Bifractal nature of chromosome contact maps

Modern biological techniques such as Hi-C permit one to measure probabilities that different chromosomal regions are close in space. These probabilities can be visualized 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 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.

Figure 1

Contact probability scaling vs multifractal analysis of a Hi-C map. (a) Hi-C map of chromosome 1 of mouse embryonic stem cells (mESC) [44]. Darker shades of red correspond to higher values of the contact probability $p_{ij}$. (b) Scaling of the average contact probability $p\left(\ell \right)$ as a function of the distance $\ell =|i-j|$. The continuous line is a best fit of a power law, Eq. (1), in the range of distances $[{10}^5,{10}^6]$, yielding $\beta \approx 1.06$. (c) Local logarithmic slope $dlnP\left(\ell \right)/d(ln\ell )$ of the contact probability. (d) Coarse graining of the Hi-C map at increasing values of the resolution $\epsilon ={10}^6,{10}^7,\text{and}2\times {10}^7\mathrm{bp}$. (e) Scaling of $Z(q,\epsilon )$ as a function of the resolution $\epsilon$ [see Eqs. (4) and (5)]. Different lines correspond to different values of $q$, increasing from top to bottom from $q=0$ to 5 at intervals of $\mathrm{\Delta }q=0.25$. (f) Local logarithmic slopes of the first momenta of $Z\left(q\right)$ ($q=0,0.25,0.5,0.75$). (g) Corresponding multifractal spectrum $K\left(q\right)$. Here and throughout the paper, spectra are obtained by a fit in log-log scale of $Z(q,\epsilon )$ vs $\epsilon$ in the range $\epsilon \in [{10}^5,{10}^7]$ unless specified otherwise. Two straight dashed lines are shown to guide the eye. The python code to compute the multifractal spectrum is available at [45].

Figure 2

Hierarchical domain model. (a) Construction of the hierarchical domain model for two different values of the parameter $a$. (b) Colored symbols show the multifractal spectrum calculated by the hierarchical domain mode at different values of $a$; the solid curves are the predictions of Eqs. (11) and (12). (c) Multifractal spectra of the first three chromosomes of mESC (colored symbols). The solid curve is a fit of Eqs. (11) and (12), resulting in $a\approx 0.413$. Here and in the following, all fits are performed using the least-squares method, unless specified otherwise.

Figure 3

Performance of null models. Points and curves show the mean-square deviation (MSD) between the multifractal spectrum computed from different sets of data and the one predicted by Eqs. (11) and (12) upon fitting the parameter $a$. Blue upward and red downward triangles denote fits of the multifractal spectrum computed from the null models expressed by Eqs. (13) and (14), respectively. In (a), fits are shown as a function of $\beta$ for fixed $N_{\mathrm{blocks}}=N/M=15$ for the first null model, and fixed $\sigma =3\times {10}^{-5}$ for the second null model. In (b), fits of the first null model are shown as a function of $N_{\mathrm{blocks}}$ for fixed $\beta =0.5$. In (c), fits of the second null model are shown as a function of $\sigma$ for fixed $\beta =0.8$. In all panels, the dark black line indicates the MSD of a fit of experimental data from mouse chromosome 1. The light orange line indicates the MSD associated with a numerical fit of the hierarchical model itself, with $a=0.45$ and $n=10$ iterations.

Figure 4

Contact probability in the hierarchical domain model. Decay of contact probability $P(\ell ;n)$ as a function of the genomic distance $\ell$ in the hierarchical model for $n=10$ and different values of the parameter $a$, shown in the figure legend. Continuous lines are the exponent predictions of Eq. (15).

Figure 5

Application of multifractal analysis to experimental Hi-C maps. (a) Exponents $\beta$ calculated from the multifractal spectrum using Eq. (15) and from a direct fit of mouse embryonic stem cells [as in Fig. 1] measured with different Hi-C methods. In particular, we analyze results of Hi-C experiments of in situ [8], TCC [48], and two independent replicates of in solution [33, 44]. Black bars indicate medians over chromosomes, colored boxes mark the first and third quartiles, and black dots are outliers. (b) Scatter plot of the data displayed in panel (a). (c) Value of $\beta$ obtained for the wild-type (wt) system and for a mutant (mut) in which either CTCF [33] or Nipbl [48] is depleted in experiments. (d) Scatter plot of the data displayed in panel (c). (e) Exponents $\beta$ obtained with the multifractal method for cell lines at different stages of human stem cell development [49]. (f) Exponents obtained by direct fit of the same data.