Computational Models for Mechanics of Morphogenesis

Computational Models for Mechanicsof Morphogenesis

Matthew A. Wyczalkowski, Zi Chen*, Benjamen A. Filas,Victor D. Varner and Larry A. Taber

In the developing embryo, tissues differentiate, deform, and move inan orchestrated manner to generate various biological shapes drivenby the complex interplay between genetic, epigenetic, and environ-mental factors. Mechanics plays a key role in regulating and control-ling morphogenesis, and quantitative models help us understandhow various mechanical forces combine to shape the embryo.Models allow for the quantitative, unbiased testing of physical mecha-nisms, and when used appropriately, can motivate newexperimental directions. This knowledge benefits biomedical research-ers who aim to prevent and treat congenital malformations, as wellas engineers working to create replacement tissues in the laboratory.In this review, we first give an overview of fundamentalmechanical theories for morphogenesis, and then focus on models forspecific processes, including pattern formation, gastrulation, neurula-tion, organogenesis, and wound healing. The role of mechanical feed-back in development is also discussed. Finally, some perspectivesare given on the emerging challenges in morphomechanics andmechanobiology. Birth Defects Research (Part C) 96:132–152,2012. VC 2012 Wiley Periodicals, Inc.

Key words: morphogenesis; development; morphomechanics; mecha-nobiology

INTRODUCTIONThe importance of mechanics indevelopmental biology is wellestablished. Biomechanical forcesare the bridge that connectsgenetic and molecular-level eventsto tissue-level deformations thatshape the developing embryo(Koehl, 1990; Taber, 1995). Inaddition, feedback from the cellu-

lar mechanical environmentaffects gene expression (Farge,2003) and differentiation (Engleret al., 2006). Hence, mechanicalforces are not only a proximalcause of morphogenesis, but theyalso play a regulatory role (Vogeland Sheetz, 2006; Wozniak andChen, 2009; Mammoto andIngber, 2010).

During the last two decades,there has been increasing appreci-ation for the value of computationalmodels in studying the mechanicsof morphogenesis (Taber, 1995;Brodland, 2006; Davidson, 2008;Rauzi et al., 2008). This interest isbeing facilitated by dramaticincreases in computing power andthe growing availability of quantita-tive experimental data, which areneeded to inform and test models(Grashoff et al., 2010; Martin,2010; Martin et al., 2010; Satoet al., 2010; Meng and Sachs,2011; Rauzi and Lenne, 2011). It isimportant to recognize, however,that mathematical models nevercan include all of the complexitiesinherent in biological systems.Rather, they are utilitarian, adhoc constructions designed to helpunderstand some aspect of thesystem.This review is intended to give

readers an overview of variousaspects of modeling efforts inmorphogenesis. We first provide abrief summary of some of the me-chanical theories for morphogene-sis, the principles of constructingcomputational models, and the

REVIE

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VC 2012 Wiley Periodicals, Inc.

Birth Defects Research (Part C) 96:132–152 (2012)

Matthew A. Wyczalkowski, Zi Chen and Larry A. Taber are from Department of Biomedical Engineering, Washington University,St. Louis, MissouriBenjamen A. Filas is from Department of Ophthalmology and Visual Sciences, Washington University School of Medicine, St. Louis,MissouriVictor D. Varner is from Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey

Supported by grants from the National Institutes of Health (F32 GM093396 (MAW), R01 GM075200 and R01 NS070918 (LAT), andT90 DA022871 (BAF)), the American Academy of Mechanics Founder’s Award from the Robert M. and Mary Haythornthwaite Founda-tion (ZC), and the American Heart Association (09PRE2060795 (VDV)).

Matthew A. Wyczalkowski and Zi Chen contributed equally to this work.

*Correspondence to: Zi Chen, Department of Biomedical Engineering, Washington University, St. Louis, MO 63130.E-mail: [email protected]

View this article online at (wileyonlinelibrary.com). DOI: 10.1002/bdrc.21013

challenges encountered therein.We then turn our attention to aselection of morphogenetic phe-nomena and describe models thathave been used to simulate theunderlying mechanisms. Finally,we provide our perspective on thechallenges and future directions inmodeling morphogenesis.Admittedly, this topic covers a

vast literature, so it is impossibleto include everything. For exam-ple, chemical and molecular bio-logical effects are crucial topics,but they are not considered indetail here. In addition, althoughall models must be supported byexperiments, only a limited set ofexperimental work that has stimu-lated or assisted relevant model-ing efforts is included in thisreview. Also, several importanttopics, such as bone developmentand cell migration, are notaddressed. We do not claim exper-tise in all areas discussed andapologize for any inappropriateomissions or misplaced emphasis.

BACKGROUND

While the role of mechanics inmorphogenesis has been recog-nized since the late 19th century,in the latter half of the 20th cen-tury its impact on the field oforganismal development waseclipsed by the prominence andsuccess of genetic and molecularbiological approaches (Davidsonand Keller, 2007; Hutson and Ma,2008; Keller et al., 2008). Recentadvances in experimental andcomputational technologies, how-ever, have revived interest in themechanical aspects of morphogen-esis. In this section, we brieflysummarize basic mechanisms ofembryonic development and pres-ent an overview of the main theo-ries and types of models used tosimulate them.

Basic MorphogeneticMechanisms

There are two principal tissuetypes in the developing embryo,mesenchyme and epithelia, whichare distinguished by the interac-tions of cells with each other and

with the extracellular matrix(ECM) (Trinkaus, 1984; Bard,1990; Davies, 2005). Mesenchy-mal tissue contains significantECM within which cells are embed-ded. Such cells form adhesive con-tacts within the ECM via focaladhesions, and exert strong trac-tion forces which pull on anddeform the matrix (Harris et al.,1980). Cellular motility typicallyinvolves proteolysis and remodel-ing of the matrix with only tran-sient cell–cell contacts (Friedl,2004). The dispersion, migration,and condensation of mesenchymalcells are guided by a variety ofsignals, including chemical gra-dients (chemotaxis), adhesiongradients (haptotaxis), and stiff-ness gradients (durotaxis) (Da-vies, 2005).Epithelia are polarized sheets of

cells with an apical side facingaway from the body (or toward aninner lumen) and a basal side con-tacting the basal lamina and ECM.The cells in an epithelium areattached to one another apicallythrough belt-like adherens junc-tions, where extracellular E-cad-herin modulates cell–cell adhesionand the cytoskeleton facilitatesthe transmission of forces throughand between cells (Lodish et al.,2004). Because of strong cell–cellcontacts, epithelia move collec-tively as sheets during morpho-genesis (Bard, 1990).In some cases, coordinated con-

traction of the apical actomyosinbands bends an epithelium byconstricting the apex and givingthe cells a wedge-like profile(Martin, 2010). Besides passivestretching, planar changes in ge-ometry can be caused by cellintercalation, as cells exchangeneighbors in the plane of the epi-thelium. This process producesconvergent extension (CE), withthe epithelium shortening alongone axis and extending in theother (Keller et al., 2008). In addi-tion, epithelia are said not to toler-ate a free edge (Trinkaus, 1984),and, in the case of embryonicwound healing and certain mor-phogenetic events, develop supra-cellular actomyosin cables at theleading edge (LE). These cables

contract to close the wound with a‘‘purse string’’ like mechanism, insome cases aided by filopodia.Other epithelial morphogeneticmechanisms include branching oftubes, fusion of sheets, and inflationof sealed vesicles via hydrostaticpressure (Davies, 2005). Bothmesenchyme and epithelia canchange shape and volume throughcellular proliferation, growth, andapoptosis.

