ARTICLE IN PRESS

Journal of Theoretical Biology 259 (2009) 646–657

Contents lists available at ScienceDirect

Journal of Theoretical Biology

0022-51

doi:10.1

� Tel.:

E-m

journal homepage: www.elsevier.com/locate/yjtbi

A new computational growth model for sea urchin skeletons

Louis G. Zachos �

Department of Paleobiology MRC-121, National Museum of Natural History, Smithsonian Institution, PO Box 37012, Washington, DC 20013-7012, USA

a r t i c l e i n f o

Article history:

Received 9 March 2009

Received in revised form

8 April 2009

Accepted 8 April 2009Available online 17 April 2009

Keywords:

Echinoid

Theoretical morphology

Voronoi

Delaunay

Simulation

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016/j.jtbi.2009.04.007

+1202 6331369; fax: +1202 786 2832.

ail address: lg_zachos@alumni.utexas.net

a b s t r a c t

A new computational model has been developed to simulate growth of regular sea urchin skeletons. The

model incorporates the processes of plate addition and individual plate growth into a composite model

of whole-body (somatic) growth. A simple developmental model based on hypothetical morphogens

underlies the assumptions used to define the simulated growth processes. The data model is based on a

Delaunay triangulation of plate growth center points, using the dual Voronoi polygons to define plate

topologies. A spherical frame of reference is used for growth calculations, with affine deformation of the

sphere (based on a Young–Laplace membrane model) to result in an urchin-like three-dimensional

form. The model verifies that the patterns of coronal plates in general meet the criteria of Voronoi

polygonalization, that a morphogen/threshold inhibition model for plate addition results in the

alternating plate addition pattern characteristic of sea urchins, and that application of the Bertalanffy

growth model to individual plates results in simulated somatic growth that approximates that seen in

living urchins. The model suggests avenues of research that could explain some of the distinctions

between modern sea urchins and the much more disparate groups of forms that characterized the

Paleozoic Era.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Sea urchins (echinoids) possess a pattern of skeletal growththat is unique among organisms. The skeleton, or test, is in realitya plated internal skeleton, covered almost everywhere inepithelium. The test consists of individual plates, arranged in apentaradial fashion, which can number from several hundred toseveral thousand for an individual adult echinoid. The plates fittogether in a mosaic to create a more or less rigid framework(Ellers et al., 1998). Growth of the test progresses in part bygrowth of each individual plate. Additionally, new plates areadded to the test throughout the lifetime of the animal.

Discrete growth of the individual plates can be distinguishedfrom the distributed growth of the overall test. The individualplates are complex structures constructed of a three-dimensionalmeshwork of calcite, termed stereom, filled (in the living form)with mesodermal tissue, termed stroma. Each plate in theechinoid corona grows by the addition of new calcite around theperimeter (Deutler, 1926) and this has been repeatedly confirmedby the use of radioisotopes (Dafni, 1984), tetracycline (Kobayashiand Taki, 1969), and other fluorescent markers to stain plates ingrowing echinoids (see Russell and Urbaniak, 2004, for a review).The skeleton can be divided into the three structurally separateparts called the corona, which makes up the major portion of the

ll rights reserved.

echinoid skeleton; the apical system of plates, located at the apexof the test and which usually consists of four or five genital platesand five ocular plates; and the peristomal and periproctal plates,which cover the peristomal and periproctal membranes asso-ciated with the mouth and anus (Fig. 1). Classically, the corona isfurther divided into five ambulacral and interambulacral areas,each consisting (in extant taxa) of two columns of plates (Hyman,1955). The earliest plates in the corona form in the larval imago orrudiment immediately before metamorphosis (Gordon, 1926), butafter that all additional plates are inserted into the corona at theedges of the ocular plates. At the cellular level, addition andresorption of stereom is made by specialized cells (Shimizu andYamada, 1980).

The echinoids can be subdivided into two broad groups basedon whether the periproct lies within the circlet of genital andocular plates of the apical system (endocyclic) or outside theapical system (exocyclic), which is operationally indicated bywhether the body has a radial (regular) or bilateral (irregular)symmetry. This division into regular or irregular is not quite thissimple (Saucede et al., 2007) and has limited value in classifica-tion, but serves to distinguish the relatively simple geometry ofthe regular form. Although many of the terms and concepts usedin this discussion are applicable to either regular or irregularechinoids, unless otherwise stated only regular echinoids, typi-cally called sea urchins, are considered.

All new plates are added on the adoral edge of the ocular platesand presence of the oculars is both sufficient and necessary forplate addition (Jackson, 1912). The corona is therefore composed

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Fig. 1. Basic terminology associated with the sea urchin skeleton. The example is

the dorsal (aboral) surface of a regular sea urchin (Goniocidaris). Each column of

ambulacral plates begins from an ocular plate and is associated with a half column

of interambulacral plates on either side. New plates are added only to the corona of

the skeleton. Several new interambulacral plates can be seen forming at the

junctions of the ocular and genital plates.

L.G. Zachos / Journal of Theoretical Biology 259 (2009) 646–657 647

of five growth zones, each associated with an ocular plate andcentered on an ambulacrum with a half interambulacrum oneither side. This concept was formalized under the name OcularPlate Rule or OPR (Mooi et al., 1994, 2005). The OPR does not referto the order of plate insertion, but new plates are not addedsimultaneously or equally around the apical system but in analternating manner (Loven, 1892). Furthermore, plates are addedin complete cycles or cohorts circling the apex, each cohortconsisting of 10 ambulacral or interambulacral plates.

