J. theor. Biol. (1998) 193, 321–334Article No. jt980704

0022–5193/98/140321+14 $30.00/0 7 1998 Academic Press

A Morphogenetic Model Accounting for Pollen Aperture Pattern inFlowering Plants

A R,* B G,* A M† P H G*

* Laboratoire Evolution et systematique, Universite Paris-Sud, 91405 Orsay Cedex, France and† Laboratoire de Paleoenvironnements et Palynologie, ISEM-CC61-Universite Montpellier II,

34095 Montpellier Cedex 5, France

(Received on 4 August 1997, Accepted in revised form on 9 March 1998)

Pollen grains are embedded in an extremely resistant wall. Apertures are well defined places where thepollen wall is reduced or absent that permit pollen tube germination. Pollen grains are produced bymeiosis and aperture number definition appears to be linked with the partition that follows meiosis andleads to the formation of a tetrad of four haploid microspores. In dicotyledonous plants, meiosis issimultaneous which means that cytokinesis occurs once the two nuclear divisions are completed. Asyncitium with the four nuclei stemming from meiosis is formed and cytokinesis isolates simultaneouslythe four products of meiosis. We propose a theoretical morphogenetic model which takes into accountpart of the features of the ontogeny of the pollen grains. The nuclei are considered as attractors actingupon a morphogenetic substance distributed within the cytoplasm of the dividing cell. This leads to apartition of the volume of the cell in four domains that is similar to the observations of cytokinesisin the studied species. The most widespread pattern of aperture distribution in dicotyledonous plants(three apertures equidistributed on the pollen grain equator) can be explained by bipolar interactionsbetween nuclei stemming from the second meiotic division, and observed variations on these patternsby disturbances of these interactions. In numerous plant species, several pollen grains differing inaperture number are produced by a single individual. The distribution of the different morphs withintetrads indicates that the four daughter cells can have different aperture number. The model providesan explanation for the duplication of one of the apertures of a three-aperturate pollen grain leadingto a four-aperturate one and in parallel it gives an explanation for how heterogeneous tetrads can beformed.

7 1998 Academic Press

Introduction

Pollen grains are produced by flowering plants toconvey their male gametes from the anthers to areceptive stigma. Once arrived, pollen grains germi-nate and elongate a pollen tube that transports thegametes through the stylar tissue to the ovules. Pollengrains are designed to complete the plant life cycle:they are constituted of two or three cells surroundedby a complex multi-layered wall. The outer part ofthis wall, the exine, is extremely resistant, imperme-able and imputrescible. In one or several well definedplaces of this wall, the apertures, exine is reduced or

absent. Apertures function as opening. The pollentube germinates through an aperture and exchange ofwater and solutes between the surrounding mediumand the pollen grain are mediated by apertures. Inaddition, in numerous species, when the exine is notflexible, apertures prevent bursting of the pollen wallby accommodating pollen volume variation. Aper-tures are thus involved in all the different functionspollen grains must perform to succeed in fertilization.Several different studies give insights into the meaningof aperture number (Dajoz et al., 1991, 1993), anddistribution on the pollen surface (Bolick, 1981;Tanikaihimoni, 1986). However, the mechanisms by

Callose

Two nucleardivisions

1 2 3 4

mmc cytoplasm

Nucleus

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which aperture number and distribution are definedare still unknown.

Pollen grains exhibit a large range of variation foraperture patterns in Angiosperms (Walker & Doyle,1975). In the present study we will focus onnon-magnoliid dicotyledons. In these species, themost common pollen grain morphology is tricolpate(three furrows uniformly placed orthogonally to theequatorial plane of the grain). Different patternsderived from the tricolpate type exist, ranging frominaperturate pollen up to more than 75 poresuniformly distributed on the pollen surface. Thedistribution of pollen apertures on the pollen grainsurface has fascinated generations of scientists, bothin biology and mathematics. This presents the stillunsolved problem formulated by the botanistTammes (1930), about the distribution of pores onpollen grains: how to set out on a sphere a definednumber of points minimizing the least distancebetween any pairs of points (this question lies nowwith mathematicians). Pores on pollen grains withlarge numbers of apertures appear to follow this rule.

Most pollen grains, however, have only one or afew apertures usually placed at highly conservedpositions on the pollen grain surface. Aperturenumber variation exists among species but also withinspecies: in 30% of species several morphs of pollengrains differing by their aperture number areproduced within a single plant (pollen heteromor-phism: Dajoz, 1990; Mignot et al., 1994). Moreover,in most of the non-heteromorphic plants, in additionto a single majority type, less than 1% of pollen grainswith different aperture patterns are recorded(Pozhidaiev, 1993). Thus, aperture number appears asa quite labile character.

