As Likely as Knot: Biopolymers under Stress

olymers were discovered--but not so named--in the eighteenth century by naturalists investigating such substances as plant cells, starch, rubber, and proteins produced in living organisms. Today, the word polymer refers to a chain-like molecule built from a large number of repeated parts. Biological proteins, for example, are generally polymers built by linking long chains of amino acids. At the University of Pennsylvania, scientists are using SDSC resources to learn how such polymers act in living tissue, where they twist and knot and become entangled with one another. Putting knots in a polymer chain or tangling several polymer chains together can make the polymers either more fragile or more stable. Michael Klein and his colleagues at Penn are discovering how and why this is so.

HYDROCARBON CHAINS

AB INITIO MODELS

EVOLUTION OF THE FRAGMENTS

ENTANGLED POLYMER CHAINS

HYDROCARBON CHAINS

"The polymer studies have resulted in some interesting discoveries," said Klein, the Hepburn Professor of Physical Sciences in the Department of Theoretical Physical Chemistry and director of the Laboratory for Research on the Structure of Matter (LRSM), in charge of the LRSM Center for Molecular Modeling. "Structural isomerism is well-studied, but it is also possible to construct isomers of molecules that differ not in their structure but in their topology. A long-chain polymer with a knot in it--even the simplest self-threaded trefoil loop--behaves unlike the same chain with no knot."

Klein is the 1999 winner of the prestigious Aneesur Rahman Prize for Computational Physics of the American Physical Society. "My research is focused on quantum and classical computer simulation of condensed matter and biophysical systems at the atomic level," said Klein, who has allocations from both NPACI and the Alliance. "I am most interested in the relationship between intra- and intermolecular interactions and physical properties." His NPACI projects include the basic study of nitric acid dissociation; a study of the nitric acid trihydrate crystals in Antarctic stratospheric clouds (where ozone destruction may occur); an investigation of the strongest "superacid" system; and the study of hydrocarbon chain radicals--which is what the remnants of broken polymer chains are called.

As generations of sailors and anglers have discovered, the break in a knotted rope almost invariably occurs at a point just outside the "entrance" to the knot. In polymer chemistry, the longer a polymer strand becomes, the more likely it is to have a knot. Fragments of DNA have been observed to contain such knots, both in experiments and computer simulations.

REFERENCES

A.M. Saitta, P.D. Soper, E. Wasserman, and M.L. Klein. 1999. Influence of a knot on the strength of a polymer strand. Nature 399: 46-48.

A.M. Saitta, and M.L. Klein. 1999. Evolution of fragments formed at the rupture of a knotted alkane molecule. J. Am. Chem. Soc. 121: 11827-11830.

A.M. Saitta, and M.L. Klein. 2000. First-principles study of bond rupture of entangled polymer chains. J. Phys. Chem. B 104: 2197-2200.

A simulation by A. Marco Saitta and Michael Klein of the University of Pennsylvania calculated the strain energy distribution in a knotted polymer strand of 36 (left) and 28 (right) carbon atoms. The strain energy localizes most on the bonds immediately outside the entrance points as the knot is tightened.

AB INITIO MODELS

In 1999, to give this empirical observation a more stringent theoretical test, Klein worked with postdoctoral researcher A. Marco Saitta and two colleagues from DuPont Central Research and Development, Paul Soper and Edward Wasserman, to model the behavior of a knot in a polyethylene strand. Polyethylene is the simplest polymer, a chain in which every link has one carbon and two hydrogen atoms. "Polyethylene is an excellent generic system to study the fundamental properties of a knotted chain," Klein said.

After obtaining a classical molecular dynamics equilibrium state for a tight trefoil knot in a 28-link polyethylene chain, Klein and colleagues modeled tensile loading on the chain using ab initio calculations based on density functional theory (DFT) and Car-Parrinello molecular dynamics (CPMD). "We found that our code ran most efficiently on SDSC's IBM SP system, because of its large memory bandwidth requirements and the excellent implementation of Fast Fourier Transforms in the IBM math libraries," Klein said. The calculations were carried out at increasing values of the chain end-to-end distance, mimicking an increasing strain on the knotted chain as though it were being held at either end by a pair of tweezers (Figure 1).

"When the strain energy content of the polymer exceeded the amount that could be accommodated as geometrical distortions from equilibrium," Saitta said, "a chain rupture occurred, always at the bond located at the entrance of the knot, regardless of the modeled location of the tweezer constraints." With an unknotted chain, breaks occurred first at the location of the constraints. This finding, published in Nature, accords with macroscopic experience of knotted ropes and lines, and it suggests that knots are topological objects that have universal properties that do not depend on their size (Figure 2).

Figure 2. Detecting a Break

Saitta and Klein calculated the schematic electron charge density immediately after a break. Carbon and hydrogen atoms are displayed with spheres corresponding to their respective covalent radii. The green area is a contour plot of the region containing most of the total electronic charge. A gap in the charge density, and thus bond-breaking, is observable between the two highlighted carbon atoms (purple).

EVOLUTION OF THE FRAGMENTS

Klein and Saitta continued their studies by modeling the further evolution of the chain after the rupture. To do so, they had to perform the CPMD calculations within the local spin-density approximation (LSDA) rather than the local density approximation, since the latter cannot deal with the unpaired electrons that occur at the broken ends of the chain. The adoption of LSDA doubled the computational size of the system. The time steps in the calculation are fractions of a femtosecond, but the simulations had to be carried out for a dozen picoseconds to depict what happens to the knot fragments. The results were published last November in the Journal of the American Chemical Society.

"What we saw was that the fragments may recombine to form cyclic alkanes, but that there is also disproportionation--further breakup," Klein said of the competing recombination and disproportionation reactions. "This poses a challenge for experimentalists using ultrafast spectroscopy. Our calculations indicate that the disproportionation effects may dominate in the early stages, because steric effects favor them against recombination. Looking at such a system experimentally may shed further light on what is proving to be a complex mixture of chemical effects."

ENTANGLED POLYMER CHAINS

A knot is simply one class of topological defect. Chain entanglements are another, and they play an important role in polymer chemistry because they are unavoidably generated during the crystallization of polymer films. Tensile strength, resistance, and other mechanical properties change dramatically in the presence of entanglements, and Klein and Saitta set out to investigate these effects, again from first principles.

They modeled the behavior and stability of two entangled polyethylene molecules. As reported in the Journal of Physical Chemistry B, they found that the presence of entanglement weakened the single-chain resistance to tension, but that, since the second chain can store half the strain energy, the net result is that the entanglement increases the resistance of the system to further insult (crazing or fracture). "As in the previous cases, the stress tends to accumulate in the central bonds owing to increased friction between crossed bonds, between bonds and atoms, and between atoms," Klein said. The rupture of entangled polymer strands thus appears to be a bond friction-driven phenomenon.

"We also observed the same disproportionation effects following bond rupture," Klein said. "Now we have begun using the Blue Horizon system at SDSC to follow these processes further as they unfold. The machine will let us increase the size of the system we study, to multiple entanglements, and to lengthen the simulation time to take account of the ultimate outcome of ultrafast, competing reactions." --MM