The nucleation of a new plate can theoretically originate in oneof two ways: either undifferentiated tissue differentiates intoskeletogenic tissue, or syncytial skeletogenic tissue is buddedfrom existing skeletogenic tissue. Coronal plates may arise fromblastema localized at the ocular plate (David et al., 1995; Mooiet al., 2005), possibly from set-aside cells (Peterson et al., 1997)derived from the larva. Regeneration in echinoderms is spear-headed by mesenchymal blastema, but this is usually associatedwith autotomy at a site prior to regeneration (Wilkie, 2001). Newplates may arise from specialized tissue, which could be alocalized reservoir of lineage restricted (committed) stem cellsor from non-localized progenitor cells (Shimizu and Yamada,1980). In Euechinoidea (modern echinoids) this tissue is localizedto the ocular plate, but there is no explanation for localization.

Interambulacral plates are added on the outside of the testusually at the junction of the ocular and genital plates, butambulacral plates are added from the interior of the test beneaththe oculars (Markel, 1981). This suggests that if the plates do arisefrom blastema, there may be separate blastema for ambulacraland interambulacral plates, and they could operate underdifferent rules. Ambulacral plates, because they arise from theinterior and are intimately associated with tube feet and the watervascular system, could be induced by neurohormones from theradial nerve cords via a mechanism like that described byThorndyke and Carnevali (2001). Interambulacral plates, at leastin some cases, seem to arise from depressions that appear at thegenital–ocular junction, and could be related to processesinvolving autotomy. In this sense, the addition of new platescould be considered a form of regeneration.

Although plates grow on all edges (ignoring changes inthickness), meridional growth (i.e., longitudinal or growthcollinear with the column) can be distinguished from growth in

width (i.e., latitudinal growth). Rate of increase in width does notoccur regularly, but is minor in the youngest plates, reaches amaximum before the largest plate, then decreases gradually to thelast plate. Overall (somatic) growth does not depend on thegrowth of all the coronal plates, but, from a certain age, only ongrowth of plates on the aboral side of the test (above the ambitus).Meridional growth also occurs predominantly on the aboral sideof the test, with the maximum rate of growth in the youngestplates. From the stage of the post-metamorphic imago up to acertain size every individual plate grows, but with increasing agethe growth becomes progressively restricted to fewer plates onthe aboral side. The length of a column of plates depends on theaddition of new plates at the apical system, but the rate ofaddition decreases with increasing age of the animal.

In regard to meridional growth, ambulacra and interambulacracoincide for most regular echinoids (Deutler, 1926), and there islittle or no translation of plates along the adradiad sutures (thosebetween the ambulacral and interambulacral plates). Compoundambulacral plates grow as whole entities in a manner similar to aninterambulacral plate. The adradial sutures, because of unequalgrowth in the ambulacra and interambulacra, deviate somewhatfrom true meridians, and the growth of the ambulacral platesdetermine to some extent that of the interambulacral plates. Theobservation that meridional growth of plates is not controlled by asimple (i.e., single) gradient from the apex to the mouth has beenreiterated in other studies (Markel, 1975, 1976, 1981; Dafni, 1984;Dafni and Erez, 1987).

2. Model description

There is a long and rich history of attempts to model thegrowth of sea urchins. Thompson (1942) discussed the shape of ageneric sea urchin and compared it to the shape of a drop ofliquid. This concept has been the basis of a number of attempts tomodel the shape of a sea urchin from a mechanical aspect.Advances in the description of form in echinoids have been almostexclusively 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 Nach-tigall, 1996). Studies of echinoid growth have been orientedtowards whole-body growth (Ebert, 1975, 1982; Lamare andMladenov, 2000; Rogers-Bennett et al., 2003), although some-times with reference to individual plate growth (Raup, 1968;Duineveld and Jenness, 1984; Telford, 1994; Abou Chakra andStone, 2008). None of these attempts have resulted in a three-dimensional model incorporating the basic aspects of bothindividual plate growth and plate addition.

2.1. Spatial reference and coordinate system

Simplicity of modeling is obtained by using a spherical frameof reference. The symmetries associated with the echinoid testlead naturally to the selection of a spherical coordinate system asthe basis of a morphologic model, using a methodology similar tothat described by Foley et al. (1990) for visualizing functions overa sphere. In addition, the aboral region of the test, where plateaddition and most plate growth occur, closely approximates aspherical surface for most regular urchins and there is little if anyloss in accuracy based on using this frame of reference.

Model methodology generates a set of points {P} on the surfaceof a sphere representing the center of growth of each plate (Fig. 2).The growth function associated with each plate has adirectionality that is dependent strictly on the relative geodesicbearings (measured along arcs of great circles) to adjacent plates.These are calculated using vector formulations (Earle, 2005)

ARTICLE IN PRESS

Fig. 2. Spherical geometry. Point P, representing the center point of a plate, on the

surface of the sphere has the Cartesian coordinates (x,y,z) and the spherical

coordinates (r,a,b), where r is the radius of the sphere, a is the angle between the x-

axis and the projection of P onto the x–y plane, and b is the depression angle

measured from the z-axis. Planes that cut the sphere through the origin intersect

the surface in great circles or geodesics. The x–y plane is a great circle representing

the equator, which corresponds to the ambitus or widest extent of the test. Planes

containing the z-axis are great circles representing meridians. Latitudes are

parallel to the equator but are not great circles.

L.G. Zachos / Journal of Theoretical Biology 259 (2009) 646–657648

replacing the computationally intensive trigonometric Napierrules for spherical navigation. Plate interrelationships arederived from a Delaunay triangulation of the points on thesurface of the sphere and plate edges derived from the dualVoronoi diagram. The plates are then mapped from the sphereonto a surface representing the distributed body plan of the test.This procedure simplifies the geometric calculations, and has theadded advantage of comparing the entire set of growth models onone spherical frame of reference.