In ‘‘higher’’ plants, microsporogenesis can besummarized in the following way: mmc (microsporesmother cells) first enclose themselves in a thickcallosic wall (Fig. 1). Then meiosis is achievedproducing a tetrad of four microspores. Microsporeswill then separate and become mature pollen grains.The function of callose in microsporogenesis is not

totally elucidated, but it is considered to act as amolecular filter and thus isolates meiocytes andmicrospores from one another and from the rest ofthe plant (Barnes & Blackmore, 1986). In the studiedspecies, meiosis is simultaneous: the partition of thecytoplasm occurs once karyokinesis is completed.During the tetrad stage, microspores build up aprimary wall and apertures are visible before the endof the tetrad stage. Aperture definition appears thento be concomitant with meiosis. The location ofapertures within the tetrad shows in addition that alimited number of patterns is observed, documentinga close link between the events of meiosis andaperture definition.

The processes by which aperture number is definedare still unknown but several different argumentsshow that the ontogeny of the pollen wall is at leastlargely, and perhaps totally, controlled by thesporophyte: in heterostylic species of Primula, theornamentation of the pollen wall is determined by thegenotype of the plant, and in Helleborus foetidus,enucleate microspores build up a normal wall (Echlin& Godwin, 1968). In heteromorphic species like Violaarvensis and Nicotiana tabacum, which are autoga-mous, the production of several pollen morphs withina single plant is stable over several generations. Thecoexistence of several pollen grains in these casesappears as a cellular differentiation process occurringduring meiosis. The variation belongs to a defined setof patterns showing strong correlation with thegeometry of the tetrad. This geometry results from thepartition of the former mmc into four daughter cells.During this stage the meiocytes are supposed to beisolated within the callosic wall. From this it is naturalto deduce, as did Wodehouse (1935) that themicro-environment within the tetrad may be respon-sible for variation of aperture number.

We developed a theoretical model aimed atunderstanding what kind of spatial relationships andwhat modifications of these relationships couldexplain both aperture position inside the tetrad andmodifications of aperture number. Differentiation

F. 1. Simultaneous meiosis. 1. The mmc are enclosed in a thick callosic wall. The two karyological divisions take place without partitionof the cytoplasm. The two axes of the second division are orthogonal. 2. This leads to a tetrahedric configuration of the four nuclei thatshare the same cytoplasm. 3. Cytokinesis occurs simultaneously with the production by each future microspore of its own callosic wall.4. The newly formed microspores are retained briefly within the callose. Apertures become visible during this stage.

a.1 a.2

b.1 b.2

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F. 2. Schematic representation of a homogeneous tetradcomposed of four microspores with three apertures. a.1 Upperview; b.1 lower view. Apertures are joined by pairs at points ofcontact of the microspores; a.2, b.2, pairs of apertures are placedin the middle of the edge of the tetrahedron formed by the fourdistal poles of the microspores.

In homomorphic species producing 4-aperturatepollen grains (Fig. 3), the 16(=4×4) apertures arejoined by pairs at the corners of the 4-aperturatemicrospores (Huyns, 1968).

In heteromorphic species producing two differentmorphs of pollen grains, all the possible combinationsof the two morphs (the two homogeneous tetrads, theheterogeneous tetrad with two microspores of threeapertures and two microspores of four, and the twotypes of heterogeneous tetrads with three microsporesof one type and one of the other) are represented innon-negligible proportions inside the tetrads of asingle anther (Mignot et al., 1995).

In heteromorphic species, as well as in homomor-phic species, the apertures are generally joined bypairs, except when the total number of aperturesinside the tetrad is odd. In that case, only one apertureis alone and seems to be turned to the outside of thetetrad (Huyns, 1968). The homogeneous tetradsobserved in heteromorphic species show the samelocation of pairs of apertures as found in homomor-phic species. The heterogeneous tetrads correspond toa mixing of the homogeneous tetrads except for thelonely aperture. This supports the idea thatgeometrical constraints during tetrad formation couldbe involved in aperture number and positiondetermination. Wodehouse (1935) pointed out thatthe constraints imposed by the geometry of the tetradare probably responsible for the pattern of aperturedistribution within the tetrad (apertures are placed atthe points of contact between microspores, withrespect to the tetrahedral distribution of themicrospores).

