Application of Two Mathematical Models to the Araceae, a Family of Plants with Enigmatic Phyllotaxis

Transcription

1 Annals of Botany 88: 173±186, 2001 doi: /anbo , available online at on Application of Two Mathematical Models to the Araceae, a Family of Plants with Enigmatic Phyllotaxis ROGER V. JEAN* and DENIS BARABEÂ { *80 Grande AlleÂe est, suite 904, QueÂbec, (QC), Canada G1R 5N1 and {Institut de recherche en biologie veâgeâtale, Jardin Botanique de MontreÂal, 4101 rue Sherbrooke est, MontreÂal, Canada H1X 2B2 Received: 19 July 2000 Returned for revision: 27 November 2000 Accepted: 21 February 2001 Published electronically: 18 May 2001 In this paper we show that two mathematical models can be of great help in the analysis of observational data, in this case the di cult and little studied phyllotactic phenomena that occur in the Araceae family. We apply the Fundamental Theorem of Phyllotaxis, together with an explanatory model of phyllotaxis, to plant specimens of this family, to obtain phyllotactic parameters and information that cannot be otherwise obtained. Most signi cant is the fact that the two models show evidence of regularities in the overwhelming diversity of the patterns observed in the Araceae (essentially Dracontium and Anthurium) characterized by discontinuous transitions. In particular, this work reveals the regularity of the behaviour of the divergence angle in the specimens analysed. Features of the in orescences of Dracontium, especially the presence of whorls, are compared to those observed in in orescences of Anthurium (characterized by the absence of whorls), and in the capitulae of Compositae (characterized by continuous transition). We question the possible meaning at the genetic level of the diversity of patterns observed at the macroscopic level. # 2001 Annals of Botany Company Key words: Phyllotaxis, Araceae, mathematical models, in orescence, development. INTRODUCTION As Arthur Eddington (the famous astronomer who veri ed Einstein's general theory of relativity) put it: `Gentlemen, you do not have a science unless you can express it in numbers'. To be more complete, at least in the case of phyllotaxis, we should add that we do not have a science unless we nd applications of it. Here we present data on Araceae, and propose a particularly meaningful application of previously developed theoretical tools (Jean, 1994). The aim of the paper is to show that the data can be analysed and given full meaning by putting them into quanti ed models. Our study underlines the crucial role of the theoretical approach in the analysis of observational data. It raises questions related to development and growth, thus redirecting attention to the specimens of plants analysed and to the observations reported. This highlights the importance of modelling, still considered by some as limited, to describing what is already known. This underlines too the rewarding advantage of multidisciplinary exchanges (e.g. the collective book by Jean and BarabeÂ, 1998a), here between theoretical and empirical biology. In nature there are two general types of phyllotactic transitions: continuous and discontinuous. A continuous transition is the phenomenon by which a structure passes during growth from a (m, n) pattern, m 5 n, to the pattern (m n, n), where the smaller number m is replaced by the sum m n, and vice versa as happens in Compositae. The transition can move continuously along consecutive integers of Fibonacci-type sequences (such as (1) and (4) introduced below) de ned by the same addition rule. In discontinuous transitions the change from one phyllotactic system (m, n) to the next is generally caused by the addition or removal of one or two parastichies (spirals) either in just one or in both families of parastichies; for example from (m, n) to(m, n 1) or to (m 1, n) (see Church, 1904, for diagrams, and Zagorska-Marek, 1994, on Magnolia). The Araceae display a great diversity of phyllotactic patterns on individual specimens and are characterized by discontinuous phyllotactic transitions. Discontinuous transitions are apparently much less frequent in the plant kingdom than continuous transitions, and are certainly more enigmatic (Jean and BarabeÂ, 1998b). The challenge in the case of the in orescences of the Araceae is to bring order to the irregularities by using models rst devised from the regularities observed in the plant kingdom in general; we will see that this can be done with a satisfying degree of success. The tools we will use are the Fundamental Theorem of Phyllotaxis (FTOP; Jean, 1994), also called the Adler-Jean theorem in the literature, and a Jean model based on the ideas of branching and minimization of entropy (Jean, 1994), which are, respectively, a descriptive and an explanatory model of phyllotaxis. The data analysed are from two sources (a) from recent observations on young in orescences of Dracontium polyphyllum (by Poisson, 1997), and (b) from mature in orescences of two species of Anthurium (by one of us). Our purpose in analysing mature in orescences of Araceae (Anthurium) is to discover up to which point their phyllotactic organization is similar to that reported for young spadices of Araceae (Dracontium). The comparison will suggest how the spadices of Araceae and their patterns in general can be a ected by growth. We will be able to /01/ $35.00/00 # 2001 Annals of Botany Company

2 174 Jean and BarabeÂÐApplying Two Mathematical Models to the Araceae check if the regularities, observed and deduced, in young spadices still hold in mature apices. We will also see how we can analyse the patterns in terms of energy (the Jean model), revealing unsuspected regularities; from this we will be able to compare the Araceae with the Compositae where transitions are continuous. Our analysis concerns the changes occurring in the phyllotactic patterns and in their entropy levels from the base to the top of the in orescence. What is important here is not to characterize the diversity of phyllotactic patterns occurring in a genus, but to study the type of transitions occurring in the in orescences. We show the importance of the spadix of the Araceae for the study of discontinuous transitions in phyllotaxis, and we report phyllotactic patterns or transitions that have not previously been reported. Data are a test for the validity of models; but models can also help to question and to organize data, to uncover their meaning, and to interpret results. There are cases reported in the literature (e.g. Jean, 1994, appendix 6) where data can be severely questioned because (a) an improperly de ned protocol was used to gather it, (b) there was insu cient understanding of the knowledge and concepts involved, and (c) there was hidden traps in data collection. The problem of recognizing patterns in regular theoretical diagrams was emphasized recently, as it was shown that one pattern can be easily mistaken for another (see Cummings and Strickland, 1998; Jean, 1999). Since the patterns seen in nature are less regular, it can be even more di cult to recognize them. Being conscious that the di culties involved in practical pattern recognition must not be underestimated, we will determine what conclusions can be inferred from the data (based on the application of two theoretical models), we will check how the data t into theoretical frames of reference, discuss the coherency of the whole data-models construct in particular with respect to whorls, and propose an interpretation of the data. MATERIALS AND METHODS In orescences of Araceae Young in orescences of Dracontium polyphyllum. The mature spadices of D. polyphyllum have a more or less cylindrical shape. Their diameters decrease gradually in the apical direction, with a narrowing at the very tip of the apical part (Fig. 1). The numbers of parastichies (m, n) (Table 1) were recorded on 17 young in orescences ranging in size from to cm, and on two mature specimens of 4.2 and 6.35 cm. As is the case for other members of the Araceae (BarabeÂ, 1995), the development of the in orescence of D. polyphyllum { { { { mm 1 mm A B FIG. 1. Schematic representation of two in orescences (No.2, No.16 of Table 1) showing the four equal regions mentioned in the text, and opposed contact parastichy pairs (arrows). A, Spadix No.2, in phase A; B, spadix No.16 in phase B. Note the presence of a sudden narrowing in region 4, and the reduction of the parastichy numbers.