Theories and Models forMorphogenesis

Several quantitative theorieshave been proposed to account formorphogenetic phenomena, wheremechanical and biochemical proc-esses interact in complex waysto sculpt a developing organism(Urdy, 2012). The role of suchtheories is to help explainobserved behavior based on bio-logically and physically plausiblemechanisms, and to make experi-mentally testable predictions(Maini, 2004). In this section, wefirst briefly discuss two early theo-ries, largely based on biochemis-try, which have had a majorimpact on the field. While the re-mainder of this review focuses pri-marily on mechanical theories andmodels, it is important to keep inmind that developmental proc-esses must simultaneously obeythe laws of mechanics, thermody-namics, and biochemistry.An early theory for morphogene-

sis was proposed by Turing(1952), who suggested that spa-tial pattern is created by reactionsbetween two morphogens—aslowly diffusing short-range acti-vator and a rapidly diffusing long-range inhibitor (Kondo and Miura,2010). Since the appearance ofthat paper, many investigatorshave used variations of this reac-tion-diffusion model to study pat-tern formation (Gierer and Mein-hardt, 1972; Murray, 2003a;Kondo and Miura, 2010). Althoughmorphogens are generally under-stood to play a prominent roleduring development (Wolpert,1969; Howard et al., 2011), froman experimental perspective theapplicability of the Turing mecha-

MECHANICS OF MORPHOGENESIS 133


nism to pattern formation hasbeen limited (Kondo and Miura,2010).The second theory involves cell

sorting. The differential adhesionhypothesis (DAH), proposed bySteinberg (1963), is based on theobservation that embryonic cells,when disaggregated and allowedto recoalesce, behave in ways thatstrikingly resemble the behavior ofimmiscible fluids. Cells of one typeaggregate so as to minimize theirsurface area, and different celltypes, when mixed together, sortthemselves into distinct homoge-neous clusters, one of which mayengulf the other. Steinberg (1963)postulated that this behavior isgoverned by differences in cell–cell adhesion, and that cell mix-tures undergo a phase separationsuch that the final configurationcorresponds to the minimum inter-facial and surface free energies(Forgacs and Newman, 2005;Lecuit and Lenne, 2007). A widerange of predictions made by theDAH have been experimentallyvalidated (Steinberg, 2007), andmany computer simulations basedon this theory or its modificationshave supported its conclusions(Brodland, 2004 and referencestherein).In both of these theories, tissue

deformation is generally regardedas a downstream consequence ofchemical patterns generated byother means. In mechanical theo-ries, on the other hand, deforma-tions themselves are the pattern,and the interplay of stress, mate-rial properties, and deformationbecomes the central consideration(Koehl, 1990). For more than acentury, investigators have specu-lated about and proposed numer-ous theories regarding physicalmechanisms of morphogenesis.Many early researchers used phys-ical simulacra of embryonic tissueto test their ideas by analogy(Weiss, 1939; Thompson, 1942).Lewis (1947), for instance, con-structed a physical model of anepithelium using brass bars andrubber bands to investigatehypotheses about the mechanicalforces driving invagination. In themodern era, computational models

have largely replaced physicalmodels, but the emphasis on me-chanical forces as the proximalcause of morphogenesis remains(Clausi and Brodland, 1993;Taber, 1995; Hutson and Ma,2008).Perhaps the most elementary

biomechanical approach is tomodel tissue as a network of one-dimensional (1D) elastic elements(springs), viscous elements (dash-pots), and contractile elements(Koehl, 1990). These models canprovide insight into basic mechani-cal behavior, but generally do notinclude other potentially importantcharacteristics, such as resistanceto shear.More commonly, mechanical

behavior of soft tissue is analyzedusing the principles of continuummechanics, whereby tissue istreated as a continuous materialrather than a collection of discreteparticles. The concepts of stress(force per unit area) and strain(relative change in length orangle) are central ideas in thetheory. These quantities mustobey equilibrium, geometric com-patibility, mass conservation, andconstitutive (stress–strain) equa-tions (Lai et al., 2009).Oster et al. (1983) and Murray

and Oster (1984) presented a con-tinuum mechanics-based theorythat can be used to simulate bothmesenchymal and epithelial mor-phogenesis. In this theory, a tis-sue is treated as a mixture of cellsand matrix. The basic continuumequations are modified to allow forcell migration and proliferation,active stresses exerted by cells onthe matrix, and passive stressesexerted by the matrix on the cells.Most applications of this theoryhave assumed that deformationsare small, which is not a validapproximation for many morpho-genetic processes.To handle arbitrarily large defor-

mations, the nonlinear growththeory of Rodriguez et al. (1994)has been used to effectively simu-late many morphogenetic proc-esses (Munoz et al., 2007; Taber,2009; Varner et al., 2010;Ambrosi et al., 2011; Filas et al.,2012). For example, active con-

traction can be simulated by nega-tive growth with a correspondingincrease in stiffness. Here, the ki-nematic (strain) equations of con-tinuum mechanics are modified toinclude volumetric growth. The ba-sic idea in this theory is that uni-form growth of tissue produces achange in volume without gener-ating stress. However, growth re-stricted by external supports orsurrounding tissue (for example,differential growth) will producestress. One limitation of thistheory is that it does not distin-guish between growth caused bycell division and that caused byhypertrophy (Taber, 1995).Models based on the Murray–

Oster and growth theoriesdescribed above treat embryonictissue as an elastic or viscoelasticsolid. Another class of continuummodels assumes that the tissuebehaves as a viscous liquid.Indeed, for the relatively long timescales of morphogenesis, embry-onic tissue can undergo perma-nent deformation analogous toviscous flow (Phillips et al., 1977;Forgacs et al., 1998), and cells insome early morphogenetic proc-esses move in eddy-like patterns.Lubkin and Li (2002) treat bothepithelia and mesenchyme as flu-ids of different viscosities, whilePouille and Farge (2008) andFleury (2011) model morphoge-netic movements in terms of clas-sical two-dimensional hydrody-namical flow.The large and complex changes

in geometry that occur duringembryogenesis pose a challengeto investigators aiming to solveproblems in morphomechanics,and finite-element methods are akey tool for obtaining solutions tocontinuum mechanical problems.In this method, a complex body isdivided into geometrically simplesubdomains (finite elements) overwhich the displacement field canbe approximated by simple func-tions. The governing equations aresolved for the entire assemblageof elements by enforcing compati-bility and continuity betweenelements (Reddy, 1993; Bathe,1996). Such tools and theorieshave been applied to a wide range

134 WYCZALKOWSKI ET AL.


of morphogenetic processes, aswe now describe.

GASTRULATION

Gastrulation is a highly orches-trated event featuring dramaticcell deformations and migrationsto establish the multilayered bodyplan of the embryo. This processestablishes the three principalgerm layers (endoderm, ecto-derm, and mesoderm). The pro-cess of gastrulation varies acrossspecies, possibly reflecting differ-ences in environment and earlyembryonic morphology (Bard,1990; Leptin, 2005).Although much has been learned

about the genetic and molecularaspects of gastrulation (Leptin,1995; Nikolaidou and Barrett,2004; Dawes-Hoang et al., 2005),less is known about the mechanicsof this process. To help elucidatethis important biomechanicalevent, many theoretical modelshave been proposed for gastrula-tion in organisms such as sea ur-chin and Drosophila. The initialstages of sea urchin gastrulationinvolve axisymmetric invaginationof a fluid-filled spherical shell(Davidson et al., 1995), while gas-trulation (or ventral furrow forma-tion) in Drosophila involves thecreation of a groove along the rel-atively flat side of an ellipsoidallyshaped embryo (Lye and Sanson,2011). These differences in geom-etry influence the mechanics con-siderably (Conte et al., 2008;Taber, 2008).In one of the earliest such inves-

tigations of morphogenesis, Odellet al. (1981) presented a two-dimensional (2D) model for an ep-ithelium that treats each cell as aviscoelastic truss-like element witha contractile apex. In a circularring of cells, a specified contrac-tion in one cell apex (simulated bya shortening of the stress-freelength) stretches neighboringcells, which themselves contract ifstretched beyond a criticalamount. With appropriately cho-sen parameter values, thisresponse produces a wave of con-traction that generates a localinvagination, which resembles

morphogenetic processes such asventral furrow formation and neu-rulation. While the authors sug-gest that this model can be usedto simulate sea urchin gastrula-tion, a cylindrical model is notappropriate for the spherical seaurchin embryo. Nevertheless, thismodel was ahead of its time, andthe mechanical feedback it incor-porates has since gained increas-ing attention in studies of morpho-genesis (Beloussov, 1998, 2008;Pouille et al., 2009; Kornikovaet al., 2010).Regional variations in mechani-

cal properties can strongly affectmorphogenetic shape change. Forexample, an epithelium thatspreads uniformly on a relativelysoft substrate may buckle if con-strained by a stiffer substrate.However, relatively little is knownabout the mechanical properties inembryos, potentially making it dif-ficult to distinguish between multi-ple mechanisms that produce sim-ilar shapes. To illustrate this point,as well as to provide guidance forfuture experiments, Davidsonet al. (1995) used spherical finite-element models to test five possi-ble mechanisms for sea urchininvagination (Fig. 1):

1. Apical constriction/basal expan-sion within a circular region;

2. cell tractoring, as cells in a ringat the outer edge of the invagi-nating region emit protrusions(rods) that contract and pullthe ring radially inward, buck-ling the cells inside the ring;

3. circumferential contraction ofan actomyosin ring surroundingthe invaginating region bucklescells inside the ring;

4. apico-basal contraction causingcells in the invaginating regionto spread and buckle due toconstraints from surroundingtissue; and

5. regional swelling in the apicallamina (located between thecells and an outer hyalinelayer), with constrained expan-sion causing the invaginatingregion to bend inward.

