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Pattern Formation in Shoots: A Likely Role for Minimal Energy Configurations of the Tunica Author(s): Paul B. Green Reviewed work(s): Source: International Journal of Plant Sciences, Vol. 153, No. 3, Part 2: The Katherine Esau International Symposium (Sep., 1992), pp. S59-S75 Published by: The University of Chicago Press Stable URL: http://www.jstor.org/stable/2995528 . Accessed: 21/07/2012 20:46 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . The University of Chicago Press is collaborating with JSTOR to digitize, preserve and extend access to International Journal of Plant Sciences. http://www.jstor.org Int. J. Plant Sci. 153(3):S59-S75. 1992. ? 1992 by The University of Chicago.All rightsreserved. 1058-5893/92/5303-0034$02.00 ROLE FORMINIMAL A LIKELY IN SHOOTS: FORMATION PATTERN OFTHETUNICA CONFIGURATIONS ENERGY PAUL B. GREEN' Departmentof BiologicalSciences,StanfordUniversity, Stanford,California94305 When simple shoots and flowersare examined,a restrictedset of patternsis found. Characterization involves three levels of scale: (1) The overall arrayis roughlyradiallysymmetrical.(2) The elements within it are usually arrangedin either straightradii or in spirallines. (3) The element itself, e.g., a leaf or petal,has a planeof bilateralsymmetrythatlies on a radiusof the overallarray.Proposedmorphogenetic mechanisms have centered on the second level, where the propagationof pattern is based on "feed forward,"an influenceof recentlymade primordiaon the site of initiationof the next. This influencehas been postulatedto be chemicalinhibitionemanatingfromthe centerof each primordium.New primordia would arise in uninhibitedregionsand then produceinhibitor.An alternatecandidatefor this influence rests on the fact that the bucklingof a constrainedinanimate sheet can also propagatepattern.Bump formationon a sheet experiencinguniform upwardpressureis influencedboth by the boundaryof the sheet and by humps alreadypresenton it. I proposethat in meristemsthe tunicaexhibitsminimal energy bucklingbehavior,and the corpussuppliesupwardpressure.Simulationsinvolvingbumpscan propagate patternswith straightor spiralconfigurations.This is feed-forwardof pattern,at the second level. When bumps also develop adaxialcreasestangentialto the marginof the dome, these creaseslink the bilateral symmetryof the appendagewith the overall radialsymmetry(levels 1 and 3). By servingas an apparent new linear anchor for bucklingof the dome, a new crease converts the influenceof a leaf on the dome from modifyinginternalcurvatureto becomingan externalboundarycondition. Creasesappearstiffand tend to form a ring. When such a ring is kept taut by internalpressure,it promotes radial symmetry, stabilizingthe whole array.The biomechanicalmodel ties togetherall threelevels of scale.Cyclicbuckling phenomena could explain why the number of patternsis restrictedand why they take the form that they do. Introduction The patterns in vegetative shoots, inflorescences, and individual flowers can differ widely in character. The patterns are both distinctive and consistent so they are heavily used in classification and in descriptionsof normal and mutant development. They are also the oldest biomathematical problem, having been well characterizedfor over 150 years, but their basis is not fully understood (Schwabe 1984; Steeves and Sussex 1989). This article will address five issues about shoot patterns:their variety, their origin during development, their propagation, their internal symmetry, and their stability. In more detail, the correspondingquestions are: (1) Despite greatapparentdiversity, only a small set of patternsof shoot structuresis commonly found among plants. Has this set been selected from a near-infinityof possibilities, or are the possibilities limited, the restrictionsbeingdevelopmental constraints?Because this second possibility will be shown to be likely, the remainingfour issues will center on mechanisms of apical development. (2) When a pattern arises de novo, what mechanisms could explain it? (3) How are established patternspropagated?These two questions encompass the long history of phyllotaxis. The broadestagreed-uponconceptis that propagation involves "feed-forward,"i.e., that the position of ' Reprintsavailablefrom the author. ManuscriptreceivedApril 1992; revisedmanuscriptreceived May 1992. S59 recently formed primordiagreatlyinfluencesthe site for the initiation of the next. What is the nature of this influence? (4) What explains the typical symmetry link between the bilaterally symmetrical appendage (leaf, stamen) and the overall radial symmetry of the array?The plane of symmetry of the appendageusually lies on a radius originatingat the center of the apex. For example, the adaxial side of each leaf "faces"the dome center;the septum between locule-pairsof anthersis on a line toward the gynoecium. What mechanism underliesthis common relationship? (5) Finally, why do the patternsbehave as local attractors?While a specificpatternis often maintained duringthe productionof scores of organs, it is occasionallyreplacedby anotherspecificpattern. For example, the vegetative shoot of snapdragonhas opposite leaves, whereasthe inflorescence produces flowersin spiral succession. The flower itself has whorls with fivefold symmetry. Shifts in pattern are seen during vegetative development, e.g., juvenile to mature foliage in ivy and eucalyptus.The patternsthus are stablewithin limits, but are capable of shifting to another pattern.How is this restrictedstabilityaccounted for? Answers to the last four issues bear on the first one, how developmental constraintscan restrict the variety of patterns. It is necessaryfirstto characterizethe patterns. Then the five issues will be addressed in turn. The analysis for each will be done in terms of observations and the correspondingtheory for mechanism.The observationswill be limited. re- S60 INTERNATIONAL JOURNAL flecting my firsthand experience. The theory treatmentalso will be briefand will centeron two perspectives. In the well-established view, diffusing substancesare thought to be criticalin organizing pattern. In the other view a major role for biophysical processes is advocated. It will be argued that, since plant appendagesarise as the folding of tunica layers, the mechanics of the buckling of sheets or plates is important. This buckling process, at the very least a late manifestation of other patterningmechanisms,will be shown to have remarkableorganizingproperties in its own right. That these propertiescan contributeplausiblemechanismspertinentto all four developmental questions is the major theme of this article. the patterns Characterizing The vast subject of phyllotactic patterns (Erickson 1983; Jean 1984; Schwabe 1984) will be greatly condensed here. Many classificationsystems use the concept of the plastochron,the time interval between initiation of successive organs. Because of the spatial regularityof pattern formation, an organ usually can be assigned an age in accord with its position. The age differences between adjacent primordia can describe a pattern. In general, there are two broad categories of pattern. As the array on the apex is viewed from above, primordia lie either on spiral lines or on straightradii. SPIRAL PATTERNS In the spiral patterns, e.g., those of the sunflower head or a pinecone, the eye is caught by two sets of spirals, one steep, the other less so. These lines are called parastichies,and the numbers in the two sets are usually consecutive terms in the Fibonacci series: 1,1,2,3,5,8, etc. The age differencebetween adjacentmembers in the two spirals is also the same pair of numbers. These specific arrangementscan be regardedas a consequence of the fact that consecutive primordia tend to arise on a "generative"or "genetic"spiral, a very slow spiralconnectingconsecutive primordia of increasingage. The angle between two consecutively made primordia, with the dome centeras the vertex, is calledthe divergenceangle. In most spiral forms it is an irrational angle, 137.51 ... .?. This is an ideal angle. In nature, the measurementsonly approximateit. This fraction of a circle is its smaller "golden section." It is found in many self-similar structuresin which a single entity (1) is divided into a largeportion (1 - X) and a small portion (X). The self-similarity equation is 1/(1 - X) = (1 - X)/X. X is 0.382. Here the whole circumference(360?)is 1, and the smaller arc, X, is (0.382 x 360?)or 137.5 1?.The largergolden section is 0.618. A rangeof divergencecould be compatiblewith OF PLANT SCIENCES a particularsimple pattern(e.g., 2,3) but personal observation in Ribes indicates that this range is not found. The divergence is consistently close to 137.50. Hence, the "ideal" divergence is regarded