Learning Lab · Polymer theory
The worm-like chain and persistence length
Persistence length as the decay constant of a correlation you can watch, not an abstract parameter.
The ideal chain of the previous lab has no memory: each segment’s direction is independent of the last. Real chains do have memory, and the worm-like chain is the minimal model that captures it. Its single parameter is the persistence length \(l_p\), the contour distance over which the chain forgets which way it was heading.
The definition is operational rather than abstract. Take the unit tangent \(\hat{t}(s)\) at each point along the chain and correlate it with the tangent a distance \(s\) further along. For a worm-like chain that correlation decays exponentially,
\[\langle \hat{t}(0)\cdot\hat{t}(s)\rangle = e^{-s/2l_p}\]in two dimensions, and \(l_p\) is simply the decay constant. The inset beside the chain is that correlation, measured from the conformation you are looking at, with the exponential drawn over it. Pairing them is the point: \(l_p\) is not a fitted abstraction, it is the width of a curve you can watch respond as you drag the slider.
Two regimes
The behaviour is controlled entirely by the ratio of contour length to persistence length, \(L/l_p\), which the readout reports.
When \(L \gg l_p\) the chain forgets its direction many times over its length, and on scales larger than \(l_p\) it looks like a random walk again, though one taking bigger and stiffer steps. This is why the ideal chain remains useful for long polymers even though it is obviously wrong locally: stiffness renormalises the step size without changing the exponent.
When \(L \lesssim l_p\) the chain cannot turn appreciably over its own length and behaves as a rod. Bending it then costs real energy.
The number that makes chromatin hard
Naked DNA has \(l_p \approx 50\) nm, about 150 base pairs. A nucleosome wraps roughly 147 base pairs of DNA in not quite two tight turns around the histone octamer. The cell is therefore bending DNA through a full circle over roughly one persistence length. Well inside the rod regime, and thermodynamically expensive; the binding energy of the histone core is what pays for it.
This is also why the chromatin fibre is not simply “DNA, but longer”. Its mechanical properties are those of the nucleosome chain, not of the double helix, and its effective stiffness is an emergent quantity that has to be measured at the scale of interest. Doing that measurement, and finding that coarse-graining introduces stiffness even where none existed, is what the coarse-graining lab is about.
Why it matters · A single length, describing how far a chain remembers its own direction, separates a floppy coil from a rod and sets the energetic price of bending chromatin.