The Fractal Path to Better Beans

Black beans, navy beans, kidney beans, pinto beans, snap, green, string, wax, butter beans - all, to a breeder, are varieties of the common bean, Phaseolus vulgaris. It's a staple crop for small and subsistence farmers the world around, yet it grows with varying success. The average yield in some developing countries—500 kilograms per hectare, or less—is often a mere 10 percent of the textbook yield.

"There's not enough fertilizer," explains Penn State plant nutritionist Jonathan Lynch. "Phosphorous, especially, is limiting."

How can you grow more beans on less phosphorous? According to Lynch and graduate student Kai Nielsen, by using fractal geometry to breed beans with better roots.

red on black “umbrella frame”

Nielsen tells of the dubious "Fractals and bean roots? What'll be next!" kinds of comments he received the first time he presented his research at a professional meeting. He shrugs. "Using fractal dimensions is not just saying roots are fractals." He is well aware that since Benoit Mandelbrot coined the term in 1975, almost everything in the natural world, from ferns to trees to clouds to flames, has been found to be sufficiently complex and irregular to warrant the label "fractal."

"The reason we started looking into fractal analysis," Nielsen says, "is that it's hard to summarize what's the advantage of a root system you'd call 'good.' There are so many parameters: branching intensities, root density with soil depth . . ."

"It wasn't just to try a sexy new technique," agrees Lynch, who had studied the bean-phosphorous connection for four years at the International Organization for Tropical Agriculture headquarters in Colombia before coming to Penn State in 1991. "It's a summary that might be useful in helping breeders.

"There's a tremendous difference in the ability of beans to grow without phosphorous," Lynch continues—a difference not between kidney bean or black, but between different plant lines, or genotypes, within each variety. "Why? From what we've seen in Colombia, it doesn't matter what kind of soil they're grown in, the efficient genotypes are efficient and the non-efficient types are not. That led us to analyze the shape of the root system."

First Lynch and horticultural ecologist Roger Koide collaborated with former Penn State mechanical engineer Andrei Jablokow on a computer model of how bean roots grow. "He's an expert modeler, but usually he models machines," says Lynch of Jablokow. "This project was interesting to him because he wanted to see modeling extended into deformable systems —living systems. He usually doesn't make models of things that change." Based on tracings Lynch had made in the tropics of bean roots growing up against glass, Jablokow's graduate student Bob Davis came up with the computer modeling software SimRoot.

Then, using SimRoot, Nielsen compared the phosphorous efficiency of the normal, umbrella-shaped bean root (an arched canopy and long taproot) to two imaginary root shapes: herringbone (taproot with many short side projections) and dichotomous (the root forks to make two smaller ones, which fork, and their forks fork . . .). "The real bean root architecture," Nielsen found, "was much more efficient."

Lynch smiles. "It's nice when your model predicts that Nature knows what it's doing. The one thing left was to say, What is it that makes this bean root different? That's how we got into fractal geometry. Fractals are all about a new way of looking at biological systems."

To take SimRoot that next step, Lynch began collaborating with Penn State mathematician Howard Weiss, "an expert on dynamical systems and fractals."

Says Nielsen, "He knows a lot about math and nothing about biology, so very good discussions evolve."

To a mathematician, says Weiss, "a fractal is essentially any object with a dimension that's not an integer." Just as the dimension of a straight line is 1 and a flat plane is 2-dimensional, a fractal has a dimension somewhere in-between: "Like .8 or 2.223," Weiss says. Fractals are bumpy lines and lumpy planes. Extremely bumpy or lumpy, says Weiss: "so deformed that they lose all of their smoothness properties, and hence cannot be studied using classical geometry or calculus." In addition, he notes, "Fractals exhibit an amazing fine structure, or detail, on arbitrarily small scales."

Like bean roots. On his computer, Nielsen displays a red, umbrella-shaped tracing of a simulated bean root. Overlaying it with a 3-dimensional grid, he counts the number of boxes that intersect the root. He repeats the count with grids of different sizes, then graphs the log of box size vs. the log of the number of boxes intersected. "If it comes out as a straight line, then the object has fractal properties," he says. And bean roots do. Their fractal dimension in a ball of soil, Nielsen says, ranges from 1.5 to 2 - a number that can be estimated for a given plant with very simple, in-the-field tests.

"That was an insight from Weiss," explains Lynch, "based on theorems in fractal geometry." (Asked to recite the theorems, Weiss says, "the fractal dimension of a set of dimension less than n-1 sitting in a space of dimension n is usually equal to the fractal dimension of the projection of the set in n-1-dimensional space.")

Continues Lynch, "The mathematicians are thinking of 14-dimensional space. We're thinking of a 1-dimensional core sample, a 2-dimensional shovel trench, the 3-dimensional shape of the root ball.

"What Kai's work has shown is that you can grow the beans for 14 days, take a soil core or stick in your shovel and expose a slice of roots and say, Okay, this bean genotype has a fractal dimension of 1.5. Too low." Weed it out.

In October, Nielsen will travel to Costa Rica to try out his core-and-shovel techniques with Douglas Beck, one of Lynch's former colleagues from the International Organization for Tropical Agriculture. Says Lynch, "We don't get too far removed from reality. We didn't plan to make a model of roots. We're just trying to grow better beans."

Kai Nielsen is a Ph.D. candidate in Horticulture, College of Agricultural Sciences, 102 Tyson Building, University Park, PA 16802; e-mail; 814-863-6165. His adviser, Jonathan Lynch, Ph.D., is assistant professor of plant nutrition; e-mail; 863-2256. This work was funded by the USDA and the College of Agricultural Sciences.

Last Updated December 01, 1995