This article deals with the physical chemical processes underlying biological self-organization by which an initially homogenous solution of reacting chemicals spontaneously self-organizes so as to give rise to a preparation of macroscopic order and form. Theoreticians have predicted that self-organization can arise from a coupling of reactive processes with molecular diffusion. In addition, the presence or absence of an external field, such as gravity, at a critical moment early in the self-organizing process may determine the morphology that subsequently develops. We have found that the formation in vitro of microtubules, a major element of the cellular skeleton, show this type of behaviour. The microtubule preparations spontaneously self-organise by way of reaction and diffusion, and the morphology of the state that forms depends on the presence of gravity at a critical moment early in the process. We have developed a numerical reaction-diffusion scheme, based on the chemical dynamics of a population of microtubules, which simulates the experimental self-organisation. In this article we outline the main features of these simulations and discuss the manner by which a permanent dialogue with experiment has helped develop a microscopic understanding of the collective behaviour.