Copyright by Louis George Zachos 2008 [PDF]

Copyright by Louis George Zachos 2008

The Dissertation Committee for Louis George Zachos certifies that this is the approved version of the following dissertation:

A New Theoretical Model for Growth of the Echinoid Test

Committee: ____________________________________ James Sprinkle, Supervisor

____________________________________ Timothy Rowe

____________________________________ Chris Bell

____________________________________ Ann Molineux

____________________________________ Rich Mooi

A New Theoretical Model for Growth of the Echinoid Test  

by Louis George Zachos, BSc; MSc; MSE Dissertation Presented to the Faculty of the Graduate School of The University of Texas at Austin In Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy The University of Texas at Austin December 2008

 

To my grandson Lucian A little urchin in his own right

 

Acknowledgments I wish to thank my supervisor Dr. James Sprinkle for his encouragement, support, inspiration, guidance, and patience throughout the period of this study. He was always available at any time for consultation but gave me a very long leash. I want to thank Dr. Ann Molineux for her support, confidence and continued advice. She was perhaps the main inspiration for me to return to academic studies. I would also like to thank Chris Bell and Tim Rowe, from the vertebrate side of the science, for all of their help and support. Through them I gained a clearer understanding of a part of paleontology that had always seemed foreign to me. Last, but certainly not least, I want to thank Dr. Rich Mooi for agreeing to serve on my committee and for being ever ready to answer (in great detail) any questioning email from me. I wish we had been closer geographically, but I never felt very far in our connection. I would like to thank all the members of the Jackson School faculty for making my time here both interesting and productive, especially Dr. Robert Loucks, who served on my examining committee; Dr. Charles Kerans for some most interesting experiences with carbonates; Dr. Kitty Milliken for trusting me not to hurt the SEM. Mr. Dennis Trombatore was always there when I was trying to find some obscure echinoid monograph from the 19th Century. There is little question but that my time at the Jackson School would have been considerably duller without Dr. Robert L. (“Luigi”) Folk, who I consider the “holotype” of a geologist.

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Dr. Tim Eakin, Division of Statistics and Scientific Computation, College of Natural Sciences, reviewed some of my math and helped me understand some of the inner workings of MATLAB®. Dr. Olaf Ellers, Bowdoin College, Brunswick, Maine reviewed the model described in this dissertation while it was still in its early stages and offered many helpful comments. Dr. Fred Hotchkiss, Marine and Paleobiological Research Institute, Vineyard Haven, Massachusetts, discussed aspects of the model and pointed out useful reference material. Dr. Jacob Dafni, Eilat, Israel, graciously sent me much of the raw data on which his studies of sea urchin growth were based. These data were very helpful in design of some aspects of the model. I especially thank Mr. Hank Stence, R.J. Peacock Canning Co., Lubec, Maine, who let me spend many enthralling hours at his facility watching sea urchin larvae metamorphose into baby sea urchins, and for all the specimens he gave me to work with. There is absolutely no question that none of this would have happened without the love of my wife Susan, her unswerving confidence in me, and her continuing support. There is really no way to even express just how crucial she was in getting me to begin, continue, and finish this whole experience.

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A New Theoretical Model for Growth of the Echinoid Test Publication No. _______________________

Louis George Zachos, Ph.D. The University of Texas at Austin, 2008

Supervisor: James Sprinkle

A new developmental model for growth of the echinoid test is based on a review of the growth patterns in regular echinoids. Echinoids are structurally composed of a tessellation of hundreds to thousands of individual plates. The two major aspects of echinoid growth are treated separately. (1) New plates are added in accordance with the Ocular Plate Rule and plate addition is hypothesized to be constitutively active but inhibited by a morphogen originating in coronal plates. Morphogen production is modeled as an inverse function of plate size and the concentration of inhibiting morphogen at a plate nucleation point is inversely proportional to the distance from surrounding plate centers. Plate addition is triggered whenever the inhibiting morphogen concentration falls below a threshold value. (2) The growth of individual plates is described using the Bertalanffy growth equation to model change in plate perimeter. vii

The geometric model is based on a spherical frame of reference, and all calculations of position and growth are modeled over the surface of the sphere (i.e., along geodesics). The data structure defined to maintain the geometric parameters is based on a spherical Delaunay triangulation of plate centers, and the edge geometry approximated by the dual Voronoi polygonalization. Echinoid plates are thus modeled as Voronoi polygons covering the sphere. Growth is modeled by the increasing radius of the sphere and the changing topology of the plates as new plates are added and existing plates grow. Final form of the complete test is generated by an affine deformation of the sphere. The growth model is implemented as the program EFORECHINOID, coded in the object-oriented programming language C++ with significant usage of the Standard Template Library (STL) for efficient coding and memory management. Most parameters are available to a user via a Graphical User Interface (GUI), and output of 3-dimensional simulations is via standard 3-D AutoCAD® Drawing Exchange Format (DXF) files. Program efficiency is O(nlogn) and reasonably parameterized growth simulations with several hundred time steps can be performed in a matter of minutes per run.

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Table of Contents  Acknowledgments ..............................................................................................................v Abstract ............................................................................................................................ vii List of Tables .................................................................................................................. xiii List of Figures ................................................................................................................. xiv Chapter 1.

Introduction ................................................................................................1

1.1 Background ..................................................................................................1 1.2 Chapter Summaries ....................................................................................11 1.3 Publication Plans........................................................................................12 1.4 Source Code ...............................................................................................14 Chapter 2.

Echinoids...................................................................................................15

2.1 Objectives ..................................................................................................15 2.2 Origin and Phylogeny ................................................................................15 2.3 Skeleton......................................................................................................21 2.3.1

Fabric and Mineralization ..............................................................21

2.3.2

Terminology ...................................................................................29

2.3.3

Symmetry .......................................................................................34

2.3.4

Plate Addition ................................................................................35

2.3.5

Homology ......................................................................................39

2.4 Life Cycle...................................................................................................43 2.4.1

Egg and Larva ................................................................................43

2.4.2

Metamorphosis and Imago Stage ...................................................44

2.4.3

Juvenile to Adult ............................................................................49

2.5 Growth .......................................................................................................49 2.5.1

Modes of Growth ...........................................................................49

2.5.2 Addition Order ...............................................................................51 ix

Chapter 3.

2.5.3

Plate Growth ..................................................................................54

2.5.4

Somatic Growth .............................................................................58

Models .......................................................................................................61

3.1 Objectives ..................................................................................................61 3.2 Theoretical Morphology ............................................................................61 3.3 Previously Applied Models........................................................................65 3.3.1 D’Arcy Thompson Model ..............................................................65 3.3.2

Deutler Model ................................................................................66

3.3.3

Raup Model ....................................................................................68

3.3.4

Moss and Meehan Model ...............................................................71

3.3.5

Seilacher Model .............................................................................73

3.3.6

Telford Model 1 .............................................................................73

3.3.7

Dafni Model ...................................................................................74

3.3.8

Ellers Model ...................................................................................76

3.3.9

Telford Model 2 .............................................................................79

3.3.10 Philippi and Nachtigall Model .......................................................82 3.4 New Developmental Model .......................................................................83

Chapter 4.

3.4.1

Plate Addition ................................................................................84

3.4.2

Plate Growth ..................................................................................92

3.4.3

Somatic Growth .............................................................................93

3.4.4

Synthesis ........................................................................................94

3.4.5

Geometric Considerations ..............................................................97

Model Implementation ..........................................................................100

4.1 Objectives ................................................................................................100 4.2 Descriptive Solid Geometry .....................................................................100 4.2.1

Spatial Reference and Coordinate System ...................................100

4.2.2

Delaunay Triangulation ...............................................................103

4.2.3

Data Model...................................................................................110

4.3 Processes ..................................................................................................112 4.3.1

Initialization .................................................................................112 x

4.3.2

Plate Addition ..............................................................................114

4.3.3

Growth .........................................................................................117

4.3.4

Deformation .................................................................................126

4.4 Parameters ................................................................................................129 Chapter 5.

Eforechinoid ...........................................................................................132

5.1 Objectives ................................................................................................132 5.2 Object Model ...........................................................................................132 5.2.1

GUI Form .....................................................................................136

5.2.2

Driver ...........................................................................................136

5.2.3

Point .............................................................................................136

5.2.4

Plate and Plate Type.....................................................................138

5.2.5

Ech_Skeleton ...............................................................................138

5.2.6

delaunayTriangle..........................................................................139

5.3 Process Flow ............................................................................................143 5.3.1

Time Steps ...................................................................................143

5.3.2

Plate Addition ..............................................................................147

5.3.3

Deformation .................................................................................147

5.3.4

Input Files ....................................................................................148

5.3.5 Output Files..................................................................................150 5.4 Performance .............................................................................................152 Chapter 6.

Results .....................................................................................................157

6.1 Objectives ................................................................................................157 6.2 Early Models ............................................................................................157 6.3 Growth .....................................................................................................157 6.4 Plate Addition ..........................................................................................163 6.5 Model Variations .....................................................................................164 6.5.1

Parameters ....................................................................................165

6.5.2

Equal Growth Rates .....................................................................166

6.5.3

Unequal Growth Rates .................................................................166

6.5.4

Variation of Threshold Value ......................................................169 xi

6.5.5

Variation of Maximum Plate Size................................................169

6.5.6

Variation of Plate Addition – All or Single .................................169

6.5.7

Variation of Initialization Configuration .....................................172

6.6 Conclusions ..............................................................................................172 References .......................................................................................................................177 Vita ..................................................................................................................................191

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List of Tables Table 2.1: Addition order of interambulacral plates ..........................................................41 Table 3.1: Models applied to growth or form of echinoids. ..............................................64 Table 6.1: Run-time parameters available to the user......................................................165

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List of Figures Figure 1.1: Ristorante il Riccio, Manciano, Tuscany, Italy .................................................2 Figure 1.2: General descriptive terminology .......................................................................4 Figure 2.1: Treatise Classification of echinoids ................................................................19 Figure 2.2: Major divisions of echinoid phylogeny ...........................................................22 Figure 2.3: Examples of stereom .......................................................................................24 Figure 2.4: Cross-plot of CaCO3 vs MgCO3 ......................................................................26 Figure 2.5: Conchoidal fracture surfaces in stereom .........................................................28 Figure 2.6: Descriptive terminology of plate edges and sutures ........................................32 Figure 2.7: Example of compound ambulacral plates .......................................................33 Figure 2.8: Symmetry axes applies to echinoids ...............................................................36 Figure 2.9: Close-up of apical system................................................................................40 Figure 2.10: The eight topologies of distinct cyclic permutations of alternating interambulacral plates in an echinoid ....................................................................42 Figure 2.11: SEM micrographs of the coronal plates of Strongylocentrotus droebachiensis less than 1 day post-metamorphosis .............................................48 Figure 2.12: SEM micrographs of the coronal plates 4-month old Strongylocentrotus droebachiensis .......................................................................................................50 Figure 2.13: Modes of growth of the echinoid corona.......................................................52 Figure 2.14: New interambulacral plates ...........................................................................55 Figure 3.1: Plate perimeter vs. cohort age .........................................................................67 Figure 3.2: Plate growth increment vs. distance from apical system.................................70 Figure 3.3: Model of echinoid plating ...............................................................................72 Figure 3.4: A family of curves ...........................................................................................77 Figure 3.5: Set of curves generated using numerical solution of Young-Laplace equation ................................................................................................................................80 Figure 3.6: C++ implementation of Ellers (1993) algorithm .............................................81 Figure 3.7: New interambulacral plate...............................................................................86 xiv