as important. It is a major clarificationthat many spiral patterns (e.g., 3,5; 55,89) can be generated by the same ideal divergence angle (Schwabe 1984). When the appendagesare largein size relative to the whole array,the number pair is small (large scales in a pinecone). When the units are small (small florets in a sunflower head), the number pair is large (89,144). When the number pair is small, the plastochronratio-relative distance of consecutively made primordia from the dome center-is large.When the pair is large,this ratio is small (fig. 1). It thus appearsthat the number of spirals, the age differencesbetween adjacent primordia,and the plastochronratio are secondary to the divergence angle. The angle can be constant while these other parameters change (Green 1987). A furthersimplificationemerges when one realizes that comparablespiral patterns can come from several ideal divergence angles; 99.50 is a second one. With this angle the divergenceis not directlyrelatedto the golden section of the dome, as before. The golden ratio still applies, however, in that a new primordium arises at the golden section of the small arc between two neighboring primordia. The new primordium arises inward of the pair, towardthe dome center and closer to the older member. This golden sectioning maneuver was also carriedout when the divergence was itself a golden section (fig.2A). It thus appears that the divergenceangle can vary while the local positioning rule is constant. Hence, the latter is given great importance. A large array of spiral patterns can be characterizedby the new primordiumarisinginterior to, and at the golden section of, the arc between older neighbors. This local activity, when repeatedover time, could integrateto give the largescale spiral patterns.The generativespiral could thus be an effect of the repeatedlocal maneuver; it is not necessarilya cause. Patternsin this class will simply be called spiral. RADIAL PATTERNS For the patternswherethe organslie on straight radii called orthostichies, a comparable local "rule" can be used for characterization.When leaves or organs arise in whorls, these are commonly staggeredor alternating.The divergence will be simply the angleof shift between members of successive whorls. Here physically adjacent membersof a whorlareassumedto have the same age and size. The new appendageforms on a radius midway on an arc between two older appendages,this arc being called the "availablecir- GREEN-PATTERN S61 FORMATION IN SHOOTS cumference."The local rule is thus that the new organ arises on the perpendicularbisector of the arc between adjacent primordia (fig. 2B). In some apices the new organs arise only one at a time in an alternatingpattern with a divergence of 1800(distichy).The correspondingavailable circumferenceis 3600 because there is only one organ at that level. This arc is bisected, with the new appendagemaximally far from previous appendages,as before (fig. 2C). This distichous pattern, zigzag in a plane, is thus viewed as a "whorlof one," e.g., corn, iris, ivy. In this group, whorled, the key local activity is bisection (0.50.5). In the other group, spiral, it is the taking of the golden section (0.618-0.382). The two large classes are thus distinguished by local behavior. Fig. 1 Spiral phyllotaxis with divergence of ca. 137.50. A, SEM of a sunflower head with new florets forming in an annularregion or "moving wave" near the center of the capitulum. The dome outer boundaryis indistinct. An involucral bract is at upper right. Bar = 1 mm. Micrographby Luis Hernandez.B, SEM of the apex of vegetativeRibeswith leaf bases and primordianumberedin sequenceof their initiation. Dome outerboundaryis relativelydistinct.An imaginary spiral line connecting ever younger leaves (increasing numbers)goes counterclockwise.It is the generativespiral. Phyllotaxisis 2,3. Bar = 100 Mm. A. Spiral ExcEvinoNs. While a majorityof shoot structures fall clearly into the above two classes, there are many exceptions. CertainEuphorbiaproduce leaves in spiral succession, but the leaves lie on straight orthostichies. Thus, spirality and orthostichy can be combined. In some plants with whorls, organ origin may not be absolutely simultaneous. Bijugateplants have truly opposite leaves in whorls of two, but the successive pairs are rotatedby an irrationalangle. Hence, whorled and spiral featurescan be combined. This is true in some types of flower development (Silene [Lyndon 1978]). Finally, some flowers have the members of adjacent whorls in line ("superposed") instead of alternating. Here the divergence is zero and no angularsectioning is carried out (Lacroixand Sattler 1988). To limit our analysis, we will be concerned primarily with two classes, the nonspiral "bisectors" and the spiral "golden sectioners." We will consider shoots B. Whorled C. Distichous 360" 137.506 Divergence Angle 2 ,2) L 5* 85..... GoldenSection Bisection Bisection of 850 of 1800 of 3600 Fig.2 The three major types of shoot organ patterns,as seen from above. Leaves are numberedin order of their origin (I is oldest). A, In all apices leaves tend to arise interior to the juxtapositionof left (L) and right (R) extremes of neighboring older leaves. In spiralphyllotaxisthe leaves arise at the smallergolden section of the whole circumference(e.g., anglebetween leaves 2 and 3); the divergenceis 137.5?.Leaves also arise at the smallergolden section betweenadjacentleaves: leaf 7 arises at 0.382 (38%)of the arc between the center of leaf 2 and leaf 4 (dotted arc). This arc is termed the availablecircumference. B, Decussate(whorled)phyllotaxis.Here the arc betweenadjacentleaf centersborderingthe dome (leaf 2) is l80? (dottedarc). The new leaf, 3, arises on the bisector of this "availablecircumference."This gives a divergenceof 90?.In a tricussatepattern this arc would be 1200,the divergence600;tetracussate:900 and 450. C, Distichous phyllotaxis.In this apex the arc bordering the dome has only one leaf center, that of leaf 2. This "available"arc of 3600is bisected by the new leaf, 3. S62 INTERNATIONAL JOURNAL OF PLANT SCIENCES Divergence Angles Abelia 00 360 450 pentamerous flowers (Anagallis) 600 Eucalyptus Hedera 137.50 1800 monocot Vinca, flowers Coleus I (Iris) dichos Helianthus Ribes Zea decussat siral I 900 99.50 Idsi Fig.3 Of the many possibledivergenceangles,only a small numberarecommon in repeatedpatternsin nature.These angles are shown as indentationsin a line to imply that each angleis a local attractorand, hence, self-stable.The threemost common types of phyllotaxisare in boxes. See fig. 2 for diagrams.Samplegenerawhose shoot or flowersconsistentlyshow characteristic angles are given below the line. Above the line are generashowing more than one leaf divergenceon the same plant. where the same developmental activity is repeated many times. 1. Thelimitedvarietyof patterns OBSERVATIONS Unorganizedgroupsof shoot structuresarerare (Schwabe1984). Whenthe regularpatternsfound in shoots are considered in terms of divergence angles, the possibilities appear to be enormous. Between the limits of 00 and 1800there is a near infinity of possible angles. The great majority of observed repetitive patterns,however, have one of a few divergences (fig. 3), 137.5? being most common (Fujita 1942). The question is, are these particularangles common because of the adaptive value of the arrangementof the mature organs, or are they common because only certain patterns can be readily generated on the apical dome? An analogy to the second case is the fact that only certain musical tones are possible on a bugle.Only certainfrequencieswill resonate.The proposal is that a two-dimensional resonance of a developmental sort similarly restrictsthe possibilities for patternproductionat the shoot apex. It is known that the common spiral pattern is well suited for the close packing,without overlap, of objects on meristem surfaces (Ridley 1986). For the presentation of many floral organs in a tight array,as in a sunflowerhead, the spiralpatternhas a demonstrableadvantage.Similarly,this divergence of 137.50 minimizes mutual shading in particularplants, like yucca and agave, where the leaves are swordlikeand are in a rosette (Niklas 1988). Featuresof the spiralpatternscan have adaptive value. The work of Niklas showed equally clearly,however, that the light-capturing advantagesof the spiralpatterndisappearedwhen long internodes were present and the leaves had petioles, etc. Thus, a greatmany plants with vegetative spiral phyllotaxis do not benefit from it in terms of light-gatheringability or in any obvious way. It is also clear that plants with nonspiral vegetative phyllotaxisare successful.Entirefamilies, mint, snapdragon,dogwood, and maple, are all decussate. The bizarre traveler's palm, which is almost a planar plant with