In biological development, the generation of shape is preceded by the spatial localization of growth factors. Localization, and how it is maintained or changed during the process of growth, determines the shapes produced. Mathematical models have been developed to investigate the chemical, mechanical and transport properties involved in plant morphogenesis. These synthesize biochemical and biophysical data, revealing underlying principles, especially the importance of dynamics in generating form. Chemical kinetics has been used to understand the constraints on reaction and transport rates to produce localized concentration patterns. This approach is well developed for understanding de novo pattern formation, pattern spacing and transitions from one pattern to another. For plants, growth is continual, and a key use of the theory is in understanding the feedback between patterning and growth, especially for morphogenetic events which break symmetry, such as tip branching. Within the context of morphogenetic modelling in general, the present review gives a brief history of chemical patterning research and its particular application to shape generation in plant development.
How plants and other organisms with cell walls achieve their shapes is a very complex problem, which will ultimately require a synthesis across many disciplines, including cell biology, biochemistry, physical chemistry and biophysics. A great deal of work is going on in each of these areas; the present review presents some of the perspectives gained from physicochemical (specifically, chemical kinetic) studies of concentration patterning in plant development. Spatial localization of molecules is critical in development; whether of differentiation factors or growth factors, chemical patterning precedes anatomical change. A great number of such spatial patterns have been characterized in recent years, in terms of the molecules themselves and where they fit in reaction networks, and in terms of where and when the patterns form, especially for higher-plant model systems (e.g. [1,2]). However, less is known about how the molecules are localized, what types of reaction are required and what role transport plays. How does pattern formation operate within the continual growth that occurs in plants? In particular, when growth regulators are localized, growth and patterning are connected directly, a feedback cycle which manifests as shape change, or morphogenesis. There is enormous scope for characterizing the growth-patterning dynamics that generate plant architectures.
To study the dynamical aspects of morphogenesis, hypotheses must be formulated in terms of mathematical models (e.g. differential rate equations), and the implications of these hypotheses tested through computation (modelling), which are in turn tested against experimental measurements (frequently in quantitative parameters involving distance and time). A number of morphogenetic mechanisms are being tested in this way. Recent collaborations between physical scientists and biologists have produced notable breakthroughs regarding auxin transport in the apical meristem [3–5] (leading to new understandings of spiral phyllotaxis) and in the root [6,7]. There is also a long tradition investigating the mechanical aspects of plant morphogenesis (e.g. [8–11]), involving leading-edge physics to capture the unique mechanical properties of plant cell walls. The anisotropic and plastic expansion of many plant cells plays a determining role in plant shape . Much of this is related to the dynamic nature of the microtubule cytoskeleton and its relation to wall formation [13,14]; notable new work predicted and verified microtubule response to cell ablation in the apical meristem .
These various mechanisms underline the central importance of dynamics (properly regulated rates) and long-range communication (e.g. by hormone transport, mechanical transduction or diffusion) in morphogenesis. Further investigation of the specific mechanisms will give a deeper understanding of how each contributes to plant form. Long-running research in chemical kinetics has particular contributions in the area of symmetry-breaking, not only how pattern first emerges in undifferentiated tissues, but also how new patterns arise from old, such as dichotomous branches or whorled structures arising from domed tips. This work has established conditions under which reactions and diffusion can form stable spatial concentration patterns; application to plant morphogenesis requires coupling the RD (reaction–diffusion) dynamics to localized growth, providing insight into the relative rates between patterning and growth necessary to translate chemical symmetry-breaking into symmetry changes in plant shape.
The chemical dynamics of patterning
The theoretical study of spatial concentration patterns began with Turing , who described how two intermediates, or morphogens, in a chemical mechanism could interact to form stable concentration patterns. One of the morphogens has to be self-enhancing, and also enhance production of the second morphogen, which in turn inhibits the first. Such RD dynamics can be found in enzyme kinetics with activation and inhibition . In addition, the two morphogens need to diffuse, at different rates. Turing showed, by linear analysis of the morphogen rate equations, the conditions necessary for pattern formation, and how the wavelength (distance between pattern repeats) is proportional to a ratio of diffusivities over reaction rate constants. Prigogine and Lefever  and later workers developed particular (non-linear) chemical mechanisms which had Turing dynamics, and which started a great deal of theoretical work in RD patterning. The requirement of different morphogen diffusivities made realization of Turing patterns in the laboratory particularly difficult, but this was solved in 1990  by using a gel reactor with decreased mobility for one of the morphogens, permitting experimental tests of much of the earlier theoretical work.
