A plausible model of phyllotaxis
Richard S. Smith*†
, Soazig Guyomarc’h†‡
, Therese Mandel‡
, Didier Reinhardt‡§
, Cris Kuhlemeier‡
,
and Przemyslaw Prusinkiewicz*¶
*Department of Computer Science, University of Calgary, 2500 University Drive NW, Calgary, AB, Canada T2N 1N4; and ‡Institute of Plant Sciences,
University of Berne, CH-3013 Berne, Switzerland
Communicated by Enrico Sandro Coen, John Innes Centre, Norwich, United Kingdom, December 6, 2005 (received for review October 22, 2005)
A striking phenomenon unique to the kingdom of plants is the
regular arrangement of lateral organs around a central axis, known
as phyllotaxis. Recent molecular-genetic experiments indicate that
active transport of the plant hormone auxin is the key process
regulating phyllotaxis. A conceptual model based on these experiments,
introduced by Reinhardt et al. [Reinhardt, D., Pesce, E. R.,
Stieger, P., Mandel, T., Baltensperger, K., et al. (2003) Nature 426,
255–260], provides an intuitively plausible interpretation of the
data, but raises questions of whether the proposed mechanism is,
in fact, capable of producing the observed temporal and spatial
patterns, is robust, can start de novo, and can account for phyllotactic
transitions, such as the frequently observed transition from
decussate to spiral phyllotaxis. To answer these questions, we
created a computer simulation model based on data described
previously or in this paper and reasonable hypotheses. The model
reproduces, within the standard error, the divergence angles
measured in Arabidopsis seedlings and the effects of selected
experimental manipulations. It also reproduces distichous, decussate,
and tricussate patterns. The model thus offers a plausible
link between molecular mechanisms of morphogenesis and the
geometry of phyllotaxis.
active transport ͉ auxin ͉ PIN ͉ polarity ͉ computer simulation
Within the variety of phyllotactic patterns found in nature, the
most intriguing and, at the same time, the most prevalent is
the spiral phyllotactic pattern characterized by the arrangement of
organs into conspicuous spirals (parastichies), where the numbers
of parastichies are consecutive elements of the Fibonacci series.
This pattern is related to the divergence angle between organs
approximating the golden angle of 137.5°. In the entire world of
developmental biology, phyllotaxis is perhaps the most striking
example of a phenomenon that can only be described by using
quantitative notions of geometry.
The regularity and mathematical properties of spiral phyllotaxis
have attracted the attention of biologists and mathematicians since
the early 19th century. They proposed conceptual, mathematical,
and computational models, which elucidated the geometric properties
of spiral phyllotactic arrangements (1) and the emergence of
phyllotactic patterns during plant development. This latter category
of models was pioneered by Hofmeister (2) and Snow and Snow (3),
who hypothesized that the creation of new primordia is inhibited by
the proximity of older primordia. New primordia, therefore, can be
formed only at a certain minimal distance from the old ones. This
general hypothesis has subsequently been refined into a number of
computational models, postulating and exploring different types of
inhibitory mechanisms such as geometric spacing (4), physical
forces (5, 6), and chemical signals (7, 8).
In the absence of molecular data, the proposed mechanisms were
more or less abstract. Recent experiments, however, provided an
insight into the molecular processes involved in phyllotaxis, pointing
to the central role of active transport of the plant hormone auxin.
When shoot apices were cultivated in the presence of auxin
transport inhibitors, the induction of lateral organs was blocked,
and the apices grew vigorously as radially symmetric structures.
Application of the natural auxin indole-3-acetic acid (IAA) to such
pin-shaped meristems induced lateral primordia, with the size and
position depending on the concentration and the position of the
applied auxin (9). Furthermore, the cellular distribution and subcellular
localization of the auxin efflux facilitator PIN1 was consistent
with a role in organ positioning (10).