The model must then account for the formation ofhomogeneous and heterogeneous tetrads and for thedistribution of apertures within the tetrad in a waycompatible with the developmental pathway leadingto the formation of microspores.

processes and generation of regular patterns areusually considered to result from complex physicalinteractions among cells (Thompson, 1917) and areusually well explained by diffusion models of amorphogenetic substance (Turing, 1952; Wolpert,1969). The mitotic spindle can be modelled bysimulating the polymerization of polarized macro-molecules in the presence of an electric field createdby the poles of the mitotic spindle being oppositelycharged during meiosis (Meggs, 1990). The modelproposed here looks at the distribution of atheoretical morphogenetic substance under theinfluence of several poles of attraction defined by thesecond meiotic division.

1.

Because the four products of meiosis are kepttogether within the tetrad, and the future aperturesare visible during this tetrad stage, it is possible todeduce from observations what the different patternsare which must be explained by the model. Theposition of apertures within the tetrad is subjected toa strict rule. In homomorphic species producingtricolpate pollen grains, the 12(=4×3) apertures arejoined by pairs in six points (Fisher distribution).More precisely (Fig. 2), pairs of apertures arepositioned at the middle of the edges of thetetrahedron formed by the four distal poles of themicrospores (Wodehouse, 1935).

F. 3. Homogeneous tetrad composed of four microspores withfour apertures. The fourth apertures are created by the duplicationof the aperture placed between ‘‘sister’’ microspores (marked withdots) in tetrads having three-aperturate microspores. In 4-apertu-rate homogeneous tetrads, apertures are also joined by pairs.

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F. 4. The meiocyte is represented by a ball with nocommunication with other objects. Four attractors, represented bypoints denoted 1 to 4, are placed in the ball. The axes going througheach pair of attractors are orthogonal.

M= 4(x, y, z) $ IN3; x2 + y2 + z2 E r25

Inside the ball four attractors are defined:

Ai =(xAi , yAi , zAi ) $ M, i=1, . . . , 4.

Attractors (A1, A2) and (A3, A4), respectively, belongto the same pair and will be considered as brotherattractors. The attractors draw the morphogeneticparticles located at any points P=(xP , yP , zP ) $ M ofthe ball with a strength of 1/dk, k=1 or k=2, whered is the distance between points Ai and P. Theattraction to a point depends on the regionalizationand on the two or three closest attractors to the pointconsidered.

b. The regionalization

Let Di be the domain of the ball defined by thepoints closest to the attractor Ai :

Di = 4P $ M/d(P, Ai )E d(P, Aj ), j$ i5.

It defines the points which will belong to themicrospore containing Ai .

In turn, this domain can be divided into three zonescorresponding to the proximity of the points withinthis area to each of the other attractors:

Dji = 4P $ Di /d(P, Aj )E d(P, Ak ), k$ i, k$ j5.

Within Di , all points are under the influence ofattractor Ai . The nearest Aj , (j$ i ) is supposed toalso be systematically acting on these points.Heterogeneity within the future microspores isintroduced by supposing that a third attractor canalso be acting on a point with respect to the positionof this point inside the domains separated by thesecond meiosis division (i.e. points placed betweenbrother attractors). Five possible regionalizationswere defined, corresponding to five options in themodel (Fig. 5):Case 1: [P $ M, P is under the influence of the twoclosest attractors (no heterogeneity).Case 2: the points belonging to the domains placedbetween brother attractors (D2

1 * D12 and D4

3 * D34 )

are influenced by the three closest attractors, thepoints belonging to the domains placed between thenon-brother attractors are under the influence of thetwo closest. This introduces an heterogeneity betweenthe domains shared by brother attractors and thedomains shared by non-brother attractors.Case 3: the points placed in the domains belonging toone pair of brother attractors (D3

4 * D43 ) are under

the influence of the three closest attractors, the otherpoints of the ball are only under the influence of thetwo closest.

2.

