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A principle curve approach to three-dimensional chromatin configuration reconstruction.

Banff International Research Station for Mathematical Innovation and Discovery

The three-dimensional (3D) configuration of chromosomes within the eukaryote nucleus is consequential for several cellular functions including gene expression regulation and is also strongly associated with cancer-causing translocation events. While visualization of such architecture remains limited to low resolution and throughput imaging modalities, the ability to infer 3D structure at increasing resolution has been enabled by recently-devised chromosome conformation capture techniques, notably Hi-C. Such assays, which utilize cross-linking, followed by restriction digestion and proximity ligation, enable identification of (pairwise) genomic loci that are spatially close via next generation sequencing. Subsequent binning yields a matrix of chromatin contact or interaction counts. Various algorithms have been advanced to operate on these contact matrices to produce reconstructed 3D configurations. Many of these are based on multidimensional scaling (MDS) following conversion of contact matrices to distance matrices. However, none of the proposed methods exploit, or actively impose, the fact that the target solution for an individual chromosome is a (smooth) one-dimensional (1D) curve in 3-space. This basic attribute of chromatin contiguity is either ignored or indirectly addressed by the imposition of constraints. Here we demonstrate the utility of principle curves in directly obtaining 1D solutions that best recapitulate the contact matrix. Our target 1D curve in 3D is a vector function with three coordinates each indexed by 1D genomic distance. Since we seek coordinate functions that are smooth with respect to genomic distance we represent each using a spline basis, parameterized such that the level of smoothness can be prescribed by a degrees of freedom (df) specification. This enables a principle curve solution to the metric scaling problem â (Frobenius norm) approximation of the contact matrix â using a readily obtained eigen-decomposition. While the suite of solutions resulting from a range of df is informative with respect to differing scales of chromatin architecture we also detail methods for selecting a single summary structure. Illustrative examples featuring chromosomes 20, 21 and 22 from IMR90 cells are showcased since the existence of orthogonal multiplex FISH imaging allows for external validation. Joint work with Trevor Hastie

Mathematics; Statistics; Computer science

video/mp4

Author affiliation: UCSF

2019-05-05

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/70062

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/