Annals of Botany 82 : 577–586, 1998
Article No. bo980716
Interactions Between Physical and Biological Constraints in the Structure of the
Inflorescences of the Araceae
B E R N A R D J E U NE* and D E N I S B A R A BE! †
* Laboratoire d’Ecologie, URA 258, UniŠersiteU Pierre et Marie Curie, BaW t. C, 2me eU tage, 4 Place Jussieu, 75 252
Paris Cedex 05, France, † Institut de recherche en biologie ŠeU geU tale, Jardin botanique de MontreU al, 4101
Sherbrooke Est, MontreU al, Canada, H1X 2B2
Received : 5 January 1998
Returned for revision : 30 March 1998
Accepted : 19 June 1998
A study of the inflorescences of Monstera and Anthurium was used to establish a relationship between biological and
physical constraints for the structure of plant organs. The physical constraint between flowers in the compact
inflorescences of Anthurium and Monstera is expressed by Aboav-Weaire’s law. The application of this law to
inflorescences indicates a linear relationship between the number of sides of a flower and the number of sides of
neighbouring flowers. However, the slope of this straight line is significantly higher for Anthurium and Monstera than
that expected in theory. This deviation from the law is attributable to a biological cause that can be estimated using
Aboav-Weaire’s law. Acting alone, the biological constraint tends to produce four-sided flowers. The equilibrium
between biological and physical constraints reduces the number of sides per flower from six (theoretical value) to 5±9
(in Anthurium) or 5±8 (in Monstera) with a variance of the measures less than that expected in theory. Furthermore,
when flower density in an inflorescence increases (towards the middle of the inflorescence in Monstera and towards
the lower section for Anthurium) the number of sides approaches six (i.e. the physical constraint dominates). When
flower density decreases (towards the top of the inflorescence) the number of sides approaches 5±5 (i.e. the biological
constraint dominates). The geometry of the inflorescences of Anthurium and Monstera is the result of the joint action
of biological and physical constraints.
#1998 Annals of Botany Company
Key words : Monstera, Anthurium, Araceae, Aboav-Weaire, inflorescence, constraint, flower.
INTRODUCTION
The qualitative and quantitative study of developmental
constraints still remains of great interest in developmental
biology (Newman, 1994). Maynard Smith et al. (1985)
distinguish two types of constraints in the evolution of
organisms : universal constraints and local constraints.
Universal constraints are direct consequences of the laws of
physics and apply to all physical systems and organisms.
Local constraints result from the biological properties of
organisms, and can be limited to a particular taxon. In the
present article, we will refer to these two types of constraints
as physical and biological, respectively. Physical constraints
(universal) are formalized in explicit mathematical laws ;
this is not yet the case for biological constraints (local). The
purpose of studying biological material is to provide
evidence of types of constraints ignored by theoreticians
that will allow them to extend the range of physical laws. As
Weaire and Rivier (1984) wrote (p. 91) : ‘ In our opinion, the
botanist and the metallurgist, like the farmer and the cowboy, should be friends ’.
What is the quantitative relationship between biological
and physical constraints in the structure of organisms ? To
answer this question, one needs a material which lends itself
to the analysis of physical and biological parameters with
respect to a specific law of physics.
The physical and mathematical laws governing the twodimensional organization of repeating structures such as
0305-7364}98}110517­10 $30.00}0
those involved in soap foam, polycrystalline nets, sunflower
capitula and plant epidermis (Aboav, 1980 a, b ; Mombach,
Vasconcellos and Almeida, 1990 ; Inglesias and Almeida
1991 ; Mombach, Almeida and Iglesias, 1993 ; Rivier et al.,
1984 ; Rivier, Schliecker and Dubertret, 1995 ; Barabe! , Jeune
and Boubes, 1996) provide a framework for the analysis of
biological constraints. For example, biologists Lewis (1931)
and Dormer (1980) analysed cell shape and the organization
of biological tissues using geometrical theory. Recently,
Aboav-Weaire’s law (Aboav, 1980 a, b) has been applied to
the study of the plant epidermis (Mombach et al., 1990,
1993). This semi-empirical law, developed to explain the
topological structure of physical materials, states that the
average number of sides mn of the neighbours of a n-sided
cell in a net is given by the equation :
(1)
mn ¯ 6®λ­(6λ­µ )}n
#
Generally, in biological structures, λ is so close to 1
(Mombach et al., 1990, 1993) that the equation becomes :
(2)
mn ¯ 5­(6­µ )}n
#
where µ is the variance of the number of sides. The
#
aforementioned authors observed that the organization of
cells within epidermal tissue obeyed Aboav-Weaire’s law
(Mombach et al., 1990, 1993). Thus, it seemed interesting to
determine whether the law could also be applied to the
shape of the flowers in Araceae inflorescences.
