Saturday, August 15, 2009

Blogging In Thompson's Footsteps

My fascination with the mathematics of biological form was first kindled by a childhood encounter with D’Arcy Thompson’s classic book On Growth and Form (henceforward oGaF). First published in 1917, this unique volume explores the intersection between biology and engineering mechanics. It is probably better known for its striking images than for its text, and for the questions it implicitly posed than for the answers it provided. This passage from a Wikipedia entry describes it nicely:

Utterly sui generis, the book never quite fit into the mainstream of biological thought. It does not really include a single unifying thesis, nor, in many cases, does it attempt to establish a causal relationship between the forms emerging from physics with the comparable forms seen in biology. It is a work in the "descriptive" tradition; Thompson did not articulate his insights in the form of experimental hypotheses that can be tested. Thompson was aware of this, saying that "This book of mine has little need of preface, for indeed it is 'all preface' from beginning to end."

The book is dense with curious facts about the connection between the Fibonacci series and the layout of seeds in sunflowers, or the ways in which simple inorganic processes can reproduce the shapes of jellyfish or the arrangement of cells in tissues. Thompson applied a mathematician’s eye to the shape of a ram’s horn and a chambered nautilus’ shell, and an engineer’s mind to the stresses and shapes in the skeletons of dinosaurs, birds and other creatures. But the most provocative part, the one responsible for the book’s continuing appeal, is the final chapter, entitled On the Theory of Transformations, or the Comparison of Related Forms. There Thompson investigates the relationship between the shapes of different species by imposing a regular mesh on one and then deforming it to match another – a device that anticipates the modern computational techniques of morphing and finite element analysis. The deformations seem regular, suggesting some deep hidden principle of morphological evolution.
And closer to home... The question elegantly posed by these images is: how does nature transform organism shapes in the course of evolution? This question depends on a still more basic one: how does nature generate organism forms in the first place? These questions are, respectively, the phylogenetic and ontogenic parts of the problem of morphogenesis.
Arguably oGaF was an anachronism, misplaced in time: a piece of 21st century biology accidentally dropped into the early 20th century, to paraphrase Edward Witten's comment about string theory. In order to get us to the point where we can begin to answer the questions implied by Thompson’s transformations, a set of distinct sciences and tools has had to be developed, each quite complex in their own right. These include molecular biology, genomics and evo devo; as well as the development of cheap computing power and computational methods of finite element analysis and computer graphics.

Nowadays, as biologists are starting to understand cells as metaphorical computers, we would say that the shape of an organism is a consequence of a “morphogenetic program” encoded in its genes and executed by its cells. Although many biologists believe this is correct in outline, we are still far from a detailed understanding of the complete morphogenetic program of any organism -- what Lewis Wolpert has called the computable embryo. However, in contrast to Thompson’s time, the foundations for that understanding are now in place, and one can reasonably predict that over the next few decades Thompson’s images will transition from tantalizing enigmas to icons of a new science of morphogenetics, encompassing genetics and mechanics, and utilizing computational modeling of morphogenesis as a fundamental tool.
The goal of this blog will be to assemble, as I have the time, bits of informational flotsam related to the goal of a full computational understanding of biological Growth and Form.

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