Turing Patterns with Pentagonal Symmetry

PHYSICAL REVIEW E, VOLUME 65, 051913

Turing patterns with pentagonal symmetry

J. L. Arago´n Instituto de Fı´sica, Universidad Nacional Auto´nomadeMe´xico, Apartado Postal 1-1010, Quere´taro 76000, Mexico

M. Torres Instituto de Fı´sica Aplicada, Consejo Superior de Investigaciones Cientı´ﬁcas, Serrano 144, 28006 Madrid, Spain

D. Gil Departamento de Paleontologı´a, Facultad de Ciencias Geolo´gicas, Universidad Complutense, Ciudad Universitaria, 28040 Madrid, Spain

R. A. Barrio Instituto de Fı´sica, Universidad Nacional Auto´nomadeMe´xico, Apartado Postal 20-364, Me´xico 01000, Distrito Federal, Mexico

P. K. Maini Centre for Mathematical Biology, Mathematical Institute, Oxford University, 24-29 St. Giles, Oxford OX1 3LB, United Kingdom ͑Received 3 November 2001; revised manuscript received 28 January 2002; published 16 May 2002͒ We explore numerically the formation of Turing patterns in a confined circular domain with small aspect ratio. Our results show that stable fivefold patterns are formed over a well defined range of disk sizes, offering a possible mechanism for inducing the fivefold symmetry observed in early development of regular echinoids. Using this pattern as a seed, more complex biological structures can be mimicked, such as the pigmentation pattern of sea urchins and the plate arrangements of the calyxes of primitive camerate crinoids.

DOI: 10.1103/PhysRevE.65.051913 PACS number͑s͒: 87.10.ϩe, 71.23.An, 73.20.At

I. INTRODUCTION pentagonal seed is formed and used as a source of morphogen in a larger disk, as one might assume that in a biological The occurrence of pentagonally symmetric organisms has system there is the possibility that a certain pattern ͑stable or been largely surveyed and studied in detail ͓1,2͔. For ex- not͒ can be frozen at a stage of development ͑calcified or ample, a pentagonal pattern is the basic pattern of the echi- biologically differentiated͒ and may serve as a seed for fu- noderm skeletons. Some radiolarians and diatoms also pro- ture development. These patterns compare notably well with vide notable examples of pentagonal patterns. However, the pigmentation pattern of Toxopneustes pileolus and, in a most mathematical models for biological pattern formation different context, still in the phylum of echinoderms, with exhibit hexagonal patterns ͓3–6͔ and the issue of selecting the plate arrangements of the calyxes of primitive camerate ͓ ͔ and stabilizing fivefold symmetric patterns has not been ad- crinoids 2 . All these results lead us to consider morphogen- dressed in these models. esis as a step by step process in the sense that once a pattern Motivated by the de novo appearance of pentagonal sym- is generated and biologically consolidated, it can serve as a seed for the next step. It also reinforces the importance of metry in the rudiment disk during the early development of ͓ ͔ regular echinoid larval forms ͑echinopluteous͓͒7͔, we care- transient or unstable patterns in biological systems 8 . fully explore the formation of radially symmetric patterns using a Turing system solved on a confined circular domain II. THE MODEL with size comparable with the characteristic wavelength, starting from random initial conditions. Since under this as- In all biological models there remain uncertainties about pect ratio the boundary controls the symmetry of the pattern, the mechanisms behind pattern formation: the study of ge- the radius of the disk ͑or the curvature of the circle͒ can be netics alone cannot provide us with such mechanisms. Per- considered as the relevant parameter. Taking into account the haps the most extensively studied mechanism for self- wavelength of the Turing equations, we may expect a pen- organized biological pattern formation is the Turing tagonal pattern if system parameters and disk radius are ad- instability ͓9–11͔ and we focus our attention on this mecha- equately tuned, such that a central plus several marginal nism. Albeit the molecular details are still unknown, the Tur- peaks are possible. We address this problem by studying nu- ing model has been shown to exhibit pigment patterns con- merically the Turing modes confined in a small size disk. As sistent with those observed in some mammals ͓12͔, seashells we shall show, the frustration induced by small disk radius ͓13͔, and marine fishes ͓14–18͔. Curved geometries have reduces the usual hexagonal symmetry of the pattern, to pro- also been introduced to model microscopic organisms such duce pentagonally symmetric pattern. We also show that they as radiolarians ͓19͔ and patterns on the hard wings of lady are stable and, under large spatial homogeneities in the initial beetles ͓20͔. Recently, it has been found that morphogens conditions, they appear over a well defined range of disk ͑the name given by Turing to the chemicals in prepattern sizes. We also simulated numerically the situation when a models͒ do exist ͓21͔, but experimental evidence that mor-

phogen patterns are set up by a Turing mechanism is yet to be found. Turing equations describe the temporal development of the concentrations of two chemicals, U and V, that diffuse at different rates, DU and DV , and react according to the nonlinear functions f and g,

