Phyllotaxis
Cris Kuhlemeier
Institute of Plant Sciences, University of Bern, CH-3013 Bern, Switzerland
Phyllotaxis, the regular arrangement of leaves or flowers
around a plant stem, is an example of developmental
pattern formation and organogenesis. Phyllotaxis is
characterized by the divergence angles between the
organs, the most common angle being 137.5-, the golden
angle. The quantitative aspects of phyllotaxis have stimulated
research at the interface between molecular
biology, physics and mathematics. This review documents
the rich history of different approaches and conflicting
hypotheses, and then focuses on recent molecular
work that establishes a novel patterning mechanism
based on active transport of the plant hormone auxin.
Finally, it shows how computer simulations can help to
formulate quantitative models that in turn can be tested
by experiment. The accumulation of ever increasing
amounts of experimental data makes quantitative modeling
of interest for many developmental systems.
A numbers problem in development
Following the stem of a tomato plant from the root junction
upward, one encounters a succession of leaves in a developmental
and temporal gradient. The bottom leaves are the
oldest and may already be senescing. Moving up the plant,
the leaves become smaller and younger until the close
packing of young leaves obstructs further access to the
tip. Located at the tip is the shoot apical meristem, a fragile,
dome-shaped tissue of 80–150 mm in diameter, where the
lateral organs – leaves at first, flowers after the onset of
reproductive development – are continuously initiated.
Organ initiation in plants is an iterative process and the
meristem can remain active over the life span of the plant.
The lateral organs are positioned in distinct patterns,
and this arrangement around the stem is called phyllotaxis.
The most prevalent patterns are distichous, spiral,
decussate and whorled patterns (Figure 1). Phyllotactic
patterns can be disrupted by experimental interference,
but, within limits, they will quickly recover and reestablish
the original arrangement [1]. On the one hand, apparently,
aberrant positioning can be corrected. On the other hand,
transitions between patterns, for instance from decussate
to spiral, occur frequently during the life of a single plant
(Figure 2a,b), indicating that developmental switches can
override the self-correction mechanism.
Models of phyllotaxis must explain its de novo
establishment in the radially symmetric embryo, the
stable maintenance of the different arrangements and
the observed transitions between phyllotactic patterns.
Most importantly, they must explain the specific divergence
angles of 1808, 908 and 137.58, and, in rarer cases,
other angles as well. This quantitative aspect makes
phyllotaxis an unusual developmental problem.
The geometry of phyllotaxis
The types of phyllotactic arrangements (Figure 1) can be
fully described by the number of organs that is simultaneously
initiated (jugacy), the angular divergences
between primordia, and their spacing along the apical–
basal axis [2]. Most common in nature are spiral patterns
with divergence angles of 137.58. This is the golden angle,
which is obtained when a circle is sectioned according to
the golden ratio of 1.618 [3,4].
Higher order patterns arise when the size of the
primordia is small relative to the circumference of
the apex. Figure 2c shows such a system with divergence
angles of 137.58 and with the leaf primordia arranged in
eight right- and 13 left-winding spirals. The numbers of
these so-called visible contact parastichies are not arbitrary,
but are given by the Fibonacci series (1, 1, 2, 3, 5, 8,
13, 21. . .), in which each term is the sum of the two previous
ones. The ratio of two consecutive Fibonacci numbers
tends to the golden ratio [3]. In simple spiral patterns,
such as seen in an Arabidopsis seedling, the average
divergence angle between successive leaves or flower primordia
can vary by several degrees from the theoretical
angle, and an angle between any two consecutive leaves
can vary by even more [5,6]. This variation in individual
angles is a logical consequence of the cellular make-up of
the meristem. Assuming an average of 25 cells in the
circumference of the Arabidopsis meristem, a lateral shift
of primordium initiation by a single cell would cause a
change in angle by 3608/25 = 14.48. By contrast, in higher
order phyllotactic patterns (Figure 2c), a deviation of <18
would disturb the pattern [3]. Another curious aspect of
higher order patterns is that the geometry of the apical
surface and the size and shape of the organs affect the
subjective perception of the pattern [7,8].
