Annals of Botany 77 : 515–527, 1996
Phyllotactic Patterns : A Biophysical Mechanism for their Origin
P. B. G R E E N*, C. S. S T E E L E† and S. C. R E N N I C H‡
* Department of Biological Sciences, † Department of Mechanical Engineering and ‡ Department of Aeronautics and
Astronautics, Stanford UniŠersity, Stanford, CA 94305, USA
Received : 22 August 1995
Accepted : 28 November 1995
The patterns seen in plant shoots and flowers, ‘ phyllotaxis ’, originate in an annular region. They are typically
propagated inward from this ring-like area. We show here that an initial undulating periodic pattern (a ‘ whorl ’ of
hump-like organs) can arise in a flat unstructured annulus. The pattern arises not from pre-localized pushes from
below, but rather as a spontaneous physical response of the expanding surface to lateral constraint. Physical
properties of a uniform formative layer (tunica) and a uniform substratum (corpus) provide the wavelength of the
undulation and hence the number of organs. Establishment of the parameters for this buckling, as well as the followthrough of organ development, is biological. We propose, however, that at the moment of periodic pattern initiation
the plant tissue simply manifests the spontaneous but complex properties of a two-layered inanimate sheet.
# 1996 Annals of Botany Company
Key words : Phyllotaxis, tunica, corpus, patterning, shoot apex, morphogenesis, biophysics, buckling.
INTRODUCTION
The characteristic patterns of plant organs, whorls and
spirals, have been of scientific interest for centuries (Jean,
1994) ; models for their mode of production continue to be
put forward (Douady and Couder, 1996). The word
production implies both the origin and propagation of
pattern. A consideration of the stability of the latter is also
appropriate. Historically, effort has centred on describing
the patterns precisely and on accounting for their propagation. The main conclusions for the latter are that pattern
is propagated inwardly from an annular formative region
and that some form of repulsion or inhibition from preexisting organs explains the positioning of new organs
(Steeves and Sussex, 1989 ; Lyndon, 1990 ; Sachs, 1991).
Obvious issues remaining open are : the nature of de noŠo
initiation of pattern, the exact nature of the inhibition
process, and the nature of the stability. This paper addresses
mainly the first issue, de noŠo formation. However, the
biophysical proposal for its solution provides plausible new
explanations for inhibition and stability. Pattern initiation
and propagation both require a primary organ initiating
process, presumably the same one. Hence our de noŠo
formation study, insofar as it advocates a new primary
event, also bears on propagation.
The de noŠo origin of pattern is best known through the
transition from the globular to heart-shaped stage of the
embryo in dicotyledonous plants. The embryo has two
opposite bulges, with gaps in between, i.e. a saddle-shape.
De noŠo formation has been documented in detail in
Anagallis where a hemispherical flower primordium, devoid
of cellular or topographical periodicity, produces a ring of
five identical sepals (Herna! ndez and Green, 1993). In these
cases a new organ first appears as hump. It consists of a
curved coherent tunica layer overlying a less organized
0305-7364}96}050515­13 $18.00}0
corpus. De noŠo formation occurs a thousand times on a
sunflower head where an apparently featureless circular
hump produces a minute five-petaled disk flower (Herna! ndez
and Green, 1993). This transition is equivalent to the
conversion of a circular annulus into a polygon. A
comparable but simpler transition is the conversion of a
linear ridge into a regular row of similar leaf primordia as
seen in Graptopetalum (Fig. 1). The ridge had been induced
to form by experimental physical treatment. A highly
regular row of close-packed new lateral roots can be
induced in radish by excess auxin (Laskowski et al., 1995).
Thus the de noŠo origin of a spatially periodic array of
organs (linear or annular) is widespread. It is common in
normal development but can occur in atypical circumstances. We will deal mainly with the annulus version of the
de noŠo appearance of periodic pattern.
The process of converting a flat annular configuration
into a predictable undulating pattern of n-fold symmetry
(breaking circular symmetry) apparently has only one
plausible general mechanism. The annulus, uniform save for
random variation, must be a ‘ wavelength-dependent amplifier ’. This means that the initial approximate uniformity is
unstable and that, out of an array of random undulatory
perturbations (e.g. of concentration or of topography), only
a restricted wavelength range will be amplified. At the centre
of the range is the intrinsic wavelength. Because of the
range, there is some latitude. A wavelength at or near the
preferred undulation will subdivide the annulus into a
whole number of wavelengths. In the embryo this number
would be two ; in Anagallis and the sunflower, five. A
wavelength is an organ (crest) plus the two half-troughs
bounding it.