On the basis of these data, Reinhardt et al. (10) proposed a
conceptual model of phyllotaxis (ref. 10; Fig. 1). According to this
model, auxin is transported acropetally toward the meristem, where
it is redirected to the primordia, which function as sinks. As a result,
auxin is depleted from the surroundings of the primordia and
reaches the organogenetic peripheral zone only at a certain minimal
distance from the two youngest primordia (P1 and P2). Auxin
accumulates at this position, where it induces a new primordium
(incipient primordium I1) that, in the course of the plastochron,
grows out and becomes a sink itself. The phyllotactic pattern thus
results from the dynamics of interaction between existing and
incipient primordia in a growing apex, mediated by the actively
transported auxin.
The mechanism proposed by Reinhardt et al. (10) is plausible in
the sense that it is consistent with the available molecular data and
captures qualitatively the inhibitory effect of the existing primordia
on an incipient primordium. However, the question of whether it is
indeed capable of generating the highly constrained geometry of
spiral phyllotactic patterns was open. We answer this question by
constructing a simulation model. It is based on data concerning the
induction of primordia by high auxin concentrations and the polar
localization of the auxin transport facilitator PIN1 in the surface
layer of the apex (10). We also include previously undescribed
experimental data focused on the distribution of auxin in the
meristem and the incipient primordia. The ensuing model shows
that the molecular mechanisms identified by Reinhardt et al. (10)
and further developed in this paper can lead to the formation of the
phyllotactic patterns observed in nature.
Experimental Results
Previously published data in refs. 9–12 formed the initial experimental
basis for the model. To meet the needs of model construction,
we also acquired additional data, focused on the localization
of auxin within the meristem and the incipient primordia.
Data Set 1: Phyllotactic Patterning Occurs in the Outer Layer of the
Shoot Meristem (L)1. The PIN1 protein is located primarily, although
not exclusively, in the external L1 layer (figure 1 A and C in ref. 10).
This localization suggests that phyllotactic patterns may be formed
essentially on the surface of the shoot apical meristem. To examine
this hypothesis, we considered transgenic plants containing the
DR5::GFP reporter construct, which is thought to reflect endogenous
auxin concentrations (12, 13). The expression of DR5::GFP in
the meristem was confined to the L1 layer (Fig. 2A). The auxin may
Conflict of interest statement: No conflicts declared.
Freely available online through the PNAS open access option.
Abbreviations: IAA, indole-3-acetic acid; L, layer of the shoot meristem; NPA, N-1-naphthylphthalamic
acid.
†R.S.S. and S.G. contributed equally to this work.
§Present address: Department of Biology, 3 Rue Albert Gockel, CH-1700 Fribourg, Switzerland.
¶To whom correspondence should be addressed. E-mail: pwp@cpsc.ucalgary.ca.
© 2006 by The National Academy of Sciences of the USA
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be retained in the L1 by auxin transporters, most importantly
AUX1, which is strictly localized to the L1 (10, 14).
Control experiments showed that even after exogenous application
of the synthetic auxin sirtinol, thought to bypass auxin transport
(15), cells in the inner layers L2 and L3 did not stain well for GFP
(Fig. 2B). One possibility is that an active product of sirtinol, for
example 1-naphthoic acid (15), could be subject to active transport
impeding its spread to L2 and L3 in a manner similar to auxin.
Summarizing these results, we limited our simulations to the L1,
considering internal layers only at the sites of primordia (see Data
Set 5).
Data Set 2: Auxin Is Readily Available Throughout the Meristem.
Further questions concern the sites of auxin synthesis, inactivation,
and turnover. These processes are surprisingly complex, because
multiple pathways contribute to auxin homeostasis (16, 17). According
to our 2003 model (10), auxin is not produced in the
meristem but imported from basal tissues. If this hypothesis were
true, inhibition of auxin transport should lead to a depletion of
auxin in the meristem. To test this hypothesis, we inhibited auxin
transport by using either the pin1 mutant background or the
transport inhibitor N-1-naphthylphthalamic acid (NPA). In the pin1
mutant, the DR5::GFP expression was indeed extremely low (Fig.