The inflorescences of Araceae are an appropriate material
# 1998 Annals of Botany Company
578
Jeune and BarabeU —Physical and Biological Constraints
to study the arrangement and shape of the flowers, observed
in a planar section, from both a physical and a biological
point of view. Several genera of this family, such as
Anthurium and Symplocarpus, have fairly regular flowers
which have already attracted the attention of botanists and
mathematicians (Barabe! , 1994 ; Barabe! and Jean, 1996 ;
Barabe! et al., 1996). The application of Aboav-Weaire’s law
to the inflorescence of Symplocarpus has already established
a quantitative relationship between physical and biological
constraints in the structure of these organs (Barabe! et al.,
1996). Continuing this work, we show here that the geometry
of inflorescences of Anthurium and Monstera is the result of
the joint action of biological and physical constraints.
However, there are two important differences between
inflorescences and the epidermis : (1) flowers are initiated
regularly at the apex of the inflorescence whereas division in
the epidermis occurs more or less randomly ; (2) flowers
appear at a given position and their surface increases
regularly. In the epidermis, there is a division of cells which
tend to maintain an optimal size during the growth of the
epidermis.
Dormer (1980) noted that (p. 5) ‘ So long as a system is
mathematically determinate there is no room for any
biological phenomenon to show itself. Biology, in the
present context, begins specifically at the point where there
are two or more mathematically admissible results, between
which the organism must choose upon some basis other
than that of geometrical necessity ’. In this perspective, since
the structure of the inflorescences of Araceae differs
qualitatively from that of the other biological structures
that have been studied in previous work, it is plausible to
expect to find new parameters in the analysis of biological
constraints. The morphological units forming the polygonal
pattern in the inflorescence are not single cells where the
osmotic pressure exerts a constant pressure on the contour
of the cell. The units of the network are tetramerous flowers
which, in the absence of physical constraints, would have a
square or rectangular form. Here there is a conflict between
a pure mathematical solution (six-sided units) and a pure
empirical solution (four-sided units). In this paper we
analyse the discrepancy between the purely mathematical
but virtual solution and the empirical solution that is really
taken by the flowers. The Aboav-Weaire law allows us to
obtain a quantitative estimate of the biological constraint
acting in the inflorescences of these genera.
others species of Anthurium were used for morphological
analysis. The value of n (number of sides of a flower) and p
(perimeter of a flower) was scored on fully developed
inflorescences. Measurements were made on photographs of
inflorescences. From a topological point of view, the number
of sides (n) is independent of the form of the flower. For
example, in Fig. 8, the flower is topologically hexagonal, yet
it is geometrically square and the array is rhombic. The
determination of vertices can present a problem at anthesis
in the case of Monstera. When there was a space between
flowers we theoretically filled up the space to obtain a threeray vertex. To minimize this problem we took measurements
on a large number of flowers (2365) belonging to four
inflorescences. Three of them had not reached anthesis, thus
it was easy to recognize a three-ray vertex on these
inflorescences. As the results are the same for all
inflorescences (Fig. 13), and considering the great number
of flowers studied, we can conclude that the determination
of vertices does not influence the results.
Statistical analysis
Analysis centred on linear regression and one- or twofactor ANOVA, after verification of the conditions of
application, notably the normality of the residuals and
homoscedasticity. The confidence intervals of averages are
calculated using the error mean square as an estimate of the
variance. Given the very large number of samples, averages
are followed not by the standard error but by approx. twice
that value, which allows a direct calculation of the limits of
the confidence interval at P ! 0±05. Comparisons of two
averages were performed using Student’s t-test.
For Aboav-Weaire’s law, we use µ the variance of
#
variable x and r as the correlation coefficient of BravaisPearson. The comparison of the slope with 5, the theoretical
value, is performed with the t-test of the conformity of a
slope to a norm. To be perfectly rigorous, the discrete
nature of variables ‘ number of sides of a flower ’ and ‘ sum
of the sides of adjacent flowers ’ does not allow the use of
this test. However, the very high values for degrees of
freedom, and the equally high values obtained for approximate variables, leaves no ambiguity as to the interpretation of the test.
