Annals of Botany 93: 423-434, 2004
© 2004 Annals of Botany Company
Spatially Quantitative Control of the Number of Cotyledons in a Clonal Population of Somatic Embryos of Hybrid Larch Larix x leptoeuropaea
1 Department of Chemistry, University of British Columbia, 2036 Main Mall, Vancouver, BC, Canada V6T 1Z1 and 2 Graduate Centre for Forest Biology, Department of Biology, University of Victoria, PO Box 1700 STN CSC, Victoria, BC, Canada V8W 3N5
* For correspondence. E-mail lionel{at}chem.ubc.ca
Received: 9 June 2003; Returned for revision: 30 September 2003; Accepted: 17 December 2003
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXLITERATURE CITED |
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• Background and Aims Many conifer embryos, both in natural seeds and in clonal populations of somatic embryos, display variability in the number of cotyledons. In hybrid larch, Larix x leptoeuropaea (synonymous with L. x marschlinsii Coaz), such variability has previously been reported in somatic embryos, together with a decrease in the average cotyledon number when benzyladenine (BA) is applied exogenously. Described here is a spatially quantitative study with the aim of throwing some light on the way cotyledon number is determined, and hence the mechanism of cotyledon formation.
• Methods Stock cultures of embryogenic tissue were maintained and later made embryogenically active by standard methods. Development through cotyledon formation was followed by optical microscopy with quantitative measurement of embryo diameter and number of cotyledons. SEMs of representative stages and cotyledon numbers were done for purposes of illustration in this account. Existing mathematics of waveforms on a disc were cast into a form suitable to compare with the quantitative data.
• Key Results The number of cotyledons is linearly related to the diameter of the apical surface of the embryo (which approximates a circular disc) at the time of first appearance of the cotyledon primordia. This linearity is a constant-spacing phenomenon between adjacent primordia. Addition of BA to the medium restricts the range of apical diameters without changing inter-cotyledon spacing. Slope/intercept ratio of the linear plot matches expectation for initiation of cotyledon pattern as a harmonic waveform on a circular disc.
• Conclusions The entire pattern of cotyledon primordia arises as a single entity coordinated by a mechanism with wave-forming properties. This is explicable by diverse mechanisms, especially either mechanical buckling (‘biophysical’) or reaction–diffusion kinetics (‘physicochemical’).
Key words: Larix x leptoeuropaea, larch, cotyledon formation, somatic embryos, pattern formation, spatial quantitation, disc waveforms.
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INTRODUCTION |
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Strategy of the project
We report here a study on quantitative spatial aspects of the initiation of whorled sets of cotyledons on the flat discoid apex of somatic embryos of a hybrid larch Larix x leptoeuropaea. Our strategy builds on work done over two decades in the group of one of the authors (Harrison et al., 1981, 1988, 1997; Harrison and Hillier, 1985) upon formation of whorled sets of hairs on the flattened growing tip of an unusually large unicellular alga, Acetabularia acetabulum (Dasycladales, Chlorophyta). Therefore, we are looking for features over-riding the distinction between unicellular and multicellular apices. We proceed expecting an ultimately affirmative response to the question: ‘When a definitive identification of the whorl-forming event is reached, will it throw light on the development of anything other than the dasyclads?’ And we adopt ‘the pattern formation viewpoint’, which envisages that ‘one must first establish such a morphogenetic sequence by identifying individual pattern-forming events leading to the final morphology.’ (quotations: Dumais and Harrison, 2000)
Our approach, at this stage, is somewhat independent of (but we hope ultimately complementary to) approaches that have led to current controversy (Barton and Poethig, 1993; Kaplan and Cooke, 1997; Bowman and Eshed, 2000) on whether cotyledons arise out of the shoot apical meristem. We look for physicochemical concepts that may form a unifying bridge between patterning mechanisms on multicellular apices with numerous spatially patterned gene activities, and those of a growing tip from which the single nucleus is some centimetres distant.
The conspicuous similarities between the phenomena studied in A. acetabulum and L. x leptoeuropaea are as follows.
