Euler diagrams through the looking glass: From extent to intent

Extension and intension are two ways of indicating the fundamental meaning of a concept. The extent of a concept, C, is the set of objects which correspond to C whereas the intent of C is the collection of attributes that characterise it. Thus, intension denotes the set of objects corresponding to C without naming them individually. Mathematicians switch comfortably between these perspectives but the majority of logical diagrams deal exclusively in extension. Euler diagrams indicate sets using curves to depict their extent in a way that intuitively matches the relations between the sets. What happens when we use spatial diagrams to depict intension? What can we infer about the intension of a concept given its extension, and vice versa? We present the first steps towards addressing these questions by defining extensional and intensional Euler diagrams and translations between the two perspectives. We show that translation in either direction leads to a loss of information, yet preserves important semantic properties. To conclude, we explain how we expect further exploration of the relationship between the two perspectives could shed light on connections between diagrams, extension, intension, and well-matchedness

Single-cell meta-analysis of SARS-CoV-2 entry genes across tissues and demographics.

Angiotensin-converting enzyme 2 (ACE2) and accessory proteases (TMPRSS2 and CTSL) are needed for severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) cellular entry, and their expression may shed light on viral tropism and impact across the body. We assessed the cell-type-specific expression of ACE2, TMPRSS2 and CTSL across 107 single-cell RNA-sequencing studies from different tissues. ACE2, TMPRSS2 and CTSL are coexpressed in specific subsets of respiratory epithelial cells in the nasal passages, airways and alveoli, and in cells from other organs associated with coronavirus disease 2019 (COVID-19) transmission or pathology. We performed a meta-analysis of 31 lung single-cell RNA-sequencing studies with 1,320,896 cells from 377 nasal, airway and lung parenchyma samples from 228 individuals. This revealed cell-type-specific associations of age, sex and smoking with expression levels of ACE2, TMPRSS2 and CTSL. Expression of entry factors increased with age and in males, including in airway secretory cells and alveolar type 2 cells. Expression programs shared by ACE2+TMPRSS2+ cells in nasal, lung and gut tissues included genes that may mediate viral entry, key immune functions and epithelial-macrophage cross-talk, such as genes involved in the interleukin-6, interleukin-1, tumor necrosis factor and complement pathways. Cell-type-specific expression patterns may contribute to the pathogenesis of COVID-19, and our work highlights putative molecular pathways for therapeutic intervention