It is obvious that any process capable of establishing fivefold symmetry on an annulus on the basis of random
perturbations would have little difficulty in propagating a
# 1996 Annals of Botany Company
516
Green et al.—Phyllotaxis : Its DeŠelopmental Origin
A
B
C
D
F. 1. The origin of linear periodic organ arrays. De noŠo leaf
formation on isolated Graptopetalum leaves. Specimen width is approx.
1±2 mm. A, B, Two SEM views of replicas of the same tissue, 6 d apart.
A row of four similar leaf primordia has arisen. C, D, Similar, but in
top view. C, Initial state. D, The upper region shows a nearly straight
row of six leaf primordia (*). The ‘ fused ’ primordia at left and right are
subdivided with the same periodicity as characterizes the whole row.
Leaf tips are crests ; gaps (shallow or deep) are troughs. Hence there
appears to be a characteristic ‘ wavelength ’ in the formative tissue.
Other primordia, below, show a range of sizes bracketing the hump size
seen in the upper row.
five-fold pattern onto a new unstructured annulus once this
periodicity already characterized an adjacent region. Thus
the study of the de noŠo formation process indirectly
addresses propagation. Further, the mutual positioning of
the new humps in the de noŠo activity addresses both the
nature of the inhibition}repulsion process and the selfcorrecting nature of stability. Presumably plants use some
form of this amplification mechanism. We will deal mainly
with one explicit proposal.
W A V E L E N G TH-DE P E N D E N T
AMPLIFICATION
There are two candidates for the wavelength dependent
amplifier. One, chemical, is reaction-diffusion and has been
well expounded recently by Harrison (1993). The other is
physical and has been advocated by Green (1992). In both
cases of selective amplification, some natural process
opposes the development of short wavelengths while another
one inhibits undulations at long wavelengths. In the chemical
scheme, diffusion opposes rapid spatial changes in concentration. In the physical scheme it is the reluctance of a
solid sheet to bend sharply which counters the production
of close-packed folds (short wavelengths).
The nature of cause and effect for both schemes, while
familiar to physical scientists, is not commonly recognized
in biology. In biology most causal chains involve many
simple transductions whereby a compound is converted
from one state to another (e.g. phosphorylated). The
complexity is in the network. Here the complexity is within
a single transduction. In the mechanism, balances of fluxes
or forces are involved that require some calculus for their
characterization. Most pertinently, solutions of differential
equations involve both the primary formative (amplifying)
function and the boundary conditions (limits and constants
of integration). This diversity for input is not found in most
transductions. This two-fold format for mechanism fits well,
however, the key questions for plant patterning : what is the
critical formative process and how do nearby established
organs influence it ? The treatment here will be primarily
qualitative.
We will assume that the primary formative activity is the
well-defined process of physical buckling (Green, 1992).
This is an explicit three dimensional process. It is the
response of a flat surface to changes in its ‘ relaxed ’ or
unstressed area. It is familiar in the wrinkling of soaked
fingertips, the buckling of wet floors, and in the production
of the saddle-shaped potato chip from a flat disk. For de
noŠo origin, the pertinent boundary conditions for the
process (the additional constants which determine a particular solution) must be non-periodic. That is, the margins
must be flat. For propagation, they will be periodic in space
(show periodic variation in slope and}or elevation). The
buckling format deals efficiently with the specification of
pattern because the mechanism is based on a reliable
spontaneous activity of the responding system. It is assumed
that a solid tissue has elastic properties and that elastically
derived structures become permanent.
This paper will develop the physical theory for buckling in
small steps. The mathematical details are in another
publication (Rennich and Green, 1996). Here we will
illustrate the principles graphically, first as they apply to a
simple linear bar, then to an annulus and a disk. The
variations possible in an annulus will be explored in detail.
They provide simulations of the natural origin of whorls.
The predictability of the number of organs per whorl,
within a wide range of parameters, suggests that the
mechanism is robust. The variations seen with extreme
values of some boundary parameters resemble unusual
Green et al.—Phyllotaxis : Its DeŠelopmental Origin
Deflection (w)
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F. 2. Bending and the role of boundary conditions in the deformation of a linear structure. In all cases one boundary can move laterally. A,
A bar, hinged (simply supported) at both ends, bends to counteract pressure from below (small arrows). B, Increased pressure produces greater
amplitude to give an obvious " wavelength undulation. C, When both ends are clamped, a whole wavelength is generated. D, With mixed end
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conditions, an asymmetrical waveform results. E, When, in addition to gentle pressure from below, in-plane compression is added to the slightly
bent bar (large arrow), the bar arches. This type of deflection (response to lateral stress) is the proposed basis of tissue patterning. F, Amplitude
is a function of the amount of compression.
configurations seen in flowers and in mutants. Both types of
observation support the plausibility of the process in nature.
Undulation in solid materials
There are two key features in de noŠo pattern formation.
