The regular initiation of leaf (or seed) primordia behind the tip of a growing shoot, called phyllotaxis, seems to have nothing in common with any pattern on sea shells, but this impression is misleading. As in shells, also on a growing shoot, new pattern elements appear in a narrow zone that shifts in the course of time due to growth. The tip of the shoot, the so-called meristem, consists of undifferentiated, rapidly dividing cells [6,7]. Only cells just leaving this zone are able to form new leaves. According to classical models, the initiation of a new leaf is inhibited by existing leaves  Therefore, a new leaf can be initiated only at a certain distance from the last formed leaf. In this way, a certain distance is maintained between the sites of leaf initiation.
In many plants, leaves are initiated along spirals. Seeds on fire cones have a corresponding arrangement. Such patterns result if not only the last, but also the next-to-last leaf or seed primordium has a repelling influence on initiation of a new one. Simulations with the standard activator-inhibitor model revealed that this mechanism cannot generate this pattern in a reliable way. The reason is that the inhibitor has to diffuse very rapidly. Thus, the inhibitor produced by the pen-ultimate leaf does not provide a reliable cue to orient the new leaf. There is a tendency to generate either an alternating (distichous) or pair-wise, 90°-rotated (decussate) patterns.
The simulations of the sea shell pattern have suggested a solution: the involvement of two inhibitory reactions. Again, a long ranging inhibitor is responsible for the pattern in space. A second inhibitor poises once formed activator maxima, causeing their breakdown. After a certain period they reappearance at a displaced position.This is illustrated in the simulation above right. After the trigger of an activation (black), one of the inhibitors (red) spreads rapidly. The other (green) spreads only slowly but remains for a longer period. New activations are possible only when the concentrations of both inhibitors become low enough (white regions).. For phyllotaxis this suggest that not the previously formed leaves but some sort of memory within the leaf forming zone is responsible for the positioning of the next leaf. In this model, the spacing around the shoot and the spacing along the shoot axis is based on different antagonists. The first spreads rapidly within the leaf forming zone, restricting the extension of the primordia. The second remains more local but his long time constant is responsible that after the initiation of a new leaf, this angular position is no longer available for a certain period.
In the simulation below, the pattern formation in the leaf forming zone is shown together with the pattern that results in the course of time. The concentration of the leaf-initiating signal is given in brown. It appears at position where the inhibitory memory (green) is least. The long ranging ihibitor (red) localizes the signal. In a wide range of parameters, there is a tendency to initiate leaves with an angular spacing of 137°, the golden angle. At larger circumferences of the leaf- or seed-initiation zone, helical arrangements are possible as well.
It is clearly visible that after the trigger of an activation (black), one of the inhibitors (red) spreads rapidly. The other (green) spreads only slowly but remains for a longer period. New activations are possible only when the concentrations of both inhibitors become low enough (white regions).If the second antagonist has a moderate diffusion, alternating or pair-wise, 90°-rotated patterns are possible as well (Meinhardt et al., 1998 ).
Recent observations have shown that the plant hormone auxin is required as a prerequisite in leaf initiation . Suppression of auxin transport suppresses leaf initiation, a situation that can be rescued by a local supply of auxin. This suggests that one of the expected antagonistic reactions results from a depletion of auxin in a wider surrounding. The figure below shows snapshots of a corresponding simulation. Auxin is shown in gray, the long lasting inhibition in red. The upper row shows the coming and going of a signal to initiate a primordium. A new primordium appears when auxin concentration is high enough and the inhibitor low enough. The primordium remains localized due to a depletion of auxin in a wider region. The blue arrow points to the site of the least concentration of the long-lasting inhibitor, foreshadowing the position where the next leaf primordia will arise.
The bottom row shows the initiation of the two subsequent primordia at displaced positions. In the simulation it is assumed that the spread of auxin is accomplished by diffusion. A more realistic model has to include its active transport.
The displacement by the golden angle is a stable configuration attained in a wider range of parameters and interactions. The simulation below is based on the same interaction as used above. As used in many botanical textbooks, the position of the activated regions are plotted on circles with shrinking radius, the most recent primordia are close to the center. The angle drawn has 137,5°. As the simulation shows, the angular separation of two successive primordia is close to the golden angle.
Leaves are flat. To account for this we have proposed long time ago that leaves are generated at a border that separates differently determined cells , in analogy to the initiation of legs and wings at compartment borders. In the simulation shown below, a regular alternation is assumed to take place at the meristem, leaving behind a periodic sequence of three cell states. If leaves can be only initiate at a particular border (e.g., m1/m2), each leaf has automatically an upper and lower surface. Shortly thereafter the first gene restricted to one leaf side has been found . Meanwhile many of the components ad/abaxial patterning are known .
More details can be found in Meinhardt, H., Koch, A.J. and Bernasconi, G. (1998). Models of pattern formation applied to plant development. In: Symmetry in Plants, (D. Barabe and R. V. Jean, Eds), World Scientific Publishing, Singapore , pp. 723-758. [Preprint PDF ]
A program that allows the formation of a helical arrangement of a leaf-forming signal can be found here.