  The principles that govern the organization of genomes, which are needed for
a deeper understanding of how chromosomes are packaged and function in
eukaryotic cells, could be deciphered if the three-dimensional (3D) structures
are known. Recently, single-cell imaging experiments have determined the 3D
coordinates of a number of loci in a chromosome. Here, we introduce a
computational method (Distance Matrix to Ensemble of Structures, DIMES), based
on the maximum entropy principle, with experimental pair-wise distances between
loci as constraints, to generate a unique ensemble of 3D chromatin structures.
Using the ensemble of structures, we quantitatively account for the
distribution of pair-wise distances, three-body co-localization and
higher-order interactions. We demonstrate that the DIMES method can be applied
to both small length-scale and chromosome-scale imaging data to quantify the
extent of heterogeneity and fluctuations in the shapes on various length
scales. We develop a perturbation method that is used in conjunction with DIMES
to predict the changes in 3D structures from structural variations. Our method
also reveals quantitative differences between the 3D structures inferred from
Hi-C and the ones measured in imaging experiments. Finally, the physical
interpretation of the parameters extracted from DIMES provides insights into
the origin of phase separation between euchromatin and heterochromatin domains.
