Downloaded by guest on September 23, 2021 morphogenesis instability data. secondary and a trends as morphological explained matches be that can pattern this intricate We that brachiopods. the show certain consider in edge also multiscale-patterned Beyond we striking variation. and mechanism, morphological interlocking in basic result the variations process parametric growth how large the highlights a in and explains forms shell and by of recovers interlocking and diversity framework the secrete modeling for they basis Our mechanistic that pattern. the shell man- uncover rigid we the other, the of each lobes by 2 both of constrained mechanics con- tle, and close geometry By the pattern instabilities. of mechanical interlocking sideration by this the regulated how on and explain created based to is framework, growth, shell mathematical spec- of a genetically physics derive be we cannot Here, that formation ified. pattern physi- of dynamic process a cal suggests which influences, or environmental injury even to other due development, irregularity significant of exhibits edge entirety shell Interlock- the the when size. throughout and maintained regularity is varying antisymmet- ing present of may patterns that lobes, ornamental fact mantle the ric 2 despite by that overlaps secreted commissure or a valve, gaps in each no other shell with each continuum the meet a When edges forms valve view: strik- 2 of the yet point closed, mundane, developmental is apparently a from an feature displays ing that shell evolved bivalved convergently a also have mollusks bivalve ago. and years million Brachiopods 540 diverged than for more (received that ancestor shell-less 2019 phyla common 11, a shell-bearing from November 2 Jablonski are David mollusks Member and Board Brachiopods Editorial by accepted and TX, Lubbock, 2019) University, 24, Tech September Texas review Rice, H. Sean by Edited Plan a Moulton E. Derek shells bivalved of interlocking puzzle morphogenetic the unlocks Mechanics www.pnas.org/cgi/doi/10.1073/pnas.1916520116 patterned environmen- by a or as attached less live such species or factors some more which external be on substrate by may It instance at regularity: for grow perfect shell perturbed, may exhibit of valves pattern not the 2 does and the shapes, edge different Also, have mantle. and rates the formed different organ, is elastic it thin how a work. explain present the not of goal does the trait is which a How- development, of during interlock. closely function valves this the both that ever, conclude why to explains tempting alone be function could it and envi- events, functional and ronmental predators understood against easily role an protective a brings organism. providing that a advantage, same trait as the a of appear is halves not it 2 may Moreover, constitute this valves contin- glance 2 a first the forms as At surprise, that gaps. commissure no edge a with the in curve other shell uous each shell the meet the of valves when development 2 perfectly the of together fact throughout fit simple i.e., shell closed; the the is of is bivalves, valves appreciated, 2 and fully the brachiopods that rarely of but shells observed the readily bra- of most the assign of features One (6). mistakenly remarkable century 19th to early the the in authors mollusks to of several chiopods evolutionary led condition an is that bivalved mollusks shell- bivalve convergence The and a brachiopods 1). from both in (Fig. diverged shell ancestor have record common they fossil that less the however, and suggests, biochemistry, (1–5) shell phylogeny, molecular B ahmtclIsiue nvriyo xod xod X G,Uie igo;and Kingdom; United 6GG, OX2 Oxford, Oxford, of University Institute, Mathematical h avso h hl r ertdsprtl y2lbsof lobes 2 by separately secreted are shell the of valves 2 The ts nionmn) Universit Environnement), etes, ` ahoosadmluk r netbaepyata pos- clocks, molecular that from phyla derived Evidence invertebrate shells. 2 calcified sess are mollusks and rachiopods | growth a,1 | ahmtclmodel mathematical li Goriely Alain , yn1 92 ilubneCdx France Cedex, Villeurbanne 69622 1, Lyon e ´ a | n R and , mollusk gsChirat egis ´ b doi:10.1073/pnas.1916520116/-/DCSupplemental hsatcecnan uprigifrainoln at online information supporting contains article This 1 the under Published is ulse eebr1,2019. 16, December published First Editorial the by invited editor guest a is S.H.R. Submission. Direct PNAS Board. a is article This to interest.y competing contributed no A.G. declare and authors paper; The the model.y research; wrote mathematical performed the R.C. of and D.E.M. analysis A.G., the research; D.E.M., designed data; R.C. analyzed D.E.M. and D.E.M. contributions: Author cellular investigate to develop- begun during have margin shell studies of the recent record at Although mantle spatiotemporal ment. the a deposi- by the as the taken of viewed form form for be the The matrix 10). thus (9, may a shell shell as the calcified of serves carbonate periostracum, calcium that the the layer of first tion organic secretes soft that thin mantle a the elas- membranous called thin organ are a anatomy by groups tic margin both and the of at secretions shells secreted the incrementally of brachiopods, mode and bivalves in between differences some Despite Background 1. found patterns instability oscillatory secondary brachiopods. multiscaled certain a on striking how the demonstrate paramet- for also accounts via and emerges variation variation demonstrate ric morphological We significant instability. mechanical how interlock- of constrained continuation and an the biaxially how as edge naturally a demonstrate emerges shell edge We shell rigid patterned lobe. ing the mantle both opposing the with the the of of mantle interaction analysis physical soft detailed the shell-secreting and a growth on during based geometry shell trait morphological this for secreting the of development. epi- behavior observations shell of the during result These mantle the modulating maintained. as interactions emerges tightly pattern genetic is