Kinematics of Cranial Ontogeny: Heterotopy,...

Kinematics of Cranial Ontogeny: Heterotopy,Heterochrony, and Geometric MorphometricAnalysis of Growth Models

CHRISTOPH PETER EDUARD ZOLLIKOFERnAND MARCIA SILVIA PONCE DE

LEONAnthropological Institute, University of Zurich, CH-8057 Zurich, Switzerland

ABSTRACT In this paper, we examine the relationship between the classical concepts ofheterotopy, heterochrony and ontogenetic allometry as descriptive and as explanatory categories inthe investigation of evolutionary developmental novelty in the hominid skull. We use concepts ofkinematic analysis of locomotion to propose a methodological framework for the kinematic analysisof cranial form change during ontogeny. We argue that a combination of geometric-morphometricmethods with graphics visualization tools currently represents the most adequate means to analyzethe kinematics of ontogeny. Using cranial growth models, we simulate how evolutionarymodifications of developmental processes impinge on morphological patterns of ontogeny, andexplore how differences in ontogenetic patterns can tentatively be traced back to underlying processdifferences. Our analyses indicate that minor alterations in growth parameters elicit complexpatterns of ontogenetic modification that are difficult to describe with the standard repertoire ofheterochronic terminology. The proposed kinematic and model-based approach is used in acomparative analysis of cranial ontogeny in Neanderthals and anatomically modern humans,indicating that early ontogenetic modification of a small set of growth parameters is a major source ofevolutionary novelty during hominid evolution. J. Exp. Zool. (Mol. Dev. Evol.) 302B:322–340, 2004.r 2004 Wiley-Liss, Inc.

INTRODUCTION

Since the early days of evolutionary thinking ithas been recognized that a prime source ofphylogenetic novelty lies in the modification ofontogeny. Haeckel’s initial definition of hetero-chrony and heterotopy as the temporal and spatialagents of evolutionary change of ontogeneticpathways (Haeckel, 1866) still represents animportant conceptual framework to explore, mea-sure, and explain how new morphologies arebrought about during evolution. How can, in anevolutionary-developmental research setting, het-erochrony and heterotopy be studied as agents ofbiological change? Researchers investigating ani-mal models or clinical instances of cranial devel-opmental modification face the challenge ofdisentangling the tremendously complex networkof cause and effect that links genes via epigeneticinteractions at various levels of organization tospatiotemporal patterns of change in cranialmorphology. Nevertheless, developmental geneti-cists find themselves in a relatively comfortablesituation compared with paleoanthropologists who

study evolutionary developmental diversificationof the hominid cranium. In addition to thenotorious material scarcity of the fossil record,fossil hominid ontogenies are no longer accessibleto direct observation and/or experimental interac-tion such that patterns of morphological diversityobserved in the fossil record must be linked withexperimental evidence from extant species to inferpotential mechanisms of evolutionary novelty viadevelopmental modification.

In this paper, we discuss two closely relatedquestions that arise in comparative morphometricstudies of fossil hominids from an evolutionarydevelopment perspective: 1) How can spatiotem-poral patterns of morphological change in thegrowing skull be optimally measured to establishand test hypotheses about underlying develop-mental processes? 2) How does evolutionary

Grant sponsor: Swiss NSF; Grant number: 31-67209.01nCorrespondence to: Christoph Peter Eduard Zollikofer, Anthro-

pological Institute, University of Zurich, Winterthurerstrasse 190,CH-8057 Zurich, Switzerland. E-mail: [email protected]

Received 8 March 2003; Accepted 25 March 2003Published online in Wiley Interscience (www.interscience.wiley.

com). DOI: 10.1002/jez.b.21006

r 2004 WILEY-LISS, INC.

JOURNAL OF EXPERIMENTAL ZOOLOGY (MOL DEV EVOL) 302B:322–340 (2004)

modification of a developmental process impingeon spatiotemporal morphological patterns ofontogeny? We study these issues using growthmodels, computer simulations and empirical evi-dence. Growth models are exploratory tools thatassume a middle position between experiment andanalysis. They combine the bottom-up approach of‘‘in silico’’ (i.e., computer-based) simulation ofdevelopmental processes with top-down analysis ofthe resulting morphological patterns of shapechange, while giving full control over all growthparameters. This approach provides general in-sights into potential correlations between devel-opmental process modification and patternmodification that can later be used to assessempirical data of spatiotemporal morphologicalchange.

The language of ontogenetic modification,from molecules to morphology

Heterochrony sensu Gould (’77) can be under-stood as a dissociation of the velocities of change insize versus change in shape between ancestor anddescendant taxa. Building upon Gould’s argu-ments, Alberch et al. (’79) proposed an operationalframework for quantitative analysis of hetero-chronic dissociation, using the concept of ontoge-netic trajectories that can be followed through age-size-shape space. Over past decades, concepts andterminology of heterochronic analysis have under-gone considerable diversification and modification(e.g., Shea, ’84, ’88; Raff and Wray, ’89; McKinneyand McNamara, ’91; McNamara, ’97). Gould’sterminology, which was originally confined tocomparative description of modifications ofshape-age trajectories, was expanded to includeheterochronic dissociation of size-age trajectories(Godfrey and Sutherland, ’95; Godfrey andSutherland, ’96; McNamara, ’97; Rice, ’97; Klin-genberg, ’98). In parallel to the expansion ofterminology, the scope of heterochronic analysiswas widened considerably and now comprisesstudies of almost any phenomenon of temporalmodification of ontogeny, from genes and morpho-genetic processes (e.g., Wilson, ’88; Slack andRuvkun, ’97) to phenes and morphological pattern(e.g., Zelditch et al., 2003). With growing empiricalevidence from molecular developmental genetics,it was recognized that the relationships betweenprocess heterochrony and pattern heterochronyare intricate and difficult to disentangle. Withinthe complex network of spatiotemporal interac-tions that connects genes with phenes, ‘‘process’’

and ‘‘pattern’’ can be defined at various levels ofscale in both space and time. For example, it wasshown that a heterochronic pattern may be causedby a heterotopic process (Raff and Wray, ’89; Cuboet al., 2000). Conversely, heterochronic dissocia-tion between developmental modules may resultin heterotopic dissociation in the structure as awhole (McNamara, 2002a). Terminology thusappears as a matter of perspective.

These issues can be exemplified with a clinicalexample (a perspective already adopted by Wilsonet al., ’88). Crouzon syndrome is a congenitalmalformation characterized by craniosynostosis(early closure of one or more cranial sutures), ahypoplastic maxilla causing upper airway obstruc-tion and pseudoprognathism, and other congenitalcranial and postcranial defects (Jones, ’88)(Fig. 1). It has been shown that premature sutureclosure, which is prevalent in both Crouzon andApert syndrome, is related to three different typesof gain-of-function mutants of FGFR (fibroblastgrowth factor receptors) genes 1 to 3 (Reardonet al., ’94; Neilson, ’95; Wilkie et al., ’95; Yu et al.,2000). These mutations affect temporal patterns ofdifferentiation, and probably also spatial migra-tion patterns, of neural crest cells (Sarkar et al.,2001; Abzhanov et al., 2003; Santagati and Rijli,2003).