Each hypothesis was carefullyevaluated relative to availabledata, and ranges of material pa-

rameters were determined thatwould be required for each modelto function as proposed. In a sub-sequent study, measurementsappeared to rule out the apicalconstriction and contractile ringmechanisms (Davidson et al.,1999).Recently, the Miodownik group

has used the continuum growththeory of Rodriguez et al. (1994)to simulate active changes in cellshape in a series of models forventral furrow formation (Munozet al., 2007; Conte et al., 2008,2009). In these models, celldimensions change by specifyingpositive or negative growth alongparticular directions, while cellwedging occurs via an apico-basalgrowth gradient. Taken together,these models include apical con-striction and basal elongation inthe invaginating mesodermalregion, and apico-basal shorteningwith transverse extension in theectoderm outside this region.The models were used to studythe effects of various combinationsof these cell shape changes, aswell as three-dimensional (3D) el-lipsoidal geometry and constraintsimposed by the surrounding vitel-line membrane (VM) and internalfluid.The 3D ellipsoidal model yielded

global shape changes similar tothose observed in experiments,and confirmed the important roleof the VM and yolk in gastrulation(Fig. 2). Importantly, invaginationin the 3D model was found to beless sensitive to variations in theratio between apical constrictionand apico-basal elongation thanthe 2D model of a transverse sec-tion, suggesting that the 3D modelis more robust to perturbations(Conte et al., 2008). Moreover,this 3D model revealed that yolkflow would be generated from thecenter of the embryo toward theanterior and posterior ends, lead-ing to global compression andexpansion of the embryo, resem-bling the motion of an accordion.Using a variation of the previous

2D model, Conte et al. (2009)investigated multiple combina-tions of invagination mechanisms,including ectodermal spreading,



Figure 1. Representations of various proposed mechanisms in finite-element models for sea urchin invagination (Davidson et al.,1995). A: General features of the models. B: Apical constriction. C: Cell tractor. D: Multicellular contractile ring surrounding invagi-nating region. E: Apico-basal contraction. F: Bending caused by gel swelling. G: Representative deformation of embryo. Modifiedfrom Davidson et al. (1995).

Figure 2. Experiments and 3D model for Drosophila gastrulation. A: Ventral (Modified from Grumbling and Strelets, 2006) andcross-sectional (Modified from Munoz et al., 2007) views of ventral furrow formation in experiments. B: Same views from finite-ele-ment model. Modified from Conte et al. (2008).



and correlated cell shape changeswith regional data on gene expres-sion. While a redundancy in mech-anisms leads to more robust devel-opment, this feature also makes itvery difficult, if not impossible, toidentify the principal driving forcesin a morphogenetic event. Addi-tional experiments, perhaps simi-lar to those suggested by Davidsonet al. (1995), can be used to ruleout some of the possible mecha-nisms (Conte et al., 2009). Muta-tions that affect certain mecha-nisms can also help determine howlocal cellular or molecular levelactivities affect global changes inshape (Davidson, 2008; Martinet al., 2010).Recently, Allena et al. (2010)

presented a similar 3D finite-ele-ment model for a Drosophilaembryo, using essentially thesame growth theory to simulatemorphogenesis. The authors simu-lated three concurrent morphoge-netic movements in early gastrula-tion: ventral furrow formation, ce-phalic furrow formation, and germband extension. Their results sug-gest that the number of active de-formation modes could be lessthan suggested by experiments,for example, apical constrictionalone, without apico-basal elonga-tion, could drive invagination.In the models discussed thus

far, any molecular underpinningsfor the most part are included onlyin a cursory manner. Two recentmodels take important steps to-

ward a more explicit accounting ofmolecular mechanisms. In the firststudy, Sherrard et al. (2010) usednew data to propose a two-stepprocess for invagination during as-cidian gastrulation. Their 2Dmodel for the embryo consists ofcoupled layers of endoderm andectoderm, and an explicit depend-ence on Rho, a cellular regulatorysignal. First, Rho-dependent apicalcontraction makes the endodermalcells wedge-shaped and pulls theectoderm around them (Fig. 3).Then, a Rho-independent basolat-eral contraction shortens andspreads the endodermal cells,while their apical sides remaincontracted and relatively stiff. Thissecond step drives invagination.In the second study, Driquez

et al. (2011) investigated howrecently identified contractile oscil-lations in an epithelium (Martinet al., 2010) can lead to a ratchet-like but sustained contractileforce. These two contractile modesare associated with the molecularsignals Snail and Fog, respec-tively, with the latter mediated bystretch. Accordingly, the model isbased on an idea similar to thestretch-activation mechanism pro-posed by Odell et al. (1981),whereby a cell apex contractswhen stretched beyond a specifiedthreshold. These researchers usedrelatively simple 1D and 2Dspring-dashpot models to repre-sent the apical side of an epithe-lium; invagination was not mod-eled explicitly. In the simulation, aregion containing a specified sto-chastic distribution of relativelysmall oscillating contractionseventually coordinates in phase tostretch a single cell beyond its crit-ical length. This triggers a sus-tained contraction of that cell, andthese responses eventually trans-form into a collective constriction.In the 2D model, this mechanismproduces a wave of contractionsimilar to that observed in vivo.

CELL REARRANGEMENT

AND PATTERN FORMATION

The last two models of the previ-ous section are examples of multi-

scale models for morphogenesis,as they consider tissue-leveleffects in response to both cellularand molecular-scale events. Multi-scale models have also been pro-posed for studies of cell rearrange-ment within tissue. Here, we con-sider specific applications to cellsorting, CE, and branching mor-phogenesis. All of these processesgenerate pattern in the embryo.

Cell Sorting

According to the DAH (Stein-berg, 1963), differences in adhe-sion strength between cells drivethe sorting and engulfment of onetissue by another. In a finite-ele-ment cell-level model, Chen andBrodland (2000) extended this hy-pothesis by including contractileforces along cell boundaries (seealso Brodland, 2002). Here, indi-vidual cells are represented astwo-dimensional polygons, withthe mechanical contributions ofapical actomyosin contraction andintercellular adhesion representedcollectively by one equivalentinterfacial tension acting tangentto the cell edge. The cytoplasm istaken as a viscous fluid. In thismodel, strengthening adhesionworks to lengthen a boundary,while contraction shortens it.The computer algorithm keepstrack of the changing shapes andpositions of the cells. In simula-tions for various boundary condi-tions, applied loadings, and mix-tures of cell types, the model pro-duces dynamic rearrangements ofcells and changes in cell geometrythat are in general agreement withexperimental observations for cellremodeling and sorting behaviors(Fig. 4) (Chen and Brodland,2000; Brodland and Chen, 2000).

Convergent Extension

During CE, epithelia narrow inone direction and extend inanother (Keller et al., 2000; Brod-land, 2006; Lye and Sanson,2011). This occurs, for example,during gastrulation, neurulation,and epiboly. In many instances,cell rearrangement in the form ofintercalation drives this process,

Figure 3. Two-step process for invagina-tion in ascidian gastrulation. Step 1: Api-cal constriction results in wedge-shapedcells in endoderm surrounded by ectoder-mal cells. Step 2: Subsequent apico-ba-sal contraction of endodermal cellsresults in invagination. Modified fromSherrard et al. (2010).



for example, cells move mediolat-erally between neighboring cells,causing the neighbors to separateand extend the tissue in the axialdirection.As a precursor to later models

developed by Brodland and col-leagues, Weliky and Oster (1990)proposed a 2D network model forcell rearrangement, with cells rep-resented by interconnected poly-gons. In this model, intracellularpressure drives swelling and pro-trusions, and tension is generatedat cell boundaries. The motion ofeach node depends on the forcesexerted on it, and cell geometryis updated to ensure geometriccompatibility. This model was usedto simulate epiboly in fish, withthe results showing that stressrelaxation could result from cellrearrangements in the epidermalenveloping layer. Soon afterwards,Weliky et al. (1991) modified thismodel to include other features ofcell motility, including directionalpersistence, tension-induced inhi-bition of protrusion, and contactinhibition. This improved modelwas then used to study tissueextension, cell rearrangement,cell–tissue boundary interactions,and parallel cell alignment.At the cellular level, some ex-

perimental data suggest that cellsemit lamellipodia which squeezebetween two neighbors to contacta more distant cell (Brodland,2006; Keller et al., 2008). Theprotrusions then contract, pullingthe originating and target cells to-gether to drive intercalation andCE. Using this idea, Brodland and

