Spiral and target patterns in bivalve nacre manifesta natural excitable medium from layer growthof a biological liquid crystalJulyan H. E. Cartwrighta,b, Antonio G. Checab, Bruno Escribanoa,1, and C. Ignacio Sainz-Dıaza

aInstituto Andaluz de Ciencias de la Tierra, Consejo Superior de Investigaciones Cientıficas, Campus Fuentenueva, and bDepartamento de Estratigrafıa yPaleontologıa, Facultad de Ciencias, Universidad de Granada, E-18071 Granada, Spain

Edited by H. Eugene Stanley, Boston University, Boston, MA, and approved May 1, 2009 (received for review January 27, 2009)

Nacre is an exquisitely structured biocomposite of the calciumcarbonate mineral aragonite with small amounts of proteins andthe polysaccharide chitin. For many years, it has been the subjectof research, not just because of its beauty, but also to discover hownature can produce such a superior product with excellent me-chanical properties from such relatively weak raw materials. Fourdecades ago, Wada [Wada K (1966) Spiral growth of nacre. Nature211:1427] proposed that the spiral patterns in nacre could beexplained by using the theory Frank [Frank F (1949) The influenceof dislocations on crystal growth. Discuss Faraday Soc 5:48–54] hadput forward of the growth of crystals by means of screw disloca-tions. Frank’s mechanism of crystal growth has been amply con-firmed by experimental observations of screw dislocations incrystals, but it is a growth mechanism for a single crystal, withgrowth fronts of molecules. However, the growth fronts composedof many tablets of crystalline aragonite visible in micrographs ofnacre are not a molecular-scale but a mesoscale phenomenon, soit has not been evident how the Frank mechanism might be ofrelevance. Here, we demonstrate that nacre growth is organizedaround a liquid-crystal core of chitin crystallites, a skeleton that theother components of nacre subsequently flesh out in a process ofhierarchical self-assembly. We establish that spiral and targetpatterns can arise in a liquid crystal formed layer by layer throughthe Burton–Cabrera–Frank [Burton W, Cabrera N, Frank F (1951) Thegrowth of crystals and the equilibrium structure of their surfaces.Philos Trans R Soc London Ser A 243:299–358] dynamics, andfurthermore that this layer growth mechanism is an instance of animportant class of physical systems termed excitable media. Arti-ficial liquid crystals grown in this way may have many technolog-ical applications.

biomineralization � chitin � interlamellar membranes � mollusc

The splendor of a pearl extends even to under a microscope;with magnification, one can see that the nacreous surfaces of

bivalve mollusks—clams, mussels, oysters, scallops, and so on—are made up of a striking arrangement of spiral, target, andlabyrinthine patterns (1); see Fig. 1. Nacre, or mother of pearl,is the iridescent material that forms an inner layer of the shellsof numerous species of mollusks as well as the pearls that manyof those same species produce. The structure of nacre is oftenlikened to a brick wall, and indeed, it is composed of bricks ofaragonite tablets (�95%) and mortar of organic so-calledinterlamellar membranes (polysaccharide and protein, �5%),but it is a brick wall built in a peculiar fashion, because first, themortar is put in place, and then the bricks grow within it. (SeeFig. 2 for a sketch of bivalve molluscan anatomy and nacrestructure.) From an examination of the extrapallial space of thebivalve mollusk, the narrow liquid-filled cavity between the softtissues and the shell of the organism, it is seen that the first visiblefeature in nacre growth is the formation of a new interlamellarmembrane in the fluid (2–4). In Fig. 3A, we show this processoccurring in a transmission electron micrograph section throughthe growth front of bivalve nacre that displays the hard shell

below and the soft body of the organism, the mantle, above theliquid-filled extrapallial space. The membrane in its moment offormation is spaced some 100 nm above an earlier-formed onebelow it; there is not space between the membrane and themicrovilli of the mantle cells above for any additional membraneto form. We must follow it back some 20 �m to find the frontwhere the process of mineralization is seen to commence, andby this stage, its spacing from the membrane beneath hasincreased to �500 nm. The core of these membranes is com-posed of rod-shaped crystallites of the polysaccharide chitin inits �-polymorph (5). This chitin must be secreted into theextrapallial liquid together with the mineral and protein com-ponents of nacre; it is thought—although not yet confirmed—that similar mechanisms are responsible for chitin as for cellulosecrystallite formation, wherein rosette structures in the cellularmembrane extrude polymer chains from closely packed pores,whence they immediately crystallize by interchain hydrogenbonding into long thin needle crystallites (6). In the case ofnacre, these crystallites of �-chitin are known to be some 20–30nm in diameter and up to hundreds of nanometers in length,consisting of hundreds of chitin polymers (7).

