Nonequilibrium Chromosome Looping via Molecular Slip Links

We propose a model for the formation of chromatin loops based on the diffusive sliding of molecular slip links. These mimic the behavior of molecules like cohesin, which, along with the CTCF protein, stabilize loops which contribute to organizing the genome. By combining 3D Brownian dynamics simulations and 1D exactly solvable nonequilibrium models, we show that diffusive sliding is sufficient to account for the strong bias in favor of convergent CTCF-mediated chromosome loops observed experimentally. We also find that the diffusive motion of multiple slip links along chromatin is rectified by an intriguing ratchet effect that arises if slip links bind to the chromatin at a preferred “loading site.” This emergent collective behavior favors the extrusion of loops which are much larger than the ones formed by single slip links.

Figure 1

Nonequilibrium chromosome looping. (a) Schematic of our model of diffusing slip links. (b) Probability of nonequilibrium loop formation in exactly solvable 1D models as a function of loop size $l$. Curves correspond to models involving (i) extrusion, (ii) diffusion, and (iii) slip links. Parameters are $k_{\mathrm{off}}^{-1}=20\min$ and (i) $v=10\mathrm{kbp}/\min$; (ii), (iii) $D=25{\mathrm{kbp}}^2/\mathrm{s}$; (iii) ${\sigma }_{\mathrm{sl}}=1\mathrm{kbp}$, and $c=1$ [11]. (c) Average loop size for models involving diffusion and slip links. Parameters are as in (b), apart from $D$ which is varied. (d) Nonequilibrium looping probability for a slip link, computed from BD simulations, with different $k_{\mathrm{off}}^{-1}=\tau$. The blue line shows an exponential fit for $k_{\mathrm{off}}^{-1}=25$ min. (e) Analysis of ChIA-PET experiments for CTCF contacts within a Mbp [12] (log-linear plot).

Figure 2

Multiple slip links and the osmotic ratchet. (a),(b) Results from 1D simulations of diffusing slip links rebinding either (a) randomly, or (b) at a loading site. Plots show the time average of the largest loop (for the case with looping weight see [29], Fig. S4). Parameters are ${\sigma }_{\mathrm{sl}}=1\mathrm{kbp}$, $L=1000\mathrm{kbp}$, $k_{\mathrm{on}}^{-1}=k_{\mathrm{off}}^{-1}=25\min$, the diffusion coefficient of a monomer is $D\sim 33.35{\mathrm{kbp}}^2/\mathrm{s}$, while $N$ is varied. There are reflecting boundary conditions at the two ends of the fiber. Typical configurations for $N=20$ are shown as insets, as “looping diagrams” showing the loop network [29]. The dotted line in (a),(b) denotes the average loop size with a single slip link; the solid line in (b) is a fit to $a+b\log N$ (see text). (c) Probability distribution of the largest loop size for different $N$ with or without loading (for the case with looping weight see [29], Fig. S4). (d) Results from 3D BD simulations of multiple slip links with loading, for $L=3000\mathrm{kbp}$, $k_{\mathrm{off}}^{-1}=25\min$. The plots show the probability distribution of the size of the largest loop for $N=1$ (with $k_{\mathrm{on}}\to \infty$), and $N=3$ (with $k_{\mathrm{on}}=10k_{\mathrm{off}}$, snapshot shown as an inset).