AOBPreview originally published online on October 11, 2004
Annals of Botany 2004 94(6):913-917; doi:10.1093/aob/mch213
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Annals of Botany 94/6, © Annals of Botany Company 2004; all rights reserved
TECHNICAL NOTE |
Statistical Recognition of Random and Regular Phyllotactic Patterns
1 Laboratoire de Cytologie Expérimentale et Morphogenèse végétale, Université Pierre et Marie Curie, Bât. N2, 4 place Jussieu, 75 252 Paris Cedex 05, France and 2 Institut de recherche en biologie végétale, Jardin botanique de Montréal, 4101 Sherbrooke Est, Montréal, Canada H1X 2B2
* For correspondence. E-mail denis.barabe{at}umontreal.ca
Received: 7 June 2004 Returned for revision: 12 July 2004 Accepted: 20 August 2004 Published electronically: 11 October 2004
ABSTRACT
• Aims A statistical method used in ecology is adapted to characterize the degree of order in phyllotactic systems.
• Scope The test consists of subdividing a planar projection of the stem apical meristem into 16 sectors and counting the number of primordia appearing in each. By dividing the sum of squared deviations by the mean number of primordia per sector the chi-square (
2) is obtained. When there are a total number of 20 primordia, if the 2 is less than 6·26, the phyllotaxis is spiral; if it is between 6·26 and 27·5 the phyllotaxis is random; and if it is greater than 27·5, the phyllotaxis is distichous or whorled (level of significance 
= 5 %). It is also possible to remove one or more sectors. If there are k sectors, the two critical values delimiting the random zone will be found in a 2 table for k – 1 degrees of freedom.
• Conclusions The method is applied to the analysis of sho mutants described by Itoh et al. in 2000 (Plant Cell 12: 2161–2174). The results obtained are in agreement with the theoretical analysis showing that a whorled or spiral phyllotactic system may contain a certain number of randomly distributed elements without losing its regular global structure.
Key words: Shoot apex, phyllotaxis, models, mutant, chi-square
INTRODUCTION
The mechanisms involved in the regulation of phyllotactic patterns have been studied intensively from an experimental and theoretical point of view (Jean, 1994
; Jean and Barabé, 1998; Reinhardt and Kuhlemeir, 2001). Phyllotactic systems are generally described by using parameters such as divergence angle, plastochrone ratio, and number of sets of opposed visible spirals (parastichies). Theoretical models are based on geometric regularities appearing at the level of shoot apical meristem (SAM). However, the discovery of genes that promote random changes in phyllotactic patterns provides a new perspective as far as the analysis of phyllotaxis is concerned (Callos and Medford, 1994; Itoh et al., 2000; Reinhardt and Kuhlemeir, 2001). The main problem with the analysis of phyllotactic mutants is the characterization of phyllotactic systems and their degree of order.
Recently Itoh et al. (2000) analysed the pattern of leaf initiation in the shoot organization (sho) mutants in rice. Leaf primordia of sho mutants were initiated at nearly random positions on the shoot apex with a mean divergence angle of 110°. To quantify the degree of order of phyllotactic patterns appearing in these mutants it was proposed to use partition entropy in the context of information theory. However, it has been demonstrated recently that partition entropy is not able to discriminate between an organized spiral system, e.g. with a divergence angle of 137·5°, and a random phyllotactic system (Barabé and Jeune, 2004). Therefore, partition entropy does not provide an adequate means to quantify the alterations in phyllotactic patterns. With the exception of distichous and whorled patterns, the partition entropy cannot give an exact representation of the degree of order in a phyllotactic system. With this formula, a spiral system will appear as disorganized even though it is a well-ordered pattern.
With respect to the problem of entropy, Jean (1994, 1998) developed a model based on a concept of hierarchy to represent the various spiral phyllotactic patterns (m, n), where (m, n) represent the visible parastichy pairs. He used an entropy-like function Eb based on the set of hierarchies to calculate their energetic cost. In this model (Jean, 1994), parameter Eb represents the production cost of each type (m, n) of spiral pattern. The value of Eb increases as the number of parastichies increases, and there is no maximum value. However, in Jean's model it is not possible to determine the entropy of the phyllotactic system without knowing the number of visible opposed parastichies (m, n).
Thus, to adequately classify phyllotactic patterns as random a statistical method allowing the degree of order in a phyllotactic system to be characterized is needed. This method should be able to quantify with precision the degree to which phyllotactic mutants are altered. To solve this problem it is proposed to adapt a method, already used in ecology to study the dispersal of plants or animals, to the field of phyllotaxis. With this method it is possible to determine, within a confidence interval, to which phyllotactic pattern a given phyllotactic organization belongs. After demonstrating how this statistical method can be adapted to study phyllotactic patterns, it is applied to the quantitative analysis of sho mutants described by Itoh et al. (2000).
