Statistical Recognition of Random and Regular Phyllotactic Patterns

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 (χ²) is obtained. When there are a total number of 20 primordia, if the χ² 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 α = 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 χ² 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.

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 E_b based on the set of hierarchies to calculate their energetic cost. In this model (Jean, 1994), parameter E_b represents the production cost of each type (m, n) of spiral pattern. The value of E_b 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

is distributed as a χ² 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

is very low, the dispersion will be regular. If the value of SS_X is very high, the dispersion will be aggregated (SS_X = sum of squared deviations).

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:

Additionally, if n and k are high enough, this distribution converges to the Poisson law of parameter n/k. When the value of n is near 20, a value that can easily be obtained experimentally, and k = 16, the approximation is valid because the probability values do not differ by more than 1 % between the two types of distribution (Table 1).

Fig. 1.

Theoretical spiral phyllotactic system subdivided into 16 sectors. Circles represent individual primordia and radiating lines represent sectors.

Table 1.

Comparison of the probability values between a binomial distribution and a Poisson distribution when n = 20 primordia and k = 16 sectors (X = number of primordia in a sector)

	X
	0	1	2	3	4
P (Binomial)	0·2751	0·3667	0·2323	0·0929	0·0291
P (Poisson)	0·2865	0·3581	0·2238	0·0933	0·0291

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).

Table 2.

χ² values for different angles of divergence (top row), different number of sectors ({16}, {8}, {4}) and different number of primordia (left column) (α = 5 %)

{16}	60	77·96	99·5	100	120	130	137·51	151·14	180	Ran
10	18·8	6	6	9·2	44·4	6	6	6	70	15·6
20	34·4	4	4	4	87·2	5·6	4	2·4	140	13·6
50	83·76	1·84	1·84	5·04	216·88	3·12	1·84	0·56	350	10·8
100	166·88	1·12	0·8	9·12	433·44	4·96	1·12	0·48	700	19·4
{8}	60	77·96	99·5	100	120	130	137·51	151·14	180	Ran
10	4·4	1·2	2·8	2·8	17·2	2·8	1·2	1·2	30	9·2
20	7·2	1·6	0·8	1·6	33·6	3·2	0·8	1·6	60	3·2
50	16·88	0·56	0·24	2·48	83·44	1·2	0·56	0·24	150	5·0
100	33·44	0·32	0·32	4·48	166·72	1·92	0·16	0·32	300	4·8
{4}	60	77·96	99·5	100	120	130	137·51	151·14	180	Ran
10	1·2	0·4	0·4	0·4	3·6	0·4	0·4	0·4	10	2
20	2	0·4	0	0·4	6·8	0	0·4	0·4	20	0
50	5·2	0·08	0·08	0·72	16·72	0·08	0·08	0·08	50	0·72
100	10·32	0·08	0·08	1·44	33·36	0	0·08	0·08	100	3·6

{16} χ² < 6·26 = spiral; 6·26 − 27·5 = random; χ² > 27·5 = distichous or whorled.

{8} χ² < 1·69 = spiral; 1·69 − 16·0 = random; χ² > 16·0 = distichous or whorled.

{4} χ² < 0·22 = spiral; 0·22 − 9·35 = random; χ² > 9·35 = distichous or whorled.

Bold indicates random phyllotaxis; italic indicates distichous or whorled phyllotaxis; normal type indicates spiral phyllotaxis.

Fig. 2.

χ² values for degrees of freedom ranging from 2 to 18. The confidence interval (α = 5 %) bounds the zone inside which the hypothesis of a random phyllotaxis is not rejected. Symbols represent phyllotactic patterns.

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.

Table 3.

χ² values for 50 primordia of which 10, 20 or 30 primordia are distributed randomly

	137·51	77·96	99·5	100	130	151·14	60	120	180
10	8·88	4·40	5·68	5·68	2·48	6·18	62	152·24	249·52
20	8·24	6·18	5·04	6·96	5·68	8·24	38·96	83·76	134·96
30	10·16	10·16	6·96	6·96	8·24	6·96	20·40	33·84	57·52

Values for which the distribution of primordia is not significantly different from a random distribution are in bold type (k = 16 sectors; α = 5 %).

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 (λ = 50/16), and calculate the χ² 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 χ² 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 χ² (28·08 > 27·50) falls outside the boundaries of the random zone (Fig. 3).

Fig. 3.

Increase of χ² values when opposed primordia are regularly added into sectors one and nine from a random distribution, such that the total number of primordia remains equal to 50. The line at χ² value of 27·5 shows the boundary between random and whorled patterns.

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.

Fig. 4.

χ² values calculated for sho mutants (sho1-1, sho2, sho3) described by Itoh et al. (2002) as a function of the degrees of freedom. In sho2P, sectors one and nine were removed from sho2 data. The confidence interval (see two lines on graph) (α = 5 %) bounds the zone inside which the hypothesis of a random phyllotaxis is not rejected.

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 λ is estimated by the sample mean , then

is approximately a standardized normal variable. The sum of squares of k independent standardized normal variables follows a χ² law with k − 1 and not k degrees of freedom because of the linear dependence of z_i:

This reduces the numbers of degrees of freedom by one. Then

follows approximately a χ² law with k − 1 degrees of freedom (Rahman, 1968).