deodands

Dynamic phyllotaxis

The Turing instability by itself, then, can’t provide an explanation for the generation of Fibonacci phyllotaxis. However there are strong indications in Turing’s later manuscripts, particularly the fragmentary Outline of the development of the daisy, that he had conceived an additional mechanism to provide that explanation.

That mechanism is that there is a small arena in which the Turing instability is at work, laying down spots in lattices and then leaving them behind as the arena follows the growth of the plant. Moreover that arena itself changes, growing in diameter, providing a continuously changing lattice to which the the theories of geometrical phyllotaxis could be applied.

The first evidence of this is in Morphogen Theory of Phyllotaxis Part I; in section 13 Turing establishes a formalism of flow matrices for the change in lattice parameters with a parameter called time, adding that

a convenient way of picturing flow matrices is to imagine the change in the lattice as being due to the leaves being carried over the surface of the lattice by a fluid whose velocity is a linear function of position. (Turing (1992) p75)

This was a way of modelling phyllotactic patterns, building on the continuous change models of Richards (1948). But this remains an essentially static picture of spots being passively transported over a changing geometry. What Turing was able to go on and create, with the aid of his new spot-generation model, was a concrete model for dynamic phyllotaxis.

In the later work, Turing typically expressed what we now think of as his reaction diffusion model in forms similar to

(*)

with

and (cf equation III.1.2 of MTP, p107 of Turing (1992)). Here U(x,t) is the morphogen and V(x,t) is the ‘poison’; the psi function represents the implicit solution of its partial differential equation in terms of the slower diffusing morphogen. The H terms represents the effect of the poison on the morphogen, the G term the morphogen’s autocatalytic nature, and the phi term its diffusive nature, parameterised by the natural wavenumber k₀.

However in Outline of the Development of the Daisy, there is a crucial extra spatiotemporal term I(x,t)U:

(**)

(note that the Saunders edition has a typo for the H and the phi). An even more revealing version of this equation is in AMT/C/27/28:

More © PN Furbank.

This is a version of the morphogenesis equation allowing for growth in the apical meristem region and the possibility of dynamic phyllotaxis. Also, below the first line, there is  a list of the numerical parameters which must be specified to allow computational solution. In addition, there is a sketch of an enlarging apical meristem region

Turing not only conceived this idea; he clearly made substantial progress with a numerical implementation of it. At one point of the Daisy draft, he comments on the number of parameters needed:

when actual computations are being carried out the number of quantities to be specified is again increased. (AMT/C/24/12: omitted from the Saunders version)

in a manner suggesting that he had already done so in practice. Moreover one of the subroutines that has survived is labelled KJELL, and AMT/C/27/C25, entitled Kjell theory, works out the algebra in Fourier space of the coefficients of an equation such as the daisy one, including the crucial growth term:

More © PN Furbank.

Also in the archives are a number of solution plots. Based on their form they are probably solutions of the reaction-diffusion equations of forms similar to (*) or (**). Whether they are directly relevant to the Fibonacci problem or as more general illustrations of morphogenesis is hard to say. My speculation is that they are attacks on the Fibonacci problem, not least as there is little evidence of any other computationally active project. One example is from AMT/K/3/8:

More © PN Furbank