# Chromosome dynamics, molecular crowding, and diffusion in the interphase cell nucleus: a Monte Carlo lattice simulation study

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## Abstract

Using Monte Carlo simulations, we have investigated the decondensation of chromosomes during interphase and the diffusive transport of spherical probe particles in the chromatin network. The chromatin fibers are modeled as semiflexible polymer chains on a fixed three-dimensional grid, taking into account their flexibility and eventual chain crossing by the aid of topoisomerases. The network thus created will obstruct the diffusion of macromolecules. A result of our simulations is that crowding of diffusing molecules leaves the dynamics of the chromosomes and the behavior of other diffusing molecules qualitatively unaffected. Furthermore, the capability of the simulated chromatin network to trap diffusing molecules over long times is lower than that measured in microrheological experiments. Microrheology is a technique that allows to determine the viscoelastic properties of a material by the motion of embedded tracer particles. Long-time trapping requires a stiff network, as only such a network quickly responds to the diffusive fluctuations of tracers and prevents them from squeezing through meshes. A high degree of crosslinking amplifies this effect. The presence of a flexible and uncrosslinked polymer simply increases the effective viscosity sensed by tracer particles. The diffusion of tracers in our simulations reveals rather viscous than elastic chromatin networks, suggesting that chromatin alone cannot account for the high elasticity of the cell nucleus.

### Keywords

Chromosome dynamics Molecular crowding Diffusion### Abbreviations

- MC
Monte-Carlo

- MSD
mean squared displacement

## Introduction

The structure of the chromatin fiber, in particular its packing topology in the cell nucleus, is critical for genome function. Gene activity is regulated by the compaction state of chromatin; DNA accessibility is determined both by the local wrapping of the DNA around the histone core and by the features of the network that is formed by the higher-order chromatin structure. Interphase chromosomes are not randomly intermingled polymer chains but occupy distinct “territories” in the nucleus. Accessing and handling the genetic information requires the transport of sophisticated molecular machinery to specific DNA sites, a nanomechanical process that is fundamentally influenced by the distribution and the folding behavior of chromatin.

Studying the dynamics of the chromatin fiber network is not straightforward. By following fluorescent tags with light microscopy, the compaction/decompaction, and mobility of individual loci (Müller et al. 2001) and vice versa, the mobility of fluorescent probes of various size has been studied (Bancaud et al. 2009; Dross et al. 2009; Görisch et al. 2004, 2005a; Guigas and Weiss 2008; Wachsmuth et al. 2000). However, current state-of-the-art techniques limit the resolution of such measurements to about 200 nm, with some exceptions from single-particle tracking (Siebrasse and Kubitscheck 2009), or superresolution techniques such as STED (Klar and Hell 1999), which, however, require particular fluorophores and have rather low sensitivity. Furthermore, the interplay of chromatin fiber dynamics and transport of macromolecules has not been studied systematically, partly due to lack of suitable experimental techniques.

Generally, biomolecules are transported by passive diffusion over distances of the order of the size of the cell (Politz et al. 2003). Such diffusional transport is affected by the presence of obstacles, rendering the diffusion characteristics dependent on the length and time scale of observation. In normal diffusion, as typically observed in homogeneous solution, the mean squared displacement (MSD) \(\langle r^2(t)\rangle\) of a molecule is proportional to the time *t*. Diffusion is called anomalous if \(\langle r^2(t)\rangle \propto t^{\alpha}\), with an anomaly parameter \(\alpha \not = 1\). Obstacles typically cause *α* to become <1; this case is called subdiffusion (Metzler and Klafter 2000).

The observation of anomalous diffusion also depends on the length scale. On a very short scale, the walker has not yet traveled far enough to encounter many obstacles by which its walk could get obstructed. The walker “sees” almost free, homogeneous space and diffusion is nearly normal. On a larger scale, the walker senses both obstacles and free space, its environment appears inhomogeneous, and diffusion becomes anomalous. On a very large scale, obstruction effects average out, diffusion becomes normal again, and the walker “remembers” a homogeneous medium. Fluorescence correlation spectroscopy showed subdiffusion in cell nuclei (Wachsmuth et al. 2000) and chromatin-induced obstruction of larger molecules (Görisch et al. 2004, 2005a), whereas smaller molecules diffused normally and no correlation between their mobility and the density of chromatin was found (Dross et al. 2009).

