Mathematical Macroevolution in Diatom Research

Janice L. Pappas has BA, BS, PhD degrees from the University of Michigan and an MA degree from Drake University. She is a theoretical and mathematical biologist and her work includes studies on diatoms and other organisms in morphometrics, morphogenesis, biological symmetry and complexity, evolutionary processes, and evolutionary ecology. Mathematics used in studies includes stochastic and delay differential and partial differential equations, orthogonal polynomials, differential geometry, probability theory, optimization theory, group theory, machine learning, information theory, and ergodic theory. Some specific studies include Morse theory and morphospace dynamics; fuzzy measures in systematics; vector spaces in ecological analysis; combinatorics and dynamical systems in macroevolutionary processes.

• List of Figures xviiiList of Tables xxxPreface xxxvAcknowledgments xxxviiPrologue -- Introductory Remarks xxxixPart I: Morphological Measurement in Macroevolutionary Distribution Analysis 11 Diatom Bauplan, As Modified 2D Valve Face Shapes of a 3D Capped Cylinder and Valve Shape Distribution 31.1 Introduction 31.1.1 Analytical Valve Shape Geometry 51.1.2 Valve Shape Constructs of Diatom Genera 81.2 Methods: A Test of Recurrent Diatom Valve Shapes 101.2.1 Legendre Polynomials, Hypergeometric Distribution, and Probabilities of Valve Shapes 121.2.2 Multivariate Hypergeometric Distribution of Diatom Valve Shapes as Recurrent Forms 151.3 Results 181.4 Discussion 221.4.1 Valve Shape Probability Distribution 221.4.2 Hypergeometric Functions and Other Shape Outline Methods 221.4.3 Application: Valve Shape Changes and Diversity during the Cenozoic 251.4.4 Diatom Valve Shape Distribution: Other Potential Studies 251.5 Summary and Future Research 261.6 Appendix 261.7 References 342 Comparative Surface Analysis and Tracking Changes in Diatom Valve Face Morphology 392.1 Introduction 392.1.1 Image Matching of Surface Features 402.1.2 Image Matching: Diatoms 412.2 Purpose of this Study 422.3 Background on Image and Surface Geometry 422.3.1 The Geometry of the Digital Image and the Jacobian 422.3.2 The Geometry of the Diatom 3D Surface Model and the Jacobian 452.3.3 The Image Gradient and Jacobian 462.4 Image Matching Kinematics via the Jacobian 472.4.1 Position and Motion: The Kinematics of Image Matching 472.4.2 Displacement and Implicit Functions 482.4.3 Displacement and Motion: Position and Orientation 492.4.4 Surface Feature Matching via the Jacobian 502.4.5 The Jacobian of Whole Surface Matching 522.5 Methods 532.5.1 Fiducial Outcomes of Image Matching of Surface Features 532.6 Results 542.6.1 Surface Feature Image Matching and the Jacobian 562.6.2 Whole Valve Images, Matching of Crest Lines and the Jacobian 602.6.3 Image Matching of more than Two Images 652.7 Discussion 712.7.1 Utility of Jacobian-Based Methods and Image Matching 722.7.2 The Image Jacobian and Rotation in A Reference Frame: Potential Application to Diatom Images 732.7.3 Deformation and Registration of Image Surfaces: An Alternative Jacobian Calculation 752.8 Summary and Future Research 772.9 References 773 Diatom Valve Morphology, Surface Gradients and Natural Classification 813.1 Introduction 813.2 Purposes of this Study 823.2.1 The Genus Navicula 833.3 Methods 843.3.1 Naviculoid Diatom Surface Analysis 843.3.2 Gradients of Digital Image Surfaces 843.3.3 Histogram of Oriented Gradients and Surface Representation 893.3.4 Application to Diatom Valve Face Digital Images 903.3.5 Support Vector Regression and Classification 903.3.6 Using HOG as Combination Gradient Magnitude and Direction Input Data for SVR 