The model determines the number and thedistribution of apertures inside the tetrad according tothe interactions among the microspore poles beforecytokinesis. The meiocyte being strongly isolatedfrom the rest of the plant by a thick callosic wall, itwill be represented as a ball with no connection withother objects. Inside the ball, the two pairs of polesof the second meiotic division will be represented astwo pairs of attractors (Fig. 4). Microspores issuedfrom the same nucleus of second division are usuallycalled ‘‘sisters’’, and the two poles of each secondmeiotic division brother poles. For convenience in thefollowing section, the four attractors are distributedin two pairs, each member of the same pair beingdenoted brother attractors. The axes which gothrough each pair of attractors are orthogonal as arethe two axes of second meiotic division in species withsimultaneous meiosis. The attractors act in thefollowing way: at any point inside the ball, eachattractor draws the units of substance. The strengthand the direction of the attraction depends on thedistance between the point and the attractors actingon this point. It is supposed that in certain regions,not all attractors have a significant action on thediffusion of the morphogenetic substance: forexample, when meiosis consists mostly in bipolarinteractions, one supposes that only the two closestattractors play a role at a given point.

a. Representation of the meiocyte and of the attractors

The meiocyte is represented by all the entire pointsinside the sphere of radius r:

D21

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Case 4: only the points placed in the domainbelonging to one of the member of the pair of brotherattractors (D3

4 ) are under the influence of threeattractors. This introduces heterogeneity between themembers of one of the pairs of brother attractors.Case 5: points placed in the domain shared by onepair of brother attractors (D2

1 * D12 ) are under the

influence of the three closest attractors like the pointsplaced in the domain belonging to one attractor (D3

4 )placed between the member of the other pair ofbrother attractors. The rest of the points areinfluenced by the two closest. This introducesheterogeneity between pairs of attractors andheterogeneity within members of one pair of brotherattractors.

c. Motion

Once the domain of influence of the attractors isdefined, the motion of the morphogenetic particlesare computed according to their position within theball following simple rules of displacement. For allpoints P $ M, the four vectors Vi =(xAi − xP ;

yAi − yP ; zAi − zP ), i=1, . . . 4, and the euclideannorms of these vectors are computed. According tothe choice made among the five possibilities describedabove, two or three vectors of smallest norm areselected. The direction of the motion of the modelledparticles of morphogenetic substance V is given by thesum of each of the selected vectors weighed by thedistance or the square of the distance between thepoint and the attracting points:

V= sn

i=1

Vi

>Vi>k , n=2 or n=3 and k=1 or k=2.

Vectors of motion m=(xm; ym; zm) are obtained atany point of the ball by normalizing the directionvectors:

if V=0 then m=0, if V$ 0 then m=V

>V> .

Then to any point of the ball can be associated a pointof arrival Pa =(xa ; ya ; za ) depending on the vector m.

F. 5. The regionalization. The planes in grey delimit the border of the domains which are closer to a pair of brother attractors thanto any other attractor. In these domains, the attraction on each point depends on the three closest attractors. Points outside the domainsare influenced by the two closest attractors. Case 2: the heterogeneity is applied in the domains shared by the two pairs of brother attractors.In Cases 3–5, the heterogeneity is restricted either to a single pair of attractors (Case 3), to a single attractor (Case 4) or to a pair of brotherattractors and an attractor (Case 5).

. ET AL.326

The motion of the modelled particles of morphogen-etic substance from Pi to Pa is ruled by a threshold s:

if vxm v Q s then xa = xi ,

if vxm v q s and if xm Q 0 then xa = xi −1,

if vxm v q s and if xm q 0 then xa = xi +1.

In the model, each point represents one unit ofvolume. Therefore several particles can accumulateon one point. At each iteration, motion is allowedonly if:

(1) the density of substance at the starting point is notzero,(2) the arrival point is not saturated with morphogen-etic substance (i.e. the density of morphogeneticsubstance at this point is below a threshold).

Starting from a uniform distribution of one unit ofsubstance at each point within the ball we iterate thealgorithm and observe how the particles of substancedistributes itself under the influence of the attractors.Because the simulated process is transient i.e. onlyacts until the first microspore walls are built, the timeavailable for movement of morphogenetic particleswithin the ball is limited. Letting the system iterateuntil all motion of morphogens within the ball hasstopped does not qualitatively change the results ofthe model.

3.

The algorithm described above builds a ‘‘vectorfield’’ inside the ball. In the areas located close to theattractors, a great number of trajectories drawn bythe ‘‘vector field’’ inside the ball join. These points areplaced on lines defining streams. The streamsconstitute a network on which morphogeneticparticles will accumulate and give rise to the figuresshown below. The number and the curvature of thestreams depend on several different parameters,parameters which control the discretization of themodel and also the power used to weight theattraction.

