complexity of the DMDGP encountered with larger and more flexible molecules by introducing a novel strategy via the Molecular Ordered Covering Problem, which optimizes the ordering of distance constraints to improve computational efficiency in DMDGP resolution. This approach utilizes a specialized Branch-and-Bound (BB) algorithm, tested on both synthetic and actual protein structures from the protein data bank. Our analysis demonstrates the efficacy of the previously proposed greedy heuristic in managing complex molecular scenarios, highlighting the BB algorithm's utility as a validation mechanism. This research contributes to ongoing efforts in molecular structure analysis, with possible implications for areas such as protein folding, drug design, and molecular modeling.

An autonomic parallel strategy for exhaustive search tree algorithms on shared or heterogeneous systems

SummaryBacktracking branch‐and‐prune (BP) algorithms and their variants are exhaustive search tree techniques widely employed to solve optimization problems in many scientific areas. However, they characteristically often demand significant amounts of computing power for problem sizes representative of real‐world scenarios. Given that their search domains can often be partitioned, these algorithms are frequently designed to execute in parallel by harnessing distributed computing systems. However, to achieve efficient parallel execution times, an effective strategy is required to balance the nonuniform partition workloads across the available resources. Furthermore, with the increasing integration of servers with heterogeneous resources and the adoption of resource sharing, balancing workloads is becoming complex. This paper proposes a strategy to execute parallel BP algorithms more efficiently on even shared or heterogeneous distributed systems. The approach integrates a self‐adjusting dynamic partitioning method in the BP algorithm with a dynamic scheduler, provided by an application middleware, which manages the parallel execution while addressing any issues of imbalance. Empirical results indicate better scalability with efficiencies above 90% for instances of an application case study for the discretizable molecular distance geometry problem (DMDGP). Improvements of up to 38% were obtained in execution speed‐ups compared to a more traditional parallel BP implementation for DMDGP.

A quantum approach to the discretizable molecular distance geometry problem

The Discretizable Molecular Distance Geometry Problem (DMDGP) aims to determine the three-dimensional protein structure using distance information from nuclear magnetic resonance experiments. The DMDGP has a finite number of candidate solutions and can be solved by combinatorial methods. We describe a quantum approach to the DMDGP by using Grover's algorithm with an appropriate oracle function, which is more efficient than classical methods that use brute force. We show computational results by implementing our scheme on IBM quantum computers with a small number of noisy qubits.

Constraint programming approaches for the discretizable molecular distance geometry problem

AbstractThe Distance Geometry Problem (DGP) seeks to find positions for a set of points in geometric space when some distances between pairs of these points are known. The so‐called discretization assumptions allow us to discretize the search space of DGP instances. In this paper, we focus on a key subclass of DGP, namely the Discretizable Molecular DGP, and study its associated graph vertex ordering problem, the Contiguous Trilateration Ordering Problem (CTOP), which helps solve DGP. We propose the first constraint programming formulations for CTOP, as well as a set of checks for proving infeasibility, domain reduction techniques, symmetry breaking constraints, and valid inequalities. Our computational results on random and pseudo‐protein instances indicate that our formulations outperform the state‐of‐the‐art integer programming formulations.

A New Algorithm for the $$^K$$DMDGP Subclass of Distance Geometry Problems with Exact Distances

The fundamental inverse problem in distance geometry is the one of finding positions from inter-point distances. The Discretizable Molecular Distance Geometry Problem (DMDGP) is a subclass of the Distance Geometry Problem (DGP) whose search space can be discretized and represented by a binary tree, which can be explored by a Branch-and-Prune (BP) algorithm. It turns out that this combinatorial search space possesses many interesting symmetry properties that were studied in the last decade. In this paper, we present a new algorithm for this subclass of the DGP, which exploits DMDGP symmetries more effectively than its predecessors. Computational results show that the speedup, with respect to the classic BP algorithm, is considerable for sparse DMDGP instances related to protein conformation.

On the optimality of finding DMDGP symmetries

The Discretizable Molecular Distance Geometry Problem (DMDGP) is a subclass of the Distance Geometry Problem, which aims to embed a weighted simple undirected graph in a Euclidean space, such that the distances between the points correspond to the values given by the weighted edges in the graph. The search space of the DMDGP is combinatorial, based on a total vertex order that implies symmetry properties related to partial reflections around planes defined by the Cartesian coordinates of three immediate and consecutive vertices that precede the so-called symmetry vertices. Since these symmetries allow us to know a priori the cardinality of the solution set and to calculate all the DMDGP solutions, given one of them, it would be relevant to identify these symmetries efficiently. Exploiting mathematical properties of the vertices associated with these symmetries, we present an optimal algorithm that finds such vertices.