its distichous phyllotaxy, survives among typical palms with spiral phyllotaxis.Grassesand many monocots are also distichous. They are clearly competitive. Selectionappearsto act on patternsonly within a specific context. Endress (1987) has made the point that flowers with a great many parts often have the spiral arrangement, while simpler whorled configurationsare associated with small numbersof organs.This latterlends itself to evolutionary specialization among parts, as is striking in orchids. In a similar vein, the distichy of grasses may be particularlycompatible with the mechanicallyeffectiveblade and sheathstructure of these plants. THEORY AND CONCLUSION A reasonableinterpretationof the above is that a regularpatternis advantageous.Some patterns may be exploited for particularfunctions.No particular divergence angle, however, has a broad selective advantage. Selection appears to act on a small set of available patterns.The three main types of pattern all can be shown in maize with the same mutant abphyl(abnormalphyllotaxis) phenotype (Greyson and Watson 1972). This shows that the various members of the set are closely related.Selection does not readilyexplain why the availableset is small or why its members have the divergence angles that they do. This indicates that the patternsreflectdevelopmental constraintsthat "allow"only certainangles.Thus the remainderof this essay will addressprocesses at the shoot apex. What developmental mecha- GREEN-PATTERN FORMATION IN SHOOTS S63 nisms could restrict the origin and propagation of pattern, limiting the "available set"? 2. Theoriginof pattern While pattern typically originates in the embryo, origin will be treated here only as it occurs in the established shoot. OBSERVATIONS-GENERAL It is relativelyrarethat one findsde novo origin of patternin a shoot. An apparentlynew pattern usually has some plausible connection with antecedent structure.For example, lateralbuds initially have decussate symmetry, but that can be tied to the bilateral symmetry of the leaf axil. A study of the origin of fivefold symmetry in the flower of Echeveriashowed that a plausible connection could be made betweenthe cyclic activity of the inflorescence meristem, which had many 900 features, and the sequential spiral development of five sepals (Green 1989). In Silene the whorled pattern in the flower first develops in spiral succession (Lyndon 1978). When pattern does arise de novo in shoots, it apparentlyhas to be whorled, because spiral patterns have a gradient of age. De novo formation of a pattern is seen in the simultaneousappearanceof a whorl of five sepals on the floralprimordiumof Anagallis. The apical dome delimits tissue that bulges to form a round floralprimordium(Greenet al. 1991). It is in the position of a lateral bud. There is no indication of any prepatternin the epidermal cells on the round, clublike structure.Within 24 h a ring of five equally spaced sepals arises (Hernandez et al. 1991). The numberof organsinitiatedcan vary. When the Anagallis primordium is treated with a cellulose synthesisinhibitor,occasionalflowershave regularsymmetry, but it is threefold or fourfold (fig. 4). The inhibitor used was unlikely to affect processes near the genome but, rather, cell wall properties (Rasmussen 1992). The size of the flower primordium was somewhat reduced. Cytokinin can increasethe size of the gynoecium in Aquilegia; carpel number increases from five to seven (Bilderback1972). It thus appearsthat the pattern-generatingprocesscan fit a roughlypreset "wavelength," bulge plus adjacent depression, onto the available annularspace. In other plants, however, cytokinin may act to reduce the wavelength. In tobacco, three ratherthan two carpels appear in a treated gynoecium of normal size (Hicks and Sussex 1970). The breaking up of a smooth annular region into a whole number of humps or appendagesis common in flower development (Sattler 1973). The model for organ identity in flowers has the whorls as annuli that breakup into organs(Meyerowitz et al. 1991). It is interestingthat in the E.!~ 3 ~~~~~~~~~~~~~~~~~~~~ Fig.4 The flowerof Anagallisarvensis.Bars = 100 Mm.A, Normal flower with five outer sepals alternatingwith five stamens. Petals appear only later. The opening of a round gynoeciumis in the center. B, A flower formed after local applicationof a cellulose synthesis inhibitor (2,6-dichlorobenzonitrile).The size of the originalroundprimordiumwas reduced,as was the numberof sepalsand stamens.The lesser numbermayreflectnear-normalbucklingoccurringin a smaller thannormal annularzone. Courtesyof Nicolas Rasmussen. 'pin" mutation of Petunia this does not happen 'fig. 5). The region of the petals, or stamens, is :ollar-like. Only the gynoecium, which is normally radially symmetrical, is unaffected. The mutation appears to eliminate the subdividing process in the outer whorls. A pin mutation in 4rabidopsisappears to involve a flaw in auxin Lransport (Okadaet al. 1991). In snapdragonthe anusualtricussatepatternoriginatesat cotyledon Formation.A mutation of the pallida gene in-reasesthe yield from 3%to 9%.At 150 this rises Lo21% (Harrison 1963). The basis of this sponLaneous,simultaneous breaking of symmetry is -learlyof general importance. OBSERVATIONS -MERISTEM STRUcrURE Patternsare describedin the surfaceplane, and it is likely the mechanism producingthem is also INTERNATIONAL S64 JOURNAL A OF PLANT SCIENCES clinal division of interior tissue is the first step in the initiation of a lateral organ (Esau 1953). This indicates a key role for the interior.In many apices, however, a bulgeappearspriorto any such division (Tiwari and Green 1991). In one of the few studiesdirectedto this point, Selkerand Green (1984) found that key shifts in microtubulealignment and periclinaldivision of the interiortissue did not anticipate bulge formation. Rather, they accompanied, or followed, reorganizationof the epidermal layer. Physiologists have long considered the shoot or leaf epidermis to be rate-limiting for growth (Kutscheraand Briggs 1987). In brief, there are alternatives. The de novo patterning mechanism could reside in the interior, the epidermis being passive. Or the specific positioning mechanismcould be in the surfacecells. Some mechanism operatingin a plane is needed. It is not certain where that plane is. THEORY Fig.5 A, A mutant of petunia informally termed "pin." Bar= 100,um.At an earlystagefourconcentricannularridges are seen. B, Later,unlikethe normalflower,the mutantdoes not producewell-definedorganswithin the outerwhorls.Collar-like structuressurroundthe gynoecium. The four rings may reflectthe four annuli in models for homeotic gene control in flowering(Coen 1991). Mutant stocks provided by CarolynNapoli, Departmentof EnvironmentalHorticulture, University of California,Davis. in this plane or parallelto it. Meristemshave long been characterized,in section, in termsof a tunica of one or more cell layersoverlyinga corpuswhere the cells are less organized(Esau 1953). The distinction is striking in the apex of tomato where only tunica cells stain heavily (ChandraSekhar and Sawhney 1985). At the ultrastructurallevel, the tunica cells have cylindrical microtubule bands, with the axis of the cylinderparallelto the surface (Sakaguchi et al. 1990). Corresponding cellulose reinforcementalignments in the plane of the dome surface correlate precisely with the phyllotactic pattern in Vinca and other plants (Green 1986). In some studies of periclinal chimeras the genotype of the epidermal layer seems important (Stewartet al. 1972; Macrotrigiano1986). In others it has been shown clearly that the genotype of the internal tissues controls the overall phenotype. This has been well documented by Hake and Freeling (1986) for the knotted mutant of maize. It is a common assumption that a peri- 1. DwFusIoN-BASED MECHANISMS. The breaking of symmetry of a flat annulus into a whole number of equivalent subregionscan be accomplished readilyby reaction-diffusiontheory (Harrison 1987, 1992). It is assumed that an activator and an inhibitor molecule diffuse within a prescribedarea. Each of the two moleculartypes has a rate of production,decay, and diffusion. Under conditions where the inhibitor diffusesmore rapidly than the activator, two interactingdifferential equations can lead to a final equilibriumpattern of the two compounds. This pattern subdivides the area. In other words, a chemical concentrationcontour map with a whole number of peaks on it results. By an unspecified mechanism, a peak of activation would lead to the formation of a bulge or organ.This well-tested model could apply in the presentcase involving floral organs (Wardlaw 1968). There are some drawbacks. The appropriatepairs of compounds have not yet been identifiedin any naturalsystem. The diffusion would have