First is the ability to subdivide space into whole numbers of
districts simultaneously. That is, the system behaves as a
‘ whole number ’ ruler. Second is that the units thus formed
interact (reposition themselves) so that a regular symmetrical
pattern results. The pattern is very stable, i.e. self-correcting
as well as self-organizing. We will illustrate the two features
graphically, in sequence. First, the bending of a simple bar
or beam will show the role of boundary conditions and the
origin of the ‘ ruler-like ’ properties. This pertains to the
simple morphogenetic transformation seen in Fig. 1. Only
one equation is used, in Cartesian coordinates. Then, in
radial coordinates, a pair of equations will be applied to an
annulus (and disk) to give more complex three-dimensional
patterning as found in shoots and flowers.
DeŠeloping a pattern along a beam or bar
A progression of diagrams dealing with a beam, shown as
a bar, will illustrate the physical bending properties pertinent
to pattern. A beam flexes when subjected to pressure from
below (Fig. 2 A). At equilibrium, the counter-force to
equalize the added stress is the resistance to bending.
Significantly, the nature of the bowing is very much a
function of the boundary conditions. If the ends are hinged
(simply supported, in engineering terms), the undulating
bend is of half a wavelength (Fig. 2 B). If the ends are fused
to a support (clamped), the bend is of one wavelength (Fig.
2 C). A hybrid is possible, showing that boundary conditions
affect the over-all deflection qualitatively (Fig. 2 D). If, in
addition, the plate is compressed in its own plane (open
arrow in Fig. 2 E), added curvature results (Fig. 2 F). The
compression, giving the bar ‘ excess length ’ relative to its
rest condition, is mechanically equivalent to ‘ tendency to
grow ’. If the compression is increased, the amplitude of
bending is simply increased (Fig. 2 F). Our treatment relies
heavily on this parallel. The upward deflection by in-plane
stress, not by a push from below, is regarded as the origin
of the hump or organ in the plant. This process can
selectively amplify small irregularities. Thus far we have
shown the role of boundary conditions and have demonstrated that in-plane compression can lead to out-of-plane
deflection.
A highly pertinent feature is the natural wavelength. This
is a ‘ ruler ’ character seen as the bar develops regularly
spaced undulations. This periodicity arises when the bar is
connected to an elastic foundation which is visualized as a
set of springs normal to it (Fig. 3 A). Now when the plate is
compressed it opposes the new stress in two ways : by
bending and by changing the length of the springs. The bar
undulates (Fig. 3 B). In the plant, we propose the tunica to
be the formative layer, the corpus to be the elastic foundation
(springs normal to the bar or tunica). Thus the cause of a
hump is an accommodation to a departure from rest length
in an essentially flat structure. The diagrams show that, as
the initial length of the bar being compressed gets longer,
the undulations subdividing it ‘ count off ’ similar districts,
as if by a ruler (Fig. 3 C, D). The unit of measurement is "
#
wavelength. Figure 3 C has five ; D has seven. Even or odd
totals are a function of the boundary conditions. As a
straight bar is extended from the length where it reversibly
deforms to 3" ‘ short ’ wavelengths (Fig. 3 E), it deforms into
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3" ‘ long ’ wavelengths (Fig. 3 F). Slight further extension,
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however, leads to 4" ‘ short ’ wavelengths (Fig. 3 G). Thus
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the periodic deflection takes the wavelength closest to that
which still satisfies the boundary conditions. The measuring
action is approximate and discontinuous. The only inflexible
relation is that an integral number of " wavelengths must be
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present at equilibrium. In a ring, of course, a whole number
of wavelengths would apply. A key feature of the mechanism
is that a tissue, treated here as a continuum, can express a
latent periodicity. This feature is analogous to the series of
Green et al.—Phyllotaxis : Its DeŠelopmental Origin
Deflection (w)
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F. 3. Subdivision of a bar : de noŠo undulation with an intrinsic wavelength. This resembles the natural phenomenon seen in Fig. 1. A, To a
straight bar an elastic foundation (springs) is added. B, Now in-plane compression (arrow at right) leads to undulation because the applied stress
is countered both by bending the bar and by stretching}compressing the springs. C, A longer straight bar, when similarly compressed, gives 2"
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wavelengths. D, A still longer bar gives 3". Note the decline in amplitude with distance from the boundaries. E, Compression of a bar gives 3"
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‘ short ’ wavelengths. F, A comparable but somewhat longer bar gives 3" ‘ long ’ wavelengths. G, A small (2 %) further increment in initial length
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of the bar gives 4" ‘ short ’ wavelengths. The buckling process thus measures length, approximately, while dealing always with whole numbers of
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half-wavelengths. With hinged boundaries, as here, the integers are odd numbers ; with clamped boundaries, they are even.
natural wavelengths in the harmonics of a vibrating string.