3 Jean and BarabeÂÐApplying Two Mathematical Models to the Araceae 175 TABLE 1. Patterns (m, n) from Poisson's observations (1997), length of the spadices, phases A and B, theoretical interval for d for spiral systems (m, n) calculated from the FTOP, in degrees, and the value of calculated from a Jean model Specimen Region 1 Region 2 Region 3 Region 4 Length of spadix (cm) No.1 (m, n) (11, 12) (10, 10) (10, 10) (7, 8) 0.247(A) d 30± ± No.2 (m, n) (11, 11) (9, 10) (9, 10) (6, 7) 0.217(A) d 36±40 36± ± No.3 (m, n) (11 12) 2(5, 6) (9, 10) (7, 7) 0.494(B) d 30± ±36 36± No.4 (m, n) 4(2, 3) 2 (4, 5) (8, 9) (5, 6) 0.282(B) d 30±45 36±45 40±45 60± No.5 (m, n) (10, 11) (9, 9) (9, 10) (7, 8) 0.380(B) d 32.7±36 36±40 45± No.6 (m, n) (10, 11) (9, 10) (9,10) (7, 8) 0.554(B) d 32.7±36 36±40 36±40 45± No.7 (m, n) (13, 14) (11, 11) (10, 11) (7, 9) 0.431(B) d 25.7± ± ± No.8 (m, n) 2(5, 6) (8, 9) (8, 8) (5, 5) 0.510(B) d 30±36 40± No.9 (m, n) (9, 10) (9, 9) (8, 9) (6, 7) 0.595(B) d 36±40 40± ± No.10 (m, n) (12, 13) (11, 12) (8, 9) (6, 7) 0.651(B) d 27.6±30 30± ± ± No.11 (m, n) (12, 13) (11, 12) (10, 10) (7, 8) 0.745(B) d 27.6±30 30± ± No.12 (m, n) 2(5, 6) (10, 10) (7, 7) (5, 7) 0.582(B) d 30±36 144± No.13 (m, n) (12, 13) (13, 13) (11, 11) (9, 9) 1.043(B) d 27.6± No.14 (m, n) 3(4, 5) (11, 11) (9, 9) (8, 9) 1.137(B) d 15±24 40± No.15 (m, n) 2(4, 5) (7, 8) (7, 8) (6, 7) 0.788(B) d 36±45 45± ± ± No.16 (m, n) (11, 13) (10, 11) (9, 10) (7, 8) 1.067(B) d 163.6± ±36 36±40 45± No.17 (m, n) (11, 11) 2(5, 6) (9, 10) (8, 8) 0.893(B) d 30±36 36± No.18 (m, n) (9, 13) (10, 11) (8, 8) (7, 8) 4.2(B) d 80± ±36 45± No.19 (m, n) (13, 15) (11,14) 2(5, 7) 3(3, 4) 6.35(B) d 166.2± ± ±77 30±

4 176 Jean and BarabeÂÐApplying Two Mathematical Models to the Araceae FIG. 2. Photographs of two typical spadices of Dracontium, specimens No. 1 and No. 2 (Table 1) (taken by G. Poisson). Arrowheads indicate a discontinuous transition. can be subdivided in two distinct developmental phases: (a) before the initiation of oral parts, and (b) during and after initiation of oral parts. The rst phase ends when the in orescence reaches a length of approx cm (4 % of the adult size). The length of mature in orescences varies from 4 to 7 cm. In the rst phase, a phyllotactic system is already established (Fig. 2), and it is preferable to study phyllotactic organization in this phase. However, it is extremely di cult to obtain material in this phase (thus only in orescences No. 1 and No. 2 (Table 1) are in phase A) ( for a complete description see Poisson and BarabeÂ, 1998). In contrast to a stem with leaves for example, where new primordia are in principle endlessly added at the tip, a spadix is a closed system. This means that for the Araceae we also have a growing tipða bare apexðbut this zone is soon transformed into one, two or three primordia (for an illustration see BarabeÂ, 1995, p. 920). The system is then closed, and lengthening of the spadix is just a matter of increasing its volume and that of the primordia which are nally transformed into tiny owers. The spadices observed by Poisson (1997) had already experienced an evolution of the phyllotactic systems along their length. Consecutive developmental stages of one in orescence should be studied to make a direct correlation between growth and phyllotactic patterns. To make such a correlation, one has to observe the same in orescence at di erent times, from its initiation until it is fully developed. Because an in orescence has to be destroyed to be observed under the microscope, such a study is impossible. Consequently, one must make observations on samples of di erent sizes, coming from di erent plants, to analyse the transitions occurring along the in orescence during growth. Development of the in orescences of the Araceae is not continuous as in the case of the vegetative apex, and it is not possible to follow up the initiation of each single primordium during the early stages of development. As has been shown in the case of other Araceae, the primordia appear more or less simultaneously on the surface of the in orescence (Barabe and Lacroix, 2000). Spadices were divided by Poisson (1997) into four equal regions, 1±4, from base to top (Fig. 1). The upper part of region 4 is characterized by an abrupt narrowing. Each region was made of several levels of primordia in close contact, at least in phase B (Fig. 1B). The number of levels included in each region varied depending on the length of the in orescence. For example, in orescence No. 2 (Fig. 1A. Table 1) has approx. three levels of primordia by region, and in orescence No. 16 (Fig. 1B, Table 1), four levels. The parastichies on the surface of the three-dimensional spadices were counted using a binocular magnifying glass. Two opposed families of parastichies were recognized in each region of each specimen, the parastichy pair making an angle close to 908, and the number of m spirals in a family and n in the other direction ( presumably the opposite direction with respect to the axis of the spadix) were counted. We will address the problem raised by the whorls reported by the observer. For two-dimensional diagrams (Fig. 1) or photographs (Fig. 2) representing typical spadices analysed by the observer, it must be emphasized that it is di cult to judge if there is e ectively any whorled pattern involved (when m ˆ n). Normally however, it is easy to determine visually, by looking at a diagram, whether a pattern is whorled or spiral. Another fact is that the observer did not unroll to