Math aids researchers in understanding protein folding

For Christodoulos Floudas, professor of chemical engineering, the watchwords of his research are "divide and conquer."

This has nothing to do with heavy-handed politics--it's a bold mathematical technique he is applying to a question that has tantalized scientists for years: how do proteins fold?

Understanding the way proteins fold--what three-dimensional shape they have--is a problem that rears its head in fields from biology and chemistry to engineering and math and has major implications for medical science. The bumps and valleys in a protein's shape determine how it reacts with other proteins.

The shapes also give drug-makers clues about how to design medicines that latch on to a bump, or a valley and interfere with the proteins involved in disease. Comprehending the way proteins fold could result in new medicines, new understanding of disease, and insights into the basic mechanisms of life.

Photo by Frank Wojciechowski Chemical Engineering Professor Christodoulos Floudas works to understand how proteins fold.

Proteins are fundamental elements of life. They send the messages, transport the chemicals, and build the structures that allow living things to exist. Scientists have known for a long time the precise chemical makeup of proteins. But that's only half the story. What really determines the function of a particular protein is the precise and intricate way it is folded.

When cells manufacture proteins, they start off as long strings of chemical blocks but quickly fold up into ringlets and loops like ribbons on a birthday present.

What puzzles scientists is that the sequence of chemicals in a protein does not offer obvious clues about the shape that the protein assumes.

For example, if one protein includes chemicals A, B, and C and they bend into a right angle, another protein may have the same ABC sequence in a straight line or a loop. For decades, scientists have tried to understand why a particular chemical sequence leads to a particular fold. Their ultimate goal is to predict the complete three-dimensional structure of a protein based solely on knowing the sequence of chemicals that go into it.

"It is a fascinating problem, because nature does it so quickly and efficiently, and we all fail to predict it," said Professor Floudas, who is one of a growing number of scientists trying to tackle the problem using mathematics, rather than experimental methods.

The mathematical approach is part of a field of applied mathematics called optimization. This is a process of looking at complex situations affected by many variables and understanding what values for the variables will produce the desired result. With proteins, the variables are the positions of atoms of the chemical units.

The chemical units in proteins, the As, Bs, and Cs, are amino acids; there are 20 amino acids that can be used in any combination. A typical protein may be madeup of hundreds of amino acids, but for the purposes of studying protein folding, scientists often look at relatively short segments, called peptides. Even a pentapeptide, a protein with just five amino acids, could fold in any of 100 billion possible structures, Professor Floudas said.

"How can you develop accurate mathematical models that capture all the interactions?" he asked.

Like many scientists, Professor Floudas thinks about protein folding in terms of how much energy is needed to hold a protein together in a particular shape. A folded protein is analogous to a person sitting crosslegged on the floor, compared to the person standing bent over at the waist; one position requires less energy than the other.

In theory, the same protein could assume many different stable, low-energy shapes. In nature, however, proteins always head down the one path that leads to the very lowest level of energy. Scientists want to know which path nature will select. Which is the lowest of the low? There are far too many to compare one at a time.

That's where the principle of "divide and conquer" comes in. Professor Floudas has figured out a way to divide the problem in half repeatedly, so that he searches for the highest energy conformations and then looks for the theoretical lowest limit on what the protein's energy could be, a number that is below the actual lowest-energy state. He then uses a mathematical process that makes the two numbers converge on the correct answer. In technical language this method is called "branch and bound global optimization approach."

Professor Floudas said it provides a "theoretical guarantee for locating the global solution to a given problem."

He has had good preliminary results with a 28-amino-acid protein and is now working on one with 58 amino acids. One validation of his method came last year when a group from Columbia University used it to make good predictions about the shape of several proteins ranging in size from 54 to 183 amino acids.

"There is strong evidence we're on the right track," Professor Floudas said.

Ken Dill, professor of pharmaceutical chemistry at the University of California, San Francisco was a keynote speaker at a conference on computational approaches to protein folding, co-organized by Professor Floudas and held at Princeton in May.

"Chris's work is an important new contribution to this field," Professor Dill said.

He added that Professor Floudas is the first to use rigorous computing methods to achieve global optimization ("global" because it doesn't just find a good answer, it finds the best answer). Professor Dill, however, believes there is still a lot of work to be done. He likens the protein-folding problem to finding a needle in a haystack: what Professor Floudas has done is find a very good way of sorting through all the hay; what is not yet clear is how good is the description of the needle itself.

In other words, do computational modelers have a good enough description of how much energy is associated with a particular shape?

Despite these unresolved issues, Professor Floudas' techniques already are finding practical applications. He is collaborating with two groups at the University of Pennsylvania that are using his optimization tools to tackle specific medical research questions, one having to do with how the immune system works and the other with the working of the protein compstatin. These collaborations combine Professor Floudas' mathematical approach with more commonly used experimental methods.

In addition, research conducted in Professor Floudas' Computer Aided Systems Laboratory addresses fundamental problems in nonlinear mixed-integer and deterministic global optimization, and their important applications in the synthesis, design, control, and operation of chemical processes.

Graduate students John Klepeis and Heather Schafroth, and postdoctoral associate Karl Westerberg are working with Professor Floudas on the protein folding problem. Dr. Westerberg, a National Institutes of Health fellow, investigates the transition states, pathways, and dynamics of protein-folding. Mr. Klepeis, a Wallace Memorial Honorific Fellow, studies the ab initio secondary and tertiary structure prediction of polypeptides. Ms. Schafroth, a National Science Foundation Graduate Research Fellow and a Gordon Wu Fellow in Engineering, investigates interactions of proteins in peptide docking.

Professor Floudas acknowledges that it is too early to claim the protein-folding problem has been solved, but he believes his approach includes several technical elements that others omit, and the results are getting better and better.

"I'm encouraged," he said, "because rigorous computational methods can advance our understanding of protein structure prediction and elucidate the dynamics of protein folding."

This story first appeared in the Sept. 27, 1999 issue of the Princeton Weekly Bulletin and is reprinted here with permission.