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Expression of pattern in plants: combining molecular and calculus-based biophysical paradigms¹

Department of Biological Sciences, Stanford University, Stanford, California 94305

Received for publication December 6, 1998. Accepted for publication April 16, 1999.

	ABSTRACT

TOP ABSTRACT INTRODUCTION TOPOGRAPHY I: TWO SIGNAL-BASED... TOPOGRAPHY II: A RESPONDING... CELL PATTERNING: ALIGNMENT... DISCUSSION LITERATURE CITED

 
Pattern formation in plant meristems occurs across a broad scale. At the topographical level (large scale), tissue folding in the meristem is responsible for the initiation of new organs in specific phyllotactic patterns and also determines organ shape. At the cellular level (small scale), oriented cell division and microtubule-based cellulose reinforcement control cell pattern and growth direction. I argue here that structural specification at each scale is highly efficient if the pertinent gene activity is manifested in two complementary biophysical categories. At large scale, one category is the tendency of the formative tissue to fold with a certain spatial periodicity determined by its material properties (e.g., bending stiffness from cellulose content). This latent tendency is formalized in a differential equation for physical buckling. The second category at this scale comprises boundary conditions that specify how the latent tendency is manifested as topography: whether tissue humps occur as whorls or Fibonacci spirals. This versatile combinatorial format accounts for the relative stability of alternative organ patterning as well as alternative organ shaping (e.g., stamens vs. carpels). It also accounts for the structural shifts seen in normal development and after mutation or chemical/physical intervention. At small scale, the latent differential activity is the tendency for groups of dividing cells to co-align their cytoskeletons. The curvature of the surface opposes this tendency. The least curved part of a new primordium is its quasicylindrical midportion. There, by aligning microtubules and cellulose coherently around the organ, a new growth direction is set. Thus large-scale buckling produces curvature variation, which, in turn, affects the localization and orientation of the cytoskeleton. This scheme for the coherent production of diverse geometrical features, involving calculus at two structural levels, is supported by complex organogenetic responses to simple physical intervention. Also, many morphological alternatives, wild type vs. mutant, reflect single changes in parameters in this differential-integral format.

Key Words: biophysics; cell pattern; cellulose reinforcement; cytoskeleton; mechanical buckling; pattern formation; phyllotaxis

	INTRODUCTION

TOP ABSTRACT INTRODUCTION TOPOGRAPHY I: TWO SIGNAL-BASED... TOPOGRAPHY II: A RESPONDING... CELL PATTERNING: ALIGNMENT... DISCUSSION LITERATURE CITED

 
The geometrical features of plants have drawn attention since antiquity on the grounds of their aesthetics and symmetry. Most famous is the crossed spiral or "Fibonacci" pattern seen, for example, in the pine cone and sunflower head. Explaining these patterns, wrote Charles Darwin, could "drive the sanest man mad" (Francis Darwin, 1897 ). Interest in patterns at the cellular level developed in two ways: (a) it was found that organs, especially in ferns, could arise from prominent apical cells with stereotyped divisions that created the organ's histology in predictable fashion (Bierhorst, 1971 ); (b) it was noted that the growth direction of tissue cells was at right angles to the hoop-like reinforcement of their primary walls by cellulose. This alignment is apparently based on the similar, i.e., transverse, orientation of microtubules in the cytoskeleton (Gunning and Hardham, 1982 ; Hardham, 1982 ; Lloyd and Barlow, 1982 ; Giddings and Staehelin, 1991 ).

Pattern formation in plants clearly spans a wide range of scale. The question arises, how can one best connect this range of phenotypical activity to gene expression as currently understood? Any satisfying explanation must not only account for the details of the geometry and its overall coherence, but must also be compatible with the pertinent developmental biology of production. This article presents a useful causal frame of reference. The tactic is to reduce phenomena at both the genomic and phenotypic ends to the bare essentials and then seek a format to make plausible paradigms connecting them. While the information available is highly fragmented, it is distinctive enough to allow a first effort.

At the phenotypical end, an outstanding feature is the prominence of characters based on angle. Phyllotactic patterns are specified by a divergence angle, the one between organs of successive developmental cycles. The spiral patterns usually have an angle that approximates the Fibonacci angle of 137.51. . .° (see below). The divergence angle in whorled patterns is the rotation observed between successive whorls. It is 60° for the iris flower with its nested whorls of three components each.

Angle also enters into the specification of organ identity. When stamens form in snapdragon, they arise as humps in a vertically undulating annulus of formative tissue. When, in the snapdragon mutant deficiens, cup-like carpels are produced instead, the annulus of formative tissue undulates in the horizontal plane, thus anticipating the vessel-like nature of carpels. Here a 90° shift in the plane of undulation appears to be an early consequence of mutation.

Angles can be well defined at the cellular level. In the fern Onoclea, leaves may have a "two-sided" apical cell, shaped like an orange segment. It makes two ranks of progeny cells at 180°. The shoot of the same plant has a cell that is triangular in surface view. This provides three ranks at 120°. The fact that in many cases the angular correlation is relatively strict, and is inherited, invites explanation. In some other cases, such as stomate formation, the orientation of cell division appears important to later function (Sylvester, Smith, and Freeling, 1996 ). However, the cell segmentation pattern does not always correlate closely with later differentiation. For example, in the corn mutant tangled, relatively normally shaped leaves arise despite considerable variation in division plane alignment and cell shape (Smith, Hake, and Sylvester, 1996 ). The explanation may be that, despite weak local coherence of cell features, the overall alignment of the reinforcing cellulose across many cells directs growth adequately.

While an explanation for inherited angular features is readily found for structures that self-assemble at the macromolecular level, the connection in higher plants, especially over the size range from microtubule pattern to phyllotaxis, is obscure.

A second challenge at the phenotypical end is the mode of development of the structure of interest. There are several major issues, rarely appreciated. First, some patterns are produced de novo, that is, without any obvious corresponding geometrical antecedents. For example, in dicots, the "featureless" globular embryo normally produces two cotyledons as it becomes heart-shaped. Similarly, in certain flowers, a ring of five sepals arises simultaneously without any (as yet) detected prepattern (Hernández et al., 1991 ). A second issue is whether the structure is stably propagated, e.g., has a particular divergence angle that varies about some apparent set value. If so, self-correction must be accounted for. Finally, many structural activities occur in alternative forms that are interconvertible. For example, spiral and whorled phyllotactic patterns can each be propagated for many cycles in an apparent self-correcting fashion. Yet within the normal life cycle, or in response to mutation or chemical treatment, whorled apices frequently convert to spirals and vice versa (Meicenheimer, 1981 ; Mingo-Castel et al., 1984 ; Kwiatkowska, 1995 ). Because these diverse perturbations can yield the same response, they likely act at different levels in a common complex pathway.

In sum, for large-scale pattern, there is de novo formation followed by the nonintuitive combination of relative stability and interconvertibility of patterns. A great simplification arises when these three phenomena are interrelated formally. This has apparently been done. Following Turing (1952) , it is possible for a formative area to have an intrinsic wavelength, that is, a tendency to undulate (in terms of some chemical or physical variable) where the wavelength, but not the position, is set. For example, fingerprints have the same undulation wavelength, but the overall patterns made by these undulations are quite distinctive. In de novo formation this wavelength is selectively amplified, from noise, to occupy the available space. Because amplification can be an interactive optimization process (Harrison, 1993 ), the undulations round off to the nearest whole number (an obvious feature of plant organ pattern). The particular whole number will be a function of the size of the zone and/or the length of undulation being amplified. Once having arisen, this number becomes a boundary condition of the formative area. Thereafter, the pattern is propagated not only because it is an overt feature of the formative area but also because it has the periodicity that is most readily amplified. Different stable whole numbers would characterize different situations depending on boundary conditions. However, large changes could shift the number, despite its relative stability. For example, a size variation could convert a "three-leafed" to a "four-leafed" clover.

To relate such activity to gene expression, schemes are needed where the input comprises reasonable gene products and the output embodies both the geometrical specifics and the various biological modes of implementation. A scheme is developed here that greatly simplifies the structural parameters. Emphasis will be on the two-dimensional geometrical features. Plants are, very roughly, patterns on cylinders. Within this constraint, the phenomenology will be split. Organ pattern and organ shape are change in topography per se (large scale). The cellular level (small scale) deals with issues of alignment: cell files and cellulose reinforcement within the topography. It will be shown that processes at the topographical level appear to affect cytoskeletal activity at the cellular level. Activities at the two levels interact to produce coherence across scales.

To begin analysis, two additional simplifications are made. First is that molecular transductions, while intrinsically complex, can be portrayed as a horizontal Y-shaped symbol (Fig. 1A). The two arms represent inputs of ligand and receptor, of enzyme and substrate, etc. The output, at the end of the stem of the Y, is one of promotion or inhibition of an activity (synthesis, cell growth, assembly of the cytoskeleton, etc.). This symbol is a variation on the usual arrow () or (T) used in many molecular summary diagrams.