913.3.7 Computational Efficiency and Cost 953.4 Diatom Valve Surface Morphological Analysis 953.4.1 SVR Model Fit of Naviculoid Taxa 953.4.2 Valve Surface Morphological Classification and Regression of Naviculoid Diatoms 963.5 Results 963.5.1 HOG Data Analysis 963.5.2 Goodness-of-Fit SVR Model Using 4D HOG Data from Naviculoid LMs 1003.5.3 SVR of Naviculoid 2D HOG Data 1023.5.4 Second Round of SVR Analysis of Remaining Naviculoid2D HOG Data 1053.5.5 Last Round of SVR Analysis of Remaining Naviculoid 2D HOG Data 1063.5.6 Classification Results from SVR Analysis of Naviculoid Taxa 1063.6 Discussion 1063.6.1 Characteristics of SVM and SVR 1073.6.2 Advantages in Using HOG Data and SVR 1083.6.3 Potential Utility of HOG Data and SVR in Diatom Research 1083.7 Summary and Future Research 1093.8 References 110Part II: Macroevolutionary Systems Analysis of Diatoms 1154 Probabilistic Diatom Adaptive Radiation in the Southern Ocean 1174.1 Introduction 1174.1.1 Diatoms in the SO 1204.1.2 Chaetocerotales and Bacillariales Speciation Rates 1204.1.3 Chaetocerotales and Bacillariales: Fe, NO3 and SiO2 Availability in the SO 1214.1.4 Modeling Diatom Adaptive Radiation 1224.2 Purposes of this Study 1234.3 Mathematical Modeling of Adaptive Radiation 1234.3.1 Quantitative Phenotypic Trait Measurement and Adaptive Radiation 1234.3.2 Adaptive Radiation: Implicit Stochastic Models 1254.3.3 Adaptive Radiation Models: Time Evolution of a Stochastic System 1274.3.4 Adaptive Radiation as an Optimal Control Problem 1284.3.5 Exit Probabilities as Boundaries for Completion of Adaptive Radiation 1304.3.6 Exit Times for the Adaptive Radiation Process 1314.3.7 Adaptive Radiation: A Study of Southern Ocean Diatoms and Niche Filling 1324.4 Methods 1334.4.1 Niche Filling and Adaptive Radiation 1344.5 Results 1344.5.1 Ecological Niche Preference, Photosynthesis Efficiency, Nutrient Enrichment or Limitation, and Adaptive Radiation 1404.5.2 Ecological Niche Preference and Photosynthesis Efficiency Rankings Representing Niche Filling as Adaptive Radiation 1414.5.3 Niche Filling and the Lyapunov modified OU Adaptive Radiation Model 1414.6 Discussion 1454.6.1 More on Specifications for Adaptive Radiation Modeling 1464.6.2 Diatom Adaptive Radiation Short-Term Trends as a Result of Niche Filling in the SO 1474.6.3 Other Potential Mathematical Modeling Regimes of Adaptive Radiation 1484.7 Summary and Future Directions 1494.7.1 Caveats in Adaptive Radiation Studies to be Considered 1494.8 References 1505 Cenozoic Diatom Origination and Extinction and Influences on Diversity 1595.1 Introduction 1595.1.1 Cenozoic Diatoms and Environmental Conditions 1605.1.2 Diatom Diversity during the Cenozoic 1615.1.3 Diversity as a Result of the Frequency of Origination and Extinction Events 1625.2 Purposes of this Study 1635.3 Methods and Background 1635.3.1 Reconstructed Diatom Origination, Extinction and Diversity during the Cenozoic 1635.3.2 Cumulative Functions and the Frequency of Cenozoic Origination nd Extinction 1655.3.3 Origination and Extinction: Heaviside Functions and Switching 1675.3.4 Origination and Extinction as a Sequence of Steps and Accumulated Switches during the Cenozoic 1685.3.5 Piecewise Continuous Switching via the Laplace Transform of the Heaviside Functions 1685.3.6 Overlapping of Origination and Extinction: A Convolution Product 1695.3.7 Non-Overlapping Origination and Extinction: A Poisson Process 1705.3.8 Test of Switch Reversibility, Cenozoic Events and a Lyapunov Function 1715.3.9 Origination and Extinction: Relation to Diversity 1725.4 Results 1735.4.1 Cumulative Frequency of Cenozoic Diatom Origination and Extinction Events 1735.4.2 Switching from Diatom Origination to Extinction over the Cenozoic 1765.4.3 Origination and Extinction Sequential Steps and Accumulated Switches during the Cenozoic 1815.4.4 Overlapping via a Convolution Product of Origination and Extinction 1835.4.5 Origination and Extinction as Poisson Processes 1845.4.6 Test of Origination and Extinction Switches: Stochastic or Deterministic Chaos? 