interlocking edges the interlocking that shell the imply cases 2 all in the Yet, of injuries. shell causing events tal owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To atr,aemnfs cossae opoueapredictable a produce to scales genetic across pattern. by manifest modulated into are probably insight factors, provides processes, model biophysical the vari- how trends wide morphological a create of By mechanism ety evolution. the in convergent variations of how example showing expla- prominent physical a a for provides generate nation model to This conspire pattern. interlocking mechanics an and process is geometry growth feature how shell this the explains been how of of model has advan- mathematical understanding A no functional and generated. is its there ancestor While clear, is shell-less years. tage in of common hundreds evolved valves a for has 2 described trait from the This that phyla closed. is 2 when seashells perfectly bivalved together in fit feature striking A Significance ee epoieagoercadmcaia explanation mechanical and geometric a provide we Here, y b NS57,LLTE(eLbrtied G de Laboratoire (Le LGL-TPE 5276, CNRS NSlicense.y PNAS PNAS | aur ,2020 7, January . y https://www.pnas.org/lookup/suppl/ | ooi eLo:Terre, Lyon: de eologie o.117 vol. ´ | o 1 no. | 43–51

EVOLUTION APPLIED MATHEMATICS A partial answer to this puzzle comes from oysters that live attached to a substratum. In these oysters, the surface of the attached valve carries the negative impression of the morphology of the substratum, while the free valve replicates in positive the form of the substratum, a phenomenon known as xenomorphism (i.e., “having a foreign form”) (Fig. 1E). No matter the irregular form of the substratum on which the oyster is attached (a stone, another shell, or an artificial substrate), the edge of the free valve closely fits with the edge of the attached valve. As the oyster grows bigger, the mantle margin of the attached valve starts to turn away from the substratum and no longer grows attached. At this stage, the shell attains what is called its idiomorphic form (i.e., “having its own form”) (17) and in some species, a zigzag- shaped commissure is generated at this stage. Our interpretation is that the xenomorphic and idiomorphic parts do not differ fun- damentally from the point of view of the growth processes. In the xenomorphic part, the form taken by the mantle margin secret- ing the attached valve is mechanically imposed by the form of the substratum, and this form is itself mechanically imposed to the mantle lobe secreting the free valve when both mantle lobes are at least temporarily in close contact while secreting the slightly opened shell. Once the shell no longer grows attached to the sub- stratum, the mechanical influence of the substratum is removed and there is only a reciprocal mechanical influence between both lobes. This reciprocal mechanical influence seems to be a gen- Fig. 1. (A) Phylogenetic relationships among brachiopods and mollusks eral characteristic of the growth of brachiopods and bivalves. For (data from refs. 5, 7, 8, and 37). (B and C) Convergently evolved shell com- example, in the case of traumatic individuals, the nontraumatic missures in fossil brachiopods (Septaliphoria orbignyana and Kutchirhynchia valve adapts its form and interlocks with the traumatic valve, no obsoleta) and bivalve mollusks (Rastellum sp. and Ctenostreon rugosum). (D) matter the abnormal form of the shell edge. An oyster with irregular interlocking pattern, Lopha sp. (Senonian, Algeria). Xenomorphic–idiomorphic transition in oysters and trauma (E) Xenomorphic oyster, Lopha sp.(Upper Cretaceous, Algeria). The attached mirroring in both bivalves and brachiopods suggest the following valve carries the negative impression of another shell, while the free valve hypothesis: Interlocking commissures are created by a combi- replicates its positive form (indicated by the arrows). (Scale bars: 10 mm.) nation of the mechanical constraints acting on each lobe and the mechanical influence of the 2 lobes on each other. In this paper, we develop a theoretical model of shell morphogenesis differential growth patterns underlying left–right asymmetries in that confirms this hypothesis and we also extract universal mor- gastropods (11) and to identify genetic and molecular bases of phogenetic rules. We show that the mechanical constraint acting shell biomineralization in both mollusks and brachiopods (12, on each lobe during growth imposes the geometric orientation 13), the morphogenetic processes underlying the diversity of of the morphological pattern while the reciprocal interaction shell shapes in both groups remain poorly known. Theoretical between lobes enforces the antisymmetry of this pattern. Both models invoking either reaction–diffusion chemical systems (14) principles are needed for perfect closure and are universal or nervous activity in the mantle epithelial cells (15), although characteristics of the growth of brachiopods and bivalves. successful in capturing the emergence of pigmentation patterns, do not explain the emergence of three-dimensional (3D) forms. 2. Mathematical Model A common default assumption in developmental biology is that A. Base Geometry. We first describe the general framework for molecular patterning precedes and triggers 3D morphogenetic the growth of bivalved shells by using the localized growth kine- processes. While this assumption might partly motivate recent matics description of refs. 18 and 19. The shell is modeled 3 studies of genetic and molecular mechanisms involved in shell as a surface r = r(s, t) ∈ R , where s is a material parame- development, only two-dimensional (2D) pigmentation patterns ter describing location along the shell edge, and t is a growth (that are molecular in nature) have been shown to map pre- “time” parameter which need not correspond to actual time but cisely with gene expression patterns (16). Marginal shell growth which increases through development. The shell is constructed in bivalves and brachiopods takes place when