a plane any spadix, to show the organization of primordia all around it. Although it is more or less easy to apply an unrolling technique on mature specimens measuring a few centimetres, it is impossible to perform these techniques on specimens only a few millimetres long. In the case of young specimens of Dracontium, Poisson (1997) used a stereo-microscope to count the number of spirals. At this scale, the most error-free method is probably to count the number of parastichies directly on the spadix, particularly if the form of the in orescence is not perfectly cylindrical, as is the case in the Araceae. In Table 1 there is no value for the divergence angle associated with the reported alternating whorls (m, m). There are many reasons for this. One is that unlike perfect whorls, the slope of a family of m spirals in a direction may not be equal to the slope of the opposed family of m spirals. This produces an interval of values for the divergence angle between whorls, around the theoretical value of 3608/(2m) (2m orthostichies) for alternating whorls. This interval cannot be given by the FTOP, but only by observation. A peculiar fact, that will be addressed in the section entitled `The divergence angles for whorls', is that when the blanks in Table 1 are lled by the theoretical values of the divergence angle for superposed whorls, i.e. 3608/m (m orthostichies), then these latter values t exactly in between the adjacent values in the table for spiral systems. The data for the 19 in orescences of Dracontium are discussed below. Mature in orescences of Anthurium. A summary of the data gathered on two mature in orescences of Anthurium upalaense Croat & Baker, and on one mature in orescence of Anthurium salvinae Hensley is displayed in Tables 2±4,

5 Jean and BarabeÂÐApplying Two Mathematical Models to the Araceae 177 TABLE 2. Phyllotactic parameters for Anthurium upalaense Region p (m, n) Type d (17, 22) Ab ± (7, 11) N ± (7, 10) Ab ± (13, 20) N ± (12,19) An ± (2, 3) N ± (12, 17) Ab ± (11, 16) Ab ± (5, 8) N ± (2, 3) N ± (5, 7) An ± (5, 7) An ± (9, 14) N ± (4, 7) N ± (4, 7) N ± (4, 7) N ± (4, 7) N ± (4, 7) N ± (4, 7) N ± (4, 7) N ± (4, 7) N ± (8, 13) N ± (8, 13) N ± (8, 13) N ± (2, 3) N ± (7, 6) N ±60 The values of m and n in (m, n) were recorded at distances of 1 cm from the base to the top of the in orescence. In this table Ab, N and An mean respectively: aberrant pattern, normal pattern [expressed by sequence (1)], and anomalous pattern [expressed by sequence (4)]. The perimeter, p, (cm) of the in orescence is also indicated. The intervals for the divergence angle (d) for spiral systems were calculated from the FTOP, in degrees, and the value of calculated from a Jean model illustrating the wide diversity of phyllotactic patterns in Araceae. The specimens were collected at the Montreal Botanical Garden, Canada (Registration Nos: 2061±68 and 2042±68). The length of these in orescences greatly exceeds that of Dracontium (Table 1), ranging from 26 to 44 cm. In the three specimens, the circumference at the base of the in orescence is approx. three-times greater than that at the top. Two parameters were measured every centimetre from the base to the top of the spadix: (1) the number of opposed contact parastichy pairs (m, n) (angle closest to 908) and (2) the circumference of the in orescence. In Tables 2±4, m is sometimes (though rarely) greater than n (instead of m 5 n, as is generally the case): this means that the chirality of the phyllotactic system changes. The data for the three large specimens of Anthurium are discussed below. The delimitation of regions. In the case of Dracontium, the spadix was divided into four equal regions, and in the case of Anthurium, the numbers of parastichies were recorded at each centimetre, as mentioned previously. Does this sectioning of the spadices have any consequence on the identi cation of the type of transition (continuous or discontinuous), on the interpretation of the data and results? The answer is no. Indeed, in the case of nearly all TABLE 3. Phyllotactic parameters for Anthurium upalaense Region p (m, n) Type d (17, 28) N ± (17, 28) N ± (17, 27) Ab ± (17, 27) Ab ± (17, 26) N ± (17, 22) Ab ± (17, 22) Ab ± (17, 22) Ab ± (16, 21) Ab ± (16, 19) Ab ± (16, 19) Ab ± (4, 5) N ± (5, 7) An ± (5, 7) An ± (2, 3) N ± (2, 3) N ± (2, 3) N ± (2, 3) N ± (2, 3) N ± (2, 3) N ± (2, 3) N ± (2, 3) N ± (2, 3) N ± (2, 3) N ± (2, 3) N ± (2, 3) N ± (2, 3) N ± (2, 3) N ± (2, 3) N ± (2, 3) N ± (2, 3) N ± (2, 3) N ± (2, 3) N ± (2, 3) N ± (2, 3) N ± (13, 20) N ± (13, 20) N ± (13, 8) N ± (3, 2) N ± (11, 8) Ab ±135 See Table 2 for details. specimens of Dracontium, if one considers the size of the ower and the length of the zone, one sees (Figs 1 and 2) that there are no more than two or three levels of owers in each region. In the larger specimens, this number can be three or four, as is also the case for the specimens of Anthurium. Therefore, by counting the number of spirals in each region we obtain a very good indication of the type of transition taking place along the in orescences, given that discontinuous transitions occur with the addition or subtraction of just one or two parastichies. The region is too small to allow the presence of a continuous transition involving a greater number of parastichies. The patterns of discontinuous transitions observed on the in orescences of Dracontium and Anthurium are similar to those observed in other genera where the numbers of spirals are counted continuously along the in orescence. The method used in the present analysis is therefore equivalent to the one generally attested to in the literature.