1. In both cases, the morphogenetic apex of any one specimen can be observed from day to day or hour to hour, because the unicell is a free-living marine organism and the somatic embryos grow from the surface of a mass of embryogenic tissue and are not concealed within megagametophytes. The morphogenetic events can be observed through the first appearance of whorled organs. Measurements can be made very close in time to the pattern-forming event.
2. In both cases, the entire set of whorled structures arises strictly simultaneously, unlike many later-developing sets of whorled structures in plants. [For L. x leptoeuropaea, we see no sign of a sequential process, or relationship to a spiral, as mentioned by Buchholz (1919).] Thus we count the emergence of all the organ primordia in a whorl as one event.
3. In both cases, the apex flattens just before the new organs appear. This allows measurement of a well-defined apical diameter.
4. In both cases, a larger apex gives rise to a whorl with more organs. This is very far from being a trivial observation. Both quantitative studies have shown that the apex diameter at the time of the whorl-pattern-forming event is very sloppily controlled, but the spacing between adjacent organ primordia is rather precisely controlled at the time of their first morphological appearance. This focuses attention on the ability of a developing organism to set up quantitative scales of distance. How do the physics and chemistry of life achieve this?
5. In both cases, the circle of centres of the organ primordia is inset from the edge of the discoid apex. In terms of the count of events, our inferences from the two cases diverge at this point. For A. acetabulum, two events were postulated to determine the inset and the spacing between adjacent organs. But the patterns of L. x leptoeuropaea fit very well, quantitatively, with still snapshots of the modes of vibration of a circular disc. This is strong circumstantial evidence that both the cotyledon primordia and the flat central region (where the shoot apical meristem will emerge) are at this time parts of a single patterning entity.
Variability in cotyledon number in the conifers
Polycotyly occurs naturally in angiosperms (e.g. species of Psitticanthus, Loranthaceae, have from three to nine cotyledons; Kuijt, 1967) but is far commoner among gymnosperms, in particular conifers; typically the initially circularly symmetric embryo is still rotationally symmetrical when capped by a whorl of cotyledons (Juguet, 2002).
The control of cotyledon number has been variously ascribed to embryo size, genetic heritability, phenotypic variability and gene expression. Although the number of cotyledons per whorl is generally characteristic of a particular taxon, it is not necessarily fixed (Butts and Buchholz, 1940). Embryo size has been observed to vary from year to year, and with it cotyledon number. In fir, Abies procera (Sorensen and Franklin, 1977), and in Cola acuminata (Oladokun, 1982), embryos were larger and had more cotyledons in good rather than poor seed years. In Pinus banksiana, cotyledon number has been shown to be heritable (Briand et al., 1998). Phenotypic variation can be brought about by altering hormone metabolism. In dicotyledonous plants, polycotyly is found in both pin1, a mutant affecting auxin transport (Aida et al., 2002), and amp1, a mutant that produces six times the normal endogenous cytokinin level (Chaudhury et al., 1993). Somatic embryo development is regulated by timed applications of exogenous plant growth regulating substances (PGRs). Somatic embryos show far greater variability in the number of cotyledons than their zygotic counterparts, and the variability can be controlled by PGRs, e.g. in vitro, by exogenous application of anti-auxin 2,3,5-triiodobenzoic zcid (Choi et al., 1997) and benzyladenine (von Aderkas, 2002). Genetic analysis of lateral organ formation during embryogenesis has been restricted to a few model dicotyledons. Polycotyly is found in some mutants, e.g. fass (Torres-Ruiz and Jürgens, 1994) and pin1 (Aida et al., 2002). Other mutations affect both the number of cotyledons initiated and the rate of leaf initiation, e.g. amp1, xtc (Conway and Poethig, 1997), häuptling and cop2 (Lehman et al., 1996). To date no cotyledon-specific genes have been isolated, reinforcing the view that events at the apex of the developing embryo involve progressive regulation and interplay of various sets of genes (Aida et al., 1997). Lateral organ formation, from cotyledons to leaf primordia, is thought to be coordinated by spatial and temporal gene expression of WUS, ANT, CUCL1, CUCL2, STM, PIN1 and CLAV (for review, see Bowman and Eshed, 2000). This model is not without controversy (Kaplan and Cooke, 1997). This centres on the timing of initiation of a shoot apical meristem capable of initiating leaves, and involves the question as to whether leaves and cotyledons are homologous. Our material and experimental design are not suited to addressing these questions. We focus on embryo size and cotyledon number to identify and characterize pattern-forming events.
