Introduction Eukaryotic genomes are organized in sets of chromosomes which are made up by a single continuous piece of DNA and associated proteins [1]. During cell division (mitosis) chromosomes adopt a compact form which is suitable for transport and which can be discerned in a light microscope. During periods of normal cell activity (interphase), chromosomes decondense. More than 100 years ago, Rabl discovered that interphase chromosomes in newt and Drosophila remain organized in distinct territories [2]. During the last twenty years similar territories of various shapes have been observed in many organisms [3], a notable exception being budding yeast whose chromosomes appear to mix freely [4],[5]. The function of these territories, the mechanism responsible for their formation, and the reasons for the differences between species are still unclear [4],[6]. In this paper we investigate, if the observed interphase structure and dynamics are the consequence of a generic polymer effect, the preservation of the local topological state in solutions of entangled chain molecules undergoing Brownian motion. This effect plays an important role for the viscoelastic properties of polymeric systems [7],[8]. In the present context, Sikorav and Jannink [9] assumed that interphase nuclei behave as equilibrated polymer solutions and estimated the disentanglement time τ d of condensing metaphase chromosomes as τ d = 1.5×10−7 (#nucleosomes)3 s, where “#nucleosomes” is the total number of nucleosomes in a chromosome. A human chromosome of typical size ≈100 mega-basepairs (Mbp) has ≈500,000 nucleosomes [1], i.e., τ d≈2×1010 s (≈500 years). From this prohibitively high estimate Sikorav and Jannink concluded that the process requires substantial topoisomerase-II (topo-II) activity. Here we reverse the argument. We suggest that interphase nuclei never equilibrate and behave like semi-dilute solutions of unentangled ring polymers which are known to segregate due to topological constraints [10]. Within these territories, individual genomic sites are highly mobile and accessible. However, the structure of interphase and metaphase chromosomes remains largely identical from a topological point of view. Thus, instead of being a problem to be overcome by evolution, slow equilibration of long chromosomes accelerates the reverse process of chromosome condensation. Experimental Evidence and Polymer Theory Nowadays, the large-length scale structure of decondensed chromosomes can be experimentally studied using Fluorescence in situ Hybridization (FISH): nucleic acids are chemically modified to incorporate fluorescent probes and specific sequences on single chromosomes can be detected [11]. In particular, it is possible to mark different portions of the genome (chromosome painting) and to determine locations of and spatial distances between targeted sites [11]. Chromosome painting indicates that chromosome territories in human nuclei have an ellipsoidal shape with radii of the order of 1 µm [4]. In contrast and as already discovered by Rabl, the interphase nuclei of organisms like newt or Drosophila are organized in elongated territories oriented between two poles of the nucleus [2],[3]. Furthemore, there are also organisms such as budding yeast whose chromosomes appear to mix freely or, at least, considerably less organized [4],[5]. The localization of territories inside the nucleus exhibits regular patterns: gene-rich chromosomes in human lymphocytes preferably locate in the nuclear interior while gene-poor chromosomes are typically found closer to the periphery [12],[13]; in contrast, in human fibroblasts positioning of territories was shown to correlate with chromosome size and not with its gene content [14]. In general, interactions between specific chromosome regions and structural elements within the nuclear envelope, such as nuclear pores or nuclear lamina, are believed to shape chromatin organization [15]. Data on the (relative) position and motion of target sites provide further insight into the organization of interphase chromosomes. In Figure 1A we show average spatial distances between targeted sites as a function of their genomic separation. The figure contains FISH data for yeast chromosomes 6 and 14 (Chr6 and Chr14, brown ○) [16], human chromosome 4 (Chr4, blue ○ and ◊) [17] and Drosophila chromosome 2L (Chr2L, orange and green ○) [18]. In the latter case, orange symbols refer to embryos in DS5 phase and green symbols to the DS1 phase which appears later in the cell cycle [19]. Two-dimensional spatial distances between sites on Chr4 measured in fibroblasts cells fixed on microscope slides [17] were here rescaled by 3/2 to obtain the corresponding 3 d distances. 