The fractal globule

The chain above starts as an open self-avoiding coil and is then squeezed by a shrinking spherical wall. Watch the circle contract. The chain has only local moves available, a corner flip and a crankshaft rotation, so it can fold but it can never pass through itself, and it has to accommodate the wall with whatever rearrangements those moves allow.

Confinement rather than attraction is what makes this affordable. An attracting chain has to find its collapsed state one favourable contact at a time, which at even a few hundred monomers takes minutes of computation. A shrinking wall compacts it geometrically: 2500 monomers reach 90% density in a few seconds, live, in a browser tab.

The distinction from the equilibrium globule matters because reaching the equilibrium globule requires the chain to thread through itself. Untangling is slow at high density, and Grosberg’s observation is that a real collapse is usually not slow enough to allow it. The chain instead falls into a kinetically trapped state and stays there: dense, unknotted, and self-similar. A chromosome, folded once during its history and never given millions of years to equilibrate, is a plausible candidate for exactly this.

The size argument is cleaner than the equilibrium one

Self-similarity plus space-filling is a strong pair of assumptions. Together they say that a subchain of \(s\) monomers is itself a small globule of the same density as the whole. So \(s\) monomers occupy a volume \(\sim R(s)^3\) with the same constant of proportionality as \(N\) monomers occupying \(R(N)^3\), and

\[R(s) \sim s^{1/3} \quad \text{for every } s\]

Compare that with the equilibrium case, which needed a screening argument, an ideal regime, and a crossover at \(N^{2/3}\). Here there is no crossover at all, and its absence is the signature. The dashed \(N^{2/3}\) marker is drawn on the plot for comparison, and the fitted \(\nu\) on either side of it should agree with each other rather than differ.

Contact probability, and why it is the headline

The fractal globule has no correlation hole. It is homogeneous rather than actively self-avoiding at short range, so unlike the self-avoiding walk it does not suppress its own near-contacts beyond what its size accounts for. The naive estimate is therefore valid:

\[P(s) \sim \frac{1}{R(s)^d} = \frac{1}{\left(s^{1/3}\right)^3} = \frac{1}{s}\]

so \(\eta = 1\) exactly, with no anomalous correction and nothing to fit.

That number is worth sitting with. When Lieberman-Aiden and colleagues published the first human Hi-C maps in 2009, the contact probability they measured fell off as \(1/s\) across roughly the 0.5 to 7 Mb range. A model with no free parameters, derived from two geometric assumptions, lands on the measured exponent. It is the single strongest piece of evidence for the fractal globule picture of chromatin, and it is why the \(s^{-1}\) reference line is drawn on the lower panel of both globule pages here.

What this demo actually measures

The collapse gets close. Once the squeeze finishes, the readouts settle at about

	crumpled (measured)	equilibrium (measured)	crumpled (theory)
\(\nu\) below \(N^{2/3}\)	0.36 – 0.37	0.46	0.333
\(\eta\)	0.81 – 0.87	1.23	1.00
density	0.91	—	~1

Those figures are stable to the second digit over tens of thousands of sweeps, and they put the two models on opposite sides of each other in both exponents. The two models are clearly separated, and each sits on the correct side of the other, which is the comparison this pair of pages exists to make.

Neither exponent is exact, and it is worth knowing why. A globule of \(N\) monomers has a radius of only \(N^{1/3}\), so at \(N = 2500\) the whole object spans about 17 lattice units. \(R(s)\) therefore has barely one decade to move in before it saturates against the size of the globule itself, and an exponent fitted over one decade is not a three-digit measurement. The shaded band on the lower panel marks the range actually used.

That saturation is also why the \(\nu\) reported above the crossover is not worth much for either model. It is fitted in the region where subchains are already comparable to the whole globule, so it wanders between runs rather than settling. In an infinite crumpled globule there is no crossover at all and \(\nu = 1/3\) everywhere; in a finite one, any subchain approaching the size of the object runs out of room, and both models flatten. The signature worth reading here is the exponent below the crossover, where the two differ by a clear and reproducible margin.

Matching one exponent is also weaker evidence than it feels like, because several mechanisms produce something close to \(1/s\) over a limited range. The model comparison page puts \(P(s)\) next to \(R(s)\), \(\tau(s)\) and the MSD precisely so that models can be judged on more than one number at a time.

Dynamics, without reptation

The equilibrium globule needed reptation because the chain was threaded through itself. The fractal globule is unknotted by construction, so it does not, and the scaling shortcut applies directly with \(\nu = 1/3\):

\[\gamma = 2\nu + 1 = \tfrac{5}{3}, \qquad \alpha = \frac{2\nu}{\gamma} = \frac{2/3}{5/3} = \tfrac{2}{5}\]

Both are standard values, obtained here from one substitution. The measured MSD exponent for chromatin loci sits in the range 0.4 to 0.5, which is close.

The one place the shortcut still needs help is hydrodynamics. The blob argument gives \(\alpha_{\text{Zimm}} = 2/3\) regardless of \(\nu\), but the accepted value for a fractal globule is nearer \(0.42\). The reason is that the structure is dense enough to screen hydrodynamic interactions as well as excluded volume, so treating a subchain as a free-draining sphere in solvent is no longer right. Recovering \(0.42\) needs the fuller treatment of Tamm and colleagues, and this page cites it rather than deriving it.

What to try

Watch the squeeze itself. The wall contracts from the coil’s own radius down to the one that gives the target density, over the number of sweeps you set. The chain crumples inward locally, which is the whole mechanism: nothing tells it to be self-similar, and it becomes so because folding locally is the only thing it can do.

Flip the toggle. The equilibrium construction has the same monomers in a sphere of comparable size, so what changes is organisation rather than compaction. Neighbouring stretches of the crumpled chain stay together in space, giving visible coloured territories; in the equilibrium walk the colour ramp is stirred right through the volume. Territorial organisation is one of the crumpled globule’s testable predictions, and chromosomes do occupy distinct territories in the nucleus.

Squeeze too fast. Drop the squeeze down to one or two thousand sweeps and watch the density readout fail to reach its target. This is not a bug and it is worth understanding. Once the wall passes a monomer, that monomer’s only legal moves are ones that take it further inward; if the wall outruns the chain’s ability to rearrange, the chain jams in an extended state instead of crumpling. Real collapses have the same competition between the rate of compaction and the rate of local relaxation. Here the jammed limit is partly an artefact of the lattice move set, so treat the slow end of the slider as the physical one.

Limitations of this page

The crumpled state here is a real simulation: a self-avoiding chain, local moves only, compacted by a moving boundary. The equilibrium state on the toggle is not. It is constructed directly as an ideal walk in a sphere, because reaching equilibrium from a collapsed state requires the chain to thread through itself, and that is precisely the slow process no browser can wait for. So the comparison is between one simulated limit and one constructed limit rather than between two runs of the same experiment.

The chain lengths are also small by the standards of this question. A few thousand monomers gives about one decade of \(R(s)\), which is enough to separate \(1/3\) from \(1/2\) but not enough to pin either to three digits.