Persistent Homology Over Directed Acyclic Graphs

We define persistent homology groups over any set of spaces which have inclusions defined so that the corresponding directed graph between the spaces is acyclic, as well as along any subgraph of this directed graph. This method simultaneously generalizes standard persistent homology, zigzag persistence and multidimensional persistence to arbitrary directed acyclic graphs, and it also allows the study of more general families of topological spaces or point-cloud data. We give an algorithm to compute the persistent homology groups simultaneously for all subgraphs which contain a single source and a single sink in $O(n^4)$ arithmetic operations, where $n$ is the number of vertices in the graph. We then demonstrate as an application of these tools a method to overlay two distinct filtrations of the same underlying space, which allows us to detect the most significant barcodes using considerably fewer points than standard persistence.Comment: Revised versio

An Epigenetic Switch Involving Overlapping Fur and DNA Methylation Optimizes Expression of a Type VI Secretion Gene Cluster

Type VI secretion systems (T6SS) are macromolecular machines of the cell envelope of Gram-negative bacteria responsible for bacterial killing and/or virulence towards different host cells. Here, we characterized the regulatory mechanism underlying expression of the enteroagregative Escherichia coli sci1 T6SS gene cluster. We identified Fur as the main regulator of the sci1 cluster. A detailed analysis of the promoter region showed the presence of three GATC motifs, which are target of the DNA adenine methylase Dam. Using a combination of reporter fusion, gel shift, and in vivo and in vitro Dam methylation assays, we dissected the regulatory role of Fur and Dam-dependent methylation. We showed that the sci1 gene cluster expression is under the control of an epigenetic switch depending on methylation: fur binding prevents methylation of a GATC motif, whereas methylation at this specific site decreases the affinity of Fur for its binding box. A model is proposed in which the sci1 promoter is regulated by iron availability, adenine methylation, and DNA replication

Algorithmic Construction of Acyclic Partial Matchings for Multidimensional Persistence

Given a simplicial complex and a vector-valued function on its vertices, we present an algorithmic construction of an acyclic partial matching on the cells of the complex. This construction is used to build a reduced filtered complex with the same multidimensional persistent homology as of the original one filtered by the sublevel sets of the function. A number of numerical experiments show a substantial rate of reduction in the number of cells achieved by the algorithm