The accessibility of a polymeric material to diffusing molecules primarily depends not only on the size of the molecules and the global density of the polymer matrix but also on the spatial arrangement of the matrix and on its dynamics. In a homogeneous immobile polymer network, molecules are retained if they are larger than the characteristic mesh size. At the same global density, a network with a very inhomogeneous mesh size distribution, i.e., with a tendency of the network fibers to lump together, can leave space for channels that allow large molecules to travel greater distances.

To assess the importance of obstructional effects in cell nuclear transport, we need to know the occupation of the nuclear space by chromatin and other macromolecules. Using a combination of fluorescence correlation spectroscopy and confocal imaging, we determined earlier that chromatin occupies about 10% of the nuclear space (Weidemann et al. 2003). However, the overall occupation of cellular compartments by macromolecules is between 5% and 40% (Ellis and Minton 2003), i.e., the volume of the cell nucleus contains up to 30% macromolecules other than chromatin. Many studies have looked at molecular crowding as the origin of anomalous diffusion (for a review, e.g., see Dix and Verkman 2008). In the cell nucleus, however, the precise contributions of the different molecular species to diffusional obstruction and anomaly and therefore the relevance of anomalous diffusion to genomic function in general are not clearly understood.

In previous papers, we developed a coarse-grained lattice model for the chromatin fiber as a very efficient tool to generate thermodynamic equilibrium conformations of the chromatin chain with varying persistence lengths and densities. This allowed us to study the diffusional properties of macromolecules in the chromatin network (Wedemeier et al. 2007), the effect of binding and sliding of proteins on DNA (Wedemeier et al. 2008), and the squeezing of proteins through dense dynamic chromatin networks (Wedemeier et al. 2009). It offers the advantage that physically consistent chain conformations can be generated in a computationally inexpensive way, since only local interactions have to be considered. In previous work on the effect of specific folding topologies on diffusion (Fritsch and Langowski 2010), we concentrated on the purely geometric obstruction effect of chromatin, assuming the simulated chromatin chain immobile and affecting diffusion only via steric volume exclusion. There we succeeded in developing a theory for the spatial arrangement of a heterogeneous polymer network in terms of correlations and derived a measure for its “clumpiness” and how this affects macromolecular transport. Simultaneously, this model allowed us to study the effect of loop-induced compartmentalization, a concept that has been discussed for some time (Bohn et al. 2007; Cremer et al. 1996; Münkel and Langowski 1998), and to compare the simulation results quantitatively with the recent intrachromosomal interaction data measured by Lieberman-Aiden et al. (2009). However, the combined effects of chromatin chain mobility, specific chromatin folding, and of crowding by other diffusing macromolecules were not included in our earlier work.

Here we have generated dynamic chromatin networks with characteristic interphase topologies and analyzed the diffusion of macromolecules in crowded and uncrowded conditions. We show that including chain dynamics allows larger macromolecules to penetrate the network, and we analyze the effect of local chain flexibility and possible self-crossing of the chromatin chain (e.g., induced by type II topoisomerases). We verify the relaxation time scales for interphase chromosomes recently estimated by Rosa and Everaers (2008) by a continuum model and show that molecular crowding by diffusing macromolecules can lead to anomalous diffusion but to a much lesser degree than crowding by the chromatin network. We also find that the stiffness of the network probed with diffusing particles is much lower than detected earlier in microrheology experiments (Tseng et al. 2004).

## Chromatin model

### Lattice discretization of the chromatin fiber

*l*= 1, \(\sqrt{2}\), and \(\sqrt{3}\) (in lattice units). The bending angle

*θ*

_{i}at the

*i*-th bond is drawn by the two segment vectors \(\vec{l}_{i-1}\) and \(\vec{l}_i\). The bending rigidity of chromatin is modeled by harmonic potential functions of

*θ*

_{i}which are installed at each bond, yielding the total internal energy of the chain

*N*and

*g*denote the number of chain bonds and the bending rigidity constant in units of

*k*

_{B}

*T*(Fritsch and Langowski 2010; Wedemeier et al. 2007). The persistence length

*L*

_{p}of the chain is the characteristic length over which the chain behaves approximately like a rigid rod and is directly related to

*g*(for a detailed derivation of the connection between the persistence length and the internal bending rigidity on the Cartesian lattice, see Fritsch and Langowski 2010). To account for steric volume exclusion (self-avoidance), the occupation of a lattice site by two or more distinct chain bonds is prohibited. For modeling a phantom chain, this rule is suspended to facilitate chain crossings.