Why are Fibonacci spirals so common in nature? Why is,
for instance, a hexagonal arrangement or a divergence
angle of 308 rare? Mathematical modeling suggests that
spiral leaf arrangements are superior for light capture, at
least under model conditions [9,10]. However, the establishment
of phyllotaxis is largely insensitive to environmental
conditions, and adaptations to the light
environment occur primarily at the post-meristematic
level through adjustment of divergence angles, leaf shape,
leaf inclination and petiole length. Thus, it seems unlikely
that phyllotactic patterning is of major adaptive significance
with respect to light capture.
Another view emphasizes the role of phyllotaxis as a
solution to a packing problem. In 1873, Hubert Airy [11]
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Corresponding author: Kuhlemeier, C. (cris.kuhlemeier@ips.unibe.ch).
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argued that the adaptive significance must lie in the bud
itself, where the patterns are most regular: ‘It is for
economy of space, whereby the bud is enabled to retire
into itself and present the least surface to outward danger
and vicissitudes of temperature’. This hypothesis is interesting
but also hard to prove experimentally.
A third idea is that the prevalence of Fibonacci spirals is
a consequence of the molecular mechanism underlying
phyllotaxis. The molecular mechanism might well constrain
the number of geometric options that is biologically
feasible. This might ultimately explain why and under
which conditions spiral patterns are preferred. In the next
sections the experimental data and their use as a basis for
quantitative models will be reviewed.
Plant stem cells
The shoot apical meristem, first observed by Caspar Wolff
in 1759 [12] (Figure 3), is a fragile, dome-shaped tissue of
80–150 mm in diameter. It remains active after the end of
embryogenesis, and all mature aerial tissues originate
from the meristem, be it an annual plant or a 1000-year
old tree.
The meristem consists of a few hundred small, rapidly
dividing cells (Figure 3). The cells in the central zone at the
tip of the dome do not differentiate, but their daughters in
the peripheral zone have the option to either retain their
identity or differentiate into the cell types of the central
axis and lateral organs. The analogy with animal stem cells
is now obvious [13]. The shoot apical meristem is an
attractive system for studying fundamental questions of
stem cell research because it is a spatially well-defined
structure and neither cell migration nor cell death complicate
matters. Cells divide symmetrically and, with the
exception of the cells on the surface, the orientation of cell
division appears random.
The stem cells are defined at the molecular level by the
expression of CLV3, a small peptide that signals to the
underlyingWUSexpressingcells.TheWUScenterpromotes
stem cell fate and can therefore be compared to a stem cell
niche [13,14]. Negative feedback between the homeobox
proteinWUSandtheCLVligand-receptorsystemisthought
to regulate the size of the stem cell population. WUS and
CLV3 are expressedin distinct subdomains of the meristem.
Genetic approaches have been essential for understanding
the interactions between WUS and CLV, but they are not
entirely conclusive in explaining how the expression
domains can be so sharply delimited in space. Interestingly,
mathematical models can explain the positioning of the
WUS domain and the phenotype of mutants and surgical
alterations with simple and realistic assumptions [15].
Figure 1. Phyllotactic patterns. (a) Distichous or alternate: one leaf per node with 1808 divergence angles (Trisetum distichophyllum). (b) Spiral: angle is 137.58
(Lysimachia dethroides). (c) Decussate: two opposite leaves per node at 908 angles between the pairs (Urtica dioica). (d) Whorled: multiple leaves per node (Galium
odoratum). Drawings by Peter Leuthold.
Figure 2. Phyllotactic transitions and higher order spirals. (a) Young vegetative Antirrhinum majus with decussate phyllotaxis. (b) Top part of older plant with spiral
phyllotaxis. (c) Mammillaria marksiana with eight right-winding and 13 left-winding visible contact parastichies. Photographs by Peter von Ballmoos.
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The analogy with animal stem cells is justified and
conceptually useful, yet should not obscure the unique
properties of plant stem cells and plant development.
Complete ablation of animal stem cells has serious consequences.
Not so in plants. Cell ablation experiments
illustrate this point. It has long been known that major
injury to the central zone has only mild effects, and growth
and leaf development continue essentially as normal and
without delay [16,17]. After laser ablation of the central
zone including all WUS (and presumably CLV3) expressing
cells, cells in the peripheral zone re-express WUS and
presumably adopt a stem cell fate [18]. Similarly, in
clavata mutants, the central zone expands through respecification
of peripheral zone cells [19]. These experiments
show that cells can switch fate and regain stem cell character.
