Entropic Elasticity in Polymers and DNA

Why Stretching a Rubber Band or a DNA Strand Fights Probability Rather Than Bonds

Press the middle of a thick natural rubber band lightly against your lower lip. Now stretch the band taut between your thumbs and touch it to your lip again. You will feel an immediate, unmistakable pulse of warmth. Hold the band stretched for ten seconds until it cools back to skin temperature, let it snap slack, and press it to your lip a third time. The rubber now feels distinctly cool. First reported by blind English natural philosopher John Gough in Manchester in 1802 and measured with sensitive thermocouples by James Prescott Joule in 1859, this thermoelastic behavior is the exact opposite of a steel spring. When you stretch a steel wire, pulling iron atoms apart increases internal crystal lattice energy $\Delta U$ and slightly cools the metal. Why does a rubber band heat up when pulled and contract forcefully when warmed?

When you stretch a rubber band or a single strand of DNA, you are not bending stiff chemical bond angles. You are fighting Ludwig Boltzmann's probability law. In 1934, Swiss physical chemist Werner Kuhn showed that a vulcanized polyisoprene rubber molecule or a dissolved biopolymer behaves as a three-dimensional random walk of $N$ freely rotating molecular segments of statistical Kuhn length $b$. At constant temperature $T$, the contractile restoring force $F$ needed to hold the two ends of a polymer chain at separation $L$ is the spatial derivative of the Helmholtz free energy $A_{\text{free}} = U - TS$:

In an ideal flexible polymer, every carbon-carbon single bond rotates freely around its axis with nearly identical internal bond energy $U$ whether the chain is crumpled or straight. That rotational freedom makes the energetic first term $(\partial U / \partial L)_T$ close to zero. Virtually all the mechanical tension comes from the second term, $-T(\partial S / \partial L)_T$. When the chain's two ends sit close together ($L \ll Nb$), the polymer backbone can writhe through billions of distinct coiled microstates $W(L)$ under constant thermal bombardment from surrounding molecules. When you pull the two ends apart toward full contour length $Nb$, only a single stretched-out configuration can span the distance ($W \to 1$, so $S = k_B \ln W \to 0$).

Stretching the rubber band forces the tangled chains into an aligned, low-entropy state. That alignment expels their lost configurational entropy $T\Delta S$ into the vibrations of the material as perceptible heat. Letting the band go allows thermal collisions to kick the chains back into their high-entropy random coil, absorbing heat from your lip. Because temperature $T$ multiplies the entropy derivative directly in Kuhn's formula, warming an entropic spring increases the violence of thermal kicks and makes the polymer pull harder.

In 1992, biophysicists Steven Smith, Laura Finzi, and Carlos Bustamante at the University of Oregon and the University of New Mexico measured this pure entropic spring inside a single biological molecule. They chemically anchored one end of a single double-stranded bacteriophage lambda DNA molecule—consisting of 48,502 base pairs with a total contour length of 16.4 micrometers—to a glass microscope slide. They attached the opposite end to a 2.9-micrometer paramagnetic polystyrene bead. Using calibrated magnetic gradients and fluid flow to pull the microscopic bead while tracking its position under an optical microscope, Bustamante's team plotted extension $L$ against piconewton force. Across forces from 0.01 to 10 piconewtons, the DNA strand resisted stretching in exact quantitative obedience to the worm-like chain entropic elasticity curve formulated by John Marko and Eric Siggia in 1995. When James Prescott Joule hung a lead weight from a vulcanized India-rubber strip in his Manchester laboratory in 1859 and warmed the rubber with a spirit lamp, the stretched band did not sag like heated iron. Instead, it contracted and lifted the lead weight upward because raising absolute temperature $T$ directly multiplies the entropic restoring force $F = -T(\partial S / \partial L)$.