All creatures great and smallThe living world is governed
by laws based on fractal geometry and on the sizes of organisms,
some scientists claim. John Whitfield looks at the debate
surrounding a biological 'theory of
|Unearthing the simple rules
for biological complexity.|
Some people see only the trees, others the whole wood. But when
Brian Enquist enters one of his Costa Rican field sites, he senses
far deeper patterns. "When I walk into the forest, I have the
feeling that, although it's very complex, there are simple rules
underlying that complexity," he says.
Enquist, an ecologist at the University of Arizona in Tucson, is
working to derive laws that can explain the workings of ecosystems,
and the biology of their constituent organisms, in terms of those
organisms' sizes - a project "as potentially important to biology as
Newton's contributions are to physics", as one commentator recently
By explaining simple scaling laws - mathematical expressions of how
organisms' biology varies with their size - in terms of the fractal
geometry of networks such as circulatory systems, Enquist and his
colleagues hope to understand patterns in metabolism, growth,
ecology and evolution right across the living world.
A string of highly cited papers in top journals, and the
Ecological Society of America's 2001 Mercer Award for young
investigators, to Enquist, attest to the impact of these ideas. But,
as with any bold new theory, some question its generality, and
others its truth. Other critics even deny the existence of the very
trends that the theory seeks to explain. Have Enquist and his
colleagues really made a stunning theoretical breakthrough, or is
the comparison with Newton hype? The debate rages, but whatever
consensus eventually emerges, just about everyone involved believes
that important biological insights will stem from this new focus on
In attempting to derive a fundamental explanation for biological
scaling laws, Enquist and his two associates - ecologist James Brown
of the University of New Mexico in Alberquerque, and physicist
Geoffrey West of Los Alamos National Laboratory, also in New Mexico
- have reopened the files on a mystery that has baffled biologists
for more than half a century.
Naturalists have long known that many aspects of organisms'
biology vary with their size. Bigger animals live at a slower pace.
They survive longer, grow more slowly, have slower heart rates, and
so on. In the 1930s, physiologist Max Kleiber of the University of
California, Davis, put a number on this trend. He showed that an
animal's metabolic rate is proportional to its body mass raised to
the power of 3/42.
This relationship has been found to hold across the living world
from bacteria to blue whales and giant redwoods, over more than 20
orders of magnitude difference in size. Scaling laws based on
exponents in which the denominator is a multiple of four apply to a
host of other biological variables, such as lifespan. "There's maybe
200 scaling laws that have quarter powers in them," says West.
|Leaves and lungs share a
similar fractal geometry.|
But for decades, scaling laws stubbornly resisted explanation.
Organisms are three-dimensional. So, from the principles of
euclidean geometry, one might expect that biological scaling laws
would operate on multiples of the third, rather than the quarter
power. Metabolic rates, for instance, might scale as the 2/3 power
of body mass - this is how body surface area, where metabolic heat
is lost, scales with its volume, which determines how much metabolic
heat a given organism can produce.
In the mid-1990s, Enquist and Brown began to search for factors
that might explain the 'missing' fourth dimension of biological
scaling. They suspected that the dynamics of organisms' internal
transport of nutrients and other resources might provide the key.
Through contacts made at the Santa Fe Institute in New Mexico, which
specializes in thorny cross-disciplinary problems, they teamed up
with West, a theoretical physicist with the mathematical expertise
to help them develop the idea.
The resulting theory frames the scaling conundrum in terms of
resource-distribution networks, such as blood vessels, the xylem
that transports water through plants and the tracheal tubes that
carry oxygen to an insect's tissues. Thus, variables such as
metabolic rate - and the effect on them of changing body size -
become a consequence of how resources are shunted around these
Much of life is designed in
a fractal-like way
|James Brown, University of New
West, Brown and Enquist's theory starts with the assumption that
evolution has made biological resource-distribution networks
maximize the area across which they can take up and release
resources and minimize the time and energy needed to transport those
resources through the organism. "Natural selection is very
powerful," says Brown. "It's difficult for organisms to deviate from
the optimum." The model also assumes that the size of networks'
terminal units is independent of body size - that a mouse's
capillaries, say, are the same size as those of an elephant.
