DNA Mechanics: Persistence Length & Worm-Like Chain

A polymer, not a rigid rod

DNA is routinely drawn as a stiff double helix, but mechanically it behaves like a flexible polymer that is constantly being bent by thermal energy at room temperature — over long enough stretches it looks less like a rod and more like cooked spaghetti, curling into a random coil. How stiff it actually is, and how it responds when you pull on its two ends, is exactly what an optical tweezer experiment measures, and this simulation reproduces it directly: a worm-like chain of DNA stretched by a virtual optical trap, with the live force-extension curve plotted against theory as it runs.

Persistence length: the ruler for stiffness

Persistence length (Lp) is the characteristic distance over which a polymer's direction, measured along its contour, forgets which way it was originally pointing — formally, the length scale at which the correlation between the tangent direction at two points decays by a factor of e. A very stiff rod has a persistence length far longer than any segment you'd ever pull on, so it looks straight over that whole segment; a very floppy chain has a persistence length far shorter than the segment, so it looks like a random coil. Double-stranded DNA's persistence length is famously about 50 nm — roughly 150 base pairs — which happens to sit in an experimentally very convenient middle ground: short enough that thermal bending is significant and measurable, long enough that DNA is still much stiffer than, say, a single-stranded polymer (whose persistence length is closer to 1 nm).

 = e^(-s / Lp)
// theta(s) = angle between tangent vectors a contour distance s apart
// Lp    = persistence length ~ 50 nm for dsDNA

The worm-like chain model

The worm-like chain (WLC) model treats the polymer as a continuously flexible, inextensible rod with a bending energy cost proportional to curvature, parameterized by exactly one stiffness number: Lp. It is the standard model for semi-flexible biopolymers — DNA, but also many cytoskeletal filaments — precisely because it needs only that one parameter to predict how the polymer's end-to-end extension responds to an applied force, which is the quantity an optical tweezer experiment directly measures.

The Marko-Siggia formula: theory meets the tweezer

Marko and Siggia (1995) derived an interpolation formula for the WLC force-extension relationship that is accurate across the entire experimentally relevant force range, from the low-force entropic regime up to forces approaching the point where the chain is nearly fully extended:

F(x) = (kB*T / Lp) * [ 1/(4*(1 - x/L)^2) - 1/4 + x/L ]

F  = applied force
x  = end-to-end extension
L  = total contour length (fully stretched length)
Lp = persistence length
kB*T = thermal energy (~4.1 pN*nm at room temperature)

At low extension the polymer is mostly resisting being pulled straight out of its thermally-favoured random coil — this is an entropic spring, and it is why the low-force part of the curve is comparatively soft: you are fighting entropy, not chemical bonds. As x approaches the contour length L the denominator (1 - x/L)^2 blows up and the formula predicts force diverging — pulling the last few percent of a DNA molecule fully straight costs disproportionately more force than the first 90%, which is exactly the steep upturn you see in the plotted curve as extension approaches 100%.

What the simulation is actually validating

The reason this simulation plots simulated force-extension data directly against the Marko-Siggia curve, rather than just animating a wiggling chain, is that the WLC model is not a cartoon — it is a quantitatively validated theory. Real optical tweezer experiments pulling on single DNA molecules match the Marko-Siggia prediction to within a few percent across four orders of magnitude of force, from sub-piconewton thermal fluctuations up to tens of piconewtons approaching the elastic limit of the B-form helix (beyond which DNA undergoes a distinct structural transition to an overstretched state, outside the WLC model's regime). Watching the live-simulated curve converge onto the theoretical Marko-Siggia line is watching the same agreement that made single-molecule DNA mechanics a foundational technique in modern biophysics.

Why 50 nm specifically matters biologically

Persistence length is not just an abstract stiffness number — it sets the scale for how DNA interacts with the proteins that package and regulate it. A nucleosome wraps roughly 147 base pairs of DNA — about one persistence length — around a histone core, which is already an energetically demanding bend given DNA's natural stiffness; this is a direct, quantitative reason nucleosome positioning and DNA looping in gene regulation are governed by polymer mechanics as much as by biochemistry, and it is why persistence length is the number every subsequent worm-like-chain calculation in molecular biology starts from.

Frequently asked questions

Try it live

Everything above runs in your browser — open DNA Mechanics and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.