The action of four attractors results in ball spacedivision: when a molecule is submitted to theattraction of two poles, the two other attractors,being in the medium plane, will not influence therelative strength of each of the two poles. A moleculewhich is farther from one attractor than from anotherwill then move toward the closest. Hence theattractors lead to the appearance of several planesdelimiting the border of domains linked to one of theattractors between which no exchange of morphogen-etic particles is possible: the space within the ball isdivided into four domains belonging to one of the

F. 6. The action of four attractors inside the ball leads to theappearance of six half-planes dividing the space of the ball into fourdomains centered on each attractor. The planes are distributed asare the cellular plates during cytokinesis in ‘‘higher’’ dicotyle-donous plants.

four attractors. A domain is constituted of * j$ iDji ,

i=1. . . 4 and j=1, . . . , 4. These four domains aredelimited by six quarter-planes intersecting in thecentre of the ball (Fig. 6). Each of them separates oneof the domains from the three other neighbouringdomains. The particles belonging to one of thesedomains cannot move to another domain but moveonly inside that domain. Whatever the regionaliza-tion, after a number of iterations, the distribution ofparticles of morphogenetic substance gather in fourzones centred on the attractors. From each zoneextends various numbers of ‘‘arms’’ constituted of thestreams defined above. The number of these ‘‘arms’’depends on the regionalization. In most cases theymeet one of the ‘‘arms’’ extending from the zones ofthe other attractors (Figs 7, 8, 9 and 10; summarizedin Table 1). In the following, the points where the‘‘arms’’ meet will be considered as indicating theposition of apertures or pairs of apertures and will becalled conjugation points.

a. The conjugation points

Depending on the regionalization (except for thepoint which is at equal distance from any attractor,denoted O), there is a variable number of points atwhich the motion vector is null. All of these points areplaced on one of the planes cited above. The ‘‘arms’’extending from the poles join at these points andtherefore the conjugation points are the ones havinga null vector of motion.Case 1: six points have a null vector of motion. Thesepoints are equidistant from each member of the sixpairs of attractors. They are arranged at the middleof the edges of the tetrahedron composed by the fourattractors (Fig. 7).

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F. 7. Result obtained when there is no heterogeneity within the ball (Case 1). The theoretical morphogenetic substance gathers in fourzones centred on the attractors. From each of the zones extends three ‘‘arms’’ connecting one attractor to the three others in six conjugationpoints placed at the middle of the axes passing through each pair of attractors.

Case 2: eight points have a null vector of motion. Thetwo supernumerary points appear by the splitting oftwo particular points in Case 1 which are placedbetween brother attractors. The supernumerarypoints are arranged at the points at equal distancefrom the three attractors acting in the respectivedomains. The four other points are the same as thefour points placed between non-brother attractors inCase 1 where no heterogeneity exists in the ball. Theyare arranged at equal distance from the four pairscomposed of non-brother attractors (Fig. 8).Case 3: seven points have a null vector of motion. Thesupernumerary point appears by the splitting of thepoint placed between one of the brother pairs ofattractors involved in the heterogeneity. The otherpoints are the same as in the first case (Fig. 9).Case 4: eight points have a null vector of motion. Thefive points which are placed in the part of the ballwhere there is no heterogeneity are the same as inCase 1. Between the brother attractors within whichthere is an heterogeneity, one (the original point ofCase 1) or two points (obtained by the splitting of thefirst) can have a null vector, depending whether theplane equidistant between the two attractors is

included in one domain or the other but in any case,the absent points are virtually present (Fig. 10).Case 5: nine points have a null vector of motion. Thesix points which are placed outside the area betweenthe members of the brother pair of attractors wherethere is heterogeneity are arranged in the same wayas in Case 2. For the three other points which areplaced in the defined area, these are arranged as inCase 4.

b. The distribution of the particles of morphogeneticsubstance inside the ball

The overall distribution of the morphogeneticsubstance inside the ball is affected by theregionalization used. Especially the number of‘‘arms’’ extending from the zones strongly depends onthe regionalization. Those arms meet in most cases 2by 2 at the points of null vectors of motion quotedabove. According to the regionalization chosen abovewe have the following distribution:Case 1: from each of the four zones extend three‘‘arms’’ connecting one zone to the three other zones.We then have six conjugation points arranged on atetrahedron as described above (Fig. 7).

