Models of Pattern Formation Applied to Plant Development1

Models of pattern formation applied to plant development1

Hans Meinhardt, Andr´e-Joseph Koch and Giuliano Bernasconi

1 Introduction interference indicate that cells communicate with each other to achieve the translation from the genetic infor- Plants are the beautiful results of a chain of complex pat- mation into the three-dimensional structure. We assume tern forming events. Pattern formation — the generation that the formation of a given structure is initiated by a of regular differences in space — occurs at several levels of particular biochemical signal that may consist of a high organization. For example, a particular group of cells at concentration of one ore more substances. If mechani- the shoot apex may receive a signal to form a leaf. The cal forces are involved that bring a particular tissue into leaf will get a polarity, developing an upper and a lower shape, it will be assumed that this is initiated by a preced- side. Some cells of the leaf may develop into stomata, ing chemical signal that caused, for instance, a rearrange- while chains of other cells may form vascular strands. In- ment of the cytoskeleton. These models are therefore dif- dividual cells may become polar. All these patterns must ferent from models assuming pattern formation directly ultimately arise from similar mechanisms although they on a mechanical level (Green and Poething, 1982 ; Oster do not act at the same structural level. et al, 1983). In this chapter we would like to discuss basic signalling Pattern formation is by no means a peculiarity of liv- systems among cells that allow to generate different cell ing systems. The formation of clouds, rivers, sand dunes, types in a defined spatial arrangement and apply them water waves or crystals are examples where the pattern to plant development. We will further show that a link- formation in the inorganic world starts from rather homo- age of several such pattern forming systems allows the geneous initial conditions. Common in these patterning reproducible generation of complex patterns. Models for processes is that minor deviations from uniformity have a the generation of the primary shoot-root axis and for the severe effect on their further growth. For instance, forma- generation of leaf primordia around the meristem will be tion of a river may have started originally from a minor elaborated. These models are certainly not complete. For depression in the landscape. The water collecting there instance, shape changes of a tissue (see Gierer, 1977a) or from the rain accelerates erosion at this location, more the tissue-specific control of cell proliferation will not be water runs toward this incipient valley and so on. Thus, discussed. Comparisons with pattern formation in the pattern formation requires a self-amplifying, or autocat- freshwater polyp Hydra, with the formation of imaginal alytic, element. However, self-amplification on its own disks in insects or the patterning on the shells of some would be insufficient for pattern formation. It would lead mollusks will reveal that in plants mechanisms are at work to a complete conversion of one state into another, just that are closely related to those in other systems that, at as a burning piece of paper will be completely converted first sight, look very different from plants. into ash (burning is an autocatalytic process). In pat- In plants, as in most other higher organisms, the adult tern forming systems, this global activation is avoided by structure develops from a single cell. The final pattern the action of an antagonist process that spreads rapidly cannot already be present in this cell in a hidden form. over a large domain. It restricts the autocatalysis to a Regulatory phenomena observed after an experimental small region. On the basis of these two principles — local

1Reprint of a chapter that appeared in: Symmetry in Plants (D. Barabe and R. V. Jean, Eds), World Scientiﬁc Publishing, Singapore; pp. 723-758

1 autocatalysis and long range inhibition — a theory of bi- complemented by the action of a fast diffusing antagonist. ological pattern formation has been elaborated (Gierer Basically, two types of antagonistic reactions are conceiv- and Meinhardt, 1972 ; Meinhardt, 1982). After a brief able. Either an inhibitory substance h is produced by introduction to this theory, we would like to show how the activator that, in turn, slows down the activator pro- inhomogeneous patterns are produced. Several applica- duction : we call this activator-inhibitor systems. Alter- tions to morphogenetic processes observed in plants will natively, a substrate s, produced everywhere in the field, then be discussed. Particular emphasis will be put on is consumed in the autocatalysis : its depletion around apical growth and on phyllotaxis. a growing activator maximum lowers the rate of the self- enhancing reaction ; these are activator-substrate systems. We shall give hereafter simple examples of these two kinds 2 Pattern formation by local of mechanisms. autocatalysis and long range inhibition 2.1 Activator-inhibitor systems One of the simplest interaction that leads to pattern for- The possibility of generating a pattern by the interac- mation is described by the following set of equations. The tion of two substances was demonstrated by Rashewsky activator a and the inhibitor h are coupled in a nonlinear (1940) and Turing (1952). In his pioneering work, Turing way. demonstrated that under certain conditions two interact- ing chemicals can generate a stable inhomogeneous pat- ∂a a2 tern (the Turing pattern) if one of the substances diffuses = Da 4a + ρa − µaa + σa (1.a) much faster than the other. This result is quite surpris- ∂t h ∂h 2 ing since diffusion is expected to smooth out concentra- = Dh 4h + ρha − µhh + σh . (1.b) tion differences rather than to generate them. Gierer and ∂t Meinhardt (1972) have shown that the crucial condition Eq.(1.a) states that a concentration change of the acti- is that the short-ranging substance must accomplish a vator per unit time (∂a/∂t) is proportional to an auto- self-enhancement while the long-ranging substance antag- catalytic production term (a2), and that the autocatal- onizes this autocatalysis. As will be shown below, sharp ysis is slowed down by the action of the inhibitor (1/h). concentration maxima, graded concentration profiles and As any biological substance, the activator decays. It is stripe-like distributions can be generated in this way. natural to assume that the number of disappearing ac- The paper of Turing marks the beginning of many tivator molecules is proportional to the number of acti- theoretical investigations, applying the concepts of chem- vator molecules present. This is expressed by the term ical reactions and diffusion to biological pattern formation −µaa. The concentration of a in a given cell may also (Lefever, 1968; Segel and Jackson 1972 ; Gierer, 1977b,; vary due to exchange of molecules with neighboring cells. Murray, 1989, Meinhardt, 1982, 1984 ; Harrison, 1993 ; The simplest way to take this exchange into account is to Koch and Meinhardt, 1994). Recently, pattern formation assume that the activator molecules diffuse between the has been observed in chemically completely defined sys- cells according to Fick’s law ; the contribution of diffu- tems (Ouyang et al, 1989 ; Castets et al, 1990; de Kepper sion to the concentration change of a is then proportional et al, 1991). It is easy to see that Turings mechanism is to 4a = ∂2a/∂x2 + ∂2a/∂y2 + ∂2a/∂z2 (with the usual also based on short range autocatalysis and long range in- notation where x, y, z are the spatial coordinates in an hibition (Meinhardt, 1984), but Tuing did not interprete orthonormal reference frame). Other modes of redistri- his result in this way. bution of molecules are conceivable as well, especially if The general principle of local autocatalysis and long communication has to be performed over long distances range inhibition can easily be translated into molecular (in plants, active transport plays a major role). For the feasible interactions. A substance a is said to be self- sake of simplicity, we do not take such terms into account. enhancing or autocatalytic if a small increase of a over The last term in Eq.(1.a) describes a small basic activator its homogeneous steady-state concentration induces a fur- production (σa). This term insures that the concentra- ther increase of a (we use the same symbol to denote a tion of the activator never sinks to zero. This is important substance and its concentration, but this should not lead for the initiation of the autocatalytic reaction at low acti- to any confusion). The self-enhancement of a has to be vator concentrations (see, for instance, Fig. 3). The term

2 (a) (b) (c)

Figure 1: Two dimensional patterns produced by the activator-inhibitor model (1). The range of the inhibitor is much smaller than the field size. (a) Initial, intermediate and final activator (top) and inhibitor (bottom) distributions. (b) Result of a similar simulation in a larger field. The concentration of the activator is suggested by the dot density. (c) Biological example of an irregular pattern : the arrangement of stomata on the bottom side of a leaf (by courtesy of Dr. M. Claviez).