2.2. Growth model

The growth of the coronal skeleton of sea urchins ischaracterized by a periodic process with two basic steps: additionof a new plate at a single nucleation point and growth of the plateby expansion of stereom crystallites outward from that point.These steps are repeated indefinitely. There must be two generalbut separate developmental pathways that periodically lead tonucleation of a new plate and result in the growth of a plate (incontrast, plates of the apical system are relatively static afterorigination in the rudiment and so do not express the samepathways as the coronal plates). Protein signaling pathways arekeys to regulation of development. Signaling proteins act asmorphogens if they meet two criteria (Fisher and Howie, 2006):they form a concentration gradient and they elicit differentresponses dependent on concentration. Tissues experience differ-ent concentrations of a morphogen dependent on their distancefrom the source of the morphogen.

2.2.1. Plate addition

The addition of new plates into the corona of a sea urchin canbe explained in terms of morphogens. The localized nature of thenucleation points implies that morphogen concentrations at thesepoints are the critical values. The periodic nature of plate additionimplies that concentration fields are varying, in turn implying

either periodic variation of source concentrations or of distancefrom a source. If we consider the source of the morphogen tooriginate at the center of a growing plate, the distance from thecenter to the fixed nucleation point continuously increases andtherefore the concentration at that locus decreases. When theconcentration falls below a critical value at the nucleation point, anew plate is added, and concentration of morphogen (generatedby the new plate) is at its maximum level, resetting the ‘‘clock’’,and the periodic nature of the process arises naturally. Thispattern of regulation is indicative of the morphogen acting as aninhibitor, suppressing addition of new plates as long as concen-tration remains above a critical level.

Because the properties of the system are not known, themorphogen concentration is approximated by the simple formulaCi ¼ ð1� At=AmaxÞk=d2 where Ci is the concentration of theinhibiting factor produced by plate i, At is the area of the plateat time t, Amax is the maximum effective area of the plate, k is anarbitrary constant representing the maximum concentration, andd is the distance (measured on the surface of the skeleton)between the plate center and the insertion point. The concentra-tion C at the point where new plates are added is the sum of the Ci

concentrations where i is evaluated over the set fPassoc ;Padj; Paddg;

where Passoc is the plate associated with the insertion point, Padj

are the coronal plates adjacent to the associated plate, and Padd areplates added at this time step (which have not yet beenincorporated into the adjacent set). The meridional distancebetween plates increases in a regular fashion as a sea urchingrows. If we assume a constant threshold level for plate initiationthis would imply that the rate of addition of new plates wouldincrease as the animal grows, an effect just opposite of what isactually observed. The threshold value is therefore considered tobe inversely proportional to the radius of the test such thatCthreshold ¼ Cthresholdð0Þ=aR where Cthreshold is the threshold concen-tration, Cthreshold(0) is the initial threshold concentration, R is theradius of the test, and a is an arbitrary constant which acts as athreshold decay factor.

The result of this process is that in a newly added plate, themorphogen source is within the critical distance and new platenucleation is inhibited. As the new plate grows, the center movesfarther from the nucleation point and the portion of themorphogen attributed to the plate decreases. The morphogenconcentration at the nucleation point is the sum of theconcentrations attributed to all the plates in the corona. However,because of the inverse square relationship of concentration todistance, only the plates in close proximity to the nucleation pointare of any significance. These are the newly added plate andpossibly the adjacent coronal plates.

2.2.2. Plate growth

Growth can be considered to be a chemical process(Bertalanffy, 1938), whereby a reacting substance a (a growthfactor) is transformed to another substance b which is conti-nuously removed from the system (consumed). The Bertalanffylinear (one-dimensional) growth function for length l is based onthis concept and has an exponential form l ¼ L� ðL� l0Þe

�kt whereL is the maximum length and l0 is the initial length, and isdependent on the age t of the plate and the transformationcoefficient k. In the model, length l is treated as the perimeter ofthe plate, the simplest linear measure. It is hypothesized thatindividual plates grow in accordance with Bertalanffy’s growthfunction with a constant rate coefficient (k) and asymptote (L) thatincreases linearly with plate cohort number (Fig. 3). The actualsize dependency is not directly tied to cohort number, but ratherto the increasing radius of the sphere such that, at the time ofplate addition, the ratio of the maximum perimeter of a plate type

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Fig. 3. Plate perimeter vs. cohort age based on growth line information reported

by Deutler (1926, pl. 4, Fig. 3) for a single column of interambulacral plates from

single specimen of Echinus esculentus. Data are interpreted from digital measure-

ments made from a scanned image and therefore dimensional units are arbitrary.

In this case, the plate cohorts correspond to the sets of plates added in each of the

10 columns shown in the Deutler figure and are more properly termed cohort sets,

although still corresponding to relative plate ages. Line patterns group growth

lines originating with successive cohort sets for clarity. Early plates reach

maximum size but later plates have not reached asymptotic sections of growth

curves. Growth rates are comparable for all plate cohorts (growth lines are nearly

parallel), but asymptotic size increases with cohort age (maximum perimeters

increase but must be extrapolated for later cohorts).

Fig. 4. Dimensions of a spherical cap. R is the radius of the sphere, drawn from the

origin. The base of the spherical cap is a circle with radius rb and height h and r is

the spherical radius or length of the geodesic from the pole to the base of the cap.