This is a manifestation of a general feature of plant
cells that is carried to the extreme in plant cell tissue
culture, where single differentiated cells can be reprogrammed
with high efficiency [20]. Might it be possible
that efficient developmental reprogramming contributes to
stem cell homeostasis in the undisturbed shoot apical
meristem as well?
The orientation of cell divisions in the shoot meristem
appears to be mostly random. The one exception allows for
a functional subdivision of the meristem into layers
(Figure 3). In dicot plants, there are usually three concentric
tissues: L1, L2 and L3. The outer L1 layer will
differentiate into the epidermis, and the L2 and L3 layers
each make variable contributions to the inner tissues [21].
In particular, the cells of the L1 layer divide almost exclusively
perpendicular to the surface. These oriented cell
divisions of the L1 cause a million-fold increase in leaf
surface area, whereas the epidermis remains a single cell
layer in depth.
As we have seen above, cells of the peripheral zone can
quickly and efficiently reprogram and acquire stem cell
fate. Switches between the L1 and L2 layers occur in only
one direction. In the rare cases that an L1 cell divides
parallel to the surface, the displaced cell loses L1 identity
and takes on L2 identity according to its position [21]. By
contrast, L2 cells seem to be unable to acquire L1 fate.
When L1 cells are destroyed by laser ablation, the underlying
L2 cells do not differentiate into L1 cells, instead they
enlarge and terminally differentiate [18]. Similarly,
expression of a cellular toxin in the L1 layer leads to
morphological defects in the epidermis but not to respecification
of underlying L2 cells [22]. Such experiments
suggest that L1 fate cannot be acquired by reprogramming
of L2 cells. An interesting possibility is that the L1 produces
a mobile signal that prevents differentiation in the underlying
layers. If this is correct, the L1 layer can be considered
a stem cell niche.
Experiments on phyllotaxis
In his 1868 textbook on general plant morphology, Wilhelm
Hofmeister [23] describes his detailed observations on
organ initiation in large numbers of plant meristems. He
concludes that each leaf arises in the largest gap between
the edges of the previous leaf primordia. Variations of this
simple concept remain the basis of all theories of phyllotaxis
and have inspired subsequent experiments. In
mechanistic terms, it suggests that existing primordia
are the source of a signal that prevents establishment of
primordium identity in the vicinity of the source.
If older primordia provide positional information,
removing them should change the position of new primordia.
Surgical experiments performed on Lupinus albus
meristems in the 1930s suggest that when the inhibiting
Figure 3. The shoot apical meristem. (a) Shoot apical meristem (v) of Brassica capitata (cabbage) as first seen by Caspar Wolff in 1759 [12]. Leaf primordia are labelled p, a,
c, d, e. (b) Meristem of tomato visualized by low vacuum scanning electron microscopy. Abbreviations: M, Meristem; P1, P2, P3 primordia. Photograph by Didier Reinhardt.
(c) Subdivision of the meristem into zones. The central zone (CZ) contains the stem cells; organ initiation takes place in the peripheral zone (PZ). (d) Subdivision of the
meristem into concentric layers, L1, L2 and L3. (e) The WUSCHEL–CLAVATA feedback loop. The CLV3 expression domain defines the stem cells.
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influence of an older primordium is removed, the new leaf
initial (I1) closest to the gap can move towards the gap.
Although this experiment has been a cornerstone of phyllotaxis
research, it needs to be interpreted with care.
Similar experiments in tomato recapitulate the original
experiments, but atypical stem elongation growth is also
evident [24]. More precise removals of I1 by infrared laser
ablation show that in most cases a new primordium forms
in the vicinity of the ablated position and then is the
starting point for a normal spiral (Figure 6c).
A biophysical patterning mechanism is unlikely
So what is the molecular nature of the inhibiting influence
exerted by pre-existing primordia? The first concept is that
the control mechanism is not based on a signaling molecule
at all. Spiral patterns exist outside biology [11,24,25], and
the regularity of phyllotaxis might simply be a consequence
of physical forces operating on a growing system.
Although genetic analysis over the past half century has
amply demonstrated the preeminence of specialized regulatory
mechanisms, the physical constraints on developmental
programs should not be neglected.