Uץ ,ϭD ٌ2Uϩ f ͑U,V͒ t Uץ

Vץ .ϭD ٌ2Vϩg͑U,V͒ t Vץ

We take the model introduced by Barrio et al. ͓17͔, obtained by observing that, in general, there is a stationary uniform solution (Uc ,Vc), given by the zeros of f and g. FIG. 1. Variation of dispersion relation, from Eq. ͑2͒, for differ- Functions are then expanded around this point in a Taylor ent parameter values. The continuous line corresponds to ␣ series, neglecting terms of order higher than cubic. The spe- ϭ0.899, ␤ϭϪ0.91, Dϭ0.516, and ␦ϭ0.01 011, to enhance the ϭ ciﬁc system we consider is mode k51 6.41 562. The dotted line corresponds to the same parameters as in the previous case but with ␦ϭ0.00 845 to enhance u ϭץ ϭ ␦ٌ2 ϩ␣ ͑ Ϫ 2͒ϩ ͑ Ϫ ͒ ͑ ͒ the mode k03 7.01 558. Finally, the dashed line corresponds to the D u u 1 r1v v 1 r2u , 1a ␣ϭ ,t same mode k03 , but obtained with parameter values 0.2334ץ ␤ϭϪ0.95, Dϭ0.12, and ␦ϭ0.01 011. v ␣rץ ͒ ͑ ͒ ϭ␦ٌ2 ϩ␤ ͩ ϩ 1 ͪ ϩ ͑␥ϩ v v 1 uv u r2v , 1b ϩ Ͻ ⇒␣ϩ␤Ͻ ,t ␤ f U gV 0 0ץ

where uϭUϪU , vϭVϪV , D ϭD, and D ϭ1. The c c U V Ϫ Ͼ ⇒␣͑␤ϩ ͒Ͼ quantity ␦ conveniently gives the size of the system, and the f UgV f VgU 0 1 0, particular arrangement of the coefﬁcients obeys conservation rules in these chemicals. There are two interaction param- ϩ Ͼ ⇒␦ ␤ϩ␣͒Ͼ DUgV DV f U 0 ͑D 0, eters r1 and r2 that, in Cartesian coordinates, control the formation of stripe or spot patterns, respectively ͓17͔. ͑ ϩ ͒2Ϫ ͑ Ϫ ͒Ͼ We investigate patterns in a two-dimensional disk with DUgV DV f U 4DUDV f UgV f VgU 0 zero-ﬂux boundary conditions, namely, ⇒͑D␤ϩ␣͒2Ϫ4D␣͑␤ϩ1͒Ͼ0, ͑3͒ ͑ ␪͒ϭ ͑ ␪͒ϭ n•“u r, n•“v r, 0, where subscripts in f and g denote differentiation. ␪෈ ␲ for all ͓0,2 ), with r at the boundary of the disk with At the onset of instability, a good approximation of the unit outward normal n. We then, in the linear regime, look critical wave vector is given by the minimum of the left part for solutions of the form ͑in the usual way͒ of the last inequality in Eq. ͑3͒, namely,

ϭ ͑␭ ͒ ͑ ͒ ͑ ␪͒ u u0 exp t ͚ CmJm kr exp im , D f ϩD g 1 ␣ϩD␤ m 2ϭ V U U V ϭ ͩ ͪ ͑ ͒ kc ␦ , 4 2DUDV 2D ϭ exp͑␭t͚͒ D J ͑kr͒exp͑im␪͒, v v0 m m ϭ ϭ␬ ␬ m where k kmn mna; mn is the nth zero of the derivative Ј of the Bessel function Jm and a is the radius of the disk. where Cm and Dm are constants and Jm are Bessel functions Observe that this equation implies that once (a,␣,␤,D) are of mth order of the first kind. By substituting these solutions fixed, one could select a pattern of a given radial symmetry in the linearized version of Eq. ͑1͒ we obtain that the disper- by varying ␦. Alternatively, solving Eq. ͑4͒ for ␦, one could sion relation is given by the solutions of ϭ␬ tune the other parameters to satisfy the condition k mn /a for a given symmetry m. ␭2ϩ͓͑1ϩD͒␦␬2Ϫ␤Ϫ␣͔␭ϩ͓͑D␦␬2ϪD␤Ϫ␣͒␦␬2 In Fig. 1 we show the dispersion relation, from Eq. ͑2͒, ϩ␣͑␤ϩ1͔͒ϭ0. ͑2͒ for a selection of different parameter values in the appropri- ate parameter domain defined by Eq. ͑3͒. The parameters ϭ␬ ϭ ͑ In order to keep the solutions as simple as possible, we en- were chosen to enhance mode k51 51 6.41 562 in what forced (0,0) to be the only spatially uniform steady state by follows, we shall consider a unitary disk, aϭ1), according to ␣ϭϪ␥ ͓ ͔ ͑ ͒ ϭ␬ ϭ setting 17 . The following conditions must be sat- Eq. 4 , and mode k03 03 7.01 558 was enhanced using isfied for diffusion-driven instability ͓10,17͔: two different sets of parameter values.