2C), whereas after the application of NPA, the DR5::GFP expression
remained high. Nevertheless, after NPA application, the
phyllotactic pattern observed in wild-type meristems disappeared,
and a ring of DR5::GFP expression was observed instead (compare
Fig. 2 D and F with Fig. 6 L and B, which is published as supporting
information on the PNAS web site).
The pin1 mutant lacks lateral organs, presumptive sites of auxin
synthesis (10). In contrast to the short-term effect of NPA, the lack
of DR5::GFP expression in pin1 meristems is likely to reflect the
cumulative effect of compromised active auxin transport over a
prolonged time. Thus, at the time scale of minutes to hours, which
is relevant for phyllotactic patterning, auxin may be readily available
throughout the meristem, even if the transport mechanism is
impeded. This observation is consistent with the recent work on
Arabidopsis embryos, according to which experimental interference
with auxin synthesis and conjugation does not greatly influence
auxin distribution (18). In conclusion, we now assume nonlocalized
auxin synthesis and turnover throughout the surface of the peripheral
zone. PIN1 function is required to maintain the unequal
distribution of auxin, which is the essence of phyllotactic pattern
formation.
Data Set 3: PIN1 Expression is Up-Regulated by Auxin. It has been
shown that auxin regulates the expression of PIN proteins in the
root (19). This result suggests that auxin may regulate the expression
of PIN1 in the shoot meristem as well. Analysis of transversal
sections (Figs. 2 E and F and 6 A–E) shows that DR5::GFP
expression peaks in incipient primordia. After sirtinol treatment,
Fig. 1. Conceptual model of the regulation of phyllotaxis by polar auxin
fluxes in the shoot meristem. Adapted from ref. 10. (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.
Fig. 2. DR5ϻGFP expression in the shoot apex. (A) Longitudinal section of a
DR5::GFP-expressing wild-type inflorescence meristem. The arrowhead indicates
local overexpression of DR5::GFP consistent with an initial. (B) Longitudinal
section of a DR5::GFP-expressing wild-type inflorescence meristem
treated with 25 ␮M sirtinol during 48 h. (C, D, F, and G) Transverse confocal
pictures, taken with comparable settings. (C) DR5::GFP expression pattern in
a pin1-7 inflorescence meristem. (D) DR5::GFP expression pattern in a wildtype
inflorescence meristem treated with 20 ␮M NPA during 24 h. (E) Top view
of a wild-type inflorescence meristem, visualized with a binocular microscope.
(F) Transverse confocal picture showing the DR5::GFP expression pattern in the
same meristem as shown in the frame indicated in E. Spots of GFP signal are
observed in the peripheral zone of the meristem where no bulge is visible
(compare to E). (G) DR5::GFP expression pattern in a wild-type inflorescence
meristem treated with 25 ␮M sirtinol during 48 h. (H) Longitudinal view of the
two first, 3-day-old, leaf primordia of a DR5::GFP-expressing seedling. Note
the strong signals at the tips of the primordia (arrowheads), and DR5::GFP
expression in the future midveins (arrows), which connect with the stem
vasculature (asterisk). P indicates the DR5ϻGFP expression in the next primordium.
(Scale bars: 25 ␮m.)
1302 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0510457103 Smith et al.
however, all cells of the L1, including those in the central zone of
the meristem, express DR5::GFP at equal levels (Fig. 2G and 6 F–J).
These observations indicate that all cells in the L1 are auxinresponsive
and that within the L1, the DR5::GFP expression levels
are likely reflecting endogenous auxin concentrations. Because the
maxima of PIN1 expression are also located at the sites of incipient
primordia (10, 12), we assume that auxin positively regulates PIN1
expression in the meristem.