Figure 3.8: Diagram showing the mechanism hypothesized to control interambulacral plate addition..........................................................................................................90 Figure 3.9: Hypothetical developmental pathways ............................................................95 Figure 4.1: Spherical geometry ........................................................................................102 Figure 4.2: Relationship between Delaunay triangles and Voronoi polygons.................104 Figure 4.3: Triangulation of a sphere...............................................................................107 Figure 4.4: Insertion of a new center point into an existing Voronoi polygon ................109 Figure 4.5: Graphic representation of the Delaunay triangulation data model ................111 Figure 4.6: Plate configuration in imago of Psammechinus miliaris...............................113 Figure 4.7: Simplified representation of morphogen inhibition of plate nucleation........116 Figure 4.8: Decomposition of a polygon into spherical triangles ....................................118 Figure 4.9: Dimensions of a spherical cap .......................................................................121 Figure 4.10: Growth algorithm ........................................................................................127 Figure 4.11: Model of plate growth in Echinus esculentus .............................................128 Figure 4.12: Examples of deformation of a spherical model ...........................................130 Figure 5.1: Class/Object diagram for EFORECHINOID ................................................134 Figure 5.2: Schematic object diagram for EFORECHINOID .........................................135 Figure 5.3: Graphical User Interface for EFORECHINOID ...........................................137 Figure 5.4: Data structure for Delaunay triangulation .....................................................141 Figure 5.5: The Right-Hand Rule for vector orientation .................................................142 Figure 5.6: Generalized process flow for EFORECHINOID ..........................................144 Figure 5.7: Set of Bertalanffy growth curves...................................................................146 Figure 5.8: Minimal (4-point) initialization file for EFORECHINOID ..........................149 Figure 5.9: Examples of output of 3-D graphics files......................................................151 Figure 5.10: Performance measures .................................................................................155 Figure 6.1: Early simulation from EFORECHINOID .....................................................158 Figure 6.2: Family of curves showing variation in plate growth rate ..............................160 Figure 6.3: Growth of text radius with age ......................................................................161 Figure 6.4: Growth of interambulacral plate perimeters with age ...................................162 xv

Figure 6.5: A sequence of images showing typical results of using equal plate growth rates and plate addition threshold values .............................................................167 Figure 6.6: A sequence of images showing the effects of high plate addition rate coupled with low plate growth rate ...................................................................................168 Figure 6.7: Examples comparing results of different plate addition threshold values with constant plate growth rate ....................................................................................170 Figure 6.8: Example showing results of using different plate addition threshold values ..............................................................................................................................171 Figure 6.9: Apical views of two initialization configurations .........................................173 Figure 6.10: Initialization configuration with closely-set plates that separate the oculargenital plate junction from the interambulacral column ......................................174

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Chapter 1: Introduction 1.1 Background The Echinoidea is a class of the phylum Echinodermata in the branch of the animal kingdom Deuterostomia, which also includes the crown groups Hemichordata and Chordata (Valentine, 2004). Common names for members of the class include sea urchins, sand dollars, sea biscuits, and heart urchins. According to Bather et al. (1900) the name echinoid originated with the Greek word Εχ ν̃ ος, meaning hedgehog and referring to their spiny character. Aristotle used the name Echinus for the sea urchin, a name applied by Linnaeus and still retained today for one of the more common genera of echinoids. The Middle English urchon is the origin of our modern word urchin. It was derived from the Old French irechon or herichon, meaning hedgehog, although the modern French word for urchin is oursin (and hérisson for the actual hedgehog), originally from the Latin ericius (meaning, of course, hedgehog). The Latin is more closely matched by the Italian word riccio, meaning both hedgehog and urchin, the latter usually distinguished as riccio di mare (no resemblance to the English, but a classy name for a restaurant; Figure 1.1). The word urchin (often preceded by street) “has come to mean a somewhat rambunctious child, usually in a run-down neighborhood, often with a dirty face and dressed in soiled clothes, who likes to sass passing grown-ups” and indicative of the generally “prickly” behavior of such children (Safire, 2006, p. 40). This meaning has been attributed to its original reference to a fairy or elf which took the form of a hedgehog and played pranks on children (Harvey, 1956) confirming what any child

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Figure 1.1. Ristorante il Riccio, Manciano, Tuscany, Italy.

2   

will tell you – that they get blamed for everything. The Germans use the word Igel to refer to the hedgehog, and Seeigel for, literally, sea hedgehogs, which confirms the close etymological relationship of echinoderms to chordates. Echinoids are characterized by the test, in reality a plated internal skeleton, covered almost everywhere in epithelium. The test consists of individual plates, arranged in a pentaradial fashion, which can number from several hundred to several thousand for an individual adult echinoid. The plates fit together in a mosaic to create a more or less rigid framework (Ellers et al., 1998), although in many Paleozoic and some modern forms the plates are imbricate and the test flexible (Jackson, 1912). The skeleton can be divided into the three structurally separate parts (Figure 1.2a) called the corona, which makes up the major portion of the echinoid skeleton; the apical system of plates, located at the apex of the test and which usually consists of four or five genital plates and five ocular plates; and the peristomal and periproctal plates, which cover the peristomal and periproctal membranes associated with the mouth and anus. Classically, the corona is further divided into five ambulacral and interambulacral areas (Figure 1.2b), each consisting (in extant taxa) of two columns of plates (Hyman, 1955). The echinoids can be subdivided into two broad groups based on whether the periproct lies within the circlet of genital and ocular plates of the apical system (endocyclic) or outside the apical system (exocyclic), which is operationally, but not quite correctly, indicated by whether the body has a radial (regular) or bilateral (irregular) symmetry. This division into regular or irregular is not quite this simple (Saucéde et al., 2007) and has limited value in 3   

Apical System

Anus

a

Mouth Corona

Periproctal Plates

Genital Plates

Peristomal Plates

Ocular Plate II I

b Interambulacra are indicated by Arabic numerals 1-5

3

2

Madreporite

I V

II

4

1 V

Lovén’s Axis

I 5

Ambulacra are indicated by Roman numerals I-V

Figure 1.2. General descriptive terminology for (a) the major regions and (b) symmetry of the echinoid skeleton.

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classification, but serves to distinguish the relatively simple geometry of the regular form. Although many of the terms and concepts used in this dissertation are applicable to either regular or irregular echinoids, unless otherwise stated only regular echinoids are considered. The individual plates themselves are complex structures constructed of a 3dimensional meshwork of calcite, termed stereom, filled (in the living form) with mesodermal tissue, termed stroma. The stereom has a variety of textures (Smith, 1980) and crystal orientations (Raup, 1965; 1966) related to the position and function of the plate. An initial set of plates is formed just before metamorphosis of the echinoid larva, consisting of the primordial genital and ocular plates, and usually five 4-plate lozenges of interambulacral plates and, at least sometimes, ten ambulacral plates. The plates of the apical system are relatively large initially, but, in most cases, become relatively smaller than the plates of the corona over time. Coronal plates are added at the adoral edges of the ocular (or terminal) plates, being displaced farther from the apical system as the echinoid grows and additional plates are added. The coronal plates may be partly resorbed, may separate from the coronal skeleton and move out onto the peristomal membrane, or may become fixed in position. Except for pores developed in the plates for tube feet, genital papillae, and the water vascular system, and openings for the mouth and anus, the skeleton is completely encompassing. The body shape of an echinoid is essentially the composite of the shapes of its individual plates.

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Discrete growth of the individual plates is distinguished from the distributed growth of the overall test. The final shape is realized only by the growth of the collection of individual plates. Each plate in the echinoid corona grows by the addition of new calcite around the perimeter (Deutler, 1926) and this has been repeatedly confirmed by the use of radioisotopes (Dafni, 1984), tetracycline (Kobayashi and Taki, 1969), and other fluorescent markers to stain plates in growing echinoids (see Russell and Urbaniak, 2004, for a review). The earliest plates in the corona form in the larval imago or rudiment immediately before metamorphosis (Gordon, 1926a; 1926b; 1929), but after that all additional plates are inserted into the corona at the edges of the ocular plates in accordance with the Ocular Plate Rule (Mooi et al., 1994). The description of growth is complicated by processes of plate compounding (Melville and Durham, 1966), resorption (Märkel, 1981), translocation (McNamara, 1987), and negative growth (Ebert, 1967). At the cellular level, addition and resorption of stereom is made by specialized cells (Shimizu and Yamada, 1980). Growth regulation is a complex, and largely unknown, process, although some insight is available from studies of aberrancies such as nonpentamerous forms (Jackson, 1927), repair to damaged areas of the test (Ameye and Dubois, 1995) and abnormal growth patterns caused by chemical pollution (Dafni, 1980, 1983). D’Arcy Thompson, in his classic work On Growth and Form (1917, 1942), discussed the shape of a generic sea urchin and compared it to the shape of a drop of liquid, i.e., a membrane in tension. That concept has been the basis of a number of 6   

attempts to model the shape of a sea urchin from a mechanical aspect. Advances in the description of form in echinoids have been almost exclusively in the realm of test shape (Ellers, 1993; Johnson et al., 2002; Telford, 1985) and generalized constructional morphology (Seilacher, 1979) and functional morphology (Philippi and Nachtigall, 1996). Studies of echinoid growth have been oriented towards whole-body (somatic) growth (Ebert, 1975; 1982; Lamare and Mladenov, 2000; Rogers-Bennett et al., 2003), although sometimes with reference to individual plate growth (Duineveld and Jenness, 1984). Raup (1968) was the first to apply a computer model to simulate echinoid discrete growth, using essentially a 1-dimensional model, but little progress has been made in that methodology since that time. Theoretical morphology is a broadly-defined term. McGhee (1999) described it as the simulation of form using a minimum number of geometric parameters, or with the simulation of the morphogenetic processes that produced a form. The word simulation can be replaced by the term model, and theoretical morphology becomes the modeling of biological form or process. Biological models can be divided into two categories, descriptive and heuristic (Ransom, 1981). Descriptive models describe growth or form, usually by application of mathematical formulas to analytically derive shape or changes in shape or morphology. Heuristic models are used to model the underlying processes that control growth and form. As in any division of a continuous series, modeling strategies may not fall strictly into one category or the other. Additionally, each strategy

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can be implemented using static techniques (paper & pencil), substitute systems, purely mathematical approaches, or computer simulation. The goal of many descriptive models is definition of a morphospace, or a multidimensional presentation of the variables that define the possible form, shape or structure of an organism or group of organisms. The dimensions can be real (length, width, etc.) or combinations of parameters manipulated as principal components. The epitome of descriptive modeling is morphometrics, described by Bookstein (1991) as the study of covariances of biological form. Descriptive models share the feature of ultimate derivation from measurements of form and shape. There are many cases in which a descriptive model is used in an attempt to explain the processes of growth and development (Roth and Mercer, 2000). A biometrical approach was applied by David and Laurin (1996) in an investigation of evolutionary processes in the echinoid Echinocardium. Problems with such applications to cladistic studies were pointed out by Rohlf (1998), although his conclusions were contested by Zelditch et al. (2001). Morphometric methods seem to have greatest applicability in studies of morphologic disparity and diversity (Villier and Eble, 2004). The approach I take falls into the second definition of modeling, the heuristic model. The methodology for graphic simulation (computational) of echinoid growth based on morphogenesis and growth of individual plates was outlined by Telford (1994). Comparable models (modeling form as well as growth) were developed for foraminifera (Tyszka and Topa, 2005), sponges (Kaandorp, 1991), corals (Kruszyński et al., 2007), 8   

and bivalves (Ubukata, 2005). A model of echinoid growth is inherently more complex than that of these other invertebrates because of the dual processes of plate addition and individual plate growth. A model for echinoids therefore must incorporate individual models for both processes, yet the two models must also be linked. A simple measure of complexity is the number of parameters, or degrees of freedom, required by the model. By this measure as well, a model of echinoid growth based on the growth of individual plates is complex. The underlying theme of this approach is that the degrees of freedom in the model parallel degrees of freedom in actual development. The mathematical constructs in the model are the heuristics or rules that represent processes in development. Processes such as diffusion, threshold inhibition, and accretive growth all have rates that are characterized by specific mathematical functions, and these functions can be expressed dynamically by changing geometric relationships among the individual plates making up the skeleton of an echinoid. A model in and of itself is of little value beyond the development of techniques. The true value of a model lies in its ability to answer, or at least attempt to answer, important questions regarding growth and development. These questions can obviously not be answered directly, therefore a model serves to test hypotheses regarding growth and development that have been generated from experiments or observations. The basis for and implementation of a new computational model for post-larval skeletal growth of echinoids is described in the following chapters. Ransom (1981) noted that although genes control the synthesis of biochemicals, it is the interaction of cells that 9   

determines the shape and pattern, the phenotype, of an organism. The new computational model will treat individual plates of the skeleton as cells (which they are) and test hypotheses regarding the individual growth trajectories of the plates, mechanisms for addition of new plates, and explanations of the patterns seen in the skeletons of echinoids. The model does not simulate any specific genus or species of echinoid, but does simulate many of the patterns of growth seen in these animals. Among the conclusions reached are hypotheses regarding: -

regulation of individual plate size and the distribution of relative plate sizes in the skeleton as a whole

-

regulation of the addition of new plates into the skeleton

-

at what level particular plates or regions can be considered homologous

-

controls on the growth patterns in separate regions of the skeleton

-

developmental polarity in patterns of plate growth and addition

-

separation of mechanisms controlling plate growth and overall (somatic) shape and form

The computational model also offers something not available via other modeling strategies – complete 3-dimensional visualization of the processes of growth and change of form. Perhaps the most convincing test of a hypothesis is the ability to construct a visual representation that can be examined from any angle for comparison to the real thing.