In its most extreme form, the biophysical theory posits
that differences in tension between the L1 surface layer
and the inner tissues lead to tissue buckling [26]. The
position of the bulges is determined by the mechanical
properties of the meristem, and once the bulges are formed,
they will grow and acquire new developmental identity.
This concept is fascinating. Differential tissue tension
provides an elegant mechanism for de novo pattern formation
based on simple engineering principles, with phyllotactic
patterning an emergent property of the system. In
biological terms, it puts the emphasis at the control of
growth, that is, at the cell wall. Local application or gene
induction of expansin, a protein that regulates wall extensibility
in vitro, induces primordia at ectopic positions
[27,28]. This suggests that wall properties can indeed
influence organ positioning, but does not prove that expansin
controls positioning. If it were the primary signal, one
would expect expansin to be differentially expressed
between the L1 and the underlying layers. Instead, the
expansin mRNA is preferentially expressed in incipient
primordia, consistent with expansin expression as the
readout of a primary signal [29].
If global tissue tension drives organogenesis, local
manipulation of that tension should affect neighboring
tissue and affect organogenesis. This is not the case.
Removal of the L1 layer from an incipient primordium
by infrared laser ablation completely abolishes its outgrowth.
However, the observed effects are strictly local.
Even the removal of large stretches of L1 tissue has no
global effects and leaves continue to form at positions with
intact L1 [30]. Therefore, there is no convincing evidence
that biophysical mechanisms constitute the principal regulatory
mechanism. However, they might well have a more
limited role in morphogenesis [31].
The patterning mechanism is based on active
transport of the plant hormone auxin
The obvious alternative to a biophysical regulatory
mechanism is chemical signaling. All evidence now points
towards the plant hormone auxin, or indole-3-acetic acid
(IAA). This simple amino acid derivative is structurally
related to the neurotransmitter serotonin and, similar to
serotonin, it is actively transported. In Arabidopsis, a
family of at least six transporters, the PIN proteins, catalyzes
auxin export from cells [32,33]. Each of the PINs has
a distinct expression pattern and a characteristic asymmetric
subcellular localization. PIN polarization can proceed
by different mechanisms [34,35], but the mechanism
most relevant for phyllotaxis is PIN polarization by auxin
itself. Auxin promotes retention of PIN in the plasma
membrane [36] and, hence, an auxin concentration gradient
across the cell could promote asymmetrical localization
of the transporter.
Inhibition of active auxin transport, either through
chemical inhibition or by mutations in the PIN1 transporter,
specifically inhibits organogenesis. This leads to the
formation of a naked meristem that grows normally but is
entirely devoid of lateral organs. The defect can be rescued
by the application of a microdroplet of auxin to the peripheral
zone of such a pin-shaped meristem [37]. Auxin
accumulates at sites of incipient organ formation [6,38] and
localized application of auxin can induce ectopic organs
[37]. Hence, auxin is required for the induction of lateral
organs.
As well as being required for lateral organ induction, it
appears that auxin also provides positional information. In
the pin1 mutant, all transcripts including the Pin1 mRNA
itself are uniformly present throughout the peripheral
zone [39,40], indicating the absence of a prepattern.
Furthermore, the position and concentration of the auxin
applied to a pin1 meristem determine the position and size
of the induced primordium. Synthetic auxins with different
transport properties induce organogenesis but the organs
are not correctly positioned [41]. Proper positioning
requires IAA. In summary, all auxins can induce organogenesis,
but correct positioning requires the endogenous
hormone.
PIN1 is predominantly expressed in the L1 surface layer
of the shoot meristem and in the incipient and young
primordia. PIN1 is induced by auxin and the expression
maxima coincide with maxima of auxin concentration
[6,38,42]. Modeling of auxin transport combined with a
comprehensive analysis of PIN1 expression indicates that
the observed subcellular polarization of PIN1 will cause
auxin to accumulate at incipient primordia [43].
The combined experimental results suggest that auxin
or, more precisely, the naturally occurring auxin IAA,
instructs the meristem as to organ patterning [18,37].
The results also suggest a simple model in which an
autoregulatory loop between auxin, PIN1 expression and
polar localization of PIN1 creates auxin maxima by transport
against concentration gradients (Figure 4a,b). The
auxin maxima can activate downstream processes through
specific receptors and the combinatorial action of members
of two large families of transcription factors (ARF and IAA/
AUX) [61,62].
