Devlin's Angle

March 2002

The Shrimp and the Mathematician

The next time you dip that succulent white shrimp into the picante sauce, remember that your tasty snack is one of Nature's marvels: a single muscle - some 40% of the total weight of the creature it belongs to - designed by nature for just one purpose: the creation of massive acceleration to propel the creature out of danger with a sudden explosive thrust that, pound for pound, makes it the envy of any Olympic sprinter.

Motion - from the beating of a heart to the familiar task of getting from point A to point B - is the essence of life. Consider: A basketball player running at full speed suddenly stops, pivots on one leg, takes two steps in another direction, then launches himself high into the air to score a basket. A fish, motionless in the water one moment, catches a sudden movement in the corner of its eye and, with a barely perceptible flick of its tail, darts off rapidly into the safety of the reeds. A cockroach scuttles across the kitchen floor to escape the sudden illumination from the overhead light you just switched on. A cormorant glides silently and elegantly above the ocean until it spots a fish in the water beneath, whereupon it suddenly swoops down in a rapid dive to secure its prey.

Evolution has equipped all living creatures with a way to move - to search for food, to seek out a mate, or to escape from danger. People and ostriches walk and run on two legs, horses and dogs on four, cockroaches on six, spiders on eight; snakes slither; fish propel themselves by pushing the water sideways with their tail; birds fly by flapping their wings to create lift and forward thrust.

How do they do it? How do the creatures that inhabit the land, the sea, and the air move? You can get some idea of the difficulty of this question from the fact that, after fifty years of well-funded research into the construction of computer-controlled machines, no one has yet been able to build a robot that can walk well on two legs. In fact, the best four- or six-legged robots do not perform anything like as well as the average dog or dung-beetle. Only the invention of the wheel - thousands of years ago - has enabled Man to build efficient transportation machines. When it comes to building machines that imitate the ways Nature solved the locomotion problem, we're still in Kindergarten.

Yet all motion comes down to just two physical principles, identified by Isaac Newton 350 years ago. One is that motion results from the application of a force. (Force = mass x acceleration.) The other is that every force produces an equal and opposite reaction. The great variety of locomotive strategies that we see around us comes not from different principles of motion but from Nature's boundless ingenuity in finding ways to apply Newton's two physical laws. Only in recent years have scientists started to understand how Nature achieves this feat, often with enormous ingenuity. For example, in the article How Animals Move: An Integrative View, by Michael Dickinson, Claire Farley, Robert Full, M.A.R. Koehl, Rodger Kram, and Steven Lehman, published in SCIENCE 288, 7 April 2000, pp.100-106, the authors point out that the old idea of a central brain directing the actions of all the muscles involved in motion is not at all accurate. Rather, the control mechanisms that govern movement are distributed throughout the organism, in many cases embedded in the design of the individual moving parts.

Other motion is a result of self-organization, where a collection of organisms, sometimes just single cells, somehow mange to communicate with each other to coordinate their activities to produce a cohesive motion of the entire collection, as if it were a single creature. (See the recent book Self-Organization in Biological Systems, by Scott Camazine, Jean-Louis Deneubourg, Nigel Franks, James Sneyd, Guy Theraulaz, and Eric Bonabeau, published by Princeton University Press.)

In all cases, arguably the most basic question about motion is, how do the movements of single cells combine to produce the motion of the entire organism or collection of which those cells are parts? This is where mathematics can help.

To appreciate the problem, think of the problem facing a group of dancers who have to coordinate their individual movements to give a pleasing performance, a football team who must act in unison to score a touchdown, or the musicians in an orchestra who must produce a perfect symphony. Each may have its leader - the dance choreographer, the football quarterback, or the orchestra conductor - but at the most basic level it's the communication between the individual performers that fuses their separate actions into a single, recognizable whole. So too with all movement of living creatures.

But how exactly is the communication and the resulting coordination achieved? In recent years, collaborations between biologists and mathematicians have started to provide answers, often with some surprising and tantalizing twists, as Dr Angela Stevens informed the audience in the special symposium on mathematical modeling of animal and plant movement at this year's AAAS meeting in Boston last month. Dr. Stevens, who works at the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany, has been using mathematics to study movement of self-organizing systems of cells. Among the curious behaviors her research had uncovered is that, sometimes, just before a group of communicating individual cells achieve perfect coordination, they generate a recognizable traveling wave pattern, not unlike the ripples that move through a line of heavy traffic on the freeway. Presumably the individual cells are communicating with each other, but how exactly are they doing so, and what produces the ripple?

Turning from the very small to the very large, mathematics has also proved useful in understanding how particular tree species propagate across a geographic region. Recent work by Mark Lewis of the University of Alberta resolves a conundrum known as Reid's Paradox: the fact that sometimes a new species of tree will spread at a rate that in botanical terms seems impossibly fast. The solution to this puzzle came not from biology but mathematics. Lewis showed how the role of chance can lead to extremely rapid plant migration.

Long recognized as a powerful tool in physics and engineering, mathematics is now finding increasing application in the biological and life sciences, often with remarkable results. The AAAS symposium in which Stevens spoke, which was organized by mathematician Hans Othmer of the University of Minnesota, gave several tantalizing glimpses of this exciting new area of scientific study - the marriage of biology and mathematics - that is helping us to understand one of the greatest of all scientific mysteries: life itself.

Other mathematics related symposia at this year's AAAS Meeting included one on the applications of Social Choice Theory to biology and another on wave patterns and turbulence, a topic much in the news following the crash of a jet airliner shortly after takeoff from New York's Kennedy Airport last fall.

Mathematicians whose national conference attendance is limited to the Joint Mathematics Meetings each January would be well advised to consider going along to next year's AAAS meeting in Denver, Colorado. Here are two good reasons. First, the mathematics talks are all designed to appeal to a wide audience of scientists, science buffs, and science journalists. That means they have a different flavor from most presentations at a mathematics conference. Second, in addition to the mathematics talks, you can wander around and enjoy a veritable smorgasbord of talks on topics in science (both natural and social), science policy, and science education.

Actually, I can think of a third good reason to attend next year's AAAS meeting: The Colorado Rockies are just an hour's drive away from the Denver location of the conference.

This month's column is adapted from a promotional article the author wrote under commission from the AAAS, with financial support provided by SIAM. I am grateful to Warren Page, Secretary of Section A (Mathematics) of the AAAS for encouraging me to write that initial article.


Devlin's Angle is updated at the beginning of each month.
Mathematician Keith Devlin ( devlin@csli.stanford.edu) is the Executive Director of the Center for the Study of Language and Information at Stanford University and "The Math Guy" on NPR's Weekend Edition. His latest book is The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip, published by Basic Books.