6 178 Jean and BarabeÂÐApplying Two Mathematical Models to the Araceae TABLE 4. Phyllotactic parameters for Anthurium salvinae Region p (m, n) Type d (1, 1) (36, 41) Ab ± (19, 21) An ± (19, 21) Ab ± (38, 45) Ab ± (38, 45) Ab ± (38, 45) Ab ± (38, 45) Ab ± (36, 47) Ab ± (35, 44) Ab ± (5, 6) N ± (35, 38) Ab ± (32, 39) Ab ± (25, 37) Ab ± (11, 19) Ab ± (22, 37) Ab ± (22, 37) Ab ± (22, 37) Ab ± (23, 36) Ab ± (23, 36) Ab ± (11, 18) N ± (11, 18) N ± (11, 18) N ± (22, 35) Ab ± (22, 35) Ab ± (22, 35) Ab ± (22, 35) Ab ± (22, 35) Ab ± (22, 35) Ab ± (22,37) Ab ± (11, 17) N ± (7, 11) N ± (7, 11) N ± (21, 32) N ± (21, 32) N ± (21, 32) N ± (21, 31) Ab ± (7, 10) Ab ± (10, 13) Ab ± (10, 11) N ± (19, 20) N ± (17, 19) An ± (7, 9) An ± (1, 1) (5, 6) N ±36 See Table 2 for details. MODELS USED IN THE ANALYSIS The Fundamental Theorem of Phyllotaxis Nature of the theorem: a descriptive model. This theorem concerns the fundamental relationship between the divergence angles d 5 1/2 (half of 3608; equivalently d ) and the phyllotactic patterns (m, n), where m and n are positive integers. It states that given a pair (m, n), we are most likely to nd that the divergence angle is within a de nite interval. The pair is known as a visible opposed parastichy pair. At the observational level, (m, n) is more precisely a conspicuous parastichy pair. There are two types of such pairs; the A-type, and the J-type (after Adler and Jean; Reick, unpubl. res.); the latter type only being consistent with van Iterson contact parastichy classi cation, and with inhibitor models of phyllotaxis. Practically speaking, a conspicuous pair is a contact parastichy pair, the oral primordia being in contact throughout the spadix (the interested reader may consult a glossary of these terms in Jean, 1994). Historically, the absence of such a theorem has been the source of many di culties encountered in the search for a sound interpretation of the data in phyllotaxis. It prevented us from having a rm mathematical grasp of it. The theorem is the mathematical foundation of the eldðthe Fundamental Theorem of Phyllotaxis (FTOP). Recently, Marzec (1999) suggested a modi cation of the FTOP so it could be applied to imperfect lattices that have undergone a warping, and he proposed that the divergence angle of a modi ed FTOP would be treated as a variable. After Adler's initial contribution to the FTOP (see Adler et al., 1997) in the rst half of the 1970s, the FTOP was independently rediscovered and used by crystallographers who came to the study of phyllotaxis in the mid 1980s. Earlier, Jean gave it a general formulation, based on properties of visible (not necessarily opposed as in Adler's approach) parastichy pairs in the context of Farey sequences, and also gave particular formulations, presentations and applications of it, involving the relevant Fibonacci-type sequences. Also, general and easily applicable algorithms allowing us to obtain in the most general cases an interval for d from (m, n) and vice versa, were directly deduced from this general FTOP (see Jean, 1994). Practical forms of the theorem. For the purpose of the present paper we now give particular formulations of the FTOP that are needed to establish the intervals for d in the tables, when m and n are the observed parameters representing consecutive terms of the Fibonacci-type sequences (1) and (4) below. For the sequence J 1; t; t 1; 2t 1; 3t 2; 5t 3;... 1 where J (the jugacy) and t are positive integers, the pair of opposed parastichies J(t, t 1) is visible only if the divergence angle is in the closed interval 3608= J t 1 ; 3608= Jt Š 2 This interval contains the limit divergence angle given by the formula d ˆ 3608 J t t 1 Š 1 3 p where t ˆ 5 1 =2. Phyllotaxis expressed by sequence (1) is called normal, meaning that it is the most usual type. For the sequence J 2; 2t 1; 2t 3; 4t 4; 6t 7;... the interval for the opposed pair J(2t 1, 2t 3) is t=J 2t 1 ; 3608 t 1 =J 2t 3 Š 5 and the limit divergence angle is d ˆ t t 1 1 Š 1 =J 6

7 Jean and BarabeÂÐApplying Two Mathematical Models to the Araceae 179 Phyllotaxis expressed by (4) is called anomalous, simply meaning the other spiral type of phyllotaxis, which is not expressed by sequence (1) and which is relatively rare. Phyllotaxes not belonging to (1) or (4) will be called aberrant, meaning that the Jean model discussed below does not allow these patterns to exist at the very origin, i.e. in Wardlaw's subdistal-organogenic region of an apex ( for a discussion of the aberrant patterns see Jean, 1998). The said model permits the existence of (1) and (4) only. Here is an illustration of how to obtain the interval for d from the pair (m, n). Consider the pattern (10, 11) which is typical for the observations reported in Table 1. The pattern is of the type (t, t 1), with t ˆ 10. Sequence (1) is concerned, giving the interval (2) with J ˆ 1, i.e. [3608/ 11, 3608/10], or [32.78, 368]. The limit divergence is obtained from (3); that is Thus if the pattern (10, 11) is observed, we can conclude with a high degree of probability that measurements of divergence angles on the specimen will fall in the interval [32.78, 368] which is around the limit The limit angle may indicate a tendency, and practically speaking this limit angle is more than just one angle in the interval. The reader is referred to Jean (1994, p. 38) for values of the endpoints of the interval for