During interphase chromosomes decondense, but fluorescent in situ hybridization experiments reveal the existence of distinct territories occupied by individual chromosomes inside the nuclei of most eukaryotic cells. We use computer simulations to show that the existence and stability of territories is a kinetic effect that can be explained without invoking an underlying nuclear scaffold or protein-mediated interactions between DNA sequences. In particular, we show that the experimentally observed territory shapes and spatial distances between marked chromosome sites for human, Drosophila, and budding yeast chromosomes can be reproduced by a parameter-free minimal model of decondensing chromosomes. Our results suggest that the observed interphase structure and dynamics are due to generic polymer effects: confined Brownian motion conserving the local topological state of long chain molecules and segregation of mutually unentangled chains due to topological constraints.

The conformational statistics of ring polymers in melts or dense solutions is strongly affected by their quenched microscopic topological state. The effect is particularly strong for non-concatenated unknotted rings, which are known to crumple and segregate and which have been implicated as models for the generic behavior of interphase chromosomes. Here we use a computationally efficient multi-scale approach to show that melts of rings of total contour length Lr can be quantitatively mapped onto melts of interacting lattice trees with gyration radii R 2 g (Lr) ∝ L 2ν r and ν = 0.32±0.01.PACS numbers: 83.80. Sg, 83.10.Rs, 61.25.he Similar to macroscopic strings tied into knots, the (Brownian) motion of polymer chains is subject to topological constraints: they can slide past each other, but their backbones cannot cross [1,2]. For linear chains, entanglements are transient and irrelevant for the equilibrium statistics: chains with a contour length exceeding the material specific Kuhn length, L ≫ l K , show Gaussian behavior with mean-square end-to-end distances R 2 (L) = l K L. The only effect of the constraints is to slow down the chain dynamics beyond a density dependent entanglement (contour) length, L e , a corresponding spatial distance or "tube" diameter, d T ∝ √ l K L e , and a characteristic entanglement time, τ e [3,4]. For loosely entangled systems, which are flexible at the entanglement scale, L e ≈ 20/(ρ K l 3 K ) 2 ≫ l K [5-7] where ρ K is the number density of Kuhn segments. The situation is different for unlinked polymer melts or solutions, where the chain conformations have to respect (long-lived) global constraints enforcing the absence of topological knots and links [8]. Experimentally prepared systems of this type have interesting materials properties [9,10]. With large (interphase) chromosomes [11][12][13][14][15][16][17] the most prominent representatives are probably found in biological systems. In this case, the relaxation times for the topological state may be of the order of centuries [12,18], making the knot-and link-free state sufficiently long lived to merit attention. The best studied and yet still controversial [15] example are melts of nonconcatenated unknotted ring polymers. Values for the characteristic exponent, ν, relating the average-square gyration radius and total contour length, R 2 g (L r ) ∝ L 2ν r , of proposed models range from ν = 1/4 for ideal lattice trees or animals [19,20], ν = 1/3 for crumpled globules [21], Hamiltonian paths [13, 22] and interacting lattice trees [19, 23], ν = 2/5 [24] from a Flory argu- * Electronic address: anrosa@sissa.it † Electronic address: ralf.everaers@ens-lyon.fr ment balancing the entropic cost of compressing Gaussian rings and the unfavorable overlap with other chains (recently refined to ν = 1/3 for the asymptotic behavior [25]), to ν = (1 − 1/(3π))/2 ≈ 0.45 [26], and ν = 1/2 for Gaussian rings, rings folded into linear ribbons [27]and swollen lattice trees [28]. There is now strong numerical evidence [14,[29][30][31][32][33] for a cr...

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.

Correlation length exponent ν for long linear DNA molecules was determined by direct measurement of the average end-to-end distance as a function of the contour length s by means of atomic force microscopy (AFM). Linear DNA, up to 48'502 base pairs (bp), was irreversibly deposited from a solution onto silanized mica and imaged in air. Under the adsorption conditions used, the DNA is trapped onto the surface without any two-dimensional equilibration. The measured exponent is ν = 0.589 ± 0.006, in agreement with the theoretical 3D value of ν = 0.5880 ± 0.0010.The persistence length ℓ p of DNA was estimated to be 44±3 nm, in agreement with the literature values. The distribution of the end-to-end distances for a given contour length s and the exponents characterizing the distribution were determined for different s. For s smaller or comparable to ℓ p , a delta function like distribution was observed, while for larger s, a probability distribution of the type x d−1 x g e −bx δ was observed with g = 0.33±0.22 and δ = 2.58±0.76. These values are compared to the theoretical exponents for Self-Avoiding Walk (SAW): namely g = γ−1 ν and δ = (1 − ν) −1 . So for d = 2, g ≈ 0.44 and δ = 4, while for d = 3, g ≈ 0.33 and δ ≈ 2.5. The derived entropic exponent γ is γ = 1.194 ± 0.129. The present data indicate that the DNA behaves on large length scales like a 3 dimensional SAW.

Understanding how topological constraints affect the dynamics of polymers in solution is at the basis of any polymer theory and it is particularly needed for melts of rings. These polymers fold as crumpled and space-filling objects and, yet, they display a large number of topological constraints. To understand their role, here we systematically probe the response of solutions of rings at various densities to "random pinning" perturbations. We show that these perturbations trigger non-Gaussian and heterogeneous dynamics, eventually leading to nonergodic and glassy behavior. We then derive universal scaling relations for the values of solution density and polymer length marking the onset of vitrification in unperturbed solutions. Finally, we directly connect the heterogeneous dynamics of the rings with their spatial organization and mutual interpenetration. Our results suggest that deviations from the typical behavior observed in systems of linear polymers may originate from architecture-specific (threading) topological constraints.