An Introduction to Distance Geometry applied to Molecular Geometry

59 Pages·2017·1.417 MB·English

by  Carlile Lavor, Leo Liberti, Weldon A. Lodwick, Tiago Mendonça da Costa (auth.)

SPRINGER BRIEFS IN COMPUTER SCIENCE
Carlile Lavor
Leo Liberti
Weldon A. Lodwick
Tiago Mendonça da Costa
An Introduction
to Distance
Geometry applied
to Molecular
Geometry
123
SpringerBriefs in Computer Science
Serieseditors
StanZdonik,BrownUniversity,Providence,RhodeIsland,USA
ShashiShekhar,UniversityofMinnesota,Minneapolis,Minnesota,USA
XindongWu,UniversityofVermont,Burlington,Vermont,USA
LakhmiC.Jain,UniversityofSouthAustralia,Adelaide,SouthAustralia,Australia
DavidPadua,UniversityofIllinoisUrbana-Champaign,Urbana,Illinois,USA
Xuemin(Sherman)Shen,UniversityofWaterloo,Waterloo,Ontario,Canada
BorkoFurht,FloridaAtlanticUniversity,BocaRaton,Florida,USA
V.S.Subrahmanian,UniversityofMaryland,CollegePark,Maryland,USA
MartialHebert,CarnegieMellonUniversity,Pittsburgh,Pennsylvania,USA
KatsushiIkeuchi,UniversityofTokyo,Tokyo,Japan
BrunoSiciliano,UniversitàdiNapoliFedericoII,Napoli,Italy
SushilJajodia,GeorgeMasonUniversity,Fairfax,Virginia,USA
NewtonLee,NewtonLeeLaboratories,LLC,Tujunga,California,USA
Moreinformationaboutthisseriesathttp://www.springer.com/series/10028
Carlile Lavor (cid:129) Leo Liberti (cid:129) Weldon A. Lodwick
Tiago Mendonça da Costa
An Introduction to Distance
Geometry applied to
Molecular Geometry
123
CarlileLavor LeoLiberti
UniversityofCampinas CNRSandÉcolePolytechnique
Campinas,SãoPaulo,Brazil Palaiseau,France
WeldonA.Lodwick TiagoMendonçadaCosta
UniversityofColoradoDenver UniversityofColoradoDenver
Denver,CO,USA Denver,CO,USA
ISSN2191-5768 ISSN2191-5776 (electronic)
SpringerBriefsinComputerScience
ISBN978-3-319-57182-9 ISBN978-3-319-57183-6 (eBook)
DOI10.1007/978-3-319-57183-6
LibraryofCongressControlNumber:2017943518
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Preface
Distance geometry (DG) is a distinct mathematical research area which includes
mathematics and computer science as fundamental components. The fundamental
problemofDGistodeterminethespatiallocations(coordinates)forasetofpoints,
inagivengeometricspace,usingdistancesbetweensomeofthem.
DG is considered to have originated in 1928, when Menger [62] characterized
severalgeometricconceptsusingtheideaofdistance[16].However,onlywiththe
resultsofBlumenthal[10],in1953,didthetopicbecomeanewareaofknowledge
knownasDG.
The main challenge of DG at that time was to find necessary and sufficient
conditions in order to decide if a given matrix is a distance matrix D. That is,
decidewhetherornotagivenmatrixDisasymmetricmatrixsuchthatthereisan
integernumberK andasetofpointsinRK,wheretheEuclideandistancesbetween
thesepointsareequaltotheentriesofthematrixD[51].Notethat,inthiscase,all
distancesareconsideredknown.
To the best of our knowledge, the first explicit mention of the fundamental
problem of DG delineated above, where not all distances are known, was given
byYemini[81],in1978.Inthiscase,theproblemmaybeharder.
Another important moment in the history of DG is related to its application to
the calculation of molecular structures, with the 1988 publication of Crippen and
Havel’sbook[15],consideredpioneersofDGintheanalysisofproteinstructures.
MoreinformationontheDGhistorycanbefoundin[52].
The first edited book fully dedicated to DG was published by Springer in 2013
[67]. ThebookbroughttogetherdifferentapplicationsandresearchersinDG.Inthe
sameyear,inJune2013,thefirstinternationalworkshopdedicatedtoDGwasheld
withspeakersfromvariousinternationalinstitutions(PrincetonUniversity,IBMTJ
Watson Research Center, University of Cambridge, École Polytechnique, Institut
Pasteur, École Normale Supérieure, SUTD-MIT International Design Center).
The event also had the support of several international scientific societies and
universities, indicating the importance of DG in many areas of knowledge (more
detailsathttp://dga2013.icomp.ufam.edu.br/).
v
vi Preface
TheinterestinDGasatopicofresearcharisesfromthewealthanddiversityof
its applications, in addition to its mathematical depth and beauty. Recent surveys
on DG highlighting the theory and applications can be found in [9, 18, 57]. For
example, applications can be found in problems from astronomy, biochemistry,
statistics,nanotechnology,robotics,andtelecommunications.
Inastronomy,theproblemisrelatedtothedeterminationofstarpositionsusing
information about the distances between some of them [60]. In biochemistry, the
problem appears in the determination of three-dimensional structures of protein
moleculesusingtheinformationobtainedfromnuclearmagneticresonance(NMR)
experiments. In statistics, there are problems related to visualization of data [22]
and dimensionality reduction [50]. In these cases, points and distances are given
in a high dimensional space Rn and the problem is how represent them in a lower
dimension,sayR2orR3,inordertohaveavisualideaofthedata.Thisapplicationis
alsolinkedtoacurrenttopicofresearchcalledBigData[2,61]. Innanotechnology,
the problem is similar to the problem in biochemistry, but on a “nano” scale
