The origin of fibonacci phyllotaxis—an analysis of Adler's contact pressure model and Mitchison's expanding apex model

Abstract

Adler's contact pressure model for Fibonacci phyllotaxis is examined theoretically. It is shown that the model, as it stands, does not account for Fibonacci phyllotaxis, since it requires, but does not provide, a mechanism for initiating new primordia with increasingly greater precision as phyllotaxis rises. Modifications are suggested which remedy this deficiency in the model; one of these modifications involves a combination of Adler's model with Mitchison's model. From a comparison of the ranges of divergence angles permitted by Adler's model against Fujita's measurements of divergence angles in plants with low phyllotaxis, it is shown that the modified contact pressure model, if based on the concept of mechanical pressures between primordia in contact, cannot account for the divergence angles found in low phyllotaxis systems. However it is shown that this deficiency can be overcome if the contact pressure effect is regarded as a chemical phenomenon, mediated by a growth inhibitor produced by the prirnordia and moving more readily in vertical directions than in other directions. Mitchison's model, which is based on the concepts of an expanding apex and primordium initiation by existing primordia, is shown to account for Fibonacci phyllotaxis only if phyllotaxis rises sufficiently slowly; to guarantee that an F n + F n+1 system can develop there must already be at least F n+1 primordia present in an F n−1 + F n system, at least F n primordia in an F n−2 + F n−1 system, and so on down to at least three primordia in a 1 + 2 system, making a total of at least F n+3 −5 primordia (where F n = nth term of the Fibonacci series with F 1 = F 2 = 1). Adler's model, modified, requires only that F n + 1 primordia be present with divergence angles in the range 120–180° to guarantee that an F n + F n + 1 system can develop.