Language of Shape (eBook)

Front Cover 1
The Language of Shape: The Role of Curvature in Condensed Matter: Physics, Chemistry and Biology 4
Copyright Page 5
Table of Contents 10
Chapter 1. The Mathematics of Curvature 14
1.1. Introductory remarks 14
1.2 Curvature 15
1.3 Differential geometry of surfaces 17
1.4 The Gauss map 19
1.5 Geodesic curvature and geodesics 20
1.6 Torsion 21
1.7 The Gauss-Bonnet theorem 23
1.8 Topology 24
1.9 A provisional catalogue of surface forms 27
1.10 A historical perspective 31
1.11 Periodic minimal surfaces 34
1.12 The Bonnet transformation: the P-surface, the D-surface and the gyroid 40
1.13 Parallel surfaces 45
1.14 Future directions 45
Appendix: A catalogue of some minimal surfaces 46
Mathematical Bibliography 53
References 54
Chapter 2. The Lessons of Chemistry 56
Inorganic Chemistry: From the discrete lattice of crystal symmetry to the continuous manifolds of differential geometry 56
2.1 The background 56
2.2 The unravelling of complex structures 57
2.3 Defects 59
2.4 The intrinsic curvature of solids 62
2.5 Hydrophobic zeolites and adsorption 65
2.6 Phase transitions, order and disorder 68
2.7 Quantitative analysis of hyperbolic frameworks: silicate densities 71
2.8 Tetrahedral frameworks: Three- or two-dimensional structures? 76
2.9 Quasicrystals 79
Organic Chemistry-The Shape of Molecules 86
2.10 The hyperbolic nature of sp3 orbitals 86
2.11 Organic sculptures: carcerands, crowns, etc. 88
2.12 Beyond graphite: fullerenes and schwarzites 91
Appendix: The Problem of Quasicrystals 93
References 97
Chapter 3. Molecular Forces and Self-Assembly 100
3.1 The background 100
3.3 The background to surface forces 109
3.4 Molecular forces in detail 111
3.5 A gallimaufry of forces 118
3.6 Self-organisation in surfactant solutions 126
Appendix A: Evolution of concepts on long range molecular forces responsible for organisation and interactions in colloidal systems 137
Appendix B: Modern concepts of self-assembly 141
Appendix C: Remarks on the nature of the hydrophobic interaction and water structure 142
References 150
Chapter 4. Beyond Flatland: The Geometric Forms due to Self-Assembly 154
4.1 Introduction: molecular dimensions and curvature 154
4.2 The local geometry of aggregates 156
4.3 The composition of surfactant mixtures: the global constraint 159
4.4 Bilayers in surfactant-water mixtures 162
4.5 Monolayers in surfactant-water mixtures 167
4.6 Geometrical physics: bending energy 170
4.7 The mesophase behaviour of surfactant- and lipid-water mixtures 173
4.8 The hyperbolic realm: cubic and intermediate phases 176
4.9 Mesostructure in ternary surfactant-water-oil systems: microemulsions 183
4.10 Block copolymer melts: an introduction 189
4.11 Copolymer self-assembly 190
4.12 Relation between material properties and structure 198
4.13 Protein assemblies in bacteria: a mesh phase 199
4.14 Self-assembly of chiral molecules 200
References 207
Chapter 5. Lipid Self-Assembly and Function In Biological Systems 212
5.1 Self-association of lipids in an aqueous environment 212
5.2 Cell membranes 226
References 245
Chapter 6. Folding and Function In Proteins and DNA 250
6.1 Overall features of protein structure 250
6.2 a-helix domains 252
6.3 a-helix / ß--sheet domains 252
6.4 ß-sheet domains 254
6.5 Membrane proteins 255
6.6 Enzymatic action 256
6.7 Protein function and dioxin poisoning 260
6.8 Geometry in hormone-receptor interactions 261
6.9 Self / non-self recognition 263
6.10 DNA folding 264
6.11 Self-assembly and crystallisation of proteins 266
References 269
Chapter 7. Cytomembranes and Cubic Membrane Systems Revisited 270
7.1 Membrane organisation 270
7.2 Recognition of hyperbolic periodic cytomembrane morphologies in electron microscopic sections 272
7.3 The structure and occurrence of cubic membranes 279
7.4 Cubic membranes in unicellular organisms: prokaryotes and protozoa 285
7.5 Cubic membranes in plants 288
7.6 Cubic membranes in fungi 297
7.7 Cubic membranes in metazoa 299
7.8 Relationships between tubuloreticular structures, annulate lamellae, and cubic membranes 327
7.9 Biogenesis of cubic membranes 330
7.10 Relationships between cubic membranes and cubic phases 334
7.11 Functionalities of cubic membranes 336
7.12 Cell space organisation and topology 337
7.13 Specific structure-function relations 340
Abbreviations 343
References 344
Chapter 8. Some Miscellaneous Speculations 352
Templating 352
8.2 Supra Self-Assembly 361
8.3 The origin of life: a role for cubosomes? 372
8.4 A final word 375
References 376
Index 378