Interestingly, and rather puzzling, during phyllotactic
patterning in the L1, PIN1 polarizes towards auxin concentration
maxima, whereas during vein formation it
seems to orient away from them [6,38,44]. How this
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happens is unclear. The molecular mechanism might
involve a protein kinase such as PINOID, which can cause
reversal of PIN1 orientation [45].
Phyllotactic transitions
Phyllotactic patterns often change during development. In
Antirrhinum, for instance, the early vegetative plant is
decussate, but later on the pattern changes to spiral
(Figure 2). There are numerous mutants that affect phyllotaxis
but most of them result in irregular leaf arrangements.
Such defects are often caused by perturbations of
the meristem, for instance in the clavata mutants where
the enlarged meristem has a tendency to fasciate [46].
‘Homeotic’ mutations, which change one regular pattern
into a different one, are scarce. This suggests that regular
phyllotactic transitions are under polygenic control.
The only well-studied homeotic mutant is abphyl1 in
maize, which changes the distichous leaf arrangement of
the wild type (one leaf per node, 1808 angles) into decussate
(908 angles between opposite leaf pairs), although not completely
– reversals to distichous are frequently observed
[47,48]. ABPHYL1 encodes an ARR protein, a negative
regulator of cytokinin action. In Arabidopsis, a septuple
arr mutant shows slight irregularities in phyllotaxis [49].
Figure 4. Models of phyllotaxis. (a,b) Experimental model of phyllotactic patterning. (c,d) Computer simulation of spiral phyllotaxis and model validation. (a) PIN1
orientation directs auxin fluxes (arrows) in the L1 layer, leading to accumulation of auxin (red color) at the initiation site (I1) in the peripheral zone. This accumulation
eventually results in organ induction. (b) Later, basipetal PIN1 polarization inside the bulging primordium (P1) drains auxin into inner layers, depleting the neighboring L1
cells. As a consequence, another auxin maximum is created in the peripheral zone at position I1 removed from primordia P1 and P2. (c) The simulation by Smith et al. [6] can
initiate spiral phyllotactic patterning starting de novo from the radially symmetric embryo. Lighter green signifies higher auxin concentration. PIN1 is depicted in red. For
the animation, see supporting movie 4 [6] at http://www.pnas.org/content/vol0/issue2006/images/data/0510457103/DC1/10457Movie4.mpg. An animation based on the
model by Jo¨ nsson et al. [53] can be found as supporting movie 1 at: http://www.pnas.org/content/vol0/issue2006/images/data/0509839103/DC1/Movie1.mov. (To view the
movie, please paste the URL into your web browser rather than clicking the URL.) (d) Comparison of the divergence angles: angles measured in Arabidopsis with standard
error bars (blue), and angles generated by the spiral phyllotaxis model (red). Reproduced, with permission, from Ref. [6]. (e) Ablation of incipient primordium I1 (white
arrowhead) leads to the initiation of a new primordium in close vicinity (red arrowhead). Reproduced, with permission, from Ref. [18]. (f) The simulation model reproduces
the effects of the ablation shown in (e). Reproduced, with permission, from Smith et al. [6].
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Several ARR proteins are direct downstream targets of
WUS, which is expressed in a radially symmetric domain
below the central zone and is probably not directly involved
in leaf positioning. If cytokinin simply increases the size of
the meristem, this changes the ratio between meristem size
and primordium size. This provides a model for phyllotactic
transitions that is in line with classical theories [7].
Additional regulators are likely to be involved in maize
and Arabidopsis.
Virtually nothing is known about the molecular
mechanisms that underlie the transitions between different
spiral systems (e.g. 5/8 to 8/13 parastichies), except
that larger meristems seem to have higher Fibonacci
numbers. A major question is how higher order patterns
such as those seen in sunflower heads are generated at all.
It seems more likely that the position of organs is determined
by their nearest neighbors in space than in developmental
time.
Quantitative approaches to phyllotaxis
The model presented in Figure 4a,b is qualitative in nature
and reflects the lack of quantitative experimental data. This
is not a specific shortcoming of this particular model but
pertains to most developmental models. However, in phyllotaxis,
qualitative models are particularly deficient
because phyllotaxis is in essence a quantitative phenomenon.