to occur through a tissue, as againstin solution wherethe theory is straightforward. Finally, while an explicit coupling to intracellular physical morphogenesis has been proposed (Harrison and Kolar 1988), the construction process generally is divorced from the patterningprocess. MECHANISMS. An alternate 2. BIOPHYsICAL proposal by Hernandez et al. (1991) addresses the origin of sepals in Anagallis. The theory is based on the de novo origin of pattern that can occur mechanically, in the solid state. The spontaneous subdivision of a disk into a whole numberof districtscan be accomplishedby minimal-energy buckling (folding) of the disk. This symmetry-breakingprocess is the response to a mismatch of stresses within the disk. A familiar GREEN-PATTERN S65 FORMATION IN SHOOTS Symmetry Breaking Unstressed Stress Gradient MinimalStrainEnergy Configuration Unstable BisectingLine A C B Fig.6 Origin of patternin a disk (potato chip). The initial flat disk has subunits all of the same size (A). When there is relative shrinkagein the center (B), the flat condition is unstable.The disk has excess rim. The saddle shape (C) is generated spontaneouslybecause it has the least strain energy,and curvatureis most evenly distributed.The disk is bisected twice by the formationof opposingcrests and troughsaroundthe rim. example is the potato chip. Upon cooking, the centershrinksmore than the periphery.This condition is unstable in a plane and, hence, the disk bends into the familiarsaddleshape(fig.6). There are two wavelengths of height around the rim, two alternatinghigh and low regions.Connecting two opposite peaks, or troughs, bisects the disk, breaking the radial symmetry. Minimal-strain energy configurationsdistributethe bending relatively evenly; sharp curvatureis avoided. Configurationsof minimal energyfor plates or sheets, simulating the tunica in this example, are calculated by satisfyingthe von Karman equations (Szilard 1974). Minimal-energybuckling is an alternative to reaction-diffusiontheory for subdividing originally homogeneous regions into a whole number of subunits. In sepal formation in Anagallis, the theory would apply to a collar-likeregion of the primordium(fig. 7). The center line of this strip is postulatedto tend to grow rapidly(have excess length) compared to the margins. It would be longer, if unconstrained. Mechanically, this is equivalent to being under compression. A coleoptile with its ends held between the fixed jaws of a vice would buckle if the jaws were closed or if the coleoptile grew. Hence, in inanimate systems "growth tendency" is simulated by compression. According to Szilard (1974) the center of a strip with a stress imbalance will undulate into hills and depressions, the wavelength (hill plus depression distance) being roughly equal to the width of the strip. The bucklingconcept shareswith reaction-diffusion theory the featurethat small random fluctuations in shape (concentration)are preferentially amplified to give a multipeak final arrangement.The bucklingmechanismdoes not, however, require morphogens, and it directly couples the patterningprocess to the initial constructionprocess. In the plant, the initial bulging would lead to a developmentalcascade,including periclinaldivisions, and more complex morphogenesis. In terms of influencingbuckling, the effect of the cellulose synthesis inhibitor on the Anagallis flower would be mainly to shorten the length of the strip and thereby reduce the number of undulations. An increase in wavelength could ex- 7-G B E ~-7 dome cent (1) (2) dome center (3) crease Fig.7 Linearbucklingin a sheet. A heatingbar (A) causes excess length down a line (B). FollowingSzilard(1974), this generatesalternatehills and depressions.Heat is equivalent to "tendencyto grow"for a tissue. When the firstpatternhas become rigidand the heatingbar is moved, a new undulation forms out of phase with the former(C). This is because the rate of change of curvaturenormal to the bar is minimized by this shift. The descendingside of a hill readilybecomes the descendingside of a depression.Folding the strip into a collar (D) shows the alternationin a whorled pattern.This simulatesthe alternationof sepals and stamensin a whorled patternin Anagallis(fig.4). An expansiongradientwithin the hump(E) can converta gentleborderinto a crease.This gives the hump dorsiventrality.The midpoint of the crease can definea plane of bilateralsymmetry.The creasecan serve as a new boundaryfor the dome. S66 INTERNATIONAL JOURNAL plain how the flo mutation in snapdragoninfluences the axillary meristem to produce the first whorl of two ratherthan five organs(Coen 1991). The effectof the "pin" mutation in Petunia could prevent the development of extra growth along the midline of the annulusand, hence, inhibit the buckling that normally breaks up the circumference into organs. In sum, the de novo origin of pattern can be explainedin terms of two alternatetheories. One establishesself-maintainingchemicalprofilesthat subdivide a tissue, forminga prepattem.The other directly folds the tissue into a whole number of humps when unusualstressesin it are relieved. 3. Propagation of pattern-feed-forward Three levels of scale are appropriateto analyze this process, the most studied aspect of phyllotaxis. At the largestlevel is the symmetry of the overall array.Stem apex cross sections, especially those of distichous and decussateplants,typically fluctuate in shape. The mean outline, however, is approximatelyround. The historicallyprominent second level of scale concernsthe patternof the units within the outline, the leaves or florets. These are often idealized as points or circles.The propertiesof the resultinglattices have been analyzed intensely (Jean 1990). The third level concerns the unit itself. A leaf soon becomes a dorsiventralstructurethat also has a plane of bilateral symmetrydenotedby the midrib.The latterplane is typically on a radius of the overall array.Similarly, floretson a sunflowerhead are at one stage rhomboidal, with the long axis of the rhombus on a radiusof the capitulum.Structureat all three levels is usually coordinated. OF PLANT SCIENCES normal propagationof pattern involves continuous feed-forwardfrom preexisting pattern. In sunflowerthe vegetative phyllotaxis is spiral (divergences 137.5?, 99.50), and the pattern pro- gressesinto higherFibonacci numberpairs as the headdevelops.In the experiments,the bractsform anew at the cuts and are not subject to influence from preexistingpattern.Apparently,an essential chain of feed-forwardof patternhas been broken. The key role of previous structure in pattern propagation, long assumed in theories of phyllotaxis, has been documentedexceptionallyclearly. The patternsseen at level 2 appearto be influenced by large-scalesymmetry at level 1. Large flat spiral arrays are common; comparable flat arrays with orthostichies are rare, apparently. Whorls of high numbers of elements, e.g., leaves in Anacharis(Green 1986), often arise on cylindrical sides of finger-likemeristems. It is thus of special interest that Hernandez (in Green 1991) found a region in a flat capitulumwhere the new florets were arrangedin an orthogonal pattern (like horizontally stacked wine bottles). Other parts of the head had the spiralpattern.The outline of the head, however, was irregular,and the unusual local patternwas at an undulation.This indicates that variation in the geometry at level 1 can influence the type of pattern found at level 2. CLOSE COUPLING OF THE LEVELSIN SMALL SIMPLE VEGETATIVE SYSTEMS.In contrast to the sunflow- er head, where the interactionbetween the three levels is relatively subtle, a close connection in simple forms is obvious by definition. A good example is a tricussate vegetative apex (Nerium OBSERVATIONS oleander,and occasional shoots of Antirrhinum). SEPARABILITY IN LARGESPIRAL The dome boundaryis a trianglelargelydefined OF THELEVELS by straightcreases at the inner base of the three SYSTEMS.Some independence of activity at the appendagesof the node (fig.8). A new organforms three levels is strikingly illustrated by work on Helianthus (Palmer and Marc 1982; Palmer and along the bisector of the arc between appendages. Steer 1985). Florets normally arise in an annular When the resultinghump bulges to form a new generativefront,alonga geneticspiral,to produce crease normal to the radius through the bump, Fibonacci spiral patterns. When straight or cir- this delimits a new triangulardome, rotated by cular cuts are made in the still undifferentiated 600 as observed in Antirrhinum(fig. 9A). Compartof the dome, however, new bractsarisealong pared with the sunflower,the outer limit of the the cut and new florets are produced at ever- dome is reduced from a diffuse annulus to alterincreasingdistances from the bracts (Hernandez nating