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1.2 Chapter Summaries Chapter 2 This chapter is an overview of the observational knowledge of echinoid growth, divided into four parts. These are theories on the origin and phylogeny of echinoids, composition and structural characteristics of the skeleton, life cycle from egg to adult, and component processes of growth. Chapter 3 This chapter is an overview of the basic concepts and background of theoretical morphology, divided into three parts. These are a general discussion of theoretical morphology as an approach to understanding growth, detailed descriptions of models that have previously been applied to the growth of echinoids, and descriptions of developmental models of growth. Chapter 4 This chapter is a detailed description of the implementation of the basic developmental growth model described in the previous chapter. It includes a description of, and the reasons for, the chosen geometric framework, justification for specific algorithms used for plate addition and growth, and discussion of the parameter space used in the actual model.

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Chapter 5 This chapter describes the structure and operation of the program that actually executes the echinoid growth model. Chapter 6 This chapter describes the implementation of the computational growth model, the parametric domain, and discussion of a series of incremental results obtained from the model. 1.3 Publication Plans The chapters of this dissertation are arranged to present a coherent description of the basis, development, and implementation of a complex model of growth in echinoids. I expect to complete three separate publications based on the information presented in this dissertation. The first paper will be submitted to the Journal of Experimental Marine Biology and Ecology. This journal, which first appeared in 1967, accepts manuscripts with an emphasis on experimental work in marine plants and animals. This first paper will deal with the mechanisms of plate addition into the echinoid corona. During the course of this study, I collected data regarding the patterns of plate addition observed in the green sea urchin, Strongylocentrotus droebachiensis. These data include tabulation of the order of addition of new plates in several hundred specimens ranging from juvenile through adult.

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In addition, I used Scanning Electron Microscopy (SEM) and Computed Tomography (CT) techniques to image early stage plates (just following addition) and specimens in early post-metamorphosis. I used this information was to develop the plate addition model used in this study. This first paper will develop the rationale for a morphogen/threshold mechanism, apply the results to models of plate addition, and compare model results with the patterns shown by the empirical data. This paper will incorporate those parts of this dissertation that deal with plate addition and will emphasize the observational results, including a significant portion of Chapter 3 (primarily sections 3.4.1 and 3.4.4), Chapter 4 (primarily section 4.3.2), and results of model runs applicable to plate addition. It will not include a detailed description of the EFORECHINOID modeling software or much of the mathematical basis (i.e., Delaunay triangulation). The second paper will be submitted to the Journal of Theoretical Biology. This journal, first published in 1961, is self-described as “the leading forum for papers that give insight into biological processes.” The journal is strongly oriented towards detailed mathematical treatments and descriptions of model development. This paper will incorporate the detailed theoretical and mathematical basis of the EFORECHINOID model, both the growth and plate insertion methodologies, a description of the modeling software, and representative modeling results. Unlike the first paper, it will not include extensive observational data on echinoid growth per se, but will strongly emphasize the

13   

theoretical basis for the model. It will incorporate much of the technical aspects of Chapters 3, 4, and 5. The third paper will be submitted to Paleobiology. This journal, first published in 1975, emphasizes the biological aspects of paleontology. This paper will discuss the general aspects of the developmental model that underlies the implementation; with a more generalized mathematical treatment than used in the first two papers. It will emphasize the results of simulations, and in this sense will be an extension to the results described in Chapter 6. 1.4 Source Code The latest version of the C++ source code for the EFORECHINOID modeling software can be obtained directly from the author. The author can be contacted via the permanent email address: [email protected].

14   

Chapter 2: Echinoids 2.1 Objectives This chapter is an overview of the observational knowledge of echinoid growth. It includes a discussion of theories on the origin and phylogeny of echinoids; and descriptions of the composition and structural characteristics of the skeleton; life cycle from egg to adult; and component processes of growth. 2.2 Origin and Phylogeny The origin of the phylum Echinodermata is as fuzzy as the origin of all the skeletonized phyla. The presence of radial, indeterminate cleavage, deuterostome gastrula, and enterocoelous development in the earliest embryonic stages are characters shared only by the phyla Echinodermata, Chordata, Hemichordata, and the enigmatic Xenoturbellida (Bourlat et al., 2006). No echinoid fossils are known before the Late Ordovician, after the first appearance of most other major echinoderm groups. Although it seems apparent that echinoids diverged from a more basal echinoderm ancestor, it is anything but apparent what that ancestor was. Originally, it was thought that echinoids descended from crinoids, and Lovén (1874, p. 80) presented a figure purporting to show the transformation of the calyx of the crinoid Poteriocrinus geometricus into the apical system of a series of echinoids, from Palaeechinus elegans to Cidaris papillata and on to Encope valienciennesi. The echinoid skeleton was described as composed of three parts: the perisomatic or interradial, the ambulacral, and the 15   

calycinal or apical (Lovén, 1883). The interradial (i.e., interambulacral) plates in echinoids were considered to be homologous to the interradial plates of cystoids. The system of apical plates was considered to be homologous to the calyx of crinoids (thus the origin of his term “calycinal” for this system of plates). The peristomal plates were considered homologous to the crinoid centrobasal (using Cyathocrinus for reference), the genitals homologous to the basals, and the oculars (his “ocellars”) to the radials. Lovén (1883) pointed out that the apical system, like the calyx of crinoids, is opposite the mouth. Carpenter (1878; 1879; 1880; 1882) also felt that the plates of the crinoid calyx and the echinoid apical system were homologous in the same manner, basing that conclusion on embryological information (origin of the genitals and basals around the right enterocoel). The remarkable echinoid Tiarechinus princeps was used as an example of the homology of the echinoid apical system to the crinoid calyx (Lovén, 1883). This species, from the Late Triassic St. Cassian beds of Italy, is of uncertain affinity (Kier, 1977). The characteristics of the apical plates of the Saleniidae were compared by Lovén to those found in Palaeocrinoidea and cystoids, and his diagrams of this (Lovén, 1883, p. 66) are striking. Reasons supporting the idea that the crinoids were the ancestors to echinoids were detailed by Clark (1909). Many of these reasons were rejected by Jackson (1912), who suggested instead that the ancestor was more likely a cystoid. The genus Bothriocidaris was first described by Eichwald (1859-60), but was unknown (fide Hawkins, 1931) to both Agassiz (1872) and Lovén (1874) and remained ignored in echinoid literature until Jackson (1896) placed it at the root of his new

16   

classification as the simplest echinoid. The view that the primitive echinoid condition is a single interambulacral column and two ambulacral columns in each of the five radial zones was supported by Jackson (1912) with an expanded discussion of Bothriocidaris. The fact that all modern echinoid juveniles have a single primordial interambulacral plate at the peristome was used by Jackson (1912) as further evidence that this was the primitive condition. This point was also stressed by Lovén (1892), although without mentioning Bothriocidaris. Mortensen (1928) denied that Bothriocidaris was an echinoid at all; rather that it was a cystoid and completely unconnected with echinoid phylogeny. Instead, an edrioasteroid (specifically the Cambrian Stromatocystites) was posited to be an ancestor to the echinoids. The multicolumn interambulacrum was in this case the primitive condition, the exact opposite of Jackson’s presumption. Even stronger support for Bothriocidaris as the ancestral form was later argued by Clark (1932), Hawkins (1929; 1931) and Jackson (1929). In a detailed redescription of the genus, Mortensen (1935) reconsidered his position by removing the genus from the cystoids and placing it in a separate subclass, Pseudoechinoidea, from which there were no descendants. He later reiterated this view (Mortensen, 1951). Data on some 31 new, well-preserved Bothriocidaris specimens was published (in Russian) by Männil (1962). These data showed that Bothriocidaris possessed a definite jaw apparatus, articulated spines, internal radial water vessel, and followed the Lovén formula for ordering of plates around the peristome. These unequivocally supported the placement of this genus in the true echinoids, albeit on a monophyletic branch that did not include any other known echinoids. This was the same conclusion reached by Durham and Melville 17   

(1957). While Bothriocidaris was not the ancestor of echinoids, it was a true echinoid. Bothriocidaris was distinguished from the other Late Ordovician echinoids by Durham (1966). These included the species described by MacBride and Spencer (1938): Aulechinus, Ectinechinus, and Eothuria (which they described as a plated holothurians but Durham regarded as an echinoid). This was evidence for an origin for the class Echinoidea prior to the Late Ordovician, casting doubt on the hypothesis that the edrioasteroid Stromatocystites was ancestral to echinoids, and suggesting instead a closer relationship to the Helicoplacoidea, described a few years earlier (Durham and Caster, 1963). While the origins of the Echinoidea are still unclear, the phylogeny within the group is more amenable to cladistic analysis and better understood. The publication of the Treatise on Invertebrate Paleontology Part U (Echinodermata3) in 1966 was a benchmark in our understanding of the phylogeny of the Echinoidea. It was a turning point between earlier phenetic classifications and the modern phylogenetic view. The history of echinoid classification was reviewed by Durham (1966), who distinguished the Treatise Classification from earlier classifications, while noting that it was similar in many respects to that of Durham and Melville (1957). The class was divided into two subclasses: Perischoechinoidea and Euechinoidea. In that scheme, Perischoechinoidea (now recognized as a paraphyletic group) included all the Paleozoic orders, only one of which, Cidaroida, survived the Permian/Triassic extinction event. Euechinoidea included all the remaining modern forms, although the ancestral lines of the Euechinoidea were projected into the Carboniferous (Durham, 1966). The major divisions of this grouping are shown in Figure 2.1.

18   

Cidaroida

Irregularia

Paleogene

Echinacea

Neogene

Cretaceous Mesozoic

Euechinoidea

Diadematacea

Cenozoic

Perischoechinoidea

Jurassic

Triassic

Silurian

Echinocystitoida

Devonian

Bothriocidaroida

Paleozoic

Carboniferous

Palaechinoida

Permian

Ordovician

Cambrian

Figure 2.1 Treatise Classification of echinoids, after Durham (1966). 19   

The Perischoechinoidea are characterized by an absence of external gills, corona rigid or flexible, with ambulacra composed of two or more columns, interambulacra or one or more columns, endocyclic periproct, simple ambulacral plates, lack of auricles in perignathic girdle, presence of lantern with grooved teeth (Durham, 1966). The Euechinoidea are characterized by presence of external gills, a usually rigid test, bicolumnar ambulacra and interambulacra, auricles in the perignathic girdle (or descent from such forms), lantern with grooved or keeled teeth (or descent from such forms), apical system with genital and ocular plates, sphaeridia, and ophicephalous pedicellaria (Durham, 1966). The Cidaroida express characters found in both subclasses, and represent the least derived living echinoids. The orientation of crystallographic axes of the calcite composing plates is phylogentically informative (Smith, 1990). Coronal plate c-axis orientation (whether perpendicular to the surface of the all the plates, perpendicular to the surface in the ambulacra and tangential in the interambulacral, or tangential to the surface of all the plates) was correlated by Smith (1990) to phylogeny of camarodont and cidaroid urchins. The correlation was shown to be in agreement with the phylogenies. All Paleozoic echinoid orders have the c-axis perpendicular to the surface of both ambulacral and interambulacral plates (Raup, 1962b), which is therefore plesiomorphic for all modern orders; the tangential conditions being derived. A new subclass, Cidaroidea, was diagnosed by Smith (1984), who retained ‘Perischoechinoidea’ as a useful term for most of the Paleozoic forms. The name Perischoechinoidea was later redefined on a more strictly cladistic basis in reference to the Paleozoic stem group with Cidaroidea either an advanced stem group or primitive Euechinoidea stem group (Smith, 2005). All other modern echinoids were placed into the 20   

crown group Euechinoidea. The usage of Perischoechinoidea is dropped in Smith’s (2005) classification and the subclass Cidaroidea restricted to the mostly post-Triassic members of the order as classically defined. In that classification, the Cidaroidea and Euechinoidea are monophyletic crown groups of Echinoidea, all other ‘perischoechinoids’ and cidaroid-like echinoids are considered to be part of the paraphyletic stem group. The basis of a distinctly developmental definition of echinoid phylogeny and classification was presented by Smith (2005). The Paleozoic stem group is characterized by a variable test structure but uniform shape, with a growth mode dominated by plate addition. The later stem group is characterized by increasing importance of plate enlargement by accretion throughout life. The Cidaroidea and Euechinoidea are characterized by a uniform test structure and variable shape, with plate accretion dominating over plate addition during growth. In the Euechinoidea the timing of plate addition shifts to early ontogeny, such that nearly all adult growth is attributed to plate accretion. This pattern of relationships is shown in Figure 2.2. The hypothesis of diagnosing the phylogeny by broad characteristics of plate addition and growth is exactly the type of question that can be tested by a computational model of those characters. 2.3 Skeleton 2.3.1 Fabric and Mineralization Individual echinoid plates are complex structures constructed of a 3-dimensional meshwork of high-Mg calcite (Smith, 1984), termed stereom, filled (in the living form) with mesodermal tissue, termed stroma. As noted earlier, stereom microstructure is both unique to 21   

Cidaroids

Stirodonts

Camarodonts

Irregulars Plate formation shifts to early ontogeny

Euechinoids Invariant number of plate columns

Plates enlarged and remolded by accretionary growth throughout ontogeny

Pa le

Perischoechinoids

oz Gr oic ou Ste p m

Va r

io us ta e x t xa i n c

t

Archaeocidaroids

Growth by plate and plate column addition

Figure 2.2. Major divisions of echinoid phylogeny (after Smith, 2005).