ρa, the source density, describes the general ability of the cells to perform the autocatalytic reaction. Slight asym- ∂a a2 metries in the source density may have a strong influence = − a ; ∂t 1 on the orientation of the emergent pattern. The equation describing the change in the inhibitor a has a steady state ∂a/∂t ≡ 0 at a=1. However, this steady state is unstable since for any concentration of a concentration h can be interpreted in the same way. The 2 inhibitor production depends in a nonlinear way on the which is a bit larger than 1, a − a will be positive and 2 the concentration of a will increase further. Other way activator concentration ρha but not on the inhibitor itself. Like the activator, the inhibitor decays (−µ h) and round, if a < 1, ∂a/∂t is negative and a becomes yet h smaller. The reason for this instability lies in the over- diffuses (Dh4h). The condition for stable pattern formation is that the inhibitor diffuses much faster (D D ) exponential autocatalytic production term in conjunction h a with an exponential decay. and that it has a higher decay rate (µh > µa) when com- pared with the activator. In contrast, if the inhibitor We now include the action of the inhibitor. Disregard- ing again diffusion, Eq.(1.b) simplifies to decays slower than the activator (µh < µa), the concentrations of a and h have a strong tendency to oscillate in ∂h = a2 − h . time. ∂t The small baseline inhibitor production σh can play If the inhibitor responds rapidly to any change of the ac- an important role : if sufficiently large, the inhibitor can tivator concentration, we have h ≈ a2 and we can express no longer drop to such a low concentration that a new the rate of change of the activator concentration as : activation becomes triggered. At low activator concen- ∂a a2 trations, the pattering system becomes asleep. It requires ≈ − a = 1 − a ; a specific trigger for the release of a new activation. As ∂t a2 we shall see below, such a mechanism can be important If a is larger than 1, then 1 − a is negative and the con- to enable the initiation of new leaves next to the apical centration will return to the steady state at a = 1. In meristem but to suppress it even if the existing leaves other words, if we include the action of the inhibitor, the obtain a large mutual distance. steady state at a = 1 is stable. A simple calculations should provide some intuition To understand why Eq.(1) generates a pattern, we why the system (1) can lead to a stable pattern. Let have to take into consideration that the inhibitor diffuses us assume, for simplicity, that the basic production rates much faster than the activator. Let us assume an array of σa and σh are zero, that we can neglect diffusion, that all cells in which a and h have everywhere their steady state other constants are equal to 1, and that the inhibitor con- value, except in one cell which has a slightly increased ac- centration is initially constant and equal to 1. Eq.(1.a) tivator concentration. This cell will also produce a little would then read bit more inhibitor than its neighbors but, since h diffuses

3 rapidly into the surroundings, it can initially be regarded strate is not used up there. This can lead to a higher as constant. As mentioned, if the inhibitor is constant, activator production at the side of an existing maximum, any deviation from the activator steady state will grow i.e., the maximum begins to move towards higher sub- further, the equilibrium being unstable. However, after strate concentrations until a new optimum position is ob- a substantial increase of the activator maximum, the in- tained. Lacalli (1981) has simulated pattern formation hibitor concentration can no longer be considered as con- during growth of the unicellular algae by such a mecha- stant. As shown above, the inhibitor leads to the stabi- nism. However, the insertion of new stomata or of new lization of the autocatalysis. Globally, this mechanism al- leaf primordia at a distance to existing structures sug- lows local autocatalytic activator increase and, due to the gests that in plants many pattern formation events are fast diffusion of the simultaneously produced inhibitor, to most likely realized by activator-inhibitor systems. a decrease of the activator concentration in the vicinity of Another difference is closely related to the feature this incipient peak. A new stable steady state is reached just described. In patterns generated by an activator- which is inhomogeneous (Fig. 1). substrate mechanism, the peaks have roughly the same extension as the space in between. If the space becomes 2.2 Activator-substrate systems much larger due to growth, the peaks split and shift into the free space : at equilibrium, peaks and separating space As mentioned, the antagonistic effect that keeps the auto- have approximately the same extension. In contrast, in catalysis under control can also result from the depletion an activator-inhibitor system, the distances between the of a substrate s that is consumed during autocatalysis. maxima can become very large, especially if intercalary The following equation provides an example (Gierer and growth is involved. The basic activator and inhibitor pro- Meinhardt, 1972) : duction rates [σa and σh in Eq. (1) ] define a threshold below which the inhibitor has to drop before a new ac- ∂a tivation is triggered. If the basic activator production is = D 4a + ρ a2s − µ a (2.a) ∂t a a a low, the required distance can be very high. ∂s 2 = Ds 4s − ρsa s + σs ; (2.b) ∂t 2.3 The dividing cell as pattern forming The autocatalysis of the activator a requires the substrate system s but s is consumed in this process (∂s/∂t ∼ −a2s). The substrate is supplied at a constant rate σs while the acti- In many developmental situations, the division of a cell is vator is removed at a rate proportional to its concentra- accompanied by a differentiation of the two daughter cells. tion (−µaa). In the inhomogeneous state, the substrate For several such systems much of the molecular machin- concentration s is low at the positions where the activator ery is already known (for review, see Way et al, 1994a). concentration a is high, and vice versa (Fig. 2). To ac- It should be emphasized that a dividing cell is a very con- complish the lateral inhibition effect, the substrate must venient system for pattern formation. As outlined above, diffuse much faster than the activator, i.e., Ds Da. pattern formation requires autocatalysis and long rang- The reaction has similarities with the so-called Brusse- ing inhibition. Many genes are known to be transcrip- lator (Prigogine and Lefever, 1968, Lefever, 1968) and tion factors that have a direct positive feedback on their is sometimes also referred as the Schnakenberg model own activation. Among these are also genes that are in- (Schnakenberg, 1979). volved in asymmetric cell divisions. The genes mec-3 and The activator-inhibitor and the activator-substrate unc-86 in C. elegans are examples (Way et al, 1994b). models have different properties that make them appro- These transcription factors are, as the rule, restricted to priate for specific applications. In an activator-inhibitor the nucleus. Thus, the condition for the autocatalysis of system, new maxima are inserted during growth if the being local would be satisfied. During a later phase in distance to existing maxima becomes too large (Fig. 3). cell division, when both nuclei are already separated but In contrast, in growing activator-substrate systems new a cytoplasmic connection still exists, the only additional maxima are formed preferentially by a split and a shift ingredient necessary for pattern formation would be a co- of existing maxima. This has the following reason. With factor required for this gene activation that can diffuse growth, the substrate concentration increases in the en- freely in the cytoplasm. This would lead to a competitive larging space between the activated regions since the sub- situation in which one of the two nuclei obtains a full acti-

4 (a) (b) (c)

Figure 2: Two dimensional patterns produced by the activator-substrate model (2). (a) Initial, intermediate and ﬁnal patterns. Upper and lower plots show the concentration of a and s respectively. A high level of a produces a pit in the distribution of the substrate s. (b) Similar simulation in a larger ﬁeld (the activator concentration is shown). Fig. 1 and 2 have been calculated with corresponding parameters ; note that nevertheless the peaks are here broader 2 2 2 and more densely packed. (c) Saturation of the autocatalysis [by replacing in Eq. 2.a the term a s by a s/(1+κaa )] leads to the formation of stripes.

Figure 3: Stages in the insertion of a new maxima during growth. The distance between the activator maxima (top) enlarges and the inhibitor concentration drops in between (bottom). Whenever the inhibition becomes too weak, a new maximum is triggered. Although the depression of the inhibitor concentration is shallow, the new maximum appears at the correct place and will be as sharp as the others since it is shaped by the rising inhibitor concentration. After reaching its steady state, the activator peak is surrounded by its own ﬁeld of inhibition. Calculated with the model (1).