RESULTS
Morphology
MATERIALS AND METHODS
Inflorescences of Monstera and Anthurium used for this
study were collected in the glasshouses of the Montreal
Botanical Garden : Monstera adansonii var. klotzschiana
(2263–86) ; Anthurium digitatum (6356–39) ; Anthurium
acaule (2040–57) ; Anthurium longilaminatum (1554–58) ;
and an unidentified Anthurium sp. (1868–48) (numbers are
the registration numbers at the Montreal Botanical Garden).
Voucher specimens were deposited at the Herbier MarieVictorin (MT). Quantitative analyses were performed on
2365 flowers of Monstera (four inflorescences) and 1129
flowers of Anthurium digitatum (one inflorescence). The
In both genera the flowers are bisexual, tetramerous, and
they form a fairly regular hexagonal net on the surface of
the inflorescence. Differences mostly concern the presence
or absence of a perianth and the shape of the gynoecium.
Anthurium flowers have four tepals and four stamens
surrounding a gynoecium with a tetragonal style (Figs 1, 2
and 8). Monstera flowers do not have a perianth. In
Monstera adansonii, flowers are composed of four stamens
surrounding a generally hexagonal gynoecium (Figs 1, 3, 4
and 9). Extrorse anthers are borne on flattened filaments.
Stamens become visible during anthesis as they emerge
between the pistils due to the elongation of filaments
(Fig. 4).
Jeune and BarabeU —Physical and Biological Constraints
579
F 1–2. Anthurium digitatum. Fig. 1. Inflorescence without spathe. Bar ¯ 2 cm. Fig. 2. Section of the inflorescence showing the hexagonal net
resulting from the packing of tetramerous flowers. Bar ¯ 2 cm.
F 3–4. Monstera adansonii. Fig. 3. Inflorescence. Bar ¯ 1 cm. Fig. 4. Section of the inflorescence showing the pentagonal or hexagonal form
of the flowers. Bar ¯ 2 mm.
F 5–7. Early stages of development in different species of Anthurium showing tepals. Bar ¯ 0±5 mm. Fig. 5. Anthurium sp. Fig. 6. Anthurium
longilaminatum. Fig. 7. Anthurium acaule. i, Inferior tepal ; 1, lateral tepal ; s, superior tepal.
580
Jeune and BarabeU —Physical and Biological Constraints
s
l
l
i
F. 8. Anthurium digitatum. Schematic representation of a section of
the inflorescence showing the patterns formed by the packing of the
flowers. Inset : scheme showing the true hexagonal form of a flower.
Black circles indicate the vertices of the polygons.
Number of sides per flower
The flowers are tetramerous and, due to this biological
constraint, tend to be geometrically four-sided (rectangular
form). On the other hand, flowers experience a contact
pressure from adjacent flowers (a physical constraint). This
contact pressure is optimally distributed for hexagonal
contours, as in honeycomb structures, and flowers thus tend
to be six-sided from a topological point of view, even if their
profile remains globally rectangular, as in Anthurium (Figs
2 and 8).
Even if we consider that the curvature of the inflorescence
is that of a sphere, the average number of sides per flower
should be, according to the Euler-Poincare! theorem, equal
to n ¯ 6®12}C where C corresponds to the total number of
flowers in the inflorescence or in the section of the
inflorescence studied. In our observations on Anthurium and
Monstera, the value of C being always greater than 200
means the average number of sides should not be less than
5±94. In the case of Symplocarpus (Barabe! et al., 1996)
where the minimum number of flowers is 60, the average
number of sides should not exceed 5±80. This value is much
higher than that of our observations (5±5). Therefore we felt
justified to neglect the effect of these minor corrections in
our results.
Anthurium. If physical constraints alone were responsible
for flower shape, an average of six sides per flower would be
F. 9. Monstera adansonii. Schematic representation of a section of
the inflorescence showing the pentagonal or hexagonal form of the
flowers. Arrowheads indicate the stigma.
observed. The measured value for the inflorescence as a
whole is slightly lower, i.e. 5±86³0±02. There is also a
significant variation in the average number of sides from the
bottom to the top of the inflorescence. The lower and
middle thirds of the inflorescence have similar averages
(5±96³0±04 and 5±91³0±03, respectively) which are not
significantly different (but not including the value 6). In the
upper third, this average decreases drastically to 5±67³0±04
(Fig. 10 A).
Flowers become smaller towards the distal end, as is
evident from the study of their perimeter. The average
perimeter of five- or six-sided flowers decreases from the
bottom (section 1) to the top of the inflorescence (section 3),
both for five-sided (from 1±45 to 1±27, bottom to top ; Fig.