View larger version (31K): [in this window] [in a new window]   Fig. 1. Two types of transduction or conversion each with two inputs. (A) Molecular steps are simplified to connect two kinds of input with the net result of promotion or inhibition. (B) The input is reduced only to parameters of the differential equation and the boundary conditions. The output is the particular solution to the differential equation. This may take the form of a concentration map if the differential equation describes a chemical process or an elevation map if the differential equation describes the deformation of a solid. (C) The two types of transduction act in tandem. Complex circuitry, including pleiotropic and combinatorial effects, can govern single input terms. In the present view, this activity is a preamble to the pattern-forming process. Differential-integral transduction will be shown to have combinatorial and pleiotropic effects of its own

	TOPOGRAPHY I: TWO SIGNAL-BASED MODELS

TOP ABSTRACT INTRODUCTION TOPOGRAPHY I: TWO SIGNAL-BASED... TOPOGRAPHY II: A RESPONDING... CELL PATTERNING: ALIGNMENT... DISCUSSION LITERATURE CITED

 
Interacting multiple monotonic fields
This widely used model embodies the concept of positional information: local activity is prescribed by local conditions in a two-step process (Kalthoff, l996 ). The first step is regionalization, i.e., the provision of positional information. The diffusion of a signal from a source produces a concentration gradient, a field. Several such fields can interact in combinatorial fashion to subdivide the formative area into ever smaller districts. Progressive subdivision corresponds in many cases to hierarchies of epistatic relationships among mutants (Theißen and Saedler, 1995 ). The second step constitutes interpretation. Cells have an inherited repertory of responses, physical and chemical, which are evoked by the signals impinging on the cell. Gradients in signals give gradients in cell behavior. These, when summed, constitute tissue behavior. Hence, this format would produce cell pattern, organ shape, and organ pattern by scaling up the same scheme. Cytoskeletal alignment could follow iso-concentration lines, in principle.

Monotonic fields have been invoked for patterning in Drosophila where spatial specification in the embryo is sequential and expression patterns clearly anticipate structural periodicity. They have also been applied to bristle pattern formation (Wigglesworth, 1940 ), a process that is not as well-defined geometrically as plant organ pattern. For plants, this perspective has been the basis of many theories of phyllotaxis where the fields are of inhibition and come from recently made primordia (Mitchison, 1977 ; Veen and Lindenmayer, 1977 ; Young, 1978 ; Chapman and Perry, 1987 ). This format has been presented as a molecular and cellular basis of plant patterning (Meyerowitz, 1996 ).

Despite the wide use of this perspective, several aspects do not fit well with the geometrical phenomenology in plants. The following discrepancies justify the consideration of other types of mechanism. (1) This mechanism assumes sequential development, in contrast with the frequent simultaneous appearance of several new organs, e.g., in whorls of developing flowers. Unlike the situation in Drosophila, the synchronous physical formation of individual organs does not appear to follow extensive previous internal molecular specification within each. (2) In many of the schemes, particularly the epistatic hierarchies, there is no explicit provision for stability or for de novo pattern formation. (3) When stability is present in phyllotaxis models using monotonic fields, it is usually for the pair of Fibonacci numbers, not the divergence angle (that between successive organs). In nature, the stability is the other way around (Schwabe, 1984 ). These apparent discrepancies are eliminated when the causal paradigm is changed from one of many interacting monotonic fields to a single undulatory field. Further, the phenomenology of interconvertible related processes is remarkably directly accounted for.

Single undulatory fields
Activities within these fields are modeled by one or more differential equations. The input-output relationship, i.e., converting "before" to "after" for the whole area, is viewed as a single complex transduction (Green, 1996 ). This is a major feature and will be employed hereafter.

Differential-integral transductions
The basic developmental step is embodied in a diagram, Fig. 1B, which has the form of a curved horizontal Y. One arm is the input for the parameters of the differential equation. The other is for particulars of the boundary conditions. The two inputs meet at the junction where the specific integration takes place. The output, a pattern, is the stem of the Y. Note that the input parameters are derived from previous transductions at the molecular level. These input pathways are often complex and lead to combinatorial and pleiotropic effects. By preceding the patterning process itself, however, they are distinct from it (Fig. 1C). This format apparently meets the criterion that the input categories are biologically plausible and that the output produces spatially periodic pattern.

The undulatory field, already mentioned, has a latent intrinsic wavelength. Wavelength is implicit in one or more differential equations. The developmental transduction, which corresponds to the integration in the model, amplifies this wavelength preferentially. This process constitutes the theme. Because the undulatory pattern must conform to the boundary conditions, integration can both fix the location of the undulations and modify them somewhat. This provides variation. An example is shown in Fig. 2 illustrating two modes of whorl formation.

View larger version (38K): [in this window] [in a new window]   Fig. 2. The basic transduction for buckling: the conversion of one topography to another. For de novo formation this process transcends three levels of structure: tissue, organ, and organ system. For propagation, it simply reiterates a preestablished pattern. Genetic input takes two forms. (I) The input pertaining to the current properties of the system that apply to the differential equation. In this case the main feature is the intrinsic wavelength (the theme). (II) The boundary conditions give details of a particular solution. The resulting integration displays both the consistent general feature (undulation) and distinct variations of its compliance with the boundary conditions

View larger version (141K): [in this window] [in a new window]   Fig. 3. Physical induction of ectopic leaves on the residual meristem of a detached Graptopetalum leaf. (A) The residual meristem is the recessed area above the abscission scar (arrowheads)(photo by K. E. Brooks). In subsequent pictures part of the overarching cells have been removed to facilitate the observation of the meristem. (B–F) Sequential scanning electron micrographs of the same growing plantlet using the replica method. (B) A residual meristem constrained by a frame of glass chips. (C) A precocious and ectopic row of leaves, now a ridge, is forming (arrowheads). (D) The former ridge is now a row of partly fused ectopic leaves. Normal leaves have started to form below and at left. (E) Normal leaves, below, now dominate; older ectopic row is at top. Void areas are simply unfilled replica sites. (F) New normal leaves above, ectopic row below. (G) A typical control plantlet (photo by K. E. Brooks). The ectopic row would be horizontal, at center (arrowheads). The image also illustrates cell patterns, i.e., the origin of the histology of a new primordium. There has been a 90° shift in cell file alignment from the horizontal flank (foreground) to the erect new stem. No shift occurs at the midline of the new stem, at right. This originally disjointed activity results in a uniform file pattern around the new stem. See Fig. 8C.

The process is very well developed for animal coat patterns (Murray, 1981 ). It is the uncontested mechanism for sea shell patterning (Meinhardt, 1995 ). It has been applied extensively to unicellular plants (Harrison et al., 1981 ; Lacalli, 1981 ; Holloway and Harrison, 1999 ), as well as phyllotaxis (Berding, Harbich, and Haken, 1983 ; Bernasconi, 1994 ). It has received experimental support through temperature and calcium concentration effects (Harrison et al., 1981 ; Harrison and Hillier, 1985 ).

However, reaction-diffusion has several impractical features. The output is not structure, but rather a prepattern for it. Also, the model involves kinetic parameters of generally unknown compounds. Putative values are generally used. There is synthesis and decay, i.e., turnover of the pertinent molecules, the cost of which is hard to estimate. It is thus difficult to find an objective criterion, at least an energy-based one, for the pertinence in nature of one patterned solution relative to another. Many of the shortcomings are eliminated if one considers a responding-system model such as mechanical buckling.

	TOPOGRAPHY II: A RESPONDING-SYSTEM MODEL (MECHANICAL BUCKLING)

TOP ABSTRACT INTRODUCTION TOPOGRAPHY I: TWO SIGNAL-BASED... TOPOGRAPHY II: A RESPONDING... CELL PATTERNING: ALIGNMENT... DISCUSSION LITERATURE CITED

 
Theory
Pattern and shape formation are treated here as the generation of a specific undulating physical topography. The physical patterning system consists of two material layers, in broad contact. They correspond to the tunica and corpus in plants. While cells are present, and contribute to material properties, the key patterning properties of the system are not contingent on cell behavior. The calculations for the structure to be formed are in terms of continuum mechanics (Szilard, 1974 ; Green, Steele, and Rennich, 1998 ).

The major input to the equation consists of two material properties characteristic of solids. These are the flexural rigidity, applying to the tunica, and the elastic constant (for displacement normal to the plane) applying to the corpus. Both parameters are reasonable "downstream" developmental consequences of gene expression. Moreover, they impart the tunica with a tendency to undulate at a specific wavelength. In the simplest case, the intrinsic wavelength is proportional to the fourth root of the quotient of these two properties. Integration gives a three-dimensional configuration (a topography) in static elastic equilibrium. Continuous conversion from an elastic to an irreversible deformed state is assumed (there is "quasi-static" physical equilibrium).

The essence of the differential equation is that in-plane stress is balanced by two opposing tendencies: reluctance to bend and reluctance to displace in the out-of-plane direction. A given imposed stress leads to an equilibrium undulatory configuration. On-off control is through the presence/absence of stress, or through the system having a wavelength larger than the size of the system. Boundary conditions are complex. They specify two of the following variables: deflection, rotation, moment, or shear at each point on the boundary. In the present case, they provide great versatility for configuration change, even from an initially flat surface.

A great advantage of buckling is that it is a minimum energy phenomenon. That means that patterns will spontaneously tend to approximate the minimal energy pattern, a direct self-correction mechanism. By the same token, conversion to a different stable pattern can be the response to a strong perturbation. One way to determine the minimal energy pattern is with the critical stress. This is the level of stress where the structure no longer effectively resists deformation. For configurations close to the minimal energy pattern, the critical stress is small, while for configurations far from it the critical stress is high. The advantage of this approach is that the critical stress can be determined for arbitrary configurations. One can thus predict among a group of configurations which one is closest to the minimal energy pattern: it is the configuration with the lowest critical stress value.