(a) (b) (c) (d) (e)

Figure 4: Pattern formation during cell division. (a) First a single cell is assumed. The activation of the autoregulatory gene is at a steady state due to the limited supply of the co-factor s (s is not shown, it remains nearly constant at all stages). (b) After doubling of the nucleus, a competition starts that will be won by one of them (c). This requires that the co-factor can be exchanged via a cytoplasmic bridge. In one of the nuclei, the activation becomes twice as high. In the other, it becomes completely suppressed. This difference can be used to trigger a specific pathway in one of the future daughter cell. Which nucleus will win can be sensitive to minute external influences. (d) After complete separation, both activations can return to the normal steady state. Alternatively, the low cell may keep the low activation. A coupling of this pattern formation with the cell cycle machinery can restrict the sensitive period to the correct time window.

5 vation while in the other the activity becomes suppressed. In botany, the most prominent example for this type of This is what has been observed for the above mentioned pattern is the initiation of new leaf primordia at the grow- gene mec-3 of C.elegans. No preceding cytoplasmic seg- ing shoot. It will be discussed in detail in section 4. regation of cytoplasmic factors has been detected. There are cases where very regular patterns are The simulation provided in Fig. 4 shows the dramatic formed although the domain has already reached a sub- concentration difference that develops in two nuclei that stantial size. The formation of feather primordia on birds are connected by a cytoplasmic bridge. (Davidson, 1983a, b) or the differentiation of ommatidia in insect eyes (Tomlinson, 1988) are examples. In these cases, the regularity is achieved by a “simulated” growth 2.4 Generation of regular and irregular (Koch and Meinhardt, 1994). Cells become competent patterns in large fields in a wave-like manner. Although many cells are already present, pattern formation can take place only in a nar- The structure produced by a reaction-diffusion system de- row zone that sweeps over the field and existing maxima pends on the size of the domain in which the pattern for- direct the position of new ones as if physical growth was mation takes place. If the range of the inhibitor is of the at work. order of the whole domain, only one activator maximum will form. If the size is comparable with the activator range, the maximum will appear at one boundary of the 2.5 Generation of stripes field. This leads to a monotonic gradient and to polarity in a previously non-polar field, a process of obvious Stripe-like patterns, i.e., structures with a long extension importance for early embryonic development. in one dimension and a short extension in the other, are In contrast, if the field is larger than the inhibitor formed at many instances during embryogenesis. Stripe- range, the domain can accommodate several activator like distributions can be generated by the mechanisms de- maxima. Their arrangement depends to a large extend scribed above if activator production has an upper bound whether the pattern formation proceeds during growth (Meinhardt, 1989). It occurs, for instance, if enzyme or not. molecules required for the autocatalysis are present only If pattern formation starts when the field has already a in a limited amount. Saturation of autocatalysis can be substantial size, the resulting activator maxima will have introduced into equation (1.a) or (2.a) by substituting a2 2 2 a somewhat irregular arrangement but a maximal and a by a /(1 + κaa ). minimal distance is maintained. The irregularity results The reason for stripe formation is easy to understand. because in an early stage of pattern formation, the mu- If activator autocatalysis saturates at relatively low con- tual inhibition of maxima is relatively week and maxima centrations, the inhibitor production is limited too and can appear close together. With increasing peak height, the mutual competition between neighboring cells is re- some of the maxima that are too close to other’s loose duced. More cells remain activated although at a lower the mutual competition and disappear. Fig. 1 provides level. Thus, an activated cell must tolerate other ac- an example. The distributions of stomata and hairs have tivated cells in its neighborhood, independently of the a corresponding pattern. If the structure is generated range of inhibition. Stripe formation requires, in addition by an activator-substrate system, the activator peak dis- to the saturation, a modest diffusion of the activator. Due tribution will be less irregular, since, as mentioned, the to this diffusion, activated regions tend to occur in large maxima can more easily be shifted (Harrison, 1982, 1993). coherent patches since, if a cell is activated, the proba- The model describes also the insertion of new peaks bility is high that the neighboring cell becomes activated during growth. Fig. 3 shows a simulation on the basis of too. It is however necessary that activated cells are close an activator-inhibitor model. With increasing distance, to non-activated cells into which the inhibitor can diffuse the inhibitor concentration becomes lower and lower until or from which substrate is obtained, otherwise no activa- the onset of autocatalysis is triggered. A corresponding tion above average would be possible. These two seem- process is the insertion of new stomata into the largest ingly contradictory requirements, coherent patches and interstices (Bünning and Sagromsky, 1948). proximity of non-activated cells, are satisfied if a stripe- Regular structures are formed if the field grows at its like pattern is formed (Fig. 2). Each activated cell is bor- boundaries. The existing fully developed maxima allow dered by other activated cells but non-activated cells are the formation of new maxima at a well-defined distance. not too far away.

6 If initiated by random fluctuations, the stripes have a repression of alternative genes has turned out to be a random orientations too. However, any asymmetry or a common mechanism to generate stable determined states. preceding pattern forming event can be used to orient the stripes such that a regular and predictable pattern is generated. 3.2 Mutual activation and stabilization of differently determined cell types 3 Cell states and differentiation Although cells have to make a particular decision to obtain the one or the other differentiation, there are numer- 3.1 Cell determination requires autocat- ous examples in botany where different cell types coexist side by side over a large extension. Leaves are polarized : alytic (autoregulatory) genes their upper side is different from the lower one. The cells For embryonic development, the signals generated by dif- of the pith, cortex and epidermis of a stem form an or- fusible molecules are necessarily transient since the pat- dered sequence of differently determined cells. Strands of tern cannot be maintained forever in the enlarging tissue. xylem and phloem cells appear in conjunction. At an appropriate stage the cells have to make use of As mentioned above, stable cell states result from self- position-specific signals, i.e., they become determined for activation and mutual competition of genes. If two (or a particular pathway by activating particular genes. Af- more) such states not only exclude each other locally but terwards the cells may maintain this determination even activate each other over longer ranges, these cell states if the evoking signal is no longer present. The activa- need each other in a close neighborhood. The local ex- tion of a particular gene has formal similarities with the clusion insures that the two states do not merge (Mein- formation of a pattern in space. In pattern formation, a hardt and Gierer, 1980). According to this model, con- morphogenetic substance has to be produced at a partic- trolled neighborhood of structures requires the following ular location but this production must be suppressed at molecular ingredients. (i) Genes (or more general feed- other locations. Correspondingly, determination requires back loops) must exist that have a positive feedback on the activation of a particular gene and the suppression their own activation. (ii) These activities are locally ex- of the alternative genes of a given developmental situa- clusive ; only one of the alternative genes can be active in tion. Based on this analogy, it has been predicted by one a given cell. (iii) Long ranging molecules provide a mu- of us (Meinhardt, 1978, 1982) that genes exist that have tual activation of those cell states that eventually become a direct or indirect autocatalytic feedback on their own neighbors. The state of a given cell depends on the help transcription. In addition, genes responsible for alterna- from different cell states in neighboring regions. tive pathways compete with each other such that only Consider a system of two cell states, A and B, that one of the alternative genes can be active within a cell. stabilize each other. The mechanism of lateral inhibition Under control of a morphogenetic gradient, a position- is somewhat indirect. It is substituted by a help for the dependent gene activation with sharp borders is possi- competing cell state. This mechanism has also a strong ble (Fig. 5). A loss of a gene function due to a mutation ability of pattern regulation. Imagine that all B cells leads to an enlargement of the neighboring regions. Mean- are removed from a system containing normally A and while, many genes with autocatalytic properties (autoreg- B cells. Due to the lack of B cells, the B state gets a ulation) have been found in Drosophila. Examples are strong support by the too many cells in the A state but the genes engrailed, even-skipped (Jiang et al, 1991), fushi the A state is no longer supported because the B cells are tarazu (Schier and Gehring, 1992) and Deformed (Regul- missing. Therefore, A cells will be converted into B cell ski et al, 1991). Examples of autoregulatory genes in until the ratio of A and B cells is again balanced. plants are deficiens and globosa that are both involved In a two dimensional array of cells, alternating stripes in the determination of floral structures (Zachgo et al., of A and B cells are especially stable since long common 1995). On theoretical grounds it is expected that the au- borders allow a very efficient mutual stabilization. It is to tocatalysis is a non-linear process. This can result from be expected that sheets of different cell types can be sta- a dimerization of the activating molecules or by multiple bilized in the same way but this has not yet been checked binding sites on the DNA. In the Deformed gene the lat- by simulations. For instance, the upper and lower sides of ter possibility is realized. Thus, the predicted principle, a leaf may need each other to differentiate correctly and a feedback of a gene on its own activity combined with to maintain this differentiation.