d, when (m, n) is expressed by other consecutive values of m and n in sequences (1) and (4). In the rare cases where the given pair (m, n) is not made of such consecutive terms, such as the aberrant pattern (9, 13) in Table 1, the reader is referred to Jean's general FTOP (1994, chapter 2) giving a general algorithm. This algorithm, valid for all cases, amounts to solving for u and v the diophantine equation j mu nv jˆ1, where m and n are relatively prime, 0 5 u 5 m, and 0 5 v 5 n. The endpoints of the interval for d 5 1/2 are then u/m and v/ n. For the cases (m, m) we have, theoretically, a divergence converging to 1808/m when the whorls are alternating, and converging towards 08 or 3608/m when they are superposed. A relevant explanatory model A brief review of the model. The model is based on a concept of hierarchy used for representing the various spiral phyllotactic patterns (m, n), and on an entropy-like function de ned on the set of hierarchies to calculate their energetic cost. The representation of the patterns (m, n) by hierarchies relates phyllotaxis to the general phenomenon of branching. A hierarchy is a branched structure made of levels of points bifurcating u or not j. The inferred representation springs from many biological factors, among which are the branching of vascular systems, and the branching patterns induced by the translocation of substances injected in leaves (Jean, 1994). It is also supported by the theory of bifurcations in Compositae proposed by the botanists Lestiboudois and Bolle Antriebe (Jean, 1994), and by Church (1904) in a similar theory for explaining Fibonacci phyllotaxis. Many predictions come out of this model, which could be con rmed or invalidated by biologists; the superposed and alternate whorls are seen to be special cases of spiral patterns, and such patterns as spiro-distichy, spiro-decussation and monostichy are obtained from the model. For each hierarchy we can calculate the values ˆ Xw X T ˆ YT kˆ1 Tˆ1 k f k log S T =X T Š 7 8 the latter parameter growing very rapidly with f(k) the number of nodes in level k of the hierarchy, and where S(T), X(T) and w represent respectively the stability, the complexity and the rhythm of growth up to level T of the hierarchy representing that growth or pattern. Parameter represents the production cost of each type (m, n) of spiral pattern. The model uses a principle of minimality expressed with a constraint parameter denoted by P r which is the number of simple and double nodes (primordia) in the rst w ˆ 2, 3, 4,... levels of each hierarchy that can be made with that number of nodes in these levels. These w levels represent a complete cycle of growth, which is repeated to form subsequent levels of the hierarchy. When a cycle is complete, the hierarchy chosen in this set of hierarchies is the one giving minimal cost. For this unique hierarchy it can be proved that w ˆ 2, and that the numbers of nodes in the successive levels then make Fibonacci-type sequences, more precisely sequences (1) and (4) only. In particular, in the set of all hierarchies it is seen that the lowest cost is given for the usual Fibonacci pattern when P r 5 6. It is impossible to give a thorough account of the model here; readers who are interested in the model itself are referred to Jean (1994, 1998). Practical formulae. According to the model, a simple node j gives rise to a double node u, and a double node u gives rise to a simple j and to a double node u, thus producing levels of nodes. There are biological reasons for this (see Jean 1994, chapter 3) [in whorled patterns (m, m) we have simple nodes only, i.e. no bifurcation]. Based on this simple rule we can calculate the parameters in formulae (7) and (8). Let us consider the pattern (m, n), where m 5 n 5 2m. The number of nodes in levels T ˆ 0, 1 and 2 are respectively m, n and m n. Necessarily we need n m double nodes, and m n m ˆ2m n simple nodes in Tˆ0 to obtain n nodes in level T ˆ 1. This gives 2 n m 2m n ˆ n nodes in level T ˆ 1. Thus, in level T ˆ 2, we have 3 n m 2 2m n ˆm n nodes, as expected. Here now are examples of calculation of, applied to the spiral system of the type (m, m 1), most frequently encountered in the Araceae. The numbers of nodes in levels T ˆ 0, 1, and 2 are respectively m, m 1, and 2m 1. We must have one double node in level T ˆ 0 to produce m 1 nodes in level T ˆ 1, so that S 1 ˆ 1/m (one double node out of m nodes). We also have X 1 ˆ 1 m 1 ˆ 1 (always), S 2 ˆ m 1 = 2m 1 (meaning that we have m 1 double nodes, out of 2m 1 nodes), and X 2 ˆ 2 2m 1, so that for the pattern (m, m 1) ˆ log m 2m 1 = m 1 Š 2m 1 log 2 9

8 180 Jean and BarabeÂÐApplying Two Mathematical Models to the Araceae where log is to the base 10. Here when m increases, increases. More generally for the pattern (m, n), where m 5 n 5 2m, wehave ˆ log m m n = n m nš m n log 2 10 As long as we have a spiral pattern (m, n), with m 5 n 5 2m, we can calculate the value of (this is the case for all spiral patterns reported here). According to the model, whorls (m, m) are special cases of spiral patterns, so the value of can also be calculated for them as long as we can recognize the spiral pattern they most probably represent. Practically speaking, the value of for a whorl in Table 1 is between the values calculated for its adjacent spiral patterns. On the other hand, according to many, true whorls do not exist (e.g. Schoute, 1938; see Jean, 1994, pp. 161±163). The model states that superposed whorls are obtained for low values of