IV.5

Self-organization and the Formation of Patterns in Plants

Paul B. Green (Stanford)

Beloussov et aZ., the introductory article (IV.1 this Chapter), pointed out con­trasting assumptions in positional information theory (PI) and in theory based on more epigenetic principles as expounded by Driesch. One key feature of PI the­ory, in addition to the mentioned use of privileged sites to establish a geometrical frame of reference for development, is that the relation between a control and the response to it need to be defined only by correlation. At any moment this gives PI theory enormous flexibility with regard to cause and effect. That is, a given posi­tional value may be coupled to virtually any kind of activity. This feature can be considered a virtue because of the obvious versatility. This can also be considered a shortcoming because it places, at least temporarily, a black box between cause and effect. Control elements ideally have an explicit connection to the responding system. Also, what happens at one time needs to lead into what happens next. This normal behavior is hard to account for explicitly with a black box present in the causal chain. The theses in IV.1 thus makes plea for a rational explanation of the self-evolving and self-correcting properties of developing systems.

1 Pattern formation mechanisms

In light of these issues, we present here a summary of our work on patterns in plants. It happens to involve: development initially without privileged sites, ex­plicit connections between control and response, and interacting self-organization features. All these aspects can be derived from an established pair of differential equations that pertain to the folding or buckling of plates (epithelia). The readiness of the transfer from physics to biology relates, almost certainly, to the fact that in plants the patterns are radial, repetitious, and well defined. The primary questions for plant pattern are: explaining its origin and its propagation. The stability of propagation is an important secondary issue.

We have taken the position that the de novo origin of patterns is the more critical process. Surely the primary event, organogenesis, is the same in de novo formation and in propagation. Even more surely, any mechanism that can produce pattern de novo, without cues, can propagate that pattern once it is present and providing periodic cues. Further, the self-organizing process is likely to be the

W. Alt et al. (eds.), Dynamics of Cell and Tissue Motion© Birkhäuser Verlag 1997

244 Green

same as the stabilizing process. These are large scale, or holistic, features that characterize the mechanism abstractly.

In pattern formation in general there is a large gap, with regard to mecha­nism, between the effect (large scale pattern, i.e. the phenotype) and the ultimate material cause (the genotype). Much effort is currently devoted to the "bottom up" approach. The tactic here is to accumulate details on the many molecular steps that pertain to a given process and then connect them in a network to account for the developmental process (e.g., Loomis & Sternberg 1995). An intri­cate network to determine the lysogeny-lysis decision in lambda phage illustrates this approach (McAdams & Shapiro 1995) . The strength of this point of depar­ture is molecular biology's process with regard to all questions of essentiality. A process is altered if a pivotal part, a switch, is altered. Specificity of molecular detail about the switches approaches completeness. The binding vs. non-binding of macromolecules and effectors is established as a ubiquitous on-off switch. The question of how the switches "all fit together", however, is still obscure for organ development. For example, lack of the gene product of deficiens in snapdragon leads to carpel production in the third whorl of the flower, replacing the normal stamens (eoen & Meyerowitz 1991). The pertinent gene has been sequenced. Thus one has a detailed molecular grip on the control, or switch, for organ identity but no obvious entry to the broader question: how could a given tissue possibly make either? The answer to that (what is being switched?) involves the genome also, but in a very different way. Thus the information currently available accounts well for control, but the connection to the response is not explicit. Sufficiency is lacking. This may be hard to obtain "from below".

The alternative "top down" approach is to start with sufficiency, at a very abstract level, and work down to the details (see Green 1987). In effect, one begins with a holistic view of the phenotype (the responding system) and works backwards toward the control elements (genes). This approach has the tactical advantage that, if development is in reality an integration through space and time, at least the analyst is doing differential, rather than integral, calculus. So, with regard to de novo pattern formation, one can ask: of what interesting differential relation could this be the integral? [This contrasts with the question: "What controls it?"] To get started, one needs an analogous process as a model, and theory appropriate to the phenomenon. Fortunately, both are available.

A humble but informative analogue of de novo pattern formation in plants is the formation of a potato chip (crisp). Here a flat featureless disk acquires a saddle shape. The final topography resembles that at the tip of an opposite-leaved plant (e.g., a maple or mint). During cooking, the rim of the potato disk hardens first. The center continues to shrink, giving "excess surface" at the periphery. The spontaneous physical solution to deal with the extra rim is to undulate into 3-D. The theory pertinent to this class of phenomena, symmetry breaking, has been expounded by Harrison (1993), for reaction-diffusion theory. The general mecha­nism requires that the initial uniform object be a wavelength-dependent amplifier. The uniform object has a characteristic wavelength (e.g., half its circumference for

IV.S Self-organization and the Formation of Patterns in Plants 245

Figure 1: De novo whorl formation. Upward contours are drawn in solid lines; depression contours are dashed. A. An initial annulus, fiat except for a random pattern of small undulations. The intrinsic wavelength equals the width of the annulus. B. In-plane com­pression at the margins generates 23 alternating humps and depressions. Compression is physically equivalent to tendency to expand (growth).

the chip). Out of an initial array of random undulations only those close to the intrinsic one will be amplified. These undulations will, to the best approximation, fill the available space with a whole number of districts (crests, troughs). Thus the chip becomes a saddle.

