The Numbers Behind the Kidney

Mathematicians find a useful niche in biological sciences

Mathematics professor Harold Layton was an unusual graduate student back in the 1980s.

While the other Duke students stuck to theorems and proofs, Layton turned his attention to something more ordinary: urine. For his dissertation he created a mathematical model of how the kidney combines water and waste to produce the familiar yellow stream.

Layton's adviser, Duke mathematics professor Michael Reed, remembered how Layton got started on the subject.

"I told him, 'Look, here's a simple model [of the kidney]. See if you can do better," Reed said. "And what resulted is a spectacular career."

Layton's career has, indeed, blossomed, say his colleagues. Teaming up with physiologists, biophysics and a computer scientist (his wife Anita), Layton has secured a steady stream (pun intended!) of funding from the National Institutes of Health and publishes regularly in the American Journal of Physiology as well as in applied mathematics journals.

 "When I was a kid, it used to be that if you got a television -- the first color television I ever had much to do with was my grandfather's -- it came with a big schematic," he said. "It had all the components and what they did and all their specs."

"We'd like to have that kind of picture, ultimately, for the kidney," he said.

The study of biological systems by mathematicians is no longer as unusual as it was when Layton was a student. According to Reed, eight of the 28 tenure-track faculty in Duke's mathematics department devote at least part of their time to problems from biology.

Layton explained why the study of the kidney lends itself to mathematics, calling it "sort of like an accounting problem."

Layton's "accounting" lies in balancing the biological equivalent of income and outgo -- keeping track of the various components of blood -- water, nitrogenous wastes, sugars and salts -- as they enter and leave the kidneys. About 1600 liters of blood enter a human's kidneys each day and about two liters of urine leave. Layton specifically looks at urine as it travels through a million tiny tubes in each kidney called nephrons “ how fast it flows, at what rate the concentrations of its components change and how those rates vary depending on the makeup of blood.

To build a model of these interrelated changes, Layton uses systems of differential equations.

Although his work, based mostly on experiments on rat kidneys, is basic science, it could lead to a better understanding of hypertension in humans, Layton said, but such application is at least a decade away.

Layton said he has learned to work with, and win the respect of, biologists and physiologists, even though he took only one biology course in college.

"At first we'd be collaborators on the papers; they would help write the description/scientific parts, and I would deal with the mathematical things," he said. "But more recently there's more give and take about 'Can you measure this?' or 'Can you look at that?'"

His principal scientific partners are Leon Moore, a professor of physiology and biophysics at Stony Brook University, andUniversity of Arizona physiologists William Dantzler and Thomas Pannabecker.

In explaining how he communicates his mathematics to scientists, Layton said, "One has to put on the Isaac Asimov suit," referring to the late and renowned science popularizer. "What would he say? How would he explain it?"

Layton found another research partner when he married his wife Anita in 2000.At first, the newlyweds did not plan to work together professionally. But that soon changed.

Anita, a Canadian citizen, was wrapping up her dissertation on numerical methods for weather models at the University of Toronto, but due to visa complications, she was having difficulty returning to Canada to defend her thesis.

Away from her own work, she began thumbing through books on the kidney and talking to Harold about his research. "I realized the method I was using to solve weather models can be applied to solve equations that arise in these [kidney] models," she said. "So I thought, 'Let's give it a try, I have nothing else to do.' We worked on it. It worked well, so we started writing papers together."

So far, they have published five papers, with four more in press.

She brings a systematic approach to computer simulations to his mathematical models, says Layton. They work together in formulating investigations and interpreting the results.

Asked what it is like to be partners in marriage and scholarship, both laughed. "It's less stressful than raising a kid," Anita said about publishing together. (The couple now has a daughter who is almost two.)

"I don't want to carry the analogy too far," Harold said. "But there's some similarity here because the paper is something you develop and it grows and it takes on a life of its own. Then you have to dress it up just right and send it off to school “ send it off to be reviewed, send it off to the world."

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In 1992, a protein was discovered that facilitates the movement of water through cell membranes. The discovery of this aquaporin protein was a breakthrough for scientists and eventually led to a Nobel Prize in Chemistry for Dr. Peter C. Agre, who recently joined Duke as vice chancellor for science and technology.

Mathematicians, however, may not have been so concerned with the discovery -- except for Duke professor Harold Layton. As an applied mathematician and "theoretical biologist" who builds mathematical models of the kidney, Layton took notice.

"The amount and quality of experimental data [about the kidney] that was available was affected in a big way by the discovery of aquaporins," Layton said.

As more and more types of aquaporins were discovered and then identified in the kidney, Layton could update his mathematical models to show water flowing across membranes -- an important mechanism in the kidney's filtration of blood and production of urine -- only in places where aquaporins are present.