Veldhuis (2006) modified theircell-level finite-element algorithmto simulate this lamellipodial activ-ity in a model for CE. The resultsillustrate possible ‘‘mechanicalpathways’’ of CE across differentlength scales, that is, lamellipodiaproduce directional, contractileforces at a subcellular level thatdrive cell intercalation, which inturn results in global convergenceand extension of epithelia.Using experiments and a similar

model, Rauzi et al. (2008) investi-gated the mechanics of germband elongation (GBE) in Dro-sophila embryos and showed thatcell intercalation could be drivenby anisotropic cortical tension.The cell networks in this modelare 2D and the cell contact linesform junctions. The total energyof the network is the sum of lineenergy (due to line tension), areaelastic energy (caused by fluctu-ating cell area), and cortical elas-tic energy (stretching of cell pe-riphery). During intercalation, two3-way vertices are brought intocontact to minimize energy, gen-erating a temporary four-way ver-tex, followed by a topologicalchange in the junction called a T1transition, which drives the cellshape changes. In the simula-tions, T1 transitions occur whenthe length between adjacent junc-tions becomes sufficiently small,and the cell network eventuallyreaches an equilibrium configura-tion where GBE is produced.Here, anisotropy of cortical ten-sion is shown to control cell shapechanges during intercalation. This

study suggests that anisotropy ofcortical tension at the subcellularlevel, regulated by myosin II ac-tivity, can drive the coordinatedcell shape changes and tissuedeformations during CE, and thatcontractile fluctuations observedin vivo can play an important rolein intercalation.

Branching

Experiments have suggestedthat mechanics plays an importantrole in the initiation and progres-sion of branching, which occursduring the formation of blood ves-sels, organs such as the lung andkidney, and mammary and sali-vary glands (Davies, 2005;Affolter et al., 2009). For example,in vitro experiments indicate thatbranching is triggered in regionsof high mechanical stress (Nelsonet al., 2005; Gjorevski and Nelson,2010). Moreover, cells cultured ona gelled substrate exert tractionson the matrix and surroundingcells, eventually forming aggre-gate clusters between which linesof tension appear (Fig. 5A). Cellsthen migrate along these lines oftension, forming cellular cords andleading to the development of anetwork structure resembling cap-illary blood vessels (Ausprunk andFolkman, 1977; Vernon et al.,1995; Vailhe et al., 2001; Murray,2003b). Such patterns areobserved for a wide variety of cellsand substrates, and since the geo-metric characteristics of the net-work are sensitive to substratestiffness, mechanical effectsappear to play a major role in thisprocess (Murray, 2003b).Manoussaki et al. (1996) pre-

sented a mechanical model forvascular network formation basedon the Murray–Oster theory formesenchymal morphogenesis(Oster et al., 1983). In this model,cells exert an isotropic tractionwhose magnitude rises, peaks,and decays with increasing celldensity. The viscoelastic matrix,which is anchored to a stiff sub-strate (the dish), carries cellsalong as it deforms, and the cellsactively crawl on the matrix in adirection biased by the orientation

Figure 4. Cell mixing and sorting in simulations of heterotypic cellular aggregates.(A) Initial configuration for both simulations. A low interfacial tension between lightand dark cell types leads to total mixing (B) while sorting occurs with high interfacialtension (C). Modified from Brodland (2002).



of matrix strain. Manoussaki et al.(1996) found that patterns formonly if the initial cell density isnot too high, and the strength ofthe traction exerted by each cellexceeds a critical value. The pat-terns generated in numerical sim-ulations of this system resemblethose observed in vitro, both intheir temporal evolution and gen-eral appearance (Fig. 5B).Many researchers have

extended this model (see Ambrosiet al., 2005, for a review). Namyet al. (2004) included the effectsof haptotaxis and the fibrous na-ture of the substrate. Combiningexperiments with finite-elementsimulations, the authors exploredthe effect of cell traction forcesand the mechanical resistance ofthe matrix on network topology,and were able to reproduce pat-terns resulting from a thicknessgradient in the substrate. Holmesand Sleeman (2000) investigatedthe role of cellular traction andmatrix viscoelasticity on the vas-cularization of tumors. Togetherwith haptotaxis, their modelincludes a diffusing tumor-gener-ated chemical signal which induceschemotaxis and mitosis in the en-dothelial cells. The model providesexcellent agreement with experi-mental results and captures manyrelevant features of angiogenesis.Tosin et al. (2006) distinguished

between mesenchymal motion ofcells, where matrix adhesion andtraction play a dominant role, andamoeboid-type motion, which is

characterized by its speed, direc-tional persistence, and chemotac-tic nature (Friedl, 2004; Ambrosiet al., 2005; Painter, 2009). Insome experimental protocols(Serini et al., 2003), cells undergoamoeboid-type motion for severalhours before anchoring to the sub-strate and proceeding with net-work formation as discussedabove. To account for the twotypes of motion and the transitionbetween them, Tosin et al. (2006)constructed a model consisting ofa cell layer and a substrate layer.The cell layer, modeled as an elas-tic fluid, is initially independent ofthe substrate to simulate amoe-boid-type chemotactic motion; acoupling term then transfersstress from the elastic substrate tothe cell layer, thus capturing thetransition to mesenchymal motion.The model yields realistic networktopologies and predicts a charac-teristic length that depends on dif-fusion and decay of a chemicalsignal, a feature that purely me-chanical models cannot capture(Ambrosi et al., 2005).Computational models have also

been proposed for the branchingof epithelial tubes in mesenchymaltissue. While the epithelium isknown to play an important role inbranching morphogenesis, Wanet al. (2008) presented a modelthat explored the possible contri-bution of active contraction of themesenchyme. Both tissue typeswere modeled as fluids of differentviscosities, with contractile cells

embedded in the mesenchyme.Using a 3D model of a sphericallobe of epithelial tissue, theauthors found that a narrow bandof contractile mesenchyme is ca-pable of generating a cleft, a criti-cal step in initiating a branchpoint, and that the relative viscos-ities of the mesenchyme and epi-thelium have a strong effect oncleft geometry.

HEAD FOLD FORMATION

The early avian embryo is initiallyorganized as a flat disc of cellswhich sits atop the yolk at the ani-mal pole of the egg (Gilbert,2000). The first structure to breakthis planar geometry is the headfold (HF), which arises as a cres-cent-shaped fold at the anteriorend of the elongating neural plate(NP) (Fig. 6A, B). The HF consti-tutes the first bounding body fold(Patten, 1971) and initiates bothheart and brain development(Stalsberg, 1969; Schoenwolf andSmith, 2000). Although research-ers have long speculated aboutthe physical forces involved in HFmorphogenesis (Balfour, 1881;Shore and Pickering, 1889; Stals-berg and DeHaan, 1968), theproblem has only recently beenquantitatively investigated (Varneret al., 2010).Varner et al. (2010) employed a

continuum mechanical frameworkin combination with the Rodriguezet al. (1994) theory for volumetricgrowth to model the mechanics ofHF morphogenesis. Their resultssuggest that, much like the prob-lem of dorsal closure (DC) in Dro-sophila (Kiehart et al., 2000), mul-tiple forces work in tandem todrive the deformations observedduring HF formation. These forcesare ectodermal in origin, andinclude several physical mecha-nisms typically associated withneurulation: CE in the NP, cellwedging along the NP border, andtransverse shortening outside theNP (Colas and Schoenwolf, 2001;Lawson et al., 2001; Moury andSchoenwolf, 1995).These driving forces were simu-

lated in a 3D model consisting oftwo parallel layers representing

Figure 5. Pattern formation in vasculogenesis. A: Endothelial cells cultured on a ma-trix gel aggregate and form cord-like structures. B: Computer simulation of this pro-cess recapitulates the observed pattern. Modified from Manoussaki et al. (1996).



the blastoderm and the VM, sepa-rated by a narrow space (Fig. 6C).With the material properties esti-mated by microindentation, themodel yields tissue morphologyconsistent with observations (Fig.6D). In addition, stress and strainfields given by the model are inreasonable agreement with exper-imental measurements, and themodel is able to predict the abnor-mal HF geometries produced bylocal dissections that relieve stress(Varner et al., 2010). This studyshows that all three driving forces,as well as contact with the VM, arerequired for proper HF formation.