How can Crouzon syndrome be described interms of spatiotemporal process modification? Again-of-function mutant in an FGF receptor generepresents molecular process heterochrony, as itleads to increased rates of signal transduction,differentiation and bone deposition in neural crestcell derivatives. At the same time, potential effectson migratory patterns must be referred to asheterotopic. From a phenotypic perspective

Fig. 1. Crouzon syndrome in an adolescent. Note maxillaryhypoplasia, and re-ossification of coronal suture followingsurgery (arrows).

KINEMATICS OF CRANIAL ONTOGENY 323

Fwhich is important regarding timing and stra-tegies of surgical correction (Richtsmeier et al.,’98; Sailer et al., ’98)Fsuture closure is aperamorphic effect, thus representing an adultfeature, but reduced midfacial growth is paedo-morphic. Finally, considering the skull as a whole,functional pseudoprognathism resulting from dis-sociation between normal mandibular (Andresenet al., 2000) and retarded midfacial growth resultsin cranial heterotopy in comparison to unaffectedsubjects.

The Crouzon example shows that heterochronicand heterotopic terminology is indeed a matter ofperspective. However, ‘‘perspective’’ does notsignify here personal taste but expresses that factthat cause and effect, or process and pattern, ofspatiotemporal variants of developmental path-ways must be investigated and compared atdifferent levels of organization and integration.

Similar arguments can be applied to questionsof evolutionary modification of cranial ontogeny.Reconstructing phylogenies with evolutionarydevelopmental data means that we must examinespatiotemporal homology relations between onto-genetic processes at various levels of scale, both intime and space (Lieberman, 2002). The definitionof homology relations between developmentalunits and processes is a complex task, since cranialdevelopment is governed by a web of genetic andepigenetic interactions rather than by a temporalsequence and/or spatial hierarchy of cause andeffect connecting genes to phenes (Lieberman,’99). Nevertheless, establishing and testing hy-potheses about ontogenetic homology relation-shipsFat least at the pattern levelFis anessential prerequisite for measuring ontogeneticmodification at the phenotypic level.

Various researchers have tackled these issuesand contributed to the foundations of a quantita-tive framework for comparative analysis of hetero-chronic and heterotopic dissociation in two- andthree-dimensional morphologies (Fink and Zel-ditch, ’95; Godfrey and Sutherland, ’96; Zelditchand Fink, ’96; Godfrey et al., ’98; O’Higgins, 2000;Zelditch et al., 2000; Roopnarine, 2001; Zelditchet al., 2003), and there is a growing number ofstudies investigating the modular temporal as-pects of ontogeny and their dissociation (Rice, ’97;Vrba, ’98; Cubo, 2000; Cubo et al., 2002; McNa-mara, 2002b; Vinicius and Lahr, 2003). Here, webuild upon these achievements and complementthem with model considerations to devise anexplicit operational framework for the compara-tive morphometric analysis of developmental

modifications during the evolution of the hominidskull.

The kinematics of morphology

In physics, during the analysis of movement aclear difference is made between kinetics andkinematics. While the former discipline studiesforces, energy and moments, the latter describesthe movement of bodies in space and time, withoutexplicit reference to the underlying processes.Analogously, ontogenetic analysis has a kineticand a kinematic aspect: we may study processes ofgrowth and development (kinetics), or the result-ing spatiotemporal patterns of morphologicalchange (kinematics). In this section, we formulatea kinematic approach to the classic notions ofheterochrony, heterotopy and allometry.

To illustrate the logical issues that have to betaken into consideration during such analyses, weuse the analysis of human locomotion (Winter,’90) as an example (Fig. 2A). In kinematic studies,one measures temporal changes in morphology bytracking the position of reference points pi on thehuman body (so-called landmarks) that werechosen to denote relevant anatomical locationson limbs (notably joints), trunk and head. It ismost reasonable and efficient to sample spatialpositions pi(xi,yi,zi) of the reference points in an

Fig. 2. Comparison between the kinematics of locomotion(A) and the kinematics of cranial ontogeny (B). A: Theposition of measurement points distributed over the body of arunning individual is tracked with a video system. Frame-by-frame analysis permits reconstruction of velocity vectors.Together, these vectors define a displacement field thatspecifies the direction and magnitude of positional changeper unit time at each point in space (human figure fromMuybridge, ’55). B: The position of anatomical landmarks in agrowing skull is sampled at different developmental stages.Shape change per unit time is quantified with growth velocityvectors, which define a displacement field that specifies thedirection and magnitude of positional change per unit time ateach point of the cranial morphology.

C.P.E. ZOLLIKOFER AND M.S. PONCE DE LEON324

external system of coordinates (e.g. that of a videotracking system) at various points in time t. Theresulting spatiotemporal trajectories P(t)

PðtÞ ¼

p1ðtÞp2ðtÞ� � �

piðtÞ� � �

pKðtÞ

0BBBBBB@

1CCCCCCA

¼

x1ðtÞ y1ðtÞ z1ðtÞx2ðtÞ y2ðtÞ z2ðtÞ� � � � � � � � �

xiðtÞ yiðtÞ ziðtÞ� � � � � � � � �

xKðtÞ yKðtÞ zKðtÞ

0BBBBBB@

1CCCCCCA

ð1Þ

of all points pi (i=1yK) describe the individual’strajectory through space. Kinematic trajectoriescan be used to derive a variety of additionalmeasurements; for example, it is possible tocalculate velocity vectors

viðtÞ ¼Dxi

Dt

Dyi

Dt

Dzi

Dt

� �¼ vxi vyi vzi

� �ð2Þ

that indicate the speed and direction of movementof each landmark in space and time (Dt can bethought of as the frame rate of the video system).The set of all velocity vectors defines a vector field,or displacement field

DðtÞ ¼ fvig ð3Þthat indicates, for each landmark, how its positionwill change from time t to time t+Dt.

Taking into account that the human body has amodular organization, D(t) can be analyzed invarious ways. For example, we may investigatehow the subject is displaced as a whole relative tothe outside world, or we may study how bodysegments move, accelerate and decelerate relativeto each other. As a further strategy, we mayexplore the structure of D(t) with statisticalmethods to decompose locomotion into yet un-recognized modules exhibiting localized kinematicproperties. In a subsequent step, we comparetrajectories between individuals. To perform thesecomparisons in a biologically meaningful way, thefollowing conditions must be met:

* Correspondence of measurement points: Datasampling must be based on homologous land-marks in all individuals.

* Correspondence of locomotor conditions: Compar-isons of running performance (e.g. speed) pre-suppose that all individuals in the sample run intothe same direction, under the same generalconditions (e.g. level run OR slope run), and usethe same stepping pattern (e.g. bipedal runningOR bipedal hopping OR quadrupedal locomotion).In other words: comparisons between magnitudes|vi| of velocity vectors presuppose that the vectorsvi are collinear.

* In a final step, we may complement kinematicanalyses with data from force platforms, straingauges and electromyograms in an attempt toreconstruct the kinetics behind the observedkinematic patterns.

Let us return to the analysis of ontogenetickinematics. To tackle questions of heterochronicand heterotopic modification, morphometric stu-dies must follow similar aims and strategies(Fig. 2B). In ontogenetic kinematics, the term‘‘movement’’ (which signifies change in spatialposition over time) is substituted with ‘‘morpho-logical change’’ or ‘‘change in form,’’ while‘‘speed’’ is substituted with ‘‘developmental rate’’or ‘‘growth rate.’’ The basic analytical strategyconsists in tracking spatial morphological changeover time, analyzing and comparing develop-mental trajectories through morphospace, andcomplementing these data with results fromexperimental developmental biology in order toinfer the processes that generate the observedpatterns of morphological change.