22   

and ubiquitous in the Echinodermata, and is an important apomorphy of the phylum. , recognized two types of stereom were recognized by Roux (1970; 1975), based on study of crinoids. These two types were termed α and β. The α-stereom form represented a regular, generally rectilinear meshwork, whereas the β-stereom form was irregular (Figure 2.3). These types were redefined somewhat anecdotally (Seilacher, 1979; 1991) in terms of biomechanical characteristics or stress-related stereom. The regular, α-stereom was considered tensional stress conducting, whereas the irregular, β-stereom compressional stress conducting (“mechanical pillow”). Four categories of stereom, rectilinear galleried, galleried, labyrinthic, and massive, were defined by Macurda and Meyer (1975) from crinoids. Further division of stereom into ten separate categories based on finer distinctions of morphology was proposed by Smith (1980). The general term for stereom-like structures is space labyrinth. A space labyrinth is a structure in space composed of a continuous surface (called a manifold) which divides the space into two parts: the region inside the surface and the region outside the surface. In echinoids, the stereom is the region inside the manifold. The morphologies of space labyrinths are described in a much more rigorous manner from an architectural viewpoint, which divides them into regular, semi-regular, and non-periodic categories (Lalvani, 1997). Architects, understandably, have much more practical concerns with the mechanics of labyrinth structures than do biologists, and thus their terminology is more meaningful when to considering the biomechanical aspects of stereom. However, it is well beyond the scope of this study to consider the plate microstructure in such detail.

23   

24   

Soft tissue (in particular, collagen, ligament, and muscle fibers) plays an important role in the final form of the fabric (Roux, 1975; Smith, 1980; 1990). It has been hypothesized that collagen secretes an inhibitor to calcite growth Nissen (1969), but exactly how tissues in the stroma interact with skeletogenesis of the crystalline stereom is unknown. A consequence of the stereom microstructure is that plate growth is linear at the scale of this study, i.e., growth consists of extension of individual trabecules around the edge of a plate. This will be an important consideration in the mathematical description of growth. Stereom is composed of the high-Mg isomorph of the mineral calcite with the variable chemical formula (Ca,Mg)CO3. Concentrations measured using X-ray diffraction techniques (Weber, 1969) ranged from 5.5 to 16.2 wt. % (6.5 to 18.7 mole %) MgCO3 for echinoid skeletal components. Systematic variations in composition were described within individual skeletal components, among different components of a single animal, among individuals of a population, and different genera from the same marine community (Weber, 1969). A cross-plot of CaCO3 and MgCO3 mole% data obtained during this study from electron microprobe analysis of four modern echinoids (Figure 2.4) shows that modern unaltered echinoid calcite is composed almost completely of these two components (with only trace amounts of other compounds, primarily SrCO3). All points plot nearly on the line represented by CaCO3 + MgCO3 = 100%. The concentration of Mg2+ varies between these four species and, as pointed out by Weber (1973), may also be sensitive to the temperature of the water in which the echinoid lives. The Mg2+ content of the calcite alters the physical

25   

10

Encope michelini Echinarachnius parma

Mole % MgCO3

Moira atropos Maretia planulata

5

0 95

90

Mole % CaCO3

Figure 2.4. Cross-plot of CaCO3 vs MgCO3 for four modern echinoids (after Zachos, 2008, fig. 5A).

26   

100

properties of the crystal to some extent. The highest percentage of Mg2+ is measured in the teeth (Märkel et al., 1977), which are the densest and hardest skeletal element in those echinoids that have a dental apparatus (Aristotle’s lantern). Fracture surfaces of stereom trabecules are characteristically conchoidal (Figure 2.5) rather than the perfect planar cleavage of pure calcite. This was attributed by Nissen (1969) to a growth-related layered structure within the crystal, but it could indicate that the Mg atoms may be distributed in a manner that imparts additional strength to the stereom. The trabeculae of asteroid ossicles are composed of long, curved, needle-like crystallites arranged in spiral lamellae (O’Neill, 1981). Similar lamellae were found in trabeculae of an echinoid plate. It was suggested that the curvature of the crystallites was important to the strength of the trabeculae, and because pure calcite cannot form smoothly curved surfaces, high Mg2+ (with a smaller ionic radius than Ca2+) is a necessary component of echinoderm calcite. Scanning transmission electron microscopy (STEM) microanalysis results from Blake et al. (1984) indicated homogeneous distribution of Mg2+ and they dismissed O’Neill’s conjecture as untenable. Smith (1990) agreed with that conclusion while reporting that Blake et al. (1984) had shown that Mg2+ was uniformly distributed in stereom down to a 20 nm level. However, Blake et al. (1984) noted difficulties with electron beam damage of calcite and stated that their data were averaged over 100 nm square areas. No other hypotheses have been offered for observed curvature of high-Mg calcite crystallites. The stereom in any given plate acts optically as a single crystal of calcite. In addition, the crystals of the plates show preferred crystal axis orientations that depend on the species and type of plate. The history of studies on the orientation of calcite in echinoderms 27   

a

b

Figure 2.5. Conchoidal fracture surfaces in stereom, Metalia spatagus: (a) one general fracture plane cuts across the fracture surface; (b) close-up showing details of fractures on individual trabeculae. 28   

was reviewed by Raup (1959) along with preliminary results he had obtained from his own analyses of c-axis orientations. The two preferred c-axis orientations in echinoid coronal plates are perpendicular (or normal) to the plate, and tangential to the plate. A particular orientation was a characteristic at the family or order level, suggesting that preferred orientations evolved independently in different lineages (Raup, 1959). This work was later expanded (Raup, 1960; 1962a), and it was suggested that crystal orientation might be related to light-sensitivity in the animal. The phylogenetic significance of crystal orientation in coronal plates was detailed by Raup (1962b). He comprehensively documented the crystal orientation of apical (Raup, 1965; Raup and Swan, 1967) and coronal (Raup, 1966) plates of all major groups of echinoids. This comprehensive data set showed that crystal orientations were more complex than originally thought. Although there were only two preferred orientations of the c-axis, some species were characterized by a combination. In addition, some species showed ontogenetic variation in orientation. Apical plates showed a confusing variability in orientations, but the variation in orientation of genital plates depends in part on larval form (Emlet, 1985; 1988). There is greater variability in the a-axis orientation of larval and adult calcite (Okazaki and Inoue, 1976; Okazaki et al., 1981), but the significance of a-axis orientation has not been determined. 2.3.2 Terminology Most morphological terminology for echinoids was exhaustively described by Hyman (1955) and Melville and Durham (1966). The surface of a post-metamorphic echinoid is divided into four regions: the periproct, the peristome, the apical system, and the corona, the

29   

latter two composing the test (see Figure 1.2 in the previous chapter). The peristome and periproct are membranous areas containing, respectively, the mouth and the anus. The periproct is located at the apex of the body in regular echinoids, directly opposite the peristome, and the line passing through the center of the periproct and peristome forms an axis of symmetry defining aboral and adoral poles. The apical system, so named because it is located at the apex (aboral pole) of the test, is composed of five genital plates and five ocular plates. In regular echinoids the periproct is surrounded by the plates of the apical system, a condition termed endocyclic. Irregular echinoids are termed exocyclic, indicative of the position of the periproct outside of the apical system and posterior to the apex. The genital plates are each penetrated by a single gonopore connected to an internal gonad. One of the genital plates is distinguished by also being penetrated by a number of small pores connected to the water vascular system. This plate is termed the madreporite, and, as the only externally obvious violation of pentameral symmetry in the skeleton of regular echinoids, it is important in defining a second symmetry axis. The ocular plates, also called terminals, are pierced by a small pore for the terminal podium of the water vascular system. Genital plates invariably border the periproct in regular echinoids, but the ocular plates may or may not touch it. When they do they are termed insert, otherwise exsert. The corona, in all modern forms, is composed of twenty columns of plates, ten of which are termed ambulacral plates and are associated with the water vascular system and penetrated by pores for the tube feet. The remaining ten columns are termed interambulacral. The columns are paired, such that there are five ambulacra, sometimes termed radials, and five interambulacra, sometimes termed interradials. The terms radial and interradial are commonly used as positional

30   

references to the ambulacra and interambulacra, particularly for plates that lie outside of the corona. Individual columns in each ambulacra or interambulacra are termed ½ columns. These terms are shown diagrammatically in Figure 1.2 (previous chapter). Sutures can in general be divided into latitudinal (horizontal and between individual plates) and meridional (near-vertical and between individual plates). The upper (towards the apex) latitudinal sutures are termed adapicad, the lower (towards the mouth) adorad. Meridional sutures between ambulacral ½ columns are termed perradiad, between interambulacral ½ columns interradiad, and between ambulacra and interambulacra adradiad. These terms are shown diagrammatically in Figure 2.6. An ambulacral plate which extends across the entire width of a ½ column is termed a primary plate, otherwise it is termed a reduced plate. Reduced plates may touch the adradial but not the perradial suture (demiplate), the perradial but not the adradial suture (occluded), or neither suture (included). The sutures between reduced plates are termed transverse. Ambulacral plates may be simple or compound. Compound plates are composed of two or more primary or reduced plates, bound together by a single tubercle, and grow as a single unit. An example is shown in Figure 2.7. The naming convention for ambulacra and interambulacra was devised by Lovén (1874). Ambulacral columns are indicated using Roman numerals I through V and interambulacral columns by Arabic numerals from 1 to 5, numbered in counterclockwise direction when facing the apical system, clockwise from the oral side. Individual ½ columns are designated a and b, also in counterclockwise direction from the apical viewpoint. The 31   

Interambulacral column

Ambulacral column

Meridional suture

Latitudinal suture Adapicad suture

Interradiad suture Adradiad suture

Perradiad suture

Adorad sutures

Figure 2.6. Descriptive terminology of plate edges and sutures.

32   

Figure 2.7. Example of compound ambulacral plates, Diplopodia steeruvitzi (from Clark and Twitchell, 1915, Plate XX, Figure 3b).

