Aberrant phyllotactic patterns in cones of some conifers : a quantitative study

The scale patterns of 6000 cones from one single tree of Pinus nigra Arn. have been examined. Apart from the main Fibonacci pattern with 8 and 13 parastichies, nine aberrant spiral patterns with Fibonacci-type sequences have been found. They are quite rare and occur with different frequencies. The parastichy quotient 8/13 of the prevalent pattern is very close to the golden ratio 0.618. In case of the black pine it appeared that the greater the deviation of the parastichy quotient m/n from 0.618, the rarer the pattern. Similar results obtained for the sample of 1506 cones collected from three individual trees of larch (Larix decidua Mill.) suggest a true correlation between the frequency of a pattern and the deviation of its parastichy quotient from the golden ratio.


Introduction
Looking at European black pine cones from below (Fig. 1), curved rows of scales running in two opposite directions can be observed, one clockwise, the other counter-clockwise.These conspicuous rows are called contact parastichies.When counted, they are found in paired numbers of the Fibonacci sequence: in open Pinus nigra cones there are usually 8 parastichies running in one direction and 13 parastichies the other way round.Personal observations of the author of the present study have shown (unpublished data) that in other species of conifers, the patterns are often seen in lower expression of the Fibonacci sequence, e.g. with 5 and 8 parastichies (Picea abies or Larix decidua) or 3 and 5 parastichies (Sequoia sempervirens).However, there are exceptions to the rule: in most species of conifers, always quite rarely but in different frequency, aberrant patterns of cones show different parastichy numbers.They belong to the "Fibonacci-type" sequences [1]; as in the main Fibonacci sequence, each number is the sum of the previous two (Fig. 2).The aim of the present study was mainly to compare the phyllotactic diversity of one single Pinus nigra tree with pooled data of other species.

Material, methods and results
In search for aberrant patterns in European black pine, 6000 cones from one single tree have been examined, almost its whole cone production of about two years.This tree was planted more than 60 years ago in a garden in Küsnacht near Zürich, Switzerland at 560 m altitude.
The open Pinus nigra cones have been selected for the study because their patterns can be neatly documented with single photographs and do not need unrolled surface techniques like vegetative shoots or magnolia floral cones [2].
The wide and distinct difference in frequency of aberrant patterns was the reason to start searching for possible correlations with mathematical or geometrical properties of the parastichy pattern.Looking at the opposite parastichy pairs (called m and n) of the various patterns, an apparent correlation of the parastichy quotient m/n with the frequency of the pattern can be found.Generally, the greater the deviation of the m/n quotient from the value of 0.618 (the golden ratio phi), the less frequent a pattern, with very few exceptions (Tab.1).Tab. 1 Deviation of m/n quotient from the golden ratio phi (= 0.618), and m:n pattern frequency in 6000 Pinus nigra cones.
The greater the deviation, the rarer the pattern, with very few exceptions.The deviation indicates the relative order of pattern frequency only within the same species.
Compared with Larix decidua cones (Tab.2), deviation is generally lower in Pinus nigra cones with all patterns in higher expression of the same sequences.
For comparison with other species of conifers, 1506 larch cones produced by three different individuals of Larix decidua Mill.growing in a forest near the location of the Pinus nigra tree in the study, were examined.It appeared that even though Larix decidua produces aberrant patterns in a much higher proportion (22%) than Pinus nigra (3%), the order of frequency of the different patterns remains mainly the same.It is identical for the first three patterns, and again the parastichy quotients m/n of the most frequent patterns are very close to the golden ratio 0.618.The m/n quotients of the very rare patterns 6:8 (bijugy of the first accessory pattern) and 6:7 are quite far removed from 0.618 (Tab.2).All Larix decidua patterns are observed in lower expression than the Pinus nigra patterns (e.g.4:7 instead of 7:11), and their parastichy quotients generally deviated more from 0.618 than those of Pinus nigra patterns.
The distribution of different phyllotactic patterns can be illustrated with a phyllotactic grid [2][3][4][5].The squares in the grid (Fig. 3) represent the individual "home address" of a given pattern.Into the grid a black line from square 0:0 to square 34:55 (outside this graph) has been drawn.We might call this line the "Fibonacci line" because it represents an m/n parastichy quotient of 0.618, or golden ratio, in good approximation.The lower expressions of the main Fibonacci sequence (black squares 1:2, 2:3, 3:5) zigzag visibly around the black line.In higher expressions, the zigzag approaches the black line more and more but will never reach it quite exactly.Thus, the pattern 8:13 (Pinus nigra) with its higher expression of the main Fibonacci sequence is nearer the black line than the pattern 5:8 (Larix decidua).It can be seen on the grid that all other patterns -observed or not -deviate less from the black line in their higher expressions (e.g.7:11) than in their lower expressions (e.g.4:7).Tab. 2 Deviation of m/n quotient from the golden ratio phi (= 0.618), and m:n pattern frequency in 1506 Larix decidua cones.
The greater the deviation, the rarer the pattern, with very few exceptions.The deviation indicates the relative order of pattern frequency only within the same species.Compared with Pinus nigra cones (Tab.1), deviation is generally higher in Larix decidua cones with all patterns in lower expression of the same sequences.Quite obviously, the possible range for aberrant cone patterns is limited.Observed aberrant patterns (red on Fig. 3 for Larix decidua, and green for Pinus nigra) are clustered around the main Fibonacci pattern with its species-specific expression.The further away from the black line, the greater the deviation of the parastichy quotient m/n from 0.618, and the rarer the pattern of a cone within the species-specific spectrum of patterns.The patterns most distant from the Fibonacci line are the Larix decidua patterns 6:8 (bijugy of the first accessory pattern) and 6:7, and the Pinus nigra patterns 6:11, 9:13 and 7:10.All of them are exceedingly rare.
There remains quite a substantial number of unexplained irregular patterns (Tab. 1, Tab. 2).In more than half of these difficult cones, one set of parastichies is quite readable while the opposite set is not.Sometimes a single parastichy may be seen to disappear or to be added to an already established pattern, disturbing and deforming the course of parastichies (Fig. S4).Usually, it is quite impossible to classify such irregular patterns (Fig. S5).Very rarely, the resulting phyllotactic transition from one pattern to another pattern can be deciphered in such cases (Fig. 4).