Experimental evidence
Many observations suggest an organizing role for physics acting at the tissue level. Experimentally, the regional application of the protein expansin, considered to affect only material properties of the cell wall, can bring on de novo organogenesis and also influence phyllotaxis (Fleming et al., l997 ). In sunflowers, direct physical intervention, which mechanically constrained the formative area, changed tissue folding patterns in predicted ways. This resulted in changes in organ shape and identity (Hernández and Green, 1993 ). In a third case, shown here in preliminary fashion, mechanical constraint in the form of a glass frame induced an ectopic row of leaves in an expanding meristem (Fig. 3).

The most direct interpretation for the above is that tissue scale processes, apparently physical, can influence details of subsequent cell behavior and presumably gene expression. Tissue effects may be causes, as well as consequences, of cellular activity. This buckling mechanism retains the many virtues of an intrinsic wavelength such as seen with reaction-diffusion, but here the essential communication is achieved physically. Stress can be viewed as the "prepatterned signal"; it is, however, more directly coupled to the response than conventional mobile signals. In fact, the signal and the response may be regarded as essentially the same process.

Buckling and organ pattern
Patterns in plant meristems can be broadly classified as whorled or spiral (Fig. 4). A whorl consists of a whole number of organs, evenly spaced in a ring. A plant with a whorled pattern has several such rings, with the organs in adjacent rings being staggered or nested. The characteristic divergence angle, that between organs of adjacent whorls, is simply 360° divided by twice the number of organs per whorl (see Fig. 4A). This gives a simple fraction of a circle. Such angles are usually stable in development and in evolution. Dicotyledonous flowers generally have whorls of four or five organs (divergence angle of 45° or 36°); monocot flowers generally have threefold symmetry (divergence angle of 60°). However, the number of organs is subject to control by chemical treatment (Rasmussen, 1991 ) and genes such as PALLIDA, ABPHYL, CYCLOIDIA. The most common whorled vegetative pattern (decussate), seen in maples and mints, has two leaves per whorl. Many plants, especially grasses and other monocots, have organs placed in a zig-zag fashion, in a plane (distichous phyllotaxis). This is viewed here as a "whorl of one" with a divergence angle of 180°. Whorled patterns in general are radially symmetrical.

View larger version (48K): [in this window] [in a new window]   Fig. 4. The close packing of similar disks in polar coordinates. (A) and (B) show placement rules for the whorled and spiral patterns. (C) In an annulus of finite width the packing of nested whorls is possible only in an infinitesimal narrow ring (shaded region). The departure from the optimum is caused by disks toward the interior overlapping and those toward the periphery becoming separated. This would correspond in the case of buckling undulations to a stretching or a shortening of the wavelength. Such a gradient would increase the total energy of the system. For buckling in a hexagonal pattern (nested whorls), energy status is sensitive to annulus breadth. For the Fibonacci spiral pattern, the packing is regular but somewhat less close (or higher energy) than hexagonal. However, this packing arrangement is insensitive to annulus width. Hence narrow annuli will be of less energy (per undulation) if the undulations are in whorls; broad annuli will be of less energy if the undulations are in the spiral pattern. Thus change in a single parameter, annulus width, which is plausibly under biological control, could be the basis for the switch between whorl and spiral patterns

The production of whorled and spiral patterns requires de novo formation and propagation (Fig. 2). Two approaches were followed in trying to understand the possible role of mechanical buckling in these processes. First, the patterns were simulated in a static fashion. This approach was used to evaluate the energy of a wide range of patterns and to determine under which conditions a given pattern would predominate. The second approach was a quasi-static simulation of pattern (i.e., several small static steps in sequence as an approximation of a dynamic process). This was used to examine the stability of the patterns during propagation.

Static simulation of pattern
Mechanical buckling can explain de novo formation of whorls and the common transition between whorls and spirals. A laterally confined flat annulus with an intrinsic wavelength, when caused to expand in-plane will produce a whole number of undulations in a whorled pattern. The fit is to the nearest whole number. When the width of the annulus is one wavelength and the annulus is many wavelengths in circumference, the undulations are humps that are approximately circular in cross section (Green, Steele, and Rennich, l996 ). A similar annulus or disk provided with adequate boundary conditions can also produce a spiral pattern (Green, Steele, and Rennich, 1998 ). However, this is not de novo formation per se since the information for the spirals has to be included in the boundaries.

Transitions between whorls and spirals are common during the vegetative and generative phases of development. Sunflower and snapdragon are good examples of whorls converting to spirals. Spirals convert to whorls during individual flower formation in Echeveria and Silene. Conversion is controlled by known genes: LEAFY, FLORICAULA, MATURA, etc. Similar switchings are controlled by known hormones (Mingo-Castel et al., 1984 ; Marc and Hackett, 1991 ; Lyndon, 1994 ).

Meicenheimer (1998) has suggested that models of plant phyllotaxis should be able to account for these common transitions. This can be addressed by asking whether there is an energetic basis for the whorl-spiral duality. Because of the minimal energy nature of mechanical buckling, the problem is reduced to finding conditions under which whorled or spiral patterns are minimal energy.

For simplicity, the alternative patterns are portrayed as closely packed circles (Fig. 4C). The correspondence between phyllotactic patterns and those of close-packed circles has long been recognized (van Iterson, 1907 ; Ridley, 1982 ; Erickson, 1983 ). For the surface of a cylinder, the optimal close-packed pattern is the well-known hexagonal array (Tarnai and Gaspar, 1995 ). For a disk, patterns based on the so-called "cyclotron spiral" represent the optimal close-packed spiral arrangements (see Ridley, 1982 ). It is clear in Fig. 4 that, for an annulus, the packing of nested whorls is ideal only in an infinitesimal zone (shaded region in Fig. 4C). A broad annulus will be, on average, less regularly packed (hence of higher energy). The spiral pattern is locally less compact than optimally packed whorls, but it is less sensitive to annulus width. The issue is whether there is a width beyond which a spiral packing can be of lower energy than packing of nested whorls. This provides a potential switch for pattern.

To make quantitative energy comparisons of different undulatory patterns on an annulus, it was necessary to generate patterned topography artificially. The appropriate set of points was produced. For the whorled pattern, a fixed number of undulations were simply placed at different radial distance as in Fig. 4C. For the spiral pattern, the position (R) of each element was determined with the following algorithm in polar coordinates:

where is a fixed divergence angle (e.g., l37.5l...°) and h is the radial rise. At each point a radially symmetrical Gaussian function, maximum at the point, was generated. The curved surfaces were summed. The energy of the resulting quasi-sinusoidal topography was then evaluated.

For whorled patterns, the energy per undulation varied only slightly as a function of circumference. Thus annuli with an exact whole number of wavelengths around the circumference were secondary minima. Within these constraints, whorled patterns were minimum energy. As annulus width increased, however, energy values increased rapidly (Fig. 5B).

View larger version (20K): [in this window] [in a new window]   Fig. 5. Energy considerations for static patterns on an annulus. Energy is measured in terms of critical stress value. (A) Spirals: sensitivity to change in divergence angle used in construction of the pattern. Because the patterns used to determine the total energy contain a finite number of undulations, the minimum observed only approximates the Fibonacci angle of 137.51° (arrow). (B) Sensitivity of whorls and spirals to the width of the formative annulus. Note the threshold value for annulus width where the energetics dictate pattern transition. Graphs prepared by Dr. Arif Karabeyoglu (Department of Aeronautics and Astronautics, Stanford University)

Quasi-static simulation of pattern
Pattern propagation is generally characterized by a positioning rule that applies throughout propagation. It relates local activity for placing a new structure to features of the pre-existing pattern. For whorled patterns, the rule is that a new organ is placed interior to, and midway between, two organs of the previous generation (Fig. 4A). These organs are identical in size, age, and distance from the center of the array. For spiral patterns, a new organ is also placed relative to two older organs. They are, however, of different previous cycles. Accordingly, they differ in size and in distance from the center. The positioning rule does not use the bisection of an angle (0.5 : 0.5) as with whorls; it rather uses the Golden Section (0.6l8:0.382) (Fig. 4B). In both cases the rule is repeated to propagate the pattern.

A major feature of propagation, rarely modeled, is that it is a moving boundary phenomenon. The formative annulus has young undifferentiated tissue at the inner margin, maturing organs at the outer boundary. Organs are forever moving outward relative to the annulus. In the steady state, seen in vegetative growth, the patterning influence continuously acts inwardly to compensate the physical outward motion. The tissue moves through boundaries that are in constant relative position on the parent axis. This reconciles unlimited cycling of the pattern with constant dimensions of the annulus. In a flower, the formative region moves inward across stationary or slowly growing tissue. In this case, the apical dome region becomes fully occupied by organs. In both cases, however, there is relative motion between the boundaries and tissue. A shift between the two types of patterning influence (in place vs. progressive) often occurs at floral induction. The extent of propagation of organ types within flowers is governed by known genes: SUPERMAN, CL;t1AVATA, etc.

A given propagating pattern can persist through vast numbers of cycles, while varying about a consistent mean divergence angle (Fujita, 1939) . This implies strong stability, as does the experimental work of Snow and Snow (l935) showing that a surgically induced spiral development reverted to the original decussate pattern.