7 (d) (e) (f)

Figure 5: Stages in the position-dependent gene activation by a morphogen gradient. A set of genes (1,2...5) is assumed whose products feed back on their own activation. In addition, all genes compete with each other. Only one of the genes can be active within one cell. Initially, gene 1 is turned on in every cell. The morphogen (left) is assumed to provoke the transition from one gene to the next. Each step requires a higher morphogen concentration. The result is an ordered sequence of gene activities in space. Although the positional information is graded, the activation of the genes is an all-or-nothing event. Regions with diﬀerent gene activities are sharply separated (for computational details, see Meinhardt, 1978, 1982)

The mechanism of mutual help of locally exclusive cell 3.3 Formation of filament-like branching states can easily be extended to more than two mem- structures bers. This allows the generation of a sequence of cell states in a self-regulating neighborhood. The concentri- The feedback of a locally active gene on the pattern that cally arranged cell types of the stem may arise in this way. has caused its activation, can lead to very complex struc- In this radial pattern, missing structures can be regener- tures. The formation of filament-like branching structures ated and, after confrontation of different cell types by will serve as an example. This kind of pattern is very graft experiments, missing intervening structures can be common in almost all higher organisms. The venation of intercalated (Warren Wilson and Warren Wilson, 1982). leaves, the trachea of insects, the blood or lymph vessels Recent molecular-genetic investigations (see Ingham and as well as neurons are examples. How can such complex Martinez-Arias, 1992) have provided direct support for patterns emerge ? this scheme in Drosophila segmentation (for details see We have seen how a local high activator concentra- Meinhardt, 1994). tion can be generated and how it can be used as the signal to cause stable gene activation if a threshold is exceeded. For the generation of vascular strands, it is as- Some properties of the lateral activation mechanism sumed that local activator maxima cause the differentia- should be mentioned. If the range of the helping tion of the exposed cells. The differentiated cells, in turn, molecules is such that only the direct neighbors are repel the signal. The signal will be shifted into a neigh- reached, the stripes are restricted to one cell diameter boring cell which will differentiate and become a part of (as it is the case in the wingless/engrailed expression in the vascular system, too. A repetition of this process — Drosophila). If the self-enhancing molecules are not dif- differentiation, shift of the signal, differentiation — leads fusible, the borders between the cell states are sharp. to strands of differentiated cells behind a wandering ac- They cannot be shifted if one region is relatively too large tivator maximum (Meinhardt, 1976, 1982). To obtain an and lineage restriction results (as indicated by the com- ordered venation, the shift of the signal has to proceed partment borders in imaginal disks of Drosophila). After into a direction away from other veins. This occurs if the initiation by random fluctuations different cell types ap- differentiated cells remove a substrate (e.g., auxin) and if pear in a balanced ratio with a salt-and-pepper distribu- the activation depends on this substrate. The shift will tion. The initiation of nerve cells within the ectoderm of always occur towards the highest substrate concentration Drosophila follows such a scheme (Campos-Ortega, 1988). (Fig. 6).

8 Figure 6: Formation of a net-like structure. The interaction of four substances is sufficient to generate a structure with branching filaments. A signal for the local elongation of the filament is generated by an activator-inhibitor system (black squares = activator). In this simulation the signal is used to differentiate cells (open squares). Dif- ferentiated cells remove a substrate (wavy lines). Since the activator-inhibitor system depends on the substrate, the activator maximum is shifted to that neighboring cell which has the largest distance from other differentiated cells, which is usually the tip of the filament. Branches are initiated along filaments if tips of growing filaments are sufficiently distant. The patterning process comes to rest if a certain density of filaments is reached (Meinhardt, 1976, 1982).

Branches are formed whenever activator maxima be- 3.4 How to generate structures at come sufficiently remote from each other during elonga- opposite positions of a field tion of filaments. Then, the inhibitor concentration can become locally so low that a new activator maxima is trig- Important steps in early plant development are the gener- gered along an existing vein. After removal of some fila- ation of bi-polarity preshadowing the shoot-root axis and ments, the system is able to regenerate the missing veins later the transition of the embryo from the globular into (or whatever it is) since in these regions, the substrate the heart (torpedo) stage. The two lobes of the “heart” is no longer removed. Its rising concentrations attract form later the cotyledons ; the cells in between give rise activator maxima from the non-injured region, in agree- to the proper shoot meristem. Several observation sug- ment with the observation in leaves (Jost, 1942). In the gest that cell-cell interactions play a decisive role in this model, minor fluctuations are decisive whether a branch pattern formation (see Jürgens,1995). will be formed towards the one side or the other. But a once formed vein has a strong influence on the forthcom- The formation of two structures at opposite ends of ing veins. Thus, the model accounts for the fact that all a field of cells is a very common elementary step in de- leaves of the same tree have in detail a different venation velopment. The formation of shoot and root in plants although the pattern is generated under control of the is only one example. Head and foot of the freshwater same genetic information. polyp Hydra are located in a similar manner. The analogy goes even further. In plants, the central shoot meristem is surrounded by the cotyledons. In hydra, the central organizing region is the hypostome, the opening of the gastric column. It is surrounded by the tentacles. For hydra many experimental data are available that cannot In the model, vein elongation proceeds away from ex- be obtained for plants since the latter have, for instance, isting veins. Therefore, the formation of anastomosis of no capability for regeneration and no markers are avail- veins is, to a large extend, an open problem. able to monitor an early decision towards the shoot or

9 Figure 7: Stages in the formation of two organizing structures at opposite positions of a field. For simplicity, the simulation is made on a circle. Two activator-inhibitor systems are assumed, responsible, for instance, for shoot and root development. The two activators (S, dark gray, and R, black) interact via the source density (SD, light gray). S increases SD and appears preferentially on a high SD. For R, it is the other way round. This leads to a repelling effect between S and R. A stable situation is reached when S and R are located at opposite positions (for details in the simulation, see Meinhardt, 1993). cotyledon pathway. Therefore, some of the models out- organizing property of the two coupled pattern forming lined below are inspired from observations and theoretical systems. In larger systems, this may require time. But it considerations for hydra (Meinhardt, 1993). is greatly facilitated by an existing asymmetry determin- The problem in generating structures at opposite po- ing which parts will be the winners or the loosers in this sitions is that a simple cross-inhibition to separate both competition. Thus, it is tempting to assume that with signals is insufficient. At an early stage when both struc- the first cell division an internal asymmetry is generated tures are close to each other, a cross-inhibition would (Fig. 4) that is used by such more elaborate mechanisms cause a strong competition between the two signals. But to cause differentiation into shoot and root. we know that both structures coexist even in very small plants or animals. Thus, the structures must repel each 3.5 Switch from shoot to floral meristem other without competing with each other. In hydra, a systematic asymmetry exists in the ani- In the development of flowering plants, usually an irre- mal that causes pieces of tissue to regenerate head and versible transition from a shoot to a flower meristem takes foot according to the original polarity. This intrinsic po- place. The precise time of this transition depends on the larity has a long time constant. It requires about 2 days age of the plant and on environmental conditions. The for polarity reversal while it takes about 6 hours to re- concept of source density used in the simulations above al- generate the head signal (Wilby and Webster, 1970). We lows straightforward molecular interactions that have this have shown that many experiments can be accounted for switching property. For the shoot meristem, we assumed under the assumption that the head signal has a posi- an activator-inhibitor system. For the flower-meristem, tive and long ranging feedback on the general ability of we assume a second activator F that produces and reacts the tissue to generate a head signal, a parameter we have to the same inhibitor. This has the consequence that called the source density (ρa, see Meinhardt and Gierer, both system mutually exclude each other. At a partic- 1974 ; Meinhardt, 1993). In other words, the head signal ular region either the shoot or the flower activator can creates in its environment a milieu that facilitates head have a high concentration. To achieve the transition, we formation. For instance, the foot could preferentially ap- assume that the shoot activator has a small activating pear at the lowest source density and have a decreasing influence on the flower activator. If the flower-activator effect on it. As shown in the simulation (Fig. 7) even if has a stronger dependence on the source density and the no initial asymmetry is present, the two emerging signals latter increases in a gradual fashion, at a certain level an shift until they reach the maximum distance. abrupt transition from the shoot- to the flower-activator In the simulation presented in Fig. 7, random fluctua- takes place. As shown in Fig. 8, this transition can be tions have been assumed for pattern initiation. The oppo- irreversible. The flower activator remains active even if site location of the two signals is accomplished by a self- the source density drops to a sub-threshold level.