Data Set 4: PIN1 Is Polarized by Auxin. Given the role of PIN proteins
as facilitators of polar transport, their localization in the cell is of
primary importance. In the L1, PIN1 is polarized toward the
incipient primordia (figure 1C in ref. 10), in which auxin concentration
appears to reach a maximum. On this basis, we hypothesize
that PIN1 in the L1 is preferentially polarized toward the neighboring
cells with the highest auxin concentration. The form and
extent of this polarization have been obtained by searching the
parameter space of the model. Although this assumption has no
direct experimental foundation, the molecular mechanism is likely
to involve two elements: (i) auxin-modulated endocytosis of PIN1
(20), and (ii) a mechanism that informs cells about auxin concentration
in their neighbors, for example, by measuring extracellular
auxin or via a receptor-ligand system.
Data Set 5: Auxin Is Exported from Primordia to Subepidermal Layers.
Strong PIN1 expression in the primordia (10) coincides with the
strands of high expression of the DR5::GFP reporter construct,
which persist during organ outgrowth and become connected to the
central vasculature (Fig. 2H; refs. 21 and 22). This observation
suggests that primordia act as sites of auxin export from the L1 to
the central vasculature. The capacity of AUX1 to retain auxin in the
L1 may be exceeded at I1, where it ‘‘escapes’’ into internal layers and
induces organ identity and organ outgrowth (10, 14). In summary,
we assume that primordia can act as sinks of auxin in the L1 despite
the high concentration and, possibly, production of auxin in the
primordia. These functions can be combined at the same site
because of the polar character of auxin transport.
Simulation Model
Key Assumptions. We distilled the experimental results and hypotheses
underlying our model into the following assumptions.
(i) Phyllotactic pattern formation is a dynamic process, driven
by the growth of the shoot apical meristem.
(ii) The L1 functions as an isolated conduit for auxin transport
in the patterning of the shoot apical meristem (Data
Set 1).
(iii) Auxin is readily available in the L1 because of flow from
the lower parts of the shoot and͞or local production (Data
Set 2).
(iv) Auxin is redistributed in the L1 by a combination of
diffusion and active transport mediated by PIN1 molecules
(Data Set 2).
(v) PIN1 concentration in a cell is up-regulated by auxin
(Data Set 3).
(vi) PIN1 localization in the cells is determined by the concentration
of auxin in the neighboring cells (Data Set 4).
(vii) Only the peripheral zone, the band of cells below the apex,
is competent to induce lateral organs in response to auxin (9).
(viii) A new primordium emerges at the location of high concentration
of auxin (see Fig. 2 E and F and Data Set 3; refs. 10
and 12).
(ix) Incipient primordia acquire new developmental identity
(23), which may modify model parameters compared to the
intervening regions.
(x) Despite high auxin levels, primordia act as sinks for auxin in
the L1, as the auxin is pumped into the internal layers,
initiating vasculature (Data Set 5).
These assumptions form the foundation for three components of
the model: a geometric model of a growing meristem (i and ii)
described in Supporting Text and Fig. 7, which are published as
supporting information on the PNAS web site, a model of cell
polarization and auxin transport (iii–vi), and an integrated model of
phyllotaxis (vii–x).
Model of Cell Polarization and Auxin Transport. At the heart of the
model lies the interaction between four factors: the overall concentration
of PIN1 within the cell, allocation of PIN1 proteins to
individual cell membranes, transport of IAA (auxin) between a cell
and its neighbors, and changes in IAA concentration. These factors
form a feedback loop, in the sense that the concentration and
localization of PIN1 proteins in a cell depend on the concentration
of IAA in this cell and its neighbors, whereas the IAA concentrations
are changing as the result of fluxes dependent on the
concentration and localization of PIN1 proteins (Fig. 8, which is
published as supporting information on the PNAS web site). To test
whether these assumptions can lead to a plausible simulation
model, capable of recreating the observed phenomena at the
cellular level (dynamics of PIN1 polarization and auxin distribution)
and whole-meristem level (emergence of a phyllotactic pattern),
we quantified these assumptions by using a set of equations.
The number of PIN1 proteins in a cell i is assumed to change
according to the formula
d͓PIN͔i
dt
ϭ production Ϫ decay
ϭ
␳PIN0
ϩ ␳PIN͓IAA͔i
1 ϩ ␬PIN͓PIN͔i
Ϫ ␮PIN͓PIN͔i, [1]
where ␳PIN0
is the base production, ␳PIN is a coefficient capturing the
up-regulation of PIN1 production by auxin, ␬PIN controls saturation
of PIN1 production at high concentrations, and ␮PIN is the decay
constant.