The decondensation of a chromosome during interphase is simulated by a gradual equilibration of an initial metaphase-like conformation of the chromatin fiber. For this equilibration, a Metropolis Monte Carlo (MC) procedure is performed during which single MC steps randomly alter the positions of chain bonds on the lattice (Carmesin and Kremer 1988; Metropolis et al. 1953). In this procedure, the internal bending rigidity is explicitly accounted for, and after a sufficiently large number of steps, the initial condensed chromatin conformation has transformed into a statistically more favorable decondensed interphase conformation (see also Fritsch and Langowski 2010).

### Lattice random walks model molecular diffusion

### Conversion of lattice units to physical units

For both the monomer displacement during the Metropolis Monte Carlo procedure and the displacements of the randomly walking molecules, the lattice units and MC time steps can be converted to physical units. Chromatin is often associated with a “30-nm fiber”. As a cylinder, its volume is \(V_{\rm cyl}= \pi d^2 L / 4\), where *d* is the diameter and *L* is length of the chromatin fiber. For our simulations, we consider chromatin a chain of beads, and then its volume is \(V_{\rm cob}= N 4 \pi R_0^3 / 3\), where *N* is the number of beads and *R*_{0} is the bead radius. For both models to be equivalent, we require the same length (*L* = 2*NR*_{0}) and the same volume, and then the bead radius is \(R_0=(3/8)^{1/2}\, d= 18\) nm. Since each bead corresponds to one lattice site, we define the lattice constant^{1}*a* = 36 nm.

Now the Monte Carlo steps can be converted to a physical time scale considering the monomer relaxation time of a dynamic polymer: In a polymer chain with the monomer size *b*, a single monomer requires the time \(\tau_0\approx \eta b^3/(k_B T)\) to travel a distance of the order of its own size (Rubinstein and Colby 2003). Up to this time scale, it does not feel the connection to other beads in the polymer, and the distance covered within a single MC step is always *r* = 36 nm. In water (*η* = 10^{ − 3} Nsm^{ − 2} at *T* = 300 K) and for monomers with *b* = 36 nm, we get *τ*_{0} = 12 \(\upmu\)s, which we define as the duration of a chain monomer displacement as well as the duration of a random walk displacement of a molecule with radius *R*_{0} = 18 nm. In order to obtain the step duration *τ* of molecules with radii *R* > *R*_{0}, *τ*_{0} is to be scaled according to the Stokes–Einstein relation \(D\!=\!k_B T/(6 \pi \eta R)\!=\!r^2/(6\tau)\), and we get *τ* = *τ*_{0}*R*/*R*_{0}.

## Simulation of interphase chromosomes

### Creating realistic chromosomes

In order to provide for a realistic simulation of molecular diffusion and chromatin dynamics, the lattice model must reproduce precisely the degree of obstruction that diffusing molecules or chromatin fibers would experience in a corresponding continuum model. In a series of test simulations, we use the concept of lattice refinement to extrapolate the continuum as the limiting case of an infinitely refined lattice. From that we deduce the correct lattice occupation for the main simulation of realistic chromosomes on an unrefined lattice.

*a*represents a cube with edge length

*a*rather than just a point in space. On a 10 × 10 × 10 lattice with

*a*= 36 nm, for instance, a site corresponds to a physical cube with edge length 36 nm (see Fig. 2, first panel from the left). Refinement of the lattice by a factor

*ξ*transforms it into a (

*ξ*× 10) × (

*ξ*× 10) × (

*ξ*× 10) lattice. The edge length of the cube is kept at 36 nm, which transforms this cube into a

*ξ*×

*ξ*×

*ξ*cube and reduces the lattice constant to

*a*

_{ξ}= 36 nm/

*ξ*(in Fig. 2, steps from the second panel to the fourth panel and from the third to the fourth).

*ξ*, and as

*ξ*approaches infinity, this lattice body becomes a perfect sphere. Figure 3 illustrates the refinement of the lattice and the increasing quality of a sphere approximation with rising

*ξ*.

Lattice refinement as illustrated drastically increases the computation effort for simulating the diffusive motion of a chain monomer, since every single lattice site that is part of a monomer has to be moved during a displacement. Furthermore, every single site has to be checked whether a displacement of the monomer leads to a double occupation. Thus, we checked how this refinement could be avoided for simulating a whole chromosome and which parameters allow us to approximate a chain of smooth beads by a single-site (*ξ* = 1) lattice chain.