NEURULATION AND BRAIN

DEVELOPMENT

Following HF development, theprocess of neurulation begins, as alongitudinal invagination in the NPcreates the neural tube. The ante-rior end of this tube then expandsto create the primitive brain, whilethe rest becomes the spinal cord.The brain next subdivides into aseries of vesicles and begins a pe-riod of rapid growth driven by

increasing cerebrospinal fluidpressure (Desmond and Jacobson,1977). Cells lining the ventriclesproliferate, differentiate, andmigrate from the inner cavity toform the characteristic layers ofthe mature brain. During the laterstages of development in mostlarge mammals, the cerebral cor-tex folds into a highly convolutedshape to accommodate this rapidexpansion.Changes in brain size and shape

are driven by the dynamic inter-play between mechanical forcesand cellular processes. For exam-ple, during early brain develop-ment, rising cerebrospinal fluidpressure increases cell prolifera-tion rates (Desmond et al., 2005),while perturbations in wall stressinduce abnormal cytoskeletal ac-tivity (Filas et al., 2011). Properbrain development requires thecareful regulation of mechanicalforces, and computational modelshave emerged as useful toolsto quantitatively test whetherhypothesized morphogenetic mech-anisms are consistent with physicallaw.

Neurulation

Neurulation has been one of themost intensively studied problemsin the developing embryo.Because of the prevalence of con-genital defects resulting fromimpaired neural tube closure (e.g.,spina-bifida and anencephaly), aswell as the complex physical proc-esses involved in shaping thetube, this problem has receivedconsiderable attention from themodeling community.As discussed earlier, Odell et al.

(1981) used a 2D model for epi-thelial invagination to simulateneurulation, showing that certainparameter values produce a closedtube. Later, Brodland and col-leagues (Clausi and Brodland,1993; Brodland and Clausi, 1994)developed 2D cell-level models forneurulation using apical constric-tion as the primary driving force.These models were used to testthe effects of varying epithelialthickness, apical force, and axialelongation, as well as the effectsof external forces on neural tubemorphology (Brodland and Clausi,

Figure 6. HF formation in chick embryo. A: Ventral aspect of the embryo illustrating the HF and NP regions. B: Out-of-plane struc-ture of the head fold shown in sagittal section. C: Three-dimensional finite-element model for HF formation (VM). D: Correspon-dence between experiment and model in sagittal section. Modified from Varner and Taber (2010).



1995). While not directly verifiedwith experiments, some elementsof neural tube morphogenesis,including the formation of hingepoints, thickening of the NP, andneural tube closure were capturedin these simulations (Fig. 7).Although neurulation involves

the coordinated bending of a cellu-lar sheet, Jacobson and Gordon(1976) showed that cells exchangeneighbors within the epithelium asit deforms. To account for this phe-nomenon, Jacobson et al. (1986)proposed a cortical tractor modelto explain the motile behavior ofepithelial cells. Expanding on ear-lier ideas for cell crawling (Oster,1984), the authors argued that in-tracellular fluid flow generatesshear stresses that affect neigh-boring cell movements. Since thecells remain attached at the apicalsurface of the epithelium, dissimi-lar flow rates can cause cell colum-narization and epithelial rollingmovements, similar to thoseobserved during neurulation. How-ever, such a mechanism has notyet been verified experimentally.More recent efforts have focused

on adapting neurulation models toinclude multiscale effects and morerealistic 3D geometries (Chen and

Brodland, 2008; Brodland et al.,2010). In these studies, triangularpatches of epithelium are subdi-vided into active and passiveregions and mapped to 3D embryoreconstructions. Morphogenesis isdriven primarily by the interfacialtension between cells. With realis-tic mechanical properties and ex-perimental geometries, thesemodels reproduce normal pheno-types for amphibian neurulationquite well. If incorrect parametersfor interfacial tension (generatedby contractile actomyosin bundles)and mediolateral tension (gener-ated by oriented lamellipodia) areused in the simulation, abnormalmorphologies result.

Primary Brain VesicleMorphogenesis

Until recently, no models forearly brain morphogenesis hadbeen proposed. Here, we discuss amodel that was used to study dif-ferences in the shape of the earlybrain tube between species (Filaset al., 2012), differences whichmay arise from variations in neuru-lation mechanisms (Lowery andSive, 2004; Harrington et al.,2009). For example, brain tubes of

amphibians and fish are initially cy-lindrical in shape with a relativelyclosed, slit-like lumen, while thoseof chicken, mouse, and humansare rounded and comparativelyopen. Despite these differences,the primary vesicles (forebrain,midbrain, and hindbrain) subse-quently form across vertebrates tosubdivide the brain tube.For amphibians and fish, the

lumen must differentially open toform the brain vesicles, and Filaset al. (2012) simulated this pro-cess using a tubular finite-elementmodel with an initially slit-likecross-section. The theory of Rodri-guez et al. (1994) was used tosimulate local circumferential con-traction at two mediolateral hingepoints, as suggested by contractileprotein expression patterns. Thetube opens into a diamond shape,consistent with normal midbraindevelopment in these species.Interestingly, Filas et al. (2012)also found that enhancing con-tractile activity through drugtreatment transforms the rela-tively round brain cross-sectionsof chick embryos into morpholo-gies reminiscent of frog and fish,which are regionally characterizedby cross-sectional shapes of dia-

Figure 7. Simulation of neurulation via apical contraction. (A) The model considers one half of a symmetric transverse strip ofectoderm. As contraction occurs in a length-dependent fashion, a neural ridge-like structure develops, followed by narrowing andthickening of the NP as hinge points develop (B, C), and closure of the neural tube (D). Modified from Clausi and Brodland (1993).



monds, triangles, and slits. Themodel showed that local contrac-tion in regions of highest actin den-sity can transform a circular tubeinto these various shapes (Fig. 8).Much like neurulation models(Odell et al., 1981; Clausi andBrodland, 1993; Chen and Brod-land, 2008), these simulations sug-gest a significant role for actomyo-sin-based contraction in regulatingearly brain morphogenesis.

Cortical Folding

Relatively little is known aboutthe forces that drive brain folding.

Cortical folding is an extremely im-portant problem, as abnormalfolding patterns are associatedwith many neurological disorders,including autism, schizophrenia,and mental retardation. Duringthe last few decades, severalmathematical models have beenproposed for the folding process.Richman et al. (1975) modeled

the cerebral cortex as a growingbilayered shell. They assumed thatthe outer layer grows faster thanthe inner layer, creating compres-sion in the outer layer that causesbuckling and the appearance offolds (Fig. 9). This hypothesis is

consistent with experiments show-ing that isolated brains fold with-out the constraint provided by theskull (Barron, 1950). With param-eters estimated from availabledata, their analysis showed thatthe predicted wavelength of thebuckling pattern is consistent withthat of normal brains. In addition,when geometry and growth rateswere adjusted according to thedata, the wavelengths matchedreasonably well those found inbrains with abnormally small folds(microgyric) or fewer folds (lissen-cephalic).More recently, Toro and Burnod

(2005) developed a 2D modelinvolving truss-like elements form-ing an elastoplastic cortical layer,with circumferential growth stipu-lated and constrained by radialfibers. This growth works in com-bination with regionally prescribedvariations in mechanical propertiesto produce folds, and the authorsinvestigated the effects of variousasymmetries. Nie et al. (2010)proposed a similar model,extended to three dimensions, inwhich folds are driven by both cir-cumferential cortical growth andradial growth of underlying tissue,consistent with recent evidencesuggesting that cortical foldinginvolves growth in multiple direc-tions (Reillo et al., 2011). How-ever, while these models producerealistic folding patterns, to ourknowledge there has been noattempt to test these mechanismsusing other data, for example,stresses and strains. Such testsare important to distinguishbetween multiple mechanisms.Recent imaging studies have

shown that folding occurs in a spa-tially and temporally phased man-ner (Neal et al., 2007; Kroenkeet al., 2009). To capture thesephenomena, Xu et al. (2010)developed a modified version ofthe differential growth model pro-posed by Richman et al. (1975),with growth included via the theoryof Rodriguez et al. (1994). In alayered circular disk representing abrain slice, circumferential growthis first prescribed in a local regionof the outer cortical layer, causingthe formation of a single bulge.

Figure 8. Brain morphogenesis driven by local apical contraction. (A–D) Cross-sec-tional views of the chick brain and F-actin distribution (green) at the midbrain (M),mid-hindbrain boundary (MH), and posterior neuroepithelium (N). Normal midbrains(A) have a relatively round shape and uniform F-actin distribution, while enhancingcontraction results in cross-sections with sharp corners and high F-actin density (as-terisk in B–D). (A0–D0) Computational models of corresponding cross-sections. Modi-fied from Filas et al. (2012).