In conifer embryos, the events preceding the appearance of cotyledons are as follows. The globular embryo grows so as to flatten its apex to a circular disc. This stage has been called ‘shoulders’ (Spurr, 1949) or ‘ring meristem’ (Juguet, 2002; von Aderkas, 2002). (We avoid ‘shoulders’ because it relates to a side view or longitudinal section. The important perspective for our study is the plan view of the disc.) Cotyledon initials arise from this flat disc (Spurr, 1949). There has been some discussion as to whether final numbers of cotyledons have been affected by fusion or splitting of cotyledon initials (Buchholz, 1919; Juguet, 1992). These studies were not of initial spacing and numbers of cotyledons. Typically, they used sampling from various stages, not direct observation of single embryos throughout development. Study of initiation in individual embryos is clearly required to characterize the pattern-forming event without later modifications.
We chose to study somatic embryos of hybrid larch (Larix x marschlinsii Coaz = L. decidua x L. leptolepis = L. x leptoeuropaea) because one of us had recently characterized the distributions of cotyledon number in maturation media without benzyladenine (BA) and with several concentration levels of BA, which changed the mean cotyledon number and distributions about the mean (von Aderkas, 2002).
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MATERIALS AND METHODS |
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Plant material
Larix x leptoeuropaea (clone 69-18) used in all experiments was provided by Marie-Anne Lelu. It had been initiated by secondary embryogenesis (Lelu et al., 1994). Larch somatic embryos were multiplied on maintenance medium and matured by transferring cultures to a medium that promoted maturation.
Maintenance.
Stock cultures were maintained on gelled, modified Murashige and Skoog (MS) medium known as MSG (Becwar et al., 1990), which was composed of MS salts supplemented with 60 mM sucrose, 9 µM 2,4-dichlorophenoxyacetic acid (2,4-D), 2·5 µM benzylamino (BA) and Gelrite (4 g L–1). Prior to autoclaving, the pH was adjusted to 5·8. Following autoclaving, 10 mM glutamine (filter-sterilized) was added. Every 14 d, the tissue was subcultured to fresh medium. Petri dishes containing the cultures were kept in the dark at 22·4 °C.
Embryogenesis.
Clumps of embryogenic tissue were removed from the maintenance medium and transferred to MSG medium supplemented with 200 mM sucrose and 4 g L–1 Gelrite, to which either 0 or 4·4 µM BAP had been added.
Microscopy, observation and measurements
Observation, photography and measurement of developing embryos in vivo were done with an Olympus SZ-Tr dissecting microscope furnished with PM-10-M camera and OSM 213120 eyepiece micrometer. As embryos appeared on the embryogenic tissue masses, they were selected individually for observation and measurements. Each embryo was followed regularly, usually day by day. A staging of the developmental sequence was devised, with particular attention to the stage at which circular symmetry was first seen morphologically to be broken. The main objective was to measure a clearly defined embryo diameter at a time as close as possible to the symmetry-breaking event.
For scanning electron microscopy, representative specimens were fixed for 1 d in 2·5 % glutaraldehyde in 0·05 M phosphate buffer. The embryos were washed in buffer three times (1 h per wash), then dehydrated in a series of ethanol solutions (30, 50, 70, 95, 95 %) for 30 min each, then left overnight in 95 % ethanol. The next morning they were transferred to 100 % ethanol before being critical-point dried (Bomar SPC-1500 Critical Point Dryer ), mounted on stubs, sputtercoated with gold (Edwards S150B Sputter Coater) and viewed on a Hitachi S-3500N scanning electron microscope in the Electron Microscope Unit of the University of Victoria.