*ξ*= 1 and the sphere at

*ξ*→ ∞ is

*a*

^{3}/(4

*π*(0.5

*a*)

^{3}/3) ≈ 2. Therefore, in order to simulate a chain of beads that occupies 10% of a predefined space, the corresponding chain of lattice sites would have to occupy approximately 20% of the lattice. With a simulation of crowding diffusants, we checked if the obstruction effect on diffusing molecules in these two problems was really comparable. At lattice refinements

*ξ*= 1, 3, 5, 7, 9, 11, and 13 on (

*ξ*× 10) × (

*ξ*× 10) × (

*ξ*× 10) lattices, we calculated the average diffusion coefficient

*D*of 200 simultaneously diffusing lattice spheres. We defined the reduced diffusion coefficient

*D*/

*D*

_{0}, where

*D*

_{0}denotes the diffusion coefficient of a freely diffusing sphere in solution. In Fig. 4, the reduced diffusion coefficients are shown in dependence of the lattice refinement. As

*ξ*increases,

*D*/

*D*

_{0}approaches a constant value of approximately 0.76, which we regard as the continuum limit of the refinement and therefore as the reduced diffusion coefficient of a perfect sphere. The value of

*D*/

*D*

_{0}at

*ξ*= 1 is 0.78, which deviates only 2.5% from the extrapolated limit. The fraction of occupied lattice sites (also shown in Fig. 4) drops from 0.2 at

*ξ*= 1 to approximately 0.1 at large values of

*ξ*. Hence, we validated the substitution of beads with lattice sites regarding the obstruction effect in a diffusant crowding situation. In another test simulation, we also verified that this substitution is valid if the lattice sites are arranged as a chain.

*L*

_{p}= 300 nm, and (c) a phantom chain with

*L*

_{p}= 300 nm. Assuming a chromatin packing ratio of 100 bp/nm, the simulated chromatin fiber is 1 mm long, and with an average bond length corresponding to 50 nm, which we had obtained from previous test simulations, the lattice chain was to be composed of

*N*= 20,240 sites. In order to make a chain of 20,240 sites occupy 20% of the simulation cube, the edge length had to be 46 lattice units. The initial conformation of our lattice chromatin (see Fig. 5d) on the 46 × 46 × 46 lattice consists of stacked rosette-like chromatin disks and resembles a metaphase chromosome. It has a diameter of 600 nm and a length of 4.4 \(\upmu\)m. For each of the chromatin models, we applied the Monte Carlo equilibration procedure on ten independent initial conformations. During every time step, 20,240 single bonds of a conformation were randomly picked for an eventual displacement. Every initial conformation ran through 10

^{7}time steps (≈ 120 s real time), and then we evaluated the resulting end conformation. Typical end conformations for the three models and the initial conformation are shown in Fig. 5.

To avoid confinement artifacts we used periodic boundaries. If a chain monomer (or a diffusing molecule) leaves the 46 × 46 × 46 lattice during an MC step at one boundary, it simply re-enters the lattice through the opposite. That way chromosomes can expand beyond the actual extent of the lattice and the nuclear space becomes virtually infinite. At the same time, the overall occupation density of the lattice is conserved.

### Decondensation dynamics

During the equilibration, we recorded the average MSD of single monomers, the MSD of the center-of-mass of the chain, and its gyration radius. The time-dependent MSD of the monomers characterizes the fastest relaxation mode of the chain and reflects the physical properties of the chain on certain time scales. In a chain polymer that is not subject to hydrodynamic forces, internal bending rigidity, or external stress, the MSD of a monomer is proportional to *t*^{0.5} up to the Rouse time *τ*_{R} at which the whole polymer has traveled a distance of its own size (Doi and Edwards 1986). For times longer than *τ*_{R}, its MSD is proportional to *t* and coincides with that of the center-of-mass. In a semi-flexible chain, the MSD of a monomer is proportional to *t*^{0.75} on time scales on which the local motion of semi-flexible chain is dominated by its internal bending rigidity (Farge and Maggs 1993). If the polymer is very dense, the local motion is governed by the self-entanglement of the polymer and the MSD is proportional to *t*^{0.25} (Doi and Edwards 1986).

*τ*

_{R}by intersecting the extrapolated MSD curves of the monomers and the center-of-mass (extrapolations not shown in Fig. 6). From that we obtain \(\tau_R\approx\text{1,000}\) s, which is well within the duration of interphase in most human cells.