Improving the sampling process in the interval Branch-and-Prune algorithm for the discretizable molecular distance geometry problem

Protein structure determination using Nuclear Magnetic Resonance (NMR) experiments is one of the most important applications of Distance Geometry, called the Molecular Distance Geometry Problem (MDGP). Using special atomic orders on the protein molecule, the MDGP can be solved iteratively using a combinatorial method, called Branch-and-Prune (BP). In order to deal with uncertainties of NMR data, there is an extension of the BP algorithm, called interval BP, where the idea is to sample values from the interval distances associated to such uncertainties. We propose a method to improve this sampling process, by reducing the interval of uncertain distances before taking the samples. All the mathematical details necessary to understand the proposal and its implementation are provided, along with some computational experiments that indicate the proposed strategy improves the interval BP algorithm.

Tuning interval Branch-and-Prune for protein structure determination

The interval Branch and Prune (iBP) algorithm for obtaining solutions to the interval Discretizable Molecular Distance Geometry Problem (iDMDGP) has proven itself as a powerful method for molecular structure determination. However, substantial obstacles still must be overcome before iBP may be employed as a tractable general-purpose alternative to existing structure determination algorithms. This work introduces an iterative variant of the iBP algorithm that leverages existing knowledge of protein structures in order to reduce the size of the effective search space by many orders of magnitude. These improvements are included in a newly released implementation of the iBP software that aims to provide a solid platform for both research and application of the iDMDGP.

Geometric Algebra to Model Uncertainties in the Discretizable Molecular Distance Geometry Problem

The discretizable molecular distance geometry problem (DMDGP) is related to the determination of 3D protein structure using distance information detected by nuclear magnetic resonance (NMR) experiments. The chemistry of proteins and the NMR distance information allow us to define an atomic order \({v_{1},\ldots,v_{n}}\) such that the distances related to the pairs \({\{v_{i-3},v_{i}\},\{v_{i-2},v_{i}\},\{v_{i-1},v_{i}\}}\), for \({i > 3}\), are available, which implies that the search space can be represented by a tree. A DMDGP solution can be represented by a path from the root to a leaf node of this tree, found by an exact method, called branch-and-prune (BP). Because of uncertainty in NMR data, some of the distances related to the pairs \({\{v_{i-3},v_{i}\}}\) may not be precise values, being represented by intervals of real numbers \({[\underline{d}_{i-3,i},\overline{d}_{i-3,i}]}\). In order to apply BP algorithm in this context, sample values from those intervals should be taken. The main problem of this approach is that if we sample many values, the search space increases drastically, and for small samples, no solution can be found. We explain how geometric algebra can be used to model uncertainties in the DMDGP, avoiding sample values from intervals \({[\underline{d}_{i-3,i},\overline{d}_{i-3,i}]}\) and eliminating the heuristic characteristics of BP when dealing with interval distances.

EXPLOITING SYMMETRY PROPERTIES OF THE DISCRETIZABLE MOLECULAR DISTANCE GEOMETRY PROBLEM

The Discretizable Molecular Distance Geometry Problem (DMDGP) involves a subset of instances of the distance geometry problem for which some assumptions allowing for discretization are satisfied. The search domain for the DMDGP is a binary tree that can be efficiently explored by employing a Branch & Prune (BP) algorithm. We showed in recent works that this binary tree may contain several symmetries, which are directly related to the total number of solutions of DMDGP instances. In this paper, we study the possibility of exploiting these symmetries for speeding up the solution of DMDGPs, and propose an extension of the BP algorithm that we named symmetry-driven BP (symBP). Computational experiments on artificial and protein instances are presented.

Recent advances on the Discretizable Molecular Distance Geometry Problem

The Molecular Distance Geometry Problem (MDGP) consists in finding an embedding in R3 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.

The discretizable distance geometry problem

We introduce the Discretizable Distance Geometry Problem in R^3 (DDGP3), which consists in a subclass of instances of the Distance Geometry Problem for which an embedding in R^3 can be found by means of a discrete search. We show that the DDGP3 is a generalization of the Discretizable Molecular Distance Geometry Problem (DMDGP), and we discuss the main differences between the two problems. We prove that the DDGP3 is NP-hard and we extend the Branch & Prune (BP) algorithm, previously used for the DMDGP, for solving instances of the DDGP3. Protein graphs may or may not be in DMDGP and/or DDGP3 depending on vertex orders and edge density. We show experimentally that as distance thresholds decrease, PDB protein graphs which fail to be in the DMDGP still belong to DDGP3, which means that they can still be solved using a discrete search.

The discretizable molecular distance geometry problem

Given a simple weighted undirected graph G=(V,E,d) with d:E→ℝ+, the Molecular Distance Geometry Problem (MDGP) consists in finding an embedding x:V→ℝ3 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.