10 Figure 8: The irreversible transition from a shoot to a floral meristem. Both systems are assumed to be under control of specific activators that are antagonized by the same inhibitor. The floral meristem is slightly activated by the shoot meristem but depends more strongly on the source density. The latter is assumed to increase with time. If a certain level is reached, an irreversible activation of the floral activator and a suppression of the shoot activator occurs.

4 Phyllotaxis tion and possible coupling between the meristem and the leaf initiation system will be discussed. Signals for leaf The investigation of the positioning of new leaves on a initiation are usually assumed to have a point- or patch- growing shoot has a long history. The idea that exist- like extension. However, the leaves are flat ; how is this ing leaf primordia inhibit the formation of other ones is flatness achieved, starting from such a point- or patch- old (Schoute, 1913). Also in more recent models (Thorn- like signal ? On many figures in textbooks showing the ley, 1975 ; Marzec and Kappraff, 1983 ; Koch et al, 1994) extension of leaf primordia it is clearly visible that they some sort of long range inhibition plays an essential role. have a long extension around the circumference of the This inhibition mechanism is easily realized in a merely shoot cylinder but only a small extension along the axis. chemical context by introducing several diffusing chem- Does the long-range inhibition have different ranges along icals affording suitable auto-catalytic reactions (Mein- the two axes ? In a third step, a case will be made that hardt, 1982 ; Bernasconi, 1994a, 1994b). Patterns closely leaves are initiated at differentiation borders and that the similar to those observed in phyllotaxis can be generated cells of a primordium have therefore from the beginning by purely physical ingredients (Levitov, 1991). For in- a different differentiation at their upper and lower side ; stance, swimming droplets of a magnetic fluid obtain a growth can then be restricted to that border. In a fourth spiral arrangement due to their mutual repulsion (Douady step we shall show that not only the inhibitory influence and Couder, 1992). of earlier formed leaf primordia determines the position However, if a leaf primordium is able to inhibit leaf of a new leaf but also that some sort of long term mem- initiation in its vicinity, why — being in the center of inhi- ory exists in those cells that descent from cells that had bition — does it not inhibit itself ? This obvious questions previously formed a leaf primordium. has rarely been asked. As outlined above, our answer is that the local self-enhancement is so strong that it can cope with this high inhibition. 4.1 Spacing of leaves by an activator- In this section we shall discuss phyllotaxis in the view inhibitor mechanism of models for pattern formation as outlined above. We will show that an approach can be made in several steps. The domain at which the formation of leaves takes place is First we show that the activator-inhibitor model on its essentially the upper edge of the growing stem idealized as own can account for the basic patterns. In such a simple a cylinder. It corresponds, in the terminology of Barabé model, the meristem, the region at which new cells are (1993), to an open system : new cells are continuously inserted is presumed to exist. But in fact, this requires added at the upper edge of the cylinder which can grow another patterning system that defines the position and indefinitely. The same model can be adapted to repro- the extend of meristem. So, in a second step, the genera- duce phyllotaxis of simple closed systems (where the pri-

11 Figure 9: Simulation with the activator-inhibitor model (1) in a two dimensional cylindrical domain. New cells added at the upper edge of the cylinder at regular time intervals δt. The cylinder is represented unwrapped (the vertical lines x = 0 and x = 1 have to be identified) and the darkness is proportionnal to the concentration of the activator a. Depending on the parameters, one gets distichous (a), decussate (b) or spiral (c) phyllotaxis. The three simulations started from homogeneous initial conditions perturbed by random fluctuations ; after a transient phase (lower part of the pictures), the system reaches a stable dynamical regime. The space along x is divided into 50 cells. The parameters used to produce these pictures are ρa = 1.0, ρh = 1.0, µa = 4.0, µh = 8.0, σa = 1.0, σh = 0.0 . The diffusion constant and time interval δt used are : (a) Da = 0.001, Dh = 0.2 and δt = 0.5 ; (b) Da = 0.0003, Dh = 0.05 and δt = 1.0 ; (c) Da = 0.001, Dh = 0.08 and δt = 0.5 . mordia are arranged on a continuous and closed surface) : be reproduced. In Fig. 9a, the peaks emerge separated by the regular phyllotactic arrangement is then achieved by a divergence angle d equal to 180o : this is an example for assuming that a morphogenetic wave moves across the a distichous phyllotaxis. If the activator maxima appear system. Cells in front of the wave are in an undifferen- in pairs, each new pair being rotated by 90o relatively tiated state and pattern formation begins in the narrow to the preceding one, one speaks of decussate phyllotaxis region determined by the position of the morphogenetic (Fig. 9b). The pattern of Fig. 9c is probably the most wave (“simulated growth” see section 2.4). However, in interesting one. The leaves are arranged on conspicuous this way, only simple closed systems can be produced, i.e., helices, the parastichies. Two equivalent helices turning systems with small parastichy numbers like (3, 5) or (5, 8). in one direction and three in the opposite one are clearly To produce structures with large parastichy numbers like visible, producing a (2, 3) phyllotactic pattern ; here the (55, 89), one needs more refined models (according to the divergence angle d is equal to 137o. Fundamental theorem of phyllotaxis (Jean, 1994), paras- The Fundamental theorem of phyllotaxis provides a tichies (55, 89) are visible and opposed if the divergence d link between the divergence angle d and the numbers lies in the range 137.45o < d < 137.53o ; this requires that (m, n) of opposed conspicuous parastichies : m and n the positioning precision is better than 0.1o ! A mecha- correspond to denominators of two successive principal nism able to achieve such a precision is discussed by Koch convergents of d/360o [see Jean (1994) or Koch et al, this et al, in this volume). volume].