Figure 9. Schematic of model for brain folding (Richman et al., 1975). A: Bilayermodel for the cerebral cortex. B: Relatively rapid growth of cortex causes buckling.From Richman et al. (1975). Reprinted with permission from AAAS.



Repeating this sequence in neigh-boring regions generates aseries of folds, similar to normaldevelopment, with predicted stressdistributions being in agreementwith experimental results (Xuet al., 2010).

HEART DEVELOPMENT

Cardiac development has beenone of the most intensively stud-ied problems in the developingembryo (Taber, 2006). Neverthe-less, the physical mechanisms thatcreate the heart remain poorlyunderstood. Most work in this areahas focused on cardiac looping,which is the first major morpho-logical sign of left–right asymme-try in the embryo.The heart begins as a relatively

straight tubular structure consist-ing of a thick layer of extracellularmatrix (cardiac jelly) sandwichedbetween an inner endocardiallayer and an outer myocardiallayer, which is a two-cell-thick epi-thelium of developing cardiomyo-cytes. During the first phase oflooping, called c-looping, the hearttube (HT) acquires the shape of ac-shaped tube with convex curva-ture normally oriented toward theright side of the embryo (Manner,2000). The dorsal mesocardium(DM) is a longitudinal structurethat initially connects the dorsalside of the heart to the foregutbefore rupturing during c-looping.Following c-looping, further defor-mation (s-looping) then createsthe basic pattern of the futureheart, with the process of septa-tion eventually dividing this curvedtube into two ventricles and twoatria (Romanoff, 1960). Here, wefocus primarily on c-looping, whichhistorically has received the mostattention from researchers in car-diac morphogenesis.It is important to realize that c-

looping involves a combination ofventral bending and rightward tor-sion (or rotation). Although thishas been known for decades(Butler, 1952), a failure to appre-ciate these two deformation com-ponents has led to considerableconfusion in interpreting experi-mental results (Manner, 2000;

Taber, 2006). Recent work hassuggested that the bending com-ponent of c-looping is intrinsic tothe HT, while torsion is drivenmainly by extrinsic forces (Taber,2006).Since the early 1900s, research-

ers have proposed numeroushypotheses for the mechanisms ofcardiac bending. The first mecha-nistic hypothesis for looping appa-rently was proposed by Patten(1922), who suggested that theHT is forced to bend simplybecause it outgrows the spacethat it occupies. This hypothesis,however, was later contradictedby the finding that the HT bendswhen isolated in culture (Butler,1952; Manning and McLachlan,1990), even when drugs are usedto inhibit any contractile woundingresponse induced by dissection(Remond et al., 2006). Otherhypotheses that have beenexplored include bending drivenby differential growth (Sissman,1966; Stalsberg and DeHaan,1969), remodeling in response tohemodynamic forces (Spitzer andLev, 1951; Hove et al., 2003),active changes in myocardial cellshape (Manasek et al., 1972;Latacha et al., 2005), and cardiacjelly swelling (Manasek et al.,1984b). However, none of thesemechanisms are consistent withall available data (Taber, 2006),and this problem is still beingdebated in the literature.Theoretical models have been

used to test some of these andother ideas. Taber et al. (1995)represented the combined HT andDM as a bilayered beam, withgrowth simulated using the theoryof Rodriguez et al. (1994). Twobending mechanisms were stud-ied. In the first, the DM isassumed to be initially under lon-gitudinal tension. Then, as it rup-tures it shortens like a rubberband and pulls the HT into a bentconfiguration with the DM locatedalong the inner curvature (asobserved experimentally). To sim-ulate this mechanism, tension wasgenerated in the DM by contrac-tion, and the HT was assumed togrow in response to stressesinduced by the bending.

The second mechanism involvesactive cell shape change. Here,contraction in the HT was assumedto occur in the circumferentialdirection, consistent with observedorientations of actin filaments inthe myocardium (Itasaki et al.,1991), and incompressibilitycauses these cells to extend axi-ally. In this model, the DM wastaken as a passive structure,which limits deformation of adja-cent cells and causes the HT tobend with the DM again lyingalong the inner curvature.Both models showed that the

proposed hypotheses are physi-cally plausible and consistent withmost data available at the time.However, later experiments withmyosin inhibitors apparently haveruled out a contractile mechanismfor bending (Remond et al.,2006), although it still remainspossible that a mechanism otherthan contraction drives changes incell shape.In this regard, Latacha et al.

(2005) speculated that actin poly-merization supplies forces thatcause regional changes in myocar-dial cell shape. This idea stemmedfrom experiments in which actinpolymerization was disrupted bothglobally and locally by variousdrugs. Based on these experi-ments, as well as known changesin cell shape during looping (Man-asek et al., 1984a; Itasaki et al.,1989; Shiraishi et al., 1992), tu-bular finite-element models weredeveloped for the isolated HT. Inthese models, cells were assumedto grow longer circumferentiallyand shorter axially near the innercurvature, while growing in bothdirections near the outer curva-ture. Predicted bending patternswere consistent with data for bothnormal and chemically perturbedlooping of isolated hearts (Latachaet al., 2005; Ramasubramanianet al., 2006).Recent experiments have sug-

gested that the torsional compo-nent of c-looping is driven largelyby (1) forces applied by theomphalomesenteric veins (OVs) atthe caudal end of the HT, with theleft OV pushing with more forcethan the right OV; and (2) com-



pressive forces applied to the ven-tral surface of the HT by a mem-brane called the splanchnopleure(SPL). The OV forces rotate the HTslightly toward the right, and theSPL enhances this rotation (Vor-onov et al., 2004). Both sets offorces are required for normallooping.To date, models for the torsional

aspect of looping have includedonly the effects of the OV forces,which experiments suggest aregenerated by a combination ofgrowth on the cranial side of theOVs and contraction on the caudalside (Ramasubramanian et al.,2006, 2008). In these models, theHT and OVs are treated as tubesconsisting of an inner layer of car-diac jelly and an outer layer ofmyocardium. Results were com-pared to measured strains andstresses that develop during nor-mal looping, as well as the effectsof various perturbations inembryos in which the SPL was dis-sected away before looping. Inone notable experiment, the left

OV, right OV, or both OVs wereremoved before looping began. Onsubsequent culture, the heartrotated toward the right, left, andright, respectively. A model usedto simulate these experiments(without modifying parameters)predicts these results and theobserved abnormal morphologyrelatively accurately (Fig. 10)(Ramasubramanian et al., 2008).Although some questions have

been answered for the loopingproblem, many remain open. Forexample, few researchers haveconsidered how these forces areregulated in time and space, ands-looping has received relativelylittle attention.

GUT LOOPING

In amniotes, the gut tube includesthree segments: the foregut,midgut, and hindgut. A primaryloop forms in the midgut on theventral side driven by elongationof the gut tube, followed by asequence of rotation, bending, and

twisting to properly position andshape the future intestine (Daviset al., 2008). Data have indicatedthat differential growth drivesaspects of this process. Althoughthere are many studies on thegenetic aspects of gut looping andthe associated left–right asymme-try (Horne-Badovinac et al., 2003;Davis et al., 2008; Taniguchi et al.,2011), the mechanics of loopingmorphogenesis remain incom-pletely understood (Savin et al.,2011).Kurpios et al. (2008) used a 2D

spring-dashpot-like model toinvestigate the asymmetric tiltingof the gut. They simulated the dy-namics of the dorsal mesenteryand gut using a model of highlydamped lattice cells driven by me-chanical forces. Each cell is repre-sented by seven interconnectednodes in a hexagonal lattice, andthe ECM is taken as an elastic me-dium with a given local volume. Inthis model, cell–cell adhesionforces, elasticity of the ECM, andintracellular forces are included to

Figure 10. Effects of atria removal on cardiac looping in chick embryo. Left and right atria removed in top and bottom panels,respectively. Initial (first column) and final (after 12 hr, second column) heart configurations in experiment, with correspondingsimulation results (third and fourth columns). Removal of the left and right atria results in leftward and rightward rotation, respec-tively, in both experiment and model. Modified from Ramasubramanian et al. (2008).



account for the damped dynamicsof the cell network. The modelyields asymmetric changes in cellshape that tilt the gut tube in theappropriate direction. More signifi-cant tilting occurs when both theepithelial cells change shape andthe mesenchymal ECM increasesvolume on the right side (e.g.,through ECM swelling).Although several studies have

been devoted to interpret the ini-tial looping (Horne-Badovinacet al., 2003; Davis et al., 2008;Taniguchi et al., 2011), the me-chanical origin of subsequent loop-ing has been described onlyrecently (Savin et al., 2011). Theembryonic gut in vertebrates has ahighly conserved number and sizeof loops in a given species, andsimilar looping patterns occuracross species from avians to

mammals. Through a series ofexperiments, Savin et al. (2011)first ruled out some possible mech-anisms that could drive gut loop-ing, such as constraints due to thebody cavity and differential growthat certain locations of the gut tube.Further dissection experimentssuggested that differential growthbetween the gut and the mesen-tery is indispensable for looping, asthe gut straightens into a long,uncoiled tube when dissectedaway, while the freed mesenterycontracts, suggesting it is undertension. Motivated by these obser-vations, the authors proposed atheory for gut looping which mini-mizes the stretching energy in themesentery, as well as the bendingenergy in the gut tube; this theorythen yields a scaling law for thewavelength of the gut loop.

