  Modern biological techniques such as Hi-C permit to measure probabilities
that different chromosomal regions are close in space. These probabilities can
be visualised as matrices called contact maps. In this paper, we introduce a
multifractal analysis of chromosomal contact maps. Our analysis reveals that
Hi-C maps are bifractal, i.e. complex geometrical objects characterized by two
distinct fractal dimensions. To rationalize this observation, we introduce a
model that describes chromosomes as a hierarchical set of nested domains and we
solve it exactly. The predicted multifractal spectrum is in excellent
quantitative agreement with experimental data. Moreover, we show that our
theory yields to a more robust estimation of the scaling exponent of the
contact probability than existing methods. By applying this method to
experimental data, we detect subtle conformational changes among chromosomes
during differentiation of human stem cells.