That is, it is a widespread physical phenomenon.
It might appear that the conversion of a straight bar to
one with 3" wavelengths of undulation constitutes a true
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breaking of symmetry, or de noŠo subdivision. This is not so.
The pressure from below gives an initial upward curvature
at the boundaries, providing specific geometrical input. The
minimal bending energy solution only insures that this
boundary bending is maximally compatible with the rest of
the bar. Because of the presence of the springs, and the bar’s
stiffness, the amplitude of the undulations falls off from the
boundaries. In effect, the minimal energy response here
‘ propagates ’, with declining amplitude, the curvature at the
boundaries. For de noŠo pattern formation the initial
curvatures must be small random undulations.
The treatment above deals only with very small curvatures ; interaction between curved regions is precluded. With
this simple treatment, concurrent undulations simply sum.
For complete self-organization, i.e. de noŠo pattern, a twoequation treatment is required (Rennich and Green, 1996).
This shift to two equations, plus the non-linear character of
the initiation process, distinguishes this treatment from
most inhibitor-based models for phyllotaxis (e.g.
Yotsumoto, 1993). The complex treatment will now be
applied, to an annulus and a disk, to show that de noŠo
pattern can be produced by buckling in a context appropriate
to the shoot meristem.
RESULTS
Origin of pattern in an annulus
For the simulation, the assumption is that a large annular
region lacking any systematic structure can be broken up,
by the minimal energy buckling response, into a finite
number of similar sub-regions. This would simulate the
origin of a whorl of new organs in the plant. The analysis
here is done for pure undulations, crests and troughs being
interchangeable. In the plant, of course, the crests (organs)
are large. The troughs are presumed to be small and to
merge, forming creases.
The first computer simulations we conducted were with
an annulus whose width was the natural wavelength and
whose mean circumference was about 25 wavelengths. The
reasoning was that a circumference large relative to the
Green et al.—Phyllotaxis : Its DeŠelopmental Origin
519
D
A
400
0
B
1000
80
E
C
200
4000
F
F. 4. Spontaneous origin of a single whorl pattern in an annulus. The annulus, with radii of length 0±8 and 1±0, is provided with a random initial
pattern of undulations. Its physical properties give it an intrinsic wavelength of 0±2. Both edges are clamped. The non-linear equations are applied
to prescribe the topographical response to compression (or growth) of the annulus. The number of iterations toward the solution is shown for
each. The stresses, in arbitrary units, were : 0, 0±510, 0±566, 0±576, 0±585, 0±645. A–F, Six sequential images are shown. Note that large undulations
arise somewhat unevenly (C,D) and then the amplitudes and spacings even out (F).
wavelength would make the geometry close to Cartesian
despite being in a radial format. The natural wavelength is
defined in terms of the fourth root of the ratio : (flexural
rigidity)}(elastic constant). The constant applies to the
springs normal to the annulus (bar in Fig. 3). Having the
width of the annulus equal the wavelength facilitates
amplification of this wavelength in the radial direction.
Both edges, inner and outer, were clamped to be kept
horizontal. This promoted the production of a simple large
crests (upwards) and troughs (downward).
The annulus was provided with an initial topography
which was random with regard to the position and phase of
myriad small undulations. The initial deformations were
ramped down so they reached zero elevation and zero slope
at the margins. At the inner margin, the mesh used for
analysis can resolve wavelengths only of a certain minimum
length. Throughout the domain, the circumferential wavelengths were kept that long or longer. Wavelengths in the
radial direction were similarly constrained. Thus the initial
undulation spectrum was uniform relative to the mesh.
Otherwise, the wavelengths were random.
Stages in the conversion of the random initial condition
into a ring which undulates in 25 wavelengths are shown in
Fig. 4 A–F. Initial deformations which by chance were near
the natural wavelength were amplified to make the early
‘ teeth ’. These stayed roughly in place, while other areas
developed their topography. Many iterations were required
before the long wavelength pattern was noticeable against
the original ‘ noise ’. The teeth mutually positioned themselves and a near uniform ring of undulations was formed.
In responding to stress, a material yields slowly to
increasing stress up to a buckling or ‘ critical ’ threshold.
After this, strain (stretch) increases much more rapidly in
response to a given increment in stress. It was found that
patterning emerged rapidly as the buckling threshold was
approached, and improved thereafter. Figure 4 includes
post-critical configurations.
Robustness of de novo periodicity production in an
annulus
The 25 undulations seen in Fig. 4 illustrate the initiation
of polygonal (n-fold) symmetry in a narrow annulus. The
initial conditions were biased. The radial dimension was
exactly the natural wavelength and the mean circumference
was about 25 times this. Both edges were clamped. We
explored the latitude in the conditions which can lead to
periodic structure in such an annulus. (1) The initial random
array was taken from a different source, with no change in
the resulting configuration. (2) The edges of the initial
deformation were not ‘ ramped down ’ to zero elevation and
slope, with no pronounced effect. (3) The natural wavelength
was varied ³20 % and the expected change in total
undulation number was seen.
