Abstract

Background and Aims

By using the technique of replicas of a developing apex it is possible to obtain a direct measure of phyllotactic parameters (plastochrone and platochronic ratio) involved in the initiation of two successive primordia at the level of the SAM. The goal of this study is to compare, in a real time setting, the value of phyllotactic parameters in distichous sytems using Begonia as a case study, with the value of the same parameters in spiral phyllotactic systems.

Methods

To determine the real-time sequence of events at the level of the SAM, replicas were made of the developing apex at different intervals using previously described techniques. Impression moulds were made at 24-h intervals. The following phyllotactic parameters were measured: plastochrone, angle of divergence, plastochrone ratio and ratio between the diameter of the leaf and the apex.

Results

The time between the appearance of two successive leaves is 15–20 d. The average value of the plastochrone ratio ( R ) is 1·3, and the ratio of the leaf to the diameter of the apex ( Γ ) is 2·5. The angle of divergence varies from 165º to 180º. The speed of advection of the primordium from the apex, varies from 0·28 to 0·37 µm d −1 .

Conclusions

The speed of advection of primordia in Begonia is lower than that of Anagalis . This is not in accordance with theoretical simulations that predict the opposite. In Begonia , the plastochrone ratio does not reflect the real time of appearance of two successive primordia. The time separating the appearance of two primordia is not directly related to the distance of these two primordia from the centre of the apex but is related instead to the enlargement of leaves.

INTRODUCTION

In the analysis of phyllotactic organization, one of the main problems is the determination of the time of appearance of primordia in the context of the three general types of phyllotactic patterns: spiral, verticillate and distichous. Phyllotactic systems are generally described by using different parameters (Fig.  1 ) such as divergence angle ( d ), plastochrone ratio ( R ) (the ratio of the distance of two successive primordia from the centre of the apex; Richards 1951 ), ratio between the radius of a primordium and the radius of the shoot apical meristem (SAM) [parameter b of van Iterson (1907) or Γ of Douady and Couder (1998) ], and number of opposed visible spirals ( m , n ) called parastichies. For a review of the significance of these parameters in phyllotaxis see Jean (1984). Among all these, parameters R or b constitute an indirect estimation of the time separating the initiation of two successive primordia ( Douady and Couder, 1992 , 1993 ).

F ig . 1.

Diagrams of SAM highlighting specific phyllotactic parameters. Primordia are numbered in order of appearance. (A) d , Divergence angle; a , b , distances from the centre of the SAM ( a / b = R ); R0 , radius of the SAM. (B) d1 , Diameter of the SAM; d2 , diameter of the leaf primordium ( d2 / d1 = Γ  ).

F ig . 1.

Diagrams of SAM highlighting specific phyllotactic parameters. Primordia are numbered in order of appearance. (A) d , Divergence angle; a , b , distances from the centre of the SAM ( a / b = R ); R0 , radius of the SAM. (B) d1 , Diameter of the SAM; d2 , diameter of the leaf primordium ( d2 / d1 = Γ  ).

Rutishauser (1998) showed empirically that spiral systems are characterized by a lower plastochrone ratio than distichous sytems. These observations correspond to predictions based on theoretical models (Jean, 1984). The experiment of Douady and Couder (1992) , which compared leaf primordia with droplets of ferrofluid moving in a magnetic field, simulated this phenomenon adequately and confirmed many of Rutishauser's observations.

In the simulations of Douady and Couder (1992 , 1993 ), the determination of phyllotactic patterns depends on only one dimensionless parameter G = VT / R0 , where V represents the speed of advection of droplets, T the periodicity of appearance of droplets and R0 the radius of the zone where particles appear. These authors suggested that, although the dimensions and time scales are not the same between their physical experiment and botanical reality, the value of the parameters can be identical allowing for a comparison between their model and botanical data.

In the model of Douady and Couder (1992 , 1993 ), it was shown that the speed of advection of droplets, which depends on T and R , is higher in distichous systems than in spiral systems. In theoretical and empirical models, it is easy to obtain theoretical values for R . However, the only way to calculate the value of V is to know the real time span separating the initiation of two leaves at the shoot apex.