The logic explaining of the "band pass" property is that some feature of the system dampens the growth of short wavelengths, another feature dampens growth of long wavelengths. In the chemical theory, short wavelength pattern is inhibited by diffusion, long wavelength by inhibitor concentration. In the physical model that we use, short wavelengths are suppressed by a solid sheet's reluctance to bend sharply. Long wavelengths are suppressed by an elastic foundation (springs normal to the structure which are reluctant to change length). Two equations are involved. Details are explained in the accompanying box by Rennich & Green (IV.6 this volume).

2 Physical model for buckling undulations

We have chosen the physical version, rather than the chemical, because it involves no turnover of components and because it provides 3-D structures directly. The physical process is one of reaching static elastic equilibrium, so we assume that development is in multiple steps. After each step, the structure is "solidified" in preparation for the next. This is a quasi-static phenomenon. The process which repeatedly puts the system out of equilibrium is a local tendency for expansion of

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surface (growth) against a constraint. This is equivalent to compression. This input of in-plane stress is countered by the two responses noted above: reluctance of a plate to bend sharply and reluctance of an elastic cushion to change its dimensions. In the plant, the proposed "plate" is a coherent layer of epidermal cells, the tunica. The elastic cushion is thought to be the disorganized cells below the tunica, the corpus.

In the de novo formation of a whorl of organs in an annulus, as occurs com­monly in flowers, there are no privileged sites. Hence there is no positional in­formation nor, apparently, localized controls of any sort. There are only circular boundary conditions and small random perturbations of initial curvature and other variables. As shown in Fig. 1, compression (growth) gives rise to a ring of 23 undu­lations of the intrinsic wavelength. The non-linear character of the response insures that these undulations mutually position themselves evenly within the available space. There is some leeway in the natural wavelength, so whole numbers always arise. This sequence fits Beloussov's scenario of no special sites initially and that, after mechanical change, special sites do arise. Here uniform constraint evoked a latent structural periodicity.

When the issue of control elements vs. responding system (so flexible in PI theory) is raised, the well defined contribution of the responding system is abun­dantly clear in this simulation. The number, and the character, of the undulations can be influenced in three ways. The following examples are from Green et al. (1996).

First, the value of a term in the differential equation itself, such as the flexural rigidity of the plate, could be changed. The natural wavelength is proportional to the fourth root of the ratio: (flexural rigidity)j(elastic constant of the foundation). Changing this ratio in the input varies the number of undulations produced in the simulation, as predicted. Thus this type of control can be algebraic.

Second, the dimensions of the formative area may be varied. One way is to enlarge the mean annulus diameter, at constant width of the annulus. Here again, simply more undulations arise, as predicted. This variation is also one where an algebraic interpretation suffices. When, however, annulus width is increased at constant outer diameter, there is an increasing difference in the fit for the undulation at the two margins. Obviously fewer undulations can fit the smaller circumference. The response to change in this dimensional input, or "control", is the production of V-shaped primordia, or even fused ridges. This provides the annulus with undulations all close to the natural wavelength. This is seen in Fig. 2 A-C.

Third, and especially interesting, is varying the boundary conditions. These are the constants of integration involving initial elevation (here kept constant at zero) and either slope or curvature at the margin. When the slope is fixed, the edge is "clamped". When the curvature is fixed, the edge is "hinged", or in engineering terms "simply supported". In Fig. 2 a left-right comparison of images involves change in boundary conditions only; the dimensions are the same. In the pair A-D the change from clamped to simply supported for the outer margin has drastic

IV.S Self-organization and the Formation of Patterns in Plants 247

A.

c.

