A theory of the regulation of phyllotaxis based on Lupinus albus

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Abstract
A theory is proposed to account for the exact regulation of phyllotaxis systems. It is consistent with a space-filling or packing interpretation of phyllotaxis, and is based on evidence showing that, at least in Lupinus albus, each leaf base is determined over a primary area which covers a constant are of the circumference of the apex, and then extends further secondarily before its turn comes to act as a contact member for younger leaf bases above. The theory is that the secondary extension of a leaf base varies as the divergence angle between its two contact leaves below, and that this gradually regulates a disturbed system. Evidence of this correlation is offered, and its cause and imperfections are discussed. To test the theory, a first diagram is constructed showing a regular phyllotaxis system with contacts 2 and 3 and with divergence 137$^\circ$, as in L. albus, and with the leaf bases represented by circles as if on the surface of an unrolled cylinder. An inner circle represents the primary area of each leaf base and an outer circle its area when later it acts as a contact member for younger leaves. Two further diagrams are then constructed which at the base are similar to the last, but higher up show a leaf that has been displaced 20$^\circ$ laterally in either direction by splitting vertically its presumptive area, as was done in the experiments on L. albus. From this level onwards the diagrams are continued in accordance with space-filling principles, by inserting the constant primary area of each leaf base in its turn into the lowest available space between the older ones and then surrounding it with a concentric zone of secondary extension proportional in width to the divergence of its contact leaves below. The diagrams return towards the previous regular arrangement about as quickly as the comparable actual apices, and finally become more regular than a normal apex. It is therefore claimed that the theory provides an escape from the chief difficulty confronting a space-filling interpretation of phyllotaxis. The changes of phyllotaxis in the more severely disturbed apices are described and interpreted, including several changes to spirals with four leaves in a turn. The shapes and development of the leaf bases are discussed in relation to regulation in this and other species.

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