DNA elasticity

At the turn of the 20 century the reality of atoms and molecules was still open to vigorous debate. The patient accumulation of indirect evidence for the molecular hypothesis finally left no room for doubt, but still molecules remained unseen actors in a strange world governed by bizarre laws (quantum theory) totally unlike the classical physics of the 19 century. By the time this picture was firmly in place, in the 1930's, it would have seemed as foolish to treat molecules by classical physics as to apply quantum theory to the motions of the planets. The two domains were viewed as entirely divorced; molecules were just too small to apply classical physics.

By mid-century, however, it became clear that enormous single molecules exist in living cells. Indeed some animals contain single molecules of DNA up to a meter in length! Biomolecules like DNA, actin, and proteins are polymers, long chains of similar repeated units. Since moreover each unit contains some dozens of individual atoms, one may wonder if the molecule itself could behave effectively as a continuum object, governed by the rules of matter in bulk. If so, then DNA would respond to applied forces as an elastic body, an enormous simplification over the complex and unintuitive underlying quantum laws.

Recently it has become possible to test this hypothesis directly in a new class of experiments on single molecules of DNA. Besides its conceptual value, the elastic picture offers the promise of detailed understanding of the binding of DNA to drugs, genetic regulation, and the compact packaging of DNA in living cells.

Testing the elastic hypothesis might seem far-fetched. DNA is a long thin rod, only 2 nm in diameter. Certainly we have no ``vise and pliers'' apparatus this small with which to stress DNA, nor any suitable microscope to observe how it deforms. Instead, experimentalists have constructed micron-scale apparatus, many thousands of times bigger than the actual length scale of interest, indirectly finding the intrinsic elastic parameters using simple ideas from statistical physics. (Even more clever, but more indirect, means had been used earlier to obtain rough estimates of these parameters.)

To see how this might be possible, imagine first a long, thin elastic object, for example a length of rubber tubing 1 cm in diameter. Scaled to this human size, the DNA in a typical experiment would be almost a kilometer long. A rubber tube resists bending, but clearly it is easier to make a 90 bend of large radius than one of small radius. Indeed, the energy cost of a 90 bend can be as small as we like if we choose a long enough segment of the tube.

The significance of this remark lies in the fact that at room temperature molecules are in constant motion. Specifically, every possible molecular motion, including bending deformations, is constantly going on, and each independent kind of motion has average energy equal to J. This seems like a tiny amount of energy, and indeed we are scarcely aware of thermal motion in everyday life. But the previous remarks imply that for any elastic rod there is a length scale beyond which the elastic cost of bending is totally negligible. This scale is called the bend-persistence length. The stiffer the rod, the longer the scale. To understand the significance of the name, return to the image of the rubber tube and imagine it is being randomly shaken. Any particular point on the tube will be pointing in a random direction, but nearby points will be pointing in roughly the same direction if they are close enough: this is ``persistence''. Points farther away than the bend-persistence length will be uncorrelated, because large-radius bends are easy to create.

For DNA the stiffness corresponds to a bend-persistence length of about 50 nm; in our tube analogy this corresponds to 25 cm, much shorter than the total tube length. Thus pure DNA in water will be a random tangled mess. One can straighten it, however, by pulling on the ends. To understand how this works, let us simplify our mental image still further. Since bends longer than 25 cm are easy to create, replace the image of the rubber rod by a chain of straight links, each about 25 cm long and completely free to pivot at the linkage points. If we pull the ends of this chain and ignore friction, it will straighten without any resistance until fully extended, and then resist further straightening (we will ignore knotting in this simplified discussion).

Remarkably, the experiment sketched above can now be performed with single DNA molecules. Researchers have succeeded in attaching one end of the molecule to a wall, and the other end to a large bead, then applying small forces to the bead by a variety of techniques (hydrodynamic drag, optical tweezers, or small magnetic fields) and observing its displacement as they vary the force. Since individual molecules are fragile, the forces must be almost unimaginably tiny, in the range of Newtons. This force corresponds roughly to the weight of a single bacterium; nevertheless the techniques just mentioned are delicate enough to apply such forces accurately.

The observed behavior (total length versus applied force) turns out to be totally different from the simple expectation just sketched, which was appropriate for a macroscopic chain. (See illustration.) The difference arises from entropic effects; they hold the key to using these experiments to learn about DNA elasticity.

The problem with the simple chain model is that it includes thermal fluctuations only incompletely. Imagine stretching the chain out to nearly its full length. There are very few possible ways for the chain to be, say, 99.9% of its full length: every link must be nearly straight. On the other hand, there are an enormous number of ways for the chain to be 10% of its full length: every link can have any angle in a vast number of combinations. In other words, the unstretched state has far more entropy than the stretched one. Statistical physics then says its free energy is effectively lower than the stretched state, even though there is no energy cost in bending each individual link. The size of this effect is again proportional to Boltzmann's constant and the temperature, which explains why we don't observe it in daily life: J is a tiny energy. For polymers, however, this effect is crucial. It says that even a freely-jointed chain resists elongation, an effect called entropic elasticity. The realization that this effect is the origin of elasticity in rubber marked the birth of polymer physics. Since the strength of the effect is proportional to the temperature, the theory predicts that a rubber band should get stiffer, and shorter, if we heat it -- a well-known effect.

It is now a simple mathematical calculation to see how the entropic elasticity effect depends on the unknown bend-persistence length of the chain, compare it to experimental data (see illustration), and find the persistence length. Dividing by J then gives the desired elastic constant for DNA, or equivalently the persistence length mentioned above, 50 nm.

We can now ask how well classical elasticity theory works to describe DNA. First, the experimental data indeed fall on the theoretical curve Moreover, further experiments at higher applied force (up to 50 times greater than mentioned above) reveal another phenomenon: in addition to getting straighter, DNA can stretch. The elastic resistance to stretching a rod is related to the bend stiffness by a formula from classical elasticity theory; DNA obeys this formula remarkably well.

Finally, it has recently become possible to take the model one important step farther. Besides resisting bending, a rubber rod resists twisting. While traditional simple polymers do not exhibit any appreciable twist resistance, complicated ones like DNA and actin do. To measure this effect, researchers have succeeded in stretching long DNA strands while keeping the ends from rotating. The extension of the molecule then depends on both the applied tension, as before, and also the amount of extra twist, if any, imposed on the molecule beyond its natural helical twist. Analyzing these experiments by a generalization of the scheme sketched above yields values for both the bend- and the twist-persistence length, two independent parameters controlling the behavior of DNA in vivo.

Bibliography:

R.H. Austin et al, ``Stretch genes,'' Physics Today Feb. 1997, pp. 32-38.

C. Bustamante, J.F. Marko, E.D. Siggia and S. Smith, ``Entropic elasticity of lambda-phage DNA,'' Science 265 (1994) 1599-600.

C. Calladine and H. Drew, Understanding DNA (Academic, 1992).

S.B. Smith, L. Finzi and C. Bustamante, ``Direct mechanical measurements of the elasticity of single DNA molecules by using magnetic beads,'' Science 258 (1992) 1122-6.

T.R. Strick, J.-F. Allemand, D. Bensimon, A. Bensimon, and V. Croquette, ``The elasticity of a single supercoiled DNA molecule,'' Science 271 (1996) 1835-1837.