Predictions from this theory weretested through experiments onchick embryos, physical models,and in silico simulations (Fig. 11).In the computer simulation, themesentery is modeled as a discre-tized isotropic membrane com-posed of a hexagonal lattice of lin-ear elastic springs, and the guttube is modeled as a membranestrip. The physical parameters(thickness, stiffness, strain, etc.)were obtained through experimen-tal measurements. Results fromboth the computational and phy-sical models match very well withexperimental observations oflooped gut tubes (Savin et al.,2011).

DORSAL CLOSURE AND

WOUND HEALING

Actomyosin cables are found in avariety of contexts, including em-bryonic wound healing and cytoki-nesis (Kiehart, 1999; Sonnemannand Bement, 2011). Here, we dis-cuss two problems that involvecontractile rings: dorsal closure(DC) in Drosophila embryos andwound healing in embryonic epi-thelia.

Dorsal Closure

During DC, the amnioserosa(AS), an eye-shaped area of epi-thelial cells, with a canthus ateach corner, is enveloped by asheet of lateral epidermis whichspreads from each side of theembryo to meet at the dorsal mid-line (Fig. 12A). The two leadingedge (LEs) of the epidermis thenfuse where they meet (Kiehartet al., 2000; Gorfinkiel et al.,2011). Forces that drive this pro-cess include contraction of the AS,tension in an actomyosin cable atthe LE, and filopodial tractionsthat pull apposing edges of theLE together in a zipper-like mechanism (Fig. 12). Laserwounding experiments, whichablate the tissue with subcellularresolution, have shown that thesemechanisms work redundantly inDC, with no single mechanismabsolutely required (Kiehart et al.,2000; Hutson et al., 2003).

Figure 11. Gut looping patterns. A: Experimental chick gut morphology after 16 daysincubation. B: A prestretched rubber sheet is stitched to the side of a straight rubbertube which, upon release, deforms into a shape mimicking the looped gut. C: A computersimulation reproduces the gut looping morphology. Modified from Savin et al. (2011).

Figure 12. A: Schematic of DC in Drosophila embryo. actin cable (AC), leading edge (LE)of the lateral epidermis, and amnioserosa (AS) are illustrated (left inset), as are the forcesgenerated by AC contraction (red), AS contraction (blue), and zipping (green) over thecourse of dorsal closure (DC). Modified from Solon et al. (2009). B: Force balance diagramat one point of the LE, showing the contributions of the AS (rASdS), the lateral epidermis(rLEdS) and the tension of the actin cable (T). Modified from Hutson et al. (2003).



Hutson et al. (2003) used laserdissection and modeling to studythe forces involved in DC. Themodel considers the force balanceat the midpoint of the LE andincludes stresses in the AS andlateral epidermis, tension in thecurved actin cable (AC), and a vis-cous drag force dependent on therate of closure (Fig. 12B). The zip-ping of the canthi was modeledkinematically. By rapidly ablatingthe AS (thereby excluding its con-tribution from the force balance)and observing the recoil of the LEand its subsequent equilibrium,the authors estimated the relativecontributions of the various forces.With appropriate parameter valuesobtained through parametric fit-ting, the resulting model repro-duced the closure of both nativeembryos and those with theAS removed, as well as for agenetic mutant (myospheroid).This model was later refined to con-sider anterioposterior and medio-lateral asymmetry (Peralta et al.,2007, 2008).A second generation model for

DC by Layton et al. (2009) consid-ers the balance of forces along theentire LE. Both the AS and the ACgenerate force by active contrac-tion, and the lateral epidermis isassumed to exert a constant anduniform tension. Zipping is speci-fied kinematically as in Hutsonet al. (2003). Control, AS removal,and mutant experiments wereused to determine parameter val-ues, and the model then predictedrelatively well the results fromadditional laser ablation experi-ments, provided that AS contrac-tion was upregulated.Solon et al. (2009) observed

that AS cells undergo pulsatile api-cal constrictions which displacethe LE. To investigate the possiblerole of this activity in DC, theauthors constructed a 2D networkmodel of the AS and LE, with thecells represented as viscoelasticpolygons. The active force gener-ated at cell boundaries is assumedto be triggered when tensionreaches a critical value, similar tothe stretch-activation mechanismof Odell et al. (1981). This modelreproduces correlations in pulsing

activity between neighboring cells,as well as pulsatile arrest followingthe loss of tissue tension fromlaser ablation.

Embryonic Wound Healing

The cytoskeletal machinery thatoperates during DC, that is, thecontractile ring and filopodia, isalso employed in the healing ofembryonic epithelial wounds(Wood et al., 2002; Garcia-Fer-nandez et al., 2009). Embryonicwounds differ from adult woundsin significant ways. For example,fewer cell types are involved, min-imal inflammation occurs, andthey heal quickly and withoutscarring (Jacinto et al., 2001; Mar-tin and Parkhurst, 2004; Sonne-mann and Bement, 2011).Martin and Lewis (1992) studied

wound healing in the skin of chickembryos. After dissecting away apatch of epidermis and a thin layerof underlying mesenchyme, theyobserved that the epidermismoved actively over the mesen-chyme until the lesion closed,leaving no scar. No filopodia wereseen, and fluorescent stainingrevealed a thick cable of actin atthe LE of the epithelium, support-ing the hypothesis that contractionof an actomyosin ‘‘purse string’’ inpart drives the closing of thewound. While these observationsform the basis for some models, itshould be noted that not all em-bryonic wounds share these fea-tures. Although an actomyosinring is observed in Xenopus (frog)embryos, for example, it appearsthat contraction of the underlyingcells and protrusive activity of theremaining epithelial cells are pri-marily responsible for wound heal-ing (Davidson et al., 2002). Signif-icantly, the wound does notbecome more round as it heals, incontrast to observations in thechick (Martin and Lewis, 1992)and mouse (McCluskey and Mar-tin, 1995).Murray and Sherratt (Sherratt

et al., 1992; Murray, 2003a) usedthe Murray–Oster theory to modelembryonic wound healing. Here,the embryonic skin consists of a

cell layer attached to a basal lam-ina, which is represented by anelastic foundation (springs). Thisstudy focuses on the aggregationand alignment of actin into an ac-tomyosin cable in the cell layerand considers the quasi-equilib-rium state when the AC halts epi-thelial expansion following wound-ing. At that point, epithelial ten-sion is balanced by elasticrestoring forces originating in thebasal lamina. With this model,Sherratt and et al. (1992) wereable to reproduce the experimen-tally observed retraction of theepithelium, and the actin concen-tration at the wound edge borea close resemblance to the ACobserved experimentally.Sadovsky and Wan (2007)

extended the Murray–Sherrattmodel to investigate the dynamicsof wound closure following thequasi-equilibrium state describedabove. In this model, the dynam-ics of tissue deformation is givenby the balance of forces originat-ing in both the epithelium and themesenchyme. Contraction in theepithelium is a function of asym-metry in a wounding signal, whichthe authors argue is most plausi-bly mechanical stress, and themesenchyme contracts at a ratewhich depends on the velocity ofthe epithelium. Utilizing a special-ized version of the Murray–Sher-ratt model and a radially symmet-ric wound, the authors reproducedthe initial and intermediate stagesof wound closure.