It is difficult to measure the time of appearance of primordia directly at the shoot apex during the exponential phase of growth. The plastochrone index ( Erickson and Michelini, 1957 ) is the main method used to determine the relative time of appearance of leaves during plant growth. Although this method is very useful in different types of morphological and physiological studies, it does not provide information on what occurs at the shoot apex during the exponential phase of growth. One way to determine the real time sequence of events at the level of the SAM is to make replicas of the developing apex at different intervals using the technique described by Williams et al . (1987) , Williams and Green (1988) and Green and Linstead (1990) . This method was successfully applied to the analysis of the SAM of Vinca major ( Williams, 1991 ), Graptopetalum paraguayense ( Tiwari and Green, 1991 ), Anagallis arvensis ( Green et al ., 1991 ; Kwiatkowska and Dumais, 2003 ) and Arabidopsis thaliana ( Kwiatkowska, 2004 ) among others. With this method it is possible to obtain a direct measure of the time (plastochrone) interval separating the initiation of two successive primordia at the apex. For example, based on our estimations, the plastochrone corresponds to 5 d in Vinca ( Williams, 1991 ) and to 2 d in Anagallis ( Kwiatkowska and Dumais, 2003 ).

Until now, Green's method ( Green and Linstead, 1990 ) has only been applied to spiral (e.g. Anagallis ) or opposite decussate (e.g. Vinca ) patterns, not distichous systems like Begonia . It is not surprising to note that there are no studies of distichous or spirodistichous patterns using the sequential replica method because in these phyllotactic patterns (e.g. Zea , Oryza and Begonia ) the very small apical meristem is nearly completely enclosed by surrounding leaves ( Haccius, 1940 ; Barabé et al ., 1992 a , b ; Charlton, 1998 ). Therefore, in contrast to spiral patterns, it is difficult to make many casts of the apex during a long period of time without damaging the apical surface or the surrounding leaves. The development of the leaf of Begonia , which has been analysed in detail in previous studies ( Barabé et al ., 1991 , 1992 a , b ; McLellan et al ., 1993 ; Jeune and Barabé, 1995 ; Thibault et al ., 1997), represents a good source of data to analyse the sequence of appearance of primordia in distichous systems. As far as is known, this study of Begonia is the first observation of a distichous or spirodistichous phyllotactic pattern in real time.

The general goal of this study is to compare the value of parameters T , R , b and V in distichous sytems using Begonia as a case study to the value of the same parameters in spiral phyllotactic systems to determine if theoretical estimates are valid. The value of T will be determined by direct observations of the developing SAM. Data for spiral patterns will be estimated from a previous study describing the development of the apex of Anagallis in real time ( Kwiatkowska and Dumais, 2003 ). This quantitative comparison will make it possible to determine if the prediction made by theoretical models corresponds to reality, and to discuss the value of theoretical models in the estimation of phyllotactic parameters.

MATERIALS AND METHODS

Specimens of Begonia scabrida A. DC. (#2184-57) were obtained from the living collection of the Montreal Botanical Garden (Canada).

Dissection and impression moulds

The shoot tip of living plants was dissected using fine forceps and insect needles (#00, #0, #1) to reveal the shoot apical meristem. Immediately following dissection, a polyvinylsilaxane polymer mixture was applied to the apex with an insect needle and left there to dry for a few minutes to take an impression and prevent any dehydration. The resulting mould was carefully removed by using fine forceps and fixed on a microscope slide with a small quantity of the same polymer. The polymer mixture used consists of one part Kerr's Reflect resin and one part Kerr's Mirror wash catalyst (Kerr UK Ltd., Peterborough, UK); these proportions ensure the desired fluidity to adequately cover all areas of the apical meristem.

Immediately after taking an impression, the shoot apex was sprinkled with water and covered with a 1·5-mL Eppendorf microtube containing some water. The entire plant material was covered with a plastic bag containing paper towels soaked in water to prevent further dehydration. Impression moulds were made at 24- or 48-h intervals. Ten trials were done on different species ( B. fagifolia , B. listida , B. roxburghii and B. radicans ). However, due to the difficulty associated with making many casts of the same apex without damaging developing primordial, only two specimens yielded data that covered one plastochrone of development.

