Keith Devlin devlin@csli.stanford.edu
The Shrimp and the Mathematician
(March 2002)


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.
_________________________________________________________________

POWRÓT