Hierarchical domain model explains multifractal scaling of chromosome contact maps

Modern experimental techniques such as Hi-C make it possible to measure the probability that different chromosomal regions are close in space. Usually, these measurements are characterized by the scaling of the contact probability as a function of the genomic distance between regions. In this work, we introduce a multifractal analysis of chromosomal contact maps. Our analysis shows that Hi-C maps display a non-trivial multifractal spectrum. We introduce a simple analytical model that describes the structure of chromosomes as a hierarchical set of domains nested in each other and we solve it exactly. The predicted multifractal spectrum is characterized by a phase transition between two phases with different fractal dimension, in excellent agreement with experimental data. These results support the view that there is no privileged level in the hierarchy of conformational domains. Within the same model, the scaling exponent of the contact probability can also be calculated. We argue that such procedure leads to a much more precise estimate of the contact probability exponent than previous approaches. By applying this method to experimental data, we demonstrate that it can capture subtle conformational differences among chromosomes that are robust in different realizations of the same experiment.