The MSD of the self-avoiding chain with persistence length 300 nm (model B) exhibits a semi-flexible polymer regime up to about 10 s and entanglement on longer times scale. We have equilibrated the conformations of model B ten times as long as the others to make the onset of the entanglement regime visible. In a similar study by Rosa and Everaers (2008), Brownian dynamics simulations were used to investigate the decondensation of 100-Mbp chromosomes. Their result for simulated chromosomes, which are comparable to our model B chains, is represented by the black dots in Fig. 6, agreeing well with our result obtained by Monte Carlo simulations. The intersection of the extrapolated MSD curves of the monomers and the center-of-mass for the model B yields a relaxation time of about 100 days, which is more than the lifetime of most human cells.

Within the first second of the decondensation, the monomer relaxation mode of the phantom chain (model C), which represents a chromosome with high topoisomerase II activity, ranges between those of models A and B. This suggests that in the compact and unequilibrated initial conformation, a high persistence is more obstructive than self-avoidance. However, after approximately 0.1 s, the model C chromosome starts to unfold rapidly, with an MSD proportional to *t*^{1.08} until about 1 s. Then it decondenses Rouse-like and slightly faster than the model A chromosome.

### Chromosome compartmentalization

*R*

^{*}of the distances \(r_{i,j}^*\) between every pair of bonds

*i*and

*j*in the simulated chromosome. Then we averaged

*R*

^{*}down to a tenfold smaller matrix

*R*in which every entry was the arithmetic average of the entries in the corresponding 10 × 10 submatrix of

*R*

^{*}. This averaging is to reduce the computation effort for the following processing of the data. From

*R*we obtained the correlation matrix

*C*in which an entry

*c*

_{ij}is the Pearson correlation coefficient of the

*i*-th row and the

*j*-th column of

*R*:

*c*

_{ij}express the spatial correlations of 500-nm sections of chromatin and take values between − 1 and 1, which mean mutual remoteness and closeness, respectively. The matrix

*C*can then be visualized as a map. We refer here to the proximity maps of single chromosomes as “auto-proximity” maps in order to distinguish them from the proximity maps between two chromosomes.

*r*

_{i}of central rows are greater than those of end rows. According to Eq. 2, this results in greater

*c*

_{ij}at the ends of the matrix diagonal.

### Chromosome territories

Similar to the auto-proximity maps of single chromosomes, we constructed the cross-proximity maps of neighboring chromosomes, whose overlap or non-overlap can be displayed this way. The computation of the correlation matrix *C* is principally the same as for the auto-proximity maps, with the only difference that the distances \(r_{i,j}^*\) are now measured between segments on distinct chromosomes. During the decondensation procedure, we simulated only one chromosome at once, but because of the periodic boundaries, we can construct a virtual second chromosome as an identical copy of the first, shifted by the period of the underlying lattice. The typical cross-proximity maps displayed in Fig. 8 correspond to the three chromatin conformations shown in Fig. 5.

### Simulated confocal images

*w*

_{xy}and

*w*

_{z}are the lateral and horizontal widths of the excitation profile, respectively.

The detected fluorescence emission intensity at a particular point is in first approximation proportional to the excitation intensity in the same spot, and therefore, the Gaussian profile also holds for the emission intensity. A two-dimensional image is obtained by scanning through a plane in the sample. Each pixel of the image corresponds to the detected signal \(I(\vec{r})\) at a particular position \(\vec{r}\), which is contributed by all excited fluorophores: \(I(\vec{r}) = \sum_{i}I_i(\vec{r})\), where \(I_i(\vec{r}) \propto I_{\rm exc}(\vec{r},\vec{r}_i)\) is the signal intensity contributed by a single fluorophore at \(\vec{r}_i=(x_i,y_i,z_i)\).

*xy*-plane at

*z*= 0 of the simulation lattice. At each site belonging to the chromatin chain, a virtual fluorophore was placed. For the computation of the image, we used the typical resolution of a confocal scanning microscope,

*w*

_{xy}= 250 nm lateral and

*w*

_{z}= 1,500 nm horizontal. The intensities were normalized with the maximum intensity, which we detect when all lattice sites in the focus are labeled with fluorophores. To quantify the chromatin density distribution contained in the images, we computed the image cross-correlation (Görisch et al. 2005b)

*w*

_{xy}and semi-major axis

*w*

_{z}) in comparison to the size of the three chromosomes from Fig. 5a–c. Each of the subfigures shows a number of projections (bright gray domain) of one of these chromosomes onto the

*xz*-plane. The single projections are shifted from each other by the lattice period (1.6 \(\upmu\)m, indicated by the black grid). One projection in the center is highlighted in dark gray. Corresponding simulated confocal images that are obtained by scanning across the

