On Size and Life

I have recently been reading the fascinating book On Size and Life, by Thomas McMahon and John Tyler Bonner (Scientific American Library, 1983). In their preface, McMahon and Bonner write

This book is about the observable effects of size on animals and plants, seen and evaluated using the tools of science. It will come as no surprise that among those tools are microscopes and cameras. Ever since Antoni Van Leeuwenhoek first observed microorganisms (he called them “animalcules”) in a drop of water from Lake Berkel, the reality of miniature life has expanded our concepts of what all life could possibly be. Some other tools we shall use—equally important ones—are mathematical abstractions, including a type of relation we shall call an allometric formula. It turns out that allometric formulas reveal certain beautiful regularities in nature, describing a pattern in the comparisons of animals as different in size as the shrew and the whale, and this can be as delightful in its own way as the view through a microscope.

Their first chapter is similar to Sec. 1.1 on Distances and Sizes in the 4th edition of Intermediate Physics for Medicine and Biology, except it contains much more detail and is beautifully illustrated. They focus on larger animals; if you want to see a version of our Figs. 1.1 and 1.2 but with a scale bar of about 10 meters, take a look at McMahon and Bonner’s drawing of “the biggest living things” on Page 2 (taken from the 1932 book The Science of Life by the all-star team of H. G. Wells, J. S. Huxley, and G. P. Wells).

In their Chapter 2 (Proportions and Size) is a discussion of allometric formulas and their representation in log-log plots, similar to but more extensive than Russ Hobbie and my Section 2.10 (Log-Log Plots, Power Laws, and Scaling). McMahon and Bonner present in-depth analysis of several biomechanical explanations for many allometric relationships. For instance, below is their description of “elastic similarity” in their Chapter 4 (The Biology of Dimensions).

Let us now consider a new scaling rule as an alternative to isometry (geometric similarity [all length scales increase together, leading to a change in size but no change in shape]), which was the main rule employed for discussing the theory of models in Chapter 3. This new scaling theory, which we shall call elastic similarity, uses two length scales instead of one. Longitudinal lengths, proportional to the longitudinal length scale ℓ, will be measured along the axes of the long bones and generally along the direction in which muscle tensions act. The transverse length scale, d, will be defined at right angles to ℓ, so that bone and muscle diameters will be proportional to d…When making the transformations of shape from a small animal to a large one, all longitudinal lengths (or simply “lengths”) will be multiplied by the same factor that multiples the basic length, ℓ, and all diameters will be multiplied by the factor that multiplies the basic diameter, d. Furthermore, there will be a rule connecting ℓ and d…d ∝ ℓ^3/2.

They then show that elastic similarity can be used to derive Kleiber’s law (metabolic rate is proportional to mass to the ¾ power), and justify elastic similarity using biomechanical analysis of buckling of a leg. I must admit I am a bit skeptical that the ultimate source of Kleiber’s law is biomechanics. In IPMB, Russ and I review more recent work suggesting that Kleiber’s law arises from general considerations of models that supply nutrients through branching networks, which to me sound more plausible. Nevertheless, McMahon and Bonner’s ideas are interesting, and do suggest that biomechanics can sometimes play a significant role in scaling.

Their Chapter 5 (On Being Large) presents a succession of intriguing allometric relationships related to the motion of large animals (running, flying, swimming, etc). Let me give you one example: large animals have a harder time running uphill than smaller animals. McMahon and Bonner present a plot of oxygen consumption per unit mass versus running speed, and find that for a 30 g mouse there is almost no difference between running uphill and downhill, but for a 17.5 kg chimpanzee running uphill requires about twice as much oxygen as running downhill. In Chapter 6 (On Being Small) they examine what life is like for little organisms, and analyze some of the same issues Edward Purcell discusses in “Life at Low Reynolds Number.”

Overall, I enjoyed the book very much. I have a slight preference for Knut Schmidt-Nielsen’s book Scaling: Why Is Animal Size So Important?, although I must admit that Size and Life is the better illustrated of the two books.

Author Thomas McMahon was a major figure in biomechanics. He was a Harvard professor particularly known for his study of animal motion, and even wrote a paper about “Groucho Running”; running with bent knees like Groucho Marx. Russ and I cite his paper “Size and Shape in Biology” (Science, Volume 179, Pages 1201–1204, 1973) in IPMB. I understand that his book Muscles, Reflexes and Locomotion is also excellent, although more technical, but I have not read it. Below is the abstract from the article “Thomas McMahon: A Dedication in Memoriam” by Robert Howe and Richard Kronauer (Annual Review of Biomedical Engineering, Volume 3, Pages xv-xxxix, 2001).

Thomas A. McMahon (1943–1999) was a pioneer in the field of biomechanics. He made primary contributions to our understanding of terrestrial locomotion, allometry and scaling, cardiac assist devices, orthopedic biomechanics, and a number of other areas. His work was frequently characterized by the use of simple mathematical models to explain seemingly complex phenomena. He also validated these models through creative experimentation. McMahon was a successful inventor and also published three well-received novels. He was raised in Lexington, Massachussetts, attended Cornell University as an undergraduate, and earned a PhD at MIT. From 1970 until his death, he was a member of the faculty of Harvard University, where he taught biomedical engineering. He is fondly remembered as a warm and gentle colleague and an exemplary mentor to his students.

His New York Times obituary can be found here.