Effect of loops on the mean-square displacement of Rouse-model chromatin

Chromatin polymer dynamics are commonly described using the classical Rouse model. The subsequent discovery, however, of intermediate-scale chromatin organization known as topologically associating domains (TADs) in experimental Hi-C contact maps for chromosomes across the tree of life, together with the success of loop extrusion factor (LEF) model in explaining TAD formation, motivates efforts to understand the effect of loops and loop extrusion on chromatin dynamics. This paper seeks to fulfill this need by combining LEF-model simulations with extended Rouse-model polymer simulations to investigate the dynamics of chromatin with loops and dynamic loop extrusion. We show that loops significantly suppress the averaged mean-square displacement (MSD) of a gene locus, consistent with recent experiments that track fluorescently labeled chromatin loci. We also find that loops reduce the MSD's stretching exponent from the classical Rouse-model value of 1/2 to a loop-density-dependent value in the 0.45–0.40 range. Remarkably, stretching exponent values in this range have also been observed in recent experiments [Weber et al., Phys. Rev. Lett. 104, 238102 (2010); Bailey et al., Mol. Biol. Cell 34, ar78 (2023)]. We also show that the dynamics of loop extrusion itself negligibly affects chromatin mobility. By studying static “rosette” loop configurations, we also demonstrate that chromatin MSDs and stretching exponents depend on the location of the locus in question relative to the position of the loops and on the local friction environment.

Figure 1

Simplified illustration of a (free-boundary-condition) Rouse polymer with LEFs and its corresponding dynamical matrix $A$. The two LEFs are modeled as non-nearest-neighbor springs and color-coded as red (dark) and green (light), and their effects on the dynamical matrix are color-coded accordingly. The subpolymers are numbered from left to right.

Figure 2

Time evolution of polymer compaction by loops. Compaction is defined as the ratio of mean-squared radius of gyration $\langle R_G^2\rangle$ [Eq. (35)] of a looped polymer to that of a Gaussian polymer without loop. The colored (nonsolid) lines represent CTCF loop extrusion simulations run on three different genomic regions using: blue (dashed), 32–38 Mb of Chr 12; green (dotted), 4.8–10.8 Mb of Chr 13; red (dash-dotted), 52–58 Mb of Chr 18. The parameters used in the CTCF LEF simulations are given in Table 1, and each polymer has 48 LEFs bound to it. The solid black line represents the loop extrusion simulation using the random loop model. The light, thin lines represent independent simulation runs. The dark, thick lines are the averages of five independent simulation runs. The medium-thick curves are exponential fits to the averages, which give steady-state $\langle R_G^2\rangle$ values of 0.44, 0.40, 0.43, and 0.28 for blue (dashed), green (dotted), red (dash-dotted), and black (solid), respectively. The fitted decay times are: blue (dashed), 1126 s; green (dotted), 1030 s; red (dash-dotted), 718 s; black (solid), 876 s.

Figure 3

Abstract representations, following Ref. [90], of three randomly chosen snapshots of loop configurations (a), (b), and (c) in the steady-state regime taken from three independent CTCF LEF simulations of a 6 Mb region of mouse Chr 12 (32–38 Mb), Chr 13 (4.8–10.8 Mb), and Chr 18 (52–58 Mb), respectively, and (d) taken from the LEF simulation using random loop model. The loop extrusion simulations are done on 600 lattice sites, each representing 10 kb of genomic length. The black, horizontal line represents stretched-out chromatin; each semicircle represents a loop extruded on the chromatin polymer by connecting monomers at the base of the loop. There are 48 loops in each loop configuration.

Figure 4

Time series of Rouse polymers' potential energy for different loop extrusion rates. The red (dashed) line shows the simulated spring potential energy versus time for a free Rouse polymer without loops, using the parameters given in Table 2. The black (solid) line shows the simulated spring potential energy versus time for a random-loop-model Rouse polymer, for 48 LEFs and the parameters given in Tables 1 and 2. The blue (dotted) and green (dash-dotted) lines show the potential energy versus time for random-loop-model polymers, populated by 48 LEFs, when all of the loop extrusion rates given in Table 1 are increased by a factor of 10 and 100, respectively, while the polymer parameters give in Table 2 are held fixed. The resolution of the line points in the plot is set to 70 s to ensure a clear view.

Figure 5

(a) Simulated MSDs of Rouse-model polymers with free boundary conditions for the classical Rouse model with no loops, shown in red (dashed), for the CTCF model with loops, shown in blue (dotted), and for the random loop model, shown in green (dash-dotted). (b) The corresponding simulated MSD stretching exponent, $\alpha \left(t\right)$, calculated according to Eq. (37) for no loops (red, dashed), for the CTCF model (blue, dotted), and for the random loop model (green, dash-dotted). The solid black line in panel (a) shows the theoretical mean MSD given by Eq. (28), and then the theoretical mean stretching exponent is calculated using Eq. (37) and shown in panel (b). In panels (a) and (b), the thin lines correspond to the results for 60 individual beads, uniformly distributed along the chromatin. The thick lines correspond to the average across all 60 beads. Each thin line is an average over 30 independent simulations with identical simulation parameters. The LEF and Rouse simulation parameters used are given in Tables 1 and 2.