*xy*-plane of the lattice are shown in the lower row of Fig. 10. The widely unfolded chromosomes of model A and model C occupy the nuclear space very homogeneously. They intermingle with neighboring chromosomes such that fluctuations in chromatin density are small. In contrast to that, the still rather condensed model B chromosome leaves portions of the nuclear space with diameters up to 1 \(\upmu\)m chromatin-free. Due to that, the laser can be focused onto certain spots at which either low or high fluorescence intensity is detectable. The image cross-correlation curves plotted in Fig. 11 express this fact quantitatively: The cross-correlation

*G*(

*r*) of model A and C hardly set themselves apart from the background signal, while the intensity for model C shows a considerable decay according to the size of the chromosome. Roughly estimated, the correlation decays at about 0.5 \(\upmu\)m, and the anti-correlation maximum occurs at 1 \(\upmu\)m. However, the correlation length of 1.16 \(\upmu\) found earlier by Görisch et al. (2005b) makes clear that chromatin agglomerations in the nucleus are significantly larger than the average-sized chromosomes simulated here, which implies that chromosomes are not homogeneously distributed within the nucleus.

## Diffusion of tracer molecules

### What affects molecular diffusion?

We have simulated the diffusion of molecules in the previously created interphase chromatin network to investigate the effects of chromatin dynamics and diffusant crowding on the molecular transport in the interphase cell nucleus. Recent simulation studies suggest that the diffusion behavior of small molecules with radii up to 10 nm does not differ between a dynamic and a static chromatin network (Wedemeier et al. 2009) and also that the influence of different network topologies cannot be resolved (Fritsch and Langowski 2010). Nuclear dynamics can only be probed with tracer molecules that are significantly influenced by the macromolecular network in the nucleus.

In simulations of molecules with radii less than 58 nm, we observed no significant trapping in static chromatin networks. In order to detect the effects of self-diffusion of the network for long times, however, we require trapping in the static network as a control. Therefore, all molecules in our diffusion simulations had the radius *R* = 58 nm. We investigated the following scenarios: a single molecule diffusing in a static chromatin network (scenario 1), 295 crowding molecules diffusing in a static chromatin network (scenario 2), and 295 crowding molecules diffusing in a dynamic chromatin network (scenario 3). Each of the diffusing molecules is represented by a quasi-spherical lattice body consisting of 33 single sites, and altogether they occupy 10% of the lattice. For comparison, chromatin occupies 20% of the lattice sites in the networks of models A and B and around 18% in the networks of model C. The slightly lower percentage in model C networks is due to some double occupations of sites by monomers. The monomers of the chromatin chains (radius *R*_{0} = 18 nm) and the diffusing molecules differ in size (*R*/*R*_{0} = 3.2). According to “Conversion of lattice units to physical units”, we adjusted the MC procedure such that one monomer performed 3.2 MC steps per MC diffusant step on average.

The fact that the diffusing molecules in the dynamic crowding scenario are larger than the monomers of the chromatin fibers prevents the formation of depletion zones around the chromatin fibers. Depletion forces are entropic forces that cause attraction between large molecules in the presence of many small molecules (Waigh 2007). If the larger molecules are very close to each other, the overall volume which the small molecules are excluded from is reduced. In turn, the gain of accessible volume increases the entropy of the small molecules and renders those states of the system more favorable in which larger molecules are generally very close. Effectively, this results in a force clustering the larger molecules. In our simulations, however, the diffusing crowder molecules are the larger ones, and consequently, no depletion attraction within the chromatin fibers occurs.

*t*) is reduced to 0.58 times that of free diffusion. Furthermore, the comparison of the MSD in scenarios 2 and 3 (Fig. 12b, c) shows that molecular diffusion in the model B network is unaffected by the dynamics of the chromosome. Consequently, measuring the diffusion of small probes cannot reveal structural and dynamic properties of condensed chromosomes. The dynamics of the models A and C chromosomes, however, enable particles to evade the trapping chromatin meshes on longer time scales.

### Diffusant crowding

*D*to a value of 0.28 of the free diffusion coefficient

*D*

_{0}(network only: reduction to 0.57

*D*

_{0}; see Fig. 13). The transition from anomalous diffusion to normal diffusion can be calculated as the intersection of the intermediate-time and long-time regime asymptotics of the MSD (Saxton 1994). In the scenarios 1 and 2, this transition occurs at about 0.08 and 0.05 s, respectively, while in a crowding scenario with the same fraction of crowders but without a network it occurs already at 0.002 s. In the static crowding scenario of models A and C chromosomes, where diffusion is anomalous at long times due to trapping (Fig. 12a), diffusant crowding skews the plateau of the MSD and hides the effect of confinement (Fig. 12b).