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Case 2: from each of the four zones extend four‘‘arms’’. Two ‘‘arms’’ connect the zones depending onthe members of the brother pairs of attractorswhereas zones shared by non-brother attractors arelinked by only one ‘‘arm’’. In this case we then haveeight conjugation points (Fig. 8).Case 3: from two of the four zones extend three‘‘arms’’, from the two others, extend four ‘‘arms’’.Two ‘‘arms’’ connect the zones shared by the pair ofbrother attractors influenced by the heterogeneity.The other zones are connected 2 by 2 by one ‘‘arm’’per pair. In this case we then have seven conjugationpoints (Fig. 9).Case 4: from three of the zones extend three ‘‘arms’’,from the other one extend four ‘‘arms’’. Between thezones belonging to the members of the pair ofattractors within which there is an heterogeneity, the‘‘arms’’ do not meet. They end at the points of zerovectors of motion defined above regardless of whetherthey are virtual or not. Elsewhere from the zonesextend arms which meet 2 by 2 in the same points asin Case 1. We then have five points of conjugation

and three ‘‘arms’’ which have no conjugates (Fig. 10).Case 5: from one of the zones extends three ‘‘arms’’,from the three others extend four ‘‘arms’’. We havethe same figure as before (no conjugation pointsbetween the zones belonging to the pairs of attractorsinvolved in the heterogeneity). In the rest of the ball,the zones belonging to the members of a pair ofbrother attractors have two conjugation pointswhereas the zones belonging to non-brother attrac-tors have one. We then have in this case sixconjugation points and three ‘‘arms’’ which have noconjugates.

c. The effect of the parameters

Two different sets of parameters having, respect-ively, quantitative effect and qualitative effectinfluenced the pattern of the streams.

(1) The radius of the ball r, the threshold whichruled the saturation in morphogenetic density and thenumber of iterations realized had quantitativeeffects. They did not affect the trajectories of the

F. 8. Result obtained when heterogeneity is introduced between the domains shared by brother attractors and the domains sharedby non-brother attractors (Case 2). From each of the zones extends four ‘‘arms’’. Two ‘‘arms’’ connect the zones extending from brotherattractors whereas zones shared by non-brother attractors are linked by one ‘‘arm’’. In this case we then have eight conjugation points.

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F. 9. Result obtained when heterogeneity is restricted to a single pair (3, 4) of attractors (Case 3). From the zones centred on attractors1 and 2 extends three ‘‘arms’’, from the zones centred on attractors 3 and 4 extends four ‘‘arms’’. Two ‘‘arms’’ connect the zones extendingfrom attractors 3 and 4. The other zones are connected 2 by 2 by one ‘‘arm’’. In this case we then have seven conjugation points.

morphogenetic particle and therefore acted only onthe number of points which reached the threshold ofsaturation. Modification of these parameters did notaffect the pattern of morphogenetic particle distri-bution. The radius of the ball only increased theresolution of the drawing, by increasing the numberof streams constituting the higher density ofmorphogenetic substance zones.

(2) Two different parameters qualitatively modifiedthe pattern of morphogenetic particle distribution, thepower used to weight the vector in the motion vectorcalculus and the threshold (s) which ruled the pathfrom one point through the other. In both cases, theoverall distribution of highest density points was notaffected (planes, points and number of ‘‘arms’’extending from the zones were not modified). Theprincipal effect of increasing the power (attraction in1/r2 in place of 1/r) is in the distribution of the streamswhich appeared less concentrated around the axespassing through the attractors. This may be explainedby the decrease in the strength of the most distantattractors in the calculus of the coordinates. Then thestreams tended to be more linear. The threshold s is

an arbitrary parameter due to the normalizationcarried out on the direction vectors (denoted v). sdetermined the ease with which the morphogeneticparticles moved from one point to another. Thebending of the streams was affected by variation of s:the higher s, the less the streams bent.

Discussion

In most of the other models of development (forexample: Drosophila, Meinhardt, 1986; Acetabularia,Goodwin & Trainor, 1985), the problem is tounderstand how spatial cues that drive subsequentdifferentiation or morphogenesis arise. In pollen grainontogeny, the spatial cues that are responsible forpollen morphology are a consequence of meiosis andare reflected by the distribution of the nuclei. Thus theproblem here is to understand how these spatial cuesare expressed and modulated to result in cytokinesisand pollen morphology determination. The biologicalelements that are involved in the processes compriseat least the components of the cytokinetic apparatus(nuclei and microtubules). What is unknown is

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whether the nuclei and the microtubules control orjust express the spatial cues. The assumptions of themodel are sufficiently lax to be compatible either witha direct control of the process by microtubules andnuclei or with the intervention of other mechanisms(such as morphogens, electric fields. . .).