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RESULTS |
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Variability in the number of cotyledons
Figure 1 illustrates the variability in number (nc) of cotyledons, with examples from nc = 2 to nc = 8 shown at a fairly advanced stage of development of the cotyledons. For embryos grown in one author’s laboratory (L.G.H.) for quantitative measurements, Fig. 2 shows a histogram of the distribution of nc. This is almost identical to the histogram reported for embryos grown in the second author’s laboratory (PvonA) in von Aderkas (2002).
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Staging adopted to locate symmetry-breaking events in the developmental sequence
Figure 3 illustrates the sequence of development of an embryo, with a notation for stages as explained in Fig. 4. For SEMs, a different embryo had to be used for each picture. In Fig. 3, examples have been shown to illustrate as closely as possible what the sequence would look like for one embryo. Hence from Fig. 3C onwards, all examples were chosen for nc = 3. Our in vivo observations under the light microscope led us to the following concept of a ‘normal’ sequence. The embryo starts as a shallow circular dome, advances to a shape close to hemispherical (sometimes topping a short cylinder) and then conspicuously flattens (Fig. 3B) without losing circular symmetry about its axis of growth. The cotyledons then appear as outgrowths from the flattened top surface (Fig. 3C). It seems most reasonable to suppose that the number, nc, of cotyledons has been established by a patterning event between the stages shown in parts B and C in Fig. 3. Usually, measurements of the spatial dimensions of the embryos have been made by following an embryo from day to day, making measurements each day. For determination of the spacing between cotyledons at the moment when their number is established, measurements were selected for stages like those shown in B and C in Fig. 3.
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Figure 4 lists the staging terminology adopted in this work. We have avoided some conventional terms such as ‘buttress’ or ‘shoulder’ in favour of terms that indicate the early circular symmetry of the embryo about its axis of growth and initiation of cotyledons out of a shape that otherwise retains this circularity.
Figure 5 shows some abnormalities displayed in the development of a minority of embryos. Figure 5A and B shows the number, nc, of cotyledons changing either by fusion of two cotyledons or splitting of one into two (usually the latter, from the time-sequence observations of LGH). These pictures illustrate that nc should be measured at the earliest possible stage of appearance of the cotyledons, if nc is to be correlated with the dimensions of the embryo at the moment of pattern formation. Figure 5C shows that the cotyledon primordia are not necessarily in contact with each other. They can be spaced out with unchanged regions of the flat top circular surface between them. Figure 5D shows the phenomena we call polygonization and facetting, in which, before appearance of cotyledons, the embryo loses circular symmetry to become a frustum of a pyramid, therefore with a polygonal top surface (in this case, five facets and pentagonal top). Figure 5E and F shows cratering of the top surface before cotyledons appeared. The crater sometimes had a rounded rim and sometimes a sharp one. In the latter case, the embryo most commonly stopped developing before cotyledons formed.
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Spatially quantitative measurements and the intercotyledon spacing
Measurements were made to determine whether or not the spacing between adjacent cotyledons was constant, as was found for the hairs of Acetabularia vegetative whorls (Harrison et al., 1981). It is not clear, a priori, whether one should expect a constant-spacing phenomenon, whatever its mechanism, to be established round the edge of the flattened circular top of the embryo (measured diameter d2) or around a smaller circle on that top surface. Figure 6 illustrates the terminology used in presenting the results. In some of the experiments, a diameter, d1, was also measured at the base of the embryo; it did not correlate significantly with the number of cotyledons. The degree of tapering of the embryo between base and flattened top is quite variable. Some embryos are almost precisely cylindrical.
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To look for a constant-spacing effect in a way that does not prejudge its precise location or its mechanism, we tested for a linear relationship between the diameter d2 of the flattened top and the number nc of cotyledons:
d2 = (/
)nc + c(1)
If such a relationship is found, the line may or may not go through the origin. If it does, the constant-spacing event is around the outside edge of the top surface. If there is a positive intercept at nc = 0, then the constant spacing is established around a smaller circle, at a distance c/2 from the outside edge. This phenomenological description should not be read as assigning a special mechanistic role to events located at that inner circle; see the section ‘How many developmental events?’ in the Discussion.
