Loop extrusion

Loop extrusion is easy to state and hard to picture. A ring-shaped protein complex called cohesin binds chromatin, holds the fibre on both sides of itself, and translocates in both directions at once. The chromatin caught between its two anchors is drawn out into a steadily growing loop. The complex reads no sequence and has no destination; it reels until something stops it. What stops it is CTCF.

The region above is not a cartoon. It is the 1 Mb stretch of mouse chromosome 15 (39.2–40.2 Mb) used in the paper, represented as 500 beads of 2 kb each, carrying the four CTCF sites found at that locus. The sliders under the animation set the strength of those four barriers individually.

The kinetics

Three stochastic moves are attempted each Monte Carlo sweep. The interesting one is binding, because the number of cohesins on the fibre is not something we set. It is something the model reaches. Writing \(N_c\) for the number of bound cohesins and \(N_f\) for the number of free sites, the binding and dissociation rates are

\[r_+ = r_\pm^0 \frac{1}{\eta + \exp[\mu(2N_c - N_f - A_0)]}, \qquad r_- = r_\pm^0 \frac{1}{1 + \exp[-\mu(2N_c - N_f - A_0)]}\]

with \(A_0 = 2N_{c0} - N_{f0}\) the asymmetry parameter, and \(\eta = N_{f0}/N_{c0} - 1\) fixed by requiring detailed balance at steady state. The exponent vanishes when exactly \(N_{c0}\) cohesins are bound; away from that point the two sigmoids push the population back. So \(\mu\) controls the size of the fluctuations about \(N_{c0}\), not the mean.

The third move is sliding. Each anchor advances one bead at intrinsic rate \(r_s^0\), giving a total sliding rate \(r_s = 2N_c r_s^0\), provided the target bead is not already occupied by another cohesin or blocked by CTCF. An anchor reaching either end of the region dissociates, taking its cohesin with it.

Move the \(N_{c0}\) slider and watch the count relax to the new target rather than jump to it. That relaxation is the point of writing the kinetics this way: residence time and loop size come out coupled, as they are in the cell, instead of being set independently by hand.

Orientation

CTCF binding sites carry a motif orientation, and in our X-inactivation model a site blocks an anchor arriving from one direction while remaining transparent to one arriving from the other. Loops that persist are therefore those anchored between a convergent pair, meaning two sites that point towards each other. The published Biophysical Journal study took the simpler route of treating CTCFs as permanently bound and blocking from both directions; the orientation toggle switches between the two.

The comparison is worth making directly. With orientation off, the four sites partition the locus into three consecutive domains. With it on, only the convergent pairs (39.216–39.350 Mb and 39.840–40.146 Mb) hold loops, and the long stretch between them has no boundary at all, which shows up immediately as a change in which squares survive on the contact map.

What to vary

The individual barrier sliders are the most informative control. Setting one site of a pair to zero removes that boundary specifically: loops that previously stalled there run past it, and the corresponding square disappears from the contact map while its neighbours remain. This is the in silico analogue of deleting a single CTCF site, and it isolates one boundary’s contribution in a way no global parameter can. In the full model the per-site probabilities are drawn from experimental ChIP-seq profiles; here they are yours to set.

Sliding rate sets processivity, meaning how far a loop grows before its cohesin unbinds. Turn it down and extrusion loses the race against dissociation, so loops never reach their boundaries and domains fail to appear even with the barriers fully in place. Structure needs both ingredients: barriers and a motor processive enough to find them.

Mean cohesin number controls crowding. Raise it and anchors begin to collide, since an anchor cannot step onto a bead another cohesin already occupies. That is one route by which domain structure becomes concentration-dependent.

Reading the contact map

Two features are worth separating. The bright square on the diagonal is the domain: loci inside one loop-anchored region meet each other more often than they meet anything outside it. The sharper spot at that square’s corner is the anchor–anchor contact, the two CTCF sites held together by stalled cohesin. Both fall out of the kinetics above; neither is written into the model.

One caveat worth stating plainly. In the published work the contact map is measured from a three-dimensional polymer whose relaxation competes with extrusion. The central result there is that structure and dynamics can disagree: the contact map can be dominated by intrachromatin attraction while the dynamics are dominated by extrusion. This page has only the one-dimensional kinetics, so its map is inferred from loop anchors. It shows you where extrusion puts contacts, not the full competition.