Discussion
Phyllotactic diversity in conifers was observed and published almost 200 years ago.When studying Picea abies cones, Alexander Braun [6] noted in 1831 that a small minority did not show the main Fibonacci sequence, but other patterns; most often the sequences 1, 3, 4, … (first accessory) and 2, 4, 6, … (bijugy of the main Fibonacci).His early observation of the most frequent aberrant patterns was confirmed by later researchers.The largest quantitative study was made by Beata Zagórska-Marek [7] in 1985 with 3200 vegetative shoots of 155 Abies balsamea trees.Her most frequent aberrants were the same two sequences 1, 3, 4, … (first accessory) and 2, 4, 6, … (bijugy).Rolf Rutishauser with his coauthors (unpublished data) studied 2055 Picea abies cones from an unknown number of trees.Again, his most frequent aberrant sequences were bijugy and first accessory, found in 71 cones.In 1994, Roger Jean [1] pooled the data from 12 750 observations on more than 650 species, not all of them conifers, and found bijugy the most frequent aberrant sequence, followed by the first accessory.In 1998, Iliya Vakarelow [8] studied both vegetative shoots and cones of conifers.In Pinus mugo, he found an influence of altitude on the phyllotaxis of the shoots but not on that of the cones, and a very much higher frequency of aberrant patterns in shoots than in cones, especially in high altitudes.
The aim of this study was to find as many different aberrant patterns as possible in cones from one single conifer, and to compare the yield of one single genet with the existing theories [1] or with the results of Beata Zagórska-Marek's large quantitative study based upon pooled genetic material from Abies balsamea shoots [7].With a single tree one can study neither intra-or interspecific differences nor environmental influences.At the same time, in this case the genetic uniformity is ensured to study phenotypic plasticity effects, and cones have an important sampling advantage: one can pick up practically all cones of a tree, whereas with vegetative shoots of larger conifers, one is limited to the accessible parts which might not represent the overall distribution of patterns.The results presented in this work show that the genetic information of one single Pinus nigra tree is sufficient to produce almost the same range of phyllotactic patterns as a very large number of conifer trees pooled together.
The statistical analysis of even a quite respectable harvest presents some difficulties.Suppose that in a sample of 6000 cones, a rare pattern occurs with a theoretical incidence of 1:6000.Then the probability to find just one single cone with this unusual pattern is only 37%.The chance to find no cone of this same pattern is almost equal, followed by a probability of about 19% to find two such cones in a 6000 cone sample.Thus, even with a sample of 6000 cones, statistically significant differences are not to be expected for very rare patterns.On the other hand, the consistently large m/n deviation of the "almost-never-reported" patterns is quite suggestive.What we can observe is at least a conspicuous trend from highest to lowest pattern frequencies if we range them according to the deviation of their m/n quotient from 0.618 (Tab. 1, Tab. 2).Quite decidedly, an ideal study would encompass several thousand cones per tree, from a considerable number of trees of the same and of other species, all in different environments and in different seasons, all trees separately analyzed: a task which is quite beyond the capacity of one single collector.
Hypotheses about the reasons for the empirically stated order of spiral pattern frequency have been made before.In computer simulations, based on the geometric model of phyllotaxis, the main Fibonacci pattern, its bijugy and the first accessory (Lucas) pattern appeared to be developmentally the most stable, which may explain their prevalence in nature [4].Roger V. Jean's entropy-based model in turn may be sorely contradicted by the existence of the 7:10 and 8:11 patterns found at first in balsam fir's vegetative shoots [7], later in magnolia flowers [2] and now also in the cones of Pinus nigra analyzed in the present study.According to the model these patterns shouldn't exist [1], but otherwise, the order of frequency of cone patterns corresponds quite well with it.Roger Jean's model and this study have the following identical results: (i) the three most frequent aberrant patterns are (in descending order) the sequences 2, 4, 6, … (bijugy), 1, 3, 4, ... (first accessory) and 3, 6, 9, … (trijugy); (ii) multijugate patterns are found in descending frequency from bijugy down to trijugy down to tetrajugy patterns (Fig. S1), and so on; (iii) accessory patterns are found in descending frequency from the sequences 1, 3, 4, … (first accessory) down to 1, 4, 5, … (second accessory) down to 1, 5, 6, … (third accessory; Fig. S2), and so on.
If we assume a correlation of pattern frequency with the deviation of their m/n quotient from the golden ratio: how would it agree with actual hypotheses about pattern formation?In recent models of phyllotaxis employing the concepts of inhibitory fields or auxin fluxes not much is being said about the reasons for empirically detectable differences in pattern frequency [9][10][11].
The fertile concept that spiral patterns are determined by the ratio of primordium size to the meristem size (P/M ratio) [2,5,12] seems to be promising in future attempts to study phyllotactic diversity.Pattern transitions are thought to result from change of available space for the primordia on the shoot apical meristem (SAM) during ontogeny [4].Many transitions have been described in Abies balsamea shoots [7] and in Magnolia flowers [2], and strikingly similar patterns have been created with computer models [4,12] when changing the ratio between primordia size and meristem circumference.Aberrant patterns might be seen as adaptation to an unusual P/M ratio right from the start.Primordia and meristem size are thought to be genetically determined [4,5,12] but might also be influenced by environment [8].
The P/M ratio can vary in both directions: primordia can be too small or too big in relation to meristem size.Therefore, in adaptation to altered P/M ratios, m/n quotients would have to show deviations in both directions from the standard value.And so they do, as some patterns have positive, other patterns have negative deviations from 0.618 (Tab.1,Tab.2).
As a consequence, the m/n hypothesis would seem to be limited to the species where the main Fibonacci sequence is found in a great majority.Species or genets with other standard patterns than the main Fibonacci sequence exist [2,3,13], and on the phyllotactic grid, their aberrant patterns should be expected in clusters around their own standard pattern, with rarest patterns at the edge of the cluster.Further studies are certainly needed.
The correlation between the frequency of spiral patterns in conifers and the deviation of their m/n quotient from the golden ratio is a hypothesis which has not been considered before in phyllotactic literature.It is a much simpler mathematical description of phyllotactic diversity than Roger Jean's entropy model [1].It might be an interesting alternative: for Fibonacci phyllotaxis, it has its important majority of results in common but does not forbid certain patterns to exist (e.g.7:10 or 8:11) which have been found in nature [2].Moreover, it accepts the existence of even more exotic patterns (e.g. the 6, 13, 19, … sequence in Picea abies [14], or the 9:13 Pinus nigra pattern in this study) which were not foreseen, and therefore not even discussed, by Roger Jean.