The representation of meiotic poles as attractorsprovides an explanation for how cytokinesis andaperture sites can be defined together. Fisherdistribution of apertures appears as a naturalconsequence of bipolar interactions between meioticpoles. In a meiocyte subjected to bipolar interactionsbetween poles, our model shows that six singularpoints with a null vector of motion and highconcentration of morphogenetic substance appearrapidly, in the same position as the pairs of apertures(Fig. 2). The distribution of the points and the overalldistribution of morphogenetic substance inside theball is slightly affected by modification of the differentparameters, confirming the geometrical constraintsruling the distributions. Small modifications of theposition of the attractors do not modify the figuresexcept for the position of conjugation points.

Bipolar interactions between meiotic poles can alsoconsistently account for the meiotic partitioning of

the cytoplasm occurring in species having simul-taneous division and tetrahedric tetrads. The exist-ence of four ‘‘attractors’’ (corresponding to themeiotic poles) induces the appearance of six planesdefining four domains in the ball between which noexchange of particles is allowed. These planes thatdivide the area of the ball into four parts areorganized as the cellular plates observed in tetradsafter simultaneous meiosis (Longly & Waterkeyn,1979). Representing poles as attractors as defined inthe model, therefore can account for the quadriparti-tioning of the meiocyte. Brown & Lemmon (1991)proposed that, in simultaneous meiosis, four cyto-plasmic domains are defined prior to cytokinesis.These domains will correspond to the four daughtermicrospores. Partition of the meiocyte in fourdaughter cells would occur at the boundaries of thesecytoplasmic domains by the positioning of thephragmoplast.

The implication of partitioning processes inaperture site definition is widely suspected in theliterature (Wodehouse, 1935; Blackmore & Barnes,1990). The relationships between pollen polarity andquadripartitioning of the meiocytes on one hand andthe reflection of pollen polarity in aperture settings on

F. 10. Result of simulation obtain when heterogeneity is restricted to a single attractor (4) (Case 3). From one of the zones extendsfour ‘‘arms’’, from the three others extends three ‘‘arms’’. We have the same figure as before (no conjugation points between the zonesbelonging to the pairs of attractors involved in the heterogeneity). In the rest of the ball, we have the same distribution as in the two otherfigures. We have in this case five conjugation points and three ‘‘arms’’ which have no conjugates.

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the other hand suggest that a common mechanismcould be responsible for both (Blackmore & Barnes,1990; Walker & Doyle, 1975). In species displayingsimultaneous division, karyokinesis and cytokinesisare not achieved by the same apparatus. Thecytokinetic apparatus involves in the phragmoplastsand in an array of microtubules based on the nuclearenvelope which acts as an extended microtubuleorganization centre (MTOC). The phragmoplastforms centripetally in the orthogonal planes placed atthe middle of the axes linking each pair of nuclei. Inseveral species of Angiosperms (Catananche caerulea,Blackmore & Barnes, 1988; N. tabacum, Orchids,Brown & Lemmon, 1991) this array comprises sixbipolar spindles connecting each of the four daughternuclei to another one. In the model the attractors aredefined by points. In ‘‘higher’’ dicotyledons, the polesof the daughter cells cannot be reduced to points butthe role of these poles is taken by the daughter cellnuclei, which can occupy up to one-third of thecytoplasm volume of each cell. However replacementof point attractors with small balls of real volumedoes not modify the distribution of the planesdescribed above. This link between cytokinesis andapertures is also confirmed by the classical correlationobserved between meiosis type, successive or simul-taneous, and the monosulcate or tricolpate conditionof the pollen grain.

The model suggests that differences in the action ofthe poles during meiosis can cause variation inaperture number. When introducing a heterogeneityin the zones of influence of the attractors as in Cases2–5, we obtained (when a third attractor acted in thedomain shared by brother attractors) the splitting ofthe conjugation point placed between the brotherattractors into two conjugation points. The distri-bution of the conjugation points agrees with thedistribution of pairs of apertures in tetrads (except forthe distance between the conjugation points).Especially the fourth aperture in species producingfour aperturate pollen grains is obtained by thesplitting of the apertures placed between sistermicrospores (Huyns, 1968). In the model, the fourthconjugation point is also obtained by the splitting ofthe points placed between members of pairs ofbrother attractors. To obtain such splitting of aconjugation point, we introduced tripolar interactionsbetween attractors. Whereas bipolar interactions aredescribed in Angiosperms, no direct evidence fortripolar interactions can be found. However in themoss Funaria, some indications of tripolar interactionduring the partitioning processes of meiosis have beendetected. In this species the partition of the meiocytein four domains occurs before the first karyological