MECHANICAL FEEDBACK IN

MORPHOGENESIS

It has become increasingly clearthat mechanical stress, in additionto driving the tissue deformationsresponsible for morphogenesis,also plays a role in the control ofcellular behavior in the embryo,including differentiation, prolifera-tion, and contractility (Vogel andSheetz, 2006; Hutson and Ma,2008; Wozniak and Chen, 2009;Levayer and Lecuit, 2012). In theDrosophila embryo, for instance,mechanical stress has been shownto affect both gene expression



(Farge, 2003; Desprat et al.,2008) and downstream signalingpathways (Pouille et al., 2009),which govern the production offorces driving gastrulation. Inter-estingly, Pouille et al. (2009)showed that gastrulation in mu-tant Drosophila embryos can berescued by local mechanical inden-tation. Results such as these sug-gest that mechanical feedback isinvolved in epigenetic control ofmorphogenesis (Brouzes et al.,2004).As discussed earlier, perhaps

the first model for morphogenesisto include mechanical feedback isthe invagination model of Odellet al. (1981), in which cells con-tract in response to stretch. Morerecently, Ramasubramanian andTaber (2008) proposed a differentmechanism, where contractionand growth are controlled bychanges in the value of a ‘‘targetstress’’ as specified by genetic ac-tivity. In their finite-element mod-els, an increase or decrease in tar-get stress triggers contraction orgrowth, respectively. Based onthis idea, models were used tosimulate bending of epithelia,including invagination in cylindricaland spherical shells.For several decades, the devel-

opmental biologist Lev Beloussovhas studied the role of mechanicalfeedback in morphogenesis. Hisexperimental observations suggestthat morphogenesis is regulatedin part by mechanical stress(Beloussov, 1998), leading tohis hyper-restoration hypothesis(HRH). According to HRH, embry-onic tissue reacts to mechanicalperturbations in a way that tendsto restore, but overshoot, the orig-inal stress (Beloussov, 1998;Beloussov and Grabovsky, 2006).This implies that embryos havethe capability to self-assemble tosome extent, presumably gov-erned by morphogenetic laws andcontrolled by mechanical feedbackloops, assuring a degree of robust-ness against perturbations (Taber,2008). Here, it is worth notingthat HRH is based on the view thattissue shortening elicits a con-tractile response, while stretchtriggers relaxation and active

lengthening. This behavior is op-posite to the stretch activationresponse postulated by Odell et al.(1981). Realizing this discrepancy,Beloussov (1998) suggested thatHRH applies to large deformations,while stretch-activated contractionoccurs when stretch perturbationsare relatively small.Although some efforts have

been geared toward obtainingquantitative support for HRH(Beloussov and Grabovsky, 2006,2007), most available experimen-tal data are qualitative. Forinstance, Beloussov et al. (1975)performed dissection experimentson amphibian embryos to mapchanges in stress distributionsduring development. Both rapid,passive deformations and slower,active deformations wereobserved. From the results ofthese experiments, Belintsev et al.(1987) proposed a phenomenolog-ical model that accounts for activeand passive deformations of cellsand mechanical feedback toaddress the collective polarizationand pattern formation in epithelia.From these studies, Beloussov andcolleagues (Beloussov et al.,1994; Beloussov, 1998; Beloussovand Grabovsky, 2006; Beloussov,2008) suggested conceptual mod-els based on HRH to explain a va-riety of morphogenetic eventssuch as gastrulation, neurulation,and CE. Experiments by Kornikovaet al. (2010), where artificiallybent explants (suprablastoporalregions of Xenopus laevis) contin-ued to bend after release of loadsalso seem to support HRH.Recently, Taber (2008) used fi-

nite-element models to explorethe theoretical limitations of HRH,as embodied in a feedback lawwhich includes an overshoot. Theproblems investigated includestretching, bending, and invagina-tion of epithelial sheets, as well asneurulation and sea urchin gastru-lation. In each model an initialperturbation, for example, a smallcontraction (presumably dictatedby genes), was used to initiate theHRH response. The results weremixed, with HRH producing realis-tic morphogenesis in some casesbut not in others. Hence, while

mechanical feedback likely isinvolved in these processes, genesalso may participate by making‘‘mid-course corrections’’ (Taber,2008) or by producing new pertur-bations at appropriate times toguide the creation of proper mor-phology.Biological systems clearly must

obey the laws of physics asexpressed mathematically. How-ever, it is not at all clear thatmathematical laws for biologyexist. Recently, Taber (2009)examined this issue using a gener-alization of HRH. The key idea wasto assume that the feedback lawdepends on the rate of change instress. For example, a slow stretchinduces growth, whereas a rapidstretch elicits a contractileresponse. Finite-element modelsfor growth of a pressurized artery,sea urchin gastrulation, woundhealing, axon stretching, andother examples improved on thepredictions of HRH, but stillyielded somewhat inconsistentresults when compared to avail-able experimental data. Therefore,the existence of mathematicallaws in biology remains an openquestion (Taber, 2009; Ambrosiet al., 2011).

CONCLUDING REMARKS

Despite numerous computationaland experimental studies, itremains incompletely understoodhow the genetic informationencoded in DNA is translated,through mechanics, into three-dimensional functional shapes.Moreover, relatively little is knownabout how epigenetic factors, inturn, interact with genetics throughmechanical feedback to regulatemorphogenesis. Clearly, molecularbiology has triggered a revolution instudies of developmental proc-esses. Many developmental biolo-gists, however, now realize that acomplete understanding of how theembryo is constructed also requiresstudies of the nuts and bolts of mor-phogenesis (i.e., mechanics),which has been largely neglectedduring the last few decades.Because development is a compli-cated process which can defy physi-



cal intuition, computational modelsfor morphogenesis are receivingincreasing attention.Although the importance of

mechanics in morphogenesis iswell established, a detailedaccounting of the stresses andstrains that drive the associatedtissue deformations is still far fromcomplete (Davidson et al., 2009).Sophisticated mathematical mod-els notwithstanding, advances inmechanobiology have been closelyconnected to technological advan-ces (Eyckmans et al., 2011), andthis will likely hold true for theforeseeable future. Such advanceshave been rapid and impressive,however. The revolution in in vivofluorescence microscopy is reveal-ing the localization and dynamicsof cellular components with un-precedented spatial and temporalresolution (Martin, 2010; Satoet al., 2010). Laser ablation tech-niques (Rauzi and Lenne, 2011,and references therein) allow forprecise mechanical perturbationsof cells and subcellular structures,and are a tool of outstanding util-ity in probing mechanical forces atthose scales. Finally, a new gener-ation of FRET-based single mole-cule stress sensors (Grashoffet al., 2010; Meng and Sachs,2011) open the possibility ofmeasuring stress with piconewtonsensitivity at the subcellular level.Taken together, our understandingof the mechanics of certain cellularand morphogenetic systems isprogressing at an acceleratingpace, and results from such modelsystems will continue to serve as aguide to the general principlesunderlying morphogenesis.For several reasons, creating

and testing computational modelsfor morphomechanical processesis a challenging endeavor. Themechanical properties of embryoscan be surprisingly variable, com-plicating the comparison bet-ween experimental and numericalresults. Critical morphogeneticevents often use multiple redun-dant mechanisms which must firstbe identified, and not all suchmechanisms need or can be incor-porated into a model. Problems indevelopment, as in much of biol-

ogy, frequently involve largenumbers of unknown parameters,and the response of the systemcan be highly nonlinear. In con-structing such models, a delicatebalance must be struck betweenrealism with impenetrable com-plexity on the one hand, andtransparency but oversimplifica-tion on the other.We perceive two major themes

in the modeling of morphogenesiswhich we feel hold particularpromise. One is the explicit incor-poration of mechanical and chemi-cal feedback into models of mor-phogenesis. Such feedback, whichhas been demonstrated in manyexperimental systems (Fernandez-Gonzalez et al., 2009; Ren et al.,2009) has long been recognized asa rich source of morphogeneticpattern (Murray and Oster, 1984).Intuition and reductionist methodstypically fail for such systems, andnew experimental approaches,computer modeling, and mathe-matical analysis will all play a rolein the description and understand-ing of the behavior of such sys-tems.The development of multiscale

models of morphomechanics(Blanchard and Adams, 2011) isperhaps even more significant.Quantitative data at multiple spa-tial scales are becoming increas-ingly available (Blankenship et al.,2006; Martin et al., 2010), andrecent models provide an excellentbeginning to capturing multiscaleeffects (e.g., Stylianopoulos andBarocas, 2007; Chen and Brod-land, 2008; Sherrard et al., 2010;Driquez et al., 2011). Still, muchremains to be done. Biologicalphenomena include the interactionbetween biochemistry, genes, andmechanics, and it appears likelythat tomorrow’s models will needto incorporate all of these aspectsto gain a comprehensive under-standing of the physical mecha-nisms of morphogenesis.

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Computational Models for Mechanics of Morphogenesis

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