the valves are open, by defining an initial curve r(s, 0) = (x0(s), y0(s), 0) (where s is both mantle lobes being retracted away from the margin of each the arclength) and a growth velocity field q(s, t) representing the valve when the shell is tightly closed. In the case of patterned rate of shell secretion such that r˙ = q (overdot represents time interlocking commissures, it is difficult to conceive of genetic derivative). In the case of bivalved shells, the field q requires and molecular processes of morphogenetic regulation that would only 2 components: a dilation rate, denoted c, which describes specify that when the margin of a mantle lobe secretes a pat- the rate of expansion of the aperture, and a coiling rate, denoted terned edge on one valve, the same complex processes must b, which is equivalent to the gradient in growth in the binormal regulate the morphogenesis of the other mantle lobe to gen- direction and dictates how tightly coiled the shell is (Fig. 2). How- erate a perfectly antisymmetric edge on the other valve, both ever, since we are interested only in the shape, we can set the patterned edges closely interlocking when the mantle is retracted dilation rate to c = 1 without loss of generality, as it is only the and the shell is closed. In other words, supposing that molecular ratio of dilation to expansion that is relevant in the shell form. patterning triggers 3D morphogenetic processes raises the ques- The key to this description is to express the velocity field in tion of the nature of the coordinating signal between both mantle a local orthonormal basis {d1, d2, d3} attached to each point of lobes and how it could be transmitted. Formulated in that way, the shell edge. Here, we choose d3 to be tangent to the shell 0 the development of closely interlocking edges and the repeated edge; i.e., r (s, t) = λ(t)d3, where the prime denotes the deriva- emergence of similar complex commissures during the evolution tive with s and λ is a scale parameter characterizing the degree of 2 different phyla are puzzling problems. of total dilation from the base curve. Defining d2 to align with

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Fig. olo tal. et Moulton the is q hinge the formulation gen- initial of this loss the without be (taken to axis an erality along gradient growth binormal linear a a through generated component velocity is growth coiling direction, binormal the curve (the midline longitudinal nesoda opooia atr ntepaeorthogonal the plane the via in generally pattern generated morphological is a surface ornamentation as Antimarginal geometric understood field. smooth velocity the growth buckled the of on pattern morphological mantle the of superposition the as Orientation. growth Ornamentation C. the to adapted framework shells. modeling bivalved Here, in same constraints (21–24). a the patterns produce use in ornamentation properties we changes of mechanical basic morphology edge). how and diverse shell explain growth, rigid man- can geometry, (the (the one shell by foundation beam framework, evolving elastic modeled this growing an elegantly Within a to be as attached can edge tle) and mantle the shells treating formation in the defor- ornamentation ornamentation underlies continued mechanism way, of basic these this This of In patterns. records of mation amplified. excess spatiotemporal be an are pat- and If patterns deformation evolve edge. the will shell development, of tern through secretion sustained next is the length to in subsequently will induce respect calcified pattern can buckled with that be the stress length and compressive mantle, of the a of to excess buckling leads an This edge. has mantle shell the it the if edge faster, However, syn- shell smooth. perfect grows the in remain margin are will as shell shell rate and the same mantle and both the chrony secreting, at itself grows is mar- it mantle that mantle the secreting If the (20). of gin deformations the mechanical as of of emerge geometry ornamentations result unknown, base largely of remain shell variations develop- the the the while underlying that processes is investigation mental our ornamentation. for antimarginal premise termed basic is The that pat- edge oscillation shell an the of of tern consist Ornamentation. typically ornamentations of 3D opods, Basis Mechanical B. the closed. the in is (of perfectly shell valves base meet 2 the the always then will curve, shell) initial smooth same the (Section have brachiopods halves rates (c in both dilation coiling seen fixed as to the due different i.e., Nevertheless, be 3.C). may valves; not halves 2 do 2 the We the mantle. for between the symmetry of a growth and assume material shell of of secretion parameter, shape growth base rate geometric the coiling single that the a is through approach emerges this shell of the benefit The 1. section 2 0 = e i.2adfrhrgoercdtisin details geometric further and 2 Fig. see ; h aegoer o iavdsel scntutdvaalclydfie rwhvlct eddfie nabs curve base a on defined field velocity growth defined locally a via constructed is shells bivalved for geometry base The (A) b {d htcnb eae oasl-iia rcs of process self-similar a to related be can that 1 , d 2 , d x 3 h rwhcnit fdlto rdarw)adaciigvlct ntebnra (d binormal the in velocity coiling a and arrows) (red dilation of consists growth The }. xs.Bvle lorqieahne in hinge; a require also Bivalves axis). y s q h eutn ufc o n av ftebvlesel ihthe with shell, bivalve the of valve one for surface resulting The (B) axis. = 2 normdlteseli obtained is shell the model our In ) hc om oaihi spiral. logarithmic a forms which 0), = bx y 0 (s xs where axis, .. hl oln requires coiling shell i.e., ); nbvle n brachi- and bivalves In 1 = o ohvle) if valves), both for x x 0 -y 0 = IAppendix, SI ln when plane n thus and h rlnt ttepitwihhsdfre u”i longer i.e., “down”; is deformed “up” has which deformed point has l the equal: at which longer arclength point the no the than are at arclengths arclength the The however, plane. antimarginal deforms, (red), the top mantle in on appearing