In developmental biology, spatial gradients have long been postulated for cellular differentiation within tissues , and these came to be understood in terms of concentration gradients of signalling molecules [21,22]. Beginning with Gierer and Meinhardt , a number of groups have worked on understanding the establishment of these signalling gradients in terms of Turing dynamics. Early work (and much modern work on animal patterning) was on tissues of fixed size. For application to plant development, however, the theory needed to be extended to growing domains. This additional feedback requires consideration of how pattern is maintained in the presence of growth, or, if not maintained, how growth rates and patterning dynamics determine pattern selection over a developmental sequence; finally, for morphogen-catalysed growth, what constraints are necessary to transform chemical pattern into plant shape? Meinhardt  modelled spiral phyllotaxis with RD equations on a uniformly growing cylinder; and Harrison et al.  computed RD pattern selection on a uniformly growing domain in their initial paper on vegetative hair whorl formation in the marine alga Acetabularia. (This giant unicellular green alga, sketched in Figure 1, allowed for the study of complex morphogenesis in the absence of cell–cell signalling, tissue layering, etc.). In a series of experimental papers, Harrison's group discovered a number of points suggesting that RD dynamics controlled the spacing of the hairs: temperature shifts changed spacing as expected for an RD mechanism [25,26]; altered Ca2+ concentration and Ca2+ inhibitors changed spacing as expected for a membrane-bound Ca2+-activated Turing morphogen [27,28] (thermodynamic parameters for these processes were also determined); direct imaging indicated Ca2+ patterning in the cell membrane as expected from the RD model , and not in the cytoplasm  as suggested by other models . Recently, molecular biology techniques have been used to study Turing patterning more directly in plants: a large collaborative project  quantitatively matched a model of the GLABRA/TRIPTYCHON reaction network underlying leaf trichome patterning (which has RD dynamics) to experimental manipulations, especially overexpression. Theoretically, there has been increasing mathematical analysis of RD patterning in growing domains (e.g. [33–35]), and study of the effects of heterogeneous transport on RD patterning (e.g. ), with application to cell differentiation in fern gametophytes  [RD dynamics can stably maintain an apical cell type during tissue growth and changing numbers of intercellular channels (plasmodesmata)].
Growth control by chemical patterning
Modelling morphogenesis: coupling chemical dynamics to growth
Whereas the work described above established RD dynamics for particular aspects of plant development, it did not directly address how chemical patterning of growth generates shape. Plant shapes are generated by differential growth rates; for example, tips and leaf blades result when localized areas grow more rapidly than surrounding regions. Localized growth has been linked to localized patterns of gene products and plant growth regulators (e.g. [3–5]). In green algae, Ca2+-activated membrane proteins [29,38] are thought to aid localized docking of vesicles carrying new wall material ; vesicle dynamics are also critical in pollen tube formation . In higher plants, growth frequently depends on localized alteration of the complex wall structure, for example reducing cellulose or pectin cross-linking to increase wall extensibility. Expansins are a class of proteins which directly effect such wall relaxation [41,42]. The molecular systems involved in wall outgrowth are therefore likely to be diverse, but share fundamental dynamical features for how pattern is maintained in the face of growth, especially when that growth is catalysed by dynamically maintained pattern. RD modelling can help determine the relative rates involved in patterning and growth, especially what is needed in the patterning-growth feedback to demarcate the clear boundaries between slow- and fast-growing regions (solid lines in Figure 1) necessary for translating pattern into form.
RD patterning and growth were first coupled directly in a feedback loop by Harrison and Kolář . They computed RD pattern formation along a curve in 2D (two-dimensional) space. Growth was catalysed in proportion to one of the RD morphogens, resulting in tip growth and branching morphogenesis. A key feature of the model was that growth ceased as the surface aged, producing boundaries between active fast-growing regions and older slow- (or non-) growing regions. One of the initial applications of this model was to the morphogenesis of Micrasterias, a unicellular green alga which develops a flattened stellate shape through repeated dichotomous branching along a growing edge. The surface-aging mechanism produced obtuse-angled branches, however, and we determined that faster feedback to shut off patterning was necessary for the acute branches characteristic of Micrasterias . The new mechanism was able to generate the diverse shapes seen across the genus, with variable tip thickness, branching number and branching angle. The key feature for controlling these shapes was a threshold value for the growth catalyst, Xth, above which Turing patterning occurred, below which it did not, defining boundaries between fast and slow growth.
Many aspects of the flattened Micrasterias shape could be modelled in two-dimensions, but a number of questions required consideration of three spatial dimensions, and are also of broader interest in higher plant development.