triangles.When a hump forms inside the and Palmer 1988). This can occur in the centrip- boundary, a geometrical feature of the new appendage soon determines the new dome boundetal direction, as before, or in the opposite direction (outward).The Fibonacci spiral pattern ary. The interconnectionscould hardlybe closer. is lost. The florets appear to "close-pack," but Thesetwo extremesof interactioncorrelatewith thereis no regulararrangement.Large-scalesymtwo modes of propagation(fig. 1). In the largemetry is lost. One can conclude that the gener- scale case, the phyllotaxisis describedas a "movative spiral,well-definedpattern,and radialsyming front." The morphogenetic activity "inmetry of the head are all unnecessaryfor floret vades" relatively inactive tissue and, while the formation. The third level can be independent; character of the previous boundary is regenerits activity is normally coordinated. ated, its absolutedimensions arenot. The activity These important experiments indicate that is thus not completely cyclic. In the small-scale GREEN-PATTERN A. C. B. dome growth IN SHOOTS FORMATION S67 D. leaf new dome demarcation growth N~~~~ Fig.8 Diagramof fully cyclic phyllotaxison a model tricussatedome. Afterthe dome has enlarged(A,B), new creases formnormalto the longestradii(dotted).These delimita new dome in C which then enlarges(D). See fig. 9A. systems (figs. 1, 9A, 9B), thereis a complete cycle. Here the configurationand dimensions of the formative region are regenerated.The dome grows, and thus the cells on it arelargelytransient(Green et al. 1991). This is called "fully cyclic" phyllotaxis. While positioningof the primordiain terms of the golden section is common to both types of cycles, takingthe bisection seems rarein "moving wave examples. Ai * .e tt tinr g CLOSECOUPLINGIN FLOWERS. Inward propa- gation of pattern with a fixed number of organs per whorl is common in flowers.It is noteworthy that in the agamous mutant of Arabidopsisthe organ number in the third whorl remains six despite the change from stamen to petal for the organ type (Yanofsky et al. 1990). Propagation of pattern can persist despite major changes in organ type. THEORY The conditions of feed-forward in pattern propagation for the two classes of pattern are elaborated readily. In the radial patterns (orthostichies) the structurenear the initiation site has a pair of elements equal in size and age as well as in distance from the dome center. Feed-forward from this bilaterally symmetrical pattern gives a bilaterallysymmetricalresult:a new primordium interiorto, and midwaybetween,a pair of older primordia ("bisection"). The original boundaryconfigurationis regenerated. In the spiral patternsthe preexistingboundary structurehas pairs of primordia that differ systematically in size and age as well as in distance from the dome center. From this asymmetrical array comes an asymmetrical response:the new primordium arises at the golden section of the arc between the pair, closer to the older (farther out) member. If the radial positioning follows a consistent plastochronratio,the outer "boundary configuration"is regenerated.I thus concludethat the two classes of patternare likely to be alternate solutions to the single issue of how cyclic feedforward activity, basically local, can regenerate apexof Antirrhinum. Note the relativelystraightcreases(broad arrows)interiorto the leaves.The creasesat theirinnermargin are linked by shelflikeconnectingtissue (long arrows)above which the new leaves will hump. When these leaves become large,they will form creases to delimit a new dome, 600 rotated, as in fig. 7. Note that leaves form near the ends of the threelongestradii on the dome. B, The spiraldome of Ribes. Divergence about 137.50. Primordia are numbered in order of their initiation. A relativelystraightcrease (broadarrow) delimits the innerbase of leaf 2. The dome boundaryis more complex than in A, the creases being of different age and height.Thereis a suggestionof connectingtissue (longarrows) completinga "ring"aroundthe dome. Micrographsby Tanya Bauriedel. the same boundaryconfigurationas before. This feature has been recognized in many modeling efforts on phyllotaxis. The question becomes, What is being fed-forward?There are two major suggestions;both have field properties: 1. INHIBITOR-FHELD.A great many plausible models assume that the recent appendages are sources of an inhibitor for initiation and that, as the formative region expands or is involved in new competent territory,leaves arise at the first available uninhibited site. It is readily seen how successive "spheres of influence" can propagate INTERNATIONAL S68 JOURNAL OF PLANT SCIENCES verted to two opposite humps, increasing stress in the disk leads to a small ridge appearingbetween the humps; then a pair of smaller bumps x x ~~~~~~~~corpus y arises on this ridge.The presenceof a pair of large F .tunica bumps "feeds-forward"into the remaining central region to generatea second pair at 900. This is "taking the bisection" and mimics decussate c X -??~~~ phyllotaxis. Here pressureis applied from below. Directional in-plane compression/tension dic0 tates the orientation of the first pair of humps. y Most significantly,no qualitative change in the A. Thinfilmdelaminationbucklingmodel input is requiredfor the "feed-forward"shift of 900 for the second pair of bumps. The humps presentin the 3-D configurationof a plate or shell thus can greatlyinfluencewhere new humps will BucklingMode Shape |_O ?/ 0 a/b form on smooth parts of the sheet. In meristems this activity could lead to the propagationof pat1st 2nd 3rd 1 -0.3 tern. A developmental progression, originating a fivefold pattern and then propagatingit, can be simulated (fig. 7). The sequenceis similar to that _ ( B. in the early flower development of Anagallis (Hernandezet al. 1991). The undulatingsurface Fig.10 The bucklingof a sheet subjectto in-planeload and is believed to originateby a tendencyof the center hence deformationas well as pressurefrom the interior(q). line of the strip to grow more than the strip's A, Generalsetup in face view and section. B, The sheet is a margins (Szilard 1974). As before, "tendency to in circle that is made to shrink very slightly the vertical digrow" translatesto in-plane compression in the rectionand extendin the horizontaldirection(thelatterstrain is 0.3 of the former).The successive formationof humps is model. This resultsin five alternatinghumps and seen in the threebucklingmodes. The ringsarecontourlines. depressions. Assuming that this topographybeThe singleinitialdome becomesa pairof largeoppositehumps; comes permanent, a lateral movement of the then a small pair forms at 900 to the first. This mimics two "tendencyto grow" line will promote expansion cycles of decussatephyllotaxis.From Chai (1990), with perin an adjacentstrip. Because of the reluctanceto mission. change curvature rapidly, the adjacent strip is expected to develop an undulation out of phase an orthogonal pattern. Symmetry sees to it that with the original. The descending portion of an new organs arise at the bisector between equiv- establishedraised bump will tend to continue its alent primordia.The same assumptionscan gen- negative slope as the growthzone moves, starting erate spiral patterns, approximating a 137.50 dia depressionin the newly expandingregion. Corvergence, and carry out the local golden section respondingly, a depression, with an ascending (Mitchison 1977; Schwabe and Clewer 1984; slope, will promote a hump. Generally, consecChapman 1988). In these models only one dif- utive parallelstrips will have humps and depresfusing compound is required.Reaction-diffusion sions in an alternatingpattern,the key featureof theory can also simulate phyllotaxis (Wardlaw orthogonal "packing." Successive equilibrium 1968; Meinhardt1982). Here an activatoras well configurationsthus can propagatepattern. This as an inhibitor is involved. pattern propagationcould occur in a collar-like Two drawbacksto these models are that the arrangement(fig. 7D). Perhaps in this fashion inhibitor has not been identified and that the ac- the new stamens in Anagallis alternatewith the tual process of producing a primordium is sep- first-formed sepals; petals arise later. In these arated from the patterningprocess. It is a nonmodels the surfaceexpansionis elastic;later simtrivial assumption that appendage formation is ulations will involve viscous yielding. "spontaneousunless inhibited." Another feature is that the new primordiaappearto become new SIMULATION. The computationalchallengeto strong sources of inhibitor instantly, an unlikely simulate feed-forwardof topographyis great. A rectangularexcerptof the entire patternhas to be assumption. used as the startingpoint. It follows that the nat2. BUCKLING. An alternate proposal is made ural boundary conditions, i.e., continuity with here on the basis of the minimal-energybuckling adjacent topography, are not met fully. Nonetheory already applied to the spontaneous gen- theless,one can constructa rectangularpatchwith eration of order. Propagation of pattern occurs two humps on it, apply appropriatestress, and in the inanimate example of Chai (1990) (fig. 10). see what position a subsequentthird bump takes. After the original circular configurationis conThe idea is that the inert plate simulatesthe load0 O__ S69 IN SHOOTS FORMATION GREEN-PATTERN Approximate Approximate Bisection GoldenSection C A 1.0 1.0 1.0 0.9 ....09 - - 0.8 ... .. 