imposed pattern pattern the oscillation with half-mode 3 Fig. a in with illustrated shell is idea 3B (the produce The Fig. edge plane. C. will shell the this the rotate to about and serves plane force appears, of the pattern moment on deformed a edge) depending the shell differ, which the may thus in (and arclengths mantle these the deforms, shell Once smooth the identical. in nearly arclength material thus are neighboring and For secretion of secretion. rate of the points rate direc- the growth by the determined the is in in shell arclength of (i.e., tion length The interlocking. plane. rotated produces a in pattern orna- same the the well of as and orientation plane antimarginal the plane 3A determine orthogonal Fig. to plane. the mentation is in problem form natural not a orna- do that shows typically seashells mentations bivalved vector of with inspection normal close plane has However, (the which growth shell plane vector of the normal direction the in to i.e., tangent margin, pointing shell the to ensapiue and amplitude, at tern’s located are factor down pattern the the mantle, and of side up lower Mathematically, between and r needed. upper is symmetry the rotation on perfectly perfect points no a a thus for that and is considers deformations, there one if shell commissure, intuitive, flat of rate is angle This coiling the needed. increased steeper is particu- the with In that occurs shell. noting base which worth shell the is the for it along parameters lar point growth material the the and development, edge, of stage the on vanish. moment down the and is up equal, the plane deformation at are arclengths of the the points mode where rotates plane same that rotated the a mantle in 3C, shown the Fig. on In ornamentation. acting of force induces which of shell, the the moment of connects a portion that calcified already region the deformable to the mantle zone, generative the in * u ec o iulsmlct eew ltprin ftesel sbigcylindrical. being deformation; as shells mantle the to of portions prior plot points we here neighboring simplicity at visual arclength for hence equal with and shape oal,asalscino hl a eapoiae saclne ihlgrtmcspiral logarithmic with cylinder a as approximated be can shell of section small a Locally, p down up, > h ouint hspolmi h rtkycmoetthat component key first the is problem this to solution The h rcs ereo oainta aacstesri depends strain the balances that rotation of degree precise The l d spcue.Ti ifrnecetsadfeeta strain differential a creates difference This pictured. as = r hw oto fabs hl ylo)adtesame the and (yellow) shell base a of portion a shows ± λ r ˙ ˆ where v, ntegoercdsrpinotie above). outlined description geometric the in l u λ = consfrtesaigo h ukigpat- buckling the of scaling the for accounts PNAS l ˆ v d ota h ifrnilsri n thus and strain differential the that so , saui etrt edtrie that determined be to vector unit a is s  lutae noclainpteni the in pattern oscillation an illustrates t and | steapiueo eeto fthe of deflection of amplitude the is ieto o xdmtra point material fixed for direction aur ,2020 7, January t 2 ietoshglgtda ela the as well as highlighted directions ieto ihlna rdet(blue gradient linear with direction ) b | h oerotation more the , o.117 vol. d r 3 0 qipdwith equipped (s) ieto)that direction) | ∗ o 1 no. nethe Once B | and s 45 )

EVOLUTION APPLIED MATHEMATICS at all points along the shell edge and at all times throughout development (SI Appendix, section 2).

E. Rule 2: In-Phase Synchrony of Ornamentation Pattern. While the coplanarity of ornamentation planes ensures that the 2 ornamen- tation patterns will appear in the same plane, it does not in itself guarantee that the 2 valves will interlock. For this to occur, we also require rule 2 of interlocking: The ornamentation patterns must coincide in phase. We now show that this synchrony is born out of the mechanical interaction of the 2 opposing mantle lobes. Following refs. 22 and 23, we treat each mantle edge as a mor- phoelastic rod (25) attached elastically via the generative zone to a foundation, the rigid calcified shell (details in SI Appendix, section 3). The 2 mantle edges interact with each other when in contact through a repulsive interaction force ensuring that the 2 mantles cannot interpenetrate. Since the 2 valves are meeting at a common plane with equiv- alent length of shell edge, and assuming that the mantle tissue of each valve has the same mechanical properties, given an excess of length that induces a mechanical pattern, the preferred buckling mode for each respective valve will be the same, if considered Fig. 3. (A) The difference between a pattern imposed in the antimarginal in isolation. The question then is what form the buckled pattern plane, with normal vector r˙, and a rotation of this plane about the d3 will take when the 2 mantle edges are not in isolation, but inter- direction. In B, an oscillatory pattern in the antimarginal plane creates an acting with each other. The problem is greatly simplified by the unbalanced strain in the generative zone, as the arclength at the valleys first rule: Since the 2 planes of ornamentation are locally aligned, is less than at the peaks. In the schematic, the green curve ld has shorter we can consider the buckling problem in a single surface. Fur- length than lu. This strain creates locally a moment around the d3 axis. (C) This moment is balanced by rotating the plane of ornamentation until the ther assuming that the curvature of the mantle along the edge arclengths are made equal and the strain is balanced. is small, we “unwrap” the common ornamentation surface and consider a planar problem. For a given excess of length due to mantle growth, we compute the possible modes of deformation describes the orientation of the pattern such that the ornamen- for the 2 mantles parametrically given by (xi (s), yi (s)), i = 1, 2, tation appears in the d3-ˆv plane (details in SI Appendix, section in the x-y ornamentation plane (SI Appendix, section 3). Once 2). Then the balance of moment can be written as a geometric these are found, we consider the total mechanical energy of the condition system, given by the sum of bending and foundation energies on   r˙ · λ˙ˆv + λˆv˙ = 0. [1] each side and the interaction energy between the two:

This is a nonlinear differential equation satisfied by the rotation (1) (2) (1) (2) E = E + E + E + E + Einteraction, [2] angle, which will depend on both the material point s and the bend bend found found development time t. where D. Rule 1: Coplanarity of Ornamentation Planes. For perfect inter- 1 k E (i) = m (s)2, E (i) = (y (s) − (−1)i δ)2. [3] locking to occur, the pattern on each individual valve must locally bend 2 i found 2 i occur in the same plane when the valve is closed. We state this

as the first rule of interlocking: The ornamentation planes of Here δ denotes the half-width of each mantle, mi is the resul- the 2 opposing valves must be aligned at all points when the tant moment acting on the growing mantle, and k describes valves are closed. This geometric rule is illustrated in Fig. 4, in which we superimpose a sinusoidal ornamentation on a bivalve. In Fig. 4A the ornamentation is truly antimarginal; i.e., there is no rotation of the plane of ornamentation. In this case, even though the pattern on the 2 valves was chosen to coincide, i.e., the sinusoidal curves are in phase, significant gaps and over- laps appear so that the valves do not interlock. Fig. 4B shows the same shell, but with a rotation of the plane of ornamen- tation. Here, a perfect interlocking is attained. Intuitively, the reason that the 2 valves can interlock is that the rotation imposed by generative zone strain causes both patterns to develop in the same plane. The argument and calculation in Section 2.C provide a geo- metric condition for the local orientation of the plane of each valve, although it is to be noted that this condition does not take into account the presence of the other valve. However, when both valves are rotated to meet in the x-y plane, rule 1 is sat- isfied. Indeed, the plane of ornamentation for the shell in Fig. 4B was computed according to the calculation described above. In Fig. 4. The first rule of interlocking: At the shell level rotating the orna- fact, we find that this is a generic feature: For a bivalved shell mentation plane is required for interlocking. A nonrotated plane of orna- growing according to the rules outlined above, and with plane mentation (A) leads to a misalignment of the ornamentation patterns and of ornamentation defined by the balance of moments Eq. 1, the thus gaps and overlaps appear when the 2 valves are closed. With rotation planes of ornamentation of each valve almost perfectly coincide (B), opposing planes agree and a perfect interlocking is attained.

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Fig. olo tal. et Moulton h opeeselwt h nrymnmzn ukigpat- buckling energy-minimizing the with shell complete The h raetto atr mre samcaia ntblt u oecs rwho h hl ertn ateadprotau.I h model, the In periostracum. and mantle secreting shell the of growth excess to due instability mechanical a as emerges pattern ornamentation The f sacntn htcaatrzstesrnt fthe of strength the characterizes that constant a is E interaction = f ((y 1 C − ). y 2 h omto fashell a of formation The ) − 2δ ) −2 , [4] Dmrhsaefre yteparameters the by formed morphospace 2D wavelength the hence pattern and ornamentation interlocking buckling the of of mode the rate parameter governs coiling mechanical which the single morphology: a shell and the governing parameters Morphospace. 2D A B. high rate. a secretion with high buckling to combined (smooth) due wavelength a arising small latter of the very form amplitude, a extreme rate): with an coiling but as (high pattern seen commissure in be of appear angle may to mecha- steep These tend very simple which a a 1C), with particular, (Fig. provides shells In commissures secretion zigzag rate. with for coiling growth nism lower of with linking shell the a to compared tude (larger pro- rate is we rate rate coiling Here growth secretion growth. mantle the mantle the to of that portional buck- assumption rate simple the the the of by make amplification governed of is generates rate pattern thus The the ling and edge. buckling produces shell mantle that patterned drives the growth that length edge—the longitudinal of to mantle excess specifically the refer we along growth growth mantle By rate. tion considering activities. as requires these actu- dis- between shell morphologies the interplay without the explain the observed To secreted thickening between seashells. be many by tinction in to e.g., shows of evidence material accretion, that empirical increase to shell clear as becomes contributing for it accretion ally possible then define direction, is and growth we it the secretion if in between but length distinction shell subtle, The is shell. accretion the of accretion † sti rvdstesmls omadi odmdlfrms iavdshells. bivalved here, most for semicircle model a good to a it is restricted and form have simplest we the but provides shape, this cross-sectional as any to applied be h rs-etoa hp saohrdge ffedm n nedorapoc may approach our indeed and freedom, of degree another is shape cross-sectional The nFg eilsrt h ag fselmrhlge sa as morphologies shell of range the illustrate we 6 Fig. In secre- and growth mantle between link the consider first We b ilhv ihronmnainampli- ornamentation higher a have will ) PNAS nti osrcin hr r ny2main 2 only are there construction, this In | aur ,2020 7, January b nti a,aselwt higher with shell a way, this In . k eto 3), section Appendix, (SI † . | o.117 vol. k and | nenergy an C, o 1 no. b low A . | 47 b

