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How Joel Cohen uses math to make sense of a complicated world
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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