33   

interambulacrum containing the madreporite is number 2, and the anterior-posterior axis (termed the Lovén axis) passes through the center of ambulacrum III and interambulacrum 5 (see Figure 1.2, previous chapter). The terminology defined above is unequivocal in designating the radial (longitudinal or meridional) position of a plate, but difficulty arises when an attempt is made to specify the zonal (or latitudinal) position. Attention has been focused almost exclusively on the plates directly associated with the peristome. The term basicoronal is used to describe the first row of plates directly abutting the peristome (Melville and Durham, 1966). These plates are distinguished from the first-formed plates in the post-metamorphic corona, which are termed the primordial plates (Melville and Durham, 1966). In most echinoids, the basicoronal plate ring consists of five pairs of ambulacral plates, and five single interambulacral (primordial) plates (David et al., 1995, p. 156). This is readily seen in many irregular echinoids, but for most regular echinoids it is not apparent beyond the earliest post-metamorphic growth stage because of resorption and remolding of the plates around the edge of the peristome. Beyond this, there is no existing terminology for zonal plate position. 2.3.3 Symmetry The basicoronal ambulacral plates of the bilaterally symmetrical irregular echinoids have either one or two tube feet and corresponding size differences (Lovén, 1874). If the plates are numbered in a-b order using the Lovén system (see Section 2.3.2), the larger plates in each ambulacral column corresponded to the sequence Ia-IIa-IIIb-IVa-Vb (or a-a-b-a-b). This sequence was later shown to apply to all echinoids (David et al., 1995) and has become 34   

known as Lovén’s Rule. Application of the rule defines a bilateral axis running through the perradiad sutures of ambulacrum III and interradiad sutures of interambulacrum 5. This axis is known as the Lovén axis. Two other axial systems, the Ubisch axis (Ubisch, 1927) and the Carpenter axis (Carpenter, 1884), were described by Hyman (1955). Ubisch (1927) defined his “Primordialebene” or primordium as an axis of symmetry running through the perradiad sutures of ambulacrum II and interradiad sutures of interambulacrum 4, based on newly metamorphosed urchins. The Carpenter axis was defined by Carpenter (1884) for use with crinoids and holothurians and is based on the line through the mouth and anus, or mouth and hydropore (madreporite in echinoids) if the anus is centrally located, rather than the symmetries of the basicoronal plates. Although the use of the Carpenter axis would bring echinoid orientation into agreement with that of other echinoderms, its failure to orient irregular echinoids according to their obvious bilateral symmetry has caused it to fall out of favor. Because of such difficulty with this point Moore and Fell (1966) redefined the Carpenter axis to correspond to the Lovén axis in echinoids, in direct contradiction with Hyman’s (1955) usage. Crystallographic data (Lucas, 1953; Raup, 1965) generally favor the Ubisch axis. The relationships among these symmetry axes are shown in Figure 2.8. 2.3.4 Plate Addition It has long been known (at least since Agassiz, 1834) that in echinoids all new coronal plates are added at the edge of the apical system. New interambulacral plates first appear at the angle between the ocular and the genital plates (Lovén, 1892). All new plates are added on the adoral edge of the ocular plates and presence of the oculars is both sufficient

35   

a

III

Madreporite b

2

3 b

a

IV

II

a b U

x is

1 a

Ca rp

en

ter

’s

Ax is

b

V

a

Loven’s Axis

4

’s A b is c h

I b

5 Figure 2.8. Symmetry axes applied to echinoids. Lovén system for numbering ambulacra and interambulacra is show. Lowercase letters are designators for the interambulacral ½ columns 36   

and necessary to plate addition, whereas the genital plates play no part (Jackson, 1912). Because of the importance of the oculars, Jackson further held that the corona is made up of five growth zones, each centered on an ambulacrum and the two half interambulacra on either side. This concept was formalized (Mooi et al., 1994) under the name Ocular Plate Rule or OPR (although there is not always a calcified ocular plate at the end of the ambulacrum in all echinoderm groups). The OPR stresses the idea that “…each so-called 'interambulacrum' is actually built by the confluence of plates from two adjacent, and otherwise ontogenetically separate growth zones" (Mooi and David, 1997, p. 308). Based on observations on the order of addition of new coronal plates, Cutress (1965, p.827) felt that “…the two columns of each interambulacrum … are more closely allied than would be supposed from their association with different oculars.” In fact, the OPR as described in a related series of papers by David et al. (1995), David and Mooi (1996), Mooi et al. (1994), Mooi and David (1997), Mooi and David (2000), Mooi et al. (2005) does not refer to the order of plate insertion. Indeed, as Agassiz (1834, p.371) so beautifully stated: “With respect to the original relations of all the plates which form these series, we should entertain a false idea if we represented them to ourselves as growing really in that vertical succession which they seem to possess. It is, indeed, at the summits of the series that the new plates are formed, but they succeed each other, like leaves in plants, spirally, from one interambulacral series to another, so that those which lie in a vertical line one upon another, do not succeed each other in the order of their first growth." At least in regard to the interambulacral plates, new plates are not added simultaneously or equally around the apical system but in an alternating manner (Lovén, 1892). Deutler (1926, p. 136) added a twist to 37   

this when he stated that the adjacent plates in a column were added simultaneously (“…gleichzeitig entstanden …”) and possessed the same growth rings. This, however, does not mean that the growth zones are exactly contemporaneous, but rather that plates belong in cohorts, a concept that was not previously considered but which plays a critical part in the growth model used here. In the sea urchin Eucidaris tribuloides new interambulacral plates are almost always added in the same sequence (Cutress, 1965): IA 3, IA 2 and 4, IA 1 and 5 (the 3-2-4-1-5 sequence). Plates are added in columns a-a-b-a-b or b-b-a-b-a in alternating sequence (see Figure 2-8 for diagram of this notation). This addition sequence is exactly the reverse of that found by Jackson (1912) for insertion of the ocular plates of E. tribuloides. Originally all exert, the oculars become insert in the sequence V-I-IV-II-III. Jackson (1912) referred to what he called the “law of ocular development” whereby ocular plates expand to contact the periproct in a definite order: I, V or V, I, IV, II, III. A series of specimens of Strongylocentrotus droebachiensis collected off the Bay of Fundy at Lubec, Maine were prepared and photographed in order to measure the addition sequence of new coronal interambulacral plates (ambulacral plates were also measured, but it is very difficult to judge the size differences among newly formed ambulacral plates). Dead specimens, of course, provide only a static snapshot of the plates from which only a relative sequence can be determined. If a strong tendency towards a particular sequence for addition of interambulacral plates were present, it would be revealed as a recurrent series. For example, if the 3-2-4-1-5 sequence were present, recurrent series would be 2-4-1-5-3,

38   

4-1-5-3-2, etc. The multiple independent series can be consolidated by rotating the individual sequences so that each begins with 1 (i.e., these example sequences can all be represented by 1-5-3-2-4). Using this consolidation there are 24 different possible sequences (beginning at position 1). The apical system of one of the specimens labeled with the sequence of plate addition is shown in Figure 2.9. Results are summarized in Table 2.1 and show that there is no single preferred sequence of interambulacral plate addition. In particular, the 3-2-4-1-5 sequence is one of the least likely to be found in S. droebachiensis. The apparent discrepancy between my results and those described by Cutress (1965) can probably be explained by the differences in ocular plate insertion between the genera Eucidaris and Strongylocentrotus. However, a summary of the ½ column order of leading and trailing plates (Figure 2.10) shows a weak preference for the a-a-b-a-b sequence characteristic of Loven’s Rule, which agrees with Cutress’ results. The alternating sequence of ambulacral plates from the original basicoronal pattern (i.e., following Lovén’s Rule) is more or less distinctly recognizable along the entire length of each ambulacrum, although more so in the older plates (closer to the peristome) than in younger (Lovén, 1883). Together, these results demonstrate that Lovén’s Rule also plays at best a minor role in the later addition of new intermbulacral plates at the apical system. 2.3.5 Homology The Extraxial-Axial Theory or EAT (Mooi et al., 1994) is the currently accepted theory of plate homologies for the Echinodermata as a whole. The theory, in essence, divides the skeleton into 2 major components: axial and extraxial, with further subdivision of

39   

40 

 

6

1

3

2

8

7

Figure 2.9. Closeup of apical system of Strongylocentrotus droebachiensis. Newly added interambulacral plates are numbered by increasing order of size, the younger cohort in red, the older in blue. Rotated insertion order is 1-2-3-4-5, which occurred in 5.4% of the cases observed. Note that in this specimen plate 10 has been split by an additional suture.

10

5

4

9

Sequence

Frequency

Sequence

Frequency

1-2-3-4-5

4

1-4-2-3-5

4

1-2-3-5-4

3

1-4-2-5-3

1

1-2-4-3-5

5

1-4-3-2-5

6

1-2-4-5-3

5

1-4-3-5-2

2

1-2-5-3-4

5

1-4-5-2-3

5

1-2-5-4-3

5

1-4-5-3-2

2

1-3-2-4-5

5

1-5-2-3-4

3

1-3-2-5-4

3

1-5-2-4-3

1

1-3-4-2-5

2

1-5-3-2-4

1

1-3-4-5-2

1

1-5-3-4-2

2

1-3-5-2-4

2

1-5-4-2-3

3

1-3-5-4-2

2

1-5-4-3-2

2

Table 2.1. Addition order of interambulacral plates, Strongylocentrotus droebachiensis. Sequence is the order of plate additions, rotated so that first addition is in the first position, Frequency is the number of specimens (out of 74) expressing the respective sequence. 41   

a a

a a

1

b a

a a

5

b a

b a

5

a b

a b

5

a b

b b

5

b b

b b

1

a

a

a

b

b

b

0.03125

0.15625

0.15625

0.15625

0.15625

0.03125

b

a

5

a

a

a

a

a

b

a

a

a

a a

b 5

b

b

b

a

0.15625

0.15625

b

a

b

a

a

a

b

b

a

b

b

b

b

b

b

b

a

a

a

b

b

b

0.0135

0.149

0.162

0.149

0.122

0.0541

b

a

a

a

a

b

b

b

b

a

0.230

0.122

Figure 2.10. (a) The eight topologies of distinct cyclic permutations of alternating interambulacral plates in an echinoid. Orientation is from the apical viewpoint, interambulacrum 1 in the lower right-hand position, pattern read in the counter-clockwise direction. The a and b positions follow the orientation shown in Figure 2.8. Number of rotational permutations is given by the number in the center of the box. Decimal values indicate expected probabilities for random occurrence of each pattern. The Lovén pattern is emphasized with a double box. (b) Observed distribution of the five youngest interambulacral plates from 74 specimens of Strongylocentrotus droebachiensis. Decimal numbers are the fraction of specimens with the given plate pattern. Graphic presentation is similar to that of Hotchkiss (1978). 42   

the extraxial skeleton into perforate and imperforate. The axial components are defined as “…all skeletal structures derived in accordance with the OPR [Ocular Plate Rule], as well as all elements (e.g., spines) associated with them.” (Mooi et al., 1994, p. 89). All other skeletal components are considered extraxial. The axial skeleton is divided into growth zones, consisting of a biserial ambulacral column and adjacent half-columns of interambulacral plates. Each growth zone is presumed to grow independently of the others. The EAT has been modified (David and Mooi, 1996) to redefine the homologies in embryological terms. In that scheme, all of the axial elements of echinoids are derived from the newly formed rudiment, while all the extraxial skeletal elements arise from within the pluteal portion of the metamorphosing larva. Most recently the embryological link has been more strongly stressed (Mooi et al., 2005). In general, questions of homology between echinoids and other echinoderm classes are not significant in the design of models for echinoid growth. However, the strong connection between the EAT and the OPR, and the canalization of growth, even conceptually, to restrictive growth zones do constrain the models. 2.4 Life Cycle The echinoid life cycle can be divided into several stages: egg, blastula, gastrula, larval (or pluteus), and post-metamorphic, although there are direct-developing forms that forego the larval stage. 2.4.1 Egg And Larva This study deals with the skeletal growth of the post-metamorphic stage. It is, however, instructive to consider the origin of the plates in the embryo. The Extraxial/Axial 43   

Theory (EAT) of echinoderm plate homologies is based on simplified interpretations from classical descriptions of comparative zoology and embryology (David and Mooi, 1996). This section reviews the results from those classical studies. The larval stage begins with a fertilized egg progressing through blastula and gastrula stages. In most echinoids, the gastrula transforms into a free-living pluteus or larval stage, termed an echinopluteus (in direct-developing echinoids the pluteus stage is skipped). The echinopluteus may be facultative (feeding) or non-facultative (non-feeding). After some period of time (varying by species), the echinopluteus begins to form a rudiment (in many details a tiny urchin). The echinopluteus may remain in this rudiment stage for some period before environmental conditions for settlement are found and the larva is recruited to the substrate. At this point, a very rapid (occurring in less than an hour) evulsion of the rudiment occurs, with resorption or apoptosis (programmed cell death) of the larval tissues. The larva metamorphoses into a tiny urchin, generally called an imago (analogous to the imago larval stages in arthropods). 2.4.2 Metamorphosis and Imago Stage The development of three plates directly over the right enterocoel of the larva of Psammechinus microtuberculatus can be distinguished in the rudiment stage (Bury, 1889). The posterior end of one of the larval skeletal rods forms a fourth plate. A fifth plate forms around the median posterior arm of a tri-radiate dorsal skeletal rod and develops the water pore characteristic of the madreporite. Bury was uncertain whether the latter two plates were associated with the right or left enterocoel. He concluded that because the five plates become 44   

the five genital plates of the post-larval form, and three are associated with the right enterocoel, the other two must be as well. Other plates begin to form around the left enterocoel, although Bury was uncertain of the exact position or number. Bury (1895) again stated his inability to determine the position and number of these plates in the pluteus. However, in the post-larval stage (imago or earliest juvenile stage) he was able to distinguish the plates based on spination. Ocular plates form on the aboral side of the terminal tube feet (his primary tentacles), alternating with five interradial plates. He stated that the oculars appeared to originate from the left side of the larva, i.e., over the left enterocoel, and that the division between the left and right body cavities lies somewhere between the genital and ocular plates. The early development of the skeleton in Psammechinus miliaris was studied by Gordon (1926a). The first plates to appear in the rudiment are the oculars, which, as shown by Bury (1895), develop external to the terminal tube feet. Contemporaneously, a single plate is developed in each interambulacrum, followed by a pair of plates aborally, and finally a fourth plate nearly at the level of the oculars. Three of the ocular plates (I, II, and III) arise de novo in the rudiment, but the other two (IV and V) are derived from the left post-oral and postero-dorsal larval spicules. The first pairs of plates of each ambulacrum then form, with the series Ia, IIa, IIIb, IVa, and Vb laid down slightly in advance of the remaining series. This is followed by the appearance of the ten buccal plates, which form in the interradial position. The buccal plates appear to form simultaneously. Gordon noted that this account of timing is typical, but that there is significant variability in the timing and order of plate formation. Genital plate 2 forms by extensions of the dorsal arch as in P. microtuberculatus. 45   