A good model should: (i) generate a wide variety of
phyllotactic patterns; (ii) generate robust patterns that are
insensitive to noise and can recover from perturbations; but
(iii) at the same time generate phyllotactic transitions, such
as the frequently observed change from decussate to spiral
(Figure 2); (iv) generate patterns de novo from a radially
symmetric embryo; and (v) be based on plausible assumptions
about the molecular mechanisms involved [6].
Most models [24,50–52] are in some way or another
guided by Hofmeister’s rule, that older primordia inhibit
the emergence of new initials in their vicinity [23]. In a
recent example, the inhibition exercised by each primordium
is inversely proportional to its distance from a given
position, and decreases exponentially with the age of that
primordium [2]. At positions where the sum of the inhibitory
effects exerted by the primordia falls below a predefined
threshold, a new primordium will arise. The model
recreates distichous and spiral patterns in a robust manner.
The introduction of a second inhibitory function that is
similar to the first, except that it decays more rapidly
over time, makes it possible to simulate decussate and
whorled systems. Variation in the short-range-inhibition
parameters also allow for phyllotactic transitions. Thus,
with only minimal assumptions about the molecular mechanisms
involved, it is possible to recreate all the important
aspects of phyllotactic patterning.
The construction of mechanistic quantitative models
that incorporate the accumulating molecular data is now
feasible. Two recent computer simulation models are based
on data on auxin and auxin transport [6,53]. Both assume
that patterning occurs in the L1 surface layer, that auxin is
uniformly available in the meristem, that polarization of
PIN proteins is based on auxin concentrations in the
neighboring cells and that the rate of auxin transport
depends on diffusion and active transport. They differ in
several details (Box 1), most importantly the algorithms
used to model cell division and the equations describing the
dependencies of PIN polarization and auxin transport on
cellular auxin concentration. Linear dependencies on
auxin concentration produced phyllotactic patterns of limited
stability [6,53]. Stability improved when strongly nonlinear
equations were used to capture PIN polarization and
active transport. Moreover, in the model by Richard Smith
et al. [2] an additional equation was introduced to increase
PIN polarization exclusively within the primordia. With
these modifications, the model can start from a radially
symmetric embryo, produce opposite cotyledons, and then
settle into a Fibonacci spiral [6]. The patterns are stable
and reproduce in vivo measured angles within one standard
deviation. The model also faithfully recapitulates the
phenotype of the pin1 mutation and the effects of selected
experimental manipulations (Figure 4).
Questions raised by the mechanistic models
Mathematics, physics and biology have all made
substantial contributions to phyllotaxis research but, until
recently, they have had surprisingly little interaction.
Mathematical models were based on simplified representations
of anatomical structures and on molecular mechanisms
with little experimental basis. Even though
technology is rapidly advancing, the lack of experimental
data remains a bottleneck. Many of the molecular components
are probably not yet known, for example, ABC
transporters, import carriers and protein kinases such as
PINOID and transcription factors. In fact, we cannot even
measure auxin concentrations with cellular resolution.
Knowing more about the nuts and bolts of the system
should enable the development of more refined models.
Even when experimental data are available, they tend
to be qualitative. The assumptions made about PIN polarization
and auxin transport are plausible, but the exact
forms of the equations used for quantitatively describing
PIN polarization and auxin transport are the outcome of
the modeling efforts. For instance, the models predict a
nonlinear dependence of PIN polarization on auxin concentration,
and a molecular mechanism for measuring
auxin in neighboring cells. Similarly, the inclusion of a
‘factor X’ that modifies PIN polarization specifically within
incipient primordia is based on the enhanced stability it
provides to the model. Such predictions serve to guide
experimentation. They illustrate the usefulness of quantitative
modeling when only limited and qualitative data
are available.
Finally, the published simulation models make the
assumption that all relevant patterning events take place
in the outer cell layer of the meristem and that the inner
tissues can be neglected. This is a useful but potentially
unjustified simplification. For instance, most of the PIN1
molecules are not located at the meristem surface but in
internal cells [40]. When auxin exits the L1 layer, it induces
vein formation, but by a mechanism that has been proposed
to depend on auxin flux rather than on concentration
[54,55]. This vein will connect to the primary vasculature of
the stem. Can we exclude the old idea [56] that the stem
vasculature dictates, or at least influences organ position?