In the present work, such a linear relationship was found (Fig. 7), both for embryos grown in control medium and for those grown in medium with 4·4 µm BA. Table 1 gives the values of the parameters in eqn (1) from the plots in Fig. 7. The spacing is not significantly affected by the concentration of BA used, although the addition of BA certainly reduces the average number of cotyledons, as observed in this work and in von Aderkas (2002). The reason for this change is a change in the distribution of embryo diameters d2, as shown in Fig. 8. Essentially, the addition of BA greatly reduces the fraction of the embryos with d2 above about 300 µm.
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For the minority of embryos that showed breaking of circular symmetry by polygonization, sectoring or facetting before appearance of cotyledons, the number of cotyledons eventually formed was not always exactly the same as the number of sides of the polygon, etc., but there was a fairly good correlation of average numbers (Table 2).
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXLITERATURE CITED |
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How many developmental events?
Three features need explanation in terms of pattern-forming theory. (1) The hemispherical embryo, stage Do(H), flattens to give a top surface in the form of a flat disc, stage An(F), with a well-defined diameter, d2. (2) Cotyledons form with a constant spacing between adjacent ones. (3) That spacing is not around any visible morphological feature of the embryo, but around a circle of smaller radius, inset a fixed (i.e. independent of nc) distance from the edge of the disc. How many developmental events need to be postulated to account for all these features: one, two or three? Chronologically, (1) happens first while (2) and (3) are simultaneous. In the previously studied case of Acetabularia, one of us has maintained (e.g. Harrison et al., 1981, 1988, 2001) that a two-stage mechanism is necessary for a system of many repeated structures (most commonly 7–22) to arise in a whorled arrangement. The first stage defines an annulus around which the second stage generates a repeating pattern. This kind of theory ascribes the origin of the inset c/2 and the spacing
to two separate events. But for small numbers of repeats in a pattern, two stages may not be necessary. For dichotomous branching, a single-stage mechanism generating wave patterns on a hemispherical tip (surface spherical harmonics) gives a good account (Harrison et al., 2001). The present example, however, with branchings ranging from one to eight parts, spans the intermediate range between small and large numbers of parts in a pattern and directly addresses the question of how complex a pattern must be for a one-stage mechanism to be inadequate.
We take the flattening of the tip as having occurred and consider features (2) and (3), the patternings on a flat disc, together. If these are produced by any kind of wave-forming mechanism, the appropriate waveforms to consider are the patterns of vibration of a circular drumskin attached to a rigid outer border. These combine circular ripples with pie-slice patterns. A priori, it is a promising possibility that the mathematics of such shapes will explain simultaneously the circle at inset c/2 and the repeated maxima around that circle at spacing . A brief description of how these waveforms will fit patterns of increasing numbers of whorled structures to increasingly large discs is given in the Appendix. This shows that a plot of d2 vs. nc should indeed be a straight line with a positive intercept. The ratio of slope to intercept is a significant quantity. The fit of one complete up-and-down ripple to the radius of the disc (second zero of the Bessel function at the edge) gives the ratio slope/intercept 0·226 (broken lines, Fig. 7A and B), compared with 0·19 and 0·21 for the experimental data (points and solid lines).
From this analysis, the answer to ‘how many developmental events?’ is that inset c/2 and spacing can be determined by any one mechanism that is able to form two-dimensional wave patterns on a circular disc. This could be, for example, chemical reaction–diffusion (Lacalli, 1981), mechanical buckling (Green, 1999) or a more complex mechano-chemical interaction. The fit of our data to waveforms on a disc is concordant with any of these types of mechanism, and strengthens the concept that the pattern of the emerging cotyledons is a single macroscopic entity governed by interactions coordinated on the spatial scale of the whole disc.
This view is supported by the absence of obvious structural features linking or separating the cotyledons in the earliest stages of their development. Gutmann et al. (1996) show longitudinal sections of somatic embryos of L. x leptoeuropaea; the embryonal mass consists of one cell type with somewhat irregular cell size and distribution. Indeed, the cotyledons are more regular in size and shape than the detailed cellular structure of the tissue from which they arise, further supporting the view that the organization of a plant is organismal and transcends the order of the cells (Kaplan and Hagemann, 1991; Cooke and Lu, 1992).