PIN1 proteins are distributed between cell membranes according
to the formula
͓PIN͔i3j ϭ
͓PIN͔ili3jb͓IAA͔ j
͸j
li3jb͓IAA͔ j
, [2]
where [PIN]i3j is the number of PIN1 proteins located in the
membrane of cell i near the wall that separates it from cell j, and
li3j ϭ lj3i is the length of the wall that separates cells i and j. The
exponentiation base b Ͼ 1 controls the extent to which PIN1 protein
distribution is affected by the neighboring cells. In the absence of
experimental data, we tested a number of alternative formulas and
found that the postulated exponential dependence of the localization
of PIN1 on the concentration of IAA results in the most stable
phyllotactic patterns.
The effect of PIN1 proteins on the efflux of auxin from cell
i to a neighboring cell j is modeled by using the formula
active_transporti3j ϭ T͓PIN͔i3j
͓IAA͔i
2
1 ϩ ␬T͓IAA͔j
2, [3]
where T is a polar transport coefficient, and ␬T is a transport
saturation coefficient. For a given number of PIN1 molecules near
the wall separating cell i from cell j, the flux of auxin from i to j is
thus assumed to increase with the concentration of auxin in cell i
and saturate with an increasing concentration of auxin in cell j. The
quadratic dependence of the flux on these concentrations is not
based on experimental data. However, in the simulations, it was
found to provide better control over the spacing of peaks in the
resulting patterns than linear relations.
Smith et al. PNAS ͉ January 31, 2006 ͉ vol. 103 ͉ no. 5 ͉ 1303
DEVELOPMENTAL
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Finally, the changes in IAA concentration in a cell i are modeled
by the equation
d͓IAA͔i
dt
ϭ production Ϫ decay ϩ diffusion
ϩ ͸j
͑Ϫactive_transporti3j ϩ active_transportj3i͒,
[4]
where the first three terms are expressed as
production Ϫ decay ϩ diffusion
ϭ
␳IAA
1 ϩ ␬IAA͓IAA͔i
Ϫ ␮IAA͓IAA͔i Ϫ ͸j
Dj͓͑IAA͔i Ϫ ͓IAA͔j͒. [5]
In the above equation, the rate of auxin production is assumed
to decrease from the maximum value ␳IAA as the auxin concentration
increases, with ␬IAA controlling saturation. Auxin decay is
controlled by the decay constant ␮IAA. Diffusion takes place directly
between neighboring cells (intercellular space is ignored), with the
cell membranes representing the main obstacle to diffusion. Each
of the diffusion coefficients Dj is proportional to the membrane
length between a given cell and its jth neighboring cell.
The above model depends on active transport of a morphogenetic
substance and, thus, is fundamentally different from reactiondiffusion
models (8, 24). To test the pattern-formation capability of
this model, we applied it to a row of cells connected at the ends to
form a ring and to a two-dimensional cellular structure (Fig. 3). In
both cases, a stable pattern of approximately equally spaced peaks
of auxin concentration emerged from an initially homogeneous
distribution with small initial perturbations. The amplitude and
spacing between these peaks could be controlled by manipulating
model parameters (Fig. 3 D–F), with decreased transport leading to
fewer peaks and increased transport leading to more peaks. Thus,
the interaction between auxin concentration and transport provides
a patterning mechanism.
Model of Phyllotactic Patterning. We initially hypothesized that
phyllotaxis in Arabidopsis is determined directly by the transportbased
patterning mechanism, operating on the growing surface of
the apical meristem. In simulations, however, we were not able to
obtain sustained spiral phyllotactic patterns by using that mechanism
alone, although patterns of irregularly spaced primordia could
easily be generated. This observation was upheld by numerous
simulations, in which we used diverse parameter values and different
formulas for polarizing PIN1. We thus concluded that additional
factors play an important role in the generation of phyllotactic
patterns in Arabidopsis. To test this hypothesis, we extended
the transport-based patterning model with the following elements,
which are related to key assumptions vii–x.