Interlamellar membrane formation is then necessarily a pro-cess of self-assembly within the extrapallial f luid; to use ourbrick-wall analogy, it is as if one empties clay, lime, sand, andwater into a tub and finds that sheets of mortar form sponta-neously within it. What prompts chitin crystallites to self-assemble into sheets? �-Chitin crystallites of similar dimensionsto the �-chitin crystallites of mollusks have been studied in vitro.When dispersed in a colloidal aqueous suspension, they form aso-called cholesteric liquid-crystalline phase in which the crys-tallites are parallel to the plane of the layers, and the crystallitesin each layer are twisted with respect to those in the neighboringlayers, with an interlayer spacing of 60–120 nm (8), comparablewith the 100 nm seen here. Similar preparations using �-chitincrystallites from a cephalopod mollusk (Todarodes pacificussquid pen) (9) and from a vestimentiferan tube worm (Riftiapachyptila tubes) (10) indicate that these too form a liquid-crystalphase in aqueous suspension. Birefringence was observed at restbetween cross-polarizers for dispersions of �0.1% �-chitin, andat 0.3% consistency, they formed hard gels (9). (Note that we areconcerned not with a molecular liquid crystal of chitin polymersbut with a colloidal or supramolecular one of chitin crystallites.)In laboratory experiments, the liquid-crystalline structure isallowed to come to equilibrium by waiting from a few days up toa year (8). In the mollusk in vivo, on the other hand, a fresh layer

Author contributions: J.H.E.C., A.G.C., and C.I.S.-D. designed research; A.G.C. and B.E.performed research; C.I.S.-D. analyzed data; and J.H.E.C. and C.I.S.-D. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

1To whom correspondence should be addressed. E-mail: bruno@lec.csic.es.

This article contains supporting information online at www.pnas.org/cgi/content/full/0900867106/DCSupplemental.

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of nacre is laid down every 1–24 h, so the system does not havetime to arrive at equilibrium, and the liquid-crystalline orderingof �-chitin is only partial. It comprises a lamellar structure, butwithin the layers there is disorder, with the crystallites forminga felt-like mesh, like a logjam on a lake, as seen in the plan viewof Fig. 3B. Simulations of the dynamics of the formation of acholesteric phase of a liquid crystal confirm that there is aninitial orientation of the material into layers followed by itsalignment within and between layers (11). The evidence leads usto conclude that, in Fig. 3, we are observing the rod-shapedcrystallites of �-chitin being compelled by their mutual interac-tions, self-organizing in the extrapallial space into a liquidcrystal.

Artificial liquid crystals are formed by altering a globalvariable—be it temperature, concentration, electric field, etc.—that causes the whole of a domain to crystallize, forming manylayers at once. We perceive from Fig. 3 that mollusks, on theother hand, perform liquid crystallization in quite a differentmanner: layer by layer. The narrowness of the molluscan extra-pallial space implies that, as chitin crystallites are secreted intothe extrapallial f luid, they can only self-assemble through theirmutual interactions into 1 new liquid-crystal layer. The continualsecretion of material from the mantle cells maintains the extra-pallial space approximately constant in depth, so that at any time1 fresh layer of the liquid crystal is being deposited, and in thisway, in time, the mollusk builds a liquid crystal within anexpanding domain. This is concurrently transformed with theaddition of proteins and aragonite into the solid biocompositethat is nacre; a process that we analyze in depth in ref. 2. Here,we concentrate on understanding the mode of liquid-crystalconstruction used by the mollusk, in which it is built up layer bylayer. This method is unique in terms of artificial liquid crystals,but is the manner in which solid crystals generally grow, and thisconnection allows us to comprehend the spiral and targetpatterns on the surface of nacre.