10 B) and six-sided flowers (from 1±60 to 1±29, bottom to
top ; Fig. 10 C). At the same time, as might be expected, sixsided flowers in the two lower sections have a larger average
perimeter than five-sided flowers (t ¯ 8±9), while towards
)!
the top of the inflorescence these two types of flowers have
similar average perimeters (t ¯ 1).
&)
Monstera. The average number of sides for flowers of the
four inflorescences analysed was 5±79³0±02, i.e. lower than
for Anthurium. As with Anthurium, there is also a significant
variation in the average number of sides from the bottom to
the top of the inflorescence, with the number of sides of
flowers in the middle section being significantly higher than
581
Jeune and BarabeU —Physical and Biological Constraints
5.9
A
6
5.85
5.9
5.8
mn
mn
6.1
5.8
5.7
5.6
5.65
5.6
5.5
4.4
2
Sections
3
6.05
B
4
5.95
3.8
5.9
3.6
5.8
3.2
5.75
5.7
4.4
2
Sections
3
C
4.2
4
mp
3.8
3.6
3.4
3.2
3
1
2
Sections
1.7
1.65 C
1.6
1.55
1.5
1.45
1.4
1.35
1.3
1.25
1.2
×
3
×
×
×
×
×
4
×
×
×
×
×
×
×
2
Sections
3
16
25
26
Sections
35
36
D
30
20
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
10
0
5
n
1
40
×
nmn
nmn
50
×
×
×
×
×
×
×
3
15
3
60
50
D
45
40
35
30
25
20
15
10
2
2
Sections
5.85
3.4
1
1
B
6
mn
mp
1
3
mp
5.75
5.7
4.2
A
6
7
8
F. 10. Anthurium digitatum. A, Average number of sides (mn) in
different sections of the inflorescence (Sections) ; B, average perimeter
(1 cm ¯ 2±7 units) of five-sided flowers (mp) in different sections of the
inflorescence (Sections) ; C, average perimeter of six-sided flowers (mp)
in different sections of the inflorescence (Sections). 1, Basal section ; 2,
median section ; 3, upper section ; D, regression of the total number of
sides of the adjacent flowers (nmn) to a n-sided flower (n) [r# ¯ 0±90].
Crosses indicate the samples. The 95 % confidence limits for an
individual value are ³2±1 units around the estimated regression line.
in the lower and upper thirds (Fig. 11 A and B). In the
middle section flowers tend to be six-sided and it is here that
the inflorescence is widest and the flowers largest. Unlike
Anthurium, flowers in the lower section of the inflorescence
2
4
6
n
8
10
F. 11. Monstera adansonii. A and B ; Average number of sides (mn)
in different sections of two inflorescences (Sections) ; C, average
perimeter (cm) of five- and six-sided flowers (mn) in different sections
of one inflorescence (Sections). 1, basal part ; 2, median part ; 3, upper
part ; D, regression of the total number of sides of the adjacent flowers
(nmn) to an n-sided flower (n) [r# ¯ 0±90] Crosses indicate the samples.
The 95 % confidence limits for an individual value are ³3±1 units
around the estimated regression line.
show the lowest average number of sides per flower. Indeed,
both five- and six-sided flowers in the middle section have an
average perimeter which is significantly greater than flowers
in the upper section (Fig. 11 C). This result is consistent with
the shape of the inflorescence (Fig. 3) where the maximum
diameter is in the middle section. It is also in this section
582
Jeune and BarabeU —Physical and Biological Constraints
T     1. Comparison of the aŠerage number of sides of
adjacent flowers to an n-sided flower (n) according to AboaŠWeaire’s law (A–W ) and estimated results (y)
n
A–W
y
3
4
5
6
7
7±2
6±6
6±2
6
5±9
3±6
4±8
5±5
5±9
6±3
mn
Monstera
∆
3±6
1±8
0±7
0±1
®0±4
A–W
y
7±1
6±6
6±3
6
5±9
5
5±4
5±7
5±9
6
A
6.5
5.5
4.5
∆
3.5
2±1
1±2
0±6
0±1
®0±1
7.5
∆ indicates the difference between the theoretical value (A–W) and
the estimated values (y). See also Fig. 12.
2
3
4
5
n
6
7
8
3
4
5
n
6
7
8
B
6.5
mn
Anthurium
7.5
5.5
4.5
that the average number of sides per flower is greatest (Fig.
