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Preface xv
Part 1: General Issues 1 1 Introduction for a Tutorial on Diatom Morphology 3 Kalina Manoylov and Mohamed Ghobara
1.1 Diatoms in Brief 3
1.2 Tools to Explore Diatom Frustule Morphology 7
1.3 Diatom Frustule 3D Reconstruction 12
1.3.1 Recommended Steps to Understand the Complex Diatom Morphology: A Guide for Beginners 13
1.4 Conclusion 15 Acknowledgements 15 References 15 2 The Uncanny Symmetry of Some Diatoms and Not of Others: A Multi-Scale Morphological Characteristic and a Puzzle for Morphogenesis 19 Janice L. Pappas, Mary Ann Tiffany and Richard Gordon
2.1 Introduction 20
2.1.1 Recognition and Symmetry 21
2.1.2 Symmetry and Growth 24
2.1.3 Diatom Pattern Formation, Growth, and Symmetry 25
2.1.4 Diatoms and Uncanny Symmetry 27
2.1.5 Purpose of This Study 28
2.2 Methods 28
2.2.1 Centric Diatom Images Used for Analysis 28
2.2.2 Centric Diatoms, Morphology, and Valve Formation 34
2.2.3 Image Entropy and Symmetry Measurement 36
2.2.4 Image Preparation for Measurement 37
2.2.5 Image Tilt and Slant Measurement Correction for Entropy Values 38
2.2.6 Symmetry Analysis 39
2.2.7 Entropy, Symmetry, and Stability 40
2.2.8 Randomness and Instability 42
2.3 Results 43
2.3.1 Symmetry Analysis 43
2.3.2 Valve Formation--Stability and Instability Analyses 49
2.4 Discussion 51
2.4.1 Symmetry and Scale in Diatoms 55
2.4.2 Valve Formation and Stability 56
2.4.3 Symmetry, Stability and Diatom Morphogenesis 57
2.4.4 Future Research--Symmetry, Stability and Directionality in Diatom Morphogenesis 58 References 59 3 On the Size Sequence of Diatoms in Clonal Chains 69 Thomas Harbich 3.1 Introduction 70
3.2 Mathematical Analysis of t he Size Sequence 73
3.2.1 Alternative Method for Calculating the Size Sequence 73
3.2.2 Self-Similarity and Fractal Structure 75
3.2.3 Matching Fragments to a Generation Based on Known Size Indices of the Fragment 76
3.2.4 Sequence of the Differences of the Size Indices 78
3.2.5 Matching Fragments to a Generation Based on Unknown Size Indices of the Fragment 80
3.2.6 Synchronicity of Cell Divisions 81
3.3 Observations 82
3.3.1 Challenges in Verifying the Sequence of Sizes 82
3.3.2 Materials and Methods 83
3.3.3 Investigation of the Size Sequence of a Eunotia sp. 84
3.3.4 Synchronicity 86
3.4 Conclusions 87 Acknowledgements 88 Appendix 3A L-System for the Generation of the Sequence of Differences in Size Indices of Adjacent Diatoms 88 Appendix 3B Probability Consideration for Loss of Synchronicity 89 References 91 4 Valve Morphogenesis in Amphitetras antediluviana Ehrenburg 93 Mary A. Tiffany and Bonnie L. Hurwitz
4.1 Introduction 93
4.2 Material and Methods 94
4.3 Observations 94
4.3.1 Amphitetras antediluviana Mature Valves 94
4.3.2 Amphitetras antediluviana Forming Valves 96
4.3.3 Amphitetras antediluviana Girdle Band Formation 101
4.4 Conclusion 101 Acknowledgments 102 References 102 Glossary 104
Part 2: Simulation 105 5 Geometric Models of Concentric and Spiral Areola Patterns of Centric Diatoms 107 Anton M. Lyakh
5.1 Introduction 107
5.2 Set of Common Rules Used in the Models 109
5.3 Concentric Pattern of Areolae 109
5.4 Spiral Patterns of Areolae 110
5.4.1 Unidirectional Spiral Pattern 111
5.4.2 Bidirectional Spiral Pattern 113
5.4.3 Common Genesis of Unidirectional and Bidirectional Spiral Patterns 113
5.5 Conversion of an Areolae-Based Model Into a Frame-Based Model 114
5.6 Conclusion 114 Acknowledgements 114 References 115 6 Diatom Pore Arrays'' Periodicities and Symmetries in the Euclidean Plane: Nature Between Perfection and Imperfection 117 Mohamed M. Ghobara, Mary Ann Tiffany, Richard Gordon and Louisa Reissig
6.1 Introduction 118
6.2 Materials and Methods 122
6.2.1 Micrograph Segmentation 123
6.2.2 Two-Dimensional Fast Fourier Analysis and Autocorrelation Function Analysis 123
6.2.3 Lattice Measurements and Recognition 123
6.2.4 Accuracy of 2D ACF-Based Calculations 125
6.2.5 The Perfection of the Unit Cell Parameters Between Different Parts (Groups of Pore Arrays) of the Same Valve and the Same Micrograph 126
6.3 Results and Discussion 126
6.3.1 Toward Standardization of the Methodology for the Recognition of 2D Periodicities of Pore Arrays in Diatom Micrographs 126
6.3.1.1 Using Two-Dimensional Fast Fourier Transform Analysis 126
6.3.1.2 Using Two-Dimensional Autocorrelation Function 131