Figure 2: Effects on pattern of varying annulus width (limits of integration) under two different sets of boundary conditions (constants of integration). In each figure the box shows boundary conditions , BC, at the large (L) and small (8) circumference. CL is clamped, 88 is simply supported, or hinged. The graph shows the radial profile of humps (solid) and depressions (dashed) . Predicted (P) and observed (0) numbers of undulations at the two margins are given. An integral number of half-wavelengths fits radially only in A and D. A-C. As width is increased, some new undulations are "fused" as V-shaped ridges (depressions) , reducing the undulation number at the smaller circumference. D­F . With both margins hinged , the natural undulation is fitted differently to the radial dimension (steep slope at both ends). When the fit of ! wavelengths is exact (D and F), this promotes formation of circumferential ridges. When the fit is poor, as in E (2! wavelengths) , the annulus is still subdivided circumferentially, but a checkerboard pattern arises. It includes more circumferential undulation.

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Figure 3: A. Propagation of pattern. An undulating annulus, as in Fig. IB, is made permanent. It is the outer one. The steep slope at its inner margin becomes the outer clamped boundary condition for a new, naIve annulus (~2). B. Compression of annulus ~2 leads to formation of the corresponding undulating pattern, exactly out of phase with ~l. This is the key cyclic feature of whorled development in plants.

effects. Apparently the flexibility of having both margins hinged now allows the one wavelength fitting across the annulus to provide a node in the center; hence, long ridges, circumferential (trenches) subdivide the annulus. One now adds radial dimensional change to this condition. At the greater width in F, four nodes fit well, and the annulus is largely tri-sectioned circumferentially. However, at an intermediate dimension, in E, the transverse fit in terms of half wavelengths is poor, and much more of the undulation takes place circumferentially. The annulus is still bisected but a "checkerboard" results. The strikingly different responses to simple dimensional changes (compare vertically within each column in Fig. 2) and to boundary conditions (left-right in Fig. 2) show that these parameters can have major, non-intuitive, consequences. The fact that the system is inherently periodic obviously contributes to this diverse behavior as a function of simple progressive change. Thus variation in dimension and boundary state are control elements, or effectors, to be added to the variables already familiar in an algebraic context.

TUrning to the issue of propagation of pattern, development at the shoot apex is centripetal. Some feature of an old large annulus must influence the new pattern on a smaller adjacent "naive" annulus inside it. Most models assume that this influence is a spatially periodic distribution of control molecules acting on a simple responsive tissue. In the present case the responsive region is not considered simple, i.e., it has latent periodicity. We investigated how the physical input of slope periodicity, restricted to the outer boundary of the naive annulus, could influence pattern development in it. We set (clamped) the slope at the outer boundary of the

IV.S Self-organization and the Formation of Patterns in Plants 249

naIve annulus to be spatially periodic. The inner boundary was clamped at zero slope. The patterned input at the margin sufficed not only to shift the phase of undulations, as is typical in whorled plants, but also to maintain the same number of undulations despite the reduced circumference (Fig. 3). In this case the key feed-forward control was the slope at the periphery, a constant of integration for the master equations.

We thus view the formative annulus of tissue at the plant apex as "an active solid body" . It can have a natural wavelength. This latent periodicity can evoke, in de novo pattern formation, a whorled pattern without a particular phase. When, of course, the formative area is physically connected to a periodic structure peripheral to itself, as is often the case in plants, that periodicity feeds in at the boundary to set the phase of phenomena in the adjacent naIve area. The phase shift in simple whorled systems is one half wavelength. We obtain it in simulations. We are now applying this approach to spiral patterns (Green 1996a).

3 Summary

The feature of de novo pattern by non-patterned mechanical influence on an an­nulus was illustrated in whorl initiation. Continuous interaction within the system was characteristic of the solution of the non-linear equations. The resulting ring of new undulations, i.e., new mechanical singularities, was able to influence the new-forming pattern in an adjacent annulus. Periodic slope caused the phase shift (alternation) so typical of organs in flowers. In this "top down" modeling, the interplay of control factors and responding system is explicit through the use of equations for buckling. A diversity of controls is available when the mechanism is embodied in differential equations. While the models are abstract, and at present devoid of molecular specificity, they are at least sufficient and explicit, in physical terms, with regard to the origin and propagation of pattern.

Subsequent work has concerned comparing the propagation of whorls and "Fibonacci" spirals, the only two patterns common in nature. In each case the propagation occurs within an annulus. Over time, tissue of its flat inner margin is converted to an undulating profile at the outer margin. The relative work required to do this can be calculated. Thus far, it appears that whorls are of least energy, provided that the intrinsic wavelength of the tissue fits the circumference exactly. Spirals require slightly more energy but are uniquely insensitive to change in an­nulus circumference. This is an attractive hypothesis to account for the prevalence of the two patterns (Green 1996b).

Acknowledgements The generous contribution of computer time for all the simu­lations, by the Maui High Performance Computer Center, Kihei, Maui, HI 96572, is gratefully acknowledged. Supported from a grant of the National Science Foun­dation to P.B.G.