To simulate the propagation of whorled and spiral patterns a quasi-static approach was used. An annulus with a width of about one wavelength was given specific boundary conditions. The inner boundary had zero deflection and moment, and the outer one was "clamped" with a sinusoidal profile characteristic of nested whorls. The whole annulus then was forced to comply, via the buckling equations, to the imposed boundary profile. Propagation was implemented by having development occur in small steps. The inner boundary was moved inward by 10% of annulus width. In-plane stress was applied to amplify the topography. The outer 10% of the annulus was removed from the formative zone. (In an actual plant this topography remains present but no longer affects the patterning occurring in the formative region.) The boundaries were restored to the original dimensions. This cycle was repeated to show that a whorled pattern can be propagated indefinitely.

Moreover, when a departure from the typical pattern (e.g., a truncated undulation) was introduced, prominence of the error diminished. During propagation, interestingly, correction did not take the form of the error being eliminated. Rather, the error simply failed to be amplified by buckling and became relatively less detectable during its movement through the annulus (Fig. 6). Self-correction was thus present. The form of the undulations was not uniform. The solutions to the buckling equations approximate the natural wavelength in the radial direction, but in the circumferential direction wavelength tends to decrease with radius. Thus, within the annulus, the aspect ratio of the undulations is changing.

View larger version (29K): [in this window] [in a new window]   Fig. 6. Phyllotactic activity treated as cyclic and self-correcting buckling of topography on a formative annulus. Primordia are shown as humps (solid contour lines) alternating with depressions (dotted lines), which become creases. Propagation of whorls on an annulus with moving boundaries (propagation on a treadmill). This is the standard condition for vegetative growth but is rarely modeled. The whorled pattern is "checkerboard." There is continuous patterning influence coming inward, continuous displacement outward. During the sequence shown, the material of the annulus is displaced outward by more than the annulus width by three stages over time. (A) (1) Regular whorl of nested undulations. (2) At bottom, the pattern has been intentionally disturbed. One peripheral primordium has been removed (arrowhead). (3) The topography of the perturbation is shown by the difference in topography between (1) and (2). In (B) and (C), note that the "error," its difference from normal, diminishes as the "treadmill" operates. This appears to be the first explicit quantitative simulation, as topography, of self-correcting vegetative propagation. Simulation by Mr. Jonathan Fay and Dr. Arif Karabeyoglu (both at the Department of Aeronautics and Astronautics, Stanford University)

Relevance
The buckling simulations have provided a specific explanation for three important phyllotactic features. First, whorls can be produced de novo but not spirals. This could explain in part why de novo pattern formation in plant yields mostly whorls. Second, a condition (i.e., annulus width) favoring the transition between whorled and spiral patterns was found. Finally, conditions could also be selected such that whorled and spiral patterns were propagated indefinitely. However, self-correction was demonstrated only for whorled patterns at this stage. Attention now shifts to the cases where differences in shape between individual primordia are emphasized.

Buckling and organ shape
Initial organ shape correlates with organ identity. I will focus on stamens that become stalk-like and carpels that become cup-like (closed vessels, ultimately). In snapdragon, stamens originate in the third whorl as five vertical undulations in an annulus (Fig. 7A). In the deficiens mutation, a comparable annulus undulates horizontally, making a wavy ribbon with five cup-like undulations that become carpels (Fig. 7B). Here the alternatives in organ form appear to reduce to a 90° variation in the plane of undulation.

View larger version (121K): [in this window] [in a new window]   Fig. 7. Buckling and organ shape. (A) Wild-type stamens (st) in snapdragon arising in the third whorl. Creases traverse the formative zone, making a ring of humps. (B) The deficiens mutant of snapdragon where the first and second whorls are composed of sepals (s) and the third whorl is made of carpels (c). The formative region of the third whorl undulates in a plane as a ribbon, forming five cup-like primordia. (C, D) A preliminary continuum mechanical simulation. A ring is confined inside a fixed circular region. When heated or "forced to grow," the excess length can be manifested as an out-of-plane undulation of five wavelengths. This simulates the normal development of stamens (as in A). When the ring develops in-plane undulation, the similarity is with carpels, showing the start of their cup-like structure (as in B). The orientation of the undulations in the model depends on the ratio of elasticity of the interior cells (corpus) and flexural rigidity of the epithelium (tunica). A change by a factor of 2 to 4 in one buckling parameter suffices to shift the plane of undulation. This mimics the change in the plane of undulation brought about by the deficiens mutation. This appears to be the first pertinent simulation. It is example 8 in Fig. 9 . The derivation was done by Dr. Charles Steele (Department of Mechanical Engineering, Stanford University) from whom details can be obtained

An action of DEFICIENS, presumably affecting the plane of buckling, would be a very early aspect of expression of this gene, occurring before visible undulation. To act as structural "switches," homeotic genes may show unusual pleiotropy in that they affect early organ-specific biophysical phenomena as well as later biochemical ones. That these two processes are separable has been shown by striking floral reversion studies (Battey and Lyndon, 1990 ).

	CELL PATTERNING: ALIGNMENT FIELDS

TOP ABSTRACT INTRODUCTION TOPOGRAPHY I: TWO SIGNAL-BASED... TOPOGRAPHY II: A RESPONDING... CELL PATTERNING: ALIGNMENT... DISCUSSION LITERATURE CITED

 
The sections above, dealing with topography, envisioned two categories of input from the genome: one for a differential equation, the other for boundary conditions. I now apply this perspective to four small-scale structural features within a single undulation (primordium): organ axis, cell files, cellulose reinforcement, and microtubular cytoskeleton. These are normally coordinated in that files and growth direction run parallel to the organ axis and reinforcement is normal to it. This coherent association is not universal. Cell patterns can be random. Or, as in ferns with single apical cells, cell division patterns can be complex but predictable. Thus, for cell pattern and presumably also cytoskeleton there is an apparent hierarchy of alternatives: unorganized vs. organized, with the latter subdivided into unidirectional and complex patterns.

The aim of this section is to see whether these "variations on a histological theme" reflect an optimization scheme parallel to that suggested for the larger scale phenomena of pattern and organ shape. No quantitative calculus-based theories have apparently been developed for this level. The proposal here is that a qualitative cytoskeletal tendency can substitute. This tendency may manifest itself as diverse stable secondary optimal configurations under various situations of organ shape and cell geometry. This is proposed to account for numerous cellular configurations seen in meristems that otherwise appear arcane.

Theory: a general tendency
The alignment of three features, i.e., cell files, cell growth direction, and cellulose microfibrils, can be reduced to cytoskeletal behavior (Green and Selker, 1991 ). The alignment of microtubules in cuboidal cells leads to a "hoop" reinforcement by cellulose. In meristems a cell typically has one characteristic alignment, although individual cell faces may differ quantitatively. This is most evident for cellulose reinforcement in the surface plane, which can be seen with polarized light (Green and Lang, 1981 ; Jesuthasan and Green, 1989 ). The correlation between reinforcement direction and microtubule alignment within a cell is relatively well established (Lang, Eisinger, and Green, 1982 ; Selker and Green, 1984 ; Sakaguchi, Hogetsu, and Hara, 1988 ). Active meristematic cells appear to have a general "hoop" arrangement for microtubules, covering four of the six faces. More developed cells, particularly epidermal cells, may have different alignments on different faces. The focus here is on the coherent hoop arrangement.

Alignment is reversibly sensitive to antimicrotubular drugs (Wasteneys and Williamson, 1989 ). Primary wall extension is normal to the reinforcement. Assuming that the division plane is typically inserted at right angles to the cytoskeletal alignment and the growth direction, as it is in proliferative divisions, aligned cytoskeletons generate a cell file.

There are many reports of small-scale features spanning the walls of cell groups. The best known is for xylem bar thickenings in adjacent tracheids (Sinnott and Block, l945 ). Here the patterns clearly transcend cell boundaries. This cooperativity among cells is supported by observations that microtubules locally cluster on opposite sides of the new wall where these thickenings form (Hepler et al., 1993 ). Moreover, the parallel hoop-like microtubular arrays in adjacent cells in meristems commonly co-align (Sakaguchi, Hogetsu, and Hara, 1988 ).

If the intracellular microtubule alignment is combined with the intercellular tendency, a tissue-level tendency emerges: cuboidal cells tend to co-align their cytoskeletons in one direction. The production of parallel cell files and unidirectional growth follows. Such a coherent aggregate is called a reinforcement field (Green, 1994 ). If the co-alignment extends continuously around the organ axis, as a collar, the reinforcement situation is viewed as a primary available optimum. (The true optimum would be perfect parallel alignment on a flat plane.) In accord with the earlier paradigm, optimality is correlated with continuity and uniformity. Individual microtubules, and arrays of them, constantly "turn over." Quantification of microtubule dynamics in living plant cells using fluorescence redistribution after photobleaching (FRAP) suggests that they perhaps maintain transverse alignment by an active self-cinching scheme (Hush et al., 1994 ). Thus, the proposal is for a dynamic, metabolism-dependent, mechanism for organizing cytoskeletal pattern among groups of cells. The suggested paradigm is that this alignment tendency will be sensitive to curvature and, when optimized, can explain many observed patterns (and the transitions among them).

