The equilibrium globule

Swelling is not the only thing a chain can do. Make the monomers attract one another, or squeeze the chain into a small box, and beyond a threshold the coil collapses into a dense globule. The demo above builds one directly: an ideal random walk confined to a sphere of radius proportional to \(N^{1/3}\). Confinement stands in for being surrounded by the rest of the globule, and the walk is ideal rather than self-avoiding for the screening reason set out below.

The toggle switches to the other route, a self-avoiding chain squeezed by a shrinking wall, which is the subject of the fractal globule page. Flipping between them on the same axes is the quickest way to see the difference: the ideal walk gives \(\nu \approx 0.46\) below the crossover and \(\eta \approx 1.23\), while the squeezed chain gives \(\nu \approx 0.37\) and \(\eta \approx 0.82\).

Overall size is immediate

At equilibrium the globule behaves like a liquid drop. Surface effects aside, the interior sits at a roughly constant density \(\rho\), set by the balance between attraction and excluded volume and not by \(N\). So \(N\) monomers occupy a volume \(N/\rho\), and

\[R(N) \sim \left(\frac{N}{\rho}\right)^{1/3} \sim N^{1/3}\]

There is nothing subtle in that: it is what constant density means. The interesting question is the internal one.

What a subchain looks like from inside

Take \(s\) consecutive monomers somewhere in the middle of the globule and ask how far apart their ends are. The tempting answer is that they are compressed like everything else, giving \(s^{1/3}\). That is wrong, for a reason worth understanding.

Our subchain is surrounded almost entirely by other parts of the same chain, at the same density as itself. Flory’s screening argument says that a chain in a melt of identical segments cannot tell its own monomers from its neighbours’ monomers. The excluded-volume repulsion that would normally swell it is screened, because pushing a neighbour away only brings another indistinguishable segment into the vacated space. There is no free energy gain in swelling, so the subchain behaves ideally:

\[R(s) \sim b\, s^{1/2}\]

This cannot continue indefinitely. The ideal estimate grows faster than \(s^{1/3}\), so at some length the subchain would have to be larger than the globule that contains it, which is impossible. Setting the two equal locates the crossover:

\[s^{1/2} \sim N^{1/3} \quad \Longrightarrow \quad s^* \sim N^{2/3}\]

Beyond \(s^*\) the subchain simply fills whatever part of the globule it occupies, and \(R(s)\) flattens off. The readouts fit \(\nu\) separately on either side of that dashed line, and it is a derived number rather than a fitted one, so watch it move as you change \(N\).

The same argument gives the contact exponent. Below the crossover the subchain is ideal, so the reasoning of the excluded volume page applies exactly, with \(\eta = d\nu = 3/2\). Above it, \(R(s)\) is constant, and a contact probability computed from a size that no longer changes cannot change either. \(P(s)\) plateaus.

Dynamics: where the shortcut breaks

The scaling shortcut gave four exponents from \(\nu\) alone. Here it fails, and the failure is instructive.

At this density the chain cannot move sideways. Every direction is blocked by other segments, so a subchain can only relax by sliding along its own contour, threading through the tube of obstacles that surrounds it. This is reptation, and it changes the first line of the argument. The relevant length is no longer the subchain’s spatial size \(s^{\nu}\) but the contour length of its tube, which grows linearly, as \(s\). Curvilinear diffusion along that tube slows in proportion to the number of monomers being dragged, so

\[\tau(s) \sim s \times s \times s = s^3\]

The mean-squared displacement is a random walk on top of a random walk. Along the tube the motion is Rouse-like, so the contour distance travelled grows as \(\ell^2 \sim t^{1/2}\). But the tube itself is shaped like an ideal chain in three dimensions, so real-space distance squared grows only like contour distance, not like its square. Composing the two,

\[\langle \Delta r^2 \rangle \sim \ell \sim \left(t^{1/2}\right)^{1/2} = t^{1/4}\]

giving \(\alpha = 1/4\), the slowest of the four models in the comparison. A chromosome that behaved this way would be effectively frozen on the timescale of transcription, which is one of several reasons to doubt that chromatin is an equilibrium globule.

What to try

Watch the colour ramp. It runs along the chain, and here the colours are thoroughly stirred through the volume: a segment from one end sits happily next to a segment from the other. Toggle to the fractal construction and that mixing disappears. The absence of mixing is the subject of the next page.

Then watch where the knee in \(R(s)\) sits as you change the chain length. It should track the dashed \(N^{2/3}\) marker, which is derived rather than fitted.

Two caveats on the numbers

The measured \(\nu\) below the crossover comes out near \(0.46\) rather than \(0.50\), because the fitting window unavoidably runs part-way into the knee. A globule of \(N\) monomers is only \(N^{1/3}\) across, so there is barely a decade of clean ideal behaviour between the bond length and the confinement radius at any chain length a browser can handle.

The measured \(\eta\) lands near \(1.23\) rather than \(1.5\) for the same reason, with the plateau above the crossover pulling the fit down. Both numbers move the right way when you enlarge the sphere with the confinement slider, which is a more useful check than taking the fitted values at face value.