division (Brown & Lemmon, 1990). This productionof lobes occurs in two different phases: first, the twoplastids adopt a tetrahedral configuration and thenthe quadripartition of the meiocyte occurs. Busby &Gunning (1988a,b) showed that the orientation of theplastids present in the cell is driven by tripolarinteractions, each plastid end being linked bymicrotubules to the two ends of the other plastid.Subsequently bipolar interactions reflected by sixbipolar spindles set at the centre of the plastids leadsto the lobing of the meiocyte. This justifies employingtripolar interaction that may disturb the usual bipolarinteractions in this model (Cases 2–5). Tripolarinteractions in the domains shared by brotherattractors lead to the duplication of the conjugationpoint placed between them when there is noheterogeneity in the ball (Case 1). However, simplegeometrical relationships, such as the ones defined inthis model cannot explain the characteristics ofregular 4-aperturate microspores: the conjugationpoints form a trapezium. The longest side links thetwo points placed between non-brother attractors.This side is parallel to the lines which link the twopoints shared by the brother attractors. The two lineswhich connect one of the two points placed betweenbrother attractors to one of the two points placedbetween non-brother attractors are of equal length.This discrepancy between the results of the model andthe observed pattern could be due to the fact that thepoles of the future daughter cells correspond to theentire nuclei and occupy volume rather than beingpoints. Obtaining regular 4-aperturate microsporesseems nevertheless not totally easy for an homomor-phic plant. The rare deviant pollen grains found inmany species show a continuum of unstable deviant4-aperturate mature pollen grains indicating thatformation of irregular microspores may occur quiteoften (Pozhidaiev, 1993; Wodehouse, 1935).

Heterogeneous tetrads composed of three mi-crospores of one type and one of the other possess aunique lonely aperture turned toward the outside ofthe tetrad (Huyns, 1968). Applying a regionalizationwhich breaks the homogeneity between members of abrother pair of attractors leads to the appearance ofthree points with no conjugate. In this event theresults from the model depicting the differentdistributions of morphogenetic substance are not inagreement with the observations of the distribution ofapertures in tetrads in Cases 4 and 5. This could bedue to the fact that the lonely aperture is not createdby interactions between poles within a tetrad but byinteraction between poles belonging to differenttetrads. Then the lonely apertures would be created inthe same way as the others and the fact that they have

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no conjugate explained by the breaking of thecommunication between tetrads once cytokinesis iscompleted. Interaction between poles belonging todifferent tetrads could be obtained by communi-cations between meiocytes. Communications betweenmeiocytes are usually recorded up to the prophase ofthe second meiosis division in some Angiospermspecies and can last longer, eventually up to thesecond telophase of meiosis in others (Blackmore &Barnes, 1988; Whelan, 1974). The cytomictic channelsare then supposed to be severed by callose deposition.However our own observations during meiosis andthe tetrad stage in N. tabacum suggest that somecommunications may persist up to the end ofcytokinesis and could be the cause of the lonelyapertures of the heterogeneous tetrads. The existenceof two different types of 4-aperturate grain in Populus(Huyns, 1968), one regular type and a second morphlooking like a 3-aperturate pollen grain with asupernumerary aperture placed in an unusualposition could also be caused by such a process.

The model described here proposes that, as for thewhole plant, the morphologies of pollen grains, andthe ontogeny of the variation of these morphologiesare tightly linked to cell division. Simultaneouspartitioning following meiosis appears to be able toexplain aperture patterns, and disturbances ofcytokinesis to be able to explain variation in thesepatterns. There are indeed at least three differentontogenetic pathways for producing apertures indicotyledons (Silene pendula; Heslop-Harrisson, 1963;Scorzonera hispanica; Barnes & Blackmore, 1986;Epilobium angustifolium, Rowley, 1975), each of themleading to tricolpate pollen grains. Derived patternsof aperture number also belong to the same set ofvariations within the non-magnoliid dicotyledonousplants. Hopefully, to explain aperture pattern and itsvariations, the phenomenon proposed here, simul-taneous cytokinesis, is present over all the differentspecies of non-magnoliid dicotyledons.

The authors wish to thank J. Shykoff for her comments.An anonymous referee also contributed to improve thispaper. This is a contribution ISEM 98-008 of the Institutdes Sciences de l’Evolution de Montpellier.

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