6.3.1.3 The Accuracy of Lattice Parameters'' Measurements Using the Proposed 2D ACF Analysis 134
6.3.2 Exploring the Periodicity in Our Studied Micrographs and the Possible Presence of Different Types of 2D Lattices in Diatoms 137
6.3.2.1 Irregular Pore Scattering (Non-Periodic Pores) 137
6.3.2.2 Linear Periodicity of Pores in Striae (1D Periodicity) 138
6.3.2.3 The Different 2D Lattices in Diatom Pore Arrays 140
6.3.3 How Perfectly Can Diatoms Build Their 2D Pore Arrays? 146
6.3.3.1 Variation of the 2D Lattice Within the Connected Pore Array of the Valve 146
6.3.3.2 Comparison of 2D Lattice Parameters and Degree of Perfection of Distinct Pore Array Groups in the Same Micrograph and Valve but With Different Rotational or Reflection Symmetry 148
6.3.3.3 The Perfection of 2D Lattices of Diatom Pore Arrays Compared to Perfect (Non-Oblique) 2D Bravais Lattices 148
6.3.4 Planar Symmetry Groups to Describe the Whole Diatom Valve Symmetries and Additionally Describe the Complicated 2D Periodic Pore Arrays'' Symmetries 149
6.3.4.1 Rosette Groups 150
6.3.4.2 Frieze Groups 151
6.3.4.3 Wallpaper Groups 153
6.4 Conclusion 153 Acknowledgment 154 Glossary 154 References 155 7 Quantified Ensemble 3D Surface Features Modeled as a Window on Centric Diatom Valve Morphogenesis 159 Janice L. Pappas
7.1 Introduction 159
7.1.1 From 3D Surface Morphology to Morphogenesis 160
7.1.2 Geometric Basis of 3D Surface Models and Analysis 163
7.1.3 Differential Geometry of 3D Surface 163
7.1.4 3D Surface Feature Geometry and Morphological Attributes 165
7.1.5 Centric Diatom Taxa Used as Exemplars in 3D Surface Models for Morphogenetic Analysis 166
7.1.6 Morphogenetic Descriptors of Centric Diatoms in Valve Formation as Sequential Change in 3D Surface Morphology 166
7.1.7 Purposes of This Study 167
7.2 Methods 168
7.2.1 Measurement of Ensemble Surface Features and 3D Surface Morphology: Derivation and Solution of the Jacobian, Hessian, Laplacian, and Christoffel Symbols 168
7.2.1.1 The Jacobian of 3D Surface Morphology 168
7.2.1.2 Monge Patch 169
7.2.1.3 First and Second Fundamental Forms and Surface Characterization of the Monge Patch 169
7.2.1.4 3D Surface Characterization via Gauss and Weingarten Maps and the Fundamental Forms 170
7.2.1.5 Peaks, Valleys, and Saddles of Surface Morphology and the Hessian 170
7.2.1.6 Smoothness as a Characterization of Surface Morphology and the Laplacian 171
7.2.1.7 Point Connections of 3D Surface Morphology and Christoffel Symbols 171
7.2.1.8 Protocol for Using Centric Diatom 3D Surface Models and Their Ensemble Surface Features in Valve Formation Analysis 173
7.3 Results 174
7.4 Discussion 184
7.4.1 Ensemble Surface Features and Physical Characteristics of Valve Morphogenesis 186
7.4.2 Factors Affecting Valve Formation 187
7.4.3 Diatom Growth Patterns--Buckling and Wave Fronts 187
7.4.4 Valve Formation, Ensemble Surface Features, and Self-Similarity 189
7.4.5 Diatom Morphogenesis: Cytoplasmic Inheritance and Phenotypic Plasticity 189
7.4.6 Phenotypic Variation and Ensemble Surface Features: Epistasis and Canalization 190
7.5 Conclusions 190 Acknowledgment 191 References 191 8 Buckling: A Geometric and Biophysical Multiscale Feature of Centric Diatom Valve Morphogenesis 195 Janice L. Pappas and Richard Gordon
8.1 Introduction 196
8.2 Purpose of Study 197
8.3 Background: Multiscale Diatom Morphogenesis 198
8.3.1 Valve Morphogenesis--Schemata of Schmid and Volcani and of Hildebrand, Lerch, and Shrestha 198
8.3.2 Valve Formation--An Overview at the Microscale 199
8.3.3 Valve Formation--An Overview at the Meso- and Microscale 200
8.3.4 Valve Formation--An Overview at the Meso- and Nanoscale 200
8.4 Biophysics of Diatom Valve Formation and Buckling 201
8.4.1 Buckling as a Multiscale Measure of Valve Formation 201
8.4.2 Valve Formation--Cytoplasmic Features and Buckling 202
8.4.3 Buckling: Microtubule Filaments and Bundles 203
8.4.4 Buckling: Actin Filament Ring 204
8.5 Geometrical and Biophysical Aspects of Buckling and Valve Formation 205
8.5.1 Buckling: Geometry of Valve Formation as a Multiscale Wave Front 205
8.5.2 Buckling: Valve Formation and Hamiltonian Biophysics 207
8.5.3 Buckling: Valve Formation and Deformation Gradients 208
8.5.4 Buckling: Multiscale Measurement With Respect to Valve Formation 210
8.5.5 Buckling: Krylov Methods and Association of Valve Surface Buckling With Microtubule and Actin Buckling 210