10.1371/journal.pcbi.1000153.g001 Figure 1 Experimental FISH data for spatial distances R 2(|N 2−N 1|) between targeted chromosome sites compared to the estimates based on the WLC model (A) and results from our simulations (B–F). Brown ○: Saccharomyces cerevisiae Chr6 and Chr14 [16]. Blue ○ and ◊: Homo Sapiens Chr4, |N 2−N 1| 4.5 Mbp, respectively [17]. Orange and green ○: Drosophila melanogaster Chr2L, DS5 and DS1 embryos respectively [18]. DS5 and DS1 are two phases of cell cycle. DS1 appears later. Black continuous line: Mean-square internal distances predicted by the WLC model, Equation 1. (A) Black dashed line: Mean-square internal distances of an ideal polymer chain inside a spherical nucleus of 5 µm radius [7]. (The exact probability distribution function of the square internal distances R 2(|N 2−N 1|) of a polymer without self-interactions obeys diffusion equation [7] with null boundary conditions (in our case the boundary is the sphere which models the nucleus).)While data for Chr4 and Chr2L show a reasonable agreement at short-length scales, the apparent large-length scale Chr4 behavior L 2/3 [29] contrasts with the observed L 2 for Chr2L. The insets show two schematic drawings of the Chr4 territory in a human nucleus (blue) and of Chr2L in Rabl phase in a Drosophila nucleus (orange). (B–E) Gray lines represent internal distances in the initial, “metaphase-like” chromosome configuration (Materials and Methods). Internal distances in simulated chromosomes have been averaged over 3 time windows of exponentially growing size: 240 s 4 Mbp for Chr4 (blue ◊). There is less data available for the dynamics of interphase chromosomes. In mammalian cells chromatin domains of ∼1 µm diameter display little or no motion in a period of several hours [23]. Cabal et al. [24] followed the motion of a marked active gene (GAL) in in-vivo yeast nuclei. They observed a mean-square displacement (msd) g 1(t = 100 s)≈0.1 µm2 for their largest observation interval, i.e., much less than the typical territory size in organisms with larger chromosomes. In particular, the authors reported anomalous diffusion with g 1(t)∼t 0.4. To rationalize this result, it is again useful to consider “worm-like” chromatin fibers in equilibrated semi-dilute solutions at typical nuclear densities. Neglecting entanglement effects, g 1(t) displays crossovers between different regimes: (1) g 1(t)∼t 0.75 up to length scales of ≈1 Kuhn length [25]; (2) g 1(t)∼t 0.5 (Rouse behavior) up to length scales of the chain radius of gyration [7]; and (3) g 1(t)∼t at larger times, when the monomer motion is dominated by the center-of-mass diffusion (cyan line, Figure 2). 10.1371/journal.pcbi.1000153.g002 Figure 2 Time behavior of the msd of the six inner beads (g 1(t), continuous lines), compared to the average square gyration radius (horizontal dashed lines) of the whole chromosome and measurements of the msd of the active GAL gene inside in vivo yeast nuclei [24] (purple dots). For comparison, g 1(t) for yeast chromosomes without topological constraints has been shown (cyan line). On short time scales, our model reproduces the typical dynamics of semi-flexible polymers with g 1(t)∼t 0.75 [25]. For the model with constraints, there is no extended Rouse regime due to the insufficient separation of Kuhn and entanglement length. Nevertheless, we observe the characteristic g 1(t)∼t 0.25 regime for entangled, flexible polymers [7]. In semidilute solutions, linear chains with a contour length exceeding a characteristic value, L≫L e, become mutually entangled, leading to confinement to a tube-like region following the coarse-grained chain contour and a drastically altered, “reptation” dynamics [7]. Estimating L e is not a trivial task. How strongly linear polymers entangle with each other depends on their stiffness and on the contour length density of the polymer melt or solution [26]. The latter is most suitably expressed in terms of the density of Kuhn segments, ρ K. In loosely entangled systems with the mean-free chain length between collisions is larger than the Kuhn length, leading to random coil behavior between entanglement points. In contrast, for filaments are tightly entangled and exhibit only small bending fluctuations between entanglement points. For a solution of chromatin fibers at a typical nuclear density of ≈0.012 bp/nm3 and a Kuhn length of 300 nm (Table 1) , i.e. the system is loosely entangled, but close to the crossover between the limiting cases. The entanglement contour length, L e, can be estimated via [26] , yielding L e≈1.2 µm or four times the Kuhn length. To a first approximation, chains can thus be considered to be flexible on the tube scale, i.e., we expect around a msd of a crossover from Rouse behavior to a g 1(t)∼t 0.25 regime characteristic of reptation [7]. Interestingly, this estimate coincides with the observations of Cabal et al. [24], who reported an intermediate effective power law g 1(t)∼t 0.4 for msd 0.05 µm2≤g 1(t)≤0.17 µm2. Using their data, we can obtain estimates for the entanglement time, τ e≃32 s, as well as for the disentanglement