L.G. Zachos / Journal of Theoretical Biology 259 (2009) 646–657 649

(ambulacral or interambulacral) to the radius of the sphere isconstant (at least in the case of regular urchins). This relation-ship is supported by the results reported by Deutler (1926),Markel (1981), and Dafni and Erez (1987) and in turn supportsthe conjecture that plate growth as well as plate addition isregulated by a morphogen gradient. Production of a constantsource of growth morphogen is hypothesized at the plate center.In this case, the morphogen acts to induce skeletal growth at arate directly proportional to its concentration and is activelytransported (via cytoplasm) to the edges of the plate whereit is consumed. Each cohort of plates is given a different andlinearly increasing maximum perimeter. Assuming that a platecan never exceed a hemisphere in extent, the maximum perimetercannot exceed 2pR, where R is the radius of the test. Themaximum perimeter, therefore, is considered to increase by afactor of the current test radius for each new plate cohort.

Growth occurs on the perimeter l of each plate and ismodeled as linear growth via the Bertalanffy equation as anincrease in perimeter. Plates are assumed to be infinitely thinand to grow isotropically, i.e., approximated by a circle on thesurface of the spherical test. The plate is therefore simplifiedto a spherical cap with the same surface area as the Voronoipolygon. The surface area of this cap is 2pRh, where R is the radiusof the sphere and h is the height of the cap (Fig. 4). The height ofthe cap is

h ¼ R 1� cosr

R

� �h i

where r is the spherical radius measured on the surface of thecap (along a geodesic of the sphere). The surface area is thengiven by

a ¼ 2pR2 1� cosr

R

� �h i

where a is the spherical area. Then, the spherical radius r iscalculated from the plate area a by

r ¼ R cos�1 1�a

2pR2

� �

and the radius rb of the base of the cap is related to the sphericalradius by rb ¼ R sin r and the perimeter l of the circle isl ¼ 2prb ¼ 2pR sin r.

At each time step every plate grows by some amountdetermined from the growth parameters, resulting in an increasein plate perimeter Dl such that ltþ1 ¼ lþ Dl. A new sphericalradius rt+1 is calculated for the plate, from which the newspherical area a of the plate can be calculated by

rtþ1 ¼ sin�1 ltþ1

2pR

a ¼ 2pR2 1� cosrtþ1

R

� �h i

The radius of the test increases as R ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSai=4p

pfor all plates i in

the test.If all plates grow at the same rate such that the relative areas of

the plates do not change (and no new plates are added), platecenter points maintain their original angular positions a and b,and the overall plate topology does not change as the testincreases in size. The growth rate of an individual plate, as definedby the Bertalanffy equation is dependent on the age t of the plateand the transformation coefficient k. Except for the initial set ofplates, for which an initial perimeter is specified, all plates areadded with the same initial perimeter, which can be considered tobe infinitesimal, such that lffi Lð1� e�ktÞ.

The poles of the sphere representing the test are fixed inangular position during growth (i.e., at 0 and p). These polarpoints are considered to represent the center points of the analand oral ‘‘plates’’. The anal (apical) plate is encircled with the 10plates of the apical system. The corona is composed of theremaining plates. Plates are added at the apical ends of eachcoronal column and grow at varying rates in accordance with ageand position. This growth and addition together result in changesin the topology of the test, represented by changes in the relativelocations of the plate center points, which appear to migratemeridionally towards the mouth. This apparent migration ofplates is only relative and, in fact, all the plates expand outward inabsolute terms. However, as the test expands, the plates do changeangular position, i.e., the depression angle b of the plate center

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L.G. Zachos / Journal of Theoretical Biology 259 (2009) 646–657650

points increases (as does the actual distance from the apexmeasured along geodesics). An important consequence of this isthe width of the plate column increases to a maximum at theequator (which represents the ambitus of the echinoid) thendecreases towards the mouth. Thus the shape of a plate varieswith position on the meridian.

Plates have a maximum width at the ambitus (equator of thesphere) and the width decreases towards either pole. The width ofa column in angular terms (a) is constant for all depression anglesb, but the length of a degree of arc in the latitudinal directionchanges such that the ratio of plate width w to height h in terms ofthe components of the displacement (change in spherical radiusDr) is cosðp=2� bÞ and because of the assumption of isotropicplate growth in model calculations the meridional component ofdisplacement Drmeridional can be calculated from the isotropicdisplacement Dr by the equation Drmeridional ¼ 2Dr=½1þcosðp=2� bÞ�. The requirement to apply this dimensional correc-tion to account for the difference in shape between thehypothetical circular plate and the actual polygonal plate is asource of departure from the reality of growth. This disparitybecomes noticeable in the basicoronal set of plates, which are notmodeled realistically in this version of the model.

2.2.3. Synthesis

The model hypothesizes that the ocular plate cells constitu-tively produce a factor which induces the addition of a new plate.New plate production is regulated by a morphogen inhibiting theplate addition inducing factor. Plate inducing activity is inhibitedabove a threshold concentration of the inhibitor, which isdependent on the stage of growth of existing plates in the corona.The use of different threshold values for ambulacral andinterambulacral plates, with different nucleation locations, resultsin differing rates of alternating plate addition. All growth zonesare calculated independently, i.e., in general, plate addition is notperfectly symmetrical.

Individual plates grow along their edges, and are separatedfrom each other by a syncytial membrane. Growth of a plate isdescribed by the Bertalanffy equation with a varying rateasymptote l ¼ Lc � ðLc � l0Þe

�kt where Lc is the maximum peri-meter for cohort c which varies by some function f of age of cohortLc ¼ f ðt0Þ where t0 is the age of the animal, as distinguished from t,the age of the plate, and k is the invariant rate constant. Theprocedures described above result in (a) a new radius R for thetest, and (b) new meridional displacements Drij for each plate i incolumn j for each time step. The displacements are additive for acolumn and their sum is equal to the length of the geodesic frompole to pole (which increases as the radius R increases).