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FIG. 3. Value of the pattern symmetry axis versus ␦Ϫ1/2, which is proportional to the radius of the disk, for simulations with param- ␣ϭ ␤ϭϪ ϭ ϭ ϭ eter values 0.899, 0.91, D 0.516, a 1, and r1 r2 ϭ0.2. The dashed lines represent noncentrosymmetric sixfold patterns. Onefold symmetry axis corresponds to mixed symmetries. Below ␦Ϫ1/2ϭ9.66, noncentrosymmetric patterns with four or less spots are obtained and for ␦Ϫ1/2Ͼ18.25 hexagonal patterns with defects are generated. Starting from ␦ϭ0.001 (␦Ϫ1/2ϭ31.623) patterns were obtained by increasing ␦ by 0.0001 and, in each case, we used 36ϫ106 time iterations to converge, with a time step of ⌬t ϭ1ϫ10Ϫ4. FIG. 2. Series of symmetries obtained in a disk with zero-flux geometry to solve the Turing equations allows us to generate boundary conditions. The values of ␦ are ͑a͒ 0.01 011, ͑b͒ 0.0075, a fivefold pattern. ͑c͒ 0.0055, ͑d͒ 0.0065, and ͑e͒ 0.0037. These correspond to select- The appearance of the central spot in some of the patterns ing modes k , k , k , k , and k , respectively. 51 61 71 32 52 in Fig. 2 may seem strange at first sight. In particular, ͑ ͒ J5(k51r) is zero at the origin and Fig. 2 a shows nonzero III. NUMERICALLY COMPUTED STEADY-STATE values there. We should, however, note that the linear analy- PATTERNS sis predicts modes under the linear regime, while the final solution depends also on the nonlinear terms and it is ex- Equations ͑1͒ are solved in polar coordinates in a unitary pected that, besides monomodes, nonlinear coupling of dif- disk (aϭ1) by a simple Euler method as described in detail ferent modes may appear. In Appendix B we perform a in Appendix A. The calculations reported here were carried modal decomposition of a pentagonal pattern to confirm this out using grid parameters Mϭ34, Nϭ68, ⌬rϭ1/34, and possibility. ⌬␪ϭ2␲/68. The parameter values ͑kept fixed in all our Since in Appendix A we obtain only necessary conditions simulations͒ are ␣ϭ0.899, ␤ϭϪ0.91, and Dϭ0.516. We for numerical stability, we performed a number of simula- used 16ϫ106 time iterations to converge, with a time step of tions with different step sizes ͑in time and space͒ in order to ⌬tϭ1ϫ10Ϫ4, which fulfills the necessary condition for nu- verify that the solution does not change under these condi- merical stability ͑A9͒. The system was always initialized us- tions. In Figs. 4͑a͒ and 4͑b͒, we show the pentagonal pattern Ϯ obtained with the same parameters as in Fig. 2͑a͒ but with ing random fluctuations ( 0.5) around the steady state Ϫ5 Ϫ5 0 ϭ 0 ϭ smaller time steps, namely, 5ϫ10 and 1ϫ10 , respec- Ui, j 0 and Vi, j 0, for all i, j. The seed of the random ͑ ͒ generator was set to the CPU time of the computer. tively. In Fig. 4 c we changed the spatial step size to M ϭNϭ68, with ⌬tϭ2ϫ10Ϫ5. Finally, in Fig. 4͑d͒, a plot of In Fig. 2, examples of patterns obtained with parameter the convergence factor ␴ for the simulations in Figs. 4͑a͒ and values r ϭr ϭ0.2, to produce a spot pattern ͓17͔, are 1 2 4͑b͒ is shown. shown. The values of ␦ correspond to increasing the size of ͑ ͒ In the range of disk sizes where we obtain pentagonally the domain and, according to Eq. 4 , different symmetries symmetric solutions, other admissible modes are possible are selected. The convergence to steady state of the pattern and we found that under several different runs, with different ␴ϭ͚ n was estimated by means of the quantity i, j(Ui, j initial conditions, it was possible to obtain these modes ͑but Ϫ nϪ1 2 Ui, j ) , at each time step. A typical value of this quantity only if we reduce the amplitude of the random fluctuations in Ϫ in all our calculations is 1ϫ10 6. In Fig. 3 a graph of the the initial conditions͒. This problem of multiple stable pattern symmetry axis value versus ␦ is shown. It can be steady-state solutions has been one of the main criticisms of seen that hexagonal and pentagonal patterns are more fre- the Turing model as a plausible explanation for robust pat- quent and the pentagonal pattern is generated in a range of ␦ tern formation. It has been shown that using different types about 16% of the explored radius. The use of the circular of boundary conditions ͓22͔ or employing domain growth