Figure 6

(a) Comparison between the simulated MSDs of Rouse-model polymers with free, periodic and fixed boundary conditions, shown as the solid, dashed, and dotted lines, respectively, for the classical Rouse polymer with no loops, shown in red (upper cluster), for the CTCF model with 48 LEFs, shown in blue (middle cluster), and for the random loop model with 48 LEFs, shown in green (bottom cluster). (b) Comparison between the corresponding simulated MSD stretching exponent, $\alpha \left(t\right)$, calculated according to Eq. (37) for no loops (red, upper cluster), for the CTCF model with 48 LEFs (blue, middle cluster), and for the random loop model with 48 LEFs (green, bottom cluster), each considered separately with the three different boundary conditions. In panels (a) and (b), each line is averaged over 30 independent simulations and averaged over 60 individual beads, which are uniformly distributed along the chromatin.

Figure 7

(a) Simulated MSDs of Rouse-model polymers with periodic boundary conditions for the CTCF model with different numbers of LEFs, shown in red (upper solid), cyan (dotted), blue (dash-dotted), magenta (middle solid), and green (bottom thick), corresponding to 0 (classical Rouse model), 24, 48, 72, and 144 LEFs, respectively. (b) The corresponding simulated MSD stretching exponent, $\alpha \left(t\right)$, for the CTCF models with 0 (red, upper solid), 24 (cyan, dotted), 48 (blue, dash-dotted), 72 (magenta, middle solid), and 144 (green, lower thick) LEFs. The dashed black line in panel (a) shows the theoretical mean MSD given by Eq. (B11). The dashed black line in panel (b) is the corresponding theoretical mean stretching exponent, calculated using Eq. (37), applied to Eq. (B11). In panels (a) and (b), each line is averaged over 30 independent simulations and averaged over 60 individual beads, which are uniformly distributed along the chromatin.

Figure 8

(a) Comparison between the MSDs versus time of polymers under periodic boundary conditions with 48 randomly located static loops (orange, dash-dotted) and periodic polymers with dynamic loops, evolving according to the CTCF model with 48 LEFs (blue, dotted). The dashed red lines represent MSDs of the polymer under periodic boundary conditions for the classical Rouse model with no loops. (b) The corresponding time-dependent stretching exponents. The LEF and Rouse simulation parameters used are given in Tables 1 and 2. In panels (a) and (b), the thin lines correspond to results of 60 individual beads, uniformly distributed along the polymer; the thicker lines are averages over the 60 beads. The result for each individual bead is averaged over 30 independent simulations with identical simulation parameters.

Figure 9

(a) Loop size distribution, (b) backbone segment length distribution, and (c) chromatin compaction distribution (as measured by the backbone length ratio) for the CTCF loop extrusion model with different numbers of LEFs of 24 (cyan, solid), 48 (blue, dashed), 72 (magenta, dotted), and 144 (green, dash-dotted). The CTCF loop extrusion parameters are given in Table 1. Distributions in panels (a) and (b) are plotted using a log-linear scale. The mean loop size in panel (a) are 107.0 (solid cyan, 24 LEFs), 90.7 (dashed blue, 48 LEFs), 77.9 (dotted magenta, 72 LEFs), and 51.8 (dash-dotted green, 144 LEFs) kb, with standard deviations of 82.6, 68.6, 57.8, and 37.0 kb, respectively. The mean backbone segment length in panel (b) are 237.3, 120.2, 80.1, and 39.9 kb, respectively, with standard deviations of 211.6, 103.4, 65.2, and 25.6 kb, respectively. The mean backbone length ratios (measuring the chromatin compaction) are 0.6254, 0.4301, 0.3243, and 0.2172, respectively, with standard deviations of 0.0524, 0.0486, 0.0404, and 0.0218, respectively. Each distribution shown is obtained from the loop configurations corresponding to 30 000 time-points (30 independent LEF simulations and 1000 time-points from each independent simulation).

