Inferring Diploid 3D Chromatin Structures from Hi-C Data

The 3D organization of the genome plays a key role in many cellular processes, such as gene regulation, differentiation, and replication. Assays like Hi-C measure DNA-DNA contacts in a high-throughput fashion, and inferring accurate 3D models of chromosomes can yield insights hidden in the raw data. For example, structural inference can account for noise in the data, disambiguate the distinct structures of homologous chromosomes, orient genomic regions relative to nuclear landmarks, and serve as a framework for integrating other data types. Although many methods exist to infer the 3D structure of haploid genomes, inferring a diploid structure from Hi-C data is still an open problem. Indeed, the diploid case is very challenging, because Hi-C data typically does not distinguish between homologous chromosomes. We propose a method to infer 3D diploid genomes from Hi-C data. We demonstrate the accuracy of the method on simulated data, and we also use the method to infer 3D structures for mouse chromosome X, confirming that the active homolog exhibits a bipartite structure, whereas the active homolog does not

Tiling by rectangles and alternating current

This paper is on tilings of polygons by rectangles. A celebrated physical interpretation of such tilings due to R.L. Brooks, C.A.B. Smith, A.H. Stone and W.T. Tutte uses direct-current circuits. The new approach of the paper is an application of alternating-current circuits. The following results are obtained: - a necessary condition for a rectangle to be tilable by rectangles of given shapes; - a criterion for a rectangle to be tilable by rectangles similar to it but not all homothetic to it; - a criterion for a generic polygon to be tilable by squares. These results generalize the ones of C. Freiling, R. Kenyon, M. Laczkovich, D. Rinne and G. Szekeres.Comment: In English and in Russian; 21 pages; 6 figures; minor improvement of exposition, Russian translation adde

Algebraic characteristic classes for idempotent matrices

This paper contains the algebraic analog for idempotent matrices of the Chern-Weil theory of characteristic classes. This is used to show, algebraically, that the canonical line bundle on the complex projective space is not stably trivial. Also a theorem is proved saying that for any smooth manifold there is a canonical epimorphism from the even dimensional algebraic de Rham cohomology of its algebra of smooth functions onto the standard even dimensional de Rham cohomology of the manifold