Application
Configurations observed on apical domes
Because the apical dome usually has a significant curvature, a complete parallel reinforcement field is rarely seen. In some cases the cell and reinforcement pattern show no overall regularity. Clear examples are the surface of the apical dome of flowering Anagallis and Vinca (Hernández et al., 1991 ; Williams, 1991 ) and the central region of the sunflower capitulum. Here, presumably, the co-alignment tendency is not effective.

On the other hand, many meristems have a distinct cellular organization. There are two major types of organized cell patterns: those with a prominent noncuboidal cell (common in ferns) and those with many cuboidal cells (common in higher plants). In the fern Onoclea, the leaf apex has a single apical cell that is two-sided in surface view (Bierhorst, 1971 ). It is shaped like an orange segment and is not cuboidal. There is no directional reinforcement seen in the surface wall of the cell, but its quadrilateral derivatives have strong cellulose alignment (Lintilhac and Green, 1976 ). The tissue-level reinforcement pattern is suboptimal for two reasons: (a) there is the presence of the nondirectionally reinforced apical cell and (b) the aligned regions are disjointed. The two reinforcement fields, produced sequentially from alternate sides of the apical cell, abut each other at 90° (Fig. 8A, 1). Thus the reinforcement pattern below the apical cell is a mosaic of two fields. The line of juxtaposition shows sharp angular discontinuity of cellulose alignment, striking in polarized light (Fig. 8A, 2). In the shoot apex of the same plant, the apical cell is three-sided. There are three reinforcement fields (Fig. 8A, 3). It is perhaps noteworthy that noncuboidal cells often divide for several cycles. Repeated division is known for some three-sided cells that produce groups of cells that make stomata (Sylvester, Smith, and Freeling, 1996 ).

View larger version (53K): [in this window] [in a new window]   Fig. 8. Observations suggesting a tessellation-optimization paradigm for expression of cell and reinforcement pattern. (A) A system with a prominent nondirectionally reinforced apical cell. Onoclea. (1) The segmentation pattern produces a left-right alternation of curved cuboidal cells, each directionally reinforced (thin lines). The directions oppose at 90° in a zig-zag line at center. (2) The apex in polarized light, compensator present (from Lintilhac and Green, l976 ). The brightness/darkness shows 90° confrontation of alignments. (3) Corresponding alignments in the three-sided apical cell of the shoot. Note approximation of overall hoop reinforcement of the organ tip. (B) Top view of higher plant apical domes. Fine lines show cellulose alignment as seen in polarized light. Cells are not shown, but cell axes are normal to cellulose alignment (1–4). Four types of cycling reinforcement fields (after Green, 1986 ). Cyclic transitions during the plastochron are shown by double arrows. Each mosaic has one or more central corridor bands, flanked by lentoid regions. Near, and sometimes also at, the center of the dome, there is sharp alignment contrast between corridors and the regions flanking them. This contrast is not present at the periphery. There the alignments become blended, particularly at the base of the curved dome, which is more cylindrical. (C) New organ formation from an aligned field. Here a 90° reorientation occurs at the periphery of the dome to generate three fields from one. The new alignments provide four directions of reinforcement for the initial, approximate, hoop reinforcement of the new organ (see box). The unused remnants of a single original field are clearly seen in Fig. 3G . The process can produce new leaves or shoots. (D) New organ formation from a random field. Here, during division, quadrilateral cells progressively optimize co-alignment in the midregion of the new hump where the shape best approximates a cylinder. This shape both localizes and sets the new reinforcement direction. This produces a primary optimal alignment field. This relationship couples the processes at the small scale (curvature-sensitive tesselation by reinforced cuboidal cells) with buckling, the larger scale processes, which produces curvature. This provides geometrical/angular coherence across scales.

In plants with alternating whorls of three leaves, the corresponding three-fold symmetrical arrangement is seen (Green, 1986 ). In plants with spiral phyllotaxis, e.g., Ribes, an equivalent asymmetrical arrangement is observed. Thus, well-defined mosaics of surface reinforcement fields are found widely among ferns and higher plants. In all cases, they comprise discontinuous approximations of the optimal available configuration: a single transversely oriented field encircling the axis. The axis is indeed encircled, but in partly disjointed fashion. There is a systematic character to the transitions between these various configurations.

Transitions between configurations
The simplest is seen on apical domes. As noted, in higher plants the disjointed mosaic of fields near the tip of the dome intergrades into a single optimal field at the base, or periphery, of the dome. However, in Onoclea signs of discontinuity of alignment persist (Fig. 8A). Perhaps cell-to-cell communication, promoting co-alignment, is slow across lines where the cells are of different age.

In the decussate plant Vinca, the alignment in the corridor region of the dome is transverse to the long axis of the corridor. Since the long axis of the ellipse changes by 90° during each plastochron, the alignment of the cells at the tip of the dome also changes. The new central alignment after the change is parallel to that at the base of the new-forming leaves, thus bridging the two former flanking alignments. The alternating orientation here is statistically canceling. As with the single apical cell of Onoclea, there is no net alignment. There is a third way of having indeterminacy at the very pole. In tricussate apices with three "corridors," a 90° alternation cannot be compatible with the three corridors. An unorganized pattern is seen where such corridors converge (Green, 1986 ; Fig. 8B).

A different type of transition occurs when a part of a single reinforcement field develops into a new axis (Fig. 8C). In vegetative phyllotaxis, this occurs at the periphery of reorganizing apical domes where they initiate primordia. As studied in Graptopetalum with polarized light for cellulose, and with electron microscopy for microtubules, the alignment revision involves two steps (Hardham, Lang, and Green, 1980 ; Green and Lang, 1981 ; Selker and Green, 1984 ). In association with bulging, there is a 90° discontinuity, a sharp shift in alignment, in a central band-like region. The cell shifts have been individually described in Vinca (Jesuthasan and Green, 1989 ) and Graptopetalum (Green, 1984 ). Such shifted regions make a new "corridor" in a mosaic because there is now a central stripe of one alignment, with two lateral regions of opposing alignment. Note that, as with ferns, this three-field arrangement is a rough approximation of a large-scale "hoop" reinforcement pattern (Fig. 8C). It is approximate because the center has alignment and because the reinforcement lines are rectangular, rather than circular. As on the established dome, these discontinuities smooth out as the tissue develops (moves relatively downward onto a more cylindrical region). A record of this geometrical revision, involving a 90° shift, is clearly seen in an image of a new stem in Graptopetalum where remnants of the original single field are prominent (Fig. 3G).

This sequence has been established in great detail for leaf and stem formation in Graptopetalum (Tiwari and Green, 1991 ). It appears to occur first for the leaf. Cell organization soon becomes random at the leaf tip. In this system leaf initiation can occur in isolation (Green and Brooks, 1978 ). In such cases the forming leaf appears to "induce" the surrounding cells to form a miniature stem under the single leaf (Green, 1984 ). It thus appears that in Graptopetalum there are four stages to stem production: ridge formation, reinforcement shifts (to generate hoop reinforcement), leaf formation, and stem formation. The last appears to be induced by the aggregate of leaves formed previously.

Using polarized light, a remarkably similar sequence was seen for leaf initiation in the distantly related aquatic plant Hippuris, formerly Anacharis (Green, l986 ). Both Graptopetalum and Hippuris produce a three-part orientation mosaic despite the fact that the geometry of the formative regions is different. In Graptopetalum the leaves form de novo atop an elongate ridge. In Hippuris they form in sequence on the flanks of a nearly cylindrical apex.

Observing cell division and growth orientation, a 90° shift in cells in a band-like region was also seen during leaf formation on explanted axillary stem tissue of watercress, Nasturtium officinale (Selker and Lyndon, 1996 ). The cell pattern was the same as for leaf formation in Graptopetalum and Hippuris, forming a three-part orientation mosaic.

The basis for the 90° shift in cellulose orientation, presumably the critical initiating discontinuity, is not known. Such shifts correlate with the long axis of newly formed daughter cells and with the orientation of the recent cross wall between the cells (Green, 1984 ). For organogenesis, the key shift may be brought on by a cytoskeletal response to directional stretch on a curved surface, a feature of buckling.

The final case of field initiation is the simple one where a part of a large unorganized field develops into a single field (Fig. 8D). This is observed histologically, for example, on the capitulum of sunflower when new organs form or on the floral primordium of Anagallis when sepals arise (Hernández et al., 1991 ). New longitudinal files simply appear midway up the primordium as it bulges out. They are not present beforehand.

A qualitative optimization model based on cytoskeletal behavior
The diverse configurations above are proposed to be alignment states, which are either at the primary available optimum (a single collar-like field) or are at secondary optima (disjointed approximations). The states are interconvertible. A qualitative algorithm based on the maximally coherent tessellation, or uniform tiling, of a surface with quadrilateral units appears capable of reproducing much of the phenomenology. The model predicts cell pattern on the basis of the inferred relative stability of cytoskeletal alignments of adjacent cells. It is a preliminary attempt at explaining cell pattern and reinforcement, based on limited data. The key observations and key assumptions are summarized below.

a) Meristematic cuboidal cells have hoop-like cytoskeletal arrays. The alignment is often particularly prominent on the external periclinal face. Here cellulose birefringence is strong and the physical forces for elongation are concentrated.

b) Noncuboidal cells have no prominent internal alignment. Such cells can divide and often do so in regular segmentation patterns. Their quadrilateral derivatives do have alignment.

c) Microtubular arrays in adjacent cuboidal cells tend to co-align. Mutual adjustments, toward improving co-alignment, occur at cell division.

d) A hierarchy of stability among the possible cell to cell interfaces is postulated.