(i) The surface of the apex is divided into the central zone,
peripheral zone, and proximal zone (located below the
peripheral zone). Auxin production depends on the zone in
which a cell is located. No auxin is produced in the central
zone. The coefficients controlling auxin production are
different in the peripheral and proximal zones, with ␳IAA(peripheral)
being greater than ␳IAA(proximal).
(ii) The peripheral zone is divided into primordia and intervening
regions (between the primordia). A primordium is
initiated when the IAA concentration in two adjacent cells
reaches a predefined threshold Th. The primordium center
is located at the midpoint of the centroids of these two cells.
Radius r increases with time (Fig. 7A) according to a
predefined function.
Primordia cells are differentiated from the intervening regions
of the shoot apical meristem, which has three consequences.
(i) The differentiated cells produce auxin at a relatively higher
rate
additional production ϭ
␳IAA͑primordium͒
1 ϩ ␬IAA͓IAA͔i
ͩ1 Ϫ
di
r
ͪ, [6]
where di is the distance of the centroid of a primordium cell
i from the primordium center.
(ii) Polarization of PIN1 proteins toward the primordium center
is increased, compared to the intervening region. This
increase is achieved by using an ‘‘auxin-concentration
equivalent,’’ [IAAЈ], instead of the current auxin concentration,
as the polarizing factor. The auxin-concentration
equivalent is defined as
͓IAAЈ͔i ϭ max͓ͭIAA͔i, ͓IAA͔maxͩ1 Ϫ
di
r
ͪͮ, [7]
where [IAA]max is the model maximum IAA concentration.
A possible molecular mechanism for the resulting bias of
PIN1 protein polarization toward the primordium center
might be a diffusing substance released by cells located at the
center. This substance would be prevented from spreading
outside the primordium by boundary cells. Another possibility
might be a mechanical induction of PIN1 polarization, caused
by stresses or strains in the primordium.
(iii) The concentration of auxin in a primordium is capped at a
maximum level [IAA]max, with the excess amount of auxin
assumed to flow into the inner tissues of the primordium
and initiate the vasculature.
Fig. 3. Pattern generation by the transport-based model. (A–D) Pattern
emergence in a sequence of 50 cells with wraparound boundary conditions
(the leftmost and the rightmost cell are considered neighbors). Taller bars
(brighter green) indicate higher IAA concentration. Simulation steps 30, 60,
70, and 80 are shown. A small amount of noise present in the initial distribution
is required to break symmetry. (E and F) Pattern dependence on model
parameters. Model parameters affect how many peaks a given number of cells
will create. Higher values of the transport coefficient result in more peaks. If
the transport coefficient is too low, no peaks will form at all. Transport
coefficient: A-D, 4.0; E, 3.0; F, 10.0. (G) Pattern formed in a simulated cellular
structure. PIN1 is depicted in red.
1304 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0510457103 Smith et al.
Simulation Results. We found that the postulated model of molecular
control mechanisms, operating on a growing shoot apical
meristem, can generate fundamental phyllotactic patterns: distichous,
decussate, tricussate, and spiral (Fig. 4 A–D; see also Movies
1–4, which are published as supporting information on the PNAS
web site). Parameter values used in the simulations are included in
Table 1, which is published as supporting information on the PNAS
web site. Sensitivity analysis indicates that the most critical component
of the model is the exponential form of Eq. 2 for PIN1
polarization. In the case of spiral phyllotaxis, the choice of the
exponentiation base b in Eq. 2 is also important (see sensitivity
analysis in Supporting Text).
In our simulations, changes to multiple parameters were needed
to obtain different phyllotaxis types. The simplest change, from
distichous to decussate, involved increasing IAA production, decreasing
the width of the peripheral zone, and increasing the size of
the central zone. The requirement for such complex changes may
explain the almost universal failure to generate mutants that change
one stable phyllotactic pattern into another (numerous unpublished
results; for discussion see ref. 25).