At the same time, genital plates 3 and 5 originate from the postero-dorsal and post-oral spicules, respectively, on the right side of the pluteus. Genital plate 1 forms de novo following the appearance of the buccal plates and rudimentary lantern, and genital plate 4 forms, also de novo, near the posterior end of the right side of the pluteus. The pattern is identical as that found by Bury (1889) for P. microtuberculatus. The coronas in the youngest post-larval specimens of the cidaroid Eucidaris metularia consist of two or three pairs of plates in each ambulacrum and four plates in each interambulacrum, one of which is the primary unpaired basicoronal plate (Mortensen, 1927). Only buccal plates are present on the peristomal membrane, and these are of unequal size and conform to Lovén’s Rule. As the juvenile grows, the ambulacral plates surrounding the peristome are loosened and pass onto the peristomal membrane. The primary interambulacral plates begin to dissolve early, and this resorption can progress to at least the sixth plates (third pair) while ambulacral plates move out onto the peristomal membrane. Plates located between the ambulacral plates (i.e., in the position of the interambulacra) on the peristomal membrane form de novo, and are not interambulacral in origin. In addition, these plates form in a single, unpaired series. The apical system is dominated by five large genital plates, one of which is pushed to one side of the regular pentagon otherwise formed by the plates. This is probably the madreporite, although in the earliest stages, the pore opening into the stone canal is not discernible. Ocular plates are relatively small. The periproct is originally naked, but several plates rapidly form to cover it.

46   

All the major biomineralized structures of Paracentrotus lividus begin to form on the 14th day during the larval period (Gosselin & Jangoux, 1998). The first genital plates 3 and 5 originate from the right postero-dorsal and post-oral larval spicules, with genital plate 2 forming on contact with the basal part of the dorsal arch, progressively surrounding the hydropore. Genital plates 1 and 4 form de novo. This is the same pattern as seen in Psammechinus microtuberculatus by Bury (1889) and P. miliaris by Gordon (1926a). Ocular plates IV and V develop from the left postero-dorsal and post-oral larval spicules, respectively; the remaining ocular plates arise de novo. Following metamorphosis, the juvenile corona is composed of five interambulacra, each with four spine-bearing plates, and five ambulacra, each with one pair of plates. Within one day of metamorphosis the coronal plates make contact and begin to form a rigid test. The ambulacral plates are asymmetric; the largest are Ib, IIb, IIIa, IVb, and Va, seemingly in contradiction to Lovén’s Rule. David et al. (1995) explained this by considering the first ambulacral pairs to consist of the Ib, IIb IIIa, IVb, and Va buccal plates and the Ia, IIa, IIIb, Iva, and Vb basicoronal plates. I obtained specimens of Strongylocentrotus droebachiensis only a few hours after metamorphosis from the sea urchin breeding facility at the R.J. Peacock Canning Co. in Lubec, Maine. Complete removal of organic material results in loss of the peristomal and apical plates (which make up most of the surface area of the test), but reveals the details of the corona just after metamorphosis (Figure 2.11). The interambulacra are composed of four plates, each with a tubercle for a primary spine. There is one complete pair of ambulacral plates, each with a single tube foot pore, plus one additional ambulacral plate in the adapicad position. Even at this early stage it is difficult to determine if the plates conform to Lovén’s 47   

a

b

c

d

Figure 2.11. SEM micrographs of the coronal plates of Strongylocentrotus droebachiensis less than 1 day post-metamorphosis: (a) Aboral view; (b) Oral view; (c) Detail of ambulacral plates between interambulacral columns, oral view; (d) Detail of interambulacral plate (with bounding ambulacral plates), oral view.

48   

Rule. This is also the case for older specimens; without the apical system in place even a 4month old specimen cannot be oriented (Figure 2.12). 2.4.3 Juvenile to Adult After only a few days the post-metamorphic echinoid looks very much like a miniature of the adult. This is especially true of the regular urchins, in which the main difference between a juvenile and an adult is in the total number and size of coronal plates. All new plates are added adjacent to the apical system: ambulacral plates originate from beneath the ocular plates, and interambulacral plates from depressions formed at the junction of the ocular and genital plates. Details of plate addition and accretionary growth will be discussed in the final section of this chapter. 2.5 Growth 2.5.1

Modes of Growth A hypothesis defining two major growth strategies as explanation of the

morphological differences between Paleozoic and post-Paleozoic echinoids was presented by Smith (2005). These growth strategies are based on the processes of plate addition and plate accretion. He held that growth in most Paleozoic echinoids was dominated by plate addition. He characterized this as a “conveyor belt” mode of growth in which plates remained unspecialized and undifferentiated throughout ontogeny. The evolutionary trend has been the predomination of plate accretion and concomitant differentiation over the process of plate addition. He also pointed out a later development, the positional fixation of some plates early in ontogeny. These developmental changes occurred in a particular order. The 49   

Figure 2.12. SEM micrograph of the coronal plates of 4-month old Strongylocentrotus droebachiensis, oral view. Orientation to Lovén axis unknown.

50   

mechanisms for plate insertion are the oldest, and are probably involved with the ubiquity of Lovén’s Rule, at least in part. There was an early mechanism for individual plate growth that resulted in fully determined plates, most of which reached a single maximum size everywhere on the test. A later mechanism allowed for essentially unlimited individual plate growth and a concomitant restriction to twenty columns. The latest mechanism is that causing fixation of some plate positions. Other, more subtle, modes of growth include differentiation (between types of plates) and inhibition of plate accretion. The term accommodation can be applied to the overall growth of the organism – where accommodation rate is equal to the sum of the rates of plate addition and accretion. Accommodation is anisotropic insofar as plates may become fixed (either at the peristome or the ambitus) and unable to migrate (in a relative sense) away from the apical system as new plates are added. These modes of growth are shown diagrammatically in Figure 2.13. 2.5.2 Addition Order Lovén’s Rule was defined for bilaterally symmetrical irregular echinoids. Although it has been applied to all echinoids, and even other echinoderms, there is no obvious reason why this should be the case (Jackson, 1927). There is a question as to whether or not there is some selective advantage to this pattern (Hotchkiss, 1995). The very fact that it is found in all echinoids (and possibly other eleutherozoans) suggests that it is rooted very deeply in the developmental gene pathways.

51   

Apical System

Initiation Point (Ocular Locus) Large Maximum Plate Size

Ambitus associated with younger plates during growth

Accommodating

Ambitus

Radial Growth

Small Maximum Plate size Fixed Endpoint

Mouth Figure 2.13. Modes of growth of the echinoid corona. 52   

The ocular plate rule (Mooi et al., 1994) delineated growth zones of plate development that would seem to require a change in nomenclature in the description of the columns. Instead of separate numbering patterns for the ambulacral and interambulacral columns, recognition of growth zones suggests a 5-fold numbering system and redesignation of the left and right interambulacral 1/2 columns associated with each ambulacral zone. Just such a nomenclature was proposed by Saint-Seine (1958), but it has never been accepted. An important corollary of the OPR is this concept of ontogenetically separate growth zones centered on the ambulacra (Mooi and David, 1997). The Extraxial-Axial Theory (EAT) of Mooi and David (1997) clearly isolates the coronal growth as requiring plate insertions in accordance with an inflexible rule, the OPR. In some sense, the insertion rule is both specified and determined. Ocular plates growing across and meeting in median suture, thereby cutting off the genital plate from contact with the corona, were observed by Jackson (1927). He found that the interambulacra are still perfectly developed even though they have no contact with a genital plate. He claimed that there was a placogenous (plate producing) zone on the adoral border of the oculars, from which the interambulacral plates originate, and that the genital plates had no part in the origin of interambulacral plates. He showed this in several examples (listing a total of 16 for Arbacia punctulata) where ocular plates unite and shut. Coronal plates may arise from a blastema localized at the ocular plate (David et al., 1995; Mooi et al., 2005), possibly from set-aside cells (Peterson et al., 1997) derived from the larva. Interambulacral plates are added on the outside of the test at the junction of the

53   

ocular and genital plates, but ambulacral plates are added from the interior of the test beneath the oculars (Märkel, 1981). This suggests that if the plates do arise from a blastema, there may be separate blastema for ambulacral and interambulacral plates, and they could operate under different rules. Ambulacral plates, because they arise from the interior and are intimately associated with tube feet and the water vascular system, could be induced by neurohormones from the radial nerve cords via a mechanism like that described by Thorndyke and Carnevali (2001). Interambulacral plates, at least in some cases, seem to arise from depressions that appear at the genital-ocular junction (Figure 2.14), and could be related to processes involving autotomy which could induce new plate formation, possibly via a mechanism similar to that described by Wilkie (2001) whereby autotomy is a precursor to regeneration. In this sense, the addition of new plates could be considered a form of regeneration. 2.5.3 Plate Growth Deutler (1926) in an oft-cited but little appreciated study described the somatic growth of Echinus esculentus by the growth of individual plates. He completely disproved Jackson’s (1912) contention that there is no trace of the earlier shape of a plate, showing quite to the contrary that nearly the entire growth history of a plate was laid out in a series of concentric growth rings that could be readily delineated by simple chemical treatment. Although plates grow on all edges, he distinguished meridional growth (i.e., longitudinal or growth collinear with the column) from growth in width (i.e., latitudinal growth). Rate of increase in width did not occur regularly, but is minor in the youngest plates, reaches a

54   

a

b

Figure 2.14. New interambulacral plates, Strongylocentrotus droebachiensis: (a) New plate forming in depression on older plate, apical system towards bottom. (b) Older plate expanding into gap, apical system to right. Arrows mark sutures between genital and ocular plates of apical system. 55   

maximum before the largest plate, then decreases gradually to the last plate. The overall (somatic) growth of E. esculentus did not depend on growth of all the coronal plates, but from a certain age only on growth of plates on the aboral side of the test (above the ambitus). Meridional growth also occurred predominantly on the aboral side of the test after a certain age, with the maximum rate of growth in the youngest plates. From the stage of the imago (which Deutler called the anlage) up to a certain size every individual plate grew, but with increasing age the growth became progressively restricted to fewer plates on the aboral side. The length of a column of plates depended on the addition of new plates at the apical system, but the rate of addition decreases with increasing age of the animal. Important aspects of growth critical to understanding the processes involved were also described (Deutler, 1926). It was shown that, in regard to meridional growth, ambulacra and interambulacra coincide, and there was no translation of plates along the adradiad sutures (those between the ambulacral and interambulacral plates). This is true of E. esculentus, and probably for most regular echinoids, but is not true for many irregular echinoids (see McNamara, 1987). He demonstrated that compound ambulacral plates grew as whole entities in a manner similar to an interambulacral plate, and furthermore that development of pores for tube feet was independent of plate growth. He also concluded that the adradial sutures, because of unequal growth in the ambulacra and interambulacra, deviated somewhat from true meridians, and he suggested that the growth of the ambulacral plates determined to some extent that of the interambulacral plates, and ultimately, he conjectured, depended on growth of the underlying water vascular system. He made one final point which, as we will later see, is critical to the model of growth developed here: as the animal grows, there is only 56   

a relative displacement of plates towards the mouth. The plates do not move closer to the mouth in an absolute sense, but in a relative way only as the surface area of the test increases. Deutler’s work was expanded upon by Märkel (1975; 1976; 1981), who described details of coronal plate growth in the four species Paracentrotus lividus, Arbacia limuli, Eucidaris tribuloides and Diadema antillarum. His results agreed with those of Deutler (1926), and he reiterated the important observation that meridional growth of plates is not controlled by a simple (i.e., single) gradient from the apex to the mouth. He also noted that whereas growth in interambulacral plates indicated adaptation to the growth of ambulacral plates (first hypothesized by Deutler, 1926) the overall shapes of the urchins were controlled primarily by the interambulacra. Measurement of calcium uptake in Tripneustes gratilla elatensis via radioactive 45Ca afforded a different approach to understanding the growth of individual plates (Dafni,1984; Dafni and Erez, 1987a). Results from individual interambulacral plates of eight specimens of progressively larger size corroborated those of Deutler (1926) and showed that the plates nearest the apex showed the highest calcification rates, with declining growth rate towards the ambitus and little or no growth subambitally; during ontogeny the ambitus line shifts to younger plates; patterns in lateral (i.e., width) growth were opposed in interambulacral and ambulacral plates; and, finally, calcification rates varied by meridional position such that plates maintained their size relations without resorption of skeleton (i.e., there is only a relative displacement of plates towards the mouth).