1845.4.7 Diversity and Its Relation to Origination and Extinction for Cenozoic Diatoms 1855.5 Discussion 1895.5.1 Diversity and the Effects from Origination and Extinction of Cenozoic Diatoms 1895.5.2 Cenozoic Events and Diatom Diversity, Origination and Extinction 1925.5.3 Origination and Extinction Related to Diversity: Markov Chain, Martingale, Ergodic Processes, and Lyapunov Functions 1935.6 Summary and Future Research 1945.7 References 1946 Diatom Food Web Dynamics and Hydrodynamics Influences in the Arctic Ocean 1996.1 Introduction 1996.2 Purposes of this Study 2006.3 Background on Arctic Ocean Diatoms 2006.3.1 Diatoms and their Relation to Sea Ice 2006.3.2 Sea Ice, Upwelling and Diatom Productivity 2016.3.3 Diatom Lipid Content as a Proxy for Biomass 2036.3.4 Diatom Biomass and the Hydrodynamics of Upwelling 2036.4 Lattice Boltzmann Model 2056.5 Lattice Boltzmann Model and Hydrodynamics 2076.5.1 Upwelling and Buoyancy 2076.5.2 Collisions and Streaming Densities of Diatom Genera during Upwelling 2096.5.3 Buoyancy and Ice 2106.5.4 Upwelling and the Splitting of the Cylindrical Rotation of Currents 2116.6 Lattice Boltzmann Model: Diatom Bloom Density, Sea Ice and Upwelling 2116.7 Lattice Boltzmann Model: Specifications for Simulation 2136.7.1 Overview of 2D LBM with Respect to Diatom Genera Lattice Nodes 2166.7.2 Buoyancy, Upwelling and Diatom Blooms in LBM via p and u 2196.8 Methods 2206.9 Results 2206.10 Discussion 2266.10.1 Arctic Diatom Food Web Dynamics: Other Potential Outcomes 2296.10.2 Diatom Blooms: Influences over Time and Space 2306.11 Summary and Future Research 2316.12 References 231Part III: General and Special Functions in Diatom Macroevolutionary Spaces 2417 Diatom Clade Biogeography: Climate Influences, Phenotypic Integration and Endemism 2437.1 Introduction 2437.1.1 Biogeography and Climate 2457.1.2 Mapping Biogeographic Patterns 2467.1.3 Biogeography as an Optimization Problem 2467.1.4 Biogeographic Pattern and Spatial Rate of Change 2477.1.5 Biogeography, Phenotypic Integration and Phenotypic Novelty 2487.2 Purposes of this Study 2497.3 Methods 2507.3.1 Freshwater Diatom Dispersal Biogeography and the Traveling Salesman Problem 2517.3.2 Freshwater Diatom Biogeographic Patterns with Respect to Climate 2527.3.3 Diatom Biogeographic Patterns and Distance Decay 2527.3.4 Endemism and Continental Area 2537.3.5 Endemism and Dispersal Distance 2537.3.6 Endemism, Phenotypic Integration and Phenotypic Novelty: The Raphe 2547.4 Results 2547.4.1 Clade Shortest Tours from Continent to Continent with Respect to Climate 2577.4.2 Magnitude of Clade Tour Stops from Continent to Continent with Respect to Climate 2587.4.3 Ecological Similarity, Biogeographical Distribution and Distance Decay 2627.4.4 Biogeographical Distribution of Freshwater Diatom Genera 2627.4.5 Endemics in each Clade and on each Continent 2657.4.6 Phenotypic Integration and Relation to Geographic Distribution 2667.5 Discussion 2677.5.1 Freshwater Diatom Clade Dispersal and Climate 2687.5.2 Distance Decay as the Pattern of Dispersal in Freshwater Diatom Biogeography 2697.5.3 Phenotypic Integration and Diatom Biogeography 2697.6 Summary and Future Research 2707.7 References 2708 Cell Division Timing and Mode of the Diatom Life Cycle 2778.1 Introduction 2778.1.1 Evolution