A central issue of ontogenetic kinematic analy-sis is data sampling. As in a kinematic study ofhuman locomotion, in a kinematic study of cranialdevelopment it is most reasonable and efficient tosample the spatial position of anatomical land-marks, which denote locations of homology be-tween the specimens in a sample. Ontogenetickinematic data can be obtained in two ways, bysampling landmark positions pi(xi,yi,zi) from thesame individual at subsequent points in time(longitudinal data) or, alternatively, by samplinghomologous data from individuals belonging to thesame population or taxon, but representing dif-ferent ages (cross-sectional data).

In both cases, the resulting data set can beimagined as an ontogenetic trajectory P(t) (seeEq. 1) through shape space (a precise definition ofshape space is provided in Appendix I). Followingthe formalism proposed in Equations (1-3), it ispossible to derive developmental velocity vectorsvi(t) that indicate the rate (temporal component;vector magnitude) and direction (spatial compo-nent) of positional change of pi at a given time t.Further, we calculate displacement fields D(t) thatindicate how the skull as a whole changes its shapeduring ontogeny, and how position and orientationof different cranial regions are modified relative toeach other.

A central task of the evolutionary analysis ofontogenetic kinematics is to compare ontogenetictrajectories between taxa. In analogy to the


kinematic analysis of locomotion, during suchanalyses, the following preconditions must be met:

* Homology of structure: Data sampling must bebased on homologous landmarks.

* Homology of pattern: If we compare ontogenetictrajectories with respect to their temporal proper-ties (heterochrony), we must verify that theyrepresent spatially homologous patterns (and,ultimately, homologous underlying processes)(Nehm, 2001; Roopnarine, 2001). In operationalterms: magnitudes |vi| along ontogenetic trajec-tories can only be compared if vectors vi arecollinear (Fig. 3). Accordingly, the postulate fromlocomotion analysis (‘‘running into the samedirection under the same conditions’’) translatesinto ‘‘moving along the same ontogenetic trajec-tory’’ (Zelditch and Fink, ’96; Godfrey et al., ’98)or, equivalently, ‘‘exhibiting similar ontogeneticdisplacement fields D(t).’’

Landmark-based geometric morphometric (GM)methods provide an ideal mathematical frame-work that analyzes the issues discussed up to thispoint in quantitative terms (see Appendix I). GMintegrates real-space properties into multivariateanalyses (Bookstein, ’91), such that a point inshape space corresponds to a specific landmarkconfiguration P(t) in physical space, and a direc-tion through shape space corresponds to onespecific pattern of correlated shape change in thelandmark configuration, i.e., a displacement fieldD(t) in physical space (Zollikofer and Ponce deLeon, 2002). Accordingly, heterotopy manifestsitself by divergence of ontogenetic trajectories

through shape space, and heterochrony in transla-tion and scaling of a descendant relative to anancestral trajectory (Fig. 3B). Note that we relaxthe collinearity condition of ‘‘pure’’ heterochrony(Fig. 3A) in favor of parallel trajectories (Fig. 3B),implying that ancestor and descendant may bedissimilar in shape but still follow similar growthtrajectories (ontogenetic pattern homology).

Heterotopic and heterochronic pattern analysiscan be extended to more complex ontogenetictrajectories through shape space (Fig. 3C). In GM

Fig. 3. A geometric-morphometric perspective on thekinematics of ontogeny (letters A and D indicate ancestraland descendant trajectories, respectively). A: ‘‘Pure’’ hetero-chrony sensu Gould. All classical categories of heterochronicvariation can be imagined as variations resulting from shift/scaling along an ancestral trajectory through shape space. B:Generalized heterochrony. Segments of ontogenetic trajec-tories of an ancestral (A) and a descendant (D) species runalong parallel lines through shape space (PC1, PC2) and areseparated by distance Dv. Under these conditions, it is possibleto establish a coordinate system (u,v), whose axis u capturesthe age-related component of shape change, independent ofthe species-specific component v. Trajectories projected onto ucan be analyzed in heterochronic terms. C: Heterotopy.Divergent ancestral (A1) and descendant (D1) trajectoriesindicate heterotopic developmental patterns in the twospecies. Heterochronic analysis is not feasible under thesecircumstances. However, as long as trajectories can be super-imposed by translation (A2 and D2), a common parameter u’can be defined which describes the time course of developmentas in A.


size-shape space, each temporally confined phaseappears as a line segment, whose characteristicdirection corresponds to a specific spatial patternof shape change.

Modeling cranial ontogeny

Having established an operational frameworkfor quantitative kinematic analysis of patterns ofontogenetic shape change, we use model simula-tions to investigate the relationship betweenontogenetic kinetics (process) and kinematics(pattern). Specifically, we explore how modifica-tions of cranial growth processes impinge on theresulting ontogenetic trajectories through age-size-shape space, and vice versa, how processinformation may be inferred from morphometricpatterns of cranial ontogeny. The modeling pro-cedure is as follows:

1) A minimal cranial ontogenetic model isdevised, which specifies spatial and temporalgrowth parameters of a morphological structure.

2) Modification of process parameters in themodel simulates evolutionary change.

3) Form change in the resulting virtual onto-genetic series is analyzed with GM methods.

4) Alterations in the course of descendantversus ancestral ontogenetic trajectories throughshape space are correlated with alterations inprocess parameters.

A minimal model of cranial ontogeny

In Equation (1), we stated that the spatiotem-poral trajectory of a moving body can be describedwith a matrix P(t) summarizing positional changeof measurement points over time. The sameformalism can be applied to describe positionalchange in K cranial landmarks pi over ontogenetictime t:

PðtÞ ¼


piðtÞ� � �

pkðtÞ

0BBBBBB@

1CCCCCCA

¼

x1ðtÞ y1ðtÞ z1ðtÞx2ðtÞ y2ðtÞ z2ðtÞ� � � � � � � � �

xiðtÞ yiðtÞ ziðtÞ� � � � � � � � �

xkðtÞ ykðtÞ zkðtÞ

0BBBBBB@

1CCCCCCA

ð4Þ

In a growth model, we need to specify explicitlyhow each landmark position pi(t) changes overtime under the influence of various growthprocesses. We introduce L growth processes uj(t),

which are summarized in a vector U(t)

UðtÞ ¼

u1ðtÞu2ðtÞ� � �

ujðtÞ� � �

uLðtÞ

0BBBBBB@

1CCCCCCA

ð5Þ

These model processes assume the role of the‘‘true’’ units of cranial growth (in empiricalstudies, they represent the hidden units of devel-opment that we try to infer from kinematic data).Processes uj(t) are defined as Gompertz (sigmoid)growth functions (Zeger and Harlow, ’87)

UðtÞ ¼

u1ðtÞ� � �

ujðtÞ� � �

uLðtÞ

0BBBB@

1CCCCA

¼

u1ðtÞ ¼ u01 � eb1a 1�e�a1 t�Dt1ð Þ� �

� � �

ujðtÞ ¼ u0j � e

b1ja 1�e

�aj t�Dtjð Þ� �

� � �uLðtÞ ¼ u0L � e

bLa 1�e�aL t�DtLð Þ� �

0BBBBBB@

1CCCCCCA

ð6Þ

that approach a finite upper limit at u1j ¼ u0jebjaj

(details are given in Appendix II, Model A). It isconvenient to summarize the growth parametersof all functions uj(t) in matrix G