051913-3 ARAGO´ N, TORRES, GIL, BARRIO, AND MAINI PHYSICAL REVIEW E 65 051913

shown ͓26͔. Much of the rest of larval growth consists of the formation and lengthening of podial buds maintaining the pentagonal symmetry of the initial lobes. These five podial buds play a major role, by migration from pole to pole, during the embryological skeletal development of regular echinoids in the framework of the axial-extraxial theory ͓27͔. Thus, the symmetry of the adult echinoid is set in this initial stage of development and, for this purpose, the first relevant event is the formation of a pentagonal pattern in the rudiment that dictates the radial symmetry of future development. Regarding ratios and relative sizes, the realistic diagrams in Figs. 5͑a͒ and 5͑b͒ compare notably well with Fig. 2͑a͒ with the exception of the central spot in the latter. In Fig. 5͑c͒ we show a simulation with the same parameters as in Fig. 2͑a͒ but the factors that control the formation of stripes ϭ ϭ or spots were changed to r1 5 and r2 3. Generally, spots are more robust and pronounced than stripes ͓17͔; actually, stripes are only formed for very small values of r2. In the simulation of Fig. 5͑c͒, the factor favoring stripes is high, they do not form because r2 is not small. The net result is a FIG. 4. Simulations of the pentagonal pattern with the same sort of modulation of amplitudes such that the central spot of ͑ ͒ parameters as in Fig. 2 a but with different spatial and temporal the pentagon, albeit present, has a very low amplitude. ͑ ͒⌬ϭ ϫ Ϫ5 ͑ ͒ ϫ Ϫ5 ͑ ͒ ϭ ϭ ⌬ step sizes. a t 5 10 , b 1 10 , c M N 68, and t Notice that taking into account how the symmetry of the ϭ ϫ Ϫ5 ͑ ͒ ␴ 2 10 , and d plot of the convergence factor for the simu- patterns are selected, from a genetic point of view, the pen- lations in ͑a͒, with continuous line, and ͑b͒ with dashed line. tagonal symmetry of the adult echinoid can be efficiently ͓ ͔ ͑ ͒ encoded using two basic parameters as highly condensed in- 23 it is possible at least on a one-dimensional domain to structions: the radius of the rudiment disk ͑or the cell num- increase greatly the robustness of certain modes. ber͒ and the characteristic wavelength of the morphogens. Let us now consider the case when a given pattern is used IV. PATTERNS OF SEA URCHINS as source of morphogens in a larger disk. That is, we simulate the influence of an initially frozen pattern in a domain Selecting the symmetry of the Turing pattern by means of where a pattern forming reaction takes place. This has been the disk radius may be relevant for capturing the essence of used to simulate the imposition of some directional prefer- the de novo appearance of the five primary podia during the ence onto the pattern formed, by means of a pattern of dif- early development of regular echinoid larval forms. At the ferentiated cells ͓28–30͔. We first consider the pentagonal first stages of larval growth, these animals develop five buds structure shown in Fig. 2͑a͒ implanted as a source of mor- of primary podia in a disk called imaginal rudiment, rudi- phogens in a disk of larger size. The procedure is as follows. ͓ ͔ ment disk, or just rudiment 7,24,25 . These five buds point The spots of the pentagonal structure shown in Fig. 2͑a͒, to the vertices of a slightly distorted pentagon and give rise ͑ ͱ␦ generated in a disk of radius r proportional to 1/ r, with to the radially symmetric adult. In Figs. 5͑a͒ and 5͑b͒, sche- ␦ ϭ Ͼ r 0.01 011), is implanted in a larger disk of radius R r matic diagrams of the realistic arrangement of the five pri- ͑ ͱ␦ ␦ ϭ proportional to 1/ R, with R 0.0006), and its amplitude mary podia in the rudiment of Eucidaris thouarsi and scaled such that the maximum equals 1. The system is then Strongylocentrotus droebachiensis sea urchin larvae are initialized with the previous pentagonal pattern as a seed at 0 ϭ 0 ϭ the center and Ui, j Vi, j 0 in the area not occupied by the spots of the pentagonal seed. These spots are considered a source of morphogen in such a way that their size is main- tained fixed and the value of the morphogen is set to 1 during all time steps. In Fig. 6͑a͒ we show the pattern that results after 800 000 iterations ͑before reaching steady state͒. The pattern compares notably well with the pigmentation pattern of the common Indo-Pacific sea urchin species Tox- opneustes pileolus ͑Alphonso urchin͒, shown in Fig. 6͑b͒, FIG. 5. Schematic diagrams showing the realistic arrangement and it may be considered as a frozen pattern when the animal of the five primary podia in the rudiment of ͑a͒ Eucidaris thouarsi reaches adulthood. The similarities between the simulated larvae ͑taken from Fig. 2 of Ref. ͓7͔͒ and ͑b͒ Strongylocentrotus pattern and the real biological pattern are supported by the droebachiensis sea urchin larvae ͓taken from Fig. 2͑f͒ of Ref. ͓25͔͔. coherence between the pigmentation pattern and the morpho- ͑c͒ Numerical calculation of the pentagonal pattern with the same logical structure of the urchin test ͓as seen in Fig. 6͑b͔͒, parameters as in Fig. 2͑a͒ but with the factors that control the for- suggesting that both pigmentation and skeleton formation ϭ ϭ mation of stripes or spots changed to r1 5 and r2 3. were simultaneous during the growth process.

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FIG. 6. ͑a͒ Transient pattern ͑after 800 000͒ iterations obtained with ␦ϭ0.0006 using a pentagonal seed as described in the text with the remaining parameter values as in Fig. 2. ͑b͒ Toxopneustes pileolus sea urchin ͑taken from echinoid website of The Natural History Museum of London, designed by A. Smith͒.