times, τ d≈τ e(L c/L e)3 [7], of the order of τ d≃2×104 s, 2×108 s (≈5 years) and 2×1010 s (≈500 years) for yeast, Drosophila and human chromosomes, respectively. Since this exceeds the life time of the entire organism (not to mention the much shorter cell cycle of most animal cells [1]), Drosophila and human chromosomes do not have the time to equilibrate during interphase. (This conclusion does not change, if we take into account entanglement relaxation via topo-II discussed in [9]. At best, this mechanism could completely remove the barrier for chain crossing, thus converting the system to a solution of phantom chains whose relaxation time is given by the Rouse time τ R≈τ e(L c/L e)2 [7]. Yeast, Drosophila and human chromosomes would relax in, respectively, 2×103 s, 106 s (≈10 days), and 2×107 s (≈250 days).) 10.1371/journal.pcbi.1000153.t001 Table 1 Summary of the relevant physical parameters for the polymer model of interphase chromosomes. Parameter Value Typical nuclear radius of a human cell [20] 5 µm Radius of the yeast (S. Cerevisiae) nucleus [22] 1 µm Length of the diploid human genomea 6×109 bp Length of the diploid Drosophila genomea 3×108 bp Length of the diploid yeast (S. Cerevisiae) genomea 2×107 bp Compaction ratio of chromatin [16] 102 bp/nm Kuhn length of chromatin [16], l K 300 nm Volume fraction of chromatin 10% a See, e.g., the website http://www.ensembl.org/index.html. While the discussion up to this point has shed some light on various aspects of the structure and dynamics of interphase chromosomes, we have so far evaded the central question, the origin of the observed chromosome territories. A priori, segregation or (micro) phase separation due to small chemical differences between chains is a common phenomenon in polymeric systems [21]. Organisms could, in principle, render different chromosomes immiscible by a labeling technique akin to chromosome painting. In practice, it is difficult to conceive a corresponding, self-organizing molecular mechanism. Here we propose that the formation of chromosome territories could be related to a different, less well-known effect, the segregation of unentangled ring polymers in concentrated solutions due to topological barriers [10],[27]. Well-separated metaphase chromosomes are clearly unentangled at the onset of interphase. Initially, decondensing chains can only rearrange locally and spread out uniformly without changing the global topological state. Up to the extremely long relaxation times for large chromosomes, interphase nuclei should therefore show a behavior similar to concentrated solutions of unentangled ring polymers. In particular, the chromosomes should remain segregated! It is instructive to compare this explanation to previously published models describing interphase chromosomes as equilibrium structures. The unexpectedly small distances on intermediate scales |N 2−N 1|>4 Mbp for Chr4 (blue ◊) were rationalized in terms of giant loops of fibers departing from an underlying (protein) backbone [17] or alternatively, in terms of random loops on all length scales resulting from specific chromatin-chromatin interactions [28]. Simulations of a multi-loop subcompartment polymer model reproduced the experimental observations on human Chr4, by imposing (and hence not explaining) confinement to a spherical territory [20],[29]. We do not exclude the possibility of such contacts. However, we claim that territories should also form, if the involved proteins are disabled. For the inverse test—to keep the linking proteins, but to equilibrate a nucleus with disabled local topology conservation—it would be instructive to investigate the structure of nuclei in long-living cells arrested in interphase and to devise ways to maximize the efficiency of topo-II. (This conclusion does not change, if we take into account entanglement relaxation via topo-II discussed in [9]. At best, this mechanism could completely remove the barrier for chain crossing, thus converting the system to a solution of phantom chains whose relaxation time is given by the Rouse time τ R≈τ e(L c/L e)2 [7]. Yeast, Drosophila and human chromosomes would relax in, respectively, 2×103 s, 106 s (≈10 days), and 2×107 s (≈250 days).) We note that a few cross-links or attachment points to a residual skeleton would be sufficient to suppress chromosome equilibration via reptation [30]. Long-lived contacts could thus stabilize the observed structures without being at their origin. How much of the experimental observations can be explained by this topology-conserving, parameter-free, minimal model of decondensing chromosomes? Unfortunately, it is difficult to derive quantitative predictions from an analytical theory due to the non-trivial initial conformation, the simultaneous presence of various crossovers (stiff/flexible, loosely/tightly entangled), and the lack of a theory describing the conformational statistics