Fig. 1
Fig.1Pinus nigra -a cone with the expression 8:13 of the main Fibonacci sequence (1, 2, 3, 5, 8, 13, 21, …).Three photographs of the same cone seen from below.In the middle and right photograph, the contours of the parastichies are marked in both directions for better visibility.The scales are numbered from the edge to the cone base (center).

Fig. 3
Fig. 3 Phyllotactic grid with aberrant patterns of Larix decidua (red squares), and Pinus nigra cones (green squares).Patterns of the main Fibonacci sequence (black squares) lie almost (but never quite exactly) on the black line which represents an m/n quotient of 0.618.Frequent aberrant patterns (6:10 and 4:7 for Larix decidua, 10:16 and 7:11 for Pinus nigra) are positioned near the black line.Rarest patterns (6:8 and 6:7 for Larix decidua, 6:11, 7:10 and 9:13 for Pinus nigra) lie far off the black line.The dotted line shows the symmetry axis of the phyllotactic grid.

Fig. 4
Fig. 4 Pinus nigra -an unusual irregular cone, starting at the cone base (center) with 8 parastichies clockwise and 11 parastichies counter-clockwise.Two parastichies (colored) are added later on in opposite directions.At the edge of the cone, the pattern reads 9:12 instead of 8:11.The numbers here mark the parastichies, not the individual scales.The scales themselves cannot be numbered to an uniform pattern.