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

Alves, Rafael; Lavor, Carlile

doi:10.1007/s00006-016-0653-2

Cited by 30 publications

(14 citation statements)

References 16 publications

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“…Very recent results [2,3] using Clifford algebra propose an alternative that avoids the sampling process in the branching phase of i BP. However, in order to apply these results to protein structure calculations, a new re-order must be defined that avoids the situation (3).…”

Section: A New Dmdgp Order For Protein Graphsmentioning

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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Section: A New Dmdgp Order For Protein Graphsmentioning

confidence: 99%

Minimal NMR distance information for rigidity of protein graphs

Lavor¹,

Liberti²,

Donald³

et al. 2019

Discrete Applied Mathematics

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“…Following the arguments given in , we first obtain the points from the intersection of the spheres centered at X 1 , X 2 , X 3 with radius d _ 1,4 , d 2,4 , d 3,4 (resulting in

\underset{P_{4}^{0}}{false}

and

\underset{P_{4}^{1}}{false}

) and centered at X 1 , X 2 , X 3 with radius

{\overset{d}{true}}_{1, 4}, d_{2, 4}, d_{3, 4}

(resulting in

\overset{P_{4}^{0}}{true}

and

\overset{P_{4}^{1}}{true}

), represented in Figure .…”

Section: Ibp and The Conformal Clifford Algebramentioning

confidence: 99%

“…See, for example, [38] for more details. Following the arguments given in [21], we first obtain the points from the intersection of the spheres centered at X 1 , X 2 , X 3 with radius d 1,4 , d 2,4 , d 3,4 (resulting in P 0 4 and P 1 4 ) and centered at X 1 , X 2 , X 3 with radius N d 1,4 , d 2,4 , d 3,4 (resulting in P 0 4 and P 1 4 ), represented in Figure 3. Once we have the starting and the ending point of an arc, we can define a rotor acting on that.…”

Section: Ibp and The Conformal Clifford Algebramentioning

confidence: 99%

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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“…As a mathematical puzzle, it was originated by Menger [11]. The field has practical applications for detecting molecular structure of proteins, since nuclear magnetic resonance techniques exist to determine distances between neighboring atoms [1].…”

Section: Introductionmentioning

confidence: 99%

“…Because of the prevalence of spheres and their intersections in this problem, [1,2] and [6] have addressed the problem by conformal geometric algebra (CGA). This is the geometric algebra of a vector space whose vectors can be interpreted as spheres, whose 2D subspaces can be interpreted as circles, and whose 3D subspaces are point pairs (the intersection of three spheres).…”

Section: Introductionmentioning

confidence: 99%

Boolean Combination of Circular Arcs using Orthogonal Spheres

Dorst

2019

Adv. Appl. Clifford Algebras

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An orthogonal sphere representation of arcs on spatial circles can be used to compactly perform Boolean combinations of such arcs. We formulate this using conformal geometric algebra, of which the oriented nature allows both minor and major arcs to be treated. Easily computable quantities discriminate the cases of relative positions. An application in the first stages of a problem in Discretizable Molecular Distance Geometry is included. We give a suggestion on how to extend this characterization by orthogonal spheres to the manifolds of arcs in the subsequent stages, using probabilistic eigenspheres of the distributions.Mathematics Subject Classification. 51B99, 15A66.

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

Cited by 30 publications

References 16 publications

Minimal NMR distance information for rigidity of protein graphs

Minimal NMR distance information for rigidity of protein graphs

Clifford algebra and discretizable distance geometry

Boolean Combination of Circular Arcs using Orthogonal Spheres

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