More significant was variation in boundary conditions
(clamped Šs. hinged) and in annulus width (with departure
from one wavelength). The four combinations of boundary
conditions (two possibilities at each edge) were each applied
to three values of annulus width (1, 1", 1" wavelengths). For
% #
each of the circular boundaries the expected number of
520
Green et al.—Phyllotaxis : Its DeŠelopmental Origin
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F. 5. Effect on patterned buckling of concurrently varying the boundary conditions and the dimensions of an annulus. The natural wavelength
is 0±2 ; maximum radius is 1±0. Three annulus widths (0±2, 0±25, 0±3, top to bottom) are used with various boundary conditions. The graphs show
deflection along radii at humps (——) or depressions (– – –). The matrix shows the two boundary conditions (BC) at large (L) and small (S)
circumference. These can be clamped (CL) or simply supported (SS). It also shows the number of wavelengths predicted (P) and observed (O)
for the larger and smaller circumferences. Positive contours are solid ; negative are dotted. A–C, Both margins clamped, increasing width. Slope
at the margins is zero. Humps are symmetrical along the radius (one wavelength). There is an increase of only one undulation as width increases
by 50 %. D–F, Outer margin hinged (simply supported), inner margin clamped. Note asymmetry along the radius : the slope is gentler near the
clamped edge. Increase of two undulations as width is increased.
undulations, based on the intrinsic wavelength, was calculated. As seen in Fig. 5 A–C in the doubly clamped condition,
buckling was highly regular and simple. There were
alternating simple crests and troughs. With 50 % increased
width there was a small increase in undulation number.
When the outer margin was hinged, buckling remained
simple and regular, but the slopes of the humps (depressions)
were greater at the outer margin, distorting the wave form
(Fig. 5 D–F). Increased width gave slightly more undulations. With the inner margin hinged, the wave form was
Green et al.—Phyllotaxis : Its DeŠelopmental Origin
A
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F. 6. Continuation of Fig. 5. A–C, Outer margin is clamped, inner is hinged. Increasing width of the annulus leads to Y-shaped ‘ fusion ’ of
undulations, reducing the number at the inner margin. Fused primordia are common in plants. D–F, Both margins are hinged (simply supported).
D, Width is 0±2. The simple hump pattern seen in Fig. 5 and A–B is replaced by undulations involving three nodes (two 1}2 wavelengths), thereby
largely splitting the annulus circumferentially. The humps are replaced by long circumferential ridges (trenches), small in number. E, The annulus
width of 0±25 does not match either 2 or 3 half wavelengths. The annulus is split circumferentially, as above, but there is now regular undulation
around the circumference. This is in alternating whorls. F, Width 0±3. 4 nodes (three¬1}2 wavelengths) are compatible with the annulus width.
The annulus is trisectioned, circumferentially, largely by long ridges (trenches) very small in number. Conclusions : undulation across the annulus
is enhanced, and circumferential undulation is suppressed, by hinged (simply supported) margins. This is especially striking when the annulus
width is an integral number of half-wavelengths. With a ‘ misfit ’ width, circumferential subdivision is combined with regular circumferential
undulation (E). This could produce, in a plant, two alternating whorls in a single step.
again distorted, but this time complex Y-shaped ‘ fusions ’ of
humps (depressions) took place, at large width, reducing the
number of undulations on the inner margin (Fig. 6 A–C).
This corresponded more closely to the reduced number that
fits there.
When both margins were hinged (Fig. 6 D–F), remarkably
522
Green et al.—Phyllotaxis : Its DeŠelopmental Origin
A
D
Width = 0.20
P
O
L
31
25
S
25
25
B
Width = 0.39
P
O
L
31
25
S
19
24
E
Width = 0.25
P
O
L
31
25
S
24
25
C
Width = 0.45
P
O
L
31
24
S
17
21
F
Width = 0.30
P
O
L
31
25
S
22
25
Width = 0.55
P
O
L
31
25
S
14
17
F. 7. Effect of increasing the initial width of an annulus upon the periodic undulations produced. Both margins clamped. Solid lines give positive
contours ; dashed lines are depression contours. The outer circumference always has a radius of 1±0. Width of the annulus is given as a fraction.
In each case the predicted (P) and observed (O) number of wavelengths for the larger (L) and smaller (S) circumferences are given. A has a width
of the intrinsic wavelength. A–C, Note that the number of undulations remains at 25 despite 1±5 fold variation in width. D–F, With still greater
width a reduction in undulations takes place as some radial ‘ ridges ’ become Y-shaped. There is an increase in variation of crest height. Sub-divided
ridges show the intrinsic wavelength.
