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Oriented Conformal Geometric Algebra and the Molecular Distance Geometry Problem

ADVANCES IN APPLIED CLIFFORD ALGEBRAS 2019年第1期29卷 9-9页

作者： Lavor, Carlile Alves, Rafael Univ Estadual Campinas UNICAMP IMECC BR-13081970 Campinas SP Brazil Fed Univ ABC UFABC CMCC BR-09210580 Santo Andre SP Brazil

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 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 , 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 can be used to select one of the two possibilities for atom i. Differently from distances , 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.

关键词： Oriented conformal geometric algebra Distance geometry branch and prune algorithm 3D protein structure

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

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ADVANCES IN APPLIED CLIFFORD ALGEBRAS 2017年第1期27卷 439-452页

作者： Alves, Rafael Lavor, Carlile Univ Campinas IMECC UNICAMP BR-13081970 Campinas SP Brazil

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 such that the distances related to the pairs , for , 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 may not be precise values, being represented by intervals of real numbers . 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 and eliminating the heuristic characteristics of BP when dealing with interval distances.

关键词： Conformal geometric algebra Distance geometry branch and prune algorithm 3D protein structure

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A new framework for sharp and efficient resolution of NCSP with manifolds of solutions

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CONSTRAINTS 2010年第2期15卷 190-212页

作者： Goldsztejn, Alexandre Granvilliers, Laurent CNRS Lab Informat Nantes Atlantique Nantes France Univ Nantes Lab Informat Nantes Atlantique Nantes France

When numerical CSPs are used to solve systems of n equations with n variables, the preconditioned interval Newton operator plays two key roles: First it allows handling the n equations as a global constraint, hence achieving a powerful contraction. Second it can prove rigorously the existence of solutions. However, none of these advantages can be used for under-constrained systems of equations, which have manifolds of solutions. A new framework is proposed in this paper to extend the advantages of the preconditioned interval Newton to under-constrained systems of equations. This is achieved simply by allowing domains of the NCSP to be parallelepipeds, which generalize the boxes usually used as domains.

关键词： Interval analysis branch and prune algorithm Preconditioned interval Newton Global constraint for under-constrained systems of equations

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A new framework for sharp and efficient resolution of NCSP with manifolds of solutions

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14th International Conference on Principles and Practice of Constraint Programming (CP 2008)

作者： Goldsztejn, Alexandre Granvilliers, Laurent CNRS Lab Informat Nantes Atlantique Nantes France Univ Nantes Lab Informat Nantes Atlantique Nantes France

ISBN: (纸本)3540859578

关键词： Interval analysis branch and prune algorithm Preconditioned interval Newton Global constraint for under-constrained systems of equations

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