11 A and B)
AboaŠ-Weaire’s law
Anthurium. For Anthurium, the distribution of the
number of sides is centred around 5±86 with a range varying
from 4 to 7. According to Aboav-Weaire’s law, physical
constraints should lead to a linear relationship with a slope
of 5 between the number of sides of a flower [independent
variable, x in eqn (3)] and the product of the number of
sides by the average number of sides of adjacent flower
[dependent variable, y, nmn of eqn (3)]. Since the number of
adjacent flowers is equal to the number of sides, this product
y is equal to the total number of sides of the adjacent
flowers :
y ¯ 5x­(6­µ ) ¯ 5x­6±142
(3)
#
(according to our measurements, the estimate of the variance
equals : µ ¯ 0±142). Empirically, we do observe a linear
#
relationship between y and x (conditioned averages of y
almost perfectly aligned ; Fig. 10 D)
yobs ¯ 8±336x®14±35
(4)
(linear correlation coefficient r ¯ 0±95). We will therefore
consider the hypothesis of a linear relationship acceptable,
despite the fact that the slope is 8±336 instead of 5. Given the
restrictions indicated previously, this value is significantly
different from 5 [t
" 100].
""#'
Comparison of estimates calculated using Aboav-Weaire’s
law with measured values (Table 1, Fig. 12 A) confirms the
previous conclusion ; with Aboav-Weaire’s law there is an
equilibrium for six-sided flowers (negative correlation),
which have, on average, six-sided neighbours. On either side
of this equilibrium there are compensations such as a flower
with less than six sides will have, on average, neighbours
with more than six sides, and Šice Šersa. In contrast, the
measured number of sides of neighbours of a three-sided
flower tends towards 3±6, while neighbours of a seven-sided
flower tend to have 6±3 sides (positive regression coefficient).
Note, however, that the difference between measured and
calculated values decreases from 3±6 (n ¯ 3) to ®0±4 (n ¯ 7)
as the number of sides of the flower increases (Table 1).
3.5
2
F. 12. Comparison of the relationship between the average number of
sides of the adjacent flowers (mn) to an n-sided flower (n) either
calculated according to the Aboav-Weaire’s law (+) or measured
experimentally (E). A, Anthurium ; B, Monstera.
Monstera. In the case of Monstera, the probability distribution is centred on 5±79 with an amplitude varying from
3 to 8. y corresponds to the following equation :
y ¯ 5x­(6­µ ) ¯ 5x­6±263 (µ ¯ 0±263)
(5)
#
#
Empirically, a linear relationship between y and x was also
observed ; as with Anthurium, conditioned averages of y are
almost perfectly aligned (Fig. 11 D) :
yobs ¯ 6±777x®5±468
(6)
(linear correlation coefficient r ¯ 0±92). The regression
coefficient of the line is 6±777, which is also high and
significantly different from 5 [t
" 29] given the
#$'$
restrictions indicated earlier.
Comparison of values calculated using Aboav-Weaire’s
law with the measured values (Table 1, Fig. 12 B) confirms
previous observations on Anthurium. There is no compensation around the value 6 for the average number of
sides of adjacent flowers. When the number of sides of a
flower decreases to 3, the average number of sides of its
neighbours tends toward 5, while for seven-sided flowers the
number of sides of its neighbours tends towards 6 (positive
correlation). We note again that the difference between
theoretical and measured values decreases from 2±1 (n ¯ 3)
to ®0±1 (n ¯ 7) as the number of sides of the flower
increases.
DISCUSSION
Variation of the number of sides
According to theory (Mombach et al., 1990), if mechanical
constraints alone were responsible for the number of sides
583
Jeune and BarabeU —Physical and Biological Constraints
5.00
0.00
4m
4b
S
4u
1
3u
–5.00
a
3m
3b
–10.00
5u
2
5b
–15.00
5m
–20.00
–25.00
5.00
5.50
6.00
6.50
7.00
7.50
b
8.00
8.50
9.00
9.50
10.00
F. 13. Relationship between the value of parameters a and b in different regressions between nmn, and n for the inflorescences of Anthurium (5),
Monstera (1, 2, 3, 4), and Symplocarpus (S). b, Basal section ; m, median section ; u, upper section. See also Table 2.
of the flowers, their probability distribution would be
approximately normal and centred on 6 with an amplitude
of 4 to 8. In fact, as we have seen, this average number is
smaller than 6, and the distribution, although approximately
normal, is slightly asymmetrical and flattened for both
Anthurium and Monstera (and also for Symplocarpus ;
Barabe! et al., 1996).
