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How Joel Cohen uses math to make sense of a complicated world
BY RENEE TWOMBLY
Inside Joel Cohen’s
head is a 3-D grid of the world, in which 100 million or so
interconnected dots pulsate irregularly, sometimes swelling or
contracting in slow-mo fashion and then rapidly reversing. To
Cohen, who is the university’s Abby Rockefeller Mauzé
Professor and head of the Laboratory of Populations, each of these
dots represents a species and it’s his life work to figure
out their dynamics and interactions.
Sometimes, Cohen takes on just a bit of this
network, as when he modeled the interconnectedness of species found
in Dutch farm soil and in a small Wisconsin lake. He has also
tackled questions of more immediate concern to most people. In a
532-page book, he laid out what he calls the “choices and
constraints” needed to figure out how many people can live
well on our planet. Most recently, he has been arguing that
modelers like him need to work side by side with bench scientists
if biology is to be understood.
Cohen is on the hunt for quantitative theories
that explain the behaviors of living populations, no matter what
they are.
He admits that while his work may seem
scattershot to many, “my mission in science is to understand
this network of life that we live in, at both a fine and a
worldwide scale of resolution,” he says. “What I do for
a living is to look for patterns that others have not yet seen. I
try to invent better tools for seeing those patterns.”
The broad reach of his research has earned
Cohen not only entrance into expected societies, such as the U.S.
National Academy of Sciences, the American Academy of Arts and
Sciences, and the American Philosophical Society, but also onto the
worldwide Board of Governors of The Nature Conservancy. In 2002,
New York Mayor Michael R. Bloomberg gave Cohen his Award for
Excellence in Science and Technology. Cohen also is a professor of
populations in the Earth Institute at Columbia University.
In his map of life, the size of each dot is
based on the biomass and importance of the species to the rest of
the network. The human dot would be huge — like New York City
on a U.S. atlas, he says. The roads leading to each dot represent
the energy consumed by the species that dot represents.
Because most species rely on other species for
their energy, or are consumed by other species in search of energy,
the dots are all interconnected. The network formed by the
connections between a species and all the other species it eats is
known as its food web. Cohen himself has kept logs of the species
of food he eats, and over time it comes to about 150 different
species of plants and animals. Humans collectively probably consume
tens of thousands of other species. “That represents a lot of
energy, and a lot of diversity, coming in,” he says, with
some amusement.
Humans, in fact, have provoked a lot of
wobbling in the global food web. “Over the last 10,000 years,
the number of humans has increased about 1,000 fold, creating a lot
more demand on other species, and providing more available
material,” Cohen says. It is easy to understand that the
roads that bring energy to the human species have multiplied, but what we don’t often consider is that the
roads that bring energy from humans to other species have increased
as well.
“All of our infectious diseases are other
species making a living off of us,” Cohen says. “Think
of the thousands of bacteria in our gut, the fungi on our skin, the
insects that suck our blood, and the diseases those insects
inject.” New microbes and viruses that prey on humans, such
as Ebola and HIV, are burgeoning around the world, and old ones
continue to thrive. Of particular interest to Cohen is
Chagas’ disease, caused by an insect-borne parasite similar
to the one responsible for African sleeping sickness. Cohen’s
mathematical model of how the disease spreads has had public health
implications for millions of poor Latin Americans. “The
network of infectious disease is incredibly dynamic,” Cohen
says.
Searching for structure
Within this dynamic, Cohen looks for coherence.
He and his lab team use the tools of mathematics to search for
patterns in the data and to test ideas. “The role of math in
biology is to take a simple idea about how a complicated system
works, understand the consequences of that idea, and see how the
expected consequences compare with actual observation,” he
says. The art of Cohen’s work is to look for features that
seem more persistent within this dynamic web, in order to deduce
the rules from which all flux is generated.
A case in point is the relationship between
species and biomass. Charted on paper, the populations of different
species in an ecosystem resemble a pyramid, called the pyramid of
numbers: at the top are a few members of large-sized species; at
the bottom are many members of smaller-sized species. Cohen and
other ecologists found, however, that the total biomass of each
species — the weight of all its members combined — is
often relatively consistent, or may be slightly larger for
bigger-bodied species. Fish species with a few big individuals at
the top of the ecosystem have roughly the same, or slightly more,
biomass as single-celled floating plant species with millions or
billions of individuals at the bottom of the food web.
Considering the relationships between species
based on their biomass leads to new ways of thinking about
ecosystems. With former postdoc Tomas Jonsson, Cohen plotted the
numbers of organisms in each species on one axis of a chart
(logarithmically scaled) and the average body weight on the other
axis (also logarithmically scaled), and connected the dots that
represent individual species by arrows to show which species are
eating which other species. Then, using observations from
ecosystems as different as a tiny lake in Michigan’s Upper
Peninsula and the topsoil of farmlands in the Netherlands, Cohen,
Jonsson and colleagues from the University of Wisconsin and the
Netherlands found that the dots and arrows fell along a trend line
with negative slope near -1.
In other words, the biomass of different
species varied far less than the average body size or the
population numbers. With current postdoc Daniel C. Reuman, Cohen
also found that this map of a community of species shows how many
predator-to-prey steps energy takes to go from the smallest to the
largest species.
In 1984, Cohen’s collaborator Stephen R.