Green et al.—Phyllotaxis : Its DeŠelopmental Origin
A
523
D
Width = 0.20
P
O
L
31
25
S
25
25
B
Width = 0.10
P
O
L
31
24
S
28
24
E
Width = 0.15
P
O
L
31
24
S
27
24
C
Width = 0.05
P
O
L
31
37
S
30
37
F
Width = 0.11
P
O
L
31
23
S
28
23
Width = 0.03
P
O
L
31
56
S
31
56
F. 8. Effect of decreasing the initial width of the annulus upon the periodic undulations produced. Conditions as in Fig. 7. The annulus in A
has a width equal to the natural wavelength. A–C, With reduction in width, the undulations vary more in height and in regularity of spacing.
D–F, The variation in height and in spacing increases ; the number of undulations increases markedly. The intrinsic wavelength loses significance.
The range of width where undulation number is basically stable (23–25) is from 0±10 to 0±3, i.e. threefold.
524
Green et al.—Phyllotaxis : Its DeŠelopmental Origin
A
B
C
D
F. 9. Four stages in the development in a constrained expanding disk. Initial random undulations are shown in A, but are not included in B–D.
A, Vertical exaggeration (VE) 181¬. B, The topography is roughly of unordered humps, as occasionally seen in the central part of abnormal
flowers. VE ¯ 308¬. C, A topography of humps arranged in ridges. VE 118¬. D, A final ‘ fingerprint ’ surface is mainly ridges. VE ¯ 43¬. This
is an apparent extrapolation of the broad annulus configuration seen in Fig. 7 F.
different results occurred. Apparently the added freedom
for bending led to a wave-form that involved three nodes
(like a horizontal S) as one wavelength was fitted to the
radial dimension (Fig. 6 D). When the width was 1"
#
wavelengths, then three half-wavelengths (four nodes) were
fitted to the width in much of the annulus (Fig. 6 F). In both
cases, undulation in the circumferential direction was greatly
reduced. Long circumferential ridges, trenches, resulted.
When, however, the radial ‘ fit ’ to " wavelength was poor,
#
much more buckling took place circumferentially. The
result was a regular ‘ checkerboard ’ pattern (Fig. 6 E).
There are parallels between the variations found in nature
and the responses seen here. The essentially simultaneous
origin of two alternating whorls can occur in flower
development in Anagallis (Herna! ndez et al., 1991). The
production of long concentric ridges is seen in a mutant of
petunia (see Green, 1992) as well as in mutant Arabidopsis
embryos (Liu, Xu and Chua, 1993). Many flower mutants
show ‘ fused ’ organs.
There is notable influence of dimensions on pattern per se.
We explored the effect of varying greatly the width of a
doubly clamped annulus. At the extremes, there were
somewhat complex effects. As the width of the annulus was
increased beyond 0±2 of its outer radius, the undulations
became ridge-like, the long axis being on a radius (Fig.
7 A–C). No striking effect, however, was seen until 0±3. Then
the ridges became less straight. Further broadening led to
the appearance of converging Y-shaped ridges (Fig. 7 D) as
also seen in Fig. 6 C. This had the effect of reducing the
number of wavelengths at the inner (smaller) circumference.
With still broader annuli, the trend continued with Yshaped converging ridges leading to further reduction in
undulation number at the inner boundary (Fig. 7 E–F). It
appears that the ridges do not break up significantly until
the annulus width approximates two wavelengths. The
subdivisions along the ridges, when seen, do appear to
reflect the intrinsic wavelength.
Decreasing the width of the annulus first leads to ridgelike primordia which parallel the annulus (Fig. 8 A–C).
Then the number of humps increases greatly, more than
doubling (Fig. 8 D–F). The natural wavelength apparently
loses significance because the humps (depressions) tend to
become isodiametric. The relative stability of the number of
undulations (24–26) over an annulus width range of more
than 2±5 times (0±11–0±3), provided that at least the inner
edge is clamped, indicates that subdivision by buckling is
robust to variation in dimensions.
