Graph theoretic and electronic properties of fullerenes, &, Biasing molecular modelling simulations

315 Pages·2016·3.95 MB·English

by  Wirz, Lukas N

Copyright is owned by the Author of the thesis. Permission is given for
a copy to be downloaded by an individual for the purpose of research and
private study only. The thesis may not be reproduced elsewhere without
the permission of the Author.
Graph Theoretic and Electronic
Properties of Fullerenes
&
Biasing Molecular Modelling
Simulations with Experimental
Residual Dipolar Couplings
Lukas N Wirz
2015
A thesis presented in partial fulfilment of the
requirements for the degree of
Doctor of Philosophy
at
Massey University, Albany
New Zealand

Contents iii
Contents
I Graph theoretical and electronic properties of fulle-
renes 1
1 Introduction 5
1.1 AShortIntroductiontoGraphTheoryandEmbeddings . . . 9
1.2 Drawingfullerenegraphs: Methodsforplanarembedding . . 13
1.3 Thelargeisomerspace . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Generationoffullerenegraphs . . . . . . . . . . . . . . . . . . 16
1.5 Transformationoffullerenegraphs . . . . . . . . . . . . . . . . 17
1.6 GeometryofFullerenes . . . . . . . . . . . . . . . . . . . . . . 23
1.7 Generatingaccurate3Dgeometries. . . . . . . . . . . . . . . . 26
1.8 Fullerenesymmetry . . . . . . . . . . . . . . . . . . . . . . . . 28
1.9 Shapes: Volumeandsurfacearea,sphericityandconvexity . 33
1.10 Topologicalandchemicalindicators . . . . . . . . . . . . . . . 37
1.11 Perfectmatchingsandtheirmolecularapplication . . . . . . . 46
1.12 Thermodynamicstabilityandthegraphenelimit . . . . . . . 51
1.13 Electronicaspectstostructureandstability . . . . . . . . . . . 56
1.14 Thegasphaseformationoffullerenes . . . . . . . . . . . . . . 57
2 ProgramFullerene: ASoftwarePackageforConstructingandAna-
lyzing Structures of Regular Fullerenes 61
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.2 GeneralstructureandhistoryoftheprogramFullerene . . . . 64
2.3 3Dstructuregeneration . . . . . . . . . . . . . . . . . . . . . . 65
2.4 Goldberg-Coxetertransformation. . . . . . . . . . . . . . . . . 68
2.5 StoneWalestransformationandvertexinsertions . . . . . . . 70
2.6 Force-fieldoptimizationsandvibrationalfrequencies . . . . . 71
iv Contents
2.7 Hamiltoniancycles . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.8 Hückelanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.9 Stabilityoffullerenes . . . . . . . . . . . . . . . . . . . . . . . . 76
2.10 TopologicalIndicators . . . . . . . . . . . . . . . . . . . . . . . 77
2.11 Volume,SurfaceAreaandDeviationfromSphericalSymmetry 80
2.12 2DGraphrepresentationsandSchlegelprojections . . . . . . 84
2.13 Databases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.14 StructureofFullereneandlibgraph . . . . . . . . . . . . . . . . 88
3 Non-face spiral fullerenes and general face-spiral algorithms 93
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.3 ResultsandDiscussion . . . . . . . . . . . . . . . . . . . . . . . 97
3.4 TheHalmaandLeapfrogTransformsofNS-Fullerenes . . . . 110
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.6 Thegeneralisedface-spiralalgorithm . . . . . . . . . . . . . . 114
4 Force fields for fullerenes and related polyhedra 119
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.2 ComputationalMethods . . . . . . . . . . . . . . . . . . . . . . 121
4.3 AGeneralHarmonicForceFieldforFullerenes . . . . . . . . 123
4.4 ResultsandDiscussion . . . . . . . . . . . . . . . . . . . . . . . 126
4.5 SummaryandConclusions . . . . . . . . . . . . . . . . . . . . 132
4.6 Aforcefieldforgeneralpolyhedralgraphsandtriangulations
ofthesphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5 Hamilton Cycles in Fullerene Graphs 137
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.3 Resultsanddiscussion . . . . . . . . . . . . . . . . . . . . . . . 140
5.4 ThenumberofHamiltoncyclesinC10+10k-D5+ (5,0)fullerene
nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.5 ThenumberofHamiltoncyclesinC12+12k-D6+ (6,0)fullerene
nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6 Gaudienes 167
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Contents v
6.2 Graphtheoreticalconsiderations . . . . . . . . . . . . . . . . . 169
6.3 Computationalmethods . . . . . . . . . . . . . . . . . . . . . . 172
6.4 Molecularstructures . . . . . . . . . . . . . . . . . . . . . . . . 173
6.5 DiscussionandConclusions . . . . . . . . . . . . . . . . . . . . 179
II Biasing molecular dynamics simulations with experi-
mental residual dipolar couplings 183
7 Introduction 187
7.1 Moleculardynamics . . . . . . . . . . . . . . . . . . . . . . . . 187
7.2 Residualdipolarcouplings . . . . . . . . . . . . . . . . . . . . 198
8 Fitting alignment tensor components to experimental RDCs, CSAs
and RQCs 217
8.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