 1) Parallel transverse alignment is most stable.

 2) Perpendicular alignment is second.

 3) Aligned to random is third.

 4) Random to random is fourth.

e) Among aggregates of dividing cells, the maximum stability possible is realized (by trial and error through the flexibility at cell division).

f) Maximum stability is a function of curvature. The available optimum configuration is co-alignment on a cylindrical surface where there is curvature only in one direction. The other configurations are secondary optima largely because uniform tessellation by rectangles is not possible on a dome surface, especially near the pole. Thus in a three-dimensional sinusoidal undulation, co-alignment is maximal in the midregion and precluded at the pole. This couples buckling phenomena at the topographical level to phenomena at the cellular level.

Two assumptions are needed to explain further coordination during organogenesis in vegetative and floral apices (Green, 1994 ):

g) Cells at the organ pole become nonmeristematic and produce a downwardly mobile activator of co-alignment. This can explain why, in vegetative systems, leaf tips are inactive morphogenetically, while the proximal part of the leaf and the apical dome tend to show strong co-alignment.

h) In flowering, co-alignment appears to be missing on smooth large domes. Perhaps some key factor is not produced (i.e., bracts may not make it). Curved distal regions of new primordia, with factor, would account for the new co-alignment arising below the tip.

Although the list of assumptions is long, they are plausible. The proposed hierarchy simply reflects the relative frequency of interfaces seen in the various situations. So the paradigm has passed only the test of relative efficiency. Note that there are two important "uphill" or nonspontaneous transitions: (a) formation of single fields from an unorganized array and (b) the formation of a three-part mosaic from a single field. Both appear associated with a local buckling process. The other transitions are spontaneous or "downhill" in that overall co-alignment is simply improved by reduction of curvature. Together, cycles of transitions can occur. Note that the anisotropy of reinforcement can influence the buckling process (influencing initial shape) so interaction between levels, in both directions, is plausible. On this important point Williamson (1991) has presented a critique, and a model, for the interplay between wall stretching and microtubule behavior.

These rules offer a framework to attempt to explain the histological sequences following physical perturbation as mentioned above: organ induction by expansin, by large-scale constraint on sunflower, and by local mechanical constraint on Graptopetalum which produced an ectopic row of histologically normal leaves (Fig. 3). This cytoskeletal scheme, combined with the buckling activity, has the potential to reduce a very wide range of phenomena to only two issues for a plane of cells: (a) what causes it to fold in a particular way and (b) how does that folding interact with cytoskeletal phenomena? These both respond to previous folding and influence future folding, completing a stable set of two-way biophysical interactions.

Some previous theory on this subject has concerned encoding the sequence of divisions (Korn, 1993 ). Other theory accounts for the striking orthogonal histological patterns in roots in terms of tensors (Hejnowicz and Karczewski, 1993 ). The latter work shares with this treatment the emphasis on orthogonality and the fundamental feature that observed patterns reflect alternative self-stable activities, governed by boundary conditions. The present model adds the variable of reinforcement direction by cellulose, the involvement of the cytoskeleton, as well as an application to the broader array of surface cell patterns seen in shoots and flowers.

	DISCUSSION

TOP ABSTRACT INTRODUCTION TOPOGRAPHY I: TWO SIGNAL-BASED... TOPOGRAPHY II: A RESPONDING... CELL PATTERNING: ALIGNMENT... DISCUSSION LITERATURE CITED

 
An apparent salient feature of the geometry of plants is the propagation of, and the transition between, a limited number of distinctive patterns. These can be regulated by the genome in a two-category format of expression, as broadly interpreted here. One category takes the form of parameters for an activity expressed as a differential equation. That is, conventional products of gene activity such as cellulose synthesis and microtubule polymerization have their influence on geometry only after being converted to biophysical parameters (e.g., bending stiffness) or dynamic activity (microtubule alignment). This can be paraphrased as a latent tendency. For organ patterning and shape, a tissue tends to buckle at a standard intrinsic wavelength. For alignment features within a primordium, the microtubular cytoskeletons, considered to be like hoops around a barrel, tend to co-align laterally to give the key tendency.

The second category is the geometry of the system as specified by boundary conditions. For example, annulus width is critical for organ pattern and curvature influences the "annealing" of microtubules. The schemes at the topographical and cellular levels are physically coupled to provide coherence of geometry from cytoskeletal features to organ pattern. It goes without saying that the genome also provides other essentials such as turgor pressure, synthesis for growth, etc.

The potential broad applicability of this perspective for organ pattern is given by examples in Fig. 9. It is a speculative interpretation of well-known structural change phenomena in terms of the buckling paradigm. It presents, at the top, an array of structural pairs that are either alternative manifestations of a developmental process (mutant vs. wild type) or unusual one-time conversions of structure. For each case, a connection is made to an apparent pertinent genetic change, or experimental treatment, regarded as the potential cause for the particular transition. These are shown at the bottom. In between, in boxes, lie the two categories of causal input (Fig. 1B) of the current paradigm in expanded form. A plausible localized site(s) of action, within the paradigm, for each switch is shown by an asterisk.

View larger version (45K): [in this window] [in a new window]   Fig. 9. Speculative application of the paradigm to diverse structural alternatives, many of which are covered in the text. The structural alternatives may be differences between wild-type and mutant or unusual one-way transitions. The switches may be mutation, chemical, or physical treatment. Ten structural alternatives (above) are connected to ten apparent controlling "switches" (below) via a differential-integral transduction (boxes in between). Each switch is connected to the structural alternatives by a line. On each line, the component of the differential equation, or of the boundary conditions, whose variation correlates with change in structure is shown by an asterisk. Note that combinatorial and pleiotropic effects exist with this type of transduction. Most of the examples are well known. This includes, as number 1, potato chip formation, which by buckling provides two wavelengths of topography, de novo. Numbers 2 and 3 are common mutants. Others are covered in the text. These include varying numbers per whorl (number 4) and the spiral-whorl transition (number 5). Number 6 reflects the work of Green and co-workers (Green, Steele, and Rennich, l996 ). In example number 7, snapdragon, the fifth stamen is often vestigial. From images of the topography (supplied by Dr. Hans Sommer, Max Planck Institute, Köln) it appears that adjacent whorls locally superimpose. Because they are out of phase there, destructive interference would be expected in the region of overlap, perhaps leading to reduced development. Number 8 is the organ shape alternative (carpels/stamens). Of special interest is the pleiotropic effect of floricaula (number 9) in the inflorescence of snapdragon where the higher number of spirals is dropped and humps convert to ridges. This transition appeared on the cover of The Plant Cell in December 1995. In continuum mechanics this pair of effects can reflect the wavelength of the patterning influence being too short for the wavelength of the formative annulus. The examples in number 10, chemical or physical induction of organ and organ pattern, are covered in the text. It is seen that many "switches" can be connected to a wide range of structural alternatives by single changes in intermediate components of a differential-integral relation. This widespread covariation is the main justification for presentation of the paradigm

Another special feature, involving differential equations, is that the constant features of the equation best re-encompass the basis of the mechanism. Because of the mathematical complexity, the unknown equation is hard to deduce from the usual tactic of correlating variation in input with variations in output (Green, 1996 ). That is, in studying diverse responses to perturbation, e.g., mutation, it is the part of the system that does not change in concert with variation of response that is revealing (and elusive). Thus, for spiral vs. whorled phyllotaxis, the buckling equations are the same. It is how the same system responds differently to different boundary conditions that is the apparent key to understanding.

On evolution
The perspective here is that natural selection acts on the alternative stable developmental activities as seen in the wild type. These can be interchanged by major perturbations either during evolution or development. When nondiscrete alternatives appear (e.g., disorganized or hybrid organs), the components for stability, such as those proposed here, are missing. Selection thus appears to act on a set of related functional alternatives which can be readily interconverted in the short run (development) or the long run (evolution). These features are readily encompassed in the differential-integral format.

With regard to D'Arcy Thompson's comment on pattern number seen in nature, these few regular patterns are few, presumably because they are stable (optimized) structures, capable of cycling. This is likely to be a small set. For Darwin, the observed patterns appear nonintuitive or maddening because their common basis is seen only with the proper mathematical paradigm.

On the organism–cell distinction
Extensive discussion has centered on the issue of the prevalence of cellular vs. organismal factors in higher plant development (Kaplan and Hagemann, 1991 ; Cooke and Lu, 1992 ). The present paradigm puts emphasis on certain continuum mechanical phenomena, which, while at the tissue level, do not require cells per se. These phenomena generate topography and thereby connect tissue to organ shape and organ arrangement. On the other hand, the production of organ-level hoop-reinforcement for the growth direction appears to be fully dependent on the cytoskeletal behavior of individual cells. The arcane geometry of the Onoclea leaf apex (Fig. 8) appears to reflect a striking compromise between the "need" for hoop reinforcement at the organ level and the restriction that it be generated on a cell-by-cell basis. The two contrasting perspectives become complementary.