[21,39]. There is a direct relationship between the application to robotics and the
calculations related to molecular geometry [23,70]. That is, given a set of robotic
arm lengths (distances), the problem is to find the locus of points that the robot
armcanreach[68].Intelecommunications,theproblemisrelatedtothepositioning
of a wireless sensor network, where the distances can be estimated by the amount
ofpowernecessaryforperformingpeer-to-peersensorcommunication.Thefurther
thesensors,themorepowerisnecessary.Sincebothsensorsknowhowmuchpower
theyused,bothsensorscancomputetheirdistance.Anexampleofthisisforrouter
positioning[24,81].
ThetheoreticalnatureandthewidevarietyofapplicationshaveresultedinDG
becomingitsownresearchareainappliedmathematics,whichincludesfundamental
concepts from mathematics (measures, norms, geometry, optimization, combina-
torics, graph theory, symmetry, uncertainty) and computer science (algorithms,
solvability,complexity).
Campinas,SãoPaulo,Brazil CarlileLavor
Palaiseau,France LeoLiberti
Denver,CO,USA WeldonA.Lodwick
Denver,CO,USA TiagoMendonçadaCosta
March2017
Acknowledgements
WewouldliketothanktheBrazilianresearchagenciesCNPq,CAPES,andFAPESP
fortheirfinancialsupport,andProf.PauloJ.S.Silva(fromUniversityofCampinas)
andProf.DouglasS.Gonçalves(fromFederalUniversityofSantaCatarina)fortheir
importantcommentsandsuggestionsthatimprovethefinalversionofthebook.
WealsothanktheBrazilianSocietyofComputationalandAppliedMathematics
(SBMAC) for the opportunity to give a Distance Geometry course that resulted in
thefirstversionofthismonographinPortuguese(http://www.sbmac.org.br/p_notas.
php).
vii
Contents
1 Introduction ................................................................... 1
1.1 NotationandBasicConcepts............................................ 1
1.2 Outline.................................................................... 4
2 TheDistanceGeometryProblem(DGP)................................... 5
2.1 DefinitionoftheDGP.................................................... 5
2.2 NumberofSolutionsoftheDGP........................................ 6
2.3 ComplexityoftheDGP.................................................. 7
2.4 DGPInstances............................................................ 10
3 FromContinuoustoDiscrete................................................ 13
3.1 ContinuousOptimizationandtheDGP................................. 13
3.2 FinitenessoftheDGP.................................................... 15
3.3 VertexOrderfortheDGP................................................ 18
4 TheDiscretizableDistanceGeometryProblem(DDGP3)................ 21
4.1 DefinitionoftheDDGP3 ................................................ 21
4.2 ComplexityoftheDDGP3............................................... 25
4.3 TheBPAlgorithmfortheDDGP3 ...................................... 26
5 TheDiscretizableMolecularDistance
GeometryProblem(DMDGP)............................................... 31
5.1 DefinitionoftheDMDGP............................................... 31
5.2 ComplexityoftheDMDGP ............................................. 34
5.3 DMDGPSymmetry ..................................................... 36
6 DistanceGeometryandMolecularGeometry............................. 41
6.1 TheDMDGPand3DProteinStructures................................ 41
6.2 OrderinginProteinMolecules .......................................... 43
6.3 ThePolynomialPerformanceoftheBPAlgorithm.................... 46
7 Conclusion..................................................................... 49
References.......................................................................... 51
ix
Chapter 1
Introduction
This monograph introduces distance geometry, based on problems related to 3D
protein structure calculations using distance information provided by Nuclear
MagneticResonance(NMR)experiments.Ourpresentationisasshortaspossible,
withexercisestohelpthereadertobetterunderstandthecontents.
Thetextisbasedonthecombinatorialstructureofdistancegeometryproblems,
differentlyfromtheclassicalapproach,whichfocusesoncontinuousmethods.This
discreteapproachisimportantfortheunderstandingofthemainconceptsinvolved
andprovidesanewwaytoconsidertheproblem. Curiously,thiscombinatorialview
aroseinthequantumcomputingcontext,whenGrover’salgorithm[3,33,42]was
proposedasamethodtosolveadistancegeometryproblemrelatedtothecalculation
ofmolecularstructures[41].
1.1 NotationandBasicConcepts
Itisassumedthatanelementaryknowledgeofanalyticgeometrywillbesufficient
tofollowthetext.Obviously,thereaderwillalsoneedtohavesomefamiliaritywith
themathematicallanguageinvolvingbasicconceptsfromlogic,sets,andfunctions.
Next, we list the main concepts that will be used in this text with the associated
notation.
(cid:129) Sets
– x2AmeansthatxisanelementofsetA.
– A (cid:2) B means that the set A is contained in B, that is, all elements in A are
elementsofB.
– A\BDfx:x2Aandx2Bgisthesetformedbytheelementsthatareboth
inAandB.
©TheAuthor(s)2017 1
C.Lavoretal.,AnIntroductiontoDistanceGeometryappliedtoMolecular
Geometry,SpringerBriefsinComputerScience,DOI10.1007/978-3-319-57183-6_1