57   

Many aspects of coronal growth were discussed by McNamara (1988), who introduced several diagrams comparing what he called growth increments to position. In these examples each plate has its own growth trajectory and can reach a different ultimate size than neighboring plates depending on age and position. 2.5.4 Somatic Growth In one of the classic papers of biology, Huxley (1932) laid the groundwork for a quantitative understanding of the growth of organisms. His simple mathematical model of allometric growth, ‫ ݕ‬ൌ ܾ‫ ݔ‬௞ , is the basis for the use of scaling laws (defined by the normalizing coefficient b and an exponential scale k) to describe the relationship between the parts and the whole of an organism (represented by x and y, respectively). The fundamental mathematical model of growth was derived by Bertalanffy (1938). It is represented by the equation ݈ ൌ ‫ ܮ‬െ ሺ‫ ܮ‬െ ݈଴ ሻ݁ ି௞௧ , where l is a linear dimension at time t, L is the maximum linear dimension and l0 is the initial value, is dependent on the age t of the organism, and the transformation coefficient k. Description of the somatic (or whole-body) growth of echinoids depends, implicitly or explicitly, on those two theoretical foundations. Because the skeleton of an echinoid, composed of individual plates, encompasses the body of the animal, plate and somatic growth are intimately linked. The basic patterns of somatic growth in a variety of echinoid species were described by Moore (1935; 1936), Moore et al. (1963), Moore and McPherson (1965), and Moore and Lopez (1966). These patterns were expressed in graphic terms, but without reference to any underlying model of growth. Likewise, Swan (1958; 1961) collected detailed growth data on 58   

Strongylocentrotus droebachiensis but did not try to apply any growth model to it. Regression models were applied to growth (Ebert, 1968), and the Bertalanffy model was first used to describe sea urchin growth by Ebert (1975). Ebert (1982) discussed the use of the Richards model (Richards and Kavanagh, 1943), which incorporates the Huxley allometric model to encompass, via an additional exponent, the Bertalanffy equation, the Gompertz growth equation, and the logistic equation. The ecological variation in growth of an irregular echinoid, Echinocardium cordatum, has been described using the Bertalanffy growth model (Duineveld and Jenness, 1984). Several growth models, both asymptotic and non-asymptotic, have been tested for describing somatic growth represented by test diameter and allometric growth of the Aristotle’s lantern of Evechinus chloroticus. No significant differences between the models were found, although the Richards model was subjectively preferred (Lamare and Mladenov, 2000). This result was foreshadowed by Ebert and Russell (1993) who concluded that there was no ideal growth model for Strongylocentrotus franciscanus. A comparison of six different growth models in a study of S. franciscanus, based on a relationship between jaw size and test diameter, found that the Bertalanffy model performed poorest, and the logistic model best Rogers-Bennett et al. (2003). However, these results were sensitive to growth at the smaller end of the size range, where the data were poorly constrained. These problems with simplified models are probably caused by the fact that somatic growth is dependent not only on nutrition and metabolism (which are theoretically the bases of the modeled relationships), but also involves biomechanical factors that control, to a greater or lesser extent, the form of the echinoid test. 59   

The shape of many genera of regular echinoids can be predicted using a force balance model, treating the test as a membrane in tension (Ellers, 1993). This methodology allows the incorporation of the known allometry in height and diameter of sea urchins during growth, a confounding factor that complicated earlier attempts at modeling somatic growth. The flexibility of the test required by the membrane model can be attributed to sutural loosening during growth (Johnson et al., 2002). Sutural loosening, due to relaxation or lengthening of connective ligaments, occurs during periods when internal coelomic pressure was positive. Presumably, the ligaments contract and the sutures tighten when coelomic pressure falls into the negative range. Such range in coelomic pressure was demonstrated by Ellers and Telford (1996).

60   

Chapter 3: Models 3.1 Objectives This chapter is an overview of the basic concepts and background of theoretical morphology. It includes a general discussion of theoretical morphology as an approach to understanding growth; detailed descriptions of models that have previously been applied to the growth of echinoids; and descriptions of developmental models of growth. 3.2 Theoretical Morphology Every organism has a morphology, which can be defined as the shape, form, and structure of the organism (generally without regard to function). Morphology is, for the organism, the outward or phenotypic expression of the genetic code that defines the individual and the species. Morphology is the basis for all historic and most modern systems of taxonomy and phylogenetics, particularly in paleobiology where actual genetic information is lacking. There is the explicit expectation that some aspects of morphology are characteristic of an individual, some of a species, and some of ever higher taxa (or, at least, more distant phylogenetic relationships). The term theoretical morphology is a composite term, with a central theme of morphology. The theoretical part of the term implies some degree of removal from the organism. By this distancing of morphology from the organism, theoretical morphology tries to explain morphology or aspects of morphology in terms of core principles of geometry, function, inheritance, growth, and development. It is a moving target. 61   

Theoretical morphology can be restricted to the simulation of biological form via “programs” of morphogenesis or growth (Reif and Weishampel, 1991). These can be actual computer programs, purely mathematical or analytic models, or physical models. Two basic kinds of morphologic models can be differentiated (Konarzewski and Kooijman, 1998). The traditional model, meant to provide an empirical description of growth, has the goal of detecting patterns of growth among organisms. These models emphasize generality and simplicity. The traditional approach is descriptive and mostly falls within the domain of biometrics. This approach is exemplified by the morphometric paradigm as defined by Bookstein (1996). The second approach is the simulation of the mechanisms that cause morphogenesis. This dichotomy of meaning was reiterated by McGhee (1999), who described theoretical morphology as either simulation of some aspect of form with a simplified set of parameters or simulation of the morphogenetic processes themselves. In the context of ecological and evolutionary modeling, differentiated two basic kinds of models can be differentiated: deterministic and stochastic (Wilson, 2000). Deterministic models attempt to capture the essence of form by way of mathematical expressions, whereas stochastic models use computers to simulate complex systems. This is a somewhat artificial distinction. Deterministic models seek to describe form by reducing biometric data to sets of mathematical equations, using the parameters of these equations to differentiate among organisms. Stochastic models are based on sets of equations that are usually intractable analytically and can only be solved by iterative 62   

methods, i.e., computer programs. They are not necessarily random as the term implies although such models can show characteristics of chaotic systems. The current trend in theoretical morphology is to exclude the descriptive and strictly morphometric approach and concentrate on modeling within the architecture of evolutionary and developmental biology, or “evo-devo” (Hall, 1992). Even in this restricted sense, the field is difficult to constrain. Modeling of developmental processes can involve simulations of developmental pathways at the molecular level or extrapolation to higher-level processes (Wilkins, 2002). Kumar and Bently (2002) took this particular definition to an even farther when they stated that developmental biology and computer science are linked by a unifying theme: construction. How are organisms constructed, and how can computer programs be constructed to simulate organismal development? This facet of theoretical morphology has its own name: computational systems biology (Kitano, 2002). A computational systems approach to modeling growth of echinoids will be developed in the next chapter. At this point, it is useful to review the range and types of models that have been applied to growth of echinoids. 3.3 Previously Applied Models The models summarized below have in common some degree of quantification of aspects of echinoid growth or develop concepts that have direct bearing on such quantification. They are presented in chronologic order, based on date of publication (Table 3.1).

63   

64 

  Plate growth Plate growth and addition Allometric growth Pneu model Dome model

Deutler (1926)

Raup (1968)

Moss & Meehan (1968)

Seilacher (1979)

Telford Model 1 (1985)

Bertalanffy equation Diffusion equation Young-Laplace equation

Body form Plate growth Plate addition Body form

Philippi & Nachtigall (1996)

Zachos(2008)

Yes

Yes

Yes

Yes (biometric)

Yes (biometric)

No

Yes (biometric)

Yes

No

No

Quantitative?

Computer

Computer

Computer

Descriptive

Computer

Conceptual

Conceptual

Descriptive

Computer

Descriptive

Conceptual

Model Type

Table 3.1. Models applied to growth or form of echinoids.

*Model type after Dera et al., (2008)

Finite element

Body form

Ellers (1993)

Young-Laplace equation

Body form

Huxley equation

Vector equations

Huxley equation

Logistic equation Quadratic growth rate

Young-Laplace equation

Mathematical Basis

Dafni (1986)

Telford Model 2 (1994)

Liquid drop

Description

D'Arcy Thompson (1917)

Model

Realistic

Complex geometrical

Complex geometrical

Complex geometrical

Complex geometrical

Simple geometrical

Parameter-Free

Complex geometrical

Complex geometrical

Complex geometrical

Simple geometrical

DEND Model Type*

3.3.1 D’Arcy Thompson model The earliest model applied to the form of an echinoid was that by D’Arcy Thompson (1917) when he compared the shape of a sea urchin to that of a drop of liquid deformed by gravity. He stated that the sea urchin is subject to similar conditions of pressure equilibration as faced by an egg, excluding external pressure, i.e., represented in the simplest form by the Young-Laplace equation:

′

where P is the internal pressure, T is the specific tension of the envelope, and and

(3.1) ′

are

the radii of curvature. Differing from the egg, however, the urchin experiences a downward pressure caused by gravity, as well as forces from the tube feet pulling the animal towards a surface. With these additional considerations, D’Arcy Thompson noted that a full analytical (i.e., mathematical) treatment of the shape was anything but trivial, but that urchins closely mimicked the shape of drops. An important consequence of this model was that, as seen in liquid drops, smaller urchins approximated spheres, and as urchin size increased the shape became increasingly flattened, the range in variation of actual urchins because of variations in the specific tension of the skeleton (and thereby offering a mechanical explanation for what would later be recognized as allometry between urchin height and diameter during ontogeny). Although D’Arcy Thompson mentioned imitating this phenomenon using water-filled rubber balls of various sizes and

65   

diagrammed a few representative urchin shapes, he did not attempt to quantify the development of sea urchin form. 3.3.2 Deutler model Prior to the development of computers (or even high speed calculators) investigators were restricted in methods used to model growth. Although not a model in the strictest sense, Deutler (1926) presented data he collected from detailed measurements of growth lines in echinoids in a set of colored graphs that were quite advanced for the time. Additional parameters extracted from his graphs, specifically change in plate perimeter vs. age, can be used to generate a set of plate growth curves (Figure 3.1). An important concept that arises from Deutler’s study is that plates are added in discrete generations or cycles. The initial set of plates can be considered the first generation. New plates are added in a manner such that a complete cycle of plates is completed around the apical axis before the next generation begins. Each cycle, therefore, constitutes a new generation, or cohort, of plates. The growth curves shown in Figure 3-1 can be accurately matched by the exponential Bertalanffy growth function (Bertalanffy, 1938), and the graph demonstrates that the asymptotes of these curves (the maximum theoretical size) increase with the age of the plate. The Bertalanffy growth function has the form: (3.2) where L is the maximum length and l0 is the initial length, is dependent on the age t of the 66   

Figure 3.1. Plate perimeter vs. cohort age based on growth line information reported by Deutler (1926) for a single specimen of Echinus esculentus. Colors group growth lines originating with successive plate cohorts. Early plates reach maximum size but later plates have not reached asymptotic sections of growth curves. Growth rates are comparable for all plate cohorts, but asymptotic size increases with cohort age.

