  The three-dimensional organization of chromatin, on the length scale of a few
genes, is crucial in determining the functional state - accessibility and the
amount of gene expression - of the chromatin. Recent advances in chromosome
conformation capture experiments provide partial information on the chromatin
organization in a cell population, namely the contact count between any segment
pairs, but not on the interaction strength that leads to these contact counts.
However, given the contact matrix, determining the complete 3D organization of
the whole chromatin polymer is an inverse problem. In the present work, a novel
Inverse Brownian Dynamics (IBD) method based on a coarse grained bead-spring
chain model has been proposed to compute the optimal interaction strengths
between different segments of chromatin such that the experimentally measured
contact count probability constraints are satisfied. Applying this method to
the {\alpha}-globin gene locus in two different cell types, we predict the 3D
organizations corresponding to active and repressed states of chromatin at the
locus. We show that the average distance between any two segments of the region
has a broad distribution and cannot be computed as a simple inverse relation
based on the contact probability alone. The results presented for multiple
normalization methods suggest that all measurable quantities may crucially
depend on the nature of normalization. We argue that by experimentally
measuring predicted quantities, one may infer the appropriate form of
normalization.