*t*.

*C*

_{vv}(

*t*) expresses the temporal decay of directional correlations in a molecular trajectory and is related to the MSD via (Jacobs and Nakanishi 1990)

*C*

_{vv}(

*t*) equals 1 at lag time

*t*= 0 and approaches zero at late lag times. The crossing of the

*t*-axis toward negative values signifies the turning back of the diffusing molecules at the encounter of an obstacle. The downward amplitudes of

*C*

_{vv}(

*t*) in the scenarios with multiple crowders (solid green and orange lines in Fig. 14) are more pronounced than in the scenario with the single diffusing molecule (dot-dashed green line). In the former scenarios, the characteristic encounter time that can be read off from Fig. 14 is ≈0.5 ms. In the latter, the strongest directional anticorrelation occurs rather at ≈1.25 ms. Further, the auto-correlation functions of the two scenarios containing chromatin networks (dot-dashed and solid green lines) mostly collapse at lag times later than ≈1 ms. This suggests that the presence of multiple highly mobile crowders in macromolecular networks is detectable only at short lag times. At later times, the effect of the mobile crowders on the trajectory of an individual molecule cannot be distinguished from that of the network.

### Nuclear elasticity

The diffusion behavior of the molecules reflects the viscoelastic properties of the chromatin networks. The fluctuation dissipation theorem says that a system at thermal equilibrium responds to small perturbations the same way as to spontaneous thermal fluctuations. This allows to relate the fluctuation spectrum of the MSD of diffusing particles to the viscoelastic response of the system. A technique utilizing this is microrheology. Here probes are injected into a system and then the motion of the probes is recorded with a tracking technique. In this sense, our diffusion simulations are microrheological and can serve to identify the nuclear viscoelasticity induced exclusively by chromatin and crowders.

The elasticity and the viscosity of a network are formally expressed by the storage modulus *G*′(*ω*) and the loss modulus *G*′′(*ω*), respectively. *G*′(*ω*) is the real part and *G*′′(*ω*) is the imaginary part of the complex shear modulus *G*^{*}(*ω*). The Fourier transform of *G*^{*}(*ω*) is the stress relaxation modulus *G*(*t*), whose Laplace transform is \(\tilde{G}(s)\). \(\tilde{G}(s)\) is related to the Laplace transform of the MSD \(\langle {r}^2(t)\rangle\), the fluctuation spectrum \(\langle \tilde{r}^2(s)\rangle\), by \(\tilde{G}(s)= k_B T/(\pi R s \langle \tilde{r}^2(s)\rangle)\). In order to get from the MSD obtained in our simulation to the moduli *G*′(*ω*) and *G*′′(*ω*), we used a method described by Mason and Weitz (1995) to avoid most of the involved Laplace and Fourier transforms: First, we computed the Laplace transform \(\langle \tilde{r}^2(s)\rangle\) of the MSD. Then we fitted \(\tilde{G}(s)\) as a function of \(\langle \tilde{r}^2(s)\rangle\) to the numerical data to get an analytic expression. And then, by substituting *i**ω* for *s*, we directly obtained the complex shear modulus *G*^{*}(*ω*).

*G*′(

*ω*) and

*G*′′(

*ω*) are plotted in Fig. 16 for the case of crowding molecules diffusing in a dynamic phantom chain network (scenario 3 and model C). A viscoelastic network rapidly relaxes via viscous diffusion when sheared slowly (low frequency

*ω*), while it does not relax when sheared rapidly (high

*ω*). Therefore, one expects

*G*′(

*ω*) >

*G*′′(

*ω*) at high frequencies and

*G*′(

*ω*) <

*G*′′(

*ω*) at low frequencies. In Fig. 16, however, the viscous modulus

*G*′′(

*ω*) dominates on all frequencies, which means that the network is rather viscous than elastic. Tseng et al. (2004) measured a high-frequency plateau of 18 Pa of the storage modulus

*G*′(

*ω*) in the intranuclear region of cells. This is significantly more elastic than what we obtained from our simulations. It suggests that the nucleus contains a more elastic network than can be constituted by a purely diffusive, non-crosslinked chromatin fiber.