G ¼

u01 a1 b1 Dt1

� � � � � � � � � � � �u0j aj bj Dtj

� � � � � � � � � � � �u0L aL bL DtL

0BBBB@

1CCCCA: ð7Þ

In a next step, we specify how processes uj(t)determine landmark positions. For each land-mark, positional change over time pi(t) is modeledas a function Fi of growth processes U(t):

piðtÞ ¼ FiðUðtÞÞ¼ FixðUðtÞÞFiyðUðtÞÞFizðUðtÞÞ

� �: ð8Þ

Assuming generalized epigenetic interactions,each process uj may potentially contribute to theposition of each anatomical landmark pi:

PðtÞ ¼


piðtÞ� � �

pkðtÞ

0BBBBBB@

1CCCCCCA


¼

F1ðu1; . . . ; uj; . . . ; uLÞ� �F2ðu1; . . . ; uj; . . . ; uLÞ� �

� � �Fiðu1; . . . ; uj; . . . ; uLÞ� �

� � �FKðu1; . . . ; uj; . . . ; uLÞ� �

0BBBBBB@

1CCCCCCA: ð9Þ

In real biological systems, we expect lessconnectivity, such that several, but not all,processes contribute to a landmark position pi(t).

In the following simulations, we consider how‘‘evolutionary’’ modifications of growth para-meters G and of functions Fi(U) impinge on theresulting spatiotemporal pattern of shape changeP(t). In a 2-dimensional implementation, we useK=5 landmarks and L=3 processes to simulategrowth in the midplane of a model skull (Fig. 4). Ina 3-dimensional implementation, we use K=9landmarks (2 bilateral pairs, 5 midsagittal) andL=4 growth processes (Fig. 4; detailed modelspecifications are given in Appendix III). It maybe anticipated here that inclusion of the thirdspatial dimension does not alter the generalfindings obtained with the 2D model. For ease ofargument, we therefore focus on the 2D model anduse the 3D model to generalize our findings.

Simulations

To explore how modifications of temporal andspatial properties of model growth processesimpinge on resulting patterns of shape change,we proceed as follows:

1) The ontogeny of ‘‘ancestral’’ and ‘‘descen-dant’’ populations is simulated according to

ancestral parameter sets G0 and P0, and modifieddescendant parameter sets Gi and Pi, respectively.

2) The landmark configurations of the result-ing cranial forms are ‘‘sampled’’ at various ages,i.e., at various points along time t, and formvariability of the pooled sample is analyzed withprincipal components analysis (PCA) of shape(Dryden and Mardia, ’98; see Appendix I).

‘‘Evolutionary’’ modifications of the modelsystem may affect any number and combinationof parameters in matrices G and P. In thefollowing simulations, our principal aim is toexplore contrasts between two extreme types ofmodification, correlated modification of entire setsof process parameters, and modification of singleparameters. In the first case (Fig. 5), all elementsin a column of matrix G are modified in the sameway (for example, all u0j are multiplied by thesame factor). In the second case, single elements ofmatrices G or P are modified.

The results of computer simulations aregraphed in Figures 5–8. To interpret theseFigures, recall that in GM analysis, a point inshape space corresponds to one specific cranialshape in physical space, and a linear trajectorycorresponds to a constant displacement field D(t)in physical space. PCA typically yields two shapefactors, PC1 and PC2. Together with size and age(i.e., time t), PC1 and PC2 constitute a 4-dimensional morphospace. For ease of visualiza-tion, we graph 2-dimensional projections of thatspace; PC2 versus PC1, and PC1 versus log-size(ontogenetic allometry). Time is represented im-plicitly by the spacing between the data pointsalong trajectories. In all analyses, the temporaldirection of trajectories is from left to right.

RESULTS

All ontogenetic trajectories through shape spacehave several characteristics in common (see PC1-PC2 graphs in Figs. 5 to 7). First, they approachadult cranial shapes as an ‘‘attractor point’’ inshape space, which reflects the basic property ofthe growth functions to asymptotically approach afinite value (see Eq. 6). Further, growth trajec-tories through shape space are typically slightlycurved. This indicates that the associated physicaldisplacement fields change over time to someextent, although the underlying growth processesremain constant. Exploration of the parameterspace of G and P indicates that an increase intemporal and spatial disparity between processesuj(t) results in increased curvature of the

Fig. 4. A minimal model of cranial growth. Cranial growthin the midsagittal plane is determined by three processes u1,u2, u3 governing growth of the braincase, the face and thecranial base, respectively; mediolateral growth is governed byprocess u4. Cranial shape is determined by a 5-landmarkconfiguration in two dimensions (circles 1-5), and by a 9-landmark configuration in three dimensions (landmarks 6a,band 7a,b lie on right and left cranial sides, respectively).


trajectories (data not shown). Visualization of theactual spatial shape change associated with thesimulated growth trajectories (Fig. 5, bottom) showsthat, given the considerable physical nonlinearitiesof shape transformation, deviation from linearity oftrajectories through shape space is moderate.

Effects of correlated parametermodifications

In a first set of simulations, we examine theeffects of modification of each parameter set {u0j},{aj}, {bj}, and {Dtj} (j=1yL). In all analyses,

Fig. 5. Modeling pure heterochrony and ontogeneticallometry. Simulation of ontogenetic modifications in a 2-dimensional cranial growth model (see Fig. 4 and Appendix IIIfor model specifications). Left graphs: ontogenetic trajectoriesthrough shape space (PC1 and PC2 represent the first twoprincipal components from PCA of shape). Correlated mod-ification of parameter sets {u0j}, {aj}, {bj}, and {Dtj} of growth

equations uj(t) (j=1..3) results in characteristic shift and/orscaling of the descendant trajectory (open circles) relative tothe ancestral trajectory (filled circles). Right graphs: ontoge-netic allometry. The TPS deformation grid at the bottomvisualizes shape change corresponding to advancement alongthe ancestral trajectory (gray arrow in the top left graph).


shape component PC1 captures E99.5% of thetotal shape variability in the sample, whilecomponent PC2 captures the remaining E0.5%.The effects of parameter modification onontogenetic trajectories are shown in Figure 5(graphs in the left column). A conspicuous com-mon feature of these simulations is the coinci-dence of ancestral and descendant trajectories.This corresponds to ‘‘pure’’ heterochrony(Fig. 3A), where descendant trajectories representvariations along the ancestral trajectory. Corre-lated changes in aj extend or shorten the ontoge-netic trajectory with respect to a common onsetpoint in shape space, and changes in bj have

similar effects. Correlated changes in Dtj extendor shorten the ontogenetic trajectory, while itsadult end acts as a fixed point. Note thatcorrelated changes in u0j have no effect on thetrajectory through shape space, as they only affectsize, not shape.

Changes in the relationship between size andshape are visualized by plotting shape componentPC1 against log-centroid size (ontogenetic allome-try, Fig. 5, graphs in the right column). Theallometric shift in the graph for u0j expresses thefact that the size-shape relationship of cranialform was changed by a general scaling factor(u0j=1.5).

