Clifford Algebra and the Discretizable Molecular Distance Geometry Problem

Alves, Rafael; Figueiredo, Weber; Petraglia, Antonio; Maculan, Nelson

doi:10.1007/s00006-015-0532-2

Cited by 37 publications

(19 citation statements)

References 26 publications

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“…cos( ω 1,4 ), …, cos( ω n −3, n ) (cosines of torsion angles in [0, 2 π ] defined by four consecutive vertices), given by [36]:…”

Section: The Discretizable Molecular Distance Geometry Problem (Dmmentioning

confidence: 99%

Minimal NMR distance information for rigidity of protein graphs

Lavor¹,

Liberti²,

Donald³

et al. 2019

Discrete Applied Mathematics

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Nuclear Magnetic Resonance (NMR) experiments provide distances between nearby atoms of a protein molecule. The corresponding structure determination problem is to determine the 3D protein structure by exploiting such distances. We present a new order on the atoms of the protein, based on information from the chemistry of proteins and NMR experiments, which allows us to formulate the problem as a combinatorial search. Additionally, this order tells us what kind of NMR distance information is crucial to understand the cardinality of the solution set of the problem and its computational complexity.

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“…cos( ω 1,4 ), …, cos( ω n −3, n ) (cosines of torsion angles in [0, 2 π ] defined by four consecutive vertices), given by [36]:…”

Section: The Discretizable Molecular Distance Geometry Problem (Dmmentioning

confidence: 99%

Minimal NMR distance information for rigidity of protein graphs

Lavor¹,

Liberti²,

Donald³

et al. 2019

Discrete Applied Mathematics

Self Cite

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“…iBP is most naturally defined as a recursive tree traversal algorithm [13,7] that uses recursively defined affine transformation matrices [24] for computing embedded coordinates. However, recent use of Clifford algebra has yielded embedding equations that are non-recursive [12], the key results of which are recalled here.…”

Section: Iterative Embedding Relationsmentioning

confidence: 99%

“…In cases where ω j / ∈ Ω , the iDMDGP nevertheless guarantees that the distance d j−3, j is available, and we may compute τ j from the cosine law for a trihedron [12]:…”

Section: Iterative Embedding Relationsmentioning

confidence: 99%

“…The iDMDGP solution space is a search tree, within which each path from the root node to a leaf node represents a distinct protein conformation. The iBP algorithm, explained in detail previously [13,5,12], recursively traverses the search tree in order to enumerate solutions to a given problem instance. If at any point in the search, the partial solution becomes infeasible with respect to the constraints, then the leaves of the current search tree node are "pruned" and iBP backtracks until a solution is identified.…”

Section: Introductionmentioning

confidence: 99%

“…This work introduces a variant of iBP that uses an iterative tree traversal algorithm based on embedding equations derived from Clifford algebra [12]. This iterative variant is ideal for iDMDGP instances that have highly repetitive vertex orders [6], as it requires no extra matrix computations for repeated atoms in the order.…”

Section: Introductionmentioning

confidence: 99%

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Tuning interval Branch-and-Prune for protein structure determination

et al. 2018

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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.

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Clifford algebra and discretizable distance geometry

Alves

Lavor

Souza

et al. 2017

Math Methods in App Sciences

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Protein structure calculations using nuclear magnetic resonance (NMR) experiments are one of the most important applications of distance geometry. The chemistry of proteins and the NMR data allow us to define an atomic order, such that the distances related to the pairs of atoms {i−3,i},{i−2,i},{i−1,i} are available, and solve the problem iteratively using a combinatorial method, called branch‐and‐prune. The main step of BP algorithm is to intersect three spheres centered at the positions for atoms i−3,i−2,i, with radius given by the atomic distances di−3,i,di−2,i,di−1,i, respectively, to obtain the position for atom i. Because of uncertainty in NMR data, some of the distances di−3,i may not be precise or even not be available. Using conformal Clifford algebra, in addition to take care of NMR uncertainties, which implies that we have to calculate sphere intersections considering that their centers and radius may not be fixed anymore, we consider a more flexible atomic order, where distances di−3,i are replaced by dj,i, where j⩽i−3. Copyright © 2017 John Wiley & Sons, Ltd.

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Clifford Algebra and the Discretizable Molecular Distance Geometry Problem

Cited by 37 publications

References 26 publications

Minimal NMR distance information for rigidity of protein graphs

Minimal NMR distance information for rigidity of protein graphs

Tuning interval Branch-and-Prune for protein structure determination

Clifford algebra and discretizable distance geometry

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