V. PATTERNS OF CRINOIDS In a second simulation we let the pattern of Fig. 6͑a͒ reach stability (6ϫ106 time iterations͒. The final pattern is shown in Fig. 8͑a͒ and it consists of a nearly twinned pentagonal ␦ structure. The disk size used in the simulation ( R ϭ0.0006) is large enough to obtain a hexagonal pattern, but in this case it has to couple with the pentagonal seed and the best way to do this is to generate a twinned structure formed by a central pentagon surrounded by distorted hexagons. This result can be related to another important and primitive taxonomic group of echinoderms: the crinoids ͓31͔.In Fig. 7 we show the typical plate diagrams of 11 calyxes of camerate crinoid fossils that we have studied ͓2,32͔. An in- teresting fact is that all these structures are variations of a fundamental arrangement that is a twinned pentagonal structure. This can be made evident as follows. By using the ͑ ͒ ͓ ͔ FIG. 7. Typical plate diagrams of camerate crinoids: a Ar- numerical approach described elsewhere 2 , we digitalize chaeocrinus microbasalis, ͑b͒ Condylocrinus verrucosus, ͑c͒ Deo- each plate diagram codifying the result using a binary sys- crinus asperatus, ͑d͒ Diamenocrinus jouani, ͑e͒ Hercocrinus el- ͑ ͒ ͑ ͒ tem: black 1 and the white background 0 . Next, we ex- egans, ͑f͒ Kyreocrinus constellatus, ͑g͒ Opsiocrinus mariana, ͑h͒ pand this binary function as a truncated Fourier-Bessel series Ortsaecrinus cocae, ͑i͒ Rhodocrinites kirbyi, ͑j͒ Sphaerotocrinus with real coefficients ͓2͔ so that a plate arrangement can be ornatus, and ͑k͒ Thylacocrinus vannioti. described by the plate model function ϭ ϭ ϭϪ ϭϪ ϭ 4 C0 0.25, C5 0.09, C10 0.42, C15 0.04, C20 0.45, ϭ ϭϪ ϭϪ ϭ ␪͒ϭ ͒ ␪͒ϩ ␪͒ S5 0.36, S10 0.1, S15 0.28, and S20 0.26, which T͑r, ͚ J5n͑kr ͓C5n cos͑5n S5n sin͑5n ͔, nϭ0 compare well with the averaged coefficients of the fossil spe- ͑5͒ cies. This good match is better appreciated if we reproduce the image using these coefficients and Eq. ͑5͒. The resulting where r and ␪ are polar coordinates and k is the wave number ͓2͔. The values of C and S were calculated for the 11 fossil species shown in Fig. 7 and it turns out that all structures are quite similar ͓2͔. An average structure can be obtained by taking the mean of the calculated constants C ϭ and S of the 11 species studied, giving C0 0.25, C5 ϭ ϭϪ ϭ ϭ ϭ 0.05, C10 0.45, C15 0.02, C20 0.48, S5 0.47, S10 ϭϪ ϭϪ ϭ 0.02, S15 0.46, and S20 0.05. Notably, this undulatory average structure is an almost perfect fivefold twinned structure, as shown in Fig. 8͑c͒, which compares well with the twinned Turing structure that results from our simulation, FIG. 8. ͑a͒ Final steady-state pattern of the simulation shown in and is shown in Fig. 8͑a͒. Fig. 6͑a͒ after 6ϫ106 time iterations. ͑b͒ Pattern obtained using Eq. In order to compare both patterns in a more proper way, ͑5͒ and the coefficients of the Fourier-Bessel analysis of ͑a͒. ͑c͒ the simulated structure of Fig. 8͑a͒ was transformed to a Average twinned structure of the arrangement of plates of camerate black and white image and the Fourier-Bessel analysis gives crinoids.

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VI. CONCLUSIONS We have explored numerically the formation of Turing patterns in a circular domain with size comparable with the characteristic wavelength. Our results show that, under large spatial homogeneities in the initial random conditions, pentagonal patterns appear and are stable over a well defined range of disk sizes ͑about 16% of the explored radius͒. Some real biological examples of pentagonal patterns in regular echinoids and criuoids are presented. Our results may offer a possible mechanism for inducing the fivefold symmetry observed in early development of these animals. By assuming that in a biological system a certain pattern can be calcified or biologically differentiated at a stage of development, we also simulate pigmentation patterns in sea urchin shells by using sources of morphogens implanted in a disk. Finally, the importance of a transient or an unstable pattern in biological systems is reinforced. FIG. 9. Schematic plate arrangement of Pycnocrinus dyeri fossil crinoid. Plates are indicated by thick white lines. Circles indicate ACKNOWLEDGMENTS the center of the plates and form a lattice of ridges decorating the We thank J. P. Adrados for help with figure preparation. plates. This lattice ͑drawn with thin white lines͒ is a twinned pen- Part of this work was financially supported by the Royal tagonal structure formed by distorted hexagons with a pentagon at Society Leverhulme Trust ͑P.K.M.͒. One of us ͑J.L.A.͒ ac- the center. The inset shows a schematic diagram of the calyx or cup of Pycnocrinus dyeri adult form from where the plate diagram was knowledges financial support from DGAPA-UNAM through obtained. Grant No. IN-108199.