There is also variation of the average along the
inflorescence. In the lower part of the inflorescence of
Anthurium where flowers are larger and contact pressures
consequently stronger, the physical constraint is greater and
the number of sides per flower is closer to 6 than in the top
part where flowers are smaller and have fewer sides. In the
case of Monstera, contact pressures and the physical
constraint should be stronger in the middle section of the
inflorescence than in the lower and upper parts in order to
explain the higher number of sides per flower. The physical
constraint is probably associated with vigour : as flower area
and flower perimeter increase, the number of sides increases
due to the increasing importance of the mechanical
constraint. A similar positive correlation between the
number of sides and the perimeter in parts of organs where
the pressure is greatest has been observed for Symplocarpus
(Barabe! et al., 1996) and in epidermal tissues (Mombach et
al., 1990).
The results show that the experimental data for both
Anthurium (slope 8±336 ; Fig. 10 D) and Monstera (slope
6±777 ; Fig. 11 D) differ significantly from theoretical values
[slope 5, eqn (3)], as was previously observed for Symplocarpus (Barabe! et al., 1996). Among the three genera
analysed (Symplocarpus, Anthurium and Monstera) the
flowers are all tetramerous. Monstera obeys the law most
closely (slope 6±78). Yet, it is possible that the pressure
between the flowers is not as great in this genus due to the
absence of a perianth. The biological constraint would thus
be predominant here as evidenced by the lower average
number of sides (5±76 for Monstera Šs. 5±86 for Anthurium
whose inflorescence is of the same elongated shape and
bears as many flowers). The paradox of a strong biological
constraint and adherence to the law is misleading, however,
because these biological constraints arise late in the
development of the flower (see the section entitled
‘ Morphology and constraints ’). Curiously, the average
number of sides is even lower than for Symplocarpus whose
flowers have a perianth, as in Anthurium (5±86).
In the case of Symplocarpus, the small number of flowers
per ellipsoid inflorescence was used to explain deviation
from the Aboav-Weaire law (Barabe! et al., 1996). Consequently, in the case of Monstera and Anthurium, where the
number of flowers studied is much greater, discrepancy
between our observations and the predictions made by the
Aboav-Weaire law is probably due to the fact that the latter
does not take into account the existence of biological
constraints or that the value of parameter λ in eqn (1) is very
different from 1.
Let us note that, in Monstera and Anthurium, the variation
of the average along the inflorescence suggests that the
equilibrium between biological and physical constraints
may fluctuate fairly easily.
The use of AboaŠ-Weaire’s law to estimate the biological
constraint
According to Aboav-Weaire’s law, as applied to the
inflorescence of Araceae, a linear relationship should exist
between the number of sides of a cell (in this case a flower)
(x) and the total number of sides of adjacent cells (y) [eqn
(2)] :
y ¯ a­bx
(7)
This linear relationship was verified in the inflorescences of
both Monstera and Anthurium, as well as Symplocarpus.
However, it is also clear that there is a relationship between
the values of parameters a and b for Monstera and Anthurium
(Fig. 13). The variance of the linear relationship is 99±6 %.
We can therefore consider :
a ¯ α­βb
(8)
This is hardly surprising. If eqn (3) is given as y ¯ a­bx, we
584
Jeune and BarabeU —Physical and Biological Constraints
T     2. Comparison of AboaŠ-Weaire’s law and experimental results
Species
Monstera
1
2
3b
3m
3u
4b
4m
4u
Anthurium
5b
5m
5u
Symplocarpus
b
µ
#
ao
at
6±07
7±45
6±97
7±40
6±35
5±48
5±52
5±65
0±21
0±38
0±32
0±19
0±29
0±37
0±08
0±17
®1±00
®9±86
®7±14
®8±99
®3±32
1±90
2±74
1±69
®0±22
®8±30
®5±52
®8±19
®1±79
3±47
2±95
2±25
7±87
9±41
7±56
5±45
0±07
0±08
0±28
0±14
®11±42
®20±60
®10±36
®2±31
®11±16
®20±39
®9±09
®13±87
k
ka
kb
n
χ#2a
χ#2b
0±78
1±57
1±62
0±81
1±54
1±57
0±22
0±57
0±54
0±56
0±59
0±57
0±60
0±57
0±48
0±52
0±51
0±53
0±56
0±54
0±57
0±53
0±45
0±49
0±57
0±58
0±62
0±59
0±63
0±59
0±51
0±55
305
409
208
298
282
255
309
299
258±1
354±4
169±5
251±6
236±9
212±2
261±7
252±5
354±8
466±4
249±3
347±2
329±9
300±6
359±1
348±3
0±26
0±21
1±27
®11±57
0±53
0±48
0±58
0±50
0±45
0±55
0±55
0±50
0±60
372
469
287
320±0
410±4
241±5
426±8
530±4
335±3
∆
a ¯ 34±70–5±92b ; r ¯®0±998 ; r# ¯ 0±996.
