Fractal and Knot-Free Chromosomes Facilitate Nucleoplasmic Transport

Chromosomes in the nucleus assemble into hierarchies of 3D domains that, during interphase, share essential features with a knot-free condensed polymer known as the fractal globule (FG). The FG-like chromosome likely affects macromolecular transport, yet its characteristics remain poorly understood. Using computer simulations and scaling analysis, we show that the 3D folding and macromolecular size of the chromosomes determine their transport characteristics. Large-scale subdiffusion occurs at a critical particle size where the network of accessible volumes is critically connected. Condensed chromosomes have connectivity networks akin to simple Bernoulli bond percolation clusters, regardless of the polymer models. However, even if the network structures are similar, the tracer’s walk dimension varies. It turns out that the walk dimension depends on the network topology of the accessible volume and dynamic heterogeneity of the tracer’s hopping rate. We find that the FG structure has a smaller walk dimension than other random geometries, suggesting that the FG-like chromosome structure accelerates macromolecular diffusion and target-search.

Figure 1

(a) Polymer configurations: Fractal globule and melted linear chain. Red-to-blue color indicates increasing monomer index. Dots represent macromolecules exploring the FG and MLC geometries. (b) Tracer MSDs in FGs. From top to bottom, $R_{\mathrm{eff}}=0.100$, 0.600, 0.630, and 0.800. The red curve shows the critical subdiffusion with $\alpha \simeq 0.5$.

Figure 2

(a) Anomalous exponent ($\alpha ={\alpha }_{\min }$) vs $R_{\mathrm{eff}}$. The borders between shaded regions show the threshold $R_{\mathrm{eff}}\mathrm{s}$ when some EV motifs close. (b) The representative eight EV motifs and their occurrence probability $P_{(m,n)}$, where $m$ and $n$ are the numbers of blocked edges and vertices. Below each motif, we denote the threshold $R_{\mathrm{eff}}$ when the motif closes. The complete list of 14 EV motifs is depicted in Fig. S7. (c) Example showing how we create the accessible lattice (AL): (Top) an obstacle geometry with two exemplified motifs, (1,4) and (3,4). If $0.5<R_{\mathrm{eff}}<0.625$, the (1,4) is open (red) but the (3,4) (blue) closed. (Bottom) The AL in red marks the open (1,4) motif. (d) Fraction of open motifs $p_{\mathrm{open}}$ as a function $R_{\mathrm{eff}}$. The dotted line shows $p_c$ for the Bernoulli bond percolation (cubic lattice).

Figure 3

(a) Rescaled MSDs using Eq. (1) for five different $R_{\mathrm{eff}}\mathrm{s}$ in FGs. This analysis yields $d_w^{\prime }=3.84$. The solid line depicts the expected scaling at $t\to \infty$. (b) Rescaled displacement PDFs $P(r|t)$ using Eq. (2) at five lag times and $R_{\mathrm{eff}}=0.63$ in FGs. The best collapse yields $d_w=3.49$ along with the corresponding fit (line). (c) Mass fraction of the largest cluster in the FG (red) and MLC (blue) at $R_{\mathrm{eff}}\approx R_{\mathrm{cr}}$. The symbol represents the results for the original ALs (circle) and the ALs after the weak edges are removed (square). Details are described in Ref. [20]. (d) Schematics explaining the difference in the walk dimensions between the FG and MLC geometries. The black, red, and green lines constitute the original AL and correspond to the spanning cluster, weak edges, and isolated small clusters, respectively. The small clusters are the fraction of the AL connected to the spanning cluster via the weak edge. (e) The walk dimensions obtained from (a) and (b) are shown for the FG and MLC, with the speculated $d_w^{\prime }=2/\alpha$ from several experiments [8, 17, 63, 64, 65].