The self-avoiding walk

The ideal chain lets a monomer sit anywhere, including on top of another monomer. Real chains cannot do that, and the consequence is not a small correction. It changes the exponent.

Flory’s estimate

The argument balances two free energies. Confining a chain of \(N\) segments to a region of size \(R\) costs entropy, and for a Gaussian chain that cost is

\[\frac{F_{\text{ent}}}{k_BT} \sim \frac{R^2}{N b^2}\]

Against it, monomers repel wherever they meet. With \(N\) monomers in a volume \(R^d\) the density is \(N/R^d\), and counting pairwise encounters gives

\[\frac{F_{\text{rep}}}{k_BT} \sim v \frac{N^2}{R^d}\]

Minimising the sum over \(R\) gives Flory’s result,

\[R \sim b\, N^{\nu}, \qquad \nu = \frac{3}{d+2}\]

which is \(3/5\) in three dimensions. The exponent is not exact, but it is remarkably close to the accepted value of about \(0.588\), and it is right for a transparent reason: the chain swells until repulsion and entropy cost the same.

The simulation above is a genuine self-avoiding walk on a cubic lattice, sampled with the pivot algorithm. A monomer is chosen at random, one arm of the chain is rotated about it by a lattice symmetry, and the move is kept only if the result still avoids itself. The \(R(s)\) panel measures the internal distance between monomers \(s\) apart, and the fitted slope should settle near \(0.6\). Switch excluded volume off and it drops to \(0.5\), which is the control.

The second measurement

Now the part that is easy to miss. Chain size and contact probability are different observables, and knowing one does not automatically give you the other.

The obvious guess is that it does. A subchain of \(s\) monomers pervades a region of size \(R(s)\). If one end were smeared uniformly through that region, the chance of finding the other end within a small contact radius would be the ratio of volumes,

\[P(s) \sim \frac{1}{R(s)^d} \sim s^{-d\nu}\]

For an ideal chain this is exact: \(d\nu = 3 \times \tfrac{1}{2} = \tfrac{3}{2}\), matching the known \(\eta = 1.5\). Switch excluded volume off in the demo and the two curves in the lower panel lie on top of each other, the measured gap sitting within about \(0.05\) of zero.

Switch it back on and they separate. The naive formula gives \(3\nu \approx 1.77\). The measured exponent is clearly steeper. Both lines come from the same simulation, and the naive curve is built from the measured \(R(s)\) rather than an assumed exponent, so the gap between them cannot be an artefact of picking the wrong \(\nu\).

What the gap means

A self-avoiding chain does more than occupy a larger volume. It actively pushes its own segments apart at short range. Around any monomer there is a region that other monomers of the same chain are unusually unlikely to enter, over and above what the overall swelling would suggest. This is the correlation hole, first analysed by des Cloizeaux.

Getting the exponent from first principles needs renormalisation-group machinery, and this page does not attempt it. The accepted asymptotic value is near \(2.2\).

Two honest caveats about the number you will actually read off. Chains of a few hundred monomers are what a browser can equilibrate, and that is not long enough to pin the second decimal place: widen the fitting window and the measured \(\eta\) moves around between roughly \(2.0\) and \(2.3\), because the curve is not a clean power law all the way from the lattice scale to the chain scale. And the ideal-chain control does not land on exactly \(1.5\) either, for a reason worth knowing. A contact is scored inside a sphere of finite radius \(a\), and while \(a\) is not much smaller than \(R(s)\) that sphere samples a visibly non-uniform part of the pair-distance distribution. The bias goes as \(0.6\,(a/R)^2\), which is why the fit starts only once \(R(s)\) has cleared \(a\) by a factor of four. The shaded band on the lower panel shows the range actually used.

What survives all of that is the comparison, because both curves are affected the same way. The gap is real, it is reproducible, and it does not depend on the contact radius you choose. Move the radius slider and the curves shift up and down together while the separation between their slopes stays put.

The practical consequence for chromatin is direct. Hi-C measures \(P(s)\), not \(R(s)\). Converting one into the other requires a model, and as this page shows, the conversion that looks obvious is only valid for a chain with no excluded volume at all. The model comparison page collects which models get which pair of exponents.