Fig. 6. Modeling heterotopy, heterochrony and ontoge-netic allometry. Simulation of ontogenetic modifications in a2-dimensional cranial growth model (see Fig. 4 and AppendixIII for model specifications; filled/open circles representancestral/descendant morphologies). Small modifications in

single local process parameters of the ancestor’s ontogeneticprogram generate complex alterations in the resultingdescendant trajectories through shape space (left graphs) aswell as in ontogenetic allometric trajectories (right graphs).Further explanations see text.


Effects of single-parameter modifications

In a second series of simulations we explore theeffects of modifications of single temporal andspatial process parameters. We consider 5 variantsof descendant populations, representing modifica-tions of temporal parameters u01, a1, b1, and dt1, aswell as variations in the function F2 (U) definingthe position of landmark #2 (see Appendix III formodel specifications). The results are graphed inFigure 6. Overall, modification of single growthparameters results in more diversity of trajec-tories than correlated modification of parametersets. Changes in u01 result in positional shift of theontogenetic trajectory while its direction andlength remain almost unaffected. This corre-sponds to the definition of generalized hetero-chrony (Fig. 3B). Modification of b1 or a1 (data notshown) results in divergence of the descendantfrom the ancestral ontogenetic trajectory (hetero-topy), and in alteration of trajectory length. Notethat the onset of the ancestral trajectory acts as afixed point of these alterations. Modification of Dt1

leads to inverse effects, as the end of the ancestraltrajectory assumes the role of a fixed point,towards which the shortened or extended trajec-tory of the descendant population converges.Finally, modification of the epigenetic interactionsbetween processes influencing the position oflandmark #2 has similar heterotopic effects aschanges in growth parameters a1 or b1.

Graphs of PC1 versus centroid size (ontogeneticallometry; Fig. 6, right graphs) show various waysof decoupling the ancestral size-shape relation-ship. In the first case (u01), decoupling is primarilyeffected via curve-shift. As an effect of modifica-tion of a1 or b1, the size-shape relationshipdiverges in a more complex way; the correspond-ing graphs in Figure 6 show that the shapetrajectory is shortened, while the size trajectoryis extended.

Exploration of the parameter space representedby G and P for 3-dimensional 4-process growthmodels yields essentially similar results. Figure 7presents an example in which uncorrelatedchanges in all parameters u0j result in parallelancestral and descendant trajectories throughshape space.

Model heterochrony and heterotopy

With regard to the concepts of heterochrony,heterotopy and ontogenetic allometry these re-sults can be summarized as follows:

* ‘‘Pure’’ heterochrony: Correlated changes ingrowth processes yield pure heterochronicchanges in ontogenetic patterns, i.e. extension orcontraction of the ancestral trajectory throughshape space (Fig. 5).

* Generalized heterochrony: Local changes in initialvalues u0j of growth processes modify the initialshape and size of the structure under considera-tion, as evinced by displacement of the onset pointof the descendant trajectory (which can be thoughtof as the end point of the preceding phase ofontogeny). With all other process parametersremaining unchanged, this results in parallelontogenetic trajectories.

* Heterotopy: Changes in timing (Dtj) and in theallometric growth characteristics (aj, bj) of singleprocesses, or in the local spatial organization ofdevelopmental modules, lead to heterotopicchanges, i.e., divergence between ancestral anddescendant trajectories.

* Process versus pattern heterochrony and hetero-topy: Modification of a single process parametertypically has a multi-pattern effect, leading todivergence and scaling of descendant relative toancestral trajectories through shape space, as wellas divergence of allometric trajectories.

Testing alterations in the model design

An important general issue that has to beaddressed in simulations is to test the robustness

Fig. 7. Parallel trajectories through shape space (4-processmodel, 3 spatial dimensions; model specifications see Fig. 4and Appendix III). A: Ancestral and descendant trajectoriesresult from equivalent growth processes (parameters aj, bj, Dtj

in C) that act upon different initial forms (parameters u0j;shaded area in parameter matrix). B: Quantifying shape withan alternative set of landmarks (compare inset graphs in Aand B) leads to slightly different results in shape space.


of results obtained with a given model systemagainst changes in the premises of the model. Totest robustness, we simulated ontogeny withalternative growth functions (see Appendix II,Models B and C). Model B implements exponen-tial decay of absolute growth rates du/dt, whilemodel C implements unlimited exponentialgrowth. The results of simulations in analogy toFigures 5–7 yield largely similar results to thoseobtained with the Gompertz growth model (datanot shown). In both cases, correlated parameteralteration yields pure heterochrony, single-para-meter modification yields heterotopic dissociation,and changes in the initial conditions result in

parallel ancestral and descendant trajectoriesthrough shape space.

Another test of robustness concerns land-mark definitions. The shape of an object canbe quantified with different sets and numbersof landmarks, such that alterations in mea-sured ontogenetic trajectories can be expected.Figures 7A and B permit comparison betweenshape trajectories resulting from alternativelandmark definitions applied to the sameontogenetic sample. It appears that even consider-able changes in landmark definitions have rela-tively moderate effects on the outcome of shapeanalysis.

Fig. 8. Cranial ontogeny in Neanderthals (filled circles)and anatomically modern humans (AMH; open/dashed circles:postnatal specimens/last trimenon fetuses). A: Ontogenetictrajectories through a subspace of shape space (PC1 and PC2account for 38% and 13% of the total shape variability in thesample, respectively). Parallel trajectories indicate a sharedpostcranial pattern of ontogenetic kinematics. B: Visualiza-tion of cranial shape change along ontogenetic trajectories (7:relative local expansion/contraction of the cranial surface)reveals discontinuity between positive/negative allometricgrowth characteristics of the viscero/neurocranial regions.

C-E: Ontogenetic allometry of the entire skull (C), the face(D) and the neurocranium (E) (regression lines are calculatedwithout perinatal specimens; solid/dashed line: Neanderthals/AMH). F: Visualization of shape difference between AMH andNeanderthal ontogenetic trajectories (the graph accounts forthe amount of shape difference perpendicular to the cranialsurface). G: Growth fields in the human cranial vault (afterEnlow, ’90). Borders between depository and resorptivegrowth fields probably coincide with contrasts betweenAMH and Neanderthal cranial morphologies (see F).


With a view on empirical studies, we maytherefore conclude that the methods proposed toanalyze ontogenetic trajectories are fairly robustin two respects; first, they are robust againstconsiderable variation in the actual temporalcharacteristics of the underlying growth processes,second, they are robust against changes in land-mark definitions.

EMPIRICAL DATA: NEANDERTHALVERSUS MODERN HUMAN ONTOGENY

Model simulations are useful tools to explore the‘‘behavior’’ of minimal cranial growth modelsunder evolutionary variation. Here, we examinehow insights gained from model systems can beapplied to empirical data. We re-analyze a data setpublished earlier that shows that differences incranial shape between Neanderthals and anato-mically modern humans (AMH) arose very earlyduring development and were maintainedthroughout postnatal ontogeny (Ponce de Leonand Zollikofer, 2001).