EVOLUTION APPLIED MATHEMATICS To study the impact of asymmetry in the model we suppose that one valve, say valve 1, has a higher secretion rate than the other one, say valve 2. The corresponding mismatch in mantle growth means that mantle 1 will have a greater (unstressed) ref- erence length, but is under the same geometric constraints as mantle 2. This mismatch induces a mechanical stress in the man- tle which is relieved by a secondary buckling instability of the entire mantle/periostracum system§. C.1. Adaptive accretion. As a first test of the model, we check that interlocking is maintained within the framework we have developed. In the base case, before any deformation, the coil- ing rates are constant for each valve, and the 2 valve edges meet at the same midplane when the valves are closed. Once a large-scale deformation occurs, the valve edges no longer meet in a single plane (the x-y plane as in the base case). Some material points along the edge will have moved in one direc- tion (to z > 0, say) while other points will have moved in the other direction (z < 0). However, the rotation of each valve about the hinge—increased rotation is needed to accommodate increased material—is a global property. Thus, the geometri- cal constraint of the presence of the opposing valve locally changes along the shell edge. The local accretion rate, i.e., local Fig. 6. Morphology variety for (symmetric) interlocking bivalved shells and coiling rate, must change in response. By analyzing the coil- sample shells illustrating the diversity of form (brachiopod species from ing geometry with such a deformation imposed, we show in SI Upper Left to Lower Right: Cererithyris arkelli; Sphenorhynchia plicatella; Appendix, section 5 that the coiling naturally adapts such that Cererithyris intermedia; Kutchirhynchia obsoleta). The simulated shells cor- the 2 shell edges still perfectly coincide, although no longer in a respond to the 4 different combinations of a low (b = 1) and a high (b = 2) single plane. coiling rate and small (k = 10) and large (k = 500) mechanical stiffness. The The next step is to reintroduce the small-scale pattern by the computational procedure is outlined in SI Appendix, section 4. (Scale bars: same process as before: A generative zone strain is induced 10 mm.) by the difference in arclength at the valleys compared to the peaks of the small-scale pattern, and thus the plane of ornamentation is defined such that the arclength is equal at value of k results in a long wavelength pattern, and vice versa, the peaks and valleys. The corresponding nonlinear ODE is while a low coiling rate produces a shallow shell, with high coil- then solved for the tilt of mantle that defines the local plane ing rate producing a steeper shell and more amplified pattern. of ornamentation (details in SI Appendix, section 5A). The For comparison, we include 4 representative shells matching the net result is that the plane of ornamentation rotates nonuni- basic characteristics of each corner of the morphospace. Since formly at each point along the shell edge compared to the by construction these shells satisfy both rules of interlocking, the base case, but the orientations still coincide locally between interlocking pattern is perfectly formed. the 2 valves. Thus rule 1 is satisfied even in the presence of asymmetry. C. Asymmetry and Secondary Ornamentation. An intriguing feature C.2. Synchrony of ornamentation with asymmetry. The concep- of our findings is that interlocking does not require symmetry tual idea of rule 2 is as before: For interlocking to occur the between the 2 valves (consider that one’s hands clasp together ornamentation patterns must be antisymmetric, a synchrony very nicely, but they also grow as almost perfect mirror images). we expect to be maintained by the mutual interaction of the Indeed, in many shells, notably in brachiopods, the 2 valves have mantles. However, the situation is more complicated by the markedly different coiling rates. In our model, rule 1 is accom- difference in mantle growth rates and requires an extension plished by a rotation of the generative zone that does not rely on of the previous mechanical model for 2 mantles geometri- the physical interaction of the opposing valve, and thus the 2 base cally constrained by each other with the additional assump- valves need not be mirror images of each other for the planes tion that they are growing at unequal rates (SI Appendix, of ornamentation to align. And once the planes align, rule 2 for section 6). antisymmetry of the pattern is accomplished by the mechanical We find that for moderate asymmetry, the interaction of the interaction of the 2 mantles. mantles is sufficient to enforce synchrony of the pattern. How- However, by linking mantle growth to secretion rate, an asym- SI Appendix metry in coiling implies also an asymmetry in mantle growth. ever, as further elucidated in , section 6, for larger Therefore, we can put our modeling framework to the test by asymmetry the mantles eventually separate due to a divergence studying the ornamentation morphology of shells with asymmet- in their reference lengths. A biomechanical coupling would be ric coiling. In particular, we are motivated by a striking feature necessary in such cases. found in some brachiopod shells, as shown in Fig. 7. These shells C.3. Asymmetry patterns. We confirm the prediction of our exhibit a secondary, long wavelength pattern, on top of which model against basic morphological trends observed in shells ‡ with the secondary pattern. In brachiopods the 2 valves cover a small wavelength primary pattern can be found . Both the the dorsal and ventral sides of the animal. Prior to the large- long and short wavelength patterns vary significantly between scale deformation, the dorsal side has the higher coiling rate species and specimens, yet remarkably, perfect interlocking is (when there is asymmetry present). Once the large-scale pat- maintained in all cases. tern appears, the following characteristics are observed: 1) The

‡We term the long wavelength pattern as secondary, as this pattern only ever appears later in development, while the small-scale ornamentation appears early and has the §In this view, the small-scale pattern is primarily focused at the thin periostracum while same characteristics as the ornamentations we have described thus far in this paper. the much thicker mantle remains effectively flat (SI Appendix, section 5 and Fig. 2).

