The metabolism of a living cell requires a permanent transfer of molecules throughout the cell and beyond its bounds. Within cell nuclei, molecules are predominantly driven by diffusion, which is influenced by the chromatin network. We propose a quantity related to the pair correlation function to measure the diffusion-relevant clumpiness of chromatin. Using Monte Carlo lattice simulations, we investigate to what extent diffusion can be anomalous due to obstruction by the chromatin network. Chromatin is simulated by a wormlike chain on a lattice, which exhibits different types of loop-induced compartmentalization on a subchromosomal level. Our simulation results show that the proposed measure of clumpiness is suitable to quantify the compartmentalization of chromatin and to connect it to diffusion anomaly parameters, critical molecule sizes for trapping and the transition lengths at which diffusion becomes normal at long times.

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 threedimensional 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 C. C. Fritsch · J. Langowski (B) Biophysics of Macromolecules, German Cancer Research Center (Deutsches Krebsforschungszentrum, DKFZ), Im Neuenheimer Feld 580, 69120 Heidelberg, Germany e-mail: joerg.langowski@dkfz-heidelberg.de 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.

We propose a kinetic Monte Carlo method for the simulation of subdiffusive random walks on a cartesian lattice. The random walkers are subject to viscoelastic forces which we compute from their individual trajectories via the fractional Langevin equation. At every step the walkers move by one lattice unit, which makes them differ essentially from continuous time random walks, where the subdiffusive behavior is induced by random waiting. To enable computationally inexpensive simulations with n-step memories, we use an approximation of the memory and the memory kernel functions with a complexity O(log n). Eventual discretization and approximation artifacts are compensated with numerical adjustments of the memory kernel functions. We verify with a number of analyses that this new method provides binary fractional random walks that are fully consistent with the theory of fractional brownian motion.