0.7 .. . . . .. . .. .. . . .. 0.8 . . . 0.7 . . .. . . . .. . . . ... .0.6 0.6 . . 025 10 ) 0.607- ....... 0.9 -- 0 0 ....... ...... .... 0.5 . .9 ......... ........ ..... .. 0 25 ){<1 __.._.t 048 ..... Y_. .. ..... 03 ..._ 0.3 025 060 ..... 048 .. - ......... 0.5 .. ... _ 0 .7O 006 22o _ .......... 0.3 04 02 k 0 02 B ~. 0.4 0.3 ..... 1 0< 0.4 0.6 0.8 1.0 ~~~~~~~~~~~~~~~~02 0 0.2 0.4 0.6 008 1-0 D Fig.11I Minimal-energybucklingproducinga new bump midway between two equivalent bumps. A, Initial configuration in 3-D. B, Contour map of the above. Note that the pair of bumps is not centeredon the horizontalmidline at 5. C, After stress application,in 3-D. D, Contour map of the above. Without the initial humps, the surfacewould bulge with a summit at X. The presenceof bumpsdisplacesthe new bump so that its center, Y, is somewhataway from the pair and on a line equidistantfrom each bump. This simulates puttinga neworganon a line that is the perpendicularbisector of the arc betweenadjacentprimordia(whorledphyllotaxis). The original bumps had their shape arbitrarilyprescribed. The broaderthird bump has its shape determinedby expansion of the sheet. bearing tunica. This latter can be one or more cell layers thick, or it could be just the very thick outermost wall (Chandra Sekhar and Sawhney 1985). The corpus would supply nonlocalized pressure from below. The pressure from below assures that bumps, not bumps and depressions, are the response. Because the entire rectangular area responds,the new hump is broaderthan the original ones that were prescribed. When the two initial bumps are equal in size and symmetrically placed, the third "induced" bump appearson a line midwaybetweenthe original two (fig. 11).This location is expected on the basis of boundaryconditions and bump position. The pair can influence position even when not centeredon the rectangle.This effecton new bump position is the proposed "bisectioning"activity for the orthogonalclass of patterns.More interesting, when one bump is significantlylargerand also set back, the initial situation resembles that for the spiral category of patterns (fig. 12). The resultis that the thirdhump ariseson a line closer to the largerand more distant hump. The effect is roughly0.6-0.4 and, hence, is nearthe "golden section." This tends to propagatethe pattern. The testing of buckling responses is an involved process. In b-rief,the initial surfaceis not characterizedby X, Y, Z coordinatesbut, rather, 0l 00. B 0. 4 0.6 0.8 10 0. 0 0.4 06 08 10 D Fig.12 Minimal-energybucklingproducinga new bump at close to the goldensection(0.6 8-0.382) betweentwo nonequivalent bumps. A, Initial configurationin 3-D. B, Contour map of the above. Note that the broaderand tallerbump is fartherfrom the right marginof the square.C, After stress application,in 3-D. D, Contourmap of the above. Without initial bumpsthe new hump would have its centerat X. Presence of the bumps displacesthe center to Y, which is away from the bumps and on a line closer to the older and larger bump. This roughlysimulatesputtinga new primordiumon a line that cuts the separationbetween two nonequivalent primordiaby the golden section. The section here is 0.6-0.4 Note that the new hump shape is influenced by the right margin.Shape changeis influencedby preexistingcurvature and by boundaries. uted along a grid of lines covering the area (fig. 11). The sinusoidalterms for each line can cancel or reinforce, to give humps or flat regions. The topographyis thus translatedinto the Fourierdomain. The solution to find the minimal energy configurationinvolves fourth derivatives, relatively easily dealt with in the Fourierformat,and is sought by successive approximation.Buckling is said to occur when a small increment in pressure yields an unusually large change in shape. Once the buckling pattern is observed, the new Fouriercoefficientsare used to constructthe new topography.The general mathematics is in Szilard (1974). The above deals with feed-forwardthroughthe curvature of preexisting bumps and is general. When, in simple apices, adjacentbumps also develop adaxial creases, a corner of the new apex is generated. Hence a leaf may have two stages of influence on the dome: first as a bump within an old boundary, and then by providing a new boundary segment. Both effects appear to promote bulgingat the same site. Both can contribute to the propagationof patternby the principlesof minimal energy buckling. 4. Symmetry links This section furtherdevelops the fact that the three levels of symmetry, apex outline, the pat- INTERNATIONAL S70 - JOURNAL dome, and makes a creasetangentialto the dome centerarea.These creasesand those seen forming on tricussate and spiral domes (figs. 8, 9B, 13) are relatively straight, i.e., are less curved than the outline of the dome. The creases extend primarily by involving new cells at the end of the crease. Points along a crease separate relatively slowly. The creases thus appearto be stable and stiff. The creases are linked by short regions of linear "connectingtissue" that are stable also (fig. 9A). Particularlyclearlyin whorled forms (fig. 8), the crease pattern serves to link the three levels of scale. It outlines the boundary of the apical dome. It positions the adaxial face of each appendage in the pattern. The midpoint of crease, plus the dome center,definesthe plane of bilateral symmetry of the appendage. ~~~~~~~- p -; OF PLANT SCIENCES THEORY .-r L Fig.13 Consecutiveimagesof the same apex of Vincamajor. Bar I100 Mm.Longarrowsshow the same characteristic anticlinal wall junction points in both images. A, Elongate ellipticaldome boundedby leaf creasesand, at left and right, by shelflikeconnectingtissue (arrowheads).Betweenthis region and the center, the dome shows a "crown,"center of curvatureinside the plant.B, A leaf has arisen2 d laterabove each connecting tissue region and formed a straightcrease (open arrowheads).The crease has its center of curvature outside the plant. The growthand bucklingdelimits the new dome. There is relativelylittle activity nearthe dome center. Micrographs by Tanya Bauriedel. tern of elements, and the element itself, can be strongly interconnectedin simple systems. This problem has been noted before (Rutishauserand Sattler 1985) but has been little analyzed. OBSERVATIONS The importance of creases has been pointed out (fig. 8). An opportunityto observe the origin and propertiesof creases came in scanning electron microscope studies of Anagallis flower primordia (Green et al. 1991). A replica and cast method allows individual flowerprimordiato be observed over time. Leaf formation on the vegetative apex follows increasedgrowthrateat the dome periphery.This is documented also for Vinca (Williams 199 1). The dome marcin arches up, overtopping the Creasesarise on peripheralregionsof the dome which have swollen to give the area a slight "crown"(fig. 13A). The region is ridgelike,along a radius of the dome. The axis of the new crease is at 900 to that of the subtle ridge on which it forms (fig. 13B). The sense of curvature in the crease is reversed.It is a furrow,not a ridge. This formation of a transverse furrow on a ridge is likely to be a minimal-energyprocess also. There is an everyday equivalent. An extendable carpenter's ruler has a crown normal to its length, keepingit stiff.If the ruleris bent so as to "break" the previous crown, a new one forms at the kink, with its axis at 900 to the former and with the sense of curvaturereversed. In the ruler there is no alternate explanation to having the tape assume a mechanically more stable (minimal-energy)configuration.It is reasonableto suggestthat crease formation in apices is a related phenomenon (fig. 13). The role of creases at the three levels is more difficult to consider in the simple spiral forms because the delineation of the dome is less obvious. It is a slow helix. Nonetheless, it is noteworthy that the bases of leaves in spiral forms not only face the dome center but are relatively straight(fig.9B) (see also Marcand Hackett 1991). Thus a modificationof the above mechanismmay apply. In complex apices with many organsin a whorl, or high numbers of spirals, the dome typically is not bounded by well-definedcreases. In sunflower the inner boundaryis always very subtle, comprising the first gentle bumps at the inner ends of the various parastichies (fig. 1). Curiously, however,the rhomboidalyoungprimordiumsoon develops a transversecrease, normal to a radius from the dome center. This crease separatesthe bract (peripheral)from the floret proper(central, toward the dome). In the nearly cylindricalapex GREEN-PATTERN FORMATION of Anacharis,the basal margin of the dome also is not abrupt(Green 1986). Sinusoidalundulating surfaces of the young leaves become ever more subtle as the dome is approached. Only in the more mature leaves is the inner sharpcrease evident. Judgingfrom the sunflowerhead, it is perhaps significant that, in forms lacking crease boundaries, activity at the three levels is more separable.Creasesmay contributeto stability. To sum up, extreme buckling, i.e., crease formation, enables one to tie togetherthe symmetry featuresat all three levels of scale. This connection is more obvious in simpler whorled apices with small numbersof organs,but partsof it seem to be general. 