Relationship between nmn and n, obtained experimentally from different sections of the inflorescences of Monstera and Anthurium ; b, regression
coefficient ; µ , variance ; ao, estimated value ; at theoretical value ; ∆ ¯ at®ao ; k, biological constraint ; ka and kb, confidence interval ; n, number
#
of flowers. b, basal section ; m, median section ; u, upper section. See also Fig. 14.
2.00
0.00
∆
–2.00
–4.00
–6.00
S
–8.00
–10.00
–12.00
–25.00
–20.00
–15.00
–10.00
–5.00
0.00
5.00
10.00
at
F. 14. Plot of the deviation (∆) of measured value ao from theoretical value at(∆ ¯ at®ao) as a function of at for the inflorescences of Anthurium,
Monstera and Symplocarpus (S). See also Table 2.
can further write : y®ya ¯ b[x®xa ] (i.e. y ¯ ya ®bxa ­bx), so
that we have : a ¯ ya ®bxa where xa and ya are the expected
values for x and y respectively. These values are xa ¯ 6 and
ya ¯ 36 for an optimal shape with zero variability, as in
honeycombs. For an optimal shape, we should therefore
have : a ¯ 36®6b, i.e.
y ¯ bx­6(6®b)
(9)
and we do indeed have a very similar equation : a ¯
34±70®5±92 b (Table 2, Fig. 13).
If we now introduce variability µ , we get the following
#
equation :
y ¯ bx­6(6®b)­µ
(10)
#
[If λ ¯ 6®b, the equation reads : y ¯ (6®λ)x­6λ­µ
#
which is also the formulation of Aboav-Weaire’s law
(Mombach et al., 1993)].
Plotting a graph of the deviation of measured value ao
from theoretical value at ¯ 6(6®b)­µ with respect to at
#
(Fig. 14), a downward bias ∆ ¯ at®ao is observed in
Monstera and Anthurium. This deviation represents the
biological constraint on the inflorescence architecture. Let
us note that the value of the deviation is calculated
independently from that of λ that was supposed to be
approximately 1 in accordance with experimental work (e.g.
Mombach et al., 1993). Therefore, to take this constraint
into account, we must write :
(11)
yt ¯ bx­6(6®b)­µ ­∆
#
where µ represents the estimated variance when there is a
#
biological constraint ∆. [Symplocarpus is in a distinct
position (Figs 13 and 14) and was not used in the calculation
of the parameters of relationship a ¯ α­βb].
Let us call µ! the variance which could be expected if there
#
was no biological constraint. Thus :
yt ¯ bx­6(6®b)­µ!
(12)
#
and the effect of the biological constraint results in a lower
variance in the measurements. Note that cell mitosis is
another biological mechanism that decreases µ (Mombach
#
et al., 1993). In the case of flowers, this is easily explained.
If we let xt be the theoretical number of flower sides when
there is no biological constraint and xo ¯ xt(1®k) the actual
number of sides, then as the flowers are tetramerous, the
impact of the biological constraint tends to lower the value
of xt. Here k will be a measure of the biological constraint
for a given flower and will range from 0 to 1. According to
the rules of variance calculations :
µ ¯ V(xo) ¯ V(xt) (1®k)# ¯ µ! (1®k)#
#
#
Jeune and BarabeU —Physical and Biological Constraints
Since µ! ¯ µ ­∆, we deduce that µ }(1®k)# ¯ µ ­∆, so
#
#
#
#
that :
±
k ¯ 1®[µ }(µ ­∆)]! &
(13)
# #
Note that this relationship is meaningful only if : µ! & µ
#
#
and 0 % k % 1, then ∆ & 0. These conditions are not fulfilled
for Symplocarpus.
In the case of Monstera and Anthurium, k can be calculated
and used to verify the hypotheses formulated about the
measurement of the biological constraint (Table 2). It is in
the apical region of the inflorescence of Monstera and
Anthurium that the biological constraint is strongest. Let us
note that this value of k was obtained using a linear
regression, and is thus an estimate rather than an absolute
value. If the distribution of the measurements is taken as
normal, we can give the estimate of the variance as a
confidence interval (then the variance estimator has a
distribution of χ#). We can therefore deduce a confidence
interval for the estimate of biological constraint k.
