I expect that everyone has wondered at some time about the variety of shapes that abound in living things. Each of these shapes has the purpose of helping that organism to survive in its habitat. There must be a reason, a mechanism, for plants and animals of macroscopic dimensions to grow to the shape they do and also govern the internal organs to be in the form they take. Let us postulate a mechanism.

    One of the first thoughts is that living things are symmetrical, for example, ehinoderms, starfish for example, have fivefold symmetry, and many animals have twofold symmetry, humans, for example. Inside plants, there are bundles of vascular tissue that are arranged around the stem in a symmetrical pattern. This reflects the external morphology of the plant. Many animals are segmented, only a few phyla such as mollusca are unsegmented. These observations indicate that there is an influence that is varying cyclicly.      In mathematics there are several functions that vary cyclicly but the one that comes to mind is the sine wave. It is also true that all curves can be modelled by superposition of sine and cosine wave curves. This is called the fourier synthesis. I analysed some leaf shapes for the harmonics and then reproduced the shape by adding the harmonics. This indicated to me that the basic idea was the correct one. Now sine waves are a solution of some differential equations. I wondered if I could find one for plants.

    The next Idea was to see if diffusion, synthesis and feedback could generate a model to give sine waves. If one has two compounds which effect each others rate of synthesis this might give something. At first I thought that each might inhibit the other but study of the equation showed that this was not correct for sine waves. If one assumes a steady state then the sine waves are spatial and not temporal.

          Let one chemical compound be A and have concentration [A] compound B have concentration [B].

rate of production of A is d[A]/dtrate of production of B is d[B]/dtA diffuses away from its site of production so it will have a concentration gradient and will obey the equation-d[A]/dx=p*d[A]/dt and also -d[B]/dx=q*d[B]/dtif rate of production of A is proportional to [B] then

d[A]/dt=r*[B] and d[B]/dt=s*[A]if we have a steady state between production and diffusion then

-d[A]/dx=p*r*[B] and -d[B]/dx=s*q*[A]so taking d/dx we have-d[A]/dx=p*r*d[B]/dx
so  d[A]/dx=p*r*s*q*[A]put prsq=kthen d2[A]/dx=k*[A]

This has the solution [A]=c*exp(û(k)*x)+d*exp(-û(k)*x)

If k is negative then the solution is a sine wave in [A]

[A]=f*sin(û(k)*x)+g*cos(û(k)*x)

[B]=(-d[A]/dx)/p*r

This shows that if cells obeyed signals from [A] or [B] and A inhibited B whilst B activated A or visa versa,  then their character would vary cyclicly. There are linear organisms like the filamentous algae and there is a cyclic nature in the occurrence of reproductive organs. Most organisms are three dimensional and these have to be modelled with this equation for three dimensions. By adding the effect of several spatial oscillations with different wavelengths any shape can be modelled. This requires that growth depends on the concentration of some of the compounds A,B,C... which are varying. Another solution is exponential decay. This could also occur and may model things like the length of an organism from a large front to a tail which slowly tapers away. Leaves, if you examine them carefully show a ripple round the edge which indicates the highest spatial frequency and is gives strong support for this idea. Since the process of growth is called morphogenesis  then these compounds are called morphogens.

    There has been a lot of work on this by biologists over the past few years and drosiphlia has been studied in detail. Scientist have been trying to find compound whose concentration varies cyclicly and some have been found. Tests  on these have shown that altering their concentration deforms the organism indicating upport for the hypothesis. 

    I hope this of interest to all visitors and let us wonder at the complexity of life and yet its beautiful simplicity.

Chris