8.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
8.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
9 Expression of molecular alignment by a weight function 227
9.1 Expansioninmagneticfieldvectors . . . . . . . . . . . . . . . 227
9.2 Expansioninsphericalharmonics . . . . . . . . . . . . . . . . 229
9.3 Restrainingintermsofmagneticfieldvectors . . . . . . . . . 232
9.4 Restrainingintermsofsphericalharmonics . . . . . . . . . . 235
9.5 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
9.6 Equivalenceofsphericalharmonicsandtensorrepresentations240
9.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
10 Alignment Blocks 245
10.1 Restrainingusingalignmentblocks . . . . . . . . . . . . . . . 246
10.2 Applicationto2F13F1fromhumanfibronectin(2CKU) . . . . . 247
10.3 Conclusionsandoutlook . . . . . . . . . . . . . . . . . . . . . . 253
References 257
Statements of Contribution to Doctoral Thesis Containing Publica-
tions 299
vi Contents
Acknowledgements
ThereareseveralpeopleIhavetothankformakingmylast3.5yearspossible
and enjoyable: Peter Schwerdtfeger, for giving me the opportunity to
cometoMasseyUniversity,andshieldingallofusfromtheadministrators.
·
Thankyouforsupportandalotoffreedom. JaneAllison,forbeing
extremelymotivating,alwaystakingtimetoansweranddiscussmyques-
·
tions, and for the cake! James Avery, for teaching me how to write
lessbadcode,howtothinkandwriteasamathematician(forchemists)and
patience when either failed. Thank you for inviting me to stay in Copen-
·
hagenfortwo-and-a-halfenjoyableandproductivemonths. Florian
SennandLukášPaštekaforusefuldiscussions,advice,andencouragement
·
when I was in need of it. Andrew Punnett for taming the bits and
electrons (as well as explaining how to) on smaller and larger computers,
·
whenever and in whichever way they misbehaved. Jayson Cosme,
Lukáš Pašteka, and Lukas Trombach for the daily fix of ping pong (table
·
tennis,tobeaccurate). AndeveryoneelseinCTCPforthethroughout
friendlyandencouragingatmosphere.
Contents vii
List of Publications
[1] Schwerdtfeger P, Wirz L, Avery J. Program fullerene: A software
packageforconstructingandanalyzingstructuresofregularfullerenes.
JComputChem2013,34:1508–1526. doi:10.1002/jcc.23278.
[2] WirzLN,TonnerR,AveryJ,SchwerdtfegerP. Structureandproperties
ofthenonface-spiralfullerenesT-C ,D -C ,D -C ,andD -C
380 3 384 3 440 3 672
and their halma and leapfrog transforms. J Chem Inf Model 2014,
54:121–130. doi:10.1021/ci4005578.
[3] SchwerdtfegerP,WirzLN,AveryJ. Thetopologyoffullerenes. WIREs
CompMolSci2015,5:96–145. doi:10.1002/wcms.1207.
[4] Wirz LN, Allison JR. Fitting alignment tensor components to ex-
perimental RDCs, CSAs and RQCs. J Biomol NMR 2015, 62:25–29.
doi:10.1007/s10858-015-9907-x.
[5] WirzLN,TonnerR,HermannA,SureR,SchwerdtfegerP. Fromsmall
fullerenes to the graphene limit: A harmonic force-field method for
fullerenes and a comparison to density functional calculations for
Goldberg–CoxeterfullerenesuptoC . JComputChem2016,37:10–17.
980
doi:10.1002/jcc.23894.
[6] WirzLN,AllisonJR. Commenton“atensor-freemethodforthestruc-
turalanddynamicrefinementofproteinsusingresidualdipolarcoup-
lings”. JPhysChemB2015,119:8223–8224. doi:10.1021/acs.jpcb.5b02801.
[7] WirzLN,Babic´ D,AveryJ,SchwerdtfegerP. Towardtightupperand
lowerboundsforHamiltoncyclesinfullerenegraphs2015.
[8] Sundholm D, Wirz LN, Schwerdtfeger P. Novel hollow all-carbon
structures. Nanoscale2015,38:15886–15894. doi:10.1039/C5NR04370K.
viii Contents
Abstract
Inthisthesistwodifferentmodels,thatislevelsofabstraction,areusedto
explorespecificclassesofmolecularstructuresandtheirproperties.
In part I, fullerenes and other all-carbon cages are investigated using
graphs as a representation of their molecular structure. By this means
thelargeisomerspace,simplemolecularpropertiesaswellaspuregraph
theoreticalaspectsoftheunderlyinggraphsareexplored. Althoughchemical
graphsareusedtorepresentotherclassesofmolecules,cavernouscarbon
moleculesareparticularlywellsuitedforthislevelofabstractionduetotheir
largenumberofisomerswithonlyoneatomtypeanduniformhybridisation
throughoutthemolecule.
In part II, a force field for molecular dynamics, that is the step wise
propagationofamolecularstructureintimeusingNewtonianmechanics,
is complemented by an additional term that takes into account residual
dipolar couplings that are experimentally measured in NMR experiments.
Addingthisforcetermleadstomoreaccuratesimulateddynamicswhichis
especiallyimportantforproteinswhosefunctionalityinmanycasescrucially
dependsontheirdynamics. Largebiomoleculesareanexampleofchemical
systems that are too large for treatment with quantum chemical methods
butatthesametimehaveanelectronicstructurethatissimpleenoughfor
accuratesimulationswithaforcefield.
Part I
Graph theoretical and
electronic properties of
fullerenes