The simulations presented in Fig. 9 are based on the 4.2 Local exclusion and long range help : activator-inhibitor system (1). As already shown in Fig. 3, a new activator maximum will be inserted whenever the The shoot meristem enables leaf ini- distance to the existing maxima exceeds a threshold. tiation in its vicinity Thus, stable activator peaks appear at the upper edge The simple simulations shown in Fig. 9 contain two im- of the growing cylinder. Each peak is supposed to induce plicit assumptions that need further elaboration. the formation of a new leaf primordium. As the cylinder grows, the peaks “recede” from the upper region, provid- • First, it is assumed that new cells are added only at ing space for new maxima. The resulting patterns of leaf the upper end of the cylinder. Thus, a more com- signals are regular. The main phyllotactic structures can plete simulation requires a pattern forming system

12 that is responsible for the shoot meristem. The lat- molecule that is produced by the shoot meristem system ter is assumed to be under the control of a separate and which has an activating influence on the leaf system. activator-inhibitor system. The generation of the The simplest version would be that the shoot meristem shoot-root axis has already been discussed above inhibitor can partially mimic the leaf inhibitor, displacing (Fig. 7). the latter molecules from their binding sites. In this way, the meristem inhibitor has an activating influence on the • Moreover, in the simulation of Fig. 9, it has been leaf system. With respect to the leaf system, the shoot assumed that the distances between once initiated meristem must have the opposite properties than those leaves do not enlarge by cell division or cell elonga- required for its own generation : a short range inhibition tion. Such a growth would lead to the insertion of (to prevent leaf initiation in the meristem) and long range further leaves between the existing leaves, in con- activation (to enable leaf formation at the border of the trast to the situation in the real plant. meristem). The simulations shown in Fig. 10 are again inspired by We shall now introduce a system for the apical meristem the observations with hypostome-tentacle relation in hy- which will have two functions, first to define an adja- dra (Meinhardt, 1993). Shown is the generation of a pair cent zone in which cell proliferation can take place and of leaves L (or, more precisely, signals to form leaves). secondly, to create a region to which leaf initiation is re- Continuing the shoot-root models given in Fig. 7, it is as- stricted (Koch and Meinhardt, 1994). sumed that the (central) shoot meristem system S has a In contrast to the problem discussed above — long-ranging influence on the source density SD. A high formation of two structures at the largest possible dis- source density is the precondition for leaf initiation. The tance — the formation of shoot meristem and leaf initia- inhibitory cross-reaction of the shoot activator onto the tion is an example for the opposite type of pattern forma- leaf activator insures that the leaves cannot emerge in the tion : formation of two structures close to each other. It meristem region itself although the source density is the is again a very general process in pattern formation. As highest there. mentioned, the hypostome of hydra with the surrounding ring of tentacles is another example. The initiation of The models provides an explanation why the leaf initi- two structures in a close neighborhood will take place if a ation zone is so sharp. The source density is graded. Since primary structure generates on longer range the precon- the pattern formation mechanism is based on a competi- dition for the other but locally excludes it. tion, only those cells that have the highest value of the Such a dependence of one pattern onto another can source density but are exposed to a sufficiently low con- be realized in several ways. The example outlined be- centration of the shoot activator can produce new leaves. low should illustrate that realistic molecular interactions In other words, the competition among the cells for pro- can be invented that are able to do this. Let us assume ducing leaves restricts their position next to the meristem. two pattern forming systems, each under control of an The cells leaving the meristem have to change their activator-inhibitor system, S for the shoot meristem and differentiation from meristematic to non-meristematic L for leaves. cells. This feature is correctly described by use of an Obviously, the meristem is required only for the initia- activator-inhibitor system for meristem formation. The tion of the leaves, not for their maintenance. This can be shape of the activator maximum is self-regulating. By modeled in two ways. Either temporary signals are gen- proliferation of cells within the peak, the peak does not erated in the initiation zone that activate, in turn, self- become broader. Instead, if cells reach the margin of the stabilizing genes (see Fig. 5). Alternatively, the pattern activator peak, they become desactivated. forming system L is in a resting state and the meristem By breeding and selection, plants that normally activates it by shifting it over a threshold. At present, form two cotyledons can become genetically modified so the experiments do not allow a decision. that three cotyledons are formed with high probability The following example for a possible realization is of (Straub, 1960). In terms of the model, the three cotyle- the second type. As mentioned above, at low activator dons indicate that either the range of the mutual inhi- concentrations the spontaneous activation can be pre- bition of the cotyledons decreased or the forbidden zone vented by a certain baseline inhibitor production. The around the shoot meristem increased, causing in this way shoot system has to bring such a resting leaf system an enlargement of the distance between the cotyledons. into the patterning mode. This requires a long-ranging The simulation of Fig. 10e shows that this is not neces-

13 Figure 10: Generation of neighboring structures : Signals for shoot meristem and leaves. (a) Stages in the generation of the signal for the shoot meristem (S, top) and the leaves (L, centre). By local autocatalysis and long range inhibition, a high S signal appears where the source density SD (bottom, see also Fig. 7) has its highest level. The source density increases under the influence of S until the L system triggers. This occurs first in a ring that subsequently desintegrates due to the competition. (b) The result is a central S signal surrounded by two L signals. (c, d) A somewhat broader S signal may lead to a larger ring-shaped L activation that may decay into three L signals. (e) Starting with a higher source density, the sequence of events may be the reverse : First a central L signal is formed. Under the influence of the somewhat later formed S signal the L signal is displaced and desintegrates into several maxima.

14 sarily connected with a change in the leaf system. A grated into this model by assuming that the bud-inducing broadening in the S distribution (for instance, due to a signal consists of the leaf signal plus a m1 specification. saturating autocatalysis) increases the size of the ring in The restriction of leaf initiation to a differentiation which leaf formation is possible and thus the distances be- border accounts in addition for several features that re- tween the leaf primordia. This is sufficient for a change main unsolved in other models of phyllotaxis. For in- from two to three L signals. stance, many plants form whorls. In whorls, the individ- ual leaves have a small distance from each other around 4.3 Segmentation in plants the stem circumference while the distance between the whorls can be large. This different spacing cannot result Leaf morphogenesis has been extensively studied by Cus- from a simple lateral inhibition mechanism. According set (1986). He shows for instance that the antero- to the model outlined above the leaves can only appear posterior polarity is fixed very early in the leaf devel- along the m1/m2 border. This determines where a whorl opment. Our aim, in this section, is to provide a model can be initiated. Thus, the distance from one whorl to the explaining how leaf polarity is established. Our system next is given by the repeated length of the nodal organi- is based on the interaction of biochemical substances and zation of the shoot, the . . . m3/m1 m2 m3/m1 ... pattern. on cell differentiation ; let us however mention that mod- In contrast, the spacing of the leaves within the whorls is els exist which explain the same facts by purely physical given by the range of the inhibition, and this can be very arguments (Green, 1992) short. Segmentation, the reiteration of polar units along the In monocotyledons, the width of a leaf may be a large body axis, is usually regarded to be involved only in an- fraction of the circumference of the stem while it has only imal development. In contrast, the spacing of leaves is a small thickness. A signal generated by an activator- mostly assumed to result from a long-ranging inhibitory inhibitor mechanism would have a more or less circular effect of the elder leaf primordia onto the formation of the shape. According to the boundary model, however, the subsequent primordia (Schoute, 1913). However, there thickness is given by the leaf formation mechanism at are several features of leaf initiation that cannot be ex- the border while the extension around the circumference plained by such a simple spacing model. Shortly after depends on the pattern forming system. Since both pro- initiation, the polar structure of leaves becomes obvious cesses are independent, the different extensions of the leaf and the leaves become flat. Their upper and lower sur- in both directions are easily described. faces obtain different features. This polarity is always correctly oriented with respect to the axis of the growing Many plants form leaves that consist of leaflets along a shoot. Moreover, in many plants axillary buds are initi- central stem. The acacia is an example. According to the ated close to a leaf at a position pointing towards the tip model, the m1/m2 border would be maintained in the of the shoot. How is this achieved ? In most models of outgrowing leaf stem. New signals (activator maxima) phyllotaxis, these features are not considered. can be generated along the stem on the m1/m2 border. Recently, we have shown that a mechanism analogous Therefore, the leaflets necessarily appear in a plane. to segmentation in animals would resolve these problems Recently, Waites and Hudson (1995) described the (Koch and Meinhardt, 1994). In the simulation shown gene phantastica that is required for dorsoventrality of in Fig. 11 it is assumed that during outgrowth a periodic leaves in Antirrhinum majus. In its absence, outgrowth sequence of (at least) three cell states is generated, to still takes place but needles are formed instead of leaves. be called m1, m2, and m3. They are arranged like belts They proposed an early dorsoventral subdivision shortly around the shoot. During growth, the repetitive structure after the determination of the primordial leaf cells. In . . . m1 m2 m3 m1 m2 m3 ... is laid down on the stem. The our view, the sequence of events is the reverse. The acti- leaf primordium is generated by an activator-inhibitor vation of phantastica corresponds to the activation of the mechanism as described above, but an additional condi- m1 belt and is expected to precede leaf initiation. This tion is imposed : a primordium can only appear on a par- sequence corresponds to the primary formation of the an- ticular border, say the m1/m2 border. The emerging leaf terior/posterior compartment border in insects that pre- is then build up with two different tissue types, m1 and cedes the formation of imaginal disks. The similarity is m2, and both cell types have necessarily the correct orien- even more striking in short germ insects with their zone of tation with respect to the apical meristem. The initiation proliferation at the posterior pole (corresponding to the of an axillary bud at the correct position is easily inte- apical meristem) : one anteroposterior border appears af-