All recent experiments and simulations are consistent with
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a phyllotactic patterning mechanism in which the
meristem is autonomous and does not receive information
from the mature tissues below. However, to definitively
answer such questions, a model is needed that integrates
the events at the surface with those in the internal layers.
Conclusions
The cells of the shoot apical meristem express many
molecular components that interact to initiate lateral
axes of growth. How should one order these components
and build meaningful mechanistic models? In developmental
biology, the approach has traditionally been
intuitive and largely qualitative. This is in contrast to
many other branches of experimental biology, such as
physiology and population genetics that have a long
tradition of quantitative approaches. A good example
is metabolic control analysis that over the past decades
has provided us with a new understanding of the regulation
of biochemical pathways [57,58]. The fascination of
phyllotaxis is that it is a numbers problem that has
always been the subject of mathematical treatment. It
is now becoming possible to create quantitative models
with realistic assumptions based on molecular data. This
is necessary if we want to order the mass of accumulating
molecular data and, at the same time, understand the
geometrical and physical constraints on development
[59].
Quantitative modeling might also help us to identify the
molecular targets for artificial and natural selection. Most
of our developmental mutants would not survive outside
the growth room because most changes in complex
regulatory networks are likely to incur severe fitness costs.
Selection for developmental novelty without loss of fitness
seems to favor selected molecular components and
often involves subtle quantitative differences [60]. Quantitative
developmental control analysis might identify such
targets and should help in understanding the evolution of
development.
Acknowledgements
I am very grateful to Didier Reinhardt, Jiri Friml, Thomas Laux, Marja
Timmermans, Henrik Jo¨nsson and Przemek Prusinkiewicz for their
insightful comments and criticisms. Work in the author’s laboratory was
supported by the Canton of Bern, the Swiss National Science Foundation
and the European Union.
References
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formation: a simulation study. Can. J. Bot. 84, 1635–1649
3 Prusinkiewicz, P. and Lindenmayer, A. (1990) The Algorithmic Beauty
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4 Jean, R.V. (1994) Phyllotaxis: A Systemic Study in Plant
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development. Plant Physiol. 139, 960–968
Box 1. Quantitative models of phyllotaxis
The models by Henrik Jo¨ nsson et al. [53] and Richard Smith et al. [6]
are similar in many aspects. In both models, organ patterning is the
result of dynamic interactions between existing and incipient
primordia in a growing apex, mediated by actively transported auxin.
Some of the key elements are:
(1) The dependence of PIN1 polarization on auxin concentration in
neighboring cells
In both models, the distribution of the PIN1 in cell i towards the
membranes facing each of the four surrounding cells j1–j4 depends on
the relative auxin concentrations in these neighboring cells (Figure I).
According to Smith et al. [6] the relationship is strongly non-linear:
the amount of PIN1 in cell i that is localized in the membrane towards
cell jn is proportional to: b½IAAŠj
, where b is a constant and the exponent
[IAA]j is the IAA concentration in the neighboring cell jn. In the
simplified example below, the auxin concentrations are set at 1, 2, 2,
and 3. If b = 3, the partitioning of PIN1 over the four sides of cell i will
be: 31
: 32
: 32
: 33
, or a threefold difference in auxin concentration
translates into a ninefold difference in PIN1 polarization. In the
simulations presented by Jo¨ nsson et al. [53], the relationship is linear
with a saturation term. In both models, an unknown short-range
signal must communicate the auxin concentration from the neighboring
cells jn to cell i. Cell walls are depicted in green and PIN1 is
depicted in red.
(2) The dependence of active auxin transport on auxin concentration
In both models, the rate of auxin transport depends on auxin
concentration. In the Jo¨ nsson model, the relationship is linear
(with a saturation term), whereas according to Smith et al. the
transport rate is proportional to the square of the cellular auxin
concentration.
(3) Auxin transport within primordia
The Smith model makes the ad hoc assumption that within
incipient primordia, PIN1 polarization is slightly different in the sense
that PIN1 orients more strongly towards the center of the primordium.
In biological terms, this is equivalent to introducing a regulatory
factor X.
(4) Cellular template
In both models, cellular growth is modeled, either by representing the
cells as circles with mechanical interactions [53], or with a cell division
algorithm [6] that mimics the division pattern seen in live apices [19].
Full descriptions of these growth and division models are presented
in the supplementary materials of Refs [6,53].
Figure I.
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