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.

A note on the Cayley-Menger determinant and the Molecular Distance Geometry Problem

Using information from protein geometry and distance data provided by Nuclear Magnetic Resonance (NMR) experiments, the Molecular Distance Geometry Problem (MDGP) can be solved by a combinatorial approach, called Branch-and-Prune (BP). The primal version of BP algorithm seeks MDGP graph realizations, while the dual BP looks for completions of associated partial distance matrices. These two algorithms are very similar when distance values are precise. In the literature, there are some proposals for extending the primal BP to take care of NMR uncertainties. Using Cayley-Menger determinant, we present a global optimization approach that also allows the dual BP to deal with NMR uncertainties.

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.

Geometric Algebra to Describe the Exact Discretizable Molecular Distance Geometry Problem for an Arbitrary Dimension

The K-Discretizable Molecular Distance Geometry Problem ( $$^{\textit{K}}\hbox {DMDGP}$$ ) is a subclass of the Distance Geometry Problem (DGP), whose complexity is NP-hard, such that the search space is finite. In this work, the authors describe it completely using Conformal Geometric Algebra (CGA), exploring a Minkowski space that provides a natural interpretation of hyperspheres, hyperplanes, points and pair of points as computational primitives, which are largely relevant to the $$^{\textit{K}}\hbox {DMDGP}$$ . It also presents a theoretical approach to solve the $$^{\textit{K}}\hbox {DMDGP}$$ using ideas from classic Branch-and-Prune (BP) algorithm in this new fashion. Time complexity analysis and practical computational results showed that the naive implementation of the CGA is not as efficient as classical formulation. In order to illustrate this, preliminary results are displayed at the end and, also, directions to future developments.

Oriented Conformal Geometric Algebra and the Molecular Distance Geometry Problem

The problem of 3D protein structure determination using distance information from nuclear magnetic resonance (NMR) experiments is a classical problem in distance geometry. NMR data and the chemistry of proteins provide a way to define a protein backbone order such that the distances related to the pairs of atoms $$\{i-3,i\},\{i-2,i\},\{i-1,i\}$$ are available, implying a combinatorial method to solve the problem, called branch-and-prune (BP). There are two main steps in BP algorithm: the first one (the branching phase) is to intersect three spheres centered at the positions for atoms $$ i-3,i-2,i$$ , with radius given by the atomic distances $$ d_{i-3,i},d_{i-2,i},d_{i-1,i}$$ , respectively, to obtain two possible positions for atom i; and the second one (the pruning phase) is to check if additional spheres (related to distances $$d_{j,i}$$ , $$j<i-3$$ ) can be used to select one of the two possibilities for atom i. Differently from distances $$d_{i-2,i},d_{i-1,i}$$ (associated to bond lenghts and bond angles), distances $$d_{j,i}$$ , $$j\le i-3$$ , may not be precise. BP algorithm has difficulties to deal with uncertainties, and this paper proposes the oriented conformal geometric algebra to take care of intersection of spheres when their centers and radius are not precise.

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.

Calculating the possible conformations arising from uncertainty in the molecular distance geometry problem using constraint interval analysis

The calculation of the 3D structure of a protein molecule is important because it is associated to its biological function. Nuclear Magnetic Resonance (NMR) experiments can provide distance information between atoms that are close enough in a given protein and the problem is how to use these distances to determine the protein structure. Using the chemistry of proteins and supposing all the distances are precise values, it is possible to define an atomic order v1, ⋅⋅⋅, vn, such that the distances related to the pairs {vi−3,vi},{vi−2,vi},{vi−1,vi} are available, and solve the problem iteratively using a combinatorial method, called Branch-and-Prune (BP). However, due to uncertainty in NMR data, the distances associated with pairs {vi−3,vi} may not be precise, which implies that there are many difficulties in applying the BP algorithm to this scenario. The use of standard interval arithmetic can be directly applied to the algorithm, but it is known that it generates overestimations. This paper proposes a new methodology to compute possible conformations on the presence of uncertainties arising from NMR distance measurements using a constraint interval analysis approach. Some numerical examples are presented.

The Interval Branch-And-Prune Algorithm for the Protein Structure Determination

A general trend of structural biology is the switch of a rigid description of protein structures, as in the first X-ray structures in the 50's, to a flexible description, in which each protein populate several distinct conformations or even a continuum of conformations. This flexibility was shown in numerous cases to have a crucial importance in the function of proteins. Most of the methods for bio-molecular structure calculations are, up to now, based on a combination of sampling and optimization, which does not allow a systematic exploration of the conformational space. But, in the case of highly flexible bio-molecules, a systematic (or global) exploration of the conformational space would be very welcome.On the other hand, the interval branch-and-prune (iBP) approach has been proposed as a method for allowing a global optimization of molecular structure under distance restraints, the so-called Distance Geometry problem. A recursive implementation of this algorithm has permitted to apply this approach on small structures of proteins in alpha-bundles for which few long-range distance restraints were known. Here, we are going to present the results obtained on a set of protein structures with sizes from 24 to 100 residues, displaying various secondary structures and topology. The distance restraints present on these structures were chosen to contain exclusively short-range information.The results obtained with various sets of distance restraints, including exact values and interval of values, will be presented, in order to experimentally evaluate the complexity of the algorithm on real-case of protein structure determination. Several procedures of acceleration will be used in order to allow a complete exploration of the tree describing the molecular Distance Geometry problem. The efficiency of the exploration will be evaluated using the self-organizing map (SOM) clustering approach, and the quality of obtained conformations with respect to Ramachandran plots and steric clashes will be evaluated.

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.

Clifford Algebra and the Discretizable Molecular Distance Geometry Problem

Nuclear Magnetic Resonance experiments can provide distances between pairs of atoms of a protein that are close enough and the problem is how to determine the 3D protein structure based on this partial distance information, called Molecular Distance Geometry Problem. It is possible to define an atomic order 1, ..., n and solve the problem iteratively using an exact method, called Branch-and-Prune (BP). The main step of BP algorithm is to solve a quadratic system to get the two possible positions for i, i > 3, in terms of the positions of i−3, i−2, i−1 and the distances di−1, i, di−2, i, di−3, i. Because of uncertainty in NMR data, some of the distances di−3, i may not be precise and the main problem to apply BP is related to the difficulty of obtaining an analytical expression of the position of atom i in terms of the positions of the three previous ones and the corresponding distances. We present such expression and although it is similar to one already existing in the literature, based on polyspherical coordinates, a new proof is given, based on Clifford algebra, and we also explain how such expression can be useful in BP using a parameterization which depends on di−3, i. The results suggest that a master equation might exist, what is generally not believed by many researchers.

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 Sensor Network in Molecular Structures of PrP(113-120) AGAAAAGA Amyloid Fibrils

The problem of locating sensors in telecommunication networks is a distance geometry problem (DGP). In such a case, the positions of some sensors are known (which are called anchors) and some of the distances between sensors (which can or cannot be anchors) are known. The DGP is to locate the positions of all the sensors. Molecular DGP (MDGP) looks sensors as atoms and their telecommunication network as a molecule for the determination of its three-dimensional (3D) structure. This Chapter defines some sensor networks for determining molecular structures of PrP(113-120) AGAAAAGA amyloid fibrils, which are unstable, noncrystalline, insoluble and hard to be determined in NMR or X-ray experimental laboratories. The amyloid fibril structure is the common structure associated with some 20 neurodegenerative amyloid diseases (including Parkinson’s, Alzheimer’s, Huntington’s, and Prions’), and other diseases such as Type II diabetes, etc. The sensor networks established in this Chapter will benefit the study of 3D molecular structures of all these diseases and will be useful in the research areas such as structural materials, computer-aided or structure-based drug design, and the computational theory of molecular dynamics, and quantum mechanics/molecular mechanics.

Solving the molecular distance geometry problem with inaccurate distance data

We present a new iterative algorithm for the molecular distance geometry problem with inaccurate and sparse data, which is based on the solution of linear systems, maximum cliques, and a minimization of nonlinear least-squares function. Computational results with real protein structures are presented in order to validate our approach.

A discrete search algorithm for finding the structure of protein backbones and side chains

Some information about protein structure can be obtained by using Nuclear Magnetic Resonance (NMR) techniques, but they provide only a sparse set of distances between atoms in a protein. The Molecular Distance Geometry Problem (MDGP) consists in determining the three-dimensional structure of a molecule using a set of known distances between some atoms. Recently, a Branch and Prune (BP) algorithm was proposed to calculate the backbone of a protein, based on a discrete formulation for the MDGP. We present an extension of the BP algorithm that can calculate not only the protein backbone, but the whole three-dimensional structure of proteins.

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.