Liquid-Crystal Layer Growth Model of NacreLayer or tangential growth of a solid crystal, analyzed by Burton,Cabrera, and Frank (12), involves 2 processes: on one hand, theaddition of material at the edges of 2-dimensional islands on thesurface of the crystal together with the occasional nucleation ofa fresh island atop an earlier one and, on the other, the continualaccretion of material at the edges of a helical ramp on the surfaceof the crystal, termed a screw dislocation (13). [Tangentialgrowth as a whole was examined by Burton, Cabrera, and Frank(12); Frank’s contribution was the analysis of screw dislocations(14)]. These 2 processes produce characteristic target and spiralpatterns seen at the molecular scale in solid crystals (15). Thesesame patterns are visible in the conspicuous features of thesurface of nacre at the mesoscale in Fig. 1. In that figure, thepatterns are arrangements of aragonite tablets in mineralizednacre. However, as we see in Fig. 3A, in growing nacre, inter-lamellar membranes extend ahead of these mineralizationfronts; the latter follow the geometry laid down by the former.This is evidenced by the forms of tablets that grow at the centersof the screw dislocations in the membranes, which mold them-selves to the spiral, as shown in Fig. 1F. Thus, the spirals andtarget patterns are in fact laid down in the interlamellar mem-branes, and we must establish that growth of an interlamellarmembrane as a liquid crystal layer by layer can give rise to thisgeometry of spirals and target patterns.

We formulate a minimal model containing the fundamentalphysics of liquid crystallization. Growth proceeds by the incor-poration of individual growth units, which are the chitin crys-tallites that compose the liquid crystal. Because this is anessentially discrete process, we employ a discrete coupled maplattice, or lattice Boltzmann model, with a square lattice ofdiscrete space and time but continuous state variables, that putsnucleation and growth on an equal footing. The surface isdivided into a randomized grid of cells, as introduced by Markusand Hess (16), to avoid anisotropic growth. Each cell has agrowth height H(i, j), initially zero, updated at the end of each

A B C

D E F

Fig. 1. Scanning electron micrographs of the growth surface of bivalve nacre. (A) General view of labyrinthine patterning. (B) Target patterns. (C) Collidinggrowth fronts and spiral. (D and E) Paired spirals. (F) The aragonite tablet at the center of a spiral follows its structure. A–D are Pteria avicula; E and F are Pteriahirundo. In all these images, we see only the mineralized portion of the nacre, consisting of aragonite tablets. The interlamellar membranes atop the uppermostmineral layer are generally lost as the soft tissue is removed, unless the sample is especially prepared to preserve soft tissue as well as biomineral, as in Fig. 3A.

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iteration of the growth. The surface is scanned on each iteration,and in each cell, there may occur either nucleation, growth, orneither as a function of the relative height of the nearestneighbors. The neighbors to each cell are assigned by proximity,defining a circle of radius R around the cell.

The essential physics we wish to capture in our model is thata fresh growth unit is incorporated into the growing surface mostprobably at a site where it is more strongly bound by virtue ofhaving most neighbors. Such sites are so-called kinks: concavitiesat the base of steps on the surface. Thus, there is no probabilityof a growth unit settling above a step, because it will slide downto occupy a more favorable site below; there is a rather lowprobability of a growth unit settling on a flat surface and a veryhigh probability of it occupying a site at the base of a step. Thisphysics we can model by setting a low probability of addition ofmaterial, which is termed nucleation, at any point over thesurface that is not elevated above its neighbors, together with acertainty of the addition of material, which is termed growth, atthe kinks of steps.

The condition for nucleation is that the cell should not stickout far above its neighbors, that the surface be sufficientlysmooth. The cell in question should have height comparable withthat of its neighbors with a difference not more than �hN. If so,there is a small probability PN of nucleating a fresh growthsurface. Nucleation is a fundamentally stochastic process, andthis is reflected in the model. If nucleation takes place, and a newgrowth unit is incorporated, the growth surface increases inheight by 1 plus or minus a small random factor �:

H�i, j�3 H�i, j� � 1 � �.

The condition for growth is that the cell must be at the edge ofa growth front; i.e., must have neighbors higher by at least �hG.If so, a growth unit is incorporated: The growth height isincreased by the mean of the higher neighbors ��, and a growthfront extends:

H�i, j�3 H�i, j� � ��k,l�

H�k, l� � H�i, j�n

� �,

where (k, l) are the coordinates for each higher neighbor, and nis the total number of higher neighbors, which depends on R.This part of the model is deterministic except for the parameter�, which adds a certain amount of noise so that the growthsurfaces are not completely flat and defect free.