As to feature (1), the transformation of a hemisphere into a flat-topped shape: this requires slow growth at the centre and fastest growth in the outermost regions, without loss of circular symmetry. We think that this is probably initiated by a separate patterning process from the later cotyledon-forming one. In that case, the three listed features need two patterning mechanisms to explain them. But, unlike the Acetabularia patterning, the first process leading to tip flattening is not being called upon to generate the inset circle around which the cotyledons are spaced. It has only to turn a hemisphere into a disc, and leave the rest to the second process.
Significance of a constant-spacing phenomenon
A constant spacing between adjacent repeats of an organ when it first becomes identifiable is a very significant phenomenon, indicating that the organism is able to set up a quantitative scale of distance. This is a common feature of biological development, but far from easy to explain definitively, although theories having this property have been known for at least half a century (Turing, 1952). Up to the present, constant-spacing effects have been explained by reaction–diffusion theory but not yet by mechanical or mechano-chemical theory. Morphogen distributions on a disc (Fig. 9), computed for Brusselator-type reaction–diffusion dynamics (Lacalli, 1981), may be compared with expressed cotyledon patterns in Figs 1E and 5A.
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That rates of reaction and diffusion can set up a measure of distance (e.g. the wavelength of a pattern) may be shown by dimensional analysis. Reaction rates provide a measure of time; a first-order rate constant (k) is in (time)–1, usual units s–1. Diffusion gives a proportionality between distance and (time)1/2; diffusivity (D) has dimensions (distance)2 (time)–1, e.g. cm2 s–1. Diffusivity and rate constant combined in the form (D/k)1/2 give (cm2 s–1/s–1)1/2 = cm. Reaction–diffusion theories can be mathematically complex; but they lead to wavelength expressions that are always variations on the theme (D/k)1/2.
Thus in the Brusselator (Prigogine and Lefever, 1968), a promising mechanism for branching processes in plants (e.g. Holloway and Harrison, 1999), reactants A and B give rise to diffusible intermediates X and Y. The kinetics (Harrison, 1993) cause X and Y to be patterned with wavelength approximately:
= 2(DX DY/a2 bcd–2 A4 B)1/4(2)
a, b, c and d are rate constants (of first, second and third orders); A and B are the concentrations of A and B. The denominator has units s–2, and the dimensional analysis is like that of (D/k)1/2, giving in cm.
What kinds of substances may be involved in patterning?
Diverse classes of biochemicals might be involved in reaction–diffusion mechanisms. X and Y could be microRNAs, transcriptional regulators or other proteins, or the smallest molecules and ions. In Acetabularia, an effect of calcium concentration on led Goodwin to propose calcium as a morphogen, while Harrison indicated it as an activator of a protein to become morphogen precursor A [see Chapter 10 of Harrison (1993) for this and other examples]. The diversity increases with the possibility of mechanical properties playing a role in patterning. We expect ultimate linkage to genes, but the link may be a very long bridge upon which our current work is at the other end. We have studied a clonal population, so that aberrant development does not disclose mutations, e.g. the ‘crater’ morphology is not related to the ‘cup-shaped cotyledon’ genes CUC1 or 2 (Aida et al., 1997). More probably, it is part of a statistical distribution of different patterns made by the same dynamic mechanism (Harrison et al., 2001; Nagata et al., 2003). In a very general sense, however, genes first expressed as chemical patterns, such as annuli, in any branching process in plants, are of potential interest. These include a few flower development genes intermediate between meristem identity and organ identity genes: FIMBRIATA (FIM) in Antirrhinum (Simon et al., 1994); UNUSUAL FLORAL ORGANS (UFO) in Arabidopsis (Ingram et al., 1995); and BLH2, a homeodomain gene in Arabidopsis, expressed in various tissues at different times in development, including expression in the floral apex in an annular pattern where the sepal whorl is about to form (M. Pidcowich, D. Godt, K. Kushalappa and G. Haughn, pers. comm.). More plan-view studies of gene expressions on apices would help in bridging the gap between genetic work and our macroscopic work. For embryos, current controversy on the relationship of lateral organ formation to the initiation of a shoot apical meristem connects to our present work only in that we adduce evidence that the cotyledon-forming region and the central region where the apical meristem will arise participate in the same patterning mechanism at the time of cotyledon initiation.