METHODS

The sample consists of 12 Neanderthal and 24AMH crania (AMH range in individual age fromlate fetal stages to adulthood, Neanderthals from2.5 years to adulthood; details see Ponce de Leonand Zollikofer, 2001). Cranial form was quantifiedwith 43 landmarks (11 midsagittal; 16 bilateralpairs), and shape variability in symmetrized speci-mens (Zollikofer and Ponce de Leon, 2002) wasanalyzed with PCA of shape (Dryden and Mardia,’98). Similar methods were applied to analyzelandmark subsets representing facial and neuro-cranial form, respectively (face: 5 midsagittallandmarks, 8 bilateral pairs; neurocranium: 7midsagittal landmark, 8 bilateral pairs). In eachanalysis, group mean differences in shape, diver-gence between group-specific ontogenetic trajec-tories through shape space, and divergencebetween allometric trajectories (obtained by mul-tivariate regression of shape on size; Zelditch andFink, ’96; Penin et al., 2002) were calculated, andthe statistical significance of between-group dif-ferences was evaluated with permutation tests(Good, ’94; permutation tests generate empiricalprobability distributions by repeated randomiza-tion of group assignment in the pooled sample).

RESULTS

Figure 8A shows cranial shape variability inshape subspace formed by the first two principalcomponents, PC1 and PC2. Neanderthals andAMH follow parallel ontogenetic trajectories alongPC1 and are separated along PC2. Group meanshapes of Neanderthals and AMH are clearlydistinct from each other (po0.01), while devia-tions between Neanderthal and AMH trajectoriesare not significant (p40.71). The apparent paral-lelism of trajectories suggests a shared postnatalpattern of shape change (unpublished data fromneonate to 2-year old Neanderthal specimenssuggest that trajectories started at similar neonatevalues of PC1). Visualization of the morphologicaldisplacement field corresponding to the trajec-tories through shape space reveals oppositegrowth characteristics of the face and the neuro-cranium (Fig. 8B); while the former grows withpositive allometry, the latter grows with negativeallometry.

In a next analytical step, the ontogeneticallometric characteristics of the skull and itssubregions are studied in more detail (Figs. 8C-E). Allometric (i.e., size-shape) trajectories appearto be biphasic, indicating an early (perinatal)phase of near-isometric growth followed by aphase of constant allometric growth (in Figs. 8C-E isometry corresponds to slope=0). During thelatter phase, Neanderthal and AMH ontogenetictrajectories do not differ significantly in slope(p>0.89) but it appears that Neanderthal trajec-tories are shifted towards larger sizes at any givencranial shape. It may be noted here that theobserved pattern of allometric ontogenetic dispar-ity cannot be described with the classic hetero-chronic terminology of size hypomorphosis versushypermorphosis, as these terms imply divergenceand/or scaling of size-shape trajectories.

Parallelism of trajectories, as evinced by Figures8A-E, implies that species-specific differencesbetween Neanderthals and AMH were alreadypresent at an early postnatal age. Note that Figure8A exhibits close similarity to the simulation inFigure 7A, where model populations differ ininitial conditions u0i, while all other growthparameters remain the same. Following a similarline of argument, one may hypothesize that theNeanderthal/AMH dichotomy mainly reflects dis-tinct initial (i.e. perinatal) growth conditions.Given the wide spectrum of possible mechanismsof divergence between ontogenetic trajectories,the Neanderthal/AMH split thus appears as a


comparatively simple evolutionary ontogeneticmodification. According to this hypothesis, post-natal ontogeny in both species represents evolu-tionary ‘‘conservatism’’ (similar growthparameters), while prenatal growth processesrepresent evolutionary novelty (generating dis-tinct perinatal cranial morphologies).

Direct comparative investigation of prenatalgrowth patterns in Neanderthals is no longerfeasible. However, it is possible to examine theavailable morphometric evidence in the light ofdevelopmental process data. Craniofacial growthis characterized by differential activities and/ordistribution of depository/resorptive skeletalgrowth fields (Enlow, ’90). Accordingly, evolution-ary modifications of growth fields can be describedin terms of heterotopic and heterochronic altera-tions, i.e., changes in spatial patterning andtemporal activity of growth fields.

To establish potential links between skeletalgrowth field activity and the Neanderthal/AMH morphological contrast, the quantitativemorphological difference across Neanderthal andAMH trajectories is visualized. The resultinggraph (Fig. 8F) reveals polarity between superiorand inferior regions of the cranial vault (note thatthis graph does not convey an immediate picture ofprenatal cranial growth fields; rather it visualizesthe contrast between AMH and Neanderthalpatterns of development before birth). The bound-ary between these regions probably coincides withthe circumcranial reversal line that separatesdepository from resorptive growth fields on theinternal surface of the braincase (Fig. 8G).Assuming that the spatial arrangement of cranialgrowth fields was largely similar in Neanderthalsand AMH (i.e., no heterotopic modification), itsuffices to postulate different temporal activity inspatially conservative growth fields, giving rise tospecies-specific perinatal morphologies. Hence,increased relative drift and displacement in theinferior vault may account for the low but poster-olaterally-expanded braincase in Neanderthalsrelative to AMH.

As suggested by the model of Figure 7, similargrowth processes acting on different initial condi-tions result in parallel ontogenetic trajectories,but also entail an array of subtle differencesbetween trajectories, such as dissociation of thesize-shape relationship (ontogenetic allometry). Inview of these model findings, many postnataldevelopmental differences between AMH andNeanderthals, such as differences in ontogeneticallometric trajectories (Fig. 8C-E) and in various

non-metric skeletal and dental developmentalfeatures (Dean et al., ’86; Tillier, ’86; Rak et al.,’94) probably have a single common cause indistinct perinatal cranial shapes. Overall, a phyle-tically ‘‘old’’ postnatal growth process acting onphyletically ‘‘new’’ perinatal cranial morphologiesmight suffice to explain a large array of postnataldevelopmental differences between Neanderthaland AMH skulls.

DISCUSSION

Aspects of kinematics

The proposed kinematic approach to cranialontogeny analyzes patterns of three-dimensionalmorphological change in terms of velocity vectorfields. In this approach, the differentiationbetween magnitude and direction of vectors is,at first instance, a technical procedure thatfacilitates comparative analysis of vector fields.However, this differentiation also relates to thebiological meaning of size and shape. The manyways in which size and shape are defined in theliterature are the principal cause of the often-cited inflation of heterochronic/heterotopicterminology and ensuing semantic confusion(McNamara, ’97; McKinney, ’99; Gould, 2000).For example, the ‘‘paradox of peramorphicpaedomorphosis’’ of human development (God-frey and Sutherland, ’96) essentially relates toquestions as to how to define size, shape, rates ofgrowth and development, and developmentalsequences in time (Rice, ’97; McNamara,2002b). As another example, consider spatialdifferences in velocity vector fields, termed‘‘pattern heterotopy’’ in this paper. From amodular perspective, the same phenomenon istermed ‘‘dissociated heterochrony’’ (McNamara,2002a). However, as pointed out by Zelditch andFink (’96), and Zelditch et al. (2000), dissociatedheterochrony may be a misinterpretation ofactual process heterotopy, resulting from over-modularization of developmental data. Overall,detailed quantitative description of patterns ofontogenetic divergence is preferable over fittingobservations to a complicated terminology, whichoften implies interpretation in terms of under-lying processes. The kinematic approach pro-posed here offers a quantitative framework thatpermits clear distinction of measurement andanalysis of form change from inference of processfrom pattern.


What can be learned from spatiotemporalgrowth models?