Figure 10

(a) Squared end-to-end distance, $R_{EE}^2$, plotted as a function of polymer contour length, measured in units of lattice number, $N$, for chromatin polymers with 24 (cyan, middle solid), 48 (blue, dashed), 72 (magenta, dotted), and 144 (green, dash-dotted) LEFs, which extrude loops according to the CTCF model. The solid black (top) line shows the theoretical scaling of $R_{EE}^2$ for the Gaussian polymer without loops. The short, dashed lines plotted next to the colored (nontop) lines indicate the theoretical scalings of $R_{EE}^2$ for the looped polymers based on the compaction fraction estimated in Fig. 9. $R_{EE}^2$ between any two loci, $i$ and $j$, is calculated by Eq. (A4). (b) Mean monomer density, as measured by $N/R_G^3$, plotted as a function of polymer contour length, $N$, for chromatin polymers with 24 (cyan, middle solid), 48 (blue, dashed), 72 (magenta, dotted), and 144 (green, dash-dotted) LEFs. The black (bottom, solid) line indicates the theoretical scaling of the 3D mean monomer density for the Gaussian polymer without loops, admitting a power law of power $-1/2$. Each colored (nonbottom) curve presented is an averaged of 300 loop configurations from 30 independent LEF simulations (10 configurations from each LEF simulation).

Figure 11

The rosette configuration investigated in Sec. 4c. Shown in blue line is the chromatin polymer itself; loop bases are highlighted as the brown circles. Each loop is 17 lattice sites long, and each backbone segment between neighboring loops is 8 lattice sites long. Thus, the fraction of the chromatin polymer inside loops is 0.68 in this configuration. There are 24 repetitions of the loop-backbone structure within the periodic chromatin polymer, making a total of 600 lattice sites, representing a genome of 6 Mb. The dashed blue line represents repetitions of the identical loop-backbone structures. Examples of each family of locus locations are labeled tip (yellow dot), base (magenta dot) and backbone (cyan dot).

Figure 12

(a) Simulated MSDs of the Rouse-model polymer with the rosette structure illustrated in Fig. 11 for three different families of locations: tip (yellow, dashed), base (magenta, dash-dotted), and backbone (cyan, dotted). (b) The corresponding simulated MSD stretching exponent for different families of locations on the rosette-structured polymer: tip (yellow, dashed), base (magenta, dash-dotted), and backbone (cyan, dotted). In panels (a) and (b), each line shows the averaged result of 720 independent simulations for beads in each family. The solid black lines in panels (a) and (b) show the theoretical MSD and stretching exponent, respectively, of a Rouse-model polymer without loops, with periodic boundary conditions, as given in Eqs. (B11) and (37).

Figure 13

(a) Comparison between the MSDs of periodic polymers without loops, subject either to uniform friction, shown in red (solid), or to lognormal-distributed (nonuniform) friction, shown in green (dashed). For the polymer subject to lognormal-distributed nonuniform friction, each bead has a fixed friction coefficient, drawn from the scaled Lognormal(0, log(2)) with mean of $\zeta$, given in Table 2. (b) The corresponding MSD stretching exponents, calculated according to Eq. (37), for the uniform friction case (red, solid) and the lognormal-distributed (nonuniform) case (green, dashed). In panels (a) and (b), thin lines represent 60 individual beads uniformly distributed along the polymer, and each thin line is an average of 30 independent simulations. Each thicker line is an average of its corresponding thin lines.

Figure 14

The hypothetical replication factory configuration discussed in Sec. 4e. The shaded region is where the friction coefficient is 10 times higher than in the unshaded region. Loci experiencing higher friction coefficients are all located at the bases of the loops. The midpoints of each loop are labeled as “tip,” and the bases of each loop are labeled as “base.” Total number of LEFs is 24. Each loop has size of 24. The backbone length between neighboring loops is 1, which is the minimal length a backbone can form given that the LEFs will block each other at distance 1. The dashed line represents repetitions of the identical loop structure.

Figure 15

(a) Shown in blue (dotted) and green (dash-dotted) lines are the mean MSDs for the loop tips and bases, respectively, of a rosette polymer in the nonuniform-friction environment, as illustrated in Fig. 14. Shown in red (solid) and magenta (dashed) lines are the mean MSDs for the loop tips and bases, respectively, of the same rosette polymer, but in a uniform-friction envioronment, i.e. without the region of ten-fold friction shown in Fig. 14. Each line is an average of 720 independent simulations for the tip beads or base beads, indifferent about the loop repetitions to which each bead belongs, as a result of symmetry. (b) The corresponding averaged stretching exponents.

Figure 16

(a) Simulated MSDs of Rouse-model polymers with fixed boundary conditions for the classical Rouse model with no loops, shown in red (dashed), for the CTCF model with loops, shown in blue (dotted), and for the random loop model, shown in green (dash-dotted). (b) The corresponding simulated MSD stretching exponent, $\alpha \left(t\right)$, calculated according to Eq. (37) for no loops (red, dashed), for the CTCF model (blue, dotted), and for the random loop model (green, dash-dotted). The solid black line in panel (a) shows the theoretical mean MSD given by Eq. (C7), and then the theoretical mean stretching exponent is calculated using Eq. (37) and shown in panel (b). In panels (a) and (b), the thin lines correspond to the results for 60 individual beads, uniformly distributed along the chromatin. The thick lines correspond to the average across all 60 beads. Each thin line is an average over 30 independent simulations with identical simulation parameters. The LEF and Rouse simulation parameters used are given in Tables 1 and 2.