and dynamics of the unentangled ring polymer melts. We have therefore resorted to Molecular Dynamics (MD) computer simulations as a tool which allows us to study the model without further approximations. The Model With a spatial discretization of 30 nm (corresponding to the bead diameter), the employed generic bead-spring polymer model [31] accounts for the linear connectivity, self-avoidance and bending stiffness of the chromatin fiber (Materials and Methods). In particular, there is an energy barrier of 70K B T to prevent chain crossing [32]. We emphasize that our description does not invoke any protein-like machinery as the nuclear matrix [33]. Furthermore, we neglect local changes of the chromatin fiber as they occur, e.g., as a result of chromatin remodeling during transcription [34], because these processes do not alter the local topological state of the fiber and are therefore irrelevant for the phenomenon we discuss. This argument does not hold for the action of topo-II whose role is precisely to (dis)entangle DNA allowing strands to cross [9],[20]. Non-directed topology changes with a particular rate could be included by suitable modifications of the energy barrier for chain crossing [35]. Similarly, it is straightforward to include (protein-mediated) interactions between specific DNA sites or effects such as confinement by or anchoring to the nuclear envelope [11],[33],[36]. However, here we concentrate on the generic case of decondensing long, internally and mutually unentangled polymers at total concentrations far above the overlap concentration. As initial states of our simulations we chose linear or ring-shaped helical structures remnant of metaphase chromosomes (Materials and Methods). Given the anisotropic shape of our “metaphase” chromosomes, we were interested to see how the decondensation is affected by the presence of other chains. The l.h.s. panel in Figure 3 shows the initial chromosome conformations in our simulations on a common scale, indicated by a typical human nuclear radius of 5 µm. For Drosophila (marked “B”, only one chromosome is shown for clarity) we assumed that 8 Chr2L model chromosomes are initially aligned along the common axis of a rectangular simulation box (nematic orientation). In the case of yeast (marked “C”) and of the human (marked “A”), we followed the decondensation of 6 respectively 4 chromosomes of equal size which were oriented randomly in the simulation box [14]. For comparison we have also studied ring shaped chromosomes (see inset of Fig. 1F) of different length under conditions corresponding to those of the human cell nucleus. 27 small rings (L c = 2×2.7 Mbp) were randomly oriented inside the simulation box, while for larger rings (L c = 2×48.6 Mbp and L c = 2×97.2 Mbp) we limited ourselves to simulations of single chains in contact with their periodic copies in adjacent simulation cells. The setup as a ring allows us to eliminate chain end effects which otherwise play an important role. 10.1371/journal.pcbi.1000153.g003 Figure 3 Initial (“metaphase-like”, left) and final (right) configurations of human Chr4 (A), of Drosophila Chr2L (B) and of yeast Chr6 and Chr14 (C) shown together with the spherical nucleus (black circle) of 10 µm in diameter and the corresponding simulation boxes (in black). For the blue configuration in A and for the configuration B, we have highlighted in red the two terminal parts up to 4.5 Mbp. In Chr4, this corresponds to the terminal 4p16.3 region [17]. (A) Simultaneous decondensation of 4 model chromosomes half the size the human Chr4. (B) Decondensation of 1 model chromosome the size the Drosophila Chr2L. The final elongated shape qualitatively resembles a Rabl-like territory. (C) Simultaneous decondensation of 6 model chromosomes the size the yeast Chr6 and Chr14. Arrows points at magnified versions of the same configurations. Lack of chromosome territoriality is evident. All simulations were performed in a constant isotropic pressure ensemble using rectangular simulation boxes with three independently fluctuating linear dimensions. The imposed pressure leads to the final density corresponding to the experimental value of ≈0.012 bp/nm3 for human nuclei or 10% of volume fraction of chromatin (Table 1). This appears a reasonable choice because the experimental density in yeast nuclei is only two times lower (≈0.006 bp/nm3, Table 1), while the rapid growing size of Drosophila embryos nuclei [19] does not allow for a univocal choice. We emphasize that the employed periodic boundary conditions do not introduce confinement to the finite volume of the simulation box. Using properly unfolded coordinates, chains can extend over arbitrarily large distances (see Figure 4 for the example of a MD simulation using a similar model but with a strongly reduced barrier for chain crossing). To give an idea of the computational effort, we consider the example of Chr4, where we simulated four model chromosomes of half of the actual length of Chr4. Each chromosome is modeled as a chain of 32,400 beads with a total contour length of 10−3 m or 97.2 Mbp. Using ≈7×104 single-processor CPU-hours on a CRAY XD1 parallel computer, we followed the dynamics over 12×106 MD time steps. The comparison to the measured single-site mobility for yeast [24] in Figure 2 suggests the value of τ ≈2×10−2 s used throughout the paper. According to this estimate, we followed the chain dynamics over 240,000 s (≈3 days) of real time. However, it is clear that more experimental data are needed to reliably fix the absolute time scale of our simulations. 