Evolving and analyzing cranial morphologies insilico showed that a minimal cranial growth modelcan replicate the wide variety of heterotopic andheterochronic phenomena proposed in theoreticalconsiderations (e.g. Gould, ’77; Alberch et al., ’79;Godfrey and Sutherland, ’96; Klingenberg, ’98)and found in empirical studies (e.g. Shea, ’88;Nehm, 2001; Roopnarine, 2001; Zelditch et al.,2003). Most notably, models show that modifica-tion of single process parameters typically resultsin complex alterations of ontogenetic trajectoriesthrough size-shape-age space (Fig. 6). From thisperspective, unexpected effects such as conver-gence of ontogenetic trajectories towards adult-hood, which were considered to representfunctional/adaptive convergence (Roopnarine,2001), can be understood as resulting from simplephase shifts between growth processes (see Fig. 6,graph representing modification of Dt).

On the other hand, the multiple and complexeffects of process modification on pattern modifi-cation even in a simple growth model point toprincipal limits of inference of process frompattern. During evolutionary studies of cranialdevelopment, it is in fact notoriously difficult toestablish such connections. First and foremost, ina spatially complex, modularly organized struc-ture such as the hominid skull, no straightforwardcausal connection exists between actual growthprocesses and observable patterns of growth anddevelopment (Lieberman, ’99). In addition, pat-terns of shape change reflect a combination ofactive local growth and passive displacement ofcranial substructures, whose relative contribu-tions to the observed displacement field cannoteasily be distinguished (Enlow, ’90).

How can heterochronic/heterotopic processand pattern analyses be brought together?Process and pattern analyses ask questions re-garding proximate and ultimate causes of ontoge-netic modification, respectively, and bothperspectives are biologically relevant in theirown right. Differences in developmental timingmight have considerable functional and adaptivesignificance for the growing organism as a whole.On the other hand, identification of proximatedevelopmental causes of ontogenetic diversityhelps relax the rigors of the ‘‘adaptationistprogramme’’ (Gould and Lewontin, ’79) and leadsto a process-oriented understanding of species-specific morphologies.

A further implication from model data that isrelevant for the empiricist concerns comparisonsof cranial morphology in phyletic and cladisticanalyses. Typically, such analyses are based onadult individuals, as one generally assumes thatonly adult specimens display the full range oftaxon-specific features. While this might be thecase for epigenetic traits and non-metric charac-ters, it is likely not to be the case for overallcranial morphology as measured with GM meth-ods. Considering Figures 5-7, it appears thatcomparisons between adults (i.e., offset points ofontogenetic trajectories), tend to mix variouscomparative criteria, such as shift, divergenceand scaling of ontogenetic trajectories, in un-known proportions. Given the importance ofidentifying phyletically valid morphological char-acters representing the underlying genetics (Lie-berman, ’99), it is essential to compare trajectoriesthrough shape space rather than points in shapespace.

Methods of data sampling and methods ofinference

One issue of principal importance in ontogenetickinematic analyses with GM methods relates todata acquisition. Whether stated explicitly ornot, defining quantitative morphological unitsalways implies hypotheses about the processesthat generate the observed morphologies.Setting cranial landmarks, therefore, is equivalentto stating hypotheses about underlyinggrowth processes, and it can be expected that thenumber and relative positioning of landmarksused to define cranial form influences the outcomeof GM analyses. How can an optimum numberand distribution of landmarks be found thatestablishes a balance between under- and over-determination of form? This issue is especiallycritical in the cranium, where the facial area isdensely populated with easily identifiable land-marks (e.g., meeting points between sutures),while the cranial vault exhibits only few suchlandmarks. Various methods have been proposedto define additional reference points, so-calledsemilandmarks, in landmark-depleted regions ofthe skull (Bookstein, ’97; Andresen and Nielsen,2001). While these methods yield data points thatare evenly distributed over entire regions of theskull, they tend to overdetermine geometric shapeat the expense of growth-oriented shape. There isno definite solution to this problem, such thatpractical solutions must combine heuristics with


iterative refinement of landmark definitions.An important first step towards testing thebiological (as opposed to the statistical) reli-ability of GM analysis is to study one and thesame sample using different landmark sets, toanalyze subsets of landmarks that representsubregions of the structure under investigation,and to use alternative methods of kinematicdata analysis.

What can be learned from Neanderthals?

Morphometric evidence supports the view thatearly modification of cranial developmental pat-terns is a major source of hominid and hominoidevolutionary diversification, while variation inpostnatal growth patterns contributes only a smallpart to the differentiation between species (Poncede Leon and Zollikofer, 2001; Lieberman et al.,2002; Rogers Ackermann and Krovitz, 2002;Williams et al., 2002). Moreover, as evinced in acomparison between adult AMH and archaic hu-mans (including Neanderthals), and betweencorresponding age classes of humans and chimps(Lieberman et al., 2002), it appears that the samesmall set of parameters is involved in generatingevolutionary novelty by changing the relative sizeand position of the face, the neurocranium and thecranial base. These parameters influence cranialbase flexion and length, relative proportions of thecranial fossae, and relative proportions of the face(Lieberman et al., 2000, 2002).

The Neanderthal data presented here fit intothis general picture. Obviously, major contrastsbetween species-specific developmental patternsmust be sought prenatally, while postnatally,Neanderthals and AMH develop along largelysimilar trajectories. The proximate developmentalcauses that generated dissociation between speciescan only tentatively be inferred. Evidence for theproposed hypothetical mechanismFdifferentialgrowth activity in conserved cranial growthfieldsFcomes from a comparative study of epige-netic traits in Neanderthal and AMH crania,indicating that relative thinning in the laterally-expanded cranial vault of Neanderthals mightreflect differential activity in resorptive versusdepository growth fields, which ultimately mayreflect species-specific differences in cerebral andskeletal growth processes (Manzi et al., ’96). Suchhypotheses need further corroboration throughcomparative analysis of early developmentalpatterns in extant species (Lieberman andMcCarthy, ’99).

The results of this and an earlier study onNeanderthal and AMH ontogeny (Ponce de Leonand Zollikofer, 2001) largely converge with theresults of a suite of studies dedicated to the samesubject, but based on different methods. Krovitz(2000) investigated facial ontogeny with EuclideanDistance Matrix Analysis (EDMA; Lele andRichtsmeier, 2001), a geometric-morphometricmethod that defines form by the matrix of allinterlandmark distances in a landmark configura-tion and uses Principal Coordinates Analysis(PCO, a variant of multidimensional scaling) tostudy shape variability. These studies, as well as arecent EDMA-based analysis of facial ontogeny ina sample comprising AMH, Australopithecus afri-canus, Pan troglodytes and Pan paniscus, suggestthat evolution through ontogenetically early dif-ferentiation is an ancient pattern of hominoidphylogeny (Rogers Ackermann and Krovitz, 2002).In another set of studies (Williams, 2000; Williamset al., 2002), Neanderthal versus human craniofa-cial ontogeny was analyzed with classical multi-variate techniques applied to sets of craniometricdistance measurements (Godfrey et al., ’98).Interestingly, these authors converge in the con-clusion that ‘‘modern humans and Neanderthalsfollow parallel shape changes from different pointsof origin’’ (Williams et al., 2002, p. 430). Alto-gether, these methodologically diverse studiespoint towards a basic pattern of ontogenetickinematicsFprenatal divergence versus postnatalhomology of ontogenetic processes.