Networks that fulfil these criteria turn out to have fractal
geometry, argue West, Brown and Enquist. This means that their
branching structure can be described mathematically by applying a
simple mathematical formula over and over again. Abandoning Euclid
and embracing fractals was the key step, says Brown. "Much of life
is designed in a fractal-like way. We were able to make that cogent
and relevant by developing rigorous quantitative models."
This fractality seems to solve the enduring puzzle of why the
scaling exponents of three-dimensional organisms are multiples of
the number four. Filling a three dimensional volume with a network
that maximizes the two-dimensional surface area available for
capturing and releasing resources creates a four-dimensional
This is difficult to visualize, but Enquist uses the analogy of how
a tree uses three-dimensional space to cram a much larger area of
essentially two-dimensional leaves into the ground space covered by
its canopy. "Fractal geometry kicks life up to another dimension,"
he concludes. In April 1997, West, Brown and Enquist unveiled their
theory by showing that they could derive Kleiber's 3/4-power scaling
law linking body mass and metabolic rate4.
The researchers have since used the theory to describe a range of
biological phenomena, such as scaling and structure in vascular
They have also expanded their thinking from the level of individual
organisms to entire ecosystems. Using their models describing the
relationship between body mass and metabolic rate, for instance, the
researchers have shown that - wherever resources such as nutrients
or water are a limiting factor - the population density of trees
scales to the -3/4 power of each individual's mass6.
The rule seems to hold for forests from the Amazon to the
The same theoretical framework also explains why, regardless of
the speed at which plants' diameter or height increases, their
growth rate scales to the 3/4 power of their body mass8.
According to this view, different plant life histories, encompassing
vastly different rates of growth and timings of sexual maturity,
simply represent different ways of following the same law for
optimal energy use9.
|From the great to the small,
one theory for all?|
So have West, Brown and Enquist derived biology's 'theory of
everything'? Enquist sees their current ideas as "an excellent first
approximation" of an eventual general explanation of resource use by
organisms. But the theory has come under fire from some researchers
who question its initial assumptions, and from others who believe
that biological reality has been lost in the quest for
Earlier this year, a group of physicists at the Massachusetts
Institute of Technology questioned the very existence of the 3/4 law
for mass and metabolic rate. Re-analysing published data sets dating
back to Kleiber's original, Peter Dodds and his colleagues concluded
that the exponents in most of them were statistically
indistinguishable from 2/310.
"The original data on metabolism don't really bear out the
prevailing idea that it's a 3/4 exponent - you get all sorts of
scalings," claims Dodds, who now works at New York's Columbia
University. Faced with a cloud of decimal places from different
studies, earlier generations of biologists fudged on 3/4, partly
because it made their slide-rule calculations easier, Dodds
Brown denies that the numbers aren't up to scratch. "Metabolic
rate is not difficult to measure," he says. "And whenever we have a
reasonable number of measurements, they all cluster about 3/4."
William Calder, an ecologist at the University of Arizona, says that
the large number of biological variables with 3/4-power scaling
confirms the trend. "It all hangs together," he says.
The more detail that one
knows about the particular physiology involved,
the less plausible these explanations
|Henry Horn, Princeton
Biologists working on fine details of the biology of particular
organisms have criticized the fractal theory's attempts at
generality. Although most agree that the cross-species correlations
look good overall, many individual species deviate widely from the
scaling rules. "The more detail that one knows about the particular
physiology involved, the less plausible these explanations become,"
says Henry Horn of Princeton University in New Jersey, a botanist
who studies patterns of tree growth. "I have a tendency to think
that nature does different things in different ways."
But criticism based on the observation that individual species
disobey scaling laws misses the point, says West. The theory is
designed to capture laws true for an idealized average organism,
rather than every individual plant and animal. West says that in
fact he expected to find that many more species deviated from the
theory's predictions. "What I find most mysterious is why in hell
does it work so well? It's a little bit frightening."