divergence (approaching 08 for J 4 0), and alternating whorls for high values (approaching 1808 when J ˆ 1) (for a visualization of this see Jean, 1994, p. 174). From (10), at least for large m and n. m n log 2 2 log t 11 RESULTS AND DISCUSSION Analysis of the divergence angle for Dracontium polyphyllum A close-up look at a typical specimen. In the case of D. polyphyllum it is impossible to determine experimentally the divergence angle of the observed systems. The reason is that it is not possible to establish observational links between the system and the time of emergence of the primordia which emerge in bunches all around the spadix below the subdistal zone of the spadix (Poisson, 1997; see also Barabe and Lacroix, 2000). The same can be said about the other specimens analysed here. On the other hand, since the phyllotactic spiral system (m, n) is known for the di erent regions, one can obtain a good approximation of the divergence angle using the FTOP. It gives the intervals for d reported in Table 1. The consideration of divergence for whorls is reported in the section entitled `The case of whorls in Dracontium'. As an example, let us consider in orescence No. 4 showing, in region 1, a system 4(2, 3), i.e. a quadrajugacy (J ˆ 4). Accepting this observation, the angle of divergence is necessarily in the interval [308, 458] around the limit divergence of /4, a value around In region 2, the FTOP indicates that the angle of divergence for the bijugate system 2(4, 5) is to be found between 368 and 458 and that the limit divergence angle is equal to approx Thus, even if the phyllotactic systems in regions 1 and 2 are visually very di erent, the transition from one system to the next means in theory a small variation of about 48 or less in the limit divergence angle. The limit divergence angle here is indicative of the fact that the variation in the real divergence angle is rather small, the two intervals being embedded in one another, and gravitating around the limit angles. In fact, this variation represents a small shift of the parastichy system during the growth of the in orescence. In the third region the same specimen shows a (8, 9) phyllotactic system. We can deduce an interval for the divergence angle: the pair (8, 9) is visible and opposed only if the divergence angle is in the interval [3608/9, 3608/ 8] ˆ [408, 458]. This interval contains the limit divergence angle given by d ˆ t 1 1 that is approx , corresponding to the sequence 51, 8, 9, 17, 26, This suggests that the angle of divergence has still increased in a regular manner less than 48. Finally in the fourth region, the system (5, 6) corresponds to the interval [608, 728] and the limit divergence angle is This represents a substantial increase with respect to the preceding regions, and we are entitled to say that the real divergence increased relatively substantially too. Theoretical considerations show that for the specimen just analysed, the divergence increases from base to top. The increase is inversely proportional to the total number of parastichies observed, a number which from region 1 to region 4 changes from 20 to 18, to 17, and nally to 11. This was predictable because the diameter of the in orescence decreases from the base to the top, and the system moves from a quadrijugate pattern (divergence angles relatively small) to a simple one (divergence angle generally large). Interestingly, the theoretical analysis of specimen No. 4 reveals that the increase of the divergence angle between regions 1 and 3 is gradual, and that the angle reached in region 4 represents an abrupt increase. This latter case corresponds to a relatively rapid decrease in the number of parastichies ( from 17 to 11). In the morphology of the in orescence of D. polyphyllum, one can observe a narrowing of the diameter of the in orescence in the apical part. This narrowing is the same for the spadices of all Araceae. The rapid drop in the number of parastichies for specimen No. 4 suggests that a morphogenetic event occurred, resulting in a sudden narrowing of the spadix in region 4 near the top, and causing a major reorganization of the parastichy system. Generalities and exceptions in Dracontium. The same type of deductions can be made for in orescences Nos 6, 10 and 15. More generally, if we except the divergence for whorls, because of the transitional status of whorls (which would in theory give intermediate values for d, between the values for the adjacent spiral systems), calculations of the intervals for d by the FTOP as noted for in orescence No. 4 show a gradual increase in divergence angle from base to top with just a few exceptions. In orescences No. 7 and No. 12 even showed a dramatic increase of the divergence angle in region 4. This increase corresponded to the presence of sequence (4) of anomalous phyllotaxis in region 4, sequence respectively represented by (7, 9) (t ˆ 3) and (5, 7) (t ˆ 2). There are three exceptions to the rule (increase of d), shown in in orescences Nos 16, 18 and 19. Again, these can be related to the presence of the anomalous phyllotactic patterns (11, 13) (t ˆ 5 in sequence (4) for specimen No. 16), and (13, 15) (t ˆ 6 in sequence (4) for specimen No. 19). In contrast to specimens Nos 7 and 12 however,