## Discussion and conclusion

In this simulation study, we have investigated the dynamics of chromosomes during the decondensation at the beginning of interphase. The motion of the chromosomes is assumed purely diffusive without any active contribution from, e.g., molecular motors. Three different models of polymer chain behavior were considered: model A, a highly flexible and self-avoiding chromatin fiber; model B, a persistent (i.e., stiff) and self-avoiding chromatin fiber; and model C, a persistent and self-penetrating chromatin fiber. Model A might be viewed as a model for active, decondensed chromatin, model B for compacted chromatin and model C as mimicking the action of type II topoisomerases, which allow DNA chains to cross. We found that model A and model C chromosomes rapidly unfold and lose their initial metaphase-like shape. After around 120 s, they occupy the nuclear space very homogeneously, such that almost no spatial correlations would be detectable in a confocal laser scanning microscope. Furthermore, we illustrated with cross-proximity maps that these kinds of chromosomes tend to intermingle with neighboring chromosomes, such that numerous inter-chromosomal contacts occur. The auto-proximity maps show that in model C chromosomes, many intra-chromosomal contacts occur between genomically very distant sites, while model A chromosomes are rather compartmentalized. The dynamics of the persistent self-avoiding chromosomes (model B), however, are much slower due to self-entanglement. After 120 s, they would still be resolvable in light microscopy. As discussed earlier by Rosa and Everaers (2008), the formation of chromatin territories can be caused by topological constraints which apply during the decondensation. Comparing the relaxation of models A and B chromosomes, we infer that the high persistence of chromatin amplifies these constraints and favors self-entanglement. We may also conclude that efficient daughter chromosome separation requires type II topoisomerases, as postulated earlier by Sikorav and Jannink (1994), at least for the compact and stiff regions.

After the decondensation of the chromosomes, we have simulated the diffusion of molecules in the resulting dynamic chromatin network. We found that crowding of the diffusing molecules leaves the diffusion behavior qualitatively unchanged, only reducing the long-time diffusion coefficient. Diffusant crowding causes subdiffusive anomaly (\(\langle r^2(t)\propto t^{\alpha} \rangle\), *α* < 1) only on very short time scales, whereas significant mesoscopic anomaly results from the obstruction by the network. In confined diffusion, crowding delays the plateau (*α* = 0) of the mean squared displacement, and a low but positive anomaly parameter *α* is detected instead. Consequently, diffusant crowding causes anomaly on time scales that are shorter than those on which the obstructive network, either dynamic or static, causes anomaly.

The monomer relaxation modes of the chromosomes are only slightly slowed down in the presence of crowding diffusants at a concentration of 10%. This underlines the requirement of chromatin entanglement for territory formation, since crowding by diffusants is presumably insufficient to keep chromosomes from decondensing.

For the diffusion in environments of the rather unfolded, persistent self-avoiding chromosomes (model B), the mean squared displacement of the probe molecules did not differ between the static and dynamic networks. Consequently, their dynamic properties cannot be distinguished by diffusion experiments. In particular, the diffusion did not reveal viscoelastic properties. In contrast, introducing dynamics to the well-decondensed A and C type chromosomes clearly facilitated the motion of diffusing molecules. The fact that we did not detect an elastic trapping regime in their mean squared displacement curves indicates that our chromatin networks are far less stiff than measured in microrheological experiments (Tseng et al. 2004). In order to provide elastic trapping, the network would have to be more rigid and more entangled, like a model B network but at the same time homogeneously distributed. If inhomogeneously distributed, chromatin can leave vast voids in the nuclear space, and diffusing molecules are likely to travel on these voids rather than within entangled network meshes. As we have found, a homogeneous distribution is in turn prevented by the entanglement, which inhibits the unfolding from compact metaphase conformations. To gain high elasticity, chromatin may change its viscoelastic properties during interphase in such a way that it can rapidly decondense and then consolidate. This could be effected, e.g., by crosslinks that induce the formation of chromatin loops (Bohn et al. 2007; Cremer et al. 1996; Lieberman-Aiden et al. 2009; Münkel and Langowski 1998). In addition to chromatin, then, one might need to invoke contributions by another, stiffer macromolecular network to account for the high elasticity of the nucleus. Such a network might also affect the folding of chromosomes.

Summing up, the key to understanding the dynamics of interphase chromosomes is their viscoelasticity. The simulations described here encourage future investigations that target the dynamic crosslinking of chromatin in order to determine to what extent chromatin is responsible for the nuclear elasticity.

## Footnotes

- 1.
The lattice constant is the physical length that corresponds to the spacing of the lattice.

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