Specimen reconstruction

Impression moulds were filled with epoxy glue (Devcon Corporation, Wood Dale, USA). Equal amounts of resin and hardener were applied to a microscope slide and gently mixed with a straight dissection needle to avoid the formation of bubbles in the mixture. The epoxy was then applied to the mould using a Pasteur pipette with a rounded tip that was shaped under a flame to a diameter of 0·34 mm.

The moulds were gradually filled by making sure the entire inner surface was covered and no air bubbles formed. Once the epoxy hardened, the impression was removed from the mould.

Microscopy

Epoxy specimens, representing a positive reproduction of the shoot apical meristem at the time of impression, were mounted on metal stubs, sputter coated with a thin layer of gold-palladium, viewed with a Jeol JSM 35 scanning electron microscope operating at an accelerating voltage of 20 or 25 kV, and photographed using 120-mm photographic film. Photographic negatives were scanned at high resolution and the resulting digital images were used to produce the photographic plates with Photoshop version 7.

The centre of leaf primordia was determined by using the median point on the axis of symmetry of the leaf. The diameter of the SAM was determined by using the area enclosed between two leaves. To determine the centre of the SAM the median point of the distance between those two primordial leaves was used.

Definitions of symbols used

  • d : angle of divergence; the angle between two successive primordia

  • r : rate of growth ln( R ) /T ; rate of displacement of a primordium

  • R0 : radius of the SAM

  • R : plastochron ratio; the ratio of the distance of two successive primordia from the centre of the apex

  • T : plastochrone (days); the ‘time’ interval between the initiation of two successive leaves

  • V : speed of advection (μm d −1 ); the speed at which a primordium becomes more distant from the centre of the SAM

RESULTS

Developmental morphology

Throughout its development, the leaf primordium of Begonia occupies a significant portion of the region of the shoot apex. The stipules at the base of the leaf give the primordium a wide area of insertion (Fig.  2 A). The larger of the two stipules is associated with the broader side of the blade (Fig.  2 B) and both stipules nearly enclose the next leaf primordium that is initiated on the apex SAM (Fig.  2 C, arrow). The blade portion of the leaf assumes an asymmetrical shape during early stages of initiation and maintains this asymmetry throughout development (Figs  2 B and 3 , days 1–5).

F ig . 2.

General features of the shoot apical meristem. (A) Top view of leaf primordium (L) flanked by two stipule primordia (arrows). (B) Later stage of leaf development showing accentuated asymmetry of the developing leaf blade. This photograph represents the same shoot apex as (A) but 5 d later. (C) Top view of a shoot apex to where the blade of the leaf primordium was removed (rL). Initiation of new leaf primordium (arrow) is visible between stipules (St) of the previous leaf. Scale bars: A, B = 54 µm; C = 42 µm.

F ig . 2.

General features of the shoot apical meristem. (A) Top view of leaf primordium (L) flanked by two stipule primordia (arrows). (B) Later stage of leaf development showing accentuated asymmetry of the developing leaf blade. This photograph represents the same shoot apex as (A) but 5 d later. (C) Top view of a shoot apex to where the blade of the leaf primordium was removed (rL). Initiation of new leaf primordium (arrow) is visible between stipules (St) of the previous leaf. Scale bars: A, B = 54 µm; C = 42 µm.

F ig . 3.

Developmental sequence of a single shoot apical meristem using the impression mould technique and showing the development of a leaf primordium (L) followed by the initation of a new leaf primordium (arrows). The date the impression mould was taken is represented on each photograph. Scale bar = 54 µm.

F ig . 3.

Developmental sequence of a single shoot apical meristem using the impression mould technique and showing the development of a leaf primordium (L) followed by the initation of a new leaf primordium (arrows). The date the impression mould was taken is represented on each photograph. Scale bar = 54 µm.

