Stratifying multiparameter persistent homology

A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter family of spaces obtained from the data. In applications data often depend on several parameters, and in this case one is interested in studying the persistent homology of a multiparameter family of spaces associated to the data. While the theory of persistent homology for one-parameter families is well-understood, the situation for multiparameter families is more delicate. Following Carlsson and Zomorodian we recast the problem in the setting of multigraded algebra, and we propose multigraded Hilbert series, multigraded associated primes and local cohomology as invariants for studying multiparameter persistent homology. Multigraded associated primes provide a stratification of the region where a multigraded module does not vanish, while multigraded Hilbert series and local cohomology give a measure of the size of components of the module supported on different strata. These invariants generalize in a suitable sense the invariant for the one-parameter case.Comment: Minor improvements throughout. In particular: we extended the introduction, added Table 1, which gives a dictionary between terms used in PH and commutative algebra; we streamlined Section 3; we added Proposition 4.49 about the information captured by the cp-rank; we moved the code from the appendix to github. Final version, to appear in SIAG

The role of small proteins in Burkholderia cenocepacia J2315 biofilm formation, persistence and intracellular growth

Burkholderia cenocepacia infections are difficult to treat due to resistance, biofilm formation and persistence. B. cenocepacia strain J2315 has a large multi-replicon genome (8.06 Mb) and the function of a large fraction of (conserved) hypothetical genes remains elusive. The goal of the present study is to elucidate the role of small proteins in B. cenocepacia, focusing on genes smaller than 300 base pairs of which the function is unknown. Almost 10% (572) of the B. cenocepacia J2315 genes are smaller than 300 base pairs and more than half of these are annotated as coding for hypothetical proteins. For 234 of them no similarity could be found with non-hypothetical genes in other bacteria using BLAST. Using available RNA sequencing data obtained from biofilms, a list of 27 highly expressed B. cenocepacia J2315 genes coding for small proteins was compiled. For nine of them expression in biofilms was also confirmed using LC-MS based proteomics and/or expression was confirmed using eGFP translational fusions. Overexpression of two of these genes negatively impacted growth, whereas for four others overexpression led to an increase in biofilm biomass. Overexpression did not have an influence on the MIC for tobramycin, ciprofloxacin or meropenem but for five small protein encoding genes, overexpression had an effect on the number of persister cells in biofilms. While there were no significant differences in adherence to and invasion of A549 epithelial cells between the overexpression mutants and the WT, significant differences were observed in intracellular growth/survival. Finally, the small protein BCAM0271 was identified as an antitoxin belonging to a toxin-antitoxin module. The toxin was found to encode a tRNA acetylase that inhibits translation. In conclusion, our results confirm that small proteins are present in the genome of B. cenocepacia J2315 and indicate that they are involved in various biological processes, including biofilm formation, persistence and intracellular growth.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Homological Algebra for Persistence Modules

We develop some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. We consider the graded module and sheaf tensor product and Hom bifunctors as well as their derived functors, Tor and Ext, and give explicit computations for interval modules. We give a classification of injective, projective, and flat interval modules. We state Kunneth theorems and universal coefficient theorems for the homology and cohomology of chain complexes of persistence modules in both the sheaf and graded modules settings and show how these theorems can be applied to persistence modules arising from filtered cell complexes. We also give a Gabriel-Popescu theorem for persistence modules. Finally, we examine categories enriched over persistence modules. We show that the graded module point of view produces a closed symmetric monoidal category that is enriched over itself.Comment: 41 pages, accepted by Foundations of Computational Mathematic

Physical constraints on accuracy and persistence during breast cancer cell chemotaxis

Directed cell motion in response to an external chemical gradient occurs in many biological phenomena such as wound healing, angiogenesis, and cancer metastasis. Chemotaxis is often characterized by the accuracy, persistence, and speed of cell motion, but whether any of these quantities is physically constrained by the others is poorly understood. Using a combination of theory, simulations, and 3D chemotaxis assays on single metastatic breast cancer cells, we investigate the links among these different aspects of chemotactic performance. In particular, we observe in both experiments and simulations that the chemotactic accuracy, but not the persistence or speed, increases with the gradient strength. We use a random walk model to explain this result and to propose that cells' chemotactic accuracy and persistence are mutually constrained. Our results suggest that key aspects of chemotactic performance are inherently limited regardless of how favorable the environmental conditions are