The probability of two loci, separated by a certain genome length, being in contact can be inferred using the chromosome conformation capture (3C) method and related Hi-C experiments. How to go from the contact map, a matrix listing the mean contact probabilities between a large number of pairs of loci, to an ensemble of three-dimensional structures is an open problem. A solution to this problem, without assuming an assumed energy function, would be the first step in understanding the way nature has solved the packaging of chromosomes in tight cellular spaces. We created a theory, based on polymer physics characteristics of chromosomes and the maximum entropy principles, referred to as HIPPS (Hi-C-polymer-physics-structures) method, that allows us to calculate the 3D structures solely from Hi-C contact maps. The first step in the HIPPS method is to relate the mean contact probability ($⟨p_{ij}⟩$) between loci $i$ and $j$ and the average spatial distance $⟨{\overline{r}}_{ij}⟩$. This is a difficult problem to solve because the cell population is heterogeneous, which means that a given contact exists only in a small unknown fraction of cells. Despite the population heterogeneity, we first prove that there is a theoretical lower bound connecting $⟨p_{ij}⟩$ and $⟨{\overline{r}}_{ij}⟩$ via a power-law relation. We show, using simulations of a precisely solvable model, that the overall organization is accurately captured by constructing the distance map from the contact map even if the cell population is highly heterogeneous, thus justifying the use of the lower bound. In the second step, the mean distance matrix, with elements $⟨{\overline{r}}_{ij}⟩\prime \mathrm{s}$, is used as a constraint in the maximum entropy principle to obtain the joint distribution of spatial positions of the loci. Using the two steps, we created an ensemble of 3D structures for the 23 chromosomes from lymphoblastoid cells using the measured contact maps as inputs. The HIPPS method shows that conformations of chromosomes are heterogeneous even in a single cell type. The differences in the conformational heterogeneity of the same chromosome in different cell types (normal as well as cancerous cells) can also be quantitatively discerned using our theory. We validate the method by showing that the calculated volumes of the 23 chromosomes from the predicted 3D structures are in good agreement with experimental estimates. Because the method is general, the 3D structures for any species may be calculated directly from the contact map without the need to assume a specific polymer model, as is customarily done.

• Received 21 May 2020
• Revised 1 January 2021
• Accepted 13 January 2021

DOI:https://doi.org/10.1103/PhysRevX.11.011051

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Published by the American Physical Society