Figure  3 represents the developmental sequence of a single shoot over a period of 15 d between the initiation of two successive leaves. The area of the SAM is tightly delimited by the base of the leaf primordium which includes the stipules. This area remains relatively unchanged as it becomes progressively enclosed by the developing stipules (days 1–8). By days 9 and 10 in this sequence, the next leaf primordium is initiated as a bump which eventually takes up most of the area of the SAM (days 10–15). The SAM will recover its surface area after the formation of the blade of the leaf (day 1). Based on detailed comparative developmental data, the time between the initiation of two primordia ranges between 15 and 20 d.

Quantitative results

Values of the different parameters under analysis are summarized in Table  1 . As noted previously the time between the appearance of two successive leaves is 15–20 d. The average value of the plastochrone ratio ( R ) is 1·3. By using this value, and the real time separating the formation of two leaves, it is possible to calculate the speed of advection (parameter V ) of Douady and Couder (1992 , 1993 ) following their formula G = VT/R0 , where T represents the plastochrone and R0 the radius of the SAM. Given that growth near the apex is considered exponential, ln R = G , and V = r × R0 , where r is the rate of growth ln( R )/ T ( Richards, 1951 ). The speed of advection, which represent the speed of removal of the primordium from the SAM, is therefore equal to 0·28–0·37 µm d −1 . The mean value of Γ , which corresponds to the ratio of the diameter of the leaf to the diameter of the SAM, varies from 1·2 to 6·3, depending on the stage of development of the leaf, with an average of 2·5. The angle of divergence varies from 165º to 180º (Table  1 ).

T able 1.

Comparison of phyllotactic parameters between Begonia scabrida and Anagallis arvensis by using real time sequences of development (μm d −1 )

T (days) ( n = 3)  R0 ( n = 13)  R ( n = 3)  r = ln( R )/ T V = r × R0 Γ ( n = 13)  d ( n = 7)  
Begonia squabrida       
15 < T < 20  21·75 µm  1·2 < R < 1·5, ( = 1·3)   0·013 < r < 0·017   0·28–0·37 µm d −1  1·2 < Γ < 6·3, ( Γ̄ = 2·5)  165–180° 
Anagallis arvensis ( n = 2)        
1·67 < T < 1·88  43 µm  = 1·1   0·051 < r < 0·057   2·19–2·45 µm d −1 ≈0·60 137·5° 
T (days) ( n = 3)  R0 ( n = 13)  R ( n = 3)  r = ln( R )/ T V = r × R0 Γ ( n = 13)  d ( n = 7)  
Begonia squabrida       
15 < T < 20  21·75 µm  1·2 < R < 1·5, ( = 1·3)   0·013 < r < 0·017   0·28–0·37 µm d −1  1·2 < Γ < 6·3, ( Γ̄ = 2·5)  165–180° 
Anagallis arvensis ( n = 2)        
1·67 < T < 1·88  43 µm  = 1·1   0·051 < r < 0·057   2·19–2·45 µm d −1 ≈0·60 137·5° 

In the case of Anagalis , the parameters were estimated from real time sequences of development (μm d −1 ) published by Kwiatwoska and Dumais (2003 , fig.  2 C). T , Plastochrone; R , plastochrone ratio; r , rate of growth (from Richards, 1951 ); V , speed of advection ( Douady and Couder, 1993 ); Γ , ratio of diameter of the leaf to diameter of the SAM (parameter b of van Iterson, 1907 ); d , angle of divergence; n , number of measures; R0 , radius of the SAM.

DISCUSSION

The value of the plastochrone ratio ( R ) in Begonia and Anagallis appears, respectively, in the range of distichous systems (1·2–1·67) and spiral systems (1·0–1·67) observed by Rutishauser (1998) for different species (Table  1 ). However, the plastochrone ratio is strongly inferior to that reported previously by Barabé et al . (1991) for B. scabrida ( R = 2·0). This can be explained by the fact that this value of two was obtained using leaves that were in their phase of elongation, instead of taking measurements during the initiation of primordia, as is the case in the present study. During the phase of elongation, the petiole of the oldest leaf is farther away from the centre of the apex than during the phase of initiation ( Barabé et al. , 1991 , fig.  2 ). According to Douady and Couder (1998) , a value of Γ superior to 1·9 leads to distichy. This is the situation in Begonia where the value of Γ is approximately four times (2·5) that of a spiral system (≈0·60) (Table  1 ). Based on these parameters there appears to be a good concordance beween the theoretical model and empirical data. However, it is not the case for the speed of advection. The speed of advection of primordia (which takes into account the plastochrone in real time) in Begonia is lower than that of Anagalis (Table  1 ). This is not in accordance with theoretical simulations that predict the opposite relationship. The parameters that were measured and deduced for B. scabrida are in the same range of values in other Begonia species such as B. fagifolia , B. listida and B. roxgurghii (D. Barabé, unpubl. res.).

