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Physicists Find Simple Solution to Great-and-Small Mystery
11 May 1999

Physicists have devised a simple mathematical explanation for a centuries-old mystery about why many biological processes change in curiously disproportional but precisely predictable ways depending on the mass of an organism's body.  In the May 13 issue of the journal Nature, a team of physicists will reveal a general theorem capable of explaining the phenomenon, known as allometric quarter-power scaling.

"The pulse rate of a mouse weighing 30 grams may be 600 beats per minute, whereas the heart of a 5-ton elephant beats only about 30 times per minute, which is an example of how pulse rate in mammals scales inversely as the quarter power of the animal's mass," says Jayanth Banavar, professor of physics at Penn State and an author of the Nature paper. "Many other biological properties vary in animals and plants as powers  of the mass derivable from the number 4," he adds.

"Because life exists in three dimensions, you might logically expect the characteristic scaling relationship to be derivable from the number 3 instead of the number 4," says Banavar, whose curiosity about the mystery was sparked by another research team's recent publication of a general mathematical model to explain allometric quarter-power scaling based on the geometry and dynamics of plant and animal circulation systems. "That model was exciting, but its detailed assumptions were quite complicated and too specific to be found everywhere, plus it still left us wondering why this scaling relationship based on the number 4 is so curiously pervasive throughout nature," Banavar says.

As physicists accustomed to the quest for simple and elegant mathematical theorems, Banavar and his colleagues set out to develop a simple theorem of allometric quarter-power scaling.  In addition to Banavar, the research team includes Amos Maritan, of the International School for Advanced Studies in Italy, and Andrea Rinaldo, of the University of Padova in Italy and the Massachusetts Institute of Technology.

The physicists approached the problem with the hunch that nature's evolutionary processes produce highly efficient circulation systems and other kinds of natural transportation networks.  They further reasoned that all transportation networks might turn out to have the same general behavior, and therefore the same underlying mathematical description, if they have achieved maximum efficiency through natural selection.  So they broadened the focus of their research to include both living and nonliving transportation networks that have been shaped by the processes of evolution.

The physicists analyzed transportation networks such as circulatory and respiratory networks within plants and animals, networks of streams in the drainage basin of a river, and networks of pipes for transporting water to homes.

The researchers say they have been able to prove an extremely simple theorem to describe any "locally connected" network of this type that is functioning at maximum efficiency.  All the nodes in such a network are linked with each other through an interconnected network.  "There is a novel mass-volume relationship underlying efficient transportation networks," Maritan says.  "For example, in order to supply water to L to the power of D houses--where L is the circulation length or the spatial extent of the colony of houses and D is the number of dimensions in which the network functions--the minimum amount of water in the pipes at any given time scales at least as L to the power of D plus 1," Maritan explains.

"This simple theorem gives us a mathematical rationale for the quarter-power scaling law that is so ubiquitous in three-dimensional systems throughout nature," Banavar says.

To explain the logic of their theorem, the researchers first note that, in a system for supplying the same amount of water to all the homes, the total amount of water flowing in the water pipes must be proportional to the number of homes in the system, or L to the power of D.  "However, an additional factor of L is necessary to account for the water filling the pipes throughout the minimum distance necessary to reach the most remote home, and this leads to the power of D plus 1," Rinaldo explains.  "You can cast it in the form of a theorem and prove it, which is what we did, but the basic idea is really that simple."

The researchers also analyzed data from networks of streams channeling precipitated water into the main stream of a river and found that allometric scaling, as predicted by their theorem, holds in that case too.

The physicists say the beauty of their theorem is that it is independent of the nature of the system.  "River networks in the drainage basin sculpted by erosion and networks in plants and animals all obey this scaling law.  Allometric scaling is built into efficient transportation networks and the pervasive ubiquity of such scaling shows the remarkable extent to which nature has been able to accommodate such networks in spite of the diversity of the systems." Banavar says.

"What's beautiful about this theorem is that it goes above the details and provides an extremely simple underlying explanation for a long-standing puzzle in a wide range of contexts," Maritan adds.

Funding for this research was provided by the Microgravity Program of the National Aeronautics and Space Administration (NASA), the North Atlantic Treaty Organization (NATO), the Italian National Institution for Nuclear Physics (INFN), the Italian Ministry of Universities and of Scientific Research and Technology (MURST), and the Petroleum Research Fund administered by the American Chemical Society.

Research contacts:

Jayanth Banavar,, 814-863-1089
Amos Maritan,, (39) 040-2240462
Andrea Rinaldo,, (617) 253-7176

PIO contact:

Barbara K. Kennedy,, 814-863-4682