48 | www.pnas.org/cgi/doi/10.1073/pnas.1916520116 Moulton et al. Downloaded by guest on September 23, 2021 Downloaded by guest on September 23, 2021 olo tal. et Moulton in the described on is 6 based that asymmetry section mechanism direction coiling plausible the buckling present a toward the already modes; in deforms hand, even bias always only other a edge impacts the requires shell This on the valve. modes, of dorsal sym- no even point by is With buckling, middle there mode geometry. the odd the as A an of trend, of (27). case metry this basis the with genetic in preference a consistent lateral lacks is animals always origin and almost direc- mechanical plants the asymmetry of asymmetry, of cases bilateral of tion known direction pop- 29 random in the a numbers displaying in same the and roughly (26) in with ulations occur shells always left-“handed” mode and odd right- an i.e., there preference; modes, lateral odd no For based is parameters. triggered mechanical be and will geometric modes on buckling different which for bility, of size correlation the positive and a asymmetry is pattern. dorsoventral large-scale there the of 2) degree and the 7C) as between (Fig. or mode” mode” “even “odd an as an either appears pattern wavelength large preference. lateral no is there (Center, latter, mode odd (Left, an deformation as mode and even an asym- as zero both (C construction appear by amplitude.) simulated; zero not is has points. origin 2-beam metry marked the a the at via at point produced simulated hollow are are (The shells squares complete is orange and regression The model, linear line. mechanical A symbol. dashed diamond the a as with plotted marked are mode odd the ing nicesdapiuerltv oselsz.T uniythis quantify To size. and shell buckling to earlier relative to amplitude lead increased would increase which an an imply stress, would mechanical asymmetry in dorsoventral in increase an a on collected are pattern: data large-scale These and (blue), the rates (B) displaying coiling pattern. shells in of large set difference the via of measure amplitude asymmetry relative an both extract we 7. Fig. B A C bevto scerycmail ihamcaia insta- mechanical a with compatible clearly is 1 Observation bevto sas ossetwt ehnclpoes as process, mechanical a with consistent also is 2 Observation Amplitude Cyclothyris o ahshell each For (A) brachiopods. in pattern large-scale and Asymmetry Burmirhynchia thierachensis . Cyclothyris vnmd oddmode even mode rd,and (red), oqihnharoyeriana Torquirhynchia Cyclothyris ag aeeghpten nbrachiopods in patterns wavelength Large ) yltyi Torquirhynchia royeriana Cyclothyris Asymmetry and sp.; umryci thierachensis Burmirhynchia Right, etlpoi orbignyana) Septaliphoria bak.Sel display- Shells (black). .I the In royeriana). T. IAppendix, SI simulated hrfr o upiigta h aepten finterlocking of patterns same is the It that mantle. surprising secreting not the biaxially therefore of a of instability consequence mechanical a constrained as simply to shells managed interlocking have bivalves secrete secretion. and shell brachiopods during that other is each conclusion Our on influence have mechanical mantle lobes reciprocal mantle the the under- both of and that edge interactions be shell mechanical may rigid the the invertebrates both with of of light groups in these stood of evolution natural and (i.e., bias these reproductive of a development induce selection). the might of that understanding characters an lin- the related advan- requires of distantly explanation in eages functional characters an similar clear, possible of emergence are repeated the structures interlocking of of tages some While during them is generate development. selection that natural unquestioned mechanisms by the from remained traits distinct of logically since promotion the has organ- However, 30). these that probability (29, in interpretation intrinsic independently the an times and isms, many function trait evolved this cer- this zigzags tiny that of that a filtering concluded presence by above and the feed particles seawater explains that from harmful bivalves particles and of food brachiopods entry in by the size evolved tain filtering prevent has as to evolved and pro- have grids (28) being commissures Rudwick zigzag-shaped into instance, that For come may posed alone. function has one its view trait to appealing this a to of how perspective According explain functional synthesis. the neo-Darwinian within the interpreted morpholo- been 20th shell have the mollusk and gies during brachiopod of Indeed, aspects considerations. most century functional complemen- to much-needed view a tary provides without origins forces developmental explanation time mechanical of biophysical This same by processes. genetic the maintained specific requiring at is mantle aperture—while the interlocking the of shell of change—growth developmental expansion simple outpacing a arising bivalved to instability form- interlocking mechanical due in a in as mechanics sculpture appears of Ornamentation shell role shells. of key features the shown common have ing we paper this In Discussion 4. perfect a exhibit simulated cases both the all in which produce geometry; interlocking. 7B, Fig. to shell in combined appearing base shells small-scale were with the components with combination computed of model in ornamentation output coiling of as plane adapted taken and 2 model pattern the mechanical small 7B at Fig. with shape in buckled model, points computed simulated the illustrate, marked To model. morphological the observed the in with predicted squares consistent trends orange and are trends. patterns model the the the as that by appear demonstrating These 7B, Fig. asymmetry. mea- different of growth several and for sures buckling mantle length to following in relative curves amplitude difference extracting bifurcation in the the difference computing the to rates, taking correspond 6) by to section measures Appendix, every asymmetry equivalent (SI the of model extract image mechanical we an the in From as available 7. are well overlaid, section a as Spear- curves with and data, a sampled as extracted 0.67 shell compute view, The of in We front 0.0001. coefficient asymmetry correlation: than a correlation against strong to rank plotted a sinusoid amplitude man’s is showing a Amplitude and 7B, 7A. fitting profile Fig. Fig. by side in pattern a shown large to asymmetry the spirals dorsoventral of extract logarithmic differ- we from fitting shell, brachiopods by each 59 of For sample species. a ent studied have we trend, u td hw htapr ftemrhlgcldiversity morphological the of part a that shows study Our by captured well are features morphological the Moreover, PNAS | aur ,2020 7, January a e notefl shell full the into fed was | o.117 vol. IAppendix, SI P | au less value o 1 no. | 49