5. Stability-patternsas localattractors OBSERVATIONS There are several categories of pattern variation within shoot development. Least dramatic is variation within a class. Plants with spiralpatterns can change their position within the Fibonacci series merely by changingthe relative size of the unit comparedwith the size of the apex as a whole. These are continuous transitions (Meicenheimer and Zagorska-Marek 1989). Such changes can be induced chemically (Schwabe 1971;Maksymowychand Erickson1977).Among whorled plants, individuals showing tricussate ratherthan the typicaldecussatepatternare common. A singleAbeliashrubcan show a very stable two, three, four, five, or six leaves per whorl, the number being consistent on a given branch, but always revertingto two in laterals. In the above variations the propertyof taking the golden section, or bisection, is preserved. In a famous experiment,Snow and Snow (1935) made a shallow diagonal cut on a decussate apex of Epilobium. A new "triangular"boundary of each half apex was established. This correlated with the transient spiral production of leaves in each. This is a discontinuous transition. Ultimately, the decussate pattern was restored. There can thus be remarkablepersistenceof a given pattern. Natural shifts from one class to another are well known. Ivy progresses from distichous to spiralupon maturity.This changecan be reversed by gibberellicacid (Marcand Hackett 1991). Eucalyptusglobulus changes from decussate to spiral. Comparableshifts among stable alternatives transformthe symmetry of the vegetative shoot to that of the inflorescenceand/or flowerin many plants as discussedabove. A shift from decussate to spiralin Epilobiumcan be induced chemically (Meicenheimer 1981). Mathematically,all patterns can be shown to intergrade(Jean 1988). In practicehowever, only selected patternscycle repeatedly.The intergrading patternswould appear to exist only as tran- S71 IN SHOOTS sients. The repeatingpatterns thus act as "local attractors." In contrastto the regulateddevelopment above standsthe responseof the sunflowerhead to cuts: there is no resumptionof pattern.It appearsthat the large systems showing moving front phyllotaxis have less self-correctingcapacity than the small, fully cyclic, forms. The subject of stability will be addressedonly in two aspects. What mechanical feature could help explain the usual high stability of the small and fully cyclic forms?What kind of formal analysis can deal explicitly with stability and "attractor"propertiesin phyllotactic systems? THEORY STABILITY OF FULLY CYCLIC FORMS. The property of stability is easiest to consider in the tricussate and tetracussateforms where the dome is polygonal and its outline shifts by only 600 or 450 per cycle. The tricussate apex is approxi- mately an equilateraltriangle (fig. 8A). Assume that new leaves always appearsimultaneously.If the development could also assurethat leaf creases at a node would act as stiff "bars"and always attain the same length (even after perturbation), the configurationwould be stable.One equilateral trianglewould be built inside another, the rotation being 600. In Abelia the squaremeristem is apparentlyas stable as the triangular(personal observation). Here, angles could vary widely despite constancy of side length. A plausible suggestion is that in both of these polygonalwhorledapices the leaf bases, and short intervening connecting tissue, act as stiff rods and form a ring. This ring would be kept taut by the tendencyof the internaltissues to expand laterally. A taut ring of four equal membersis a square.Thus the same internalpressure consideredto be involved in bucklingcould contributeto stabilityat the largestlevel of scale. This "taut ring" hypothesis might also work in simple spiral forms even though the ring of leaf creases is a slow helix. Perhapsconnectingtissue makes the dome periphery act as a functional ring,despite the spiralnatureof the dome boundary (fig. 9B). Such postulated taut rings could not exist in the largespiralforms such as the sunflowerhead. The periphery of the dome is not bounded by obvious creasesand connectingtissue. This could explain why there appears to be little restoring activity once the geometry of the head is distributed. The cutting experiments indicate that pattern,at level 2, is not self-restoring.The regularity in many capitula could thus reflectmere continuation of the previous vegetative pattern,where the ring stability was possible. The established pattern would be fed forwardwith fidelity only as long as geometry at level 1 remained normal. S72 INTERNATIONAL JOURNAL FORMAL ANALYSIS OF STABILITY. The taut ring concept may give stability at level 1. Minimalenergybucklingis a plausiblecomponent to contributeto stabilityat level 2. These are qualitative suggestions. How could such ideas be involved in a formal treatment of self-stability? Self-regulationin generalinvolves a balanceof two opposing processes that normally are equal and therebymaintaina constantvalue of the controlled variable.That value is an "attractor."An example is the level of water in a tank that is being filled at constant flow and simultaneously is losing waterthrougha hole in the bottom. Input rate is constant; output rate is a function of the height of the water. Departuresfrom the equilibrium level are corrected.The appropriatedifferential equation is dh/dt = A - Bh, where h is height, A is the inflow rate, and B a constant linkingoutflowto height.At equilibriumh = A/B, the attractor value. This explains stability of a scalar,h. How can such self-stabilizingprinciples be extended to phyllotacticsystems that are twodimensional, are cyclic, and show rotation? A major step in this direction was made by Ungar (in Green 1991) for a very simple model of an apical dome. The dome is a triangle that alternatelygrows by having one side extend and then demarcatesa leaf by drawinga new line from a particularvertex of the triangle,intersectingthe opposite side. The small triangle is the leaf; the remaininglarge triangleis the new dome for the next cycle. A new vertex, made at the previous intersection,is used for the next cycle. The system is fully cyclic; the components in it are transient because a given line borderingthe dome lasts at most three cycles. The phyllotaxis is spiral. The model addressesthe problemof how a cyclic "engine" (same configurationis repeated)can be stable despitethe transientnatureof the components in it. A remarkablefact is that if the initial triangle is a golden triangle (sides 1, 1, 0.618), it will maintainits shape, size, and rotationthroughany number of cycles. It will produce "leaves" with a divergence of 1440 indefinitely, provided several condition-sare met. The intersection to demarcatethe new leaf must "cut"the opposite side by a certainfraction,n, and the relativeextension of the line that grows must be I/n. In the present example n must equal 0.618, the larger"golden section." The golden ratio thus appears in the structureitself and in two operations needed to make it in cyclic fashion. What is remarkableis that this algorithm, when repeatedly applied to an initial triangle of any shape, will gradually produce the golden triangle.Thus the golden triangle, and the divergence of 1440, acts as an attractor (a universal one) when this algorithm is applied. Ungar translatedthis pencil and paperscheme OF PLANT SCIENCES into matrix algebra.The triangleis a vector. The operations are put in a matrix. The term n appears in the matrix. The term n appears also in the Eigenvector, the stabilized entity (dome shape). In this way one can prove the stability