Fig. 5. Relationship between Delaunay triangles and Voronoi polygons. Center

points of polygons are connected to form triangles, each leg is perpendicular to an

edge of the Voronoi polygon, and each edge bisects the distance between two

center points. Legs of the triangles, although perpendicular to the polygon edges,

do not necessarily intersect the corresponding edge. The areas of the polygons are

optimized in the sense that every point within the polygon is closer to its center

point than to the centers of any other polygon.

3. Implementation

3.1. Delaunay triangulation

A critical aspect of the modeling program is a method tomaintain a table of all plate adjacencies, maintain a list of nodesassociated with each plate edge, and ensure the co-location ofnodes associated with adjacent plates while at the same timeallowing for growth of plates, addition of new plates, andtranslocation of plates. The method can be generalized as thesolution to the problem of determining, for a finite set of points(plate centers) in a continuous space (the test), the association ofall locations in the space with the closest member of the set ofpoints. This solution partitions the space into a set of regions(presumptive plates) equivalent to Voronoi polygons. The Voronoicriterion is: given a set of n points fp1 . . . png on the unit sphere,these points can be used to generate a partition of the sphere S

into n convex non-overlapping spherical polygons P (althoughthey may share edges). A point p on the sphere in the interior (noton an edge) of polygon Pk satisfies the inequality jp� pkjojp� plj

for all lak. The strict inequality means that for any fixed p there isonly one k such that the relationship holds. Therefore, thepolygons do not overlap and every point p on the sphere is in atleast one polygon. The partition of the sphere is such that everypolygon Pk contains in its interior those points p 2 S which arecloser to pk than to any other generating point. A point pedge on anedge between two polygons Pk and Pl satisfies the equation jp�pkj ¼ jp� plj: A point pnode on the intersection of the edges of threepolygons Pk, Pl, and Pm satisfies the three equations:jp� pkj ¼ jp� plj; jp� pkjj ¼ jp� pmj; jp� plj ¼ jp� pmj. The dualprocedure, known as Delaunay triangulation, connects the set ofpoints such that any two joined points share an edge of a Voronoiregion, and results in a set of triangles that also completelypartitions the space (Okabe et al., 2000). Conceptually, if thecoronal skeleton is defined by centers of growth of every plate,then the Voronoi regions around each center define the topologyof the associated plate such that every point inside the plate iscloser to its center than to the center of any other plate. Therefore,the Voronoi regions define the maximal extent of each plate in thecoronal mosaic (Fig. 5).

Two structural archetypes that can be used to understand thestability of spatial structures are the lattice and the platestructures (Wester, 1993). A triangulated convex polygon repre-sents a pure lattice structure, and a trivalent (no more than threefaces meet at a point) convex polyhedron represents a pure platestructure. Kinematic stability is imparted by the distribution ofbar forces (acting along the lattice) and edge forces (acting at thejunctions of plates). There is a dualism between these twoarchetypes and plates can be substituted with a lattice viatriangulation (Wester, 1996). A Voronoi polygonalization repre-sents a pure plate structure by this definition, and the dualDelaunay triangulation a pure lattice structure. Wester (2002)

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L.G. Zachos / Journal of Theoretical Biology 259 (2009) 646–657 651

used the skeleton of the sea urchin as an example of a perfect andpure plate structure. He stated that because of the manner inwhich urchins grow, by both plate growth and addition, that theskeleton could be structurally stable only if it were a pure platestructure. There is then, besides the convenient geometriccharacteristics, a functional reason to model a sea urchin usingVoronoi polygons.

Spherical Delaunay triangulation can suffer from various formsof degeneracy, including identical, collinear, and coplanar points(Saloman, 1999). Degeneracy conditions are handled as specialcases in the model program and corrected if they occur. Theseforms of geometric degeneracy are not just mathematicallyimportant; they also appear to represent ‘‘forbidden’’ features innatural systems. When such degeneracies do arise in naturalsystems, they are virtually instantaneous occurrences (an exampleis when four plates intersect at a point) and do not persist.Another biologically significant aspect of the geometric model isthat when the number of regions is large and the points relativelyevenly distributed, the Voronoi polygons tend to approach regularhexagonal shapes, a feature characteristic of echinoid coronalplate patterns.

The implementation of spherical Delaunay triangulation wasbased on the algorithm described by Augenbaum and Peskin(1985). The algorithm incorporates initialization of the triangula-tion and insertion of new points as its two major steps.A minimum of four non-collinear points is required to initiatethe triangulation, although for the echinoid model a completetriangulation representing an initial set of plates is read from afile. Insertion of new points accesses several methods for testingfor degeneracy, testing for broken triangles, calculation of newtriangles, and creation of adjacency and node set tables. Distanceand bearing between plates is calculated as well. In general, vectormethods are used in place of trigonometric calculations in orderto optimize computer runtime. Points are stored with coordinatesin both three-dimensional Cartesian and polar form for conve-nience in calculations.

Although the use of Voronoi polygons to represent plates is asimple and effective methodology for modeling coronal plategrowth, it causes significant difficulties in the modeling of plateaddition. It is not possible in an unconstrained Delaunaytriangulation to model realistically the small plates as they areadded. Addition of a new plate into the set of existing polygonscreates a new polygon by division of an existing polygon. Thenewly added plate breaks the symmetry of the plates located at ornear the insertion point until it grows away from the point.Unrealistically large plates at the head of each column are artifactsof this process.