On combinatorial and pleiotropic responses
It is not widely recognized that the phenomena of combinatorial and pleiotropic interactions exist not just at the molecular level, but also within the differential-integral format in the solid state. A quantitative change in a single boundary parameter for buckling can lead to two concurrent changes in manifestation. For example, in aberrant spiral phyllotaxis (floricaula) (Carpenter et al., 1995 ; Fig. 9) there are two consequences. One set of spirals and the hump-like character of primordia are lost together. This is pleiotropy. On the other hand, to obtain fused primordia by buckling, a change in both the width of the formative area and its boundary conditions (hinged vs. clamped) can be necessary (Green, Steele, and Rennich, 1996 ). This is combinatorial action. It is clear that similar phenomenology can have a macroscopic as well as a molecular basis.

There is no doubt that the genome specifies an astronomical number of events that occur predictably in both time and space. In morphogenesis, the final manifestation is geometrical and physical: a shape, a pattern, an alignment. Most current suggestions for the specification emphasize the role of myriad interactions that are (a) at the molecular level and (b) are reducible to issues of promotion and inhibition. This is often thought to scale up, with many interactions, to account for details at higher levels of organization. This article suggests that much of such activity is a prelude to other transductions that can be understood only by involving the supermolecular level of organization explicitly. The tactic here has been to start with phenomena at the epithelial level. One can then scale up to patterning and organ form and down to cell pattern and reinforcement direction. Because the transductions that bring on change take the form of differential-to-integral, the pertinent gene products need to be converted to biophysical parameters. The boundary conditions at both structural levels (curvature in two and three dimensions) overlap, allowing for coordination between levels.

Thus by adding calculus and biophysics, the features of stability, coherence, and geometrical specificity can be added explicitly to the well-known essential genetic controls pertinent to patterning and morphogenesis. This article has pointed out that the added context has three salient features. (a) The pertinent analytical part of the context, e.g., the set of latent tendencies can be as challenging and essential to understanding as the nature of the molecular control itself. The tangible part of the context, the parameters for the pertinent equation and boundary conditions, are products of the genome. (b) The switch, effective in the particular context, is usually a change in the boundary conditions. In some cases, parameters or values in the equation itself may change to act as a switch (e.g., to change wavelength). In either event, these parameters are likely to be "far downstream," i.e., considerably derived from products of conventional expression sought in screens. (c) An unknown context is difficult to seek. A common tactic, used here, is to consider what is sufficient to account for the demonstrable complexity of the normal process itself. In this case, for patterning the buckling analogy was explored. For explaining mechanisms where calculus is heavily involved, this tactic appears to be necessary. It is added to, and contrasts with, the extensive information derived from the perturb-and-assay interventions (against constant background), which deal with what is essential to cause the developmental outcome to change (see Green, 1996 ). Adding the gene-produced biophysical context to the well-known promotion–inhibition phenomenology broadens the view of genetic control of a process to provide a more general grasp, approximating genetic basis.

View larger version (75K): [in this window] [in a new window]   Paul B. Green (1931–1998)

	FOOTNOTES

² Paul Green died of cancer before completing the final version of this manuscript. Minor revisions of the text and the figures were done by Jacques Dumais. Correspondence should be sent to Jacques Dumais, Department of Biological Sciences, Stanford University, Stanford, CA 94305.

 The family of the late Paul B. Green expresses their gratitude to Jacques Dumais for completing preparation of this manuscript for submission. Jacques Dumais would like to acknowledge the contributions of Mr. Jonathan Fay, Dr. Arif Karabeyoglu, and Dr. Charles Steele. He would also like to thank the numerous colleagues and former students of Paul Green for their comments on earlier versions of this manuscript and Mr. Ted Manning for preparing Figs. 1 , 2 , 8 , and 9 . Finally, he is indebted to the Green family for their suggestions and encouragements. Part of this research was supported by NSF grant IBN-9603705.

	LITERATURE CITED

TOP ABSTRACT INTRODUCTION TOPOGRAPHY I: TWO SIGNAL-BASED... TOPOGRAPHY II: A RESPONDING... CELL PATTERNING: ALIGNMENT... DISCUSSION LITERATURE CITED

 
Battey, N. H., and R. F. Lyndon. 1990 Reversion of flowering. Botanical Review 56: 162–189.

Berding, C., T. Harbich, and H. Haken. 1983 A pre-pattern formation mechanism for the spiral-type patterns of the sunflower head. Journal of Theoretical Biology 104: 53–70.[CrossRef]

Bernasconi, G. P. 1994 Reaction-diffusion model of phyllotaxis. Physica D 70: 90–99.[CrossRef][Web of Science]

Bierhorst, D. W. 1971 Morphology of vascular plants. Macmillan, New York, NY.

Carpenter, R., L. Copsey, C. Vincent, S. Doyle, R. Magrath, and E. Coen. 1995 Control of flower development and phyllotaxy by meristem identity genes in Antirrhinum. Plant Cell 7: 2001–2011.

Chapman, J. M., and R. Perry. 1987 A diffusion model of phyllotaxis. Annals of Botany 60: 377–389.[Abstract/Free Full Text]

Cooke, T. J., and B. Lu. 1992 The independence of cell shape and overall form in multicellular algae and land plants: cells do not act as building blocks for constructing plant organs. International Journal of Plant Sciences 153: S7–S27.[CrossRef][Web of Science]

Darwin, F. [ed.] 1897 The life and letters of Charles Darwin: including an autobiographical chapter. Appleton, New York, NY.

Erickson, R. O. 1983 The geometry of phyllotaxis. In J. E. Dale and F. L. Milthorpe [eds.], The growth and functioning of leaves, 53–88. Cambridge University Press, Cambridge.

Fleming, A. J., S. McQueen-Mason, T. Mandel, and C. Kuhlemeier. 1997 Induction of leaf primordia by the cell wall protein expansin. Science 276: 1415–1418.[Abstract/Free Full Text]

Fujita, T. 1939 Statistische Untersuchungen über den Divergenz-winkel bei den schraubigen Organstellungen. Botanical Magazine, Tokyo 53: 194–199.

Giddings, T. H., and L. A. Staehelin. 1991 Microtubule-mediated control of microfibril deposition: a re-examination of the hypothesis. In C. W. Lloyd [ed.], The cytoskeletal basis of plant growth and form, 85–99. Academic Press, New York, NY.

Green, P. B. 1984 Shifts in plant cell axiality: histogenetic influences on cellulose orientation in the succulent Graptopetalum. Developmental Biology 103: 18–27.

———. 1986 Plasticity in shoot development: a biophysical view. In D. H. Jennings and A. J. Trewavas [eds.], Plasticity in plants, 211–232. Cambridge University Press, Cambridge.

———. 1994 Connecting gene and hormone action to form, pattern, and organogenesis. Journal of Experimental Biology 45: 1775–1788.

———. 1996 Transductions to generate plant form and pattern: An essay on cause and effect. Annals of Botany 78: 269–281.[Abstract/Free Full Text]

———, and K. Brooks. 1978 Stem formation from a succulent leaf: its bearing on theories of axiation. American Journal of Botany 65: 13–26.[CrossRef][Web of Science]

———, and J. M. Lang. 1981 Toward a biophysical theory of organogenesis: birefringence observations on regenerating leaves in the succulent, Graptopetalum paraguayense E. Walther. Planta 151: 413–426.[CrossRef][Web of Science]

———, and J. M. L. Selker. 1991 Mutual alignments of cell walls, cellulose, and cytoskeletons: their role in meristems. In C.W. Lloyd [ed.], The cytoskeletal basis of plant growth and form, 303–322. Academic Press, New York, NY.

———, C. S. Steele, and S. C. Rennich. 1996 Phyllotactic patterns: a biophysical mechanism for their origin. Annals of Botany 77: 515–527.[Abstract/Free Full Text]

———, ——— and ———. 1998 How plants produce pattern: a review and a proposal that undulating field behavior is the mechanism. In R. V. Jean and D. Barabé [eds.], Symmetry in plants, 359–392. World Scientific, Singapore.

Gunning, B. E. S., and D. R. Hardham. 1982 Microtubules. Annual Review of Plant Physiology 33: 651–698[Web of Science]

Hardham, D. R. 1982 Regulation of polarity in tissues and organs. In C. W. Lloyd [ed.], The cytoskeleton in plant growth and development, 377–403. Academic Press, London.

———, J. M. Lang and P. B. Green. 1980 Reorganization of cortical microtubules and cellulose deposition during leaf formation in Graptopetalum paraguayense. Planta 149: 181–195.

Harrison, L. G. 1993 Kinetic theory of living pattern. Cambridge University Press, New York, NY.

———, and N. A. Hillier. 1985 Quantitative control of Acetabularia morphogenesis by extracellular calcium: a test of kinetic theory. Journal of Theoretical Biology 114: 177–192.[CrossRef]

———, J. Snell, J. Verdi, D. E. Vogt, G. D. Zeiss, and B. R. Green. 1981 Hair morphogenesis in Acetabularia mediterranea: temperature-dependent spacing and models of morphogen waves. Protoplasma 106: 211–221.