10.1371/journal.pcbi.1000153.g004 Figure 4 Human Chr4 territories are less stable if the energy barrier against chain crossing is switched off. The swelling from the initial “metaphase” configuration is monitored through the time behavior of the gyration radius [7], where rl (t) is the position vector of the lth bead and r cm(t) is the center of mass of the configuration at time t. Without barrier, chromosomes swell easier and have larger size (green and red lines, (A)). Comparison amongst internal distances between two sites located at N 1 and N 2 Mbp from one chosen end of the fiber and avalaible experimental data reflects this behavior (B). We have averaged over 3 time windows of exponentially growing size: 240 s 1 Mbp). (We note that the relation between the square of the gyration radius and the mean square internal distances of a polymer R 2(|N 2−N 1|), [7], is compatible with chromosome territories being compact objects with . However, the reverse conclusion [20],[29] is incorrect: globular polymer conformations also follow , but do not have a fractal structure where the same exponent characterises the entire chain conformation (see, for example the dashed line in Figure 1). Simple confinement alone cannot explain the chain structure.) This behavior seems to be robust, since all our simulation data for linear chains and rings of different size beautifully collapse onto each other. Similar, quasi-fractal structures were reported in [10]. Taken together this suggests that our ring samples are relatively well-equilibrated and that (in agreement with our working hypothesis) long, unentangled linear chains initially relax to a very similar structure. However, we still require an explanation for the deviations between this apparently quite robust prediction and the experimental data in Figure 1D. Reptation theory [30] would suggest that the further equilibration of linear chromosomes proceeds by a very slow escape of the terminal parts of the chain from their initial environment. Qualitatively, this effect is directly observable in Figure 3 where we have marked the terminal parts of our model chromosomes in red. Interestingly, the experimental data for the spatial distances between sites with genomic distances in the Mbp range on human Chr4 were determined in the ≈4.5 Mbp 4p16.3 region which is located at the end of p-arm [17]. A good way to quantify the consequences is to measure R 2(|N 2−N 1|,N 1 = 0), i.e., mean-square spatial distances between the chain ends and points along the fiber (Figure 1E and Figure 5). These distances show a stronger time dependence than results averaged over the entire chain. In particular, they follow the WLC prediction up to much larger contour length distances before crossing over to the bulk averages. The point of departure from the WLC prediction can be used to estimate up to which distance from the end the chains are equilibrated after a certain time. (The temporal behavior of the ratio between the escaped portion of the chain and the whole contour length L c at short times t is compatible with the power-law ∼t 1/2 predicted by reptation theory ([30], data not shown).) The comparison to the experimental data in Figure 1E suggests that the 4p16.3 region of the human Chr4 was nearly equilibrated in the experimental situation. We emphasize that we expect spatial distances between marked sites in the interior of long chromosomes to fall onto the corresponding simulation data in Figure 1D and 1F. This is at least qualitatively supported by a remark in [39] where van den Engh et al. report the more centrally located 6p21 region on human Chr6 to be more compact than the 4p16.3 region near the end of Chr4. 10.1371/journal.pcbi.1000153.g005 Figure 5 Mean square spatial distances R 2(|N 2−N 1|) between a site of the fiber located at N 2 Mbp from one chosen end of the chain and the end (here located at N 1 = 0): comparison between simulated and the avalaible experimental data on Drosophila Chr2L (left) and yeast Chr6 and Chr14 (right). Gray lines represent internal distances in the initial, “metaphase-like” chromosome configuration (Materials and Methods). Internal distances have been averaged over 3 time windows of exponentially growing size: 240 s
