Single Polymers: They Vibrate Harmoniously, Unknot Unpredictably

July 09, 1997

A vibrating guitar string and a tiny strand of DNA have something surprising in common.

If you dunk the microscopic DNA molecule in water and then pluck it, it obeys the same simple law of motion as a plucked guitar string. That in itself is not so surprising, says physicist Steven Chu, but what intrigues him is the remarkable precision of the molecule's adherence to this 200-year-old law. Previously, scientists had thought that such molecules would exhibit much more complex behavior.

Chu and a student made this discovery when they immersed individual strands of DNA in water to study the vibrations of polymers, the large spaghetti-like molecules that figure in everything from plastics to synthetic fabrics to the DNA in living cells. Researchers previously had been limited to studying the motion of polymers in bulk, by the millions and billions. Chu's laboratory is providing some of the first detailed studies of the behavior of individual polymers, a subject that industrial engineers consider important in designing improved methods for producing plastics.

In a second, related study, Chu and two doctoral students also discovered that individual polymers unravel in highly individual and unpredictable ways when exposed to swift currents of water. The first study, reported in the July 10 issue of the journal Nature, was conducted with former student Steven Quake, now an assistant professor of physics at the California Institute of Technology. It suggests that simple, linear approximations of polymer motion may be much more accurate than most scientists had thought.

Guitar analogy

The idea that the motions of a polymer can be described by a set of frequencies corresponding to a fundamental tone and its higher harmonics, similar to the vibrations of a musical string, is an old one. But most researchers have considered this simple "linear theory" to be only a rough description of the actual motion. They have felt that, in the real world, polymers submerged in a solution and capable of forming knots make polymer behavior complex and less easy to predict.

In the Nature paper, "We decided to take the analogy to the guitar string very seriously and see how well it held up. It turned out to be much closer than we expected," Chu said. To study individual polymer vibrations, the scientists started with strands of DNA 20 microns long immersed in water that had been labeled with a fluorescent dye. (A strand of human hair is about 25 microns across.) Next, they attached tiny plastic spheres about one-third of a micron in diameter to each end of the spaghetti-like molecules. Then, using a laser tool called optical tweezers that Chu helped develop while at Bell Labs, they gripped the spheres at each end of a DNA strand and pulled them far enough apart to stretch the molecules to about three-quarters of their full length.

Normally, DNA is too tiny to see with an optical microscope. But the dyed strands showed up clearly, so the scientists were able to videotape their vibrations. The thermal agitation of the water molecules, called Brownian motion, acted like tiny fingers plucking at the strand. When the researchers analyzed these movements, they found that they could be described by the motion of a set of independent harmonic tones to an accuracy of better than 1 percent. They carried their analysis up to the eighth harmonic.

Harmonic motion first was described by the French mathematician Dalembert in the 1700s. He discovered that the motion of a string held taut at both ends could be fully described by superimposing a series of simple sine waves with wavelengths that fit evenly into its length. In 1954 American scientist Bruno Zimm suggested that the motion of a polymer in solution can actually be explained by Dalembert's mathematical description.

Normally, a scientist would not even try to use such a linear theory to describe the movement of a polymer in solution. If you move one segment of a single strand that is submerged in water, that movement generates water waves that then exert forces on all the other segments. The force exerted on closer segments is greater than that exerted on segments farther away. Since the distance of the segments depends on the instantaneous configuration of the entire polymer, the mathematics to solve this problem is intractable.

To simplify the math, Zimm replaced actual distances between segments with average distances. Strictly speaking this assumption is not mathematically rigorous. In his treatment he left out a number of complicated effects: for example, his model allowed the polymer to pass through itself as a "phantom-like" strand. Despite the shaky derivation, the basic conclusion that polymer motions can be described by a linear set of equations may still be correct, Chu said.

Twenty years later Pierre-Gilles de Gennes, professor of the CollËge de France and winner of the Nobel Prize for his contributions to polymer science, emphasized that collective polymer motion was far more complicated than had been assumed by Zimm and others. As an alternative, he developed a "scaling" theory that describes the dynamics of a polymer without having to linearize the equations that describe its motion.

"Because we can actually see the molecules move, we can directly observe the higher-order vibrations for the first time. When we started this work, we sided with de Gennes and felt that polymer motion cannot be perfectly linear. But we looked very hard for non-linearity and found no evidence for it," Chu added. The researchers are continuing their search for a breakdown of the harmonic model.

How DNA strands unravel

Chu's second study, performed with graduate students Thomas T. Perkins and Douglas E. Smith, was published in the June 27 issue of the journal Science. It shows that identical polymers in identical conditions act quite differently, indicating that small random conditions play an unexpectedly important role in way polymers unravel.

In that experiment Chu and his two doctoral students observed how immersed DNA strands unravel when exposed to microscopic currents. Such currents, or flow fields, occur as a fluid passes through any constriction or nozzle. Understanding how polymers deform in these fields is necessary to understand how polymers can reduce drag in pipelines and how they behave during processes such as injection molding.

The researchers manufactured microscopic currents by using microfabrication techniques to create flow channels that were only 650 microns wide and 220 microns deep. Fluorescently labeled DNA molecules flowed down one channel until they reached the center of the cell where they moved into a cross current. The researchers videotaped the molecules as they reacted to the cross-current by unraveling to a greater or lesser extent.

Despite taking great care to use identical strands of DNA in identical flow conditions, the experimenters observed a great diversity in the way that they unraveled. "We have found that random thermal fluctuations in the initial starting point of the elongation get magnified into dramatic differences in the way each molecule unravels," Chu said.

Until now theorists have described the elongation of these molecules according to a "mean-field" theory that assumes the description of the average behavior is adequate. Since all the previous polymer experiments probed the behavior of a large collection of molecules, an average or so-called "mean-field" theory was developed that fit the experimental data.

The new results show that any "averaged" theory is incorrect. As an analogy, consider all the animals in a zoo. If someone did a series of experiments that only measured the average size, weight and number of legs of the animals, he would get a distorted picture. For example, he might find that the average number of legs on the animals is 2.7, and then look for a theory of animal development that could explain his finding. Only by looking at each individual animal can you get a true sense of the diversity of species.

Similarly the observation of thousands of individual, but identical, molecules showed the researchers that the elongation process follows a number of different scenarios and the rate at which they unwind depends dramatically on the initial shape of the partially coiled polymers.

The unexpected outcome of the experiment was that apparently identical polymers exhibit what de Gennes has called "molecular individualism." As the Nobel laureate states in a comment on the Stanford paper in the same issue of Science, "Normally, the average coil shape is enough to describe many features. But not here."

This individualism apparently arises from small random conditions. "We have found that random thermal fluctuations in the initial starting point of the elongation get magnified into dramatic differences in the way each molecule unravels," Chu said. "In the 'nature versus nurture' debate, no one pays attention to the importance of tiny random influences. Perhaps such influences play an important role in how our lives evolve as well."

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Stanford University

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