of shape and size: there is an Eigenvectorwith an Eigenvalue of 1.0. This appears to be the first time that the golden proportionof a cyclingstructure has been linked to an essential golden proportion in the stable process generatingit. Thus self-similarityof geometryhas been linkedto selfstabilityof activity. Stabilityfeaturesmay explain the prevalence of self-similar structuresand the golden ratio in plants. It has not yet been possible to generalizein this format. If more than one side grows, or if n is not 0.618, the special propertiesare lost. In the latter case the divergenceis kept, but size goes to zero or infinity. Goodall (1991) has published a general analysis of such "cut-grow"algorithms. Less restrictedmodels can perpetuateany divergence angle. There is thus a largegap between the formal modeling and the desired end point: explaining why the two main classes of phyllotaxis can show "local attractor"properties at several divergences. Nonetheless, an appropriatemathematical formatis availablethat might show that "sectioning"ratios of 0.618 or 0.5 are essential to stability.Futurestudy of apices shouldprovide biological/physical meaning to the terms in the matrix. The presentanalysis of five issues can be summarized. The divergences favored in nature are probablyselected becausethey are stable, not because the final organarrangementis highly adaptive. The patterns can originate by a minimalenergy buckling process, an inherently reliable mechanism. Patterncan be propagatedby a local positioning maneuver that places the new primordium at the bisection or golden section of the arc between primordianear the site. This can be done by the same minimal energy process. The link between the overall symmetry of the array and the bilateralityof the new primordium can be embodied in a tangentialcrease. It forms as a minimal energy phenomenon. If these creases form a polygon and serve as part of a taut ring scheme, they will contribute to stability at the largest level of scale. Thus stability at all levels of scale can be provided by schemes involving minimal energy. Stability, while producingnew structurein a pattern,is consideredto be the key feature in phyllotaxis. Discussion The mechanism for patterningin shoots generally has been assumed to reside in the interior cells, the tunicarespondingpassively(e.g., Clowes 1961). The best-known candidate for a mechanism is reaction-diffusion,or inhibitor-diffusion, GREEN-PATTERN FORMATION which could operate in any plane parallelto the surface. At chemically unique sites there would ensue a localized upwardthrust,possibly accompaniedby a periclinaldivision. This would be the first physical response to a chemical patterning mechanism. The pattern of inhibitors and oriented division has not been demonstrated;the mechanism for coupling concentrationprofile to directional physical bulging has not been specified. The main proposal here differs in two ways. The patterningprocess is thoughtto be primarily in the tunica. The process may be biophysical. The argument is based on the observation that inanimate sheets can generate order spontaneously and can propagatea pattern with the angular shifts characteristicof phyllotaxis (fig. 7). In these simulations there is no possibility of localized upwardthrustbecausethe influencefrom below is uniform pressure. This is also true, of course, in the organized cyclic whorl formation in the unicellularalgaAcetabularia.Hence, a key role for the surface may be general. Historically,the natureof feed-forwardof pattern has had many proposals(see Schwabe[1984] for referencesfor this paragraph).Almost from the beginninga causal role for the generativespiral itself as a self-extendingperiodic function was considered unlikely. In the sunflower head this would requireaction at a greatdistance from the last-formed primordium, over a short period of time. Church(1904) proposed, more reasonably, that each parastichywas a self-extendingperiodic line function and that new primordia formed where these functions intersected. Curiously, Plantefol selected only certain parastichiesto be key linear periodic functions, somehow coordinated at the apex. A shift to consideringthe feedforward as a two-dimensional process, not directly involving lines, started with Hofmeister and was taken up by Snow and Snow. They postulated feed-forward from the complex boundary of the dome, a primordiumarising in the "first available space." Space per se did not suggestmechanism. Othersproposedthat the effects of a nearby organ related not to its boundariesbut ratherto its whole structure,particularly its center, as a source of inhibitor or morphogen. Such models can propagatepattern readily. In this context the present proposal is also a two-dimensional model, but one where the body of an established primordium, as well as its boundary,feeds forwardonto the dome. The curvature of a young hump can influence the minimal-energyconfigurationof the dome of which it is a part. Also, as with the model of Snow and Snow, the boundary of the primordium is important. Once a crease has formed, this acts as partof the anchoringboundaryfor the next buckling of the dome. These two influences act in IN SHOOTS S73 sequence.Whenthe primordiumis small, its general curvature would be included in the dome configuration.When large, its adaxial tangential crease would form the new dome boundary. In whorled apices these influences would alternate in time. In spiral apices both would always be present. The proposalcombines the perspectives of Schuepp (1914), who considered folding important, and Snow and Snow, who emphasized boundaries.It adds the idea that cyclic minimalenergy configurationsare seen on the tunica. An emphasis on feed-forward,boundary configurations,and facilitated buckling has characterized previous analyses by Green ([1985]; well reviewed in Lyndon [1990]), but the minimalenergyconceptwas not present.The majorparameters were cellulose alignment and a postulated feed-forwardstretchingaction by the appendages on the dome (Green 1989). Such stretchingappears to be significant in the subsequent establishment of new hoop reinforcementby cellulose in appendages(for stamens, see Hernandezet al. [1991]), but it seems unessential to pattern formation as such. The reasons for the shift in emphasis to simple plate behavior are (1) in the sunflowerhead there is no prominent stretch in advanceof primordiumformation,(2) in the simulations here which can propagatepattern there is no requirementfor directionalpropertiesof the plate, and (3) the previous alignment-basedtheory did not address the mechanism for self-stability. Cellulose alignment and stretch phenomena do occur on the dome and are viewed as essential for extending the shoot axis and establishing new elongation directions in laterals. To explain pattern per se, however, the present mechanism based on the simpler assumptions of minimal energy is more general. The minimal-energyconcept in buckling implies a maximization of entropy.Jean (1990) has put forward a theory that Fibonacci sequences can be evaluated in terms of their information content and, hence, entropy. The more commonly found spiralsequenceshave lowerentropy accordingto his definition. Jean's entropy refers to progressionsthat the lattices of leaf positions could take as Fibonacci patterns reach higher numbers. Energyof buckling,on the other hand, refers to the process producing individual primordia. The relation between the two applications of thermodynamicsis not yet clear. Bucklingphenomenologyat the tissue level apparently can account for much about the local activity and its coordination in surface tissue. Coordinationwithin interior tissues by a tensor has been proposed by Hejnowicz for root histology (see Green and Selker 1991). Both his and my proposal support the contention of Kaplan and Hagemann (1991) that the explanation of plant pattern involves principles not evident in S74 INTERNATIONAL JOURNAL cell theory and, hence, transcendingit. It is not necessary to envision an opposition of cellular versus organismal mechanisms. One can considera causalloop. Organismalfeatures,e.g., stress fields, bring on humps in a tissue and also cause discrete changes, such as 900 shifts in reinforcement direction, that can only take place in cells (Green 1984; Selker 1990). The cellularactivities then integrateto organizenew large-scalefeatures of the appendage,i.e., give it its hoop reinforcement. The new organ is then involved, as a boundarycondition, in the next set of stressfields. Intra-and supracellularprocessesare both in the loop. The present buckling mechanism is proposed to be an important aspect of plant shoot morphogenesis.The main structuralcomponents are humps and creases. 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