In the simulation of Douady and Couder (1992) for a divergence angle of 180°, two primordia move quickly away from each other during a period of time sufficient to allow the formation of a new primordium in the opposite position. The time separating the formation of two primordia corresponds to a distance relationship between them and the SAM, and is characterized by the plastochrone ratio. This view of phyllotactic dynamics forms the basis of many theoretical models such as those of van Iterson (1907) , Richards (1951) , Jean (1994) and Guerreiro and Rothen (1998) . In contrast, in Begonia , the latest formed primordium does not initially ‘move away’ from the centre of the SAM (Fig.  4 ). This is represented by the low value of the speed of advection. To have a speed of advection equivalent to that of the spiral system of Anagalis , the plastochrone ratio would have to be e 2·18 ( = 8·85). Obviously, a plant with such a plastochrone ratio has not been reported yet. The long delay between the formation of two successive primordia is due to the formation of an extensive area of insertion of the leaf around the apex, which corresponds to a high value of Γ , a low value of G , and to a leaf that is not displaced from the SAM initially. This is different from theoretical models that predict a strong positive correlation between Γ , R and G . This result indicates that, although the simulation of Douady and Couder (1992 , 1993 ) is analogous to a growing apex in a temporal framework, it does not correspond to the geometry and the growth dynamics involved in a distichous system. Here there is no correlation between the plastochrone ( T ) and the plastochrone ratio ( R ) in real time. Among the phyllotactic parameters that were studied, it is Γ that allowed for a better characterization of distichous and spiral systems.

F ig . 4.

Comparison of theoretical model ( Douady and Couder, 1992 ) and Begonia in terms of speed of advection ( V ) and growth of leaf ( g ) around SAM at different times ( T1 and T2 ).

F ig . 4.

Comparison of theoretical model ( Douady and Couder, 1992 ) and Begonia in terms of speed of advection ( V ) and growth of leaf ( g ) around SAM at different times ( T1 and T2 ).

In the distichous system of Begonia (it is probably the case for many distichous sytems) the establishment of the phyllotactic system is governed by local biological mechanisms linked to the developmental state (diameter) of the last formed leaf and not to its position (distance) with respect to the preceding leaf. This mode of development is difficult to explain using strictly biophysical mechanisms such as the presence of a physical undulating field (buckling) hypothesized by Green et al . (1998) . However, the distichous phyllotaxis of Begonia may probably be integrated more easily in models where auxin is required for organ initiation and positioning ( Kuhlemeier and Reinhardt, 2001 ; Smith et al ., 2005 ). Following Reinhardt et al . (2003) , one may speculate that only the last formed leaf is effective as a sink of auxin resulting in a distichous phyllotactic pattern. The preceding interpretation remains speculative and more developmental studies are needed to explain the phyllotaxis of Begonia .

It is important to note that in order to make the impression moulds of the apex of Begonia , the stipules of the developing leaf which enclose the apex must be removed. This manipulation probably induces some form of physiological stress at the level of the SAM that leads to a slowing down of growth, and the appearance of a delay in the development of incomimg primordia. However, considering that the apex is a well-integrated entity, it is believed that such a pertubation is not sufficiently strong to affect the general temporal scale that distinguishes between distichous and spiral phyllotactic systems.