METHODS
The following method is derived from a statistical test frequently used in ecology to analyse spatial dispersal of plants or animals (Fowler et al., 1998; Lefranc et al., 2001).
Let x be the number of taxa counted in a quadrat having a negligible surface area in relation to the territory under study, and k be the number of quadrats. If the dispersion of taxa is random, the distribution of x will follow a Poisson law. Therefore the normal approximation of this law implies that the ratio
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2 having k – 1 degrees of freedom (see Appendix). A confidence interval bounds the zone inside which the null hypothesis is not rejected (hypothesis of random dispersal of taxa). Outside the limits of this interval, if the value of
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This test can easily be adapted to analyse phyllotactic patterns. A planar projection of the shoot apical meristem (SAM) is subdivided into k equal sectors (k = 2, 4, 8, 16, ...), each one containing x foliar primordia (Fig. 1). If all primordia (n) in SAM are dispersed randomly, x will follow a binomial distribution:
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Since we know the total number of primordia (n) and the number of sectors (k), it is of course possible to use the normal approximation of the exact binomial distribution to calculate the expectation. However, the degree of precision obtained by using the binomial law does not compensate for the increase in calculation necessary to obtain this precision. As it has been shown above, the Poisson law, estimated by using the mean number of primordia appearing in each sector, is completely satisfactory for our purposes.
RESULTS
Table 2 comprises angles of divergence corresponding to distichous (180°) or whorled patterns (60°, 120°), spiral patterns (77·96°, 99·5°, 137·51°, 151·14°), belonging to normal and anomalous phyllotactic series in the sense of Jean (1994) and hypothetical spiral patterns (100°, 130°). When a SAM containing more than 20 primordia is divided into 16 sectors (d.f. = 15) (Fig. 1), the different phyllotactic patterns are readily distinguishable (Table 2). For example, when d.f. = 15, the random phyllotactic pattern is the only one appearing in the zone of random dispersion (Fig. 2). The dots located below the confidence interval correspond to a uniform dispersion (spiral phyllotaxis), and those located above the confidence interval to an aggregated dispersion (distichous or whorled phyllotaxis).
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However, when the SAM is divided in four or eight sectors, aggregated or uniform patterns are observed inside the limits of the random phyllotactic zone. To obtain a statistically significant delimitation between different types of patterns, the number of sectors must be 16 with 20 or more primordia (Table 2). However, when there are 16 sectors and more than 50 primordia it becomes difficult to statistically separate uniform phyllotactic patterns charaterized by even numbers such as 100° from random patterns. This is not the case for spiral patterns characterized by angles of divergence of 77·96°, 99·5°, 137·51° or 151·14°.
The preceding results show it is possible to statistically differentiate between distichous or whorled patterns (aggregated dispersion), spiral patterns (uniform dispersion) and random patterns (random dispersion). However, is it still possible to separate these three types of phyllotactic patterns when they are mixed with randomly distributed primordia? This question has been addressed by determining the number of randomly distributed primordia that could be incorporated in a uniform spiral phyllotaxy to statistically obtain a random phyllotactic pattern. With a total number of primordia equal to 50, it appears that the number of randomly distributed primordia needed to modify a particular phyllotactic pattern varies depending on the angle of divergence (Table 3). For example, when ten primordia are randomly distributed in a phyllotactic pattern characterized by an angle of divergence of 137·51°, this pattern cannot be statistically distinguished from a random pattern (Table 3). In other cases (e.g. 60°, 77·96°, 99·5°, 130°) the addition of 20 randomly distributed primordia does not change the pattern significantly. This indicates that a regular phyllotactic system may tolerate a certain amount of disorganization without loosing its global characteristics. The distichous (180°) and whorled patterns of 120° are particularly stable because the presence of 30 randomly distributed primordia does not alter the general type of phyllotaxis.
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This same question can be addressed by adding primordia to defined sectors within an otherwise random pattern. Consider, for example, the distichous pattern found in maize (Jackson and Hake, 1999
) and rice (Itoh et al., 2000). Let us assume a random distribution of 50 primordia, calculated with the Poisson law of parameter (
= 50/16), and calculate the 2 values when primordia are progressively added in opposed sectors one and nine (distichous position), and concomitantly deleted in other sectors at random. It is found that the 2 values increase proportionally to the number of primordia added in sectors one and nine. Starting with a completely random pattern, <20% of primordia (X = 8) are required in a distichous position (sectors one and nine) to obtain a regular phyllotactic system, because the value of the 2 (28·08 > 27·50) falls outside the boundaries of the random zone (Fig. 3).