Carpenter and other researchers from the University of Wisconsin
catalogued the open-water ecosystem of Michigan’s Tuesday
Lake, estimating the population of 56 different species that ranged
from microscopic phytoplankton to several species of fish. In this
environment, the largest species (a fish) was a trillion times the
weight of the smallest but was outnumbered by a factor of 10
billion. After graphing each species’ average body mass
against its abundance (again, on logarithmic scales), Cohen found
that nearly all species fell near a straight diagonal line drawn
from the rare heavy species to the common light ones.
In 1985, his Wisconsin colleagues replaced the
biggest fish with the even larger, and predatory, largemouth bass
and, in 1986, again catalogued the lake’s ecosystem. Cohen
and his colleagues found the ecological pattern remained much the
same. It was like a different cast of characters acting out the
same play.
More recently, Dutch researchers asked Cohen to
collaborate on a study that tested whether soil that was
intensively farmed looked ecologically different from soil that was
either conventionally or organically farmed. The linear pattern
again held across the samples. It was remarkably similar to the
overall pattern at Tuesday Lake, Cohen says, but the line sloped
downward more steeply in the more intensively farmed plot, compared
to organic soil.
“Now we want to understand the connection
between the slope and elevation of the line and the physical and
chemical management of the farm soil,” Cohen says. “The
elevation and the slope of the line could be useful summary
indicators of the quality of land use, of how people affect a
natural system.”
From fish to famines
Understanding populations of lake and farm
dwellers is useful, but the big questions Cohen gets asked are
often about people. It’s often a variation on: How many
humans is too many?
We’ve all heard the dire warnings about
shortages of food, wars over water, and epidemics of disease that
will befall our planet once we hit a certain number of billions.
But Cohen argues that the future cannot be simply plotted as an
extension — arithmetically, geometrically or logarithmically
— of the past.
Compared with human history prior to World War
II, the world’s population growth rate since 1950 has been,
and still is, unprecedented, Cohen says. Since 1600, the human
population has increased more than tenfold, from about half a
billion to well past 6 billion. The increase of 800 million people
in the last decade of the 20th century exceeded the total
population in 1600. Within the lifetime of some people now alive,
world population has tripled, and within the lifetime of everyone
older than 40 years old, it has doubled, he says.
Yet his 1995 book on global population, How Many People Can The Earth Support?, is neither “an alarmist tract nor a
cornucopian lullaby,” Cohen says. It is, instead, a study of
contingencies. Cohen says that in the process of writing the book,
“I came to question the question. ‘How many people can
the earth support?’ is not a question in the same sense as
‘How old are you?’ It cannot be answered by a numbers.”
But that doesn’t mean that people have
not tried. Estimates of the earth’s human carrying capacity
range from fewer than 1 billion to more than 1 trillion, he says.
Most frequently, estimates fall between 4 billion and 16 billion,
with a median estimate of 12 billion. Cohen says this
“enormous spread follows from widely varying concepts,
methods and assumptions.”
One dramatic example, he says, is the United
Nation’s prediction that if human populations continued to
grow at 1990 rates in each major region, then the population would
increase more than 130-fold in 160 years, from about 5.3 billion in
1990 to about 694 billion in 2150. “But those figures are
extremely sensitive to the future level of average
fertility,” Cohen says. If, hypothetically, couples bore only
as many children as needed to replace themselves, world population
would level off at about 8.4 billion by 2150.
And while everyone recognizes that the
finiteness of the earth guarantees that ceilings on human numbers
do exist, Cohen says that human choices, now and in the future,
will decide where those limits fall. Questions of wealth,
technology, politics, economics, demographics, environment, and,
above all, values — what people want from life — can be
approached as “population problems” to help quantify
what is conditional and probable, Cohen says.
“The earth’s capacity to support
people is determined partly by the processes that the social and
natural sciences have yet to understand, and partly by choices that
we and our descendants have yet to make,” says Cohen.
In short, there’s no easy answer, and
anybody who offers one is probably wrong.
Doing math in a cell
Today, Cohen can be found encouraging the use
of the mathematical methods he’s long applied to groups of
organisms to tackle the questions that take place within those
organisms.
“Mathematical analysis will ultimately
reveal biological processes better than any microscope ever
has,” Cohen says. “Coping with the hyper-diversity of
life at every scale of spatial and temporal organization will
require fundamental conceptual advances in mathematics.
Understanding how the cell came together is tremendously exciting,
and we ought to really engage mathematics to the fullest extent
that we can to help in that quest.”
In an essay published in Public Library of Science Biology in December, Cohen suggested applying math to
understand how cells work, how the brain is linked to behavior and
emotion, how heritable features are transferred between species,
and how environmental conditions interact with biochemical
processes.
Consider the complex network of gene
interactions, proteins and signaling processes within and between
cells. “This network is probably incomprehensible without
some mathematical structure that has yet to be invented,”
Cohen says. Math will not only benefit the study of biology, but
working on difficult biological problems will likely stimulate
mathematicians to create new math — just as physics and
astronomy have in the past.
At Rockefeller, and other institutions, that
means educating mathematicians and biologists to be comfortable
with each other’s thought processes and very different
languages. It also means being wrong. The great thing about working
with mathematical models, after all, Cohen says, is the opportunity
to disprove them.
“I expect to be proved wrong. I announce
to my colleagues that I’ve found a pattern and then lots of
them set out to disprove it. In the meantime, though, I have fun
pushing the pattern as far as the data will go. I consider my most
successful modeling work to be something that looked good enough to
push people to go and find out if it was true,” Cohen says.
“What I really hope is that my work
stimulates people to find out what actually happens.”
SIDEBAR: Where estimation meets litigation
March 18, 2005
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