(i) What maintains clefts between branched tips?
(ii) There is a greater diversity of potential patterns in 3D (three-dimensional) space; how are these selected? (e.g. a dichotomous branch in 2D space could, in 3D space, still be a branch; or it could be a section through a flattened tip; or a section through a whorl, such as a flower or the Acetabularia hairs.)
To address these questions, we developed a finite element model of a surface in 3D space, with RD pattern controlling outgrowth . The model successfully generated extended tip growth, a fundamental type of morphogenesis (Figure 2A), as well as breaking this symmetry to give dichotomous branching (Figure 2B). The first question could be answered simply by geometry: slow growth in the saddle orthogonal to fast growth in the tips maintained dichotomous branch morphology . Further work  was required for the other questions. The chemical spacing of RD patterns is central to answering the second question: the pattern that fits into a growing domain determines the shape generated. In three dimensions, fits of wave patterns to the geometry are more complicated than a simple wavelength between peaks; for the roughly hemispherical tips seen in much of development, patterns are given by surface spherical harmonic functions; for a particular growing domain size, increasingly complex patterns (and changing symmetries) are analogous to shortening wavelength, and are controlled by the same RD parameters. For example, in Figure 2, the increasing complexity from A to B to C was effected by decreasing morphogen diffusivity. For morphogenesis which depends on chemical patterning, increasingly complex shapes can be caused by decreased diffusivity or increased reaction rates; or, with growth-pattern feedback, to high relative growth rates and weak shut-off of patterning (e.g. lower Xth, in our model). For the last question, it was proposed  that a ‘morphogenetic template’ exists at the onset of Micrasterias development that determines the branching plane. Our modelling showed that this template is not likely to be geometric: for RD mechanisms, successive branches tend to be orthogonal to one another (Figure 3A), even on ellipsoidal initial shapes with up to 16:1 axial ratios. The template may more probably be chemical: a prepattern in a precursor to the growth catalyst (Figure 3B) forces successive branches to be in-plane (Figure 3C). Interestingly, branching plane is maintained better in the ‘wing’ lobes than the polar lobe (Figure 3D); this is also observed in Micrasterias cells. In the model, this stems from polar compared with wing differences in the prepattern (Figure 3B), which is a natural harmonic for hemispherical tips and may underlie the polar–wing differences in vivo.
Three-dimensional computations of morphogenesis catalysed by chemical patterns
Establishing a plane for successive dichotomous branches
Experimentally, the quantitative techniques used on Acetabularia are being extended to multicellular plants. Harrison and von Aderkas  showed that, in conifer embryogenesis, cotyledons (seed leaves) form with a constant spacing (cotyledon number, which is variable, depends on embryo diameter). Cotyledon patterns fit well to Bessel functions, indicating that they are determined by a wave pattern (potentially RD-generated) on a flattened disc. Future theoretical work will address the changes in patterning as the embryo changes from dome shaped to flattened tip, and work continues to experimentally perturb cotyledon spacing.
Work on quantitative models is increasing our understanding of the dynamics and long-range communication underlying plant morphogenesis. Further development and testing of these models will clarify the roles that different mechanisms play in particular cases of development, for example in the relative roles of diffusional compared with directed transport mechanisms. In addition to clarifying particular mechanisms, combined models will be needed to provide a more complete picture of many morphogenetic processes, for example combining chemical pattern formation with mechanically accurate surface deformation. The present review has focused on the application of RD patterning to plant growth and morphogenesis. RD theory is particularly well developed for understanding de novo pattern formation, the spacing between pattern elements and transitions between successive patterns. Mathematically formulated models can reveal underlying principles which are mechanism-independent. For example, although a particular RD mechanism was illustrated in the present paper, the limits on the patterning-growth feedback apply generally to any patterning mechanism. Models bridge from molecular-level data to larger-scale processes, not only revealing the pattern-forming properties inherent in particular aspects of plant biochemistry and biophysics, but also revealing mathematical commonalities which unify and systematize the dynamical underpinnings of morphogenesis.
Experimental Plant Biology: Why Not?!: 4th Conference of Polish Society of Experimental Plant Biology, an Independent Meeting held at Jagiellonian University, Krakow, Poland, 21–25 September 2009. Organized and Edited by Kazimierz Strzałka (Jagiellonian University, Krakow, Poland).
Thanks to T.C. Lacalli and J. Dumais for stimulating discussions and comments on the manuscript.
Work is funded by the Natural Sciences and Engineering Research Council of Canada, and the British Columbia Institute of Technology.
In memory of L.G. Harrison