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A case study: sho mutants
In the sho mutants studied by Itoh et al. (2000)
, leaf primordia were initiated at nearly random positions on the shoot apex with a mean divergence angle of 110°. To quantify the degree of order of phyllotactic patterns appearing in these mutants, the method described above was applied to the data previously published by Itoh et al. (2000, fig. 6). For the purposes of the present analysis, each mutant is considered as a single apex. If data were taken on several apices, an ANOVA test would be necessary to determine their homogeneity. Of course, in the case of heterogeneity the present method is not applicable. Here, only an example of the application of this statistical test is shown. With the SAM divided into 16 sectors, the phyllotactic patterns appearing in mutants correspond statistically either to an aggregative dispersion (distichous or whorled pattern; sho2) or a random dispersion (sho1-1 and sho3). The aggregative pattern is not surprising considering that the wild-type pattern is distichous, and is suggestive of a developmental constraint exerted by the initial phyllotactic pattern on the types of patterns that can appear in phyllotactic mutants. To test this, sectors one and nine, which comprise the greatest number of primordia, were removed from sho2 data in the analysis. The newly obtained pattern (sho2P) corresponds to a random phyllotaxis (Fig. 4). It appears that phyllotactic mutants experience a shift from a distichous to a random phyllotactic organization. Therefore, in the same phyllotactic system, one may find regularly or randomly distributed elements. This is in agreement with the present analysis which shows that a distichous, whorled or spiral phyllotactic system may contain a certain number of randomly distributed elements without losing its regular global structure.
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DISCUSSION
The present analysis presents only a few examples of divergence angles. A more complete table, taking into account a greater range of angles of divergence could be produced. However, the purpose of this study was not to determine the confidence interval for each angle but only to show that different patterns can be distinguished by using a very straightforward statistical analysis. This statistical test is simple and accurate enough to determine a confidence interval for different phyllotactic patterns.
In the preceding section it has been demonstrated that it is possible to statistically separate three general types of phyllotactic patterns by using a probability distribution. Of course, in many cases, it may be easy to qualitatively distinguish a whorled phyllotactic pattern from a spiral or random pattern, particularly when whorls do not include many primordia. However, it is more difficult to distinguish between a random pattern and a spiral pattern by analysing their degree of order. For example, as has been demonstrated previously, it is not possible to distinguish a spiral pattern from a random pattern by using the formula of partition entropy (Barabé and Jeune, 2004). On the other hand, the use of a simple probability test may distinguish between a random pattern and a spiral pattern. A statistical analysis can also bring out a pattern that is not visible a priori on a planar representation of the SAM. For example, in the case of sho mutants, the predominace of a distichous phyllotactic pattern is not recognizable by visual observation of the schematic alignment of primordia.
The method described above should be viewed as complementary to already existing phyllotactic models. By using this method, it is possible to statistically determine to which phyllotactic pattern (distichous, whorled, spiral or random) a particular type of phyllotactic organization belongs. Subsequently, one can use a given deterministic theoreotical model to quantitatively analyse the relationship between phyllotactic parameters such as angle of divergence, plastochrone ratio and number of parastichies in the case of regular systems.
The method presented above is a statistical tool that can be used to analyse empirical results. It does not constitute a theoretical model. However, let us note that there are no stochastic general models of phyllotaxis. All theoretical models developed up to now are deterministic and based on regularities appearing in phyllotactic patterns (Jean, 1994; Jean and Barabé, 1998). It is hoped that the statistical tool presented here will open new avenues of research for the development of a probabilistic model of phyllotaxis.
APPENDIX
If x follows a Poisson law, where its parameter is estimated by the sample mean 
, then
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2 law with k – 1 and not k degrees of freedom because of the linear dependence of zi:
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2 law with k – 1 degrees of freedom (Rahman, 1968).
ACKNOWLEDGEMENTS
We would like to thank Professor Christian Lacroix, Professor David Morse and Mr Stuart Hay for their valuable comments on the manuscript. This research was supported by grants from the Natural Sciences and Engineering Research Council of Canada to D.B.
REFERENCES
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Reinhardt D, Kuhlemeier C. 2001. Phyllotaxis in higher plants. In: McManus MT, Veit BE, eds. Meristematic tissues in plant growth and development. Boca Raton, FL: CRC Press, 172–212.
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