EVOLUTION APPLIED MATHEMATICS structures have evolved repeatedly among brachiopods and The fact that physical processes are key in shell morphogen- bivalves, an evolutionary trend which is a predictable outcome esis does not imply that genetic and molecular processes are of the physics of the growth process. It is also worth noting that irrelevant. For example, both the amplitude and wavelength we have restricted our study to self-similar shell growth (prior to of ornamentation may vary considerably among oyster species, emergence of any large-scale pattern) and with small-scale pat- possibly because of species-specific combinatorial variations in terns appearing at right angles to the shell margin. While it is control parameters such as commarginal growth rate or stiffness a suitable assumption for most bivalves and brachiopods, there of the mantle. Given that these parameters may be genetically are species that deviate from self-similarity or with ribs appear- modulated, our approach might open the door to future stud- ing oblique to the shell margin. In such shells interlocking is ies aiming at understanding how biochemical and biophysical consistently maintained, suggesting that the process we propose processes across scales could conspire to regulate the develop- is robust with respect to these perturbations as well. Accordingly, ment and variations of morphologies among different species. we hypothesize that mechanical forces also play the same role in The interplay between predictable patterns and unpredictabil- these systems. However, to model these forces explicitly would ity of specific outcomes in large part defines biological evolution require introducing an additional torsional component in the (34). Cells, tissues, and organs satisfy the same laws of physics as generative zone¶ and/or deviating from the self-similar growth nonliving matter, and in focusing on the noncontingent and pre- that we have utilized in our geometric construction. While such dictable rules that physical processes introduce in development steps are certainly feasible, and conceptually all of the same and in the trajectories that are open to morphological evolution, ideas outlined in our paper would still apply, modeling such cases we shift the focus from the Darwinian perspective of “the sur- would introduce additional computational complexity and is left vival of the fittest” to a more predictive one of “the making of as future work. the likeliest.” There are other striking examples in nature of organisms While buckling and wrinkling instabilities have long been with matching of body parts, such as the closed mouth of the viewed as only detrimental in engineering, an increasing number snapdragon flower (29, 31), the interacting gears of the plan- of studies, often inspired by biology, have shown the potential thopper insect Issus (30), or dental occlusion in vertebrates (32). contribution of this physical phenomenon to smart applica- The role of mechanics in the morphogenesis of such structures tions (35). Interlocking structures are ubiquitous in man-made could be the subject of fruitful future inquiries. Among mollusks, structures where they serve as physical connections between the hinge in bivalves is also formed by a series of interlock- constitutive parts in such diverse areas as building or biomed- ing teeth and sockets on the dorsal, inner surface of the shell. ical engineering, and their presence in nature is a source of In this case too, the hinge teeth are secreted by 2 lobes of the inspiration for biomimetic engineering (36). Our study shows mantle which are retracted from the hinge line when the shell that brachiopods and bivalves have made good use of mechan- is tightly closed and when teeth and sockets interlock in each ical instabilities to secrete their interlocking shell since about other. The morphologies of these hinge teeth (e.g., taxodont, 540 million years ago; in this light perhaps the growth of heterodont, schizodont ...) have traditionally provided the basis these invertebrates could be inspirational in biomimetic research of bivalve classifications, but recent molecular phylogenies (33) for the development of self-made interlocking structures at show that these characters do not always bear a coherent phy- many scales. logenetic signal, which could be explained by the fact that ahistorical physical processes play an important role in their Data Availability. All materials, methods, and data needed to development. evaluate the conclusions are present in the main text and/or SI Appendix.

ACKNOWLEDGMENTS. We thank E. Robert (Universite´ Lyon 1, France) and ¶In terms of the plane of ornamentation, our model considers a rotation about the J. Thomas (Universite´ de Bourgogne, Dijon, France) for providing us an tangent d3 direction; an oblique pattern could be produced by also rotating about the access to paleontological collections. This work was made possible by a d2 direction, which would create a “slant” to the antimarginal ornamentation. visiting position from LGL-TPE, Universite´ Lyon 1.

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