In dicotyledonous plants, such as Arabidopsis, the embryo produces
two cotyledons that arise near-simultaneously at angles close
to 180°. From this decussate starting situation, a spiral phyllotaxis
with divergence angles of Ϸ137.5° gradually emerges (26, 27). Our
model recreates this pattern de novo within the standard error (Fig.
5). The spiral phyllotactic pattern is perpetuated if the simulation
continues (Table 2, which is published as supporting information on
the PNAS web site).
Further support for the model is offered by recreating mutant
phenotypes and manipulated plants. Specifically, the pin1 phenotype
is characterized by the absence of primordia (28) and low
DR5::GFP expression (Fig. 2C). Removing production of PIN1 in
the model reproduces the inhibition of organ formation (Fig. 4E).
However, the concentration of auxin appears to be higher than in
the pin1 mutant and more similar to NPA-treated apices (Fig. 2 C
and D).
Local application of auxin in the peripheral zone of a pin1 mutant
produces an isolated primordium (figure 3A in ref. 10), whereas
local application of auxin at the tip of the apex results in the
formation of a ring-shaped primordium (figure 3C in ref. 10). These
experimental results are reproduced by the model (Fig. 4 F and G).
In contrast, our model does not capture the dependence of the
primordium size on the amount of auxin; furthermore, the simulated
ring consists of many separate primordia, as observed in the
pinoid mutant (figure 3F in ref. 10). We attribute these shortcomings
to the simplistic nature of the geometric algorithm defining
position and scope of primordia in our model, rather than failure
of the postulated molecular patterning mechanism.
Laser ablation of incipient primordium cells causes the emergence
of a displaced primordium in the vicinity of the ablated site
(figure 3 in ref. 29). This effect is reproduced in our model (Fig. 4
H and I).
Conclusions
We constructed a simulation model of shoot apical meristems that
reproduces the phyllotaxis of Arabidopsis vegetative shoots. Our
simulation model supports the basic tenet of the conceptual model
proposed by Reinhardt et al. (10), according to which phyllotaxis
results from the dynamics of interaction between existing and
incipient primordia in a growing apex, mediated by actively transFig.
4. Simulated shoot apical meristems. (A–D) The arrangement of primordia
into phyllotactic patterns: distichous (A), decussate (B), tricussate (C),
and spiral (D). (E–G) Simulated pin1 mutant (E), primordium formed in the pin1
mutant after localized application of auxin in the peripheral zone (F), and
primordium ring formed in the pin1 mutant after localized application of
auxin at the tip of the apex (G). Arrows indicate the site of auxin application.
(H and I) Simulated results of cell ablation: control apex (H) and the apex in
which five cells shown in black have been removed (I). Note the shift in the
position of the primordium initial (cells with high auxin concentration, shown
in bright green) near the ablation site.
Fig. 5. Comparison of the divergence angles: angles measured in Arabidopsis
with standard error bars (blue), and angles generated by the spiral phyllotaxis
model (red).
Smith et al. PNAS ͉ January 31, 2006 ͉ vol. 103 ͉ no. 5 ͉ 1305
DEVELOPMENTAL
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ported auxin. New primordia emerge in the areas of high auxin
concentration and maintain a distance from each other by depleting
auxin in their proximity.
An important aspect of our model is the transport-based patterning
mechanism, which involves a putative positive feedback
loop between the concentration of auxin, the localization of PIN1,
and the polar transport of auxin. We showed that this mechanism,
combined with diffusion, can ‘‘break symmetry’’ of an approximately
uniform auxin distribution and form a spatial pattern of
auxin concentrations that is stable over time (Fig. 3). The magnitude
and spacing of concentration maxima depend on model
parameters. This patterning mechanism is fundamentally different
from reaction-diffusion in that it postulates a controlled redistribution
of an existing morphogen (auxin) in space, whereas reactiondiffusion
postulates a controlled local production of two or more
morphogens. The transport-based patterning is also different from
canalization, which postulates a sustained polar transport of auxin
along the gradient of its concentration (14, 30–33). In contrast, in
our model, auxin is preferably pumped toward the neighboring cell
with the highest auxin concentration.