Fig. 6. Graphical representation of the Delaunay triangulation data model. The

central node is indexed, and is associated with the three surrounding nodes in

counterclockwise (CCW) order. The lines connecting these nodes are edges of three

Voronoi polygons. The centers of these polygons, also ordered in a CCW direction,

are connected to form a Delaunay triangle. The indices of the centers and the nodes

on opposite sides of the central node correspond to one another. This same

relationship is repeated for every node and every center on the sphere, which is

completely covered by both triangles and polygons. These relationships are used to

construct two arrays, one for nodes and one for centers, which form the core of the

data model.

3.2. Initialization

The model is sensitive to the initial configuration of plates. Theinitial plate arrangement described by Gordon (1926) forPsammechinus miliaris was the basic model used for initialization.This initial configuration is valid for cidaroid echinoids as well(Mortensen, 1927) and includes the 10 apical plates (oculars andgenitals), 5 four-plate lozenges of interambulacral plates, and fivetwo-plate columns of ambulacral plates. The actual configurationof this initial model can be varied somewhat by altering thespecific locations of the plate centers and thus their interplatedistances, although the plate adjacencies do not change. Theprimary effect of this variation is a change in the locations of theplate insertion points and thus the widths of the columns. Otherconfigurations, using different numbers of plates, can also beused. There is no implied symmetry, so the number of columnscan also be varied by using a different number of ocular plates.

3.3. Deformation

Although the growth model proceeds with the skeletonrepresented over the surface of a sphere, the shape of outputgraphic files can be modified by applying an affine transformationto the sphere, a process called deformation. There is nothing in thedevelopmental model that suggests any particular shape for thebody of an urchin, and the fact that nearly all regular urchins canbe shown to approximate a drop-like shape regardless of thedetails of the plate structure strongly suggests that the shape ispurely mechanical in origin.

The membrane model was shown by Ellers (1993) to modelthe profile of most urchins. The numerical solution to the Young–Laplace equation he described cannot be used directly to transforma given spherical coordinate. It is possible to use curve-fittingtechniques to estimate the functional form of the equation for anygiven solution, but the non-linear form of the solutions addssignificant complexity to the problem. However, by using thenumerical method to generate a set of discrete points with highdensity, satisfactory transformation of the coordinates can beaccomplished by simple linear interpolation. The membranemodel is generated once and the solution of the z-coordinate interms of depression angle b, normalized to a unit sphere, is storedas a look-up table. The translation of any point can be rapidlycalculated by finding the values in the table that bracket the point,and interpolating.

3.4. Data model

The essential results of Delaunay triangulation are twoassociated arrays, one of the center point triangles and the otherof the connecting nodes where the primary index of the array isthe same as the enclosing Delaunay triangle (Fig. 6). Deriveddirectly from these arrays are (a) an adjacency matrix of all platesadjacent to any given plate, (b) the ordered edges of the plates,

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(c) the ordered nodes composing the edges, (d) the distancesbetween plate centers and nodes, and (e) the relative bearingsbetween adjacent plates.

This data model has many important characteristics withregard to efficiency in model execution, a key concern in modeldevelopment. Because of the dual nature of a Delaunay triangula-tion (based on the center points) and a Voronoi polygonalization(based on the nodes), the complete 3D structure of the model iscontained in these two data objects. In addition, once thetriangulation is initialized, additional points can be added andthe triangulation updated in O(n) time. The adjacency of any twoplates (whether they share an edge) and the distances andbearings between any two plates can be readily calculated, and

Fig. 7. Class/object diagram for the growth model. See text for explanation of the

classes and their implementation in C++.

Fig. 8. Generalized process flo

any change in relative position of center points immediatelyresults in the new plate edge configurations.

3.5. Object model

The object model is composed of seven C++ classes, althoughadditional classes in the Standard Template Library (STL) of C++(Ammeraal, 1997) are used intensively. The GUIForm classcontains the attributes required to initialize and interface with(in this case) the Microsofts Windowss Graphical User Interface(GUI). The utility class driver acts as a controller between the GUIand the other program classes. A Universal Modeling Language(UML) diagram of the object model, following the conventionsdescribed by Lee and Tepfenhart (2001) is shown in Fig. 7.

The basic geometric data element of the model is defined bythe Point class. All other higher-level objects are composed of Point

objects. The Plate class is a particular collection of Point objectsthat represent the center point of a skeletal plate as well as the setof nodes corresponding to points defining the plate edges. It isdependent on the PlateType class for specific plate definitions.

The core of the model is based on the Ech_Skeleton class whichincludes the attributes and methods required for manipulatingthe Plate objects that comprise the model echinoid, as well asmost of the methods used to create output files. The underlyingdata model is defined in the delaunayTriangulation class, whichincludes the attributes and methods required to maintain thethree-dimensional locations, extents, and interrelationships of allthe elements of the model.

In simple terms, an instance of a modeled sea urchin is anEch_Skeleton, composed of a large number of Plates of different

w for the growth model.

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L.G. Zachos / Journal of Theoretical Biology 259 (2009) 646–657 653

PlateTypes, each plate being composed of a center Point and nodePoints and edges, where the edges connect node Points accordingto a pattern defined by delaunayTriangle. Growth is controlled bythe driver routine using the parameters entered by the userthrough the GUIForm.

Object-oriented programs (OOP) are not readily flowcharted,since flowcharts fail to capture many of the properties of OOP, e.g.,encapsulation, modularity, or inheritance. Considering that themany functions of the program are complexly interleaved,generalized process flow (Fig. 8) varies according to choice ofparameters.

3.6. Parameters

At its basic level, the model requires an initial startingconfiguration, parameters defining the process of plate addition,

Table 1Input parameters for echinoid growth model.

