Recent results on assigned and unassigned distance geometry with applications to protein molecules and nanostructures
type: Item
instance of: scholarly article
title: Recent results on assigned and unassigned distance geometry with applications to protein molecules and nanostructures
zbMATH Open document ID: 1429.51011
author: Simon J. L. Billinge
author: P. M. Duxbury
author: Douglas S. Gonçalves
author: Antonio Mucherino
author: Carlile Lavor
published in: Annals of Operations Research
publication date: 2019-03-06T00:00:00Z
review text: In this review paper, the authors provide an update on the state of the art reviewed in their paper on the topic of assigned (aDGP) and unassigned (uDGP) distance geometry problems [4OR 14, No. 4, 337--376 (2016; Zbl 1364.51005)]. A summary of the definitions, notations and results is given. Moreover, they discuss various important advances that have emerged over the years 2016--2018, for example generalization of the mathematical description to a new class of problems called vector geometry problems (VGPs), which can be assigned (aVGP) or unassigned (uVGP). In the following, we formally state these problems. For this purpose, we need some additional definitions. Let \(V\) be a set of \(n\) objects and \(x : V \rightarrow \mathbb{R}^K\) be the function that assigns positions (coordinates) in a Euclidean space of dimension \(K\) to the objects belonging to the set \(V\), whose elements are called vertices. The function \(x\) is referred to as a \textit{realization}. Moreover, for any \(u,v \in V\), we denote its corresponding edge weight by \(d(u, v)\) or associated vector by \(\mathbf{s}(u, v)\). For DGP, let \(\mathcal{D} = (d_1, d_2, \ldots, d_m)\) be a finite sequence consisting of \(m\) distances, called a \textit{distance list}. Distances in \(\mathcal{D}\) can be represented either with a nonnegative real number (when the distance is exact) or by an interval. On the other hand, for VGP, let \(\mathcal{D}_s = (\pm \mathbf{s}_1, \pm \mathbf{s}_2, \ldots, \pm \mathbf{s}_m)\) be a finite sequence consisting of \(m\) interparticle vectors. Note that for every vector \(\mathbf{s}_i\) in the list, its negative \(-\mathbf{s}_i\) also appears. Let us denote the set of all possible unordered pairs \(\lbrace u, v \rbrace\) of vertices in \(V\) by \(\hat{E}\). Then an injective function \(\ell : \lbrace 1, 2, \ldots, m \rbrace \rightarrow \hat{E}\) is called \textit{assignment function}. Finally, we can define the corresponding problems. 1. Given a list \(\mathcal{D}\) of…
reviewed by: Niko Tratnik
Mathematics Subject Classification ID: 51P05
Mathematics Subject Classification ID: 92E10
Mathematics Subject Classification ID: 82D80
Mathematics Subject Classification ID: 51K99
zbMATH DE Number: 7032593
zbMATH Keywords: distance geometry
zbMATH Keywords: graph rigidity
zbMATH Keywords: molecular conformations
zbMATH Keywords: nanostructures
describes a project that uses: iBP
describes a project that uses: CNS
MaRDI profile type: Publication
full work available at URL: s10479-018-2989-6
OpenAlex ID: W2887171800
Wikidata QID: Q62562102
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Recent results on assigned and unassigned distance geometry with applications to protein molecules and nanostructures (DOI)
DOI: 10.1007/S10479-018-2989-6

Recent results on assigned and unassigned distance geometry with applications to protein molecules and nanostructures (DBLP publication ID)
DBLP publication ID: journals/anor/BillingeDGLM18