Given that R and R0 remain constant during the exponential phase of growth, if the sequential replica method leads to an increase of the plastochrone value, T0T1 = α T0 with α > 1. Therefore the unperturbed speed of advection V0 will be reduced proportionally:  

formula
If it is supposed that the speed of advection is of the same order of magnitude as Anagallis then for plastochrone values, 15 < T1 < 20, and advection speeds, 2·19 < V0 < 2·45 and 0·28 < V1 < 0·37, 5·52 < α < 8·75 and 1·71 < T0 < 3·38 will be obtained.

This indicates that the theoretical unperturbed plastochrone ( T0 ) would need to be <4 d to invalidate the present results. Recent observations of unaltered leaves measuring 1 cm in length as a starting point on specimens of B. cabrida growing in the greenhouses of the Montreal Botanical Garden showed that the plastochrone is not <10 d (D. Barabé, unpubl. res.). Therefore this observation supports the developmental results presented in this paper.

CONCLUSIONS

Our results show two points concerning the interpretation of theoretical models in relation to distichous phyllotactic systems: (1) the plastochrone ratio does not always reflect the real time of appearance of two successive primordia; (2) the duration between the appearance of two primordia is not directly related to the distance of these two primordia from the centre of the apex but to the enlargement of leaves. These points should therefore be taken in consideration in the modeling of phyllotactic organization.

ACKNOWLEDGEMENTS

The authors would like to thank Charles Bertrand and Linda Dumont for their technical support for this project. This research was supported by individual discovery grants from the Natural Sciences and Engineering Research Council of Canada to Denis Barabé and Christian Lacroix.