West puts much of the disagreement down to a cultural difference
between physicists and biologists. "If Galileo were a biologist, he
would have written a big fat tome on the details of how different
objects fall at different rates," he says. Enquist agrees.
"Biologists tend to lack a philosophical bent to look for
generalities," he says.
But growing numbers of biologists are now gravitating towards the
new theoretical framework. "I've gone from being an atheist to an
agnostic," says botanist Karl Niklas of Cornell University in
Ithaca, New York. Niklas, the author of the newtonian comparison,
now collaborates with Enquist. If the theory holds up, he says, it
has "the potential to explain virtually everything" that relates to
Another enthusiast, ecologist Mark Ritchie of Utah State
University in Logan, has adapted the theory to explain how animals
of different sizes share out the resources in an ecosystem. He has
modelled communities of foraging animals as a fractal network of
resource exploitation. From this, he has predicted how many species
can coexist in a particular place - in other words, he is trying to
explain biodiversity from fundamental principles. The model gives a
good fit to the patterns of species richness seen on the Serengeti
and the Minnesota prairie11.
In unpublished work, Ritchie has moved on to looking at how the area
of an animal's home range scales with body mass. For carnivores, he
has found that the amount of land needed rises sharply with
increasing body size, because their prey is relatively rare. This
makes predators especially vulnerable to habitat fragmentation.
Even critics such as Horn do not argue with the data that West,
Brown and Enquist are producing. "The relationships are so tight
that there have to be some very powerful generalizations behind the
mechanics," he says. But Horn doubts the significance of fractal
geometry. Although trees look like fractals, he says, their growth
is more complicated than the iteration of mathematical formulas,
involving a complex balance between tissue formation and death.
I don't think ecological
patterns are a consequence of
|Karl Niklas, Cornell
Jayanth Banavar, a theoretical physicist at Pennsylvania State
University in University Park, believes biological generalizations
can be explained without resorting to fractals. Two years ago,
Banavar and his colleagues published a simpler model, which argued
that Kleiber's 3/4 rule is a consequence of any network in which
resources flow outward from a source to their uptake sites12.
Considering the total length of the network, the number of sites at
which resources can leave it, and the total amount of resources in
the network at any one time, they argued that the 3/4 rule falls
naturally from the calculations. "The three-quarters is inherently
built into any directed network - you don't need fractality, or
optimality, or anything," says Banavar.
But many experts on scaling are sceptical of Banavar's approach -
including those who are critical of West, Brown and Enquist. West
adds that Banavar's theory does not explain the wide variety of
scaling phenomena embraced by the fractal theory. "It's not
correct," he claims. "And even if it was correct it only explains
one piece - the metabolic rate."
|Seeing the wood for the
trees: Brian Enquist is searching for a grand
Meanwhile, like a fractal, West, Brown and Enquist's theory is
branching out in all directions. "We were aware that we'd uncovered
the tip of an iceberg," says Brown, "but not how big it was and
where it might lead."
Brown, Enquist, West and their collaborators now have their
sights set on a mind-boggling range of problems. They believe that
scaling laws can be applied to the molecules and subcellular
organelles involved in respiration and cell division. A paper
co-authored by Brown and published last week includes trends in body
temperature - the next most important biological property after
size, says Brown - in the theory13.
Eventually, he hopes to explain relationships between temperature,
body size and rates of evolutionary change.
Brown is also intrigued that animal cells grown in a culture dish
increase their metabolic rate and rate of division - as if they were
aware of no longer being part of a massive unit. The same goes for
cancer cells, which have rebelled against the larger unit. He is
similarly fascinated by the observation that cancer cells can modify
resource- distribution networks, stimulating blood vessel
Whether West, Brown and Enquist have hit the nail on the head
with their fractal approach, Niklas is convinced that simple and
general rules governing diverse biological phenomena lie waiting to
be discovered. "I don't think that the ecological patterns that we
see surfacing in fossils and living organisms and across the
continents are a consequence of chance," he concludes.
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