Direct investigation of early patterns of devel-opment is only beginning (Lieberman andMcCarthy, ’99; Jeffery and Spoor, 2002) and willhelp substantiate hypotheses about the role ofontogenetic divergence as a source of evolutionarynovelty. Exploring the network of cause and effectthat connects observed patterns of spatiotemporalshape change with underlying growth processesand processes of evolutionary modification is aniterative task. The endeavor of identifying devel-opmental units, analyzing their ontogenetic kine-matics, and studying their evolutionary variabilitymust follow a multidisciplinary approach, combin-ing kinematic morphometric analysis with modelconsiderations and with insights from the ‘‘ki-netic’’ disciplines of developmental biology.

ACKNOWLEDGMENTS

We would like to thank Miriam Zelditch andDan Lieberman for many insightful observationsthat helped improve this manuscript. Useful


comments were also provided by an anonymousreviewer.

APPENDIX I: GEOMETRICMORPHOMETRICS

In GM, the form of each specimen is quantifiedby the position of anatomical landmarks, whichdenote locations of biological homology betweenspecimens in the sample. The two components ofform, size (Centroid Size; Bookstein, ’91) andshape are statistically independent of each other,which greatly facilitates tracking ontogeneticallometry along trajectories through age-size-shape space. A specimen’s shape is expressed asthe multidimensional deviation of all landmarksfrom the corresponding landmarks of a referenceconfiguration. The reference configuration istypically the sample mean and is evaluated withgeneralized-least-squares superimposition meth-ods (Rohlf and Slice, ’90). In contrast to size,which is a scalar, shape is a multivariate measure,i.e., a vector. Each specimen’s shape can beimagined as a location (i.e., a position vector) inmultidimensional shape space (linearized Pro-crustes space; Dryden and Mardia, ’98). Tovisualize patterns of shape variation in thesample, Principal Components Analysis (PCA) isused. This method permits extraction of statisti-cally independent factors of shape variation thataccount for the largest, second largest andsuccessively smaller proportions of shape varia-bility in a sample. PCA is mainly used as adimension reduction technique. A major strengthof GM analysis is that it preserves a direct linkbetween data in shape space and data in physicalspace. Most notably, vectors in shape spacecorrespond to displacement fields in physicalspace. This permits visualization of ontogenetictrajectories in terms of cranial shape transforma-tion (Zollikofer and Ponce de Leon, 2002; seeFig. 8B, F).

APPENDIX II: GROWTH FUNCTIONS

Growth and static allometry

In the present context, Huxley’s allometricgrowth model represents a suitable starting point,as it postulates a process-oriented background ofmorphological patterns of allometry (Huxley, ’32).In this model, growth processes uj(t) and uk(t) (asdefined in Equation 5) exhibit proportionality of

specific growth rates

duj

dt

1

uj¼ a

duk

dt

1

uk; ðA1Þ

where the specific growth rate of u is defined asthe absolute growth rate (du/dt) in relation to theactual amount of u, and factor a is a coefficient ofepigenetic interaction between uj and uk. Uponintegration, Equation (A1) yields the static allo-metric equation

uj ¼ c � uak; ðA2Þ

where c is a scaling factor, and a is the allometriccoefficient. Note that upon integration of Huxley’sallometric process equation time t vanishes, suchthat the resulting allometric pattern equationdescribes a time-independent correlation betweenunits uj and uk.

The time course of u(t) follows from integrationover the time course of its specific growth rate.Various functions describing this latter processhave been proposed (Zeger and Harlow, ’87).In computer simulations, we use the followingmodels:

Model A: Gompertz growth

Integration of a function describing exponentialdecay of the specific growth rate over time

du

dt

1

u¼ b � e�at ðA3Þ

yields the Gompertz growth function

uðtÞ ¼ u0 � eba 1�e�atð Þ; ðA4Þ

which saturates at

uðt ! 1Þ ¼ u0 � eba: ðA5Þ

The point of inflexion is at t=0, where the slopereaches a maximum value of u0b.

The concept of allometry was expanded byJolicoeur to a series of units u1, u2,y, uj,y, uL,such that we may think of sets of growth functions

ujðtÞ ¼ u0j � ebjaj

1�e�ajtð Þ

; j ¼ 1 . . .L

� �; ðA6Þ

where multivariate allometric exponents aj/ak linkany two variables uj and uk (Jolicoeur, ’63). Toallow ‘‘evolutionary’’ variability in the way growthprocesses interact with each other, we relax thecondition of allometry and define individualprocesses uj(t) as

ujðtÞ ¼ u0j � ebjaj

1�e�ajðt�DtjÞð Þ

;

�ðA7Þ


where oDtj accounts for temporal shifts betweenprocesses.

Models B and C

Two alternative models also satisfy allometricEquations (A1) and (A2). In model B, growth ratesdepend only on time, not on the actual amount ofxi. Assuming exponential decay of growth rates

duj

dt¼ bj � e�at; ðA8Þ

integration over time yields exponential satura-tion functions of the form

ujðtÞ ¼ u0j þbj

aj1 � e�aj t�Dtjð Þ

� �: ðA9Þ

Model C assumes time-dependent specific growthrates of the form

duj

dt� 1

uj¼ bj

1

t: ðA10Þ

Integration yields a growth function exhibiting‘‘allometry’’ in time:

ujðtÞ ¼ u0j � tbj : ðA11Þ

We may relax the allometric constraints andintroduce a parameter for time shift, accountingfor temporal modularity of ontogenetic processes

ujðtÞ ¼ u0j � ðt � DtjÞbj: ðA12Þ

Equation (A12) describes a single phase of themultiphase growth model proposed by Vrba (’98)and further studied by Vinicius and Lahr (2003).In our view, this model is biologically less realisticthan models A and B, as it implies unboundedgrowth for t - N.

APPENDIX III: MODEL SPECIFICATIONS

Growth in the midplane of a model skull issimulated with 3 processes u1(t), u2(t) and u3(t).The parameter set defining the ‘‘ancestral’’ con-dition for model A is summarized in matrix G0:

G0 ¼u01 a1 b1 Dt1

u02 a2 b2 Dt2

u03 a3 b3 Dt3

0@

1A

¼1 1 1 01 1 2 01 1 1:5 0

0@

1A: ðA13Þ

In the ‘‘ancestor,’’ the spatial positions P0(t) of the5 cranial landmarks are influenced by these

processes in the following way (Fig. 4):

P0ðtÞ ¼

x1ðtÞ y1ðtÞx2ðtÞ y2ðtÞx3ðtÞ y3ðtÞx4ðtÞ y4ðtÞx5ðtÞ y5ðtÞ

0BBBB@

1CCCCA

¼

0 00 u3ðtÞ

�u1ðtÞ 0�u1ðtÞ u1ðtÞ�u2ðtÞ 0

0BBBB@

1CCCCA: ðA14Þ

This corresponds to a low-connectivity model (cf.Eq. 9). In evolutionary modifications of P(t),function y2 (t) contains process interaction

y2ðtÞ ¼u2 þ u3

2: ðA15Þ

Growth of a 3-dimensional skull model issimulated with L=4 processes and K=9 land-marks. Matrix G0 has the following ancestralform:

G0 ¼

1 1 1 01 1 2 01 1 1:2 01 1 1:1 0

0BB@

1CCA; ðA16Þ

and landmark positions are determined as follows:

P0 ¼

0 0 00 u2þu3

2 0�u1 0 0�u1 u1 0�u1 0 �u4�u1

2u1þu2

2�ðu1þu3þu4Þ

3

0BBBBBB@

1CCCCCCA: ðA17Þ

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