LITERATURE CITED

Barabé
D
Brouillet
L
Bertrand
C
Symétrie et phyllotaxie: le cas des Bégonias
Annales des Sciences Naturelles Botanique Paris, 13ième série
 , 
1991
, vol. 
11
 (pg. 
33
-
37
)
Barabé
D
Brouillet
L
Bertrand
C
Organogénie de la feuille du Begonia radicans Vellozo et du B. scabrida . A.DC. (Begoniaceae)
Canadian Journal of Botany
 , 
1992a
, vol. 
70
 (pg. 
1107
-
1122
)
Barabé
D
Daigle
S
Brouillet
L
On the interpretation of the asymmetrical leaf of Begonia by D'Arcy Thompson
Acta Biotheoretica
 , 
1992b
, vol. 
40
 (pg. 
329
-
332
)
Charlton
WA
Jean
RJ
Barabé
D
Pendulum symmetry
Symmetry in plants
 , 
1998
Singapore
World Scientific
(pg. 
61
-
87
)
Douady
S
Couder
Y
Phyllotaxis as a physical self-organized process
Physical Review Letters
 , 
1992
, vol. 
68
 (pg. 
2098
-
2101
)
Douady
S
Couder
Y
La physique des spirales végétales
La recherche
 , 
1993
, vol. 
24
 (pg. 
26
-
35
)
Douady
S
Couder
Y
Jean
RJ
Barabé
D
The phyllotactic patterns as resulting from self-organization in an iterative process
Symmetry in plants
 , 
1998
Singapore
World Scientific
(pg. 
539
-
570
)
Erickson
RO
Michelini
FJ
The plastochrone index
American Journal of Botany
 , 
1957
, vol. 
44
 (pg. 
297
-
305
)
Green
PB
Linstead
P
A procedure for SEM of complex shoot structures applied to the inflorescence of snapdragon ( Antirrhinum )
Protoplasma
 , 
1990
, vol. 
158
 (pg. 
33
-
38
)
Green
PB
Havelange
A
Bernier
G
Floral morphogenesis in Anagallis : scanning electron micrograph sequences from individual growing meristems before, during, and after the transition to flowering
Planta
 , 
1991
, vol. 
185
 (pg. 
502
-
512
)
Green
PB
Steele
CR
Rennich
SC
Jean
RJ
Barabé
D
How plants produce patterns: a review and a proposal that undulating field behaviour is the mechanism
Symmetry in plants
 , 
1998
Singapore
World Scientific
(pg. 
359
-
392
)
Guerreiro
J
Rothen
F
Jean
RJ
Barabé
D
Universal results from a simple model of phylotaxis
Symmetry in plants
 , 
1998
Singapore
World Scientific
(pg. 
571
-
600
)
Haccius
B
Untersuchungen über die Bedeutung der distichie für das Verständnis der zerstreuten Blattstelung bei den Dikotylen
Botanisches Archiv
 , 
1940
, vol. 
40
 (pg. 
58
-
150
)
van Iterson
G
Mathematische und mikroskopisch-anatomische Studien über Blattstellungen, nebst Betraschtungen über den Schlenbau der Miliolinen.
 , 
1907
Jena
Gustav-Fisher
Jean
RV
Phyllotaxis: a systemic study in plant morphogenesis
 , 
1994
Cambridge
Cambridge University Press
Jeune
B
Barabé
D
Allometry and geometry of the leaf of Begonia
Acta Biotheoretica
 , 
1995
, vol. 
43
 (pg. 
205
-
215
)
Kuhlemeier
C
Reinhardt
D
Auxin and phyllotaxis
Trends in Plant Science
 , 
2001
, vol. 
6
 (pg. 
187
-
189
)
Kwiatkowska
D
Surface growth at the reproductive shoot apex of Arabidopsis thaliana pin-formed 1 and wild type
Journal of Experimental Botany
 , 
2004
, vol. 
55
 (pg. 
1021
-
1032
)
Kwiatkowska
D
Dumais
J
Growth and morphogenesis at the vegetative shoot apex of Anagallis arvensis L
Journal of Experimental Botany
 , 
2003
, vol. 
54
 (pg. 
1585
-
1595
)
McLellan
T
The roles of heterochrony and heteroblasty in the diversification of leaf shapes in Begonia dregei (Begoniaceae)
American Journal of Botany
 , 
1993
, vol. 
80
 (pg. 
796
-
804
)
Reinhardt
D
Pesce
E-R
Stieger
P
Mandel
T
Baltensperger
K
Bennett
M
, et al.  . 
Regulation of phyllotaxis by polar auxin transport
Nature
 , 
2003
, vol. 
426
 (pg. 
255
-
260
)
Richards
FJ
Phyllotaxis: its quantitative expression and relation to growth in the apex
Philosophical Transactions of the Royal Society
 , 
1951
, vol. 
235
 (pg. 
509
-
564
)
Rutishauser
R
Jean
RJ
Barabé
D
Plastochrone ratio and leaf arc as parameters of a quantitative phyllotaxis analysis in vascular plants
Symmetry in plants
 , 
1998
Singapore
World Scientific
(pg. 
171
-
212
)
Smith
RS
Guyomarc'h
S
Mandel
T
Reinhardt
D
Kuhlemeier
C
Prusinkiewicz
P
A plausible model of phyllotaxis
Proceedings of the National Academy of Sciences of the USA
 , 
2005
, vol. 
103
 (pg. 
1301
-
1306
)
Thibeault
G
Barabé
D
Brouillet
L
Apparition de l'asymétrie foliaire chez les plantules du Begonia subvillosa et du B. fagifolia (Begoniaceae)
Canadian Journal of Botany
 , 
1997
, vol. 
75
 (pg. 
1079
-
1097
)
Tiwari
SC
Green
PB
Shoot initiation on a Graptopetalum leaf: sequential scanning electron microscopic analysis for epidermal division patterns and quantification of surface growth (kinematics)
Canadian Journal of Botany
 , 
1991
, vol. 
69
 (pg. 
2302
-
2319
)
Williams
MH
A sequential study of cell divisions and expansion patterns on a single developing shoot apex of Vinca major
American Journal of Botany
 , 
1991
, vol. 
68
 (pg. 
541
-
546
)
Williams
MH
Green
PB
Sequential scanning electron microscopy of a growing plant meristem
Protoplasma
 , 
1988
, vol. 
147
 (pg. 
77
-
79
)
Williams
MH
Vesk
M
Mullins
MG
Tissue preparation for scanning electron microscopy of fruit surfaces: comparison of fresh and cryopreserved specimens and replicas of banana peel
Micron and Microscopia Acta
 , 
1987
, vol. 
18
 (pg. 
27
-
31
)

Comments

0 Comments