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 in- hibition 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) distribu- tions. (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.