Looping probabilities in model interphase chromosomes

Abstract

Fluorescence in-situ hybridization (FISH) and chromosome conformation capture (3C) are two powerful techniques for investigating the three-dimensional organization of the genome in interphase nuclei. The use of these techniques provides complementary information on average spatial distances (FISH) and contact probabilities (3C) for specific genomic sites. To infer the structure of the chromatin fiber or to distinguish functional interactions from random colocalization, it is useful to compare experimental data to predictions from statistical fiber models. The current estimates of the fiber stiffness derived from FISH and 3C differ by a factor of 5. They are based on the wormlike chain model and a heuristic modification of the Shimada-Yamakawa theory of looping for unkinkable, unconstrained, zero-diameter filaments. Here, we provide an extended theoretical and computational framework to explain the currently available experimental data for various species on the basis of a unique, minimal model of decondensing chromosomes: a kinkable, topologically constraint, semiflexible polymer with the (FISH) Kuhn length of l(K) = 300 nm, 10 kinks per Mbp, and a contact distance of 45 nm. In particular: 1), we reconsider looping of finite-diameter filaments on the basis of an analytical approximation (novel, to our knowledge) of the wormlike chain radial density and show that unphysically large contact radii would be required to explain the 3C data based on the FISH estimate of the fiber stiffness; 2), we demonstrate that the observed interaction frequencies at short genomic lengths can be explained by the presence of a low concentration of curvature defects (kinks); and 3), we show that the most recent experimental 3C data for human chromosomes are in quantitative agreement with interaction frequencies extracted from our simulations of topologically confined model chromosomes.

Figure 1

Mean-square internal distances R²(|N₂ − N₁|) between targeted interphase chromosome sites at N₁ and N₂ mega-basepairs (Mbp) from one chosen end of the fiber: comparison between currently available FISH data and our model for interphase chromosomes (for details, see (19)). (Brown squares) FISH data for yeast chromosomes (8). (Violet squares) FISH data for human chromosomes (7). The short length-scale distances (up to ≈4 Mbp) were measured on the (nearly equilibrated) end region 4p16.3 on human chromosome 4 (19). (Black lines) Exact wormlike chain (WLC) expression, Eq. 1: l_K = 300 nm (solid line) and l_K = 56 nm (dashed line). (Red line) Model human chromosomes. (Cyan line) Model human chromosomes with N₁ = 0. (Green line) Model yeast chromosomes.

Figure 2

Rescaled interaction frequencies X(|N₂ − N₁|)/κ between chromosome sites at N₁ and N₂ mega-basepairs (Mbp) from one chosen end of the fiber can be quantitatively detected by chromosome conformation capture (3C). (Brown squares) 3C interaction frequencies between sites exploring the entire contour length (L_c ≈ 0.3 Mbp) of chromosome 3 in yeast Saccharomyces cerevisiae (10). (Violet squares) 5C interaction frequencies between sites exploring the (silenced) human β-globin locus (12). (Orange squares) Hi-C interaction frequencies between sites on human chromosome 1 (13). (Black lines) Interpolation formula for a WLC proposed by Shimada and Yamakawa (22), Eq. 5: l_K = 300 nm (solid line) and l_K = 56 nm (dashed line). Due to the unknown efficiency of the cross-linking reaction, 3C does not measure the absolute value of the interaction frequencies and we have shifted the experimental data according to a convention discussed in 3C Interaction Frequencies X(L) and Chain Looping Probabilities J(L; r_min, r_c).

Figure 3

Potentials used to model the stiffness of the fiber with a Kuhn length l_K ≈ 300 nm (k_θ = 5 k_BT): U_stiff (θ) = k_θ/2 θ² (Eq. 8, red line); U_stiff (θ) = k_θ (1 − cos θ) (Eq. 9, green line); $U_{\text{stiff}}\left(\theta \right)=k_{\theta }/2{\theta }^2{\left(\tanh \left[{\left(2{\mathrm{E}}_{\mathrm{s}}/{\mathrm{k}}_{\theta }{\theta }^2\right)}^{10}\right]\right)}^{1/10}$, and E_s = 5.2 k_BT (Eq. 10, blue line).

Figure 4

Kuhn length of the fiber as a function of the stiffness constant k_θ (in temperature units) for the three bending potentials, as shown in Eqs. 8–10. (Black line) Large-k_θ analytical expression obtained for potentials from Eqs. 8 and 9 (l_K ≈ 60βk_θ nm). (Inset) The probability Π_no–kink(L) of observing no kink along the contour length L of a chromatin fiber is affected by the choice of the potential modeling the stiffness of the fiber: results from the exact evaluation of Eqs. 11 and 13 (lines) and for simulated fibers after switch among the potentials from Eqs. 8–10 (squares). Colors as in Fig. 3.

Figure 5

Looping probabilities J(L;r_min, r_c) between chromatin loci in wormlike 30-nm fibers with l_K = 300 nm (8,19). (Top) Theoretical predictions (solid lines) derived from our interpolation expression for the probability density of internal distances p_L^I(r) (Eq. 5, together with Eqs. 2 and 3) agree with measured looping probabilities (squares) in equilibrated model 30-nm fibers. (Bottom) Experimental 3C interaction frequencies in yeasts (10) may be reproduced only assuming unrealistically high contact radii. Solid lines are the same as in top panel.

Figure 6

By using a softer chromatin fiber with l_K = 56 nm (3), theoretical predictions for looping probabilities J(L;r_min, r_c) (solid lines) derived from our interpolation expression for the probability density of internal distances p_L^I(r) (Eq. 5, together with Eqs. 2 and 3) reproduce well 3C interaction frequencies in yeasts (10) (top). However, average square internal distances between chromatin sites measured by FISH (8) are clearly off when compared to the theoretical WLC expression from Eq. 1 (bottom).

Figure 7

Minimal concentrations of kink along the chromatin fiber increase looping probabilities at short length-scales to up to ≈2 orders of magnitude (top). However, FISH is rather insensitive to kinks (bottom). In both panels, the Kuhn length l_K of the chromatin fiber is = 300 nm. Colors as in Fig. 3.

Figure 8

(Top) Looping probabilities J(L;r_min, r_c) (left y axis) and looping free energy cost ΔG(L;r_min, r_c) (right y axis) between chromatin sites on model human chromosomes. The large length-scale behavior is compatible with the power law ∼ L⁻¹. (Black line) Shimada and Yamakawa (22) theoretical expression from Eq. 5. (Bottom) In agreement with our finding for model yeast chromosomes (Fig. 7, bottom), FISH is insensitive to kinking. Here, average square internal distances for model human chromosomes have been calculated by assuming the existence of a centromere-hinge, as explained in our previous work (19). Colors as in Fig. 3.

Figure 9

Looping probabilities J(L;r_min, r_c) (top) and average square internal distances (bottom) between chromatin sites on model human chromosomes and ring polymers of different linear sizes collapse to single curves. Here, we have reported simulations results corresponding to the fiber stiffness potential U_stiff(θ) = k_θ(1 − cos θ) from Eq. 9. (Bottom panel, green solid line) Corresponding to the line plotted in Fig. 8, bottom. We remark that noticeable discrepancies between the curves are likely to be simulation artifacts, which are due to different choices in the initial configurations (19).