of Diatom Cell Division Dynamics 2788.1.2 Diatom Life Cycle as a Dynamical System 2788.1.3 Diatom Cell Division and Growth Rate 2798.1.4 Diatom Cell Size Diminution during Mitosis 2808.1.5 Diatom Cell Division during Meiosis 2818.1.6 Diatom Cell Division after Meiosis 2828.2 Purposes of this Study 2838.3 Background on the Diatom Cell Cycle 2838.3.1 Diatom Life Cycle Timing: Stages 2838.3.2 Diatom Life Cycle Timing: Switches 2848.3.3 Diatom Life Cycle Timing: Cell Behavior 2858.4 Modeling the Diatom Life Cycle: Timing of Stages and Switches 2868.4.1 Delay Differential Equations 2868.4.2 Solutions to DDEs 2868.4.3 Mackey-Glass System of DDEs 2878.4.4 Mackey-Glass System: Stage 1 of the Diatom Life Cycle 2888.4.5 Mackey-Glass System: Stage 2 of the Diatom Life Cycle 2898.4.6 Mackey-Glass System: Stages 3 and 4 of the Diatom Life Cycle 2898.4.7 Mackey-Glass System: The Diatom Life Cycle Switches 2908.5 Methods 2918.6 Results 2928.7 Discussion 3008.7.1 Cell Size Control and the Diatom Cell Cycle Structure 3038.7.2 Potential Alterations to the Mackey-Glass System when Applied to the Diatom Life Cycle 3048.7.3 Mackey-Glass Systems: Utility and Applications 3058.7.4 Potential Additional Analyses of Results from Mackey-Glass Systems 3058.8 Summary and Future Research 3058.9 References 3069 Diatom Morphospaces, Tree Spaces and Lineage Crown Groups 3139.1 Introduction 3139.1.1 Euclidean Spaces are Subspaces of Hilbert and Banach Spaces 3159.1.2 From Geometrical to Topological Spaces as Mathematical Morphospaces 3159.2 Occupied and Unoccupied Morphospace 3169.3 Purposes of this Study 3189.4 Morphospace Structure and Dynamics 3189.4.1 Morphospace Networks and All Possible Morphologies 3189.4.2 Networks, Hierarchy and Morphospace 3199.5 Phylogeny Structure and Phylogenetic Dynamics 3209.5.1 Phylogenetic Trees and Mapped Traits 3209.5.2 Phylogenetic Trees and the Geometry of Tree Spaces 3219.6 Measuring Occupied Morphospace: Clustering Coefficients 3229.7 A Brief Background on Diatom Morphospaces 3239.8 Mathematical Morphospaces in the Context of a Diatom Phylogeny 3249.9 Methods 3259.9.1 Input Data for Morphospace Analysis 3259.9.2 Diatom Lineage Crown Groups Embedded in a Metric Space 3269.9.3 Diatom Submorphospaces Embedded in a Metric Morphospace 3339.9.4 Clustering Coefficients as Measures of Occupied Morphospace 3349.10 Results 3349.11 Discussion 3369.11.1 Trees, Networks and Morphospaces 3459.11.2 Probabilistic Distances in Lineage Crown Group Morphospace 3459.11.3 Diatom Novelties Versus Repetitive Forms in Occupied Morphospace 3459.11.4 Diatom Teratologies and Mutagenicities: Influences on Morphology 3469.11.5 Understanding Diatom Evolution via Morphospace and Phylogenetic Analyses 3479.12 Summary and Future Research 3479.12.1 What is Morphological Data? 3479.12.2 Tempo and Mode of Phylomorphogenetic Spaces 3489.13 References 348Part IV: Macroevolutionary Characteristics of Diatoms 35510 Diatom Morphological Complexity Over Time as a Measurable Dynamical System 35710.1 Introduction 35710.1.1 Complexity and Evolution 36010.2 Diatom Morphological Complexity 36010.3 Purposes of this Study 36110.4 Characterizing Morphological Complexity 36110.5 Information and Morphology 36210.6 Information and Complexity 36310.7 Markov Chains and their Properties 36410.7.1 Markov Chains and Lyapunov Exponents 36610.8 Ergodicity and Chaoticity 36710.8.1 Entropy Rates 36810.9 Kolmogorov Complexity and Entropy 37010.10 Methods 37210.10.1 Transition Probability Matrix and Properties of a Markov Chain 37210.10.2 Measuring Morphological Kolmogorov Complexity 37510.10.3 Diatom Morphological Complexity over Geologic Time and Comparison of Cretaceous and Cenozoic Taxa 37510.11 Results 37610.12 Discussion 38510.13 Summary and Future Research 38910.13.1 Is Morphological Complexity Related to Morphological Symmetry? 