Parameter Number of elements Value domain

Plate growth rate 6 plate types 0.001–1.0

Maximum perimeter factor 6 plate types 0.5–2pThreshold decay factor 2 plate types 1.0–3.0

Initial test radius 1 Various

Shape deformation ratio 1 0.01–100.0

Number of time steps (iterations) 1 50–1000

Initial configuration 1 input file 4–1000 plates

The value domain lists the respective range of values for realistic simulations.

Fig. 9. Example output, based on a growth model with 42 initial plates, equal threshold

deformation applied. Shading represents different plate types. (A) Selected growth stage

is slightly reduced in size to fit illustration). (B) Same growth stages as in A, normalized t

an adoral direction.

parameters defining the processes of plate growth and, if desired,parameters defining the deformation to be applied to thespherical model (Table 1). Programmatically there are additionalrequirements for file names and directories, but these are notconsidered here.

The Bertalanffy growth equation is dependent on an exponentwhich is the product of the growth rate coefficient and amonotonically increasing age. The natural measure of age in thecomputer model is the iteration or time step, and this choiceconfines the value of the rate coefficient to a range of values thatresult in realistic growth curves. The Bertalanffy equation isapplied in the simulation to the perimeter of individual plates, sothat the equation also implicitly depends on a value for themaximum perimeter of a plate. Choice of too high a growth rateresults in chaotic behavior of the model because the underlyingassumptions (which were used to simplify the model) are validonly when the intermediate steps of growth are close inmagnitude and changes in the overall geometry of the platesdoes not significantly change topology between time steps.

The initial configuration represents the urchin at time t ¼ 0. The‘‘standard’’ configuration consists of a total of 42 polygonsrepresenting five genital, five ocular, 10 ambulacral, and 20interambulacral plates, along with a mouth and anal ‘‘plate’’ (theplating of the peristomal and periproctal membranes is notmodeled). The initialization parameters therefore are the coordinatesof the 42 plates, along with their plate type, and an initial radius forthe sphere. The algorithms for plate addition are hard-coded into theprogram, but the threshold values for the morphogens responsiblefor ambulacral and interambulacral plate nucleation as well as thelocation of the points of plate addition can be varied.

values for plate addition, and 40 iterations, with Young–Laplace membrane model

s at 1st, 13th, 27th, and 40th iteration showing increasing size with time (last stage

o a unit radius to show details of plate addition and apparent migration of plates in

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Fig. 11. Growth of interambulacral plate perimeters (normalized to test radius)

with increasing age. Plate growth is shown only for 2nd, 5th, and 10th plate

cohorts for clarity. Plate cohorts are added approximately every 20th iteration in

this example, but individual plates in a cohort are added at different ages (i.e.,

columns grow independently).

L.G. Zachos / Journal of Theoretical Biology 259 (2009) 646–657654

The growth of plates requires a separate set of parameters foreach type of plate (there are six defined types: mouth, anus,genital, ocular, ambulacral, and interambulacral). The parametersare the rate constant k, the maximum perimeter for the firstcohort, and the factor for incremental increase in maximumperimeter with cohort number.

The deformation of the sphere, using the (Ellers, 1993)membrane model, requires three parameters: the radius of apicalcurvature, the apical pressure, and the gradient of pressure withdepth. Because the numerical approximation is normalized, theradius of apical curvature is set to the value 1.0, and the ratio ofgradient to pressure is input as a single non-dimensional value.Although not really representative of the physical parameters forwhich they are named, a judicious choice of the gradient topressure ratio (between 100 and 0.01) can result in a variety ofshapes that approximate that of many urchins.

4. Simulation and results

The somatic growth of the test in an example model is shownin Fig. 9A. The addition and growth of individual plates are mostclearly seen when the test is normalized to a unit radius (Fig. 9B).The figured models display the application of Young–Laplacedeformation to the original spherical reference system.

4.1. Plate addition

The morphogen/inhibition algorithm used in the model resultsin one of the more striking features of an echinoid test: thealternation of plate columns. The degree of offset between platecolumns can be varied but is dependent only on the choice ofparameters used in the algorithm. Although other naturalmechanisms of plate alternation are certainly possible, the modelconfirms that the proposed mechanism is a viable solution thatcould be tested experimentally.

Fig. 10. Expansion of test radius with age, spherical model. (A) Sigmoidal growth

curve, stair-step pattern as test approaches maximum size is an artifact of periodic

plate additions at critical sizes. (B) Sigmoidal growth curve, high growth rate

caused model to reach maximum size relatively quickly. (C) Growth curve at slow

growth/slow addition rates, model run ended in exponentially increasing period of

growth.

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Fig. 12. Model performance. (A) Pure growth model with no plate additions, 42

plates. Linear increase in run time with increased number of iterations, using the

spherical model and the Young-Laplace deformation model. (B) Pure growth model

with no plate additions, varying numbers of plates. O(n log n) increase in run time

with increased number of plates n, spherical model. (C) Growth with plate

additions, 42 initial plates. Variable increase in run time associated with increase

in threshold value, 100 iterations, spherical model. Numbers are the total number

of plates at end of respective run. Approximate O(n log n) increase in run time with

increased rate of plate additions (measured by threshold value).

L.G. Zachos / Journal of Theoretical Biology 259 (2009) 646–657 655

4.2. Growth

The somatic growth of a natural echinoid test, as measured bythe radius, can generally be graphed as a sigmoidal or logisticcurve (Rogers-Bennett et al., 2003). Growth increases at anexponentially increasing rate for some period of time thendecreases exponentially until the radius reaches a constant value.This pattern can be seen in the growth simulations (Fig. 10A, B).The length (in iterations) of a model run is arbitrary, i.e., there areno programmatic requirements. Depending on the parametersused in the simulations, growth may reach a maximum sizebefore the model computation is terminated (Fig. 10B), or may bestopped while still in the exponentially increasing region of thecurve (Fig. 10C).

