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. Contact maps are characterized by self-similar blocks along the diagonal, corresponding to a hierarchy of tightly packed chromosomal domains. Statistical properties of these maps, and thus of chromosomal interactions, are usually summarized by the scaling of the contact probability as a function of the genomic distance. This approach is fraught with limitations, since such scaling is usually rather noisy. Here, we introduce a multifractal analysis of chromosomal contact maps. Our analysis reveals that Hi--C maps are bifractal, i.e. they are 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 domains nested in each other and we solve it exactly. The predicted multifractal spectrum is in excellent quantitative agreement with experimental data. 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.
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.
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