Let the limits of the confidence interval be µ a and µ b (at
#
#
risk α), i.e. µ a ¯ SS}χ#a/ and µ b ¯ SS}χ#( −a/ ), the limits of
#
#
" #
#
the confidence interval of k will be ka and kb obtained by
replacing µ by µ a and µ b, respectively, in eqn (13) (Table 2).
#
#
#
Variations in the value of biological constraint k along the
axis of the inflorescence appear to be progressive even if,
overall, there are some significant differences as shown in
our study of the average number of sides per flower.
Morphology and constraints
In Anthurium and Monstera inflorescences there is a
physical constraint between flowers tending to produce sixsided flowers, which works against a biological constraint
tending to produce four-sided flowers. The result of the
interaction between biological and physical constraints is
described with a modification in the Aboav-Weaire law.
If we consider the average number of sides per flower, the
biological constraint is strongest in Anthurium. In the lower
and middle sections of the inflorescence flowers are
tetramerous and originally square-shaped, although from
a topological point of view they are actually six-sided on
average. The appearance of hexagons on mature Anthurium
inflorescences can be explained by the following process.
Very early in development, even before perianth parts are
completely differentiated, flowers take on their characteristic
tetragonal shape (Figs 5–7). The two lateral tepal primordia
are larger than the inferior or superior primordia, and they
grow perpendicularly to the axis of the inflorescence (Figs
5–7). As the flower primordia are slightly out of line, it
makes it easier for an additional side to appear at the
intersection of the inferior and superior tepals of two
adjacent flowers. The stronger the pressure between adjacent
flowers, the closer the average number of sides will be to 6
(Fig. 8).
In Monstera, the situation is different. The primordia
have a fairly rounded shape during the first stages of
development (Fujita, 1942). During growth, the pressure
exerted by flowers on each other rapidly leads to the
appearance of hexagonal patterns, which explains the
paradox previously mentioned (i.e. that a weaker physical
585
constraint is associated with a better fit to Aboav-Weaire’s
law). This development pattern resembles that of Symplocarpus (Barabe! , 1994). However, since each flower produces
four stamens, this tendency toward tetramerism is more
obvious in areas where flowers are smaller (in the lower and
upper parts of the inflorescence), and the contact pressures
consequently weaker.
In conclusion, there is a physical constraint between
flowers in the compact inflorescences of Anthurium and
Monstera. This constraint can be expressed by a linear
relationship between the number of sides of a flower and
that of its neighbours. However, the slope of this straight
line is significantly higher than expected in theory. This
discrepancy explains the fact that neighbouring flowers tend
to have the same number of sides, although theory states
that there is an equilibrium around 6 : a flower with less than
six sides should be surrounded by neighbours with more
than six sides, and Šice Šersa. In Anthurium and Monstera
high values for this deviation appear in specific sections of
the inflorescence, and not at random. We eliminate this
discrepancy by introducing a biological constraint into
Aboav-Weaire’s law which acts to oppose the physical
constraint [eqn (11)]. This biological constraint is the
morphogenesis of the tetramerous flowers. Acting alone, it
would produce four-sided flowers. The equilibrium between
biological and physical constraints reduces the average
number of sides per flower from 6 (theoretical value) to 5±9
(in Anthurium) and 5±8 (in Monstera). Furthermore, the
number of flower sides has a graded distribution. When
flower density increases (towards the middle of the
inflorescence in Monstera and towards the lower section for
Anthurium) the number of sides tends towards 6 (physical
constraint dominant), but when flower density decreases
(towards the top of the inflorescence) the number of sides
tends toward 4 (biological constraint dominant).
The equilibrium between these two constraints has been
described only at an advanced stage of the morphogenesis
of the inflorescence. In a future study, we intend to analyse
the evolution of this relationship over time during morphogenesis.
A C K N O W L E D G E M E N TS
We would like to thank Drs Christian Lacroix and David
Morse and Mr Jacques Dumais for their valuable comments
on the manuscript. We wish also to thank Ms Brigitte
Vimard for data collection and Mr Denis Lauze! for the
photographs of the inflorescences. This work was supported
by grants from the Fonds FCAR (Fonds pour la formation
des chercheurs et l’aide a' la recherche, Que! bec), the NSERC
(Natural Sciences and Engineering Research Council of
Canada) to D. B., and from the IRBV (Institut de recherche
en biologie ve! ge! tale), as part of its seminar program, to B. J.
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