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plate, and the rate constant k. This in turn indicates that plates can be grouped into cohorts (the set of plates associated with a cycle of plate addition around the corona), each with independent maximum growth parameters L. The instantaneous growth rate k, in contrast, appears to be nearly constant for all cohorts. 3.3.3 Raup model There can be little question that the echinoid growth model described by Raup (1968) was an important benchmark in the application of digital computing to the problem of growth, all the more so because it was published four decades ago and has not been expanded (or even duplicated) during that period of time. A computer model of plate growth in regular echinoids was described by Raup (1968). He simplified his model to a small set of factors that determine plate geometry in two dimensions. First, he assumed that addition of new plates could be simulated by a logistic function, a sigmoidal curve represented by the equation: (3.3) where P is the number of plates at time t, and a, b, and c are arbitrary constants. The resulting curve was sampled to give either an exponentially increasing rate, constant rate, or exponentially decreasing rate of plate insertion vs. time. He readily admitted that there was no evidentiary basis for the choice of a logistic curve to model plate addition, but the logistic model is an explicit result in some studies of population growth. The fact that a logistic model of plate insertion works in the Raup model suggests that growth of the 68   

echinoid corona can be considered in the context of a population of individually growing plates, at least in a statistical sense. Secondly, he modeled the growth rate of plates as a parabolic function of distance from the apical system: (3.4) where r is the growth rate, x the distance from the apical system, and a, b, and c are constants, again sampling only a portion of the curve to set the initial rate, determined whether or not the rate becomes negative (indicating resorption), and whether or not the rate increases distally. As an example, using the data he collected from Strongylocentrotus pallidus (Raup, 1968, Fig. 5A), the growth curve calculated from a least squares fit is: 0.0008

0.0667

1.5734

and is shown in Figure 3.2. Raup (1968) felt that the fact that the growth rate decreases and then appears to increase again in the oldest (most distal from the apex) plates was significant. However, it should be recognized that this is an artifact of the methodology, i.e., if the growth rate is modeled by a decaying exponential instead of a parabolic function, the rate does not increase distally (Figure 3.2). By using only these two functional descriptions of plate addition and growth, Raup (1968) was able to simulate a variety of plate patterns resembling those seen in

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2.5

Measured Values

2.0

Growth Increment (mm)

Parabolic Fit Exponential Fit

1.5

1.0

0.5

0

0

10

20

30

40

50

60

Distance from Apical System (mm)

Figure 3.2. Plate growth increment vs. distance from apical system (in millimeters). Data points are from Raup (1968, Text-Fig. 5a). Red curve is least squares fit of parabolic function y = ax 2 +bx + c to points (after Raup, 1968, TextFig. 7b). Green curve is least squares fit of exponential decay function y = ae-λx .

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actual echinoids. The exact details of his computer program were not given by Raup (1968), but it is relatively straightforward to program a similar model (extended to three dimensions). The results match many aspects of the restricted 2-dimensional models figured by Raup (1968). One representative model using a spherical reference frame is shown in Figure 3.3. 3.3.4 Moss and Meehan model A significant amount of detailed biometric data collected from two regular and one irregular species of echinoids were tabulated by Moss and Meehan (1968). The emphasis was on comparing allometric growth among the various measured features but they stated a concept important to the understanding of growth in echinoids: it is the growth of the soft tissues of the animal that provide the primary expansive forces. The plates are embedded in mesoderm (indeed, each plate within a syncytium or multinucleated mass of cytoplasm contained within a single cell membrane), and plate growth and location is passive and compensatory to the overall expansion of the test. Moss and Meehan (1968) were also able to clarify another conceptual problem that had confounded biologists for over a century: the apparent migration of plates from the apical system to the edge of peristome. The problem was to explain this apparent movement of plates towards the mouth while the diameter of the peristome increased. Many biologists evoked resorption of plates at the edge of the peristome, although Deutler (1926) demonstrated that the apparent translocation of plates was only relative

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Figure 3.3. Model of echinoid plating based on Raup (1968) model methodology, displayed over spherical reference frame, with both ambulacral and interambulacral plates represented. Triangular plates in apical area are required for initiation.

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and that, in fact, all the plates expand outward in absolute terms. By moving the coordinate system to the center of the test (rather than at the center of the peristome), this absolute expansion became readily apparent (Moss and Meehan, 1968). 3.3.5 Seilacher model The expression “pneu” was applied by Seilacher (1979) to describe the shape and form of an echinoid test. It is derived from the word pneumatic, meant to highlight the similarity in shape of a sea urchin and a fluid-filled balloon (although pneumatic implies the fluid is air). He expanded on this theme (Seilacher, 1991) as an example of selforganization. He held that not only the shape but even the patterns of plates in the echinoid skeleton are controlled by his pneu principle, which, however, was never stated in quantitative (i.e., testable) terms. The pneu model implies that, if lacking any internal rigid structure, the body of an organism will conform to that of a membrane in tension. An extension of the model implies that a rigid shell will conform to a dome structure where buckling is minimized. Seilacher (1991) even concluded that the phylogenetic history of echinoids confirmed his model, although again without any quantitative support. 3.3.6 Telford model 1 In contrast with Seilacher’s pneu or tension model, Telford (1985) advanced the hypothesis that the echinoid skeleton acted not only as a membrane, but also as a dome to support compressive stresses on the test. He explored this concept by careful 73   

examination of height/diameter ratios and plate sutures in a wide variety of echinoids. Plate thickness increased from apex to ambitus, and he found that radial sutures were thicker and more tightly bound by collagen than circumferential sutures. These are characteristics expected for a dome structure in compression, where tensional forces act circumferentially at the base (equivalent to the ambitus of the echinoid skeleton). Development of radial ribs (thickening of meridional plate sutures) prevents bending of the plates under vertical compression whereas increased collagen binding prevents expansion of the sutures under tension. 3.3.7 Dafni model An argument for the biomechanical control of sea urchin morphology was presented by Dafni (1986). He demonstrated that height:diameter ratios were affected by substrate type, concluding that it was the strength of adherence of tube feet to the bottom that caused the test to be flatter on hard substrates. By either finding damaged specimens or surgically causing damage, he was able to show that loss or atrophy of internal musculature also affected test shape. Earlier studies (Dafni, 1980; 1983) had demonstrated significant malformation of the test because of pollution effects on calcification (and by corollary, the strength of the test). A soap-bubble analogy to plate patterning was first proposed by Raup (1968), but Dafni (1986) made the conjecture that because plates are formed within a syncytial membrane, the calcification is secondary to growth of the membrane “capsule”, and this elastic capsule actually does respond to adjacent plates in the same fashion as a bubble. The plate sutures, in this case, can be 74   

modeled by the equation: (3.5) where

is the radius of curvature of a suture, and

and

are the radii of the adjacent

plates. He used this relationship to explain the somewhat wavy sutures seen in plates of Tripneustes gratilla elatensis, but such interpretation requires reversal in the size relationships of plates. Although curvature of plate sutures, particularly in the case of thin plates, is commonly seen in urchins, there is little evidence that it follows the relationship shown in the equation above, and therefore the utility of the soap-bubble model is uncertain. Using detailed measurements of calcification rates in individual plates of the sea urchin T. gratilla elatensis, Dafni and Erez (1987a) were able to model plate growth as a function of age and distance from the apex. They related calcification rate to plate number (counting from the apex or youngest plate) by a power function in the form of the Huxley (1932) equation of allometry (rewritten here for clarity): (3.6) where

is the calcification rate of plate N,

the initial calcification rate, and k is a

gradient coefficient (for T. gratilla elatensis the gradient coefficient k had a mean value of -0.762±0.238). They also found that there was a relationship between the initial rate and the diameter D of the urchin:

75   

(3.7) Replacing

: (3.8)

where a and b are regression coefficients, which for T. gratilla elatensis were a = 11.58 and b = -0.053. Calcification was measured (effectively) by weight and, if the minor increase in plate thickness is ignored, the calcification rate is proportional to the rate of increase in plate area. Rewriting in terms of rate increase in plate perimeter does not alter the form of these equations. A family of curves using representative values for these parameters is shown in Figure 3.4. The graph shows that the Dafni model proposes that older plates (those with a higher plate number) have a steeper decline in growth rate than younger plates. This is the pattern that is observed in sea urchins, where the oldest plates (on the oral surface) are the smallest and the newest plates, near the apical system, are growing the fastest. 3.3.8 Ellers model The water-drop model of D’Arcy Thompson (1917) was successfully quantified by Ellers (1993) and called the membrane model (a term first introduced to echinoid studies by Telford, 1985). He based the model on engineering principles used for thinshelled domes (Timoshenko, 1940). His starting point was the same Young-Laplace equation used by D’Arcy Thompson, but in a discrete form:

76   

Values are plate number

Growth Rate

10 9 8 7 6 5 4 3 2

Diameter

Figure 3.4. A family of curves calculated from the relationships described by Dafni and Erez (1987a). Plates are numbered from the apical system (older plates have higher plate numbers). Growth rate declines more steeply for older plates.

77   

(3.9) where p is the local internal equilibrium pressure at a point on the surface, N is the constant specific tension of the surface, and

and

are the local radii of curvature at

the point. The internal pressure at a point is represented by: (3.10) is the internal pressure at the apex,

where

The physical expressions of apical pressure

is the gradient of pressure with depth z. and pressure gradient

are dependent on

the interpretation of the rigid skeleton as a flexible membrane. These terms were reduced to the ratio ⁄

. Apical pressure was considered to represent an urchin’s coelomic

pressure, but coelomic pressure not only varies but can be negative (Ellers and Telford, 1992). The membrane model only works, however, if

is strictly positive, so Ellers

(1993) used a weighted average pressure as an estimate of consider

, but it is reasonable to

as representing the maximum coelomic pressure of an echinoid. The

pressure gradient

can be considered to be a proxy for biomechanical pressures

associated with podia, spines, muscles, and ligaments (Ellers, 1993). The skeleton represented as a shell is a surface of rotation. Therefore, the outward pressure p is the same for any point on the surface at the same latitude or depression angle β, where

cos , and the problem can be reduced to 2-dimensional form. The

non-linear ordinary differential equation resulting from the Young-Laplace equation,

78   

which has no known closed form expression, is solved numerically, using an algorithm derived from Timoshenko (1940) and outlined by Ellers (1993). A more detailed derivation of the finite element equations required to model the shape of the surface was given by Royles et al. (1980), and FORTRAN source code implementation of the algorithm by Sofoluwe (1983), although the Ellers (1993) algorithm is more elegant and simpler to apply. Application of this method results in families of curves approximating the profiles of regular sea urchins. A representative set of curves is shown in Figure 3.5. A short C++ implementation of the algorithm, based on Ellers (1993), is shown in Figure 3.6. 3.3.9 Telford model 2 Telford (1994) argued that simple structural models of echinoid growth were not relevant to studies of morphogenesis. He instead proposed that a graphical simulation based on individual plate growth in three dimensions could describe growth. He distinguished this methodology from what he called structural models, which he further categorized into pneu, dome, and plate structure models. The dome model explained the shape of a sea urchin mechanically. The dome shape was described as a response to stresses, both tensional and compressional but mostly external, in the echinoid skeleton (Telford, 1985). As we have seen, Dafni (1986; 1988) expanded this concept by considering stress resulting from forces applied by the tube feet, jaws, and internal ligaments. Ellers (1993) concluded that the shape of a sea 79   

Base

Figure 3.5. Set of curves generated using numerical solution of Young-Laplace equation based on algorithm from Ellers (1993). These curves approximate the shape of a liquid-filled membrane and a sea urchin test.

80   

// Ellers1993.cpp // This program is based on the published algorithm // in Ellers, 1993, Proc. R. Soc. London B 254, 125-129 #include #include #include #define PI (4.0*atan(1.0)) using namespace std; int main(int argc, char* argv[]) { double x, z, s, u, r, gammap, delx, delz; ofstream datfile; if (argc == 3) { r = atof(argv[1]); gammap = atof(argv[2]); } else { r = .02; //Default value gammap = 50; //Default value } delx = .0001; delz = .0001; datfile.open("C:\\temp\\dome.txt", ofstream::out | ofstream::trunc); if (datfile == NULL) return false; x = delx; z = (delx*delx)/(2*r); u = delx/r; while (u < sin(50*PI/180)) { s = (2/r)*(1 + gammap*z); u = u + delx * (s - u/x); z = z + delx * u/sqrt(1-u*u); x = x + delx; datfile