Experimental data show that PIN1 proteins in the L1 layer of a
wild-type Arabidopsis meristem are generally oriented toward the
primordium initial, and the position of primordia coincides with the
maxima of auxin concentration. The transport-based patterning
mechanism is consistent with these data but, according to our
simulations, does not generate phyllotactic patterns by itself. Therefore,
we postulate an additional factor: differentiation of cells within
primordia. The primordia cells are characterized by increased auxin
concentration, enhanced polarization of PIN1 proteins in the L1
toward primordium centers, and export of auxin from the L1
toward the internal layers. All of these characteristics have a strong
experimental basis (Fig. 2; ref. 10).
Our model suggests that phyllotaxis is not governed by a single
mechanism, but represents a combined effect of several factors.
This complexity may be needed in nature to generate phyllotactic
patterns in the presence of noise. The small number of cells around
the peripheral zone of the Arabidopsis apex limits the precision in
primordia placement to Ϸ15° (360° divided by Ϸ24 cells). A robust
mechanism is thus needed to initiate and maintain phyllotactic
patterns despite a relatively large departure in the placement of
individual primordia from their mathematically ideal positions. The
problem of generation of phyllotactic patterns in the context of an
irregular geometry was not considered in previous models, which
assumed continuous or uniformly discretized space.
Assumptions of our model include the localization of auxin
sources and the molecular mechanism of PIN1 protein localization
in a cell. According to the model of Reinhardt et al. (10), auxin is
produced outside the shoot apical meristem and is acropetally
transported into the meristem through the L1. We were not able to
recreate spiral phyllotactic patterns under these conditions and
assumed a uniform production throughout the L1 in the peripheral
zone instead, with an additional boost in the primordia. Also, our
model postulates localization of PIN1 toward the neighboring cells
with the highest auxin concentration but leaves open the question
of what molecular mechanism may produce this localization.
The answers to these questions may lead to the integration of the
model of phyllotaxis with a model of vasculature formation in
the leaf and stem. Although both processes are mediated by auxin,
the proposed mechanism of PIN1 polarization involved in phyllotaxis
is almost opposite to the canalization mechanism proposed for
veins (14, 30, 32, 33). It is thus interesting how these different
mechanisms may be reconciled in the growing plant.
Materials and Methods
All Arabidopsis thaliana lines were in the Columbia background.
The pin1 mutant allele used was pin1-7. DR5::GFP refers to the
DR5rev::GFP line described in ref. 12. Treatments with NPA and
sirtinol, visualization by confocal microscopy, and determination of
divergence angles are described in Supporting Text. Simulation
models were implemented by using the VV programming environment
(34). Further technical details are given in Supporting Text.
We thank Colin Smith (University of Calgary) for providing and
supporting the modeling software VV; Dr. Jirˇı´ Friml (Universita¨t Tu¨bingen,
Tu¨bingen, Germany) for providing anti-PIN1 antibodies and
stimulating discussions; Dr. Jim Hanan for critically reading the manuscript;
Drs. Jan Traas and Christophe Godin for making results available
before publication; Drs. Enrico Coen, Henrik Jo¨nsson, Eric Mjolsness,
and Elliot Meyerowitz for discussions; and Christopher Ball and Rebecca
Alder for expert care of the plants. This work was supported by the
Natural Sciences and Engineering Research Council of Canada
(NSERC) and Informatics Circle of Research Excellence postgraduate
scholarships (to R.S.S.), Swiss National Science Foundation Grant
3100A0-105807 and European Union Project no. 12878 “MechPlant” (to
C.K.), and Human Frontier Science Program Research Grant RGP0013͞
2001 and NSERC Discovery Grant RGP 130084 (to P.P.).
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1306 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0510457103 Smith et al.