Combinatorial chromatin modification patterns in the human genome revealed by subspace clustering

Chromatin modifications, such as post-translational modification of histone proteins and incorporation of histone variants, play an important role in regulating gene expression. Joint analyses of multiple histone modification maps are starting to reveal combinatorial patterns of modifications that are associated with functional DNA elements, providing support to the ‘histone code’ hypothesis. However, due to the lack of analytical methods, only a small number of chromatin modification patterns have been discovered so far. Here, we introduce a scalable subspace clustering algorithm, coherent and shifted bicluster identification (CoSBI), to exhaustively identify the set of combinatorial modification patterns across a given epigenome. Performance comparisons demonstrate that CoSBI can generate biclusters with higher intra-cluster coherency and biological relevance. We apply our algorithm to a compendium of 39 genome-wide chromatin modification maps in human CD4+ T cells. We identify 843 combinatorial patterns that recur at >0.1% of the genome. A total of 19 chromatin modifications are observed in the combinatorial patterns, 10 of which occur in more than half of the patterns. We also identify combinatorial modification signatures for eight classes of functional DNA elements. Application of CoSBI to epigenome maps of different cells and developmental stages will aid in understanding how chromatin structure helps regulate gene expression

Approximating L^2-signatures by their compact analogues

:Let G be a group together with an descending nested sequence of normal subgroups G=G_0, G_1, G_2 G_3, ... of finite index [G:G_k] such the intersection of the G_k-s is the trivial group. Let (X,Y) be a compact 4n-dimensional Poincare' pair and p: (\bar{X},\bar{Y}) \to (X,Y) be a G-covering, i.e. normal covering with G as deck transformation group. We get associated $G/_k$-coverings (X_k,Y_k) \to (X,Y). We prove that sign^{(2)}(\bar{X},\bar{Y}) = lim_{k\to\infty} \frac{sign(X_k,Y_k)}{[G : G_k]}, where sign or sign^{(2)} is the signature or L^2-signature, respectively, and the convergence of the right side for any such sequence (G_k)_k is part of the statement

New Ramsey Classes from Old

Let C_1 and C_2 be strong amalgamation classes of finite structures, with disjoint finite signatures sigma and tau. Then C_1 wedge C_2 denotes the class of all finite (sigma cup tau)-structures whose sigma-reduct is from C_1 and whose tau-reduct is from C_2. We prove that when C_1 and C_2 are Ramsey, then C_1 wedge C_2 is also Ramsey. We also discuss variations of this statement, and give several examples of new Ramsey classes derived from those general results.Comment: 11 pages. In the second version, to be submitted for journal publication, a number of typos has been removed, and a grant acknowledgement has been adde

Simplicial matrix-tree theorems

We generalize the definition and enumeration of spanning trees from the setting of graphs to that of arbitrary-dimensional simplicial complexes $\Delta$, extending an idea due to G. Kalai. We prove a simplicial version of the Matrix-Tree Theorem that counts simplicial spanning trees, weighted by the squares of the orders of their top-dimensional integral homology groups, in terms of the Laplacian matrix of $\Delta$. As in the graphic case, one can obtain a more finely weighted generating function for simplicial spanning trees by assigning an indeterminate to each vertex of $\Delta$ and replacing the entries of the Laplacian with Laurent monomials. When $\Delta$ is a shifted complex, we give a combinatorial interpretation of the eigenvalues of its weighted Laplacian and prove that they determine its set of faces uniquely, generalizing known results about threshold graphs and unweighted Laplacian eigenvalues of shifted complexes.Comment: 36 pages, 2 figures. Final version, to appear in Trans. Amer. Math. So

Comprehensive analysis of the chromatin landscape in Drosophila melanogaster.

Chromatin is composed of DNA and a variety of modified histones and non-histone proteins, which have an impact on cell differentiation, gene regulation and other key cellular processes. Here we present a genome-wide chromatin landscape for Drosophila melanogaster based on eighteen histone modifications, summarized by nine prevalent combinatorial patterns. Integrative analysis with other data (non-histone chromatin proteins, DNase I hypersensitivity, GRO-Seq reads produced by engaged polymerase, short/long RNA products) reveals discrete characteristics of chromosomes, genes, regulatory elements and other functional domains. We find that active genes display distinct chromatin signatures that are correlated with disparate gene lengths, exon patterns, regulatory functions and genomic contexts. We also demonstrate a diversity of signatures among Polycomb targets that include a subset with paused polymerase. This systematic profiling and integrative analysis of chromatin signatures provides insights into how genomic elements are regulated, and will serve as a resource for future experimental investigations of genome structure and function

Various L2-signatures and a topological L2-signature theorem

For a normal covering over a closed oriented topological manifold we give a proof of the L2-signature theorem with twisted coefficients, using Lipschitz structures and the Lipschitz signature operator introduced by Teleman. We also prove that the L-theory isomorphism conjecture as well as the C^*_max-version of the Baum-Connes conjecture imply the L2-signature theorem for a normal covering over a Poincar space, provided that the group of deck transformations is torsion-free. We discuss the various possible definitions of L2-signatures (using the signature operator, using the cap product of differential forms, using a cap product in cellular L2-cohomology,...) in this situation, and prove that they all coincide.Comment: comma in metadata (author field) added

Young wall realization of crystal graphs for U_q(C_n^{(1)})

We give a realization of crystal graphs for basic representations of the quantum affine algebra U_q(C_n^{(1)}) using combinatorics of Young walls. The notion of splitting blocks plays a crucial role in the construction of crystal graphs

Brick polytopes, lattice quotients, and Hopf algebras

This paper is motivated by the interplay between the Tamari lattice, J.-L. Loday's realization of the associahedron, and J.-L. Loday and M. Ronco's Hopf algebra on binary trees. We show that these constructions extend in the world of acyclic $k$-triangulations, which were already considered as the vertices of V. Pilaud and F. Santos' brick polytopes. We describe combinatorially a natural surjection from the permutations to the acyclic $k$-triangulations. We show that the fibers of this surjection are the classes of the congruence $\equiv^k$ on $\mathfrak{S}_n$ defined as the transitive closure of the rewriting rule $U ac V_1 b_1 \cdots V_k b_k W \equiv^k U ca V_1 b_1 \cdots V_k b_k W$ for letters $a < b_1, \dots, b_k < c$ and words $U, V_1, \dots, V_k, W$ on $[n]$. We then show that the increasing flip order on $k$-triangulations is the lattice quotient of the weak order by this congruence. Moreover, we use this surjection to define a Hopf subalgebra of C. Malvenuto and C. Reutenauer's Hopf algebra on permutations, indexed by acyclic $k$-triangulations, and to describe the product and coproduct in this algebra and its dual in term of combinatorial operations on acyclic $k$-triangulations. Finally, we extend our results in three directions, describing a Cambrian, a tuple, and a Schr\"oder version of these constructions.Comment: 59 pages, 32 figure