In biochemistry remote from genetics but likely to be close to developmental dynamics, significant classes of substances include: (1) for mechanical buckling, cell wall constituents that govern mechanical properties; (2) for mechano-chemistry, the cytoskeleton, e.g. actin and actin-associated proteins; (3) for reaction–diffusion, integral membrane proteins such as those heading signal-transduction cascades (for possible relation of these to Brusselator A and X, see Harrison et al., 1997), and also enzymes affecting rate constants, e.g. Brusselator a, b, c and d; (4) various types of plant growth regulators (PGRs). We had hoped that BA would turn out to be a -controller; but Figs 7 and 8 show that it reduces the average number of cotyledons by restricting the size range of the embryos, without changing . Other PGRs should, however, be tried.
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ACKNOWLEDGEMENTS |
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We thank the following: for culture maintenance, the UBC Chemistry Department’s Biological Service, Elena Polishchuk, Director, and Susanna Ip, and at UVic, Andrea Coulter, Stephen Burgess, Lindsay White and Heather Milligan; for assistance with the SEM, Dr C. Singla; for comments on the manuscript of this paper, David M. Holloway, and for research grants to both of us, NSERC Canada.
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APPENDIX |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXLITERATURE CITED |
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Matching the data for d2 vs. nc to Bessel function waveforms on a disc
[For a clear introductory text on classical wave theory in one, two and three dimensions, see Chapter 3 of Kauzmann (1957); for application of Bessel functions to morphogenetic patterns in unicellular algae, see Lacalli (1981).] These are essentially products (in the mathematical sense) of undulating radial functions Jn (r), having circular maxima, minima and zeros (nodal circles) with simple sine wave functions of angle
around the disc:
U(r,) = A Jn (xnl r /a)sin(n)(A1)
where U is the displacement that has wave character, e.g. actual up-or-down distance for a vibrating drumskin, or increase or decrease of a chemical concentration from a spatially uniform value for reaction–diffusion dynamics; r, are circular polar coordinates with origin at the centre of the disc; A is a time-dependent amplitude, oscillatory for a drumskin vibration, growing monotonically in time to a final steady value for a reaction–diffusion pattern; a is the outer radius of the disc, corresponding to d2/2; xnl r/a = x is the independent variable for the Bessel function, Jn, representing radius scaled so that the radius a corresponds to the lth zero of the function of order n (Fig. 10A); and the angular function has a number of maxima, n, around the circle corresponding to nc for the cotyledon pattern.
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To clarify the nature of the function Jn, the first cycle of this function is plotted in Fig. 10 for n = 1, 2, 3 and 10. If one chooses the boundary condition that the edge of the disc is at the second zero (l = 2) of Jn, each plot ends at the disc outer radius. Evidently, as n increases, patterning is moving towards the outer regions of the disc, with an increasing flat unpatterned space in the centre. (Compare the pattern for n = 10 with the shapes shown for nc = 6 and 8 in Fig. 1E and F.) A conspicuous feature of Fig. 10B–E is that, as the radius of the disc grows, the width of the patterned region stays almost constant. In particular, the distance from the first maximum to the second zero remains close to 6 on the scale of the radial variable x (distance between the two dotted lines). This distance corresponds to c/2. (Fig. 7 has diameters as ordinates, and intercept c; Fig. 10F, rotated 90°, has radii as ordinates, and intercept c/2.)
Thus the mathematics of harmonic waveforms on a disc accounts simultaneously for the inset circle and the spacing of cotyledons around it as illustrated in Fig. 6, provided that whatever wave-making mechanism is beginning to operate at developmental stage An(F) has a boundary condition U = 0 at the edge of the disc, i.e. that edge is a nodal circle; and the fit is quantitatively good (in terms of slope/intercept in Fig. 7) if the mechanism selects the second nodal circle. For any postulated mechanism, mechanical or physicochemical, it would be a substantial theoretical project to determine which nodal circle would be selected.
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LITERATURE CITED |
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