Green et al.—Phyllotaxis : Its DeŠelopmental Origin
Origin of pattern on a disc
Increasing the width of the annulus naturally leads to the
question of pattern formation on a disc. This is of special
interest because the alternating whorl pattern of an entire
vibrating disc, a drum head (Chen and Zhou, 1993 ; Soedel,
1993) can resemble the symmetry of a complete flower
(alternating whorls). It is noteworthy that in the drum-head
no natural wavelength is present because there is no elastic
foundation. The treatment there is linear, using Bessel
functions. Therefore, an interesting issue is : what patterned
response does a whole disc make when an intrinsic
wavelength and non-linearity are present ? One can imagine
a pattern of concentric whorls, but any simple alternation
tendency has to be confounded by the variation in
circumference (as in Fig. 7). In many flowers, e.g., cucurbits,
whorls of five in the periphery change to a whorl of three for
the innermost carpels (as seen in the cross section of a
cantaloupe). Alternatively, the tendency to ‘ nest ’ similar
equal-sized organs over ever decreasing circumferential
distances might be solved by shifting to a spiral pattern
where circumferential spacing may be less tightly coupled to
the radial dimension. Accordingly, simulations comparable
in every way with those for annuli were applied to a disc.
The disc had an initial array of random undulations (Fig.
9 A). When compression was kept below the buckling (or
‘ critical ’) threshold, the disc showed an apparently disorganized pattern of humps in the response (the initial
variation is not shown). See Fig. 9 B. This structure resembles
the topography of the central region of a flower which
develops extra carpels, e.g. the fasciated mutant of tomato
(Szymkowiac and Sussex, 1992) Also, in sunflower, florets
which arise without connection to previous pattern develop
in this way (Herna! ndez and Palmer, 1988). In the simulation,
the periphery did not show the regular pattern of an annulus
of comparable diameter. In fact, the periphery seemed less
undulatory than the bulk of the disc.
When compression was increased beyond the critical
threshold, the disc developed curved parallel ridges, as in a
fingerprint (Fig. 8 C, D). The pattern was not obviously
reminiscent of either whorls or spirals. Thus, in our hands
at least, a regular whorled pattern (whole number of organs
in a ring) can be generated readily in one step in an annulus
but not on a disc. In the plant, an annular generative region
is in fact observed. We thus conclude that buckling within
an annulus, with at least one edge clamped, is a plausible
mechanism for whorl initiation in the plant. The tunica
would be the entity resisting flexure, the corpus is physically
coupled to it and would be the entity resisting dimensional
change normal to the undulation. We have given the model
annulus uniform properties with well defined nodes at both
boundaries. In nature, probably only the outer boundary is
well defined and the physical properties grade off gradually
to make the inner boundary. This is unlikely to diminish the
effectiveness of the mechanism.
525
DISCUSSION
De novo pattern formation
The question of the origin of periodicity during development
has been well examined in a book by Held (1992). We
endorse the position of Harrison (1993) that a holistic
mechanism, where the global situation influences local
activity, is essential to understanding the establishment of
initial symmetry. Here we add to his general argument a
specific example of a wave-length dependent amplifier that
is physical in nature. The general biophysical approach for
pattern has been developed extensively by Oster, Murray
and Harris (1983), Goodwin (1994), and others. There is a
fine review by Dillon, Maine and Othmer (1994). As noted
in the Introduction, this study bears on three issues : the
initiation event, the nature of inhibition, and self stability.
Also, this study deals with the linkage between causality in
morphogenesis and advanced mathematics. These matters
will be discussed in turn.
The initiation process
Most models for plant pattern are based on patterned
controls and assume that the organ initiating event is
momentary. It involves the instantaneous production of a
new organ as a point or as a well-defined district (Douady
and Couder, 1996). This assumption succeeds beautifully
in many cases, specifically for propagation. We argue here
that, instead of the critical event initiating periodicity being
an ‘ abrupt activation at a point ’, it could equally well be a
‘ gradual undulation of an area ’. A key feature is that the
organ forming tissue can have a latent physical periodicity,
evoked by spatially periodic or, most importantly for de
noŠo pattern formation, by spatially non-periodic phenomena. In contrast to many models, here the periodicity in the
surface does not arise from periodic prelocalized upward
pushes from the interior (Esau, 1953). Rather, it arises de
noŠo from the condition where a constrained uniform sheet
(a) has excess surface (‘ wants to expand ’) and (b) is
physically coupled to a uniform elastic substratum.
The inhibition feature
The inevitable mutual repulsion feature is, in our physical
model, the intrinsic reluctance of the solid sheet itself to
change its curvature sharply. Thus the proposed inhibition
aspect resides in the physical responding system ; it is not a
key metabolic activity of diffusing controls acting inside the
formative region. In our scheme the controls need reside
only at the boundary of the formative area.
Stability
The third issue mentioned in the Introduction was the
basis of stability. There is a substantive difference between
the form of stability found in many simple models based on
localized chemical inhibition and that observed in plants.