In this way, as a growth front grows out, its height may vary,and if it encounters another front of different height, followingthe rules above, defects may be spontaneously incorporated intothe growth. Two growth fronts that meet may join seamlesslywith each other, terminating growth; they may completelyoverlap each other, forming an edge dislocation; or they mayoverlap along part of the edge and join along the rest, nucleatinga spiral defect as growth continues. All of these behaviors arecomparable in the model—Fig. 4—with nacre itself (Fig. 1).

In our model the set of parameters comprises:

� Stochastic term, typically �0.05–0.1, that varies the heightof the growth cell on nucleation and growth, otherwiserenormalized to be unity. This term is responsible fordefects. Physically this growth-height variation is relatedto the fact that the growth units in the case of nacre are

nacreprismatic layer

periostracum

interlamellar membraneintertabular membrane

aragonite tablet

mantle

100 µm

20 µm

extrapallial spacemantle cells with microvellosities

5 µm

= adoral boundary of the mineralization front= adoral boundary of the interlamellar membrane

growth direction

Fig. 2. Sketch of bivalve molluscan anatomy indicates the position of the liquid-filled interlamellar space between the mineralized shell and the mantle partof the soft body of the organism and illustrates with successive amplifications the brick and mortar structure of nacre. The growth surface, on which the patternsin Fig. 1 are observed, is between the mantle and the shell; i.e., into the page.

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nonidentical; they are rods of differing lengths. An addi-tional factor in the appearance of height variations leadingto defects in all layer growth is the presence of impurities.

�hN Threshold of elevation with respect to the neighbors belowwhich a cell’s neighborhood is considered sufficiently flatfor nucleation of a fresh layer to occur. A larger value of�hN implies that more nucleation will take place, typicallyset similar in magnitude to �.

PN Probability of nucleation occurring in a given cell in which,according to �hN, it may occur; typical value 10�3 to 10�6.PN in combination with �hN determine the frequency ofnucleation. If PN is zero, screw dislocations leading tospiral growth are the only growth mechanism; if PN islarge, the surface is rough.

�hG Threshold of difference in height with the neighbors forthere to be an active neighbor. There must be at least 1active neighbor for a site to be a kink and so to have growthin a cell. Physically, this parameter in combination with �determines the frequency of screw dislocations. Typicalvalues to produce spiral growth are 0.4–0.8. If �hG is large,spiral growth is suppressed and growth proceeds by layers.

If �hG is small, growth fronts that encounter one anothercross to produce edge dislocations.

R Radius of the circle of neighbors, typically set between 3and 10. The model begins with a randomized grid (16): Apoint is set at random within each cell at the beginning ofthe simulation. Subsequently the neighbors of the cell aresought by tracing a circle of radius R about this point.Owing to the initial random positioning of the center ofthe circle, this procedure produces a varying numberof neighbors per cell. As with �, the randomizationintroduced with the grid here is related to the nonidenticalnature of the chitin crystallites that are the growth unitsfor the nacre. A larger radius R implies wider terracesbetween growth steps. This is physically related to thesurface diffusion of the growth units. If we look at nacrefrom different species, we indeed find that some havesteeper, narrower terraces, and others have shallower,wider ones.

In Fig. 4, we display the evolution in time of 3 differentscenarios in our model, to be compared with the various

Fig. 3. Cross-sectional and surface views of the interlamellar membranes of bivalves. (A) Transmission electron micrographs show the interlamellar membranesbeing laid down before mineralization in Pinctada radiata; the orientation of the sample is as in Fig. 2. The blowups provide more detail of the membranes andshow that the initial spacing of �100 nm when a fresh layer is laid down gradually increases to �500 nm in mineralized nacre over a distance of some 20 �m.We have surmised elsewhere (2) that this increase in layer spacing accompanies the coating of the chitin liquid-crystal layers with proteins before mineralizationand is likely caused by an additional repulsion term in the force balance between membranes introduced by the electrostatic charges on the amphiphilic structureof the mature membrane. This change in the force balance from the original cholesteric liquid crystal to the mature membrane covered with protein would serveto repel the membrane from its neighboring interlamellar membrane (2). (B) Scanning electron micrograph of the interlamellar membrane of Anodonta cygneashows its logjam structure when viewed from above. When interpreting this image, one must bear in mind various points. This is a completed interlamellarmembrane seen in plan view; we are not just looking at chitin, but at the result of a later stage of nacre formation in which the chitin has become coated withprotein. This stabilizes the layer, which is why we are able to see it in this micrograph, for which the sample has inevitably been subjected to drying, which mustdisintegrate the uncoated initial state of the chitin layer.

