Anomaly and Scaling Behaviors of Diffusion in Fractal Globules

Abstract

Anomalous diffusion in random polymeric geometries, including fractal globules, is studied as a model of biomolecular transport in the cell nucleus. The fractal globule is a non-equilibrium condensed polymer state sharing core configurational characteristics with the actual chromatin structure; its topological traits, including entanglement-free regional condensation, are thought to be deeply related to dynamics, such as polymeric loci reorganization and obstructed diffusion. By computational routes, we assess the anomaly of tracer particle diffusion obstructed by fractal globules and other random geometries, quantified in terms of extensive dynamic observables. As a function of the effective tracer radius, a broad spectrum of anomalous diffusion is identified, including long-lived anomalous diffusion at the critical condition implying the existence of a scale-invariant spatiotemporal structure. The morphology and local patterns of the space accessible by tracers are found to be intimately connected to dynamics, and we suspect the long-lived anomaly has a structural origin. The duality transformation of the space provides further insights into the geometric origin of the anomalous diffusion; by means of it the space is mapped onto percolation geometries, specifically a Bernoulli type, which provide scaling theories of dynamic observables. Through the scaling analysis of mean squared displacements and probability density functions, we report non-universal scaling behaviors between different random geometries, especially fractal globules and equilibrium globules. They are characterized by non-universal walk dimensions, while the analysis consistently demonstrates the walk dimension is much smaller for the case of obstructed diffusion in fractal globules. From this observation, we speculate the cell nucleus is spatially organized in such a way that it can shelter asymptotically faster diffusion of biomolecules than in equilibrated polymer condensates.