39010.14 References 39011 Diatom Surface Symmetry, Symmetry Groups and Symmetry Breaking 39911.1 Introduction 39911.1.1 Geometry as a Basis for Form, Surfaces and Symmetry 39911.1.2 Inverse Functions as a Basis for Symmetry and Stability 40011.2 Symmetry of 3D Organismal Surfaces 40011.2.1 Shape versus Surface Symmetries 40111.2.2 Geometry of Non-Flat 3D Surfaces: Bidirectional Curvature and Its Relation to Twists and Writhes 40211.2.3 Knots: Geometry and Topology of Closed Curved Surfaces 40311.2.4 From Hyperbolic Geometry and Surfaces to Hyperbolic Knots 40411.2.5 Closed Helices, Hyperbolic Knots and Mobius Surfaces 40511.3 Symmetry Groups 40511.3.1 Diatom Surface Symmetry Groups: Cyclic, Reflective, Dihedral, Glide, Scale, and Knot 40611.3.2 States of Symmetry 40711.4 Purposes of this Study 40911.5 Methods 41011.5.1 Systems of Parametric 3D Equations for Exemplar Diatom Surface Models 41111.5.2 Symmetry Groups: Cyclic, Reflective, Dihedral, Glide, Scale, and Knot 41111.5.3 From Partial Derivatives to Ordinary Derivatives to Assess Stability 41111.5.4 Inverse Jacobian Eigenvalues and Surface Symmetry Analysis 41111.5.5 Stability and Inverse Jacobian Eigenvalues 41411.5.6 Diatom Surface Symmetries and Symmetry Group Assessment 41511.5.7 Vegetative Size Reduction and Symmetry Breaking 41611.5.8 Relative Stability and Symmetry 41611.5.9 Symmetry Gradients 41611.6 Results 41711.7 Discussion 42811.7.1 Diatom Surface Symmetries and the Intricacies of Assessment 42811.7.2 More on Diatom Surface Symmetries and Handedness 42911.7.3 Diatom Vegetative Reproduction and Symmetry Breaking 42911.7.4 Symmetry Breaking, Vegetative Reproduction, Size Reduction, and Stability 43011.7.5 Eigenvalues and Variance: Instability and Fluctuating Asymmetry 43011.7.6 Symmetry Groups and Evolutionary Dynamics: Symmetry in Diatoms and Adaptation 43111.8 Summary and Future Research 43111.9 References 43212 Evolvability of Diatoms as a Function of 3D Surface Phenotype 43712.1 Introduction 43712.1.1 3D Surface Properties -- An Overview 43812.1.2 From Differential Geometry to the Characterization of 3D Surfaces 43812.1.3 The Phenotype Characterized via a 3D Surface and Its Geometric Characteristics 44012.1.4 From Geometric Phenotype to Evolvability 44112.1.5 Evolvability and Phenotypic Novelty 44112.1.6 Evolvability and Diatoms 44212.1.7 Diatom Exemplar Phenotypic and Valve Plication Characteristics 44312.1.8 Diatom Architecture and the Geometric Phenotype 44412.2 Purposes of this Study 44512.3 Methods 44512.3.1 Phenotypic Inertia 44612.3.1.1 Measurement of Phenotypic Inertia: Christoffel Symbols 44612.3.2 Phenotypic Robustness 44712.3.2.1 Measurement of Phenotypic Robustness: the Hessian 44812.3.3 Phenotypic Stability 44912.3.3.1 Measurement of Phenotypic Stability: the Laplacian 44912.3.4 Evolvability of each Diatom Genus: Actinoptychus, Arachnoidiscus and Cyclotella 45012.3.5 Evolvability Among Diatom Genera 45212.3.6 Contribution of Phenotypic Inertia, Robustness and Stability to Evolvability 45212.3.7 Phenotypic Novelty Measurement 45412.3.8 Evolvability and Phenotypic Novelty 45512.4 Results 45512.4.1 Phenotypic Inertia, Robustness and Stability 46012.4.2 Evolvability across Genera 46312.4.3 Evolvability and the Fitness Function Components of Phenotypic Inertia, Robustness and Stability 46512.4.4 Phenotypic Novelty and Comparison to Evolvability 46512.5 Discussion 46712.6 Summary and Future Research 46912.7 References 470Epilogue -- Findings and the Future 475Index 479