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phenomena in nacre of Fig. 1. In each case, the simulation beganat time 0 with a flat surface of zero height. In Fig. 4A, we showthe evolution of target patterns; we see the system after 87 and93 iterations. Each terrace expands laterally, and, once theuppermost terrace has expanded sufficiently, a fresh terracenucleates on it. This is similar to what is seen in Fig. 1B in nacreand corresponds to island growth, the first process of Burton–Cabrera–Frank (12). In Fig. 4B, we show, after 300 and 310iterations, colliding growth fronts and an evolving spiral similarto those seen in nacre in Fig. 1C. As the spiral rotates and growsout, it meets other growth fronts coming from the top right. Thefronts of the same height annihilate each other by joiningseamlessly as they meet. Last, in Fig. 4C, we show, after 230 and250 iterations, paired spirals similar to those in the nacre of Fig.1 D and E. Fig. 4 B and C corresponds to spiral growth, whichis the second process of Burton–Cabrera–Frank. Those in Fig.4B are single screw dislocations, whereas the paired spirals in Fig.4C are Frank–Read sources (26).

These time evolutions can be seen in nacre samples by lookingdown through the nacre layers, as the third dimension ofspace—into the material—is equivalent to time. However, al-though that procedure takes us back in time, it does not indicatewhich events are coeval, as the model does. From our model, wecan see that Wada (1) was correct in noting the parallel betweennacre growth and the Burton–Cabrera– Frank (12) dynamics [infact, Wada (1) singled out the spiral growth mechanism that wasFrank’s contribution (14)] of layer growth, albeit, for a liquidcrystal rather than the solid crystal Wada had envisaged.

DiscussionFrom our model of layer growth, we can comprehend why thesespiral and target patterns, familiar from such diverse instances asthe Belousov–Zhabotinsky reaction in chemistry and ventricularfibrillation of the heart in biology, appear in nacre: All of thesesystems are describable in physical terms as excitable media (17).Excitable media form a rather general class of systems in whichthere are elements that are quiescent until excited into an activestate by some stimulus, after which, they are unresponsive tofurther stimuli during some refractory period before returning totheir initial quiescent, excitable state. This very simple physics issufficient to produce complex spatiotemporal patterns of targetsand spirals, which is why these patterns are observed so generallyin many different fields. Here, the excitable paradigm holds too,and we can describe the growth of nacre as an excitable medium:A point on the surface of the last-formed layer is quiescent, butreceptive to the nucleation of a new layer above it, given a largeenough perturbation, that is to say, a fluctuation that concen-trates sufficient material nearby to nucleate a new surface. Theedges of layers, the kinks of steps, where new material isincorporated, are active, whereas just after a new layer hasformed, there is a refractory period in which another new layercannot be formed at that point, because material is preferentiallyincorporated at the growing edge of the layer nearby. Thus, thegrowth dynamics of our model of the formation of interlamellarmembranes of nacre corresponds to an excitable medium.

The growth model we put forward here is for the formation ofa liquid crystal of chitin that is the first component of nacre tobe laid down. But nacre—by which we mean the completed

A B Cii ii

i

ii

i i

Fig. 4. Our model of liquid-crystal layer growth reproduces the features seen in micrographs of nacre growth and shows—what cannot be seen in nacresamples—how they evolve in time. (A) Target patterns—see Fig. 1B—after (i) 87 and (ii) 93 iterations; parameters � � 0.08, �hN � 0.1, PN � 10-3, �hG � 0.8, R � 10.(B) Colliding growth fronts and spiral—see Fig. 1C—after (i) 300 and (ii) 310 iterations; parameters � � 0.06, �hN � 0.06, PN � 5 10�6, �hG � 0.6, R � 4. (C) Pairedspirals—see Fig. 1 D and E—after (i) 230 and (ii) 250 iterations; parameters � � 0.08, �hN � 0.09, PN � 10-4, �hG � 0.8, R � 5. The color scheme is shown alongsideeach subfigure; lighter is higher. These 3 cases can be found in supporting information (SI) Movie S1, Movie S2, and Movie S3.