9 Jean and BarabeÂÐApplying Two Mathematical Models to the Araceae 181 there is thus rst a dramatic decrease in d, followed by a gradual increase for specimens Nos 16 and 18, while for specimen No. 19 there is a gradual decrease throughout. It must be emphasized that specimens Nos 18 and 19 represent fully developed in orescences, precisely those in orescences which show two aberrant patterns. Notice again that the measures were taken before initiation of ower parts (phase A) for two spadices only (Nos 1 and 2). This alone is su cient to explain the exceptions, given the fact that the measures must be taken in phase A for minimum distortion and for maximum signi cance. The presence of the aberrant systems (9, 13) and (11, 14) in region 1 of in orescence No. 18 and in region 2 of specimen No. 19 respectively (the only two mature specimens), indicates that development can bring important deformations in the pattern initiated in the distal zone of the spadix, where the pattern is meaningful. In the 19 specimens analysed, the divergence angle d generally increases from the base to the tip of the spadix for the spiral systems (in approx 90 % of cases). We will stress in the discussion of whorls in Dracontium that this increase still occurs if we ll the blanks in Table 1 for d for the whorls (m, m) by the divergence 3608/m for superposed whorls. The only exceptions (in Nos 16, 18 and 19 again) are associated with the presence of anomalous sequences, aberrant patterns, and mature spadices. We observe the presence of anomalous and aberrant patterns in ve specimens only (Nos 7, 12, 16, 18, and 19). For specimens Nos 7 and 12 this results in an abrupt increase in d. We observe also that not only do we have discontinuous transitions in the Araceae, but also transitions between normal [sequences in (1)] and anomalous [sequences in (4)] sequences. Summary of the regularities in Dracontium. To summarize, it must be emphasized that in the relatively erratic behaviour of the various patterns (m, n) along the spadices of Dracontium there are, in fact, four phenomena of regularity based on observation, and ve phenomena of regularity evidenced by the two models. That is, generally speaking, we have: (a) Observed: R1, the number of parastichies decreases from base to top; R2, the spiral systems are generally of the type (m 1, m); R3, the patterns (m, m) are reported as alternating whorls; R4, there is a narrowing of the spadices in region 4. (b) Deduced: R5, the length of the interval for the angle of divergence given by the FTOP increases from base to top ( from region 1 to region 4); R6, the angle of divergence increases from base to top; R7, the divergence angles are smaller than 508, while for the usually-encountered spiral patterns in plants the divergence angles are greater than 758, and in most cases greater than 1008 (e.g. the Fibonacci angle ); R8, the entropy decreases from base to top; R9, the spiral systems are generally spiro-superposed [that is of the type (t, t 1) for t 4 5 in sequence (1)], patterns which are in between regular spiral patterns [small values of t in (1)] and superposed whorls [large values of t in (1)]. The deduced regularities are put in evidence by the two models, and could not be put in evidence otherwise. According to FTOP, R1 generally implies R5, R6 and R7. The other model implies R8 and R9. The few exceptions to R6, R7 and R8 point to in orescences Nos 7, 12, 16, 18 and 19, but, in general, only for one level of observation. This stresses the robustness of the regularities mentioned, at least in young spadices. Analysis of phyllotactic parameters for Anthurium Table 2 deals with a mature specimen of Anthurium whose spadix is divided into 26 regions, each 1 cm. We investigated whether the phyllotactic organization of a mature spadix of Anthurium is comparable to that of young in orescences of Dracontium. Tables 3 and 4 contain similar data on two other mature specimens of Anthurium which lead to similar conclusions. In in orescences No. 2 and No. 3 (Tables 3 and 4) there is a greater variation in the number of parastichies than in in orescence No. 1 (Table 2), ranging respectively from 78 to 22, from 45 to 8, and from 22 to 6. This is, of course, correlated with the length of the spadix (44, 39 and 26 cm respectively). Sometimes (m, n) does not entirely change over a distance, leaving the intervals for d and the values of unchanged, even if the perimeter p of the spadix decreases regularly. But the circumference of the spadix may also stay unchanged over some distance leaving m n unchanged, as in Anthurium No. 3 (Table 4). Also the fall in m n becomes abrupt near the tip of the spadix, going from 20 to 13 parastichies in Table 2, and from 33 to 21 in Table 3 (regions 36 and 37). In Table 4, however, there is no such abrupt fall. In many cases, as we reach region 4 in Dracontium, the narrowing of the spadix also produces a sudden fall in the number m n of parastichies. From the base to the top of the in orescences the phyllotactic organization changes irregularly, resulting in the erratic behaviour of the theoretical interval for the divergence angle (Tables 2±4). Also the length of the interval for d is smaller in the case of aberrant patterns (Ab) than in other cases; and a reversal in chirality of the phyllotactic organization can be observed, for example between regions 25 and 26 in Table 2, and regions 36 and 37 in Table 3. However, on the specimens of Anthurium mentioned in Tables 2±4, one can recognize important regularities with just minor deviations. From the base to the top of the apex, there is a: r1, (R1) gradual decrease of the parastichy number m n; r2, gradual decrease in the perimeter p of the spadix; r3, (R8) gradual decrease of the parameter.