The range in growth for the set of plates composing the test of arepresentative model at any given stage of growth is shown in Fig. 11,in which plate perimeter has been normalized to test radius forclarity. This pattern can be compared with the pattern empiricallyderived from the Deutler (1926) biometric data (Fig. 3), keeping inmind that the Deutler data are derived from a single specimen at asingle point in its growth series. This demonstrates that the model issimulating plate growth as expected. These results confirm that,even with the simplifying assumptions, the underlying Delaunaypattern and Bertalanffy growth of individual plates are sufficient forrealistic formation of coronal plate mosaics.

4.3. Performance

Performance of the pure growth model (no plate additions)shows a linear increase in processing time as the number ofiterations increases (Fig. 12A). This gives a baseline performancemeasure confirming linear response for a constant n, which ismeasured as the number of plates. Although processing times willvary by machine and configuration, on a 2.19 GHz Intel Duo ProCPU and 2.00 GB RAM laptop a model with 42 plates and 100iterations executed in 43 s for a spherical model and 135 s if theYoung–Laplace deformation is computed for the output files.

Performance of pure growth models with different numbers ofplates and a constant number of iterations (Fig. 12B) is demon-strated to be O(n log n). This confirms that the performance of thealgorithm used for spherical Delaunay triangulation is also noworse than O(n log n), better than the O(n2) time typical for two-dimensional Delaunay triangulation algorithms. A representativepure growth model with 1000 plates completed 50 growthiterations in 1073 s or 17.9 min.

A typical model with plate addition is more difficult to evaluatebecause the number of plates increases with the number ofiterations, but the overall processing time increases geometricallywith increasing plate addition threshold (Fig. 12C). The increase ofthe plate addition threshold (which reduces its inhibitory action)results in increased numbers of plates added to the corona as theiterations increase. The exact number is a property of the initialplate configuration, the intrinsic growth rates of the plates, andthe threshold values (which can differ for ambulacral andinterambulacral plates) for new plate addition. A representativemodel with 42 initial plates, intrinsic growth rates of 0.03 for bothambulacral and interambulacral plates, and plate addition thresh-old of 20,000 for both types of plates completed 100 growthiterations in 1259 s or 21 min. This particular model ended with atotal of 1209 plates in the test.

5. Conclusions

The success of the model serves as a proof of concept that it ispossible to simulate major aspects of growth of sea urchins based

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on a simplified set of developmental and geometric assumptions.The model suggests mechanisms for growth and addition of newplates that can be tested experimentally, although probably noteasily. Particularly significant would be isolation of tissuesresponsible for new plate addition and determination of mechan-isms that activate them. Whereas there have been a numberof studies of somatic growth in sea urchins, there are a fewthat detail the growth of individual plates, and none of thesein a comprehensive manner that is applicable to model valida-tion.

As configured, growth simulations generated by the computa-tional model depart from the pattern seen in living sea urchins ateither pole of the reference frame. A problem with the addition ofnew plates (at the apical pole) was noted earlier, and is caused bya natural distinction between plates of the apical system andplates of the corona that is not implemented in the model. Theproblem with basicoronal plate configurations (at the oral pole)appears to be an artifact of the simplifying assumption of isotropicplate growth (and the programmatic methods for its implementa-tion within the geometric constraints of the model).

Intriguingly, the effects of these problems on the growthsimulations are most apparent when simulating the growth ofmodern sea urchins. Modern sea urchins are all characterized by10 ambulacral and 10 interambulacral plate columns and for themost part by a relatively large and static apical system. Therestriction in number of plate columns results in plates that arerelatively wider than tall, i.e., which depart from the idealizedcircular shape. In contrast, extinct disparate groups of sea urchinsknown only from the Paleozoic Era are characterized by greatlyreduced (or non-existent) apical systems, more than 20 totalcolumns of plates, and plates that are generally isometric inoutline. It is often stated that the modern sea urchins are the lesscomplex forms, but in terms of the computational modeldescribed here the Paleozoic forms can be considered to be theleast complex.

Current research is underway along two parallel paths:modification of the model to correct for the problems notedabove; and extension of the model to simulate the variety ofPaleozoic-style sea urchin forms. Better simulation of plateaddition and growth around the apical system can probablybe obtained by altering the underlying spherical Delaunaytriangulation (and associated data structure) from an uncon-strained model to a constrained model (Hjelle and Dæhlen, 2006).This will explicitly distinguish apical from coronal plates withlittle loss in generalization of the growth model. Solution of theproblem of anisotropic plate growth will need to be evaluated interms of the cost in added complexity (which could be substantialfrom the underlying developmental aspect) versus benefit in morerealistic simulation. On the positive side, relaxation of thisconstraint may lend itself readily to simulated growth of Paleozoicsea urchins.

Acknowledgments

The author is grateful to Jim Sprinkle, Chris Bell, Ann Molineux,Rich Mooi, and Tim Rowe who served on his dissertationcommittee and for the many hours they spent in helping toimprove the results of this study. Olaf Ellers made many helpfulcomments on an early version of the model, Jacob Dafni graciouslysent the author much of the raw data regarding his studies of seaurchin growth, and David Raup helped to improve the manuscriptwith his comments. This work was supported by the GeologyFoundation of the Jackson School of Geosciences at the Universityof Texas at Austin, and by the Smithsonian Institution FellowshipProgram.

Appendix A. Supplementary material

Supplementary data associated with this article can be foundin the online version at doi:10.1016/j.jtbi.2009.04.007.

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