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product: mother of pearl, or a pearl itself—is no longer a liquidcrystal but a complex solid. Some of the complexity of the seriesof processes the liquid crystal then undergoes to become nacre,we lay out in ref. 2. The problem of supramolecular assembly inbiology—how does an organism assemble itself: After the cellsmake the molecules that are to form part of the structure, howdo they form into tissue; f lesh and bone?—is, as yet in general,unsolved. Nacre is of particular interest and is by far the mostintensively studied nonhuman organomineral biocomposite. Itlies at the intersection of 2 groups of supramolecular biologicalstructures: fibrous composites and biominerals. It is of interestnot just because it is beautiful, but also because it is strong: withjust 5 percent polysaccharide and protein, its work of fracture isup to 3,000 times higher than that of inorganic aragonite (18).The importance of fluid dynamics in the organization of su-pramolecular biological structures, both in transporting materi-als and in their self-assembly, is beginning to be recognized (19).Such is the case with biomineralization processes such as boneand fish otolith formation, and it is likely that further exampleswill come to light as our knowledge of the so-called ‘‘softchemistry’’ involved in biological mineral deposition improves.The extrapallial liquid is fundamental in the formation of nacre,and it is probable that we shall, in future, find that there areactive fluid dynamical processes involved in the transport ofchitin crystallites, proteins, and mineral to produce nacre.

Nacre is secreted by some species of 4 classes in the phylumMollusca: Bivalvia, Gastropoda, Cephalopoda, and, to a minorextent, Tryblidiida. Although the material has many points incommon between the classes—hence its general classification asnacre—at the level of details, there are important differences. Inthe present case, we have concentrated on bivalve nacre. Theother large class of nacre-secreting mollusks, the gastropods,possesses an additional feature not present in bivalves, thesurface membrane, interposed between the mantle and the shell.This organizes the mineralization of gastropod nacre so that itgrows in towers, rather than in the fronts of bivalve nacre (20).Nonetheless, gastropod interlamellar membranes have a similargeometry to those in the bivalves, and screw and edge disloca-tions are noted in gastropod nacre (2); they are simply not so

clearly visible at the growth surface as in bivalves, on account ofgastropod mineralization being in towers rather than fronts thatfollow the leading edges of the membranes as in bivalves. Itseems likely that our liquid-crystal growth model, possibly withsome variations, should be applicable in gastropods too.

Today, layer-by-layer growth of materials is of great techno-logical interest; but the synthetic process used is often a ratherprimitive one consisting of building up layers by dipping thematerial into first one component then another to produce acorresponding layering of so-called artificial nacre (21). Nature,on the other hand, is more sophisticated in using a self-organizedsystem of liquid crystallization to assemble nacre layer by layer.Liquid crystals have long been surmised to play an importantrole in the organization of biological tissues (22–24); yet, withouta model for their formation, a key section of the argument thatbiological systems make use of liquid crystals for the self-organization of supramolecular structure has been lacking (25).Here, we have shown how such a biological liquid crystal may beconstructed and that this self-assembly is occurring through anunfamiliar process—unfamiliar, that is, to humans, becausemollusks have been using it since the Cambrian—of liquidcrystallization layer by layer. It may be of advantage in manytechnological applications to follow nature’s way.

Materials and MethodsScanning Electron Microscopy. Shells of living specimens of Pteria avicula,Pteria hirundo, and Anodonta cygnea were fixed with 2.5% glutaraldehydein a 0.1 M cacodylate buffer. After CO2 critical-point drying samples wereobserved intact with a Leo Gemini 1530 field emission scanning electronmicroscope.

Transmission Electron Microscopy. We had access to original material ofPinctada radiata of the late H. Nakahara. This was prepared at Meikai Uni-versity (Chiba, Japan) by M. Kakei according to the protocol described in ref. 4.We used a Philips CM20 transmission electron microscope.

ACKNOWLEDGMENTS. We thank Prof. M. Kakei (Meikai University, Chiba,Japan) for providing material of Prof. H. Nakahara. This work was sup-ported by Research Projects CGL2007-60549 and CGL2008-06245-CO2-01(Ministerio de Educacıon y Ciencia) and Research Group RNM190 (Junta deAndalucıa).

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