Hejnowicz, Z., and J. Karczewski. 1993 Modelling of meristematic growth of root apices in a natural co-ordinate system. American Journal of Botany 80: 309–315.[CrossRef][Web of Science]

Hepler, P. K., A. L. Cleary, B. E. S. Gunning, P. Wadsworth, G. O. Wasteneys, and D. H. Zhang. 1993 Cytoskeletal dynamics in living plant cells. Cell Biology International 17: 127–142.[CrossRef][Web of Science]

Hernández, L. F., and P. B. Green. 1993 Transductions for the expression of structural pattern: analysis in sunflower. Plant Cell 5: 1725–1738.[Abstract]

———, A. Havelange, G. Bernier, and P. B. Green. 1991 Growth behavior of single epidermal cells during flower formation: sequential scanning electron micrographs provide kinematic patterns for Anagallis. Planta 185: 139–147.

Holloway, D. M., and L. G. Harrison. 1999 Algal morphogenesis: modelling interspecific variation in Micrasterias with reaction-diffusion patterned catalysis of cell surface growth. Philosophical Transactions of the Royal Society of London 354: 417–433.[CrossRef]

Hush, J., P. Wadsworth, D. A. Callaham, and P. K. Hepler. 1994 Quantification of microtubule dynamics in living plant cells using fluorescence redistribution after photobleaching. Journal of Cell Science 107: 775–784.[Abstract]

Jean, R. V. 1994 Phyllotaxis. Cambridge University Press, New York, NY.

Jesuthasan, S., and P. B. Green. 1989 On the mechanism of decussate phyllotaxis: biophysical studies on the tunica layer of Vinca major. American Journal of Botany 76: 1152–1166.

Kalthoff, K. 1996 Analysis of biological development. McGraw-Hill, New York, NY.

Kaplan, D. R., and W. Hagemann. 1991 The relationship of cell and organism in vascular plants: are cells the building blocks of plant form? BioScience 41: 693–703.[CrossRef][Web of Science]

Korn, R. W. 1993 Apical cells as meristems. Acta Biotheoretica 41: 175–189.[CrossRef][Web of Science]

Kwiatkowska, D. 1995 Ontogenetic changes of phyllotaxis in Anagallis arvensis L. Acta Societatis Botanicorum Poloniae 64: 319–325.[Web of Science]

Lacalli, T. C. 1981 Dissipative structures and morphogenetic pattern in unicellular algae. Philosophical Transactions of the Royal Society of London B294: 547–588.

Lang, J. M., W. R. Eisinger, and P. B. Green. 1982 Effects of ethylene on the orientation of microtubules and cellulose microfibrils of pea epicotyl cells with polylamellate cell walls. Protoplasma 110: 5–14.

Lintilhac, P. M., and P. B. Green. 1976 Patterns of microfibrillar order in a dormant fern apex. American Journal of Botany 63:726–728.

Lloyd, C. W., and P. W. Barlow. 1982 The co-ordination of cell division and elongation: the role of the cytoskeleton. In C. W. Lloyd [ed.], The cytoskeleton in plant growth and development, 203–228. Academic Press, London.

Lyndon, R. F. 1994 Control of organogenesis at the shoot apex. Tansley Review Number 74. New Phytologist 128: 1–18.[CrossRef][Web of Science]

———. 1998 The shoot apical meristem, its growth and development (Developmental and cell biology series). Cambridge University Press, Cambridge.

Marc, J., and W. P. Hackett. 1991 Gibberellin-induced reorganization of spatial relationships of emerging leaf primordia at the shoot apical meristem in Hedera helix L. Planta 185: 171–178[Web of Science]

Meicenheimer, R. D. 1981 Changes in Epilobium phyllotaxy induced by N-1-naphthylphthalamic acid and -4-chlorophenoxyisobutyric acid. American Journal of Botany 68: 1139–1154.[CrossRef][Web of Science]

———. 1998 Decussate to spiral transitions in phyllotaxis. In R. V. Jean and D. Barabé [eds.], Symmetry in plants, 125–143. World Scientific, Singapore.

Meinhardt, H. 1995 The algorithmic beauty of sea shells. Springer-Verlag, Berlin.

Meyerowitz, E. M. 1996 Plant development: local control, global patterning. Current Opinion in Genetics and Development 6: 475–479.[CrossRef][Web of Science][Medline]

Mingo-Castel, A. M., C. Gomez-Campo, M. E. Tortosa, and A. M. Pelacho. 1984 Hormonal effects on phyllotaxis of Euphorbia lathyris L. Botanical Magazine, Tokyo 97: 171–178.[CrossRef][Web of Science]

Mitchison, G. J. 1977 Phyllotaxis and the Fibonacci series. Science 196: 270–275.[Abstract/Free Full Text]

Murray, J. D. 1981 A pre-pattern formation mechanism for animal coat markings. Journal of Theoretical Biology 88: 161–199.[CrossRef]

Odell, G. M., G. Oster, P. Alberch, and B. Burnside. 1981 The mechanical basis of morphogenesis. I. Epithelial folding and invagination. Developmental Biology 85: 446–462.[CrossRef][Web of Science][Medline]

Rasmussen, N. 1991 Studies on the determination of organ pattern and organ identity in flower development. Ph.D. dissertation, Biological Sciences, Stanford University, Stanford, CA.

Ridley, J. N. 1982 Packing efficiency in sunflower heads. Mathematical Biosciences 58: 129–139.[CrossRef][Web of Science]

Sakaguchi, S., T. Hogetsu, and N. Hara. 1988 Arrangement of cortical microtubules at the surface of the shoot apex in Vinca major L.: observations by immunofluorescence microscopy. Botanical Magazine, Tokyo 101: 497–508.[CrossRef][Web of Science]

Schwabe, W. W. 1984 Phyllotaxis. In P. W. Barlow and D. J. Carr [eds.], Positional controls in plant development, 403–440. Cambridge University Press, Cambridge.

Selker, J. L., and P. B. Green. 1984 Organogenesis in Graptopetalum paraguayense E. Walther: shifts in orientation of cortical microtubule arrays are associated with periclinal divisions. Planta 160: 289–297.[CrossRef][Web of Science]

———, and R. F. Lyndon. 1996 Leaf initiation and de novo pattern formation in the absence of an apical meristem and pre-existing patterned leaves in the watercress (Nasturtium officinale) axillary explants. Canadian Journal of Botany 74: 625–641.

Sinnott, E. W., and R. Block. 1945 The cytoplasmic basis of intercellular patterns in vascular differentiation. American Journal of Botany 32: 151–156.[CrossRef][Web of Science]

Smith, L. G., S. Hake, and A. W. Sylvester. 1996 The tangle1 mutation alters cell division orientations throughout maize leaf development without altering leaf shape. Development 122: 481–489.[Abstract]

Snow, M., and R. Snow. 1935 Experiments on phyllotaxis III. Diagonal slits through decussate apices. Philosophical Transactions of the Royal Society of London B225: 63–94.

Steinberg, M. S., and T. J. Poole. 1981 Strategies for specifying form and pattern: adhesion-guided multicellular assembly. Philosophical Transactions of the Royal Society of London B295: 451–460.

Sylvester, A. W., L. Smith, and M. Freeling. 1996 Acquisition of identity in the developing leaf. Annual Review of Cell and Developmental Biology 12: 257–304.[CrossRef][Web of Science][Medline]

Szilard, R. 1974 Theory and analysis of plates, classical and numerical methods. Prentice Hall, Englewood Cliffs, NJ.

Tarnai, T., and Z. Gaspar. 1995 Packing of equal circles in a square. Acta Technica 107: 123.

Theißen, G., and H. Saedler. 1995 MADS-box genes in plant ontogeny and phylogeny: Haeckel's ‘biogenetic law’ revisited. Current Opinion in Genetics and Development 5: 628–639.[CrossRef][Web of Science][Medline]

Thompson, D'Arcy W. 1942 On growth and form. Cambridge University Press, Cambridge.

Tiwari, S. C., and P. B. Green. 1991 Shoot initiation on a Graptopetalum leaf: sequential scanning electron microscopic analysis for epidermal division patterns and quantitation of surface growth (kinematics). Canadian Journal of Botany 69: 2302–2319.[CrossRef]

Turing, A. M. 1952 The chemical basis for morphogenesis. Philosophical Transactions of the Royal Society of London B237: 37–72.

van Iterson, G. 1907 Mathematische und mikroskopisch-anatomische Studien über Blattstellungen, nebst Betraschtungen über den Schalenbau der Miliolinen. Gustav Fischer Verlag, Jena.

Veen, A. H., and A. Lindenmayer. 1977 Diffusion mechanism for phyllotaxis. Plant Physiology 60: 127–139.[Abstract/Free Full Text]

Wasteneys, G. O., and R. E. Williamson. 1989 Reassembly of microtubules in Nitella tasmanica: quantitative analysis of assembly and orientation. European Journal of Cell Biology 50: 76–83.[Web of Science]

Wigglesworth, V. B. 1940 Local and general factors in the development of "pattern" in Rhodnius proxilus (Hemiptera). Journal of Experimental Biology 17: 180–200.[Abstract]

Williams, M. H. 1991 A sequential study of cell divisions and expansion patterns on a single developing shoot apex of Vinca major. Annals of Botany 68: 541–546.

Williamson, R. E. 1991 Orientation of cortical microtubules in interphase plant cells. International Review of Cytology 129: 135–205.[CrossRef]

Young, D. A. 1978 On the diffusion theory of phyllotaxis. Journal of Theoretical Biology 71: 421–432.[CrossRef][Web of Science][Medline]

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