Those models provide stability of contact pattern (e.g. 3, 5
phyllotaxis) throughout a range of allowable divergence
526
Green et al.—Phyllotaxis : Its DeŠelopmental Origin
angles (Erickson, 1983 ; Douady and Couder, 1996). Plants
apparently show the inverse : stability of divergence angle
through a wide range of contact patterns. For example,
Kiwatowska (pers. comm.) found a mean variation in
divergence angle of 0±7° when, in the 2, 3 phyllotaxis of
Anagallis, the allowable variation was 60°. Many models,
based on abrupt activation at a point, or the provision of
non-overlap of new finite primordia, have minimal roles for
boundary conditions. Hence a wide variety of divergences
can occur with given contacts. We anticipate that models
using complex boundary conditions, such as ours, will
explain the stability of the form observed in nature.
On cause and effect
The present work bears on the role of mathematics in the
analysis of cause and effect in development. Analysis of
most processes breaks down into two issues : controls and
the responding system. For pattern formation there is the
additional issue of whether either is self-organizing. As to
the first matter, controls in most causal or transduction
chains, to be recognized as such, need to have only a
functional link with the response. For example, a chemical
neurotransmitter can control the production of action
potentials in a nerve ; auxin concentration can control
extension rate. The functional relation is often simple
proportionality. The coupling can, however, be monotonic
or bell-shaped as with the auxin influence on growth. The
key point is that the cause and effect relation is algebraic
and it is only operational ; it can reflect a non-explicit
functional connection. Such a format obviously can be a
useful starting point for refinement of the causal relationship.
For many features of propagation of phyllotactic pattern,
the above causal frame appears adequate. The controlling
factors can be viewed as inhibitory substances. The
responding system would be a universal and spontaneous
tendency to make an organ. Organogenesis thus occurs
when inhibition falls below a given threshold. The relation
between controls and the relatively undefined responding
system is algebraic, a correlation. Extending this view to the
origin of phyllotactic pattern, however, requires that the
controls be self-organizing. This feature, involving advanced
calculus, is in fact available in reaction-diffusion theory.
Here two chemicals, an activator and an inhibitor, interact
in a pair of differential equations to produce a de noŠo
pattern of control substances (Harrison, 1993). This
plausible scenario does not, however, exhaust the possibilities. The self-organizing feature may reside in the
responding system, not the controls, as demonstrated here.
In such a system, the non-patterned controlling elements
(the input) can take three forms. All are the result of
antecedent metabolic activity and development. (a) Control
factors may act through changing the constants in the
master differential equation, e.g. altering the flexural rigidity
of the sheet. Such variation leads to the predictable variation
in number of undulations (data not shown). (b) They may
vary the dimensions of the formative area. The non-intuitive
difference between the ridges in Fig. 6 D and F and the
checkerboard in Fig. 6 E is due solely to a difference in width
of the annulus. (c) They may vary the boundary conditions.
The presence of Y-shaped primordia (Šs. I-shaped) in Fig.
6 C is due to the hinged condition at the inner margin of the
annulus. Such forked primordia are not seen when, with the
same dimensions, the inner margin is clamped (Fig. 5 C, F).
It is clear that here the relation between controls and the
responding system is much more specific than it is in the
correlations sufficient in the algebraic framework. In the
self-organizing format of reaction-diffusion, the control
compounds (activators) are patterned within the responding
system. Once again the relation to the responding system
(the physical response) is a correlation only. Our scheme has
the pertinent control factors at the margin of the responding
system and there is an explicit relation connecting them to
the geometrical response. We do not vary the function. Each
response presented here is a particular integration resulting
from specific values of two kinds of controls : the dimensions
over which the integration occurs and the constants of
integration at the boundaries. It is seen that the latent
periodicity can emerge in a wide variety of patterns,
depending on values for the control factors.
Most models which recognize the two layered structure at
the apex assume that patterning arises in the deeper layer
(corpus) via chemical diffusing controls and that the outer
layer is passive. We suggest on the other hand that the origin
of patterning may be physical, not chemical, and involves
both layers, not just one.
We show that the buckling mechanism is workable, and
in fact robust, for the initiation of whorls. Further, some of
the unusual responses occurring at extreme values for the
control parameters resemble configurations seen in mutants
of flowers and embryos. The present proposal has its closest
precursor in the work of Otto Schu$ epp (1916) who
considered excessive growth of the tunica to be the basis of
shoot patterning. He refers to the ‘ sich faltenden
Oberfla$ chenschicht ’ (self-folding surface layer). His idea
that the physics of the expanding surface tissue, as against
pre-localized upward pushes from the corpus, is instrumental in hump formation is developed further here. The
formal quantitative connection between tissue properties
and latent periodicity was unknown in Schu$ epp’s time. We
speculate that it underlies the origin and propagation for
spiral as well as whorled patterns.
A C K N O W L E D G E M E N TS
Dedicated to the illustrious scientific career, and the
transition to emeritus status, of Professor Andreas Sievers
of the University of Bonn, Germany. Supported by a grant
from the National Science Foundation to P.B.G.
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