10 182 Jean and BarabeÂÐApplying Two Mathematical Models to the Araceae The parameters m n, and vary proportionally to p. In these specimens of Anthurium, m n and behave similarly to those in Dracontium. The length of the intervals for d is rather stable, the values of d being generally tightly squeezed around the respective limit divergence angles. Generally there is a di erence of less than 18 between the endpoints of the intervals for d in the case of specimen No. 3 of Anthurium (Table 4), while for specimen No. 1 (Table 2) which is smaller, the discrepancy between the endpoints is larger. In the case of Anthurium, R5, R6 and R7 do not hold; R1, R4 and R8 still hold; R2, R3, and R9 do not. This points out the great stability of R1 (r1) and R8 (r3)ðvalid for both mature and young spadicesðwhich in turn depend on the variation of the circumference of the spadix (r2). Di erences between Anthurium and Dracontium In Anthurium, the behaviour of the divergence angle d obtained from the FTOP, varies considerably from one centimetre to the next, even when we consider normal phyllotaxis only (N). This remark applies to all three specimens of mature Anthurium (except when (m, n) does not change, as is the case for regions 14±34 in Table 3). This is an important di erence from Dracontium where the divergence behaves regularly as discussed earlier. Concerning aberrant phyllotaxes, the patterns reported in Table 2 for Anthurium are generally either normal or anomalous. In addition, there are four aberrant cases [cases not belonging to sequences (1) and (4)], namely (17, 22), 2(7, 10), (12, 17), and (11, 16). Their presence here can be associated with distortions generated by the growth of the spadix. In Dracontium, there are only two aberrant cases, occurring in the two longest and mature in orescences No. 18 and No. 19. On Anthurium No. 3 (Table 4), which is 44 cm long, 16 di erent aberrant patterns are observed compared to ve on in orescence No. 2 (Table 3), which is 39 cm long. The number of di erent aberrant patterns seems to be correlated with the length of the spadix. The meaningful di erences between the mature in orescences of Anthurium and the young in orescences of Dracontium, are therefore as follows: D1: Almost no whorls (mˆn) were observed in all three specimens of mature Anthurium, while a quarter of the regions in Dracontium are reported to be whorls. D2: The value of the parameter J [in sequences (1) and (4)] is greater than 1 in about half of the regions of the three mature Anthurium specimens, while in Dracontium it is generally equal to 1 (84 % of the regions if we except whorls). D3: The behaviour of the value of the divergence is irregular in Anthurium, while it is rather regular in Dracontium. D4: We have only two aberrant patterns in the 19 in orescences and the four regions of Dracontium (i.e. in about one region in 40), but in mature Anthurium, aberrant patterns occur regularly along every centimetre of the spadices (i.e. in about two regions in ve). Di erences D3 and D4 can be attributed to distortions experienced during growth from the tiny spadix to the mature one considered here. As has been shown in the genus Symplocarpus, as the length/width ratio of the in orescence increases, the number of parastichies decreases (Barabe and Jean, 1996), meaning a change in the phyllotactic pattern. The occurrence of D4 could be tested further by comparing the phyllotactic organization of in orescences of Anthurium in early stages of development with mature in orescences of Anthurium. A functional relationship between d and (m, n) The FTOP shows a relationship between (m, n) and an interval for the divergence d, when a particular sequence is concerned. The theorem does not reveal the relative behaviour of the respective variations of d and m n, or the function that can link the two parameters in natural systems, such as the Araceae, where the two parameters move from one sequence to another. The FTOP is descriptive and based on the regularity of the lattices expressing the theoretical patterns. On the other hand, we have noticed in Table 1 that as m n decreases by moving from one sequence to another, d increases. There is thus some kind of constraint linking the two parameters; and we need a model, based on some constraints, which can deal with the relative behaviour of the two parameters. This behaviour is indeed indicative of some constraints exerted on the natural phyllotactic systems. The Jean model does reveal the said behaviour, via the parameter, as we will now point out brie y. Let us consider, for example, the patterns of the type (t, t 1) in in orescence No. 10 of Dracontium (Table 1). From region 1 to region 4, t decreases from 12 to 6. Then the Jean model shows that d increases (see Jean, 1994, p. 174: follow the dotted line at the bottom of diagram summarizing the model). Now, as t decreases, the parameter P r ˆ m n (the total number of parastichies) in the model decreases too (the reader can check in the same source, on p. 139, the sequences starting with 1 and t ˆ 2, 3, 4, 5,... in Table 6.1). So the value of d is inversely proportional to (m n). Moreover is proportional to (m n) (e.g. formula (11) above), thus is inversely proportional to d. It generally appears that the model works in agreement with reality in the spadices of Dracontium. The case of whorls in Dracontium An approximation of for whorls. In the cases of whorls, the value of cannot be calculated in practice, thus there are gaps in Table 1. However, to gain an indication of the value we can look at the adjacent spiral patterns. For example, in in orescence No. 5 in Table 1, we can see that the whorl (9, 9) is squeezed between (10, 11) and (9, 10) followed by (7, 8). There is a decrease in the values of. So for (9, 9) the value of is expected to be between 5.72 and We can say that the pattern (9, 9) is a transitional one between the two spiral patterns (10, 11) and (9, 10), obtained by an asymmetry breaking process. There is a