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 morpho- gen 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- 1063-651X/2002/65͑5͒/051913͑9͒/$20.0065 051913-1 ©2002 The American Physical Society ARAGO´ N, TORRES, GIL, BARRIO, AND MAINI PHYSICAL REVIEW E 65 051913 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 non- linear 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, ob- tained 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 pa- rameters 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. 051913-2 TURING PATTERNS WITH PENTAGONAL SYMMETRY PHYSICAL REVIEW E 65 051913 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.