APPENDIX A: FINITE-DIFFERENCE SCHEME pattern is shown in Fig. 8͑b͒. Both undulatory patterns ͑av- eraged and mimicked͒ are pentagonal twinned structures By using subscripts to denote differentiation, in polar co- with the property that the sides of the central pentagon are ordinates the Laplacian reads also sides of antinode hexagons surrounding this pentagon. Our Turing model thus captures the essence of these 1 1 ץ ϩ ץ ϩ ץ2ϭٌ rr ␪␪ r . structures that evolution has explored in crinoids, which are r2 r fluctuations of a basic pentagonal twinned structure. This average structure is a simple and predictable geometrical pat- The system ͑1͒ is then tern that evolution may have occupied in the morphospace of early crinoid skeletons. Hence, this is another example that 1 1 2 u ϭD␦ͩ u ϩ u␪␪ϩ u ͪ ϩ␣u͑1Ϫr v ͒ϩv͑1Ϫr u͒, corroborates the claim of the more general work by Thomas t rr 2 r 1 2 ͓ ͔ r r et al. 33 , where it is established that viable design elements ͑ ͒ available for animals, to use as skeletons, have been fully A1a exploited in early ages. Evidently, evolution also tried the 1 1 ␣r twinned structure itself, as can be seen in Fig. 9, where the ϭ␦ ϩ ϩ ϩ␤ ͩ ϩ 1 ͪ ϩ ͑␥ϩ ͒ ͩ ␪␪ ͪ vt vrr v vr v 1 uv u r2v , schematic plate arrangement of Pycnocrinus dyeri camerate r2 r ␤ crinoid fossil ͓31,32͔ is shown. In the figure, plates are indi- ͑A1b͒ cated by thick white lines. The centers of the plates are in- ϭ ϭ dicated by circles and form a lattice of ridges decorating the with zero-flux boundary conditions ur vr 0 at the bound- plates. Observe that this lattice ͑drawn with thin white lines͒ ary of the disk (rϭa). is a twinned pentagonal structure formed by distorted hexa- We use the simple Euler ͑forward difference͒ approxima- gons with a pentagon at the center, quite similar to the ones tion to approximate ut , ur , vt , vr , and central differences to shown in Fig. 8. Notice also that plates constitute the Dirich- approximate urr , u␪␪ , vrr , and v␪␪ . Using equally spaced ͓ ͔ ␪ ϭ ⌬ ϩ ⌬ let domains of the lattice of ridges 34 . This plate arrange- points along r, , and t we thus denote ri ( r)/2 i r, ␪ ϭ ⌬␪ ϭ ⌬ ϭ ment is also observed in Glyptocrinus decadactylus camerate j j , and tn n t, where i 0,1,...,M, j crinoid fossil ͓31,32͔. ϭ0,1,...,N, ⌬rϭ1/M, and ⌬␪ϭ2␲/N. We start from r The plate arrangement shown in Fig. 9 was directly ob- ϭ(⌬r)/2 in order to avoid the singularity of the Laplacian at tained from the calyx or cup of the published images of fossil the origin. The center of the disk is obtained by using bound- crinoids ͓31,32͔. A schematic diagram of the calyx is shown ary conditions such that central points are joined in pairs in the inset of Fig. 9; the stem of the fossil is represented by separated by ␪ϭ␲/2, that is, the three white lines at the bottom of the figure and white ␪ ͒ϭ ␪ ͒ filled circles indicate the starting point of the arms. U͑r0Ϫ1 , j ,tn U͑r0 , jϩN/2 ,tn ,

051913-6 TURING PATTERNS WITH PENTAGONAL SYMMETRY PHYSICAL REVIEW E 65 051913 for jϭ0,1,...,(N/2)Ϫ1, and ␦⌬ nϩ1 n D t n n n U ϭU ϩ ͑U ϩ Ϫ2U ϩU Ϫ ͒ i, j i, j 2 (i 1,j) (i, j) (i 1,j) ␪ ͒ϭ ␪ ͒ ͑⌬r͒ U͑r0Ϫ1 , j ,tn U͑r0 , jϪN/2 ,tn , ␦⌬ D t n n n for jϭN/2,...,N. ϩ ͑U ϩ Ϫ2U ϩU Ϫ ͒ ⌬␪͒2 (i, j 1) (i, j) (i, j 1) n n ␪ ␪ ri͑ If U(i, j) and V(i, j) denote U(ri , j ,tn) and V(ri , j ,tn), respectively, we then solve D␦⌬t ϩ ͑ n Ϫ n ͒ϩ␣⌬ n ͑ ͒ ⌬ U(iϩ1,j) U(i, j) tU(i, j) . A4 ri r D␦⌬t nϩ1ϭ n ϩ ͑ n Ϫ n ϩ n ͒ Ui, j Ui, j U(iϩ1,j) 2U(i, j) U(iϪ1,j) ͑⌬r͒2 The truncation errors arise from the Taylor series expan- sions used to approximate derivatives D␦⌬t ϩ ͑ n Ϫ n ϩ n ͒ U(i, jϩ1) 2U(i, j) U(i, jϪ1) ϩ r ͑⌬␪͒2 Un 1ϪUn ⌬t i i, j i, j ϭ ϩ ͑␰͒ ͑ р␰р ϩ⌬ ͒ ⌬ ut utt t t t D␦⌬t t 2 ϩ ͑Un ϪUn ͒ϩ⌬t͓␣Un ϩVn r ⌬r (iϩ1,j) (i, j) (i, j) (i, j) i and Ϫ␣ n ͑ n ͒2Ϫ n n ͔ ͑ ͒ r1U(i, j) V(i, j) r2U(i, j)V(i, j) , A2a n Ϫ n ϩ n Uiϩ1,j 2Ui, j UiϪ1,j ␦⌬t 2 nϩ1ϭ n ϩ ͑ n Ϫ n ϩ n ͒ ͑⌬r͒ Vi, j Vi, j V(iϩ1,j) 2V(i, j) V(iϪ1,j) ͑⌬r͒2 ͑⌬r͒2 ϭu ϩ u ͑␰͒͑rϪ⌬rр␰рrϩ⌬r͒, ␦⌬ rr 12 rrrr t n n n ϩ ͑V ϩ Ϫ2V ϩV Ϫ ͒ ⌬␪͒2 (i, j 1) (i, j) (i, j 1) ri͑ with similar expressions for ur and u␪␪ . ␦⌬t From these equations, we can calculate the truncation er- ϩ ͑ n Ϫ n ͒ϩ⌬ ͓␤ n Ϫ␣ n ⌬ V(iϩ1,j) V(i, j) t V(i, j) U(i, j) ri r ror introduced by the finite-difference discretization of Eq. ͑A3a͒, ϩ␣ ⌬ n ͑ n ͒2ϩ ⌬ n n ͔ ͑ ͒ r1 tU(i, j) V(i, j) r2 tU(i, j)V(i, j) , A2b ⌬t ͑⌬r͒2 Tn ϭ u ͑r ,␪ ,␨ ͒ϩ u ͑␯ ,␪ ,t ͒ with zero-flux boundary conditions i, j 2 tt i j ijn 12 rrrr ijn j n n ϭ n n ϭ n ͑⌬␪͒2 ⌬ UMϩ1,j UM, j , Ui,Nϩ1 Ui,N , r ϩ u␪␪␪␪͑r ,␰ ,t ͒ϩ u ͑␹ ,␪ ,t ͒, 12 i ijn n 2 rr ijn j n n ϭ n n ϭ n VMϩ1,j VM, j , Vi,Nϩ1 Vi,N . which is O(⌬t)ϩO(⌬r2)ϩO(⌬␪2)ϩO(⌬r). By assuming As initial conditions we use random small fluctuations that the involved derivatives are bounded, the truncation er- 0 ϭ 0 ϭ ror goes to zero as ⌬t, ⌬r, and ⌬␪ go to zero. Consequently, around the steady state Ui, j 0 and Vi, j 0, for all i, j. Given the nonlinear terms in Eqs. ͑A2͒ a rigorous analysis the numerical method is consistent with the partial differen- ͑ ͒ of the stability of the difference scheme is not an easy task. tial equation A3a . In practice, however, since important sources of instability To determine when the method is stable, we find the equa- ϭ Ϫ are the higher-order terms ͓35͔, necessary conditions and a tion satisfied by the error e U u. By definition of the trun- good estimation of ⌬t are obtained by investigating the con- cation error, the exact solution u satisfies sistency and stability of Eqs. ͑A2͒ as if they were uncoupled. Thus, we consider the equations D␦⌬t nϩ1ϭ n ϩ ͑ n Ϫ n ϩ n ͒ ui, j ui, j u(iϩ1,j) 2u(i, j) u(iϪ1,j) ͑⌬r͒2 1 1 ϭ ␦ͩ ϩ ␪␪ϩ ͪ ϩ␣ ͑ ͒ ut D urr u ur u, A3a D␦⌬t r2 r n n n ϩ ͑u ϩ Ϫ2u ϩu Ϫ ͒ ⌬␪͒2 (i, j 1) (i, j) (i, j 1) ri͑ 1 1 D␦⌬t v ϭ␦ͩ v ϩ v␪␪ϩ v ͪ ϩ␤v. ͑A3b͒ ϩ ͑ n Ϫ n ͒ϩ␣⌬ n ϩ⌬ n t rr 2 r ⌬ u(iϩ1,j) u(i, j) tu(i, j) tTi, j . r r ri r

As above, the ﬁnite-difference discretization of Eq. ͑A3a͒ By substracting Eq. ͑A4͒ from this equation, after grouping is common terms, we obtain

051913-7 ARAGO´ N, TORRES, GIL, BARRIO, AND MAINI PHYSICAL REVIEW E 65 051913

␦⌬ ␦⌬ ␦⌬ TABLE I. Normalized magnitudes of the coefﬁcients for the ϩ D t D t D t en 1ϭͩ 1Ϫ2 Ϫ2 Ϫ ϩ␣⌬t ͪ en Fourier-Bessel decomposition of the pattern in Fig. 4͑c͒. (i, j) ⌬ ͒2 ⌬␪͒2 r ⌬r (i, j) ͑ r ri͑ i m/n 1234567 D␦⌬t D␦⌬t D␦⌬t ϩͩ ϩ ͪ n ϩͩ ͪ n e ϩ e Ϫ 0 0.372 0.021 1.000 0.058 0.013 0.004 0.015 ͑⌬ ͒2 r ⌬r (i 1,j) ͑⌬ ͒2 (i 1,j) r i r 1 0.003 0.006 0.012 0.004 0.002 0.001 0.000 2 0.002 0.016 0.004 0.003 0.002 0.002 0.001 D␦⌬t D␦⌬t ϩͩ ͪ n ϩͩ ͪ n Ϫ⌬ n 3 0.002 0.010 0.002 0.003 0.002 0.002 0.002 e ϩ e Ϫ tT . ⌬␪͒2 (i, j 1) ⌬␪͒2 (i, j 1) i, j ri͑ ri͑ 4 0.002 0.010 0.000 0.002 0.002 0.002 0.002 ͑A5͒ 5 0.594 0.057 0.012 0.017 0.014 0.011 0.011 6 0.002 0.008 0.003 0.000 0.000 0.001 0.001 ϭ ͉ n ͉ nϭ ͉ n ͉ р 7 0.002 0.007 0.003 0.000 0.000 0.001 0.001 If we deﬁne Tmax max T (i,j) and E max e(i,j) , for 1 i рM and 0р jрN, from Eq. ͑A5͒ we get

␦⌬ ␦⌬ ␦⌬ Equations ͑A7b͒ and ͑A8b͒ can be written in a more trans- ϩ D t D t D t ͉en 1͉рͯ1Ϫ2 Ϫ2 Ϫ ϩ␣⌬tͯEn parent way if we take into account that, since we are avoid- (i, j) ͑⌬ ͒2 ͑⌬␪͒2 r ⌬r r ri i ϭ ϭ ϭ ⌬ ing the origin r 0, min(ri) r0 ( r/2). By substituting this into Eqs. ͑A7b͒ and ͑A8b͒ and rearranging the terms we D␦⌬t D␦⌬t D␦⌬t D␦⌬t ϩͯ ϩ ͯEnϩͯ ͯEnϩͯ2 ͯEn obtain ⌬ ͒2 r ⌬r ⌬ ͒2 ⌬␪͒2 ͑ r i ͑ r ri͑ ϩ⌬ ͑ ͒ tTmax . A6 1 ⌬tр , ͑A9a͒ 4D␦ 4D␦ The absolute value sign can be removed provided that all the ϩ terms under absolute value sign are nonnegative. Since D ͑⌬r͒2 ⌬r͑⌬␪͒2 Ͼ0 and ␦Ͼ0, we assume that

␣у0, ͑A7a͒ 1 ⌬tр , ͑A9b͒ 4␦ 4␦ D␦⌬t D␦⌬t D␦⌬t ϩ ϩ͉␤͉ 1у2 ϩ2 ϩ , ͑A7b͒ ͑⌬r͒2 ⌬r͑⌬␪͒2 ⌬ ͒2 ⌬␪͒2 r ⌬r ͑ r ri͑ i so that, after removing absolute value signs in Eq. ͑A6͒ and respectively. Both inequalities provide necessary conditions simplifying, we obtain for the stability of Eq. ͑A2͒. The values used in all our calculations are Dϭ0.516, ⌬rϭ1/34, ⌬␪ϭ2␲/68, and ␤ϭϪ0.91. Thus by consider- ͉ nϩ1͉р nϩ␣⌬ nϩ⌬ ϭ͑ ϩ␣⌬ ͒ nϩ⌬ e(i, j) E tE tTmax 1 t E tTmax . ing, for example, the value of ␦ to obtain the pentagonal

It can be proved ͑see, for instance, theorem 1.2.1 of Ref. ͓35͔͒ that the above sequence of inequalities with nonnegative numbers is equivalent to

͉ n ͉р ␣n⌬t͑ 0ϩ ⌬ ͒ϭ ␣n⌬t ⌬ e(i, j) e E n tTmax e n tTmax , where we used the fact that E0ϭ0. Thus, as a ﬁxed value of → ͑ ͒ tn , the error goes to zero as Tmax 0, and Eq. A4 is stable under the assumptions ͑A7͒. The above stability analysis is also valid for the discretization of Eq. ͑A3b͒, with the only difference that if ␣у0 ͓assumption ͑A7a͔͒, then, according to the ﬁrst condition in Eq. ͑3͒ ␤ is negative (␤рϪ␣) ͓17͔. Thus, the discretization of Eq. ͑A3b͒ is stable under the assumptions

␤р0, ͑A8a͒

␦⌬t ␦⌬t ␦⌬t 1у2 ϩ2 ϩ ϩ͉␤͉⌬t. ͑A8b͒ FIG. 10. Reconstructed image using the complex coefﬁcients ⌬ ͒2 ⌬␪͒2 r ⌬r ͑ r ri͑ i tabulated in Table I and Eq. ͑B1͒.

051913-8 TURING PATTERNS WITH PENTAGONAL SYMMETRY PHYSICAL REVIEW E 65 051913 pattern in Fig. 2͑a͒ (␦ϭ0.01 011) we get ⌬tр0.009 326 and 2 kmn ⌬tр0.004 791, respectively. C ϭ mn ␲ ͒2Ϫ 2 ͒ 2 ͓͑kmna m ͔͓Jm͑kmna ͔

a 2␲ APPENDIX B: MODAL DECOMPOSITION ϫ ͵ ͵ ͑ ␪͒ ͑ ͒ Ϫim␪ ␪ u r, rJm kmnr e d dr. The principal modes involved in the Turing patterns simu- 0 0 ͉ ͉ lated in this work can be revealed by means of a modal By examining the magnitude Cmn of these coefficients, we decomposition. In particular, the solution of Eqs. ͑1͒, u(r,␪), can determine the principal modes of the function u(r,␪). can be expanded in a Fourier-Bessel series as We performed the Fourier-Bessel decomposition of the pattern in Fig. 4͑c͒. In Table I, the normalized magnitudes ͉ ͉ ϭ ϭ ϱ ϱ Cmn for m 0,1,...,7andn 1,2,...,7,aretabulated. It ͑ ͒ ͑ ͒ ͑ ␪͒ϭ ͑ ͒ im␪ ͑ ͒ can be seen that the more relevant modes are 03 , 51 , and u r, ͚ ͚ CmnJm kmnr e , B1 ͑ ͒ mϭ0 nϭ1 01 . Since J0 is nonzero at the origin, the enhancement of modes k03 and k01 account for the central spot in the pentagonal pattern. ϭ␬ where kmn mn /a, as defined in Sec. II. In order to verify our result, we use the obtained ͑com- Using the orthonormality and completeness of plex͒ coefficients, to reconstruct the pattern using Eq. ͑B1͒. im␪ Jm(kmnr)e , the complex coefficients Cmn are given by The result is shown in Fig. 10.

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