Modeling chromatin fractal structure and dynamics: Crosslinked single chain under active forces

Chromatin folding during interphase is organized in a compact, fractal-like manner with minimal entanglements. Experimental studies estimate a mass (Hausdorff) fractal dimension of d_f ≈ 2.77, similar-yet not equal-to the prediction of the fractal globule model d_f = 3. While the fractal globule model describes a densely packed linear polymer with no crosslinking, a variety of crosslinks are ubiquitous in in vivo chromatin architecture, originating from, e.g., complexes such as cohesin and condensin associated with the structural maintenance of the chromosome family, lamin-A proteins, and other bridging proteins. Here, we exploit the analogy of chromatin to collapsed polymers below the Θ temperature and use Monte Carlo simulations for a self-interacting random-walk-based model to generate ensembles of chromatin-like structures. The interaction potential includes self-avoidance and long-range van der Waals attractive interactions. We show that the attractive interaction strength can be tuned such that the resulting structures capture well both the fractal packing and connectivity characteristics of chromatin as observed in Hi-C experiments. Furthermore, we introduce crosslinks to obtain a spectral dimension d_s = 1.2 to successfully recover the experimentally observed Rouse model subdiffusive behavior, where the mean square displacement (MSD) grows as ∼ t^ν with ν = 1 - d_s/2 ≈ 0.4. We numerically evaluate the MSD using both previously derived analytical expressions and Langevin dynamics simulations, under both active (normal cell) and thermal (ATP-depleted) conditions in the nucleus. Furthermore, we compute the previously uncharacterized topological dimension d_l of the model chromatin-like network, demonstrating that the different fractal dimensions obey the theoretically expected inequalities 1 ≤ d_s ≤ d_l ≤ d_f ≤ d = 3.