15 Figure 11: The modular construction of a plant and its simulation. (a) Cross-section through the growing tip of a shoot. The apical shoot meristem A is a tissue in which rapid cell division occurs. At its periphery the primordia P appear that will grow into leaves L. Axillary buds B diﬀerentiate somewhat later, in proximity of a leaf. The shoot can be regarded as a periodic repetition of an “elementary module” M formed by a node N and internode I region ; every nodal-internodal segment bears a leaf L and an axillary bud B. In our model it is assumed that each module M results from the iteration of (at least) three subunits, m1, m2 and m3. It obtains in this way an intrinsic polarity. (b) Simulation of plant growth. The stem of the plant is idealized as a cylinder that is represented here unwrapped. The apical meristem A contributes to the stem elongation by addition of new cells. These diﬀerentiate so as to produce the repetitive sequence . . . m1 m2 m3 m1 m2 m3 ... indicated here by three grey levels in the background. The m1/m2 border acts as a precondition for the formation of a signal inducing leaf initiation. Due to lateral inhibition, leaves are placed along spirals with a (2,3) phyllotaxis.

ter the other and their orientation is perpendicular to polar character of one of the resulting boundaries is used the direction of growth. Different in both systems is the to generate the polar structure of the leaf ; this is similar actual positioning of the initiation site along the crucial to the compartment borders that generate the precondi- border. According to our views, in leaves this is deter- tion for the generation of a polar limb (Meinhardt, 1983, mined by the autocatalysis/lateral inhibition mechanism. 1986b). Imaginal disks are positioned at the intersection of the anteroposterior border with a second border at a particular dorsoventral position (Meinhardt, 1983, 1986). There- 4.4 The role of a long term memory in fore, while leaves may produce a spiraled arrangement, phyllotaxis the imaginal disks emerge at a particular dorsoventral level. In the previous sections we proceeded from simple to more complex models of phyllotaxis, taking step by step into The model proposed is related to the node-internode consideration the role of the meristem and the nodal na- concept (Lyndon, 1990) according to which the leaves ture of shoot growth. are derived only from the nodal regions. Different plants Although the mechanism of local self-enhancement use different strategies to generate this periodic pattern. and long range inhibition is able to reproduce essential While in Sambucus a single cell layer gives rise to both features of phyllotaxis, thorough numerical simulations structures (Zobel, 1989a, 1989b), in Silene four layers of show that it is not yet sufficient to give a satisfactory de- cells are associated with each leaf, two of them form the scription. The emerging structures are not stable against nodal and two the internodal cells (Lyndon, 1990). The perturbations. For instance, small additional fluctuations main difference of the model we propose is that an alter- of a and h can cause a transition from a spiraled to a de- nation of (at least) three elements is required and that the cussate pattern. Moreover the results depend strongly on

16 the number of cells forming the meristem border. At last, acts as a long lasting memory of the positions of former despite numerous attempts, the spiral phyllotaxis with activator maxima. A new activator peak will be trig- divergence angle close to 99.5o has not been observed in gered as soon as the combined inhibitory action of h and the numerical simulations based on the simple activator- s sinks below a certain threshold. Fig. 12 shows a spatio- inhibitor model. temporal plot of the memory field s. The time delay In the following, we show that these problems disap- between two consecutive peaks, the plastochrone T , and pear under the assumption that not only the leaf primor- their angular distance, the divergence angle d, converge dia inhibit each other but that also some sort of long term rapidly to constant values. memory exists in the progeny of those cells that formed The system described by equations (3) exhibits all once a leaf primordium (Bernasconi, 1994a, 1994b). the essential features of phyllotaxis (Bernasconi, 1994b). To simplify the modeling we regard now only the (one- The process is robust and stable with respect to random dimensional) ring of cells next to the meristem in which fluctuations. The regular phyllotactic pattern is rapidly pattern formation takes place. In the two-dimensional achieved even if the system starts from random initial model (1) discussed above, the inhibitory influence of the conditions or from two opposite maxima representing the receding peaks on the apical meristem vanishes gradu- two cotyledons. After a strong perturbation, the system ally as their distance to the tip increases. In the one- restores a normal pattern, passing through a transient dimensional system, the decreasing effect of elder leaves phase in which the system searches for a dynamically sta- on the apical meristem is implemented by a pulsating ble regime (see next section). activator-inhibitor system. Such oscillations take place The characteristic parameters of the phyllotactic pat- if in (1) the activator decays more rapidly than the in- tern, the divergence angle d and the plastochrone T hibitor (i.e., if µa > µh). Soon after formation of an depend on the molecular interpretable parameters of activator peak, it vanishes again due to the accumulat- Eq. (3), i.e., the decay rates, diffusions coefficients, the ing inhibitor. It leaves behind a long lasting trace of its source term and the saturation constant. In Fig. 13, d ephemeral existence in the form of a slowly vanishing in- has been plotted as a function of the diffusion coefficient hibitory substance. Ds of the long lasting inhibitor s. For large values of Ds, To simulate the phyllotaxis on a ring of cells we em- only distichous phyllotaxis is obtained (d = 180o). For ploy two inhibitors (Bernasconi, 1994b ; Bernasconi and lower values of Ds, the distichous phyllotaxis becomes Koch, 1995). The first, h, has a long range as discussed unstable, leading to a first bifurcation. For a given value above. It accomplishes the lateral inhibition. The sec- of the parameter Ds, several distinct values of the di- ond inhibitor, s, has an even longer time constant and vergence are sometimes possible. Which of the possi- only a moderate diffusion (if any). It describes the fad- ble pattern is actually obtained depends on the initial ing inhibition of earlier primordia. The following set of concentration of the three substances a, h and s. Tak- equations provides an example of such a double inhibitor ing smaller and smaller values of Ds, more and more mechanism. alternative values of d appear. The two main branches o converge toward the golden angle ΦG ≈ 137.5 and to- o ∂a ∂2a a2 ward the Lucas’ angle ΦL ≈ 99.5 . These two angles can = D + − µ a + σ (3.a) o −1 a 2 a a be respectively written as ΦG = 360 · 1/(2 + τ ) and ∂t ∂x h(s + κa) √ Φ = 360o · 1/(3 + τ −1) where τ = ( 5 + 1)/2 is the ∂h ∂2h L 2 golden mean. Other branches exist in this diagram which = Dh 2 + a − µhh (3.b) ∂t ∂x all converge toward noble numbers, i.e, to numbers which 2 ∂s ∂ s can be written as (a+bτ −1)/(c+dτ −1), with |ad−bc| = 1. = Ds 2 + a − µss . (3.c) ∂t ∂x The increasing complexity of this branching tree for To understand how the system (3) works, consider a decreasing values of Ds is understood in the following nearly homogeneous initial state. Due to autocatalysis way. The less s diffuses, the weakest is the “repulsion” of and lateral inhibition, the activator-inhibitor interaction the activator peaks and the more peaks can be placed in leads to one ore more activator peaks. Since µa > µh, the sensitive ring of cells below the apex. these peaks are unstable and disappears fairly rapidly. This model and the model considering the nodal na- However, these temporary activator peaks cause local and ture of leaf initiation outlined in the previous section long lasting increase of the second inhibitor s. The latter might appear very different. However, in both models

17 Figure 12: Phyllotaxis simulated on a ring of cells with two different inhibitors according to the model (3) ; the ring is unwrapped. Shown are space-time plots of the inhibitor s responsible for the “memory” ; (a) shows the rapid convergence toward a (2, 3) phyllotaxis out of random initial conditions (divergence angle d ≈ 138o). In (b) the same (2, 3) structure is obtained out of different initial conditions : an initial state with approximately bilateral symmetry simulates the presence of two cotyledons. (c) Here, a (1, 4) phyllotaxis (d ≈ 99o) is achieved. The parameters used for (a) and (b) are µa = 30.0, µh = 10.0, µs = 1.0, σa = 1.0, κa = 0.05, Da = 0.300, Dh = 4.0 and Ds = 0.001. The ring is divided into 100 cells. The simulation in (c) differs only for κa = 0.0 and Da = 0.06. the inhibition around the circumference of the sub-apical the stem causes a slight variation in the positions of the region and the inhibition from elder leaves are accom- subsequent primordia (Snow and Snow, 1933). However, plished by separate mechanisms. However, the equiva- these disturbances vanish gradually and the divergence lence of both models is not yet fully explored. angles converge to the normal value of 136.3o. In similar A process formally very similar to the spiral phyl- experiments the region has been removed in which the lotaxis can be observed in the pigmentation pattern on next primordium is expected to appear. The subsequent the shells of some tropical snails. Rows of dots are formed regulation proceeds in a similar way but occasionally a re- along oblique lines or at staggered positions. Fig. 14 pro- versal of the winding direction of the genetic spiral occurs. vides two examples. As happens in plants, shells enlarge As shown in Fig. 15 the restoration of the original diver- by growth restricted to a narrow zone. The once gener- gence is reproduced by the model including the possible ated pattern is fixed and thus represents a time record of direction reversal. In agreement with the experiments, the events that took place at the (linear) zone of pattern the results are to a large degree independent of the size formation. The model for shells has been developed in- of the excised regions and of the precise position of the dependently (Meinhardt, 1995) but it is almost identical cuts (Bernasconi and Koch, 1995). to the model of phyllotaxis described above. This shows The excised region itself produces sometimes a regu- once again that mechanisms in pattern formation can be lar phyllotactic pattern, if a critical size is exceeded. In very universal. agreement with the theoretical expectation, due to the small field size the first primordia have a 180o spacing. 4.5 Altered phyllotaxis after experimen- As the excised domain grows up to its normal size, the divergence angles converge to the normal value of 136.3o tal interference (Fig. 15). In their experiments, Snow and Snow (1933, 1935) per- In Epilobium hirsutum, leaves appear in alternating turbed normal phyllotaxis and observed the subsequent pairs (decussate phyllotaxis). In a related set of experi- regulation. In the following, we show that our model ac- ments, Snow and Snow (1935) split the apices by a vertical counts for their observations. cut 45o to the plane of leaf pairs. Such an apex is thus Lupinus albus is a plant with a (2, 3) phyllotaxis and subdivided into two nearly symmetric parts. Most of the a divergence angle d close to the golden angle (d = split apices regenerated and developed spiral phyllotaxis. 136.3 ± 1.88o). An isolation of the most recently formed A minority regenerated a decussate pattern or produced primordium by a tangential cut parallel to the axis of other arrangements showing, for instance, two fused pri-

18 Figure 13: Divergence angles d observed in model (3). The divergence angle d in the stable regime is plotted as a function of the diffusion constant Ds. For small values of Ds, the stable regime is not unique ; for a given parameter o set, different initial conditions can lead to patterns on the golden branch (ΦG ≈ 137.5 ) or on the Lucas branch o (ΦL ≈ 99.5 ). The reaction-diffusion system (3) has many other branches, but they are too short to be visible on this diagram. Only divergence angles relative to spiral patterns have been reported. For the simulations of this diagram we have assumed that the inhibitor h diffuses instantly in the system. This simplifies the computations (see Koch and Bernasconi, in this volume, or Bernasconi, 1994b). The parameters used here are µa = 30.0, µh = 15.0, µs = 1.0, σa = 2.0, κa = 0.25 and Da = 0.08. The meristem border contains 100 cells.

(a) (b)

Figure 14: A patterning formally analogous to phyllotaxis on the shell of a snail. Shells enlarge by growth restricted at an edge. Thus, like in phyllotaxis, the pattern is a time record of events on the growing edge. The model is based on the assumptions of one autocatalytic substance and two inhibitors nearly identical to (3). The diﬀusion of the long-lasting inhibitor determines whether activation appear in staggered positions (a) or along oblique lines (b); activator is black, long lasting inhibitor is gray (from Meinhardt, 1995).

19 Figure 15: Simulation of primordium isolation. (a) The original divergence angle is nearly equal to 136.3o ; the region where a primordium is about to appear (between the arrows) is removed from the apex. (b) The cut apex regenerates the missing part ; after a short transient phase, the divergence angle converges again to 136.3o. (c) The small removed part of the meristem also regenerates, producing a pattern with divergence angle equal to 136.3o. Note that the structures in (b) and (c) have opposite winding directions.

(b) (c)

(a)

Figure 16: Regulation of the phyllotactic pattern after a split of a decussate apex. (a) The initial structure is decussate. An incision is made on the apex so as to produce two symmetric halves (arrows show the points where the meristem border is cut). (b) and (c) In agreement with the results of Snow and Snow (1935), a spiral phyllotaxis emerges on each half-apex although the initial size of the meristem is restored. mordia. These results demonstrated that the divergence This is remarkably similar to an observation of Snow and angle is not based on an intrinsic property of a given plant Snow : they report that some split meristems produced species but depends on the size of the apex and on the a single leaf and two fused leaves in alternating sequence dynamics of growth. (Snow and Snow, 1935). We simulate the splitting experiments by separating an initially decussate apex into two symmetric or nearly symmetric parts. Each half evolves then independently 5 Conclusion and the two meristems grow up to their normal size. The simulations are in good agreement with the observations. The general principles of pattern formation — local auto- In each of the two regenerating apices, the phyllotaxis catalysis and long ranging inhibition — provide an expla- evolves in most cases to Fibonacci spiral phyllotaxis. De- nation of many basic observations in plant patterning : pending on the time and position of the cut, the spirals the generation of polarity as required for the early em- of the two apices may wind in the same or in opposite di- bryo or the initiation of structures with a somewhat ir- rections (Fig. 16). Occasionally, the cut apex restores its regular spacing as observed in stomata. To account for original decussate phyllotaxis. This is especially frequent more complex patterns, a superimposition of several pat- when one fragment is substantially larger than the other. tern forming reactions must be involved. The initiation of In our simulations we have also observed a periodic alter- structures at opposite positions of the ﬁeld (shoot-root) nation of a normal primordium and two fused primordia. or at a close neighborhood (meristem-leaf primordia) can

20 be accomplished in this way. Coen S. and Meyerowitz E.M. (1991) : The War of Whorls : Genetic Phyllotaxis has been approached in several steps. The Interactions Controlling Flower Development. Nature 353, 31- 37. activator-inhibitor model on its own accounts for the basic Cusset G. (1986) : La Morphogenèsedu Limbe des Dicotylédones. spacing patterns. The addition of a separate meristem- Can. J. Bot. 64, 2807-2839. atic system allows generation of a sensitive zone in which Davidson D. (1983a) : The Mechanism of Feather Pattern Devel- leave initiation can take place. The separation of the in- opment in the Chick. I. The Time of Determination of Feather Position. J. Embryol. Exp. Morph. 74, 245-259. hibition around the circumference and of earlier formed Davidson D. (1983b) : The Mechanism of Feather Pattern Devel- leaves makes the phyllotactic pattern more robust and ac- opment in the Chick. II. 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