An Efficient Exhaustive Search for the Discretizable Distance Geometry Problem with Interval Data

Abstract

The Distance Geometry Problem (DGP) asks whether a simple weighted undirected graph can be realized in a given space (generally Euclidean) so that a given set of distance constraints (associated to the edges of the graph) is satisfied. The Discretizable DGP (DDGP) represents a subclass of instances where the search space can be reduced to a discrete domain having the structure of a tree. In the ideal case where all distances are precise, the tree is binary and one singleton, representing one possible position for a vertex of the graph, is associated to every tree node. When the distance information is however not precise, the uncertainty on the distance values implies that a three-dimensional region of the search space needs to be assigned to some nodes of the tree. By using a recently proposed coarse-grained representation for DDGP solutions, we extend in this work the branch-and-prune (BP) algorithm so that it can efficiently perform an exhaustive search of the search domain, even when the uncertainty on the distances is important. Instead of associating singletons to nodes, we consider a pair consisting of a box and of a most-likely position for the vertex in this box. Initial estimations of the vertex positions in every box can be subsequently refined by using local optimization. The aim of this paper is twofold: (i) we propose a new simple method for the computation of the three-dimensional boxes to be associated to the nodes of the search tree; (ii) we introduce the resolution parameter ρ, with the aim of controling the similarity between pairs of solutions in the solution set. Some initial computational experiments show that our algorithm extension, differently from previously proposed variants of the BP algorithm, is actually able to terminate the enumeration of the solution set by providing solutions that differ from one another accordingly to the given resolution parameter.

A Distance Geometry Procedure Using the Levenberg-Marquardt Algorithm and with Applications in Biology but Not only

“…To assess the performance of PLM as a refinement tool, we compare it with the Spectral Projected Gradient (SPG) algorithm [3] for minimizing the function (7) over C. Notice that SPG was already successfully used in previous works as a local solver for DGP [17].…”

“…For SPG, we used the same parameters as in [17], and stopped the iterations when d k ≤ ε. The maximum number of iterations was set to 2,000 for both SPG and PLM.…”

MD-jeep: a New Release for Discretizable Distance Geometry Problems with Interval Data

“…Once a tree leaf node is reached, the solution to the DDGP instance can be simply obtained by extracting the positions x v from the function z. The use of the boxes B v can then be useful to verify how different this latest found solution is from other possible solutions that the algorithm will encounter in the further exploration of the tree (for more details, see [11] and the resolution parameter recently introduced in BP).…”

A study on the impact of the distance types involved in protein structure determination by NMR

“…In previous works on the DDGP, NMR data were simulated from known molecular models extracted from PDB files [2]. As research went on (see for example [4,11,17,23]), the considered DDGP instances approached more and more the genuine NMR data. Initially proposed for instances containing exact distances only [12], the BP framework was extended to interval distances in [11]; it was then associated to a coarsegrained representation (to better deal with uncertainty) in [23], and to a multi-threading-like approach (to deal with larger instances) in [17].…”

“…As research went on (see for example [4,11,17,23]), the considered DDGP instances approached more and more the genuine NMR data. Initially proposed for instances containing exact distances only [12], the BP framework was extended to interval distances in [11]; it was then associated to a coarsegrained representation (to better deal with uncertainty) in [23], and to a multi-threading-like